· Princeton Mathematical Series EDITORS: LUIS A. CAFFARELLI, JOHN N. MATHER, and ELIAS M. STEIN I. The Classical Groups by Hermann Weyl 3. An Introduction to Differential Geometry

Princeton Mathematical Series EDITORS: LUIS A. CAFFARELLI, JOHN N. MATHER, and ELIAS M. STEIN

I. The Classical Groups by Hermann Weyl 3. An Introduction to Differential Geometry by Luther Pfahler Eisenhart t4. Dimension Theory by W Hurewicz and H. Wallman 6. The Laplace Transform by D. V Widder 7. Integration by Edward J. McShane 8. Theory of Lie Groups: I by C. Chevalley 9. Mathematical Methods of Statistics by Harald Cramer

10. Several Complex Variables by S. Bochner and W T. Martin 11. Introduction to Topology by S. Lefschetz 12. Alegebraic Geometry and Topology edited by R. H. Fox, D. C. Spencer, and A. W.

Tucker 14. The Topology of Fibre Bundles by Norman Steenrod 15. Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod 16. Functionals of Finite Riemann Surfaces by Menahem Schiffer and Donald C. Spencer 17. Introduction to Mathematical Logic, Vol I by Alonzo Church 19. Homological Algebra by H. Cartan and S. Eilenberg 20. The Convolution Transform by I. I. Hirschman and D. V Widder 21. Geometric Integration Theory by H. Whitney 22. Qualitiative Theory of Differential Equations by V V Nemytskii and V V Stepanov 23. Topological Analysis by Gordon T. Whyburn (revised 1964) 24. Analytic Functions by Ahlfors, Behnke, Bers, Grauert et al. 25. Continuous Geometry by John von Neumann 26. Riemann Surfaces by L. Ahlfors and L. Sario 27. Differential and Combinatorial Topology edited by S. S. Cairns 28. Convex Analysis by R. T. Rockafellar 29. Global Analysis edited by D. C. Spencer and S. Iyanaga 30. Singular Integrals and Differentiability Properties of Functions by E. M Stein 31. Problems in Analysis edited by R. C. Gunning 32. Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss 33. Etale Cohomology by J. S. Milne 34. Pseudodifferential Operators by Michael E. Taylor 36. Representation Theory of Semisimple Groups: An Overview Based on Examples by

Anthony W Knapp 37. Foundations of Algebraic Analysis by Masaki Kashiwara, Takahiro Kawai, and Tatsuo

Kimura. Translated by Goro Kato 38. Spin Geometry by H. Blaine Lawson, Jr., and Marie-Louise Michelsohn 39. Topology of 4-Manifolds by Michael H. Freedman and Frank Quinn 40. Hypo-Analytic Structures: Local Theory by Franr;ois Treves 41. The Global Nonlinear Stability of the Minkowksi Space by Demetrios Christodoulou

and Sergiu Klainerman 42. Essays on Fourier Analysis in Honor of Elias M. Stein edited by C. FejJerman, R.

FejJerman, and S. Wainger

ESSAYS ON FOURIER ANALYSIS IN HONOR OF ELIAS M. STEIN

EDITED BY

Charles Feffennan, Robert Feffennan

and Stephen Wainger

PRINCETON UNIVERSITY PRESS

1 PRINCETON, NEW JERSEY

www.MathSchoolinternational.com

Copyright © 1995 by Princeton University Press Published by Princeton University Press, 41 William Street,

Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press, Chichester, West Sussex

All Rights Reserved

LIbrary of Congress Cataloging.In-Publication Dala

E"ays on Fourier Analysis in honor of Elia' M. Stein / edited by Charles

Fefferman, Robert Fefferman, and Stephen Wainger.

p. crn.-(Princeton rnathematical series: 42)

Proceedings of the Princeton Conference in Harmonic

Analysis, held May 13-17, 1991.

Includes bibliographical references.

ISBN 0-691-08655-9 (alk. p.1per)

I. Fourier analysis-Congresses. I. Stein, Elias M., 1931-.

n. Fefferman, Charles, 1949-. m. Fefferman, Robert, 1951-. IV. Wainger, Stephen, 1936-. V. Series.

QA403.5.P761993

515'.2433--<1c20 92-43054

CIP

Princeton University Press books are printed on acid-free paper

and meet the guidelin... for permanence and durability of the Committee

on Production Guidelin.., for Book Longevity of the Council

On Library Resonrc...

Printed in the United State, of America

10 9 8 7 6 5 4 3 2 I

CONTENTS

INTRODUCTION CHARLES PEFFERMAN, ROBERT PEFFERMAN, AND STEPHEN

WAINGER vii

ONE Selected Theorems by Eli Stein. CHARLES FEFFERMAN

TWO Geometric Inequalities in Fourier Analysis. WILLIAM BECKNER 36

THREE Representing Measures for Holomorphic Functions on Type 2 Wedges. AI- BOGGESS AND ALEXANDER NAGEL 69

FOUR Some New Estimates on Oscillatory Integrals. JEAN BOURGAIN 83

FIVE Dilations Associated to Flat Curves in JR" • ANTHONY CARBERY, JAMES VANCE, STEPHEN WAINGER, AND DAVlD WATSON 113

SIX Nonexistence of Invariant Analytic Hypoelliptic Differential Operators on Nilpotent Groups of Step Greater than Two. MICHAEL CHRIST 127

SEVEN

Operateurs Bilineaires et Renormalisation • R. R. COIFMAN, S. DOBYINSKY, AND Y, MEYER 146

EIGHT Numerical Harmonic Analysis. R. R. COIFMAN, Y, MEYER, AND V. WICKERHAUSER 162

NINE Some Topics from Harmonic Analysis and Partial Differential Equations • ROBERT A. PEFFERMAN 175


vi

TEN Function Spaces on Spaces of Homogeneous Type. YONGSHENG HAN AND GtJlDO WEISS

ELEVEN The First Nodal Set of a Convex Domain. DAVID JERISON

TWELVE On Removable Sets for Sobolev Spaces in the Plane. PETER W. JONES

THIRTEEN Oscillatory Integrals and Non-Linear Dispersive Equations. CARLOS E. KENIG

FOURTEEN Singular Integrals and Fourier Integral Operators. D. H. PHONG

FIFTEEN Counterexamples with Harmonic Gradients in R 3 • THOMAS H. WOLff

CONTENTS

211

225

250

268

286

321

INTRODUCTION

This is the proceedings of the Princeton Conference in Hannonic Analysis, which was held from May 13 through May 17, 1991. This conference was a very special event because it celebrated Elias M. Stein's extraordinary impact on mathematics, on the occasion of his sixtieth birthday. Through his brilliant research, books, lectures, and teaching, E. M. Stein has changed the way central problems in analysis are approached. The articles presented here are a clear reflection ofthis and a tribute to his stellar contribution.

We would like to take this opportunity to express our gratitude to those who made the conference possible. We thank the National Science Foundation for funding and the Princeton Mathematics Department for both its hospitality and its financial support. It is a pleasure to acknowledge the contributions ofM. Christ and M. Machedon, who provided us with financial assistance through their PYI grants. Thanks go to J. Ryfffor his many helpful suggestions and his encouragement, and to Scott Kenney and the staff of the Princeton Mathematics Department for the great amount of work they did to make things run smoothly. Finally, we wish to thank the speakers and the many participants in the conference for their energy and enthusiasm, which made possible its success.

Charles Fefferman Robert Fefferman Stephen Wainger

A:


1

Selected Theorems by Eli Stein

Charles Fefferman

INTRODUCTION

The purpose of this survey article is to give the general reader some idea of the scope and originality of Eli Stein's contributions to analysis. His work deals with representation theory, classical Fourier analysis, and partial differential equations. He was the first to appreciate the interplay among these subjects, and to perceive the fundamental insights in each field arising from that interplay. No one else really understands all three fields; therefore, no one else could have done the work I am about to describe. However, deep understanding of three fields of mathematics is by no means sufficient to lead to Stein's main ideas. Rather, at crucial points, Stein has shown extraordinary originality, without which no amount of work or knowledge could have succeeded. Also, large parts of Stein's work (e.g., the fundamental papers [26], [38], [41], [44], [59] on complex analysis in tube domains) don't fit any simple one-paragraph description such as the one above.

It follows that no single mathematician is competent to present an adequate survey of Stein's work. As I attempt the task, I am keenly aware that many of Stein's papers are incomprehensible to me, while others were of critical importance to my own work. Inevitably, therefore, my survey is biased, as any reader will see. Fortunately, S. Gindikin provided me with a layman's explanation of Stein's contributions to representation theory, thus keeping the bias (I hope) within reason. I am grateful to Gindikin for his help, and also to Y. Sagher for a valuable suggestion.

For purposes of this article, representation theory deals with the construction and classification of the irreducible unitary representations of a semisimple Lie group. Classical Fourier analysis starts with the LP-boundedness of two fundamental


' "2 3

,I,~,:'~,; ,:,;::< iCHAPTER 1 SELECTED THEOREMS BY ELl STEIN

operators, the maximal function x h

I +rex) = sup 2h l If(y)1 dy, h>O x-h

and the Hilbert transfonn f(x)dy

Hf(x) = lim -11 E-+O+ rr Ix-YI>E x - y

Finally, we shall be concerned with those problems in partial differential equations

that come from several complex variables.

COMPLEX INTERPOLATION

Let us begin with Stein's work on interpolation of operators. As background, we

state and prove a classical result, namely the

M. Riesz Convexity Theorem. Suppose X, Yare measure spaces, and suppose T is an operator that carries functions on X to functions on Y. Assume T is bounded from LPO(X) to L'0(y), andfrom LPl(X) to L'I(Y). (Here, PO,PI,rO,rl E

[1,00].) Then T is bounded from U(X) to Lr(Y)for .1 = ..!.... + (1-/), ! =P PI Po r

.!.... + (1-/) 0 < t < 1. '1 ro'-

The Riesz Convexity Theorem says that the points (i, ~) for which T is bounded from LP to L' fonn a convex region in the plane. A standard appli

cation is the Hausdorff-Young inequality: We take T to be the Fourier transfonn 2

on Rn , and note that T is obviously bounded from L I to L 00, and from L2 to L •

Therefore, T is bounded from LP to the dual class LP' for 1 ~ P ~ 2. The idea of the proof of the Riesz Convexity Theorem is to estimate Jy (Tf) . g

for fEU and gEL". Say f = Fei'" and g = Gei'/t with F, G ~ 0 and

,p, 1/J real. Then we can define analytic families of functions fz, gz by setting

fz = Faz+bei"', gz = Gcz+dei'/t, for real a, b, c, d to be picked in a moment.

Define

(1) (Z) = l (Tfz)gz·

Evidently, is an analytic function of z. For the correct choice of a, b, c, d we have

and Igzlr~ = Ig( when Rez = 0;(2) Ifzl PO= Ifl P

(3) Ifzl PI = Ifl P and Igzl'; = Ig( when Rez = 1;

and gz = g when z = t.(4) fz = f

Iii

From (2) we see that II fz II LPo, IIgz II L'~ ~ C for Re z = O. So the definition (1) and the assumption T: LPO ---+ L'0 show that

(5) 1(z)1 ~ C' for Rez = O.

Similarly, (3) and the assumption T: LPI ---+ L'1 imply

(6) I(z) I ~ C' for Re z = 1.

Since is analytic, (5) and (6) imply 1(z)! ~ C' for 0 ~ Re z ~ 1 by the

maximum principle for a strip. In particular, 1(t)j ~ C'. In view of (4), this

means that IJy(Tf)gl ~ C', with C' detennined by II filL" and IIgllu" Thus, T is bounded from LP to L' , and the proofof the Riesz Convexity Theorem

is complete.

This proof had been well-known for over a decade, when Stein discovered an

amazingly simple way to extend its usefulness by an order of magnitude. He real

ized that an ingenious argument by Hirschman [H] on certain multiplier operators

on LP (Rn) could be viewed as a Riesz Convexity Theorem for analytic families

of operators. Here is the result.

Stein Interpolation Theorem. Assume Tz is an operator depending analytically on z in the strip 0 ~ Re z ~ 1. Suppose Tz is bounded from LPO to L'0 when Re z = 0, andfrom LP' to L" when Re z = 1. Then Tr is bounded from LP to L' where .1 = ..!.... + (1-/) ! = .!.... + (1-/) and 0 < t < 1.

, P Pl Po' r '1 ro -

Remarkably, the proof of the theorem comes from that of the Riesz Convexity

Theorem by adding a single letter of the alphabet. Instead of taking (z) =

fy(T fz)gz as in (1), we set (z) = Jy(Tzfz)gz. The proof of the Riesz Convexity Theorem then applies with no further changes.

Stein's Interpolation Theorem is an essential tool that penneates modern Fourier

analysis. Let me just give a single application here, to illustrate what it can do.

The example concerns Cesaro summability of multiple Fourier integrals.

We define an operator TaR on functions on Rn by setting

-TaRf(~) (= 1

2)aI~I ~ R2 + f(~)·

Then

(7) II TaRfIILP(IR") ~ Capllfllu(R,"), if I~ -~I < a

I


5 4 .",:t

CHAPTER J

cases p = I and p = 2. Inequality (7), due to Stein, was the first non-trivial progress on spherical summation of multiple Fourier series.

REPRESENTATION THEORY I

Our next topic is the Kunze-Stein phenomenon, which links the Stein Interpolation Theorem to representations of Lie groups. For simplicity we restrict attention to G = SL(2, lR), and begin by reviewing elementary Fourier analysis on G. The irreducible unitary representations of G are as follows:

The principal series, parametrized by a sign u = ±1 and a real parameter t; The discrete series, parametrized by a sign u = ± 1 and an integer k ::: 0; and The complementary series, parametrized by a real number t E (0, 1).

We don't need the full description of these representations here. The irreducible representations of G give rise to a Fourier transfonn. If f is a

function on G, and U is an irreducible unitary representation of G, then we define

l(u) = Lf(g)Ugdg,

where dg denotes Haar measure on the group. Thus,! is an operator-valued function defined on the set of irreducible unitary representations of G. As in the Euclidean case, we can analyze convolutions in tenns of the Fourier transfonn. In fact,

(8) f---;;g = ! .g

as operators. Moreover, there is a Plancherel fonnula for G, which asserts that

2 2AIIfIl L 2 (G) = f IIf(U) II Hilbert-Schmidtd/l (U)

for a measure /l (the Plancherel measure). The Plancherel measure for G is known, but we don't need it here. However, we note that the complementary series has measure zero for the Plancherel measure.

These are, of course, the analogues of familiar results in the elementary Fourier analysis of lRn • Kunze and Stein discovered a fundamental new phenomenon in Fourier analysis on G that has no analogue on jRn. Their result is as follows.

Theorem (KuDze-Stein Phenomenon). There exists a uniformly bounded rep

resentation UlJ, T of G, parametrized by a sign CT = ±1 and a complex number r

in a strip Q, with the following properties.

(A) The Ua,T all act on the same Hilbert space H,

SELECTED THEOREMS BY ELI STEIN

(B) ForfixedCT = ±1,g E G, and;;, TJ E H,thematrix.element «UlJ,T)g~,TJ) is an analytic function of r E Q.

(C) The UlJ, T for Re r = ! are equivalent to the representations ofthe principal series.

(D) The U+l,T for suitable r are equivalent to the representations of the complementary series,

(See (14) for the precise statement and proof, as well as Ehrenpreis-Mautner [EM] for related results.)

The Kunze-Stein Theorem suggests that analysis on G resembles a fictional version of classical Fourier analysis in which the basic exponential ~ ~ exp(i~.

x) is a bounded analytic function on a strip I1m ~ I ~ C, unifonnly for all x.

As an immediate consequence of the Kunze-Stein Theorem, we can give an analytic continuation of the Fourier transform forG. In fact, we set !(u, r) = fGf(g)(UlJ.r)gdgforu = ±1,r E Q.

L 1Thus, f E (G) implies !(CT,.) analytic and bounded on Q. So we have continued analytically the restriction of ! to the principal series. It is as if the Fourier transform of an L I function on (-00, (0) were automatically analytic in a strip. If f E L 2(G), then !(CT, r) is still defined on the line {Re r = ~}, by virtue of the Plancherel formula and part (C) of the Kunze-Stein Theorem. Interpolating between Ll(G) and L2(G) using the Stein Interpolation Theorem, we see that f E U(G) (1 ~ P < 2) implies !(u, .) analytic and satisfying an LP' -inequality on a strip Qp. As p increases from 1 to 2, the strip Qp shrinks from Q to the line {Re r = !}. Thus we obtain the following results.

Corollary 1. If fEU (G) (1 ~ p < 2), then! is bounded almost everywhere with respect to the Plancherel measure.

Corollary 2. For 1 ~ p < 2 we have the convolution inequality II f * g 1IL2(G) ~ Cp II fllu(G) IIg 1I£2(G)'

To check Corollary 1, we look separately at the principal series, the discrete series, and the complementary series. For the principal series, we use the LP'inequality established above for the analytic function r ~ /(CT, r) on the strip Qp. Since an LP' -function analytic on a strip Q p is clearly bounded on an interior line {Re r = ~}, it follows at once that ! is bounded on the principal series. Regarding the discrete series UlJ,k we note that

lip'

(9) (L: /llJ,k 1I!(UlJ,k) liP' ~ Ilfllu(G))lJ,k


7 6 : 'I'I"

CHAPTER 1

for suitable weights f.lCf,k and for 1 ~ p ::: 2. The weights /-to.k amount to the Plancherel measure on the discrete series, and (9) is proved by a trivial interpolation, just like the standard Hausdorff-Young inequality. The boundedness of the /lj(U".k)/I is immediate from (9). Thus the Fourier transform j is bounded on both the principal series and the discrete series, for f E LP(G) (l ~ p< 2). The complementary series has measure zero with respect to the Plancherel measure, so the proof of C~rollary 1 is complete. Corollary 2 follows trivially from Corollary 1, the Plancherel fonnula, and the elementary fonnula (8).

This proof of Corollary 2 poses a significant challenge. Presumably, the Corollary holds because the geometry of G at infinity is so different from that of Euclidean space. For example, the volume of the ball of radius R in G grows exponentially as R -+ 00. This must have a profound impact on the way mass piles up when we take convolutions on G. On the other hand, the statement of Corollary 2 clearly has nothing to do with cancellation; proving the Corollary for two arbitrary functions f, g is the same as proving it for IfI and Igi. When we go back over the proof of Corollary 2, we see cancellation used crucially, e.g., in the Plancherel formula for G; but there is no explicit mention of the geometry of G at infinity. Clearly there is still much that we do not understand regarding convolutions on G.

The Kunze-Stein phenomenon carries over to other semisimple groups, with profound consequences for representation theory. We will continue this discussion later in the article. Now, however, we tum our attention to classical Fourier analysis.

CURVATURE AND THE FOURIER TRANSFORM

One of the most fascinating themes in Fourier analysis in the last two decades has been the connection between the Fourier transfonn and curvature. Stein has been the most important contributor to this set of ideas. To illustrate, I will pick out two of his results. The first is a "restriction theorem," i.e., a result on the restriction /Ir of the Fourier transform of a function f E LP(IR") to a set r of measure zero. If p > 1, then the standard inequality / E LP' (]Rn) suggests that / should not even be well-defined on r, since r has measure zero. Indeed, if r is (say) the x-axis in the plane IRz, then we can easily find functions! (Xl, XZ) = lf1(xd1/r(xz) E LP(]Rz) for which /Ir is infinite everywhere. Fourier transforms of f E LP(]R2) clearly cannot be restricted to straight lines. Stein proved that the situation changes drastically when r is curved. His result is as follows.

Stein's Restriction Theorem. Suppose r is the unit circle, 1 ~ P < ~, and

! E CO'(IR2). Then we have the a priori inequality /I/Irllu ~ Cp/lfllu(1R2), with CP depending only on p.

'.,,' , , SELECTED THEOREMS BY ELI STEIN"1,

Using this a priori inequality, we can trivially pass from the dense subspace Co

to define the operator f 1-----+ fir for all! E LP (IRz). Thus, the Fourier transform of f E LP (p < ~) may be restricted to the unit circle.

Improvements and generalizations were soon proven by other analysts, but it was Stein who first demonstrated the phenomenon of restriction of Fourier transforms.

Stein's proof of his restriction theorem is amazingly simple. If /.L denotes uniform measure on the circle r c ]R2, then for f E Co(]Rz) we have

Z(10) r,fl = r(ff.l)(j) = (f * /l, f) =5 Ilfllull! * /lllu"Jr JIR2

The Fourier transform /l(~) is a Bessel function. It decays like I~ I-! at infinity, a fact intimately connected with the curvature of the circle. In particular, il E L q

for 4 < q ~ 00, and therefore IIf * /lllu' ~ Cpllfllu for 1 =5 p < ~,by the usual elementary estimates for convolutions. Putting this estimate back into

(10), we see that Ir I/Iz ~ C p II fIIi>, which proves Stein's Restriction Theorem. The Stein Restriction Theorem means a lot to me personally, and has strongly influenced my own work in Fourier analysis.

"The second result of Stein's relating the Fourier transform to curvature concerns the differentiation of integrals on lR" .

Theorem. Suppose f E LP(lRn ) with n ?: 3 and p > n~l' For X E IRn and

r > 0, let F (x, r) denote the average of ! on the sphere ofradius r centered at x. Then limr--..o F(x, r) = f(x) almost everywhere.

The point is that unlike the standard Lebesgue Theorem, we are averaging ! over a small sphere instead of a small ball. As in the restriction theorem, we are seemingly in trouble because the sphere has measure zero in lRn , but the curvature of the sphere saves the day. This theorem is obviously closely connected to the smoothness of solutions of the wave equation.

The proof of the above differentiation theorem relies on an

Elementary Tauberian Theorem. Suppose that limR-->O *lOR F(r)dr exZ

ists drand It r I~~ I < 00. Then limR-->O F(R) exists, and equals limR-+O *f: F(r)dr.

This result had long been used, e.g., to pass from Cesaro averages of Fourier series to partial sums. (See Zygmund [Z].) On more than one occasion, Stein has shown the surprising power hidden in the elementary Tauberian Theorem. Here we apply it to F(x, r) for a fixed x. In fact, we have F(x, r) = I f(x + ry)d/-t(y),

with f.l equal to normalized surface measure on the unit sphere, so that the Fourier transforms of F and f are related by F(~, r) = f(l;)/l(rl;) for each fixed r.


CHAPTER 1 SELECTED THEOREMS BY ELI STEIN 9',·" i '.

8 the unit disc. Recall that F belongs to HP (0 < P < 00) if the normTherefore, assuming I E L 2 for simplicity, we obtain

I.. (f' 1:, F(x. 'll' d}X ~ f ' [L I :, F(x.,)I'd+, = f ,[I..I:, F(~, 'll' d~] d, ~ I.. f ,I :, ~(,n"li«ll'd'd~

~ I.. (f'I:, ~('~)I' d' lli«ll'd~ (con,tl f.. li«ll' d~= < 00

(Here we make crucial use of curvature, which causes fl to decay at infinity, so that 2

the integral in curly brackets converges.) It follows that 1000 rl f, F(x, r)1 dr < 00 for almost every x E jRn. On the other hand, *It F(x, r)dr is easily seen to be the convolution of I with a standard approximate identity. Hence the usual

Lebesgue differentiation theorem shows that limR-->O *It F(x, r)dr = I(x)

for almost every x. So for almost all x E jRn, the function F (x, r) satisfies the hypotheses of the

elementary Tauberian theorem. Consequently,

1 l R

lim F(x, r) = lim - F(x, r)dr = I(x) r-->O R-->O R 0

almost everywhere, proving Stein's differentiation theorem for I E L 2(jRn).

To prove the full result for I E LP(jRn), P > n~l' we repeat the above argument with surface measure f.L replaced by an even more singular distribution on jRn .

Thus we obtain a stronger conclusion than asserted, when I E L 2. On the other hand, for I E L He we have a weaker result than that of Stein, namely Lebesgue's

differentiation theorem. Interpolating between L 2 and L 1+e , one obtains the Stein

differentiation theorem. The two results we picked out here are only a sample of the work of Stein

and others on curvature and the Fourier transform. For instance, J. Bourgain has

dramatic results on both the restriction problem and spherical averages. We refer the reader to Stein's address at the Berkeley congress [128] for a survey

of the field.

HP-SPACES

Another essential part of Fourier analysis is the theory of H P-spaces. Stein transformed the subject twice, once in a joint paper with Guido Weiss, and again in a joint paper with me. Let us start by recalling how the subject looked before Stein's work. The classical theory deals with analytic functions F (z) on

IIFIIHP == limr-->1_<!0211" !F(reilJ)lPdl:l) ~ is finite. The classical H P-spaces serve two main purposes. First, they provide growth

conditions under which an analytic function tends to boundary values on the unit circle. Secondly, H P serves as a substitute for LP to allow basic theorems on

Fourier series to extend from 1 O. To prove theorems about

F E H P, the main tool is the Blaschke product

nveilJvB(z) = Zv (11) z 1 - zvz '

where {zv} are the zeroes of the analytic function F in the disc, and l:Iv are suitable phases. The point is that B(z) has the same zeroes as F, yet it has absolute value 1 on the unit circle. We illustrate the role of the Blaschke product by sketching the proof of the Hardy-Littlewood maximal theorem for H p. The maximal theorem

says that IIF*llu :::: CpliFI/HP forO < p < 00, where F*(l:I) = sUPzer(lJ) !F(z)l, and r(l:I) is the convex hull of eilJ and the circle of radius! about the origin.

This basic result is closely connected to the pointwise convergence of F(z) as

z E rcl:l) tends to eilJ . To prove the maximal theorem, we argue as follows.

First suppose p > 1. Then we don't need analyticity of F. We can merely assume that F is harmonic, and deduce the maximal theorem from real variables. In fact, it is easy to show that F arises as the Poisson integral of an LP function I on

the unit circle. The maximal theorem for I, a standard theorem of real variables,

saysthatllMfllu :::: Cpll/llu,whereM/(l:I) = SUPh>o(2k I:~hh 1/(t)ldt). Itis quite simple to show that F*(l:I) < CM/(l:I). Therefore IIF*lIu :::: CIIMfllu < C'I!FIIHP, and the maximal theorem is proven for HP (p > 1).

If p :::: 1, then the problem is more subtle, and we need to use analyticity of F(z). Assume for a moment that F has no zeroes in the unit disc. Then for o < q < p, we can define a single-valued branch of (F(z))q, which will belong

to H ~ since F E H p. Since p == ~ > 1, the maximal theorem for H Pis already

known. Hence, maxzer«(I) I(F(z))ql E L ~ ,with norm

1I!.

71: (max I(F(z))ql) q dl:l :::: Cp,qllFql1 ~ I!. = Cp.ql!FlI~p.-11" zer(lJ) H q

That is, II F* II LP :::: Cp,q II F II HP, proving the maximal theorem for functions without zeroes.

To finish the proof, we must deal with the zeroes of an F E H P (p:::: 1). We bring in the Blaschke product B(z), as in (11). Since B(z) and F(z) have the same zeroes and since IB(z)1 = 1 on the unit circle, we can write F(z) = G(z)B(z)

with G analytic, and /G(z)1 = !F(z)1 on the unit circle. Thus, I/G!IHP = I/FIIHP,


';. ,~,

10 CHAPTER I I) SELECTED THEOREMS BY ELl STEIN:.:./ 11

Inside the circle. G has no zeroes and IB(z)1 :::: 1. Hence IFI :::: IGI. so

II max IF(z)llIu :::: II max IG(z)llIu S CpllGIIHP = CpIlFIIHP, ur~) ur~

by the maximal theorem for functions without zeroes. The proof of the maximal theorem is complete, (We have glossed over difficulties that should not enter an expository paper.)

Classically H p theory works only in one complex variable. so it is useful only for Fourier analysis in one real variable. Attempts to generalize H p to several complex variables ran into a lot of trouble. because the zeroes of an analytic function F(zl ... Zn) E HP fonn a variety V with growth conditions. Certainly V is much more complicated than the discrete set of zeroes {zv} in the disc. There is no satisfactory substitute for the Blaschke product. For a long time, this blocked all attempts to extend the deeper properties of H P to several variables.

Stein and Weiss [13] realized that several complex variables was the wrong generalization of H P for purposes of Fourier analysis. They kept clearly in mind what H P spaces are supposed to do. and they kept an unprejudiced view of how to achieve it. They found a version of H P theory that works in several variables.

The idea of Stein and Weiss was very simple. They viewed the real and imaginary parts of an analytic function on the disc as the gradient of a harmonic function. In several variables, the gradient of a harmonic function is a system u = (UI, Uz, ... , un) of functions on IRn that satisfies the Stein-Weiss Cauchy-Riemann equations

au} aUk " aUk(12) L.. ax =0.= ax}' k kaXk

In place of the Blaschke product, Stein and Weiss used the following simple observation. Ifu = (UI • , . un) satisfies (12), then lu\P = (UT + u~ +... + u~)! is subharmonic for p > ~::::i. We sketch the simple proof of this fact. then explain how an H P theory can be founded on it.

To see that lul P is subharmonic, we first suppose lui =I 0 and calculate Ll.(/il/P) in coordinates that diagonalize the symmetric matrix ( ~ ) at a given point. The result is

(13) MliW) = pluIP-z{lw/zlulz - (2 - p)liilz}'

'h £!!! dWIt Wk = BXt an Vk = UkWk.

Since L~=l Wk = 0 by the Cauchy-Riemann equations, we have

Z

:::: (n - 1) I: \wjlZ = (n - l)[wlz - (n - O!wkl z,IWkl 2 = II: Wj

j#j#

i.e., IWklz :::: n~l Iw1 2 • Hence Ivlz :::: (maxk IWkI2)luIZ :::: (n;;-l )lwlZlulZ, so

the expression in curly brackets in (13) is non-negative for p ~ ~::::i. and litjP is subharmonic.

So far, we know that lul P is subharmonic where it isn't equal to zero. Hence for 0< r < r(x) we have

(14) 'u(xW :::: AVly-xl=:rlu(yW,

provided lu(x)! #- O. However. (14) is obvious when ju(x)1 = o. so it holds for any x. That is, lul P is a subharmonic function for p ~ ~::::i, as asserted.

Now let us see how to build an H P theory for Cauchy-Riemann systems, based on subharmonicity of lul P , To study functions on IRn- 1 (n ~ 2), we regard IRn- 1

as the boundary of lR+ = {(XI ... xn)lxn > 0), and we define HP(IR+) as the space of all Cauchy-Riemann systems (UI, Uz, ... ,un) for which the norm

lIulltp = sup ( IU(xI ... xn-l,tWdxI ... dxn_1 1>0 JRn-1

is finite. For n = 2 this definition agrees with the usual H P spaces for the upper half-plane.

Next we show how the Hardy-Littlewood maximal theorem extends from the disc to lR+.

Define the maximal function M(u)(x) = sUPly_xl<t lu(y, 01 for x E IRn- l .

Then for U E HP(lR+), :=7 < P. we have M(u) E U(lRn- l ) with norm flRn-I (M(u)ydx :::: C/lullt•.

As in the classical case, the proof proceeds by reducing the problem to the maximal theorem for LP (p > 1). For small h > 0, the function Fh(x, t) = lu(x, t + h)1 Z=7 (x E lRn- 1

, t ~ 0) is subharmonic on lR+ and continuous up to the boundary. Therefore,

(15) Fh(X, t) :::: P.I.(fh),

where P.I. is the Poisson integral and h(x) = Fh(x,O) = lu(x, h)1 ~::7. By definition of the H P-norm, we have

1 l!h(xWdx- S lIulltp, with p = (n-l)(16) -- P > 1. /Rn-l n - 2

On the other hand. since the Poisson integral arises by convolving with an approximate identity. one shows easily that

(17) sup PJ.(fh)(Y, t) :::: CJ;(x)Iy-x/<t

with

H(x) = supr-(n-I) [ Ifh(Y)ldy (x E jRn-I). r>O J1x-YI<r


.. ,I

12 CHAPTER J SELECTED THEOREMS BY ELl STElN 13 The standard maximal theorem of real variables gives $

'r~!:; the harmonic conjugate v. Stein and I showed in [60] that the same thing happens

{ unp ~ Cp ( Ifhl P, J.Il.n-1 J.R.n-1

since jj > 1. Hence (15), (16), and (17) show that

1 ( SUP Fh(Y, t»)P dy ~ Cpllull~p, I.e. xERn-~ Iy-xl <t

(18) 1 (sup lu(y, t)I)P dy ~ Cpllull~p. xERn-1 Iy-xl+h<t

The constant Cp is independent of h, so we can take the limit of (18) as h --+ 0 to obtain the maximal theorem for HP. The point is that subharmonicity of lui ~::i substitutes for the Blaschke product in this argument.

Stein and Weiss go on in [13] to obtain n-dimensional analogues of the classical theorems on existence of boundary values of H P functions. They also extend to IR~ the classical F and M Riesz theorem on absolute continuity of H I boundary values. They begin the program of using H P (lR~) in place of U(lRn-I), to extend the basic results of Fourier analysis to p = 1 and below. We have seen how they deal with the maximal function. They prove also an H P-version of the Sobolev theorem.

It is natural to try to get below p = ~:::i, and this can be done by studying higher gradients of harmonic functions in place of (12). See Calder6n-Zygmund [CZ].

A joint paper [60] by Stein and me completed the task of developing basic Fourier analysis in the setting of the H P-spaces. In particular, we showed in [60] that singular integral operators are bounded on HP(lR~) for 0 < p < 00. We proved this by finding a good viewpoint, and we found our viewpoint by repeatedly changing the definition of H p. With each new definition, the function space H P

remained the same, but it became clearer to us what was going on. Finally we arrived at a definition of H P with the following excellent properties. First of all, it was easy to prove that the new definition of HP was equivalent to the Stein-Weiss definition and its extensions below p = ~:::i. Secondly, the basic theorems of Fourier analysis, which seemed very hard to prove from the original definition of HP(IR~), became nearly obvious in terms of the new definition. Let me retrace the steps in [60].

Burkholder-Gundy-Silverstein [BGS] had shown that an analytic function F = u + iv on the disc belongs to HP (0 < P < 00) if and only if the maximal function u*(B) = SUPZEf(6') lu(z)1 belongs to LP (Unit Circle). Thus, H P can be defined purely in terms of harmonic functions u, without recourse to

in n dimensions. That is, a Cauchy-Riemann system (UI, U2, •.• un) on IR~ belongs to the Stein-Weiss H P space (p > ~:::i) if and only if the maximal function u"(x) = sUPly_xl<t lun(y, t)1 belongs to LP(lRn - 1). (Here, the nth function Un plays a special role because lR~ is defined by (xn > 0).) Hence, HP may be viewed as a space of harmonic functions u(x, t) on IR~. The result extends below p = ::::i if we pass to higher gradients of harmonic functions.

The next step is to view H P as a space of distributions f on the boundary lRn

-I. Any reasonable harmonic function u(x, t) arises as the Poisson integral of a distribution f, so that u(x, t) = flJt * f(x), flJt = Poisson kernel. Thus, it is natural to say that f E H P (IRn - l ) if the maximal function

(19) f"'(x) == sup IflJt * f(Y)1 Iy-xl<t

belongs to LP. Stein and I found in [60] that this definition is independent of the choice of the approximate identity flJt, and that the "grand maximal function"

(20) Mf(x) = sup sup IflJl * f(Y)1 (rp,)EA Iy-xl</

belongs to LP, provided f E H p. Here A is a neighborhood of the origin in a suitable space of approximate identities. Thus, f E H P if and only if f* E U for some reasonable approximate identity. Equivalently, f E H P if and. only if the grand maximal function Mf belongs to LP. The proofs of these various equivalencies are not hard at all.

We have arrived at the good definition of H P mentioned above. To transplant basic Fourier analysis from LP (1 < p < (0) to HP (0 < P <

00), there is a simple algorithm. Take Calder6n-Zygmund theory, and replace every application of the standard maximal theorem by an appeal to the grand maximal function. Only small changes are needed, and we omit the details here. Our paper [60] also contains the duality of HI and BMO. Before leaving [60], let me mention an application of HP theory to LP -estimates. If a denotes uniform surface measure on the unit sphere in IRn , then f t----+ (fx r a * f is bounded

n on LP(lR ), provided n ~ 3 and I*- 41 :s 4 - n~I' Clearly, this result gives infonnation on solutions to the wave equation.

The proof uses complex interpolation involving the analytic family of operators

Ta : f t----+ (-ll.) j a * f (a complex),

as is clear to anyone familiar with the Stein interpolation theorem. The trouble here is that (-ll.) 1 fails to be bounded on L I when a is imaginary. This

makes it impossible to prove the sharp result (I *- 41 = 4 - n~l) using


14 15 CHAPTER 1

LP alone. To overcome the difficulty, we use HI in place of L I in the inter

polation argument. Imaginary powers of the Laplacian are singular integrals,

which we know to be bounded on H 1• To show that complex interpolation works

on H1, we combined the duality of H 1 and HMO with the auxiliary function

f#(x) = sUPQu ,b, fQ If(Y) - (meanQf)ldy. We refer the reader to [60] for

an explanation of how to use f#, and for other applications.

Since [60], Stein has done a lot more on H P , both in "higher rank" settings, and

in contexts related to partial differential equations.

REPRESENTATION THEORY n

Next we return to representation theory. We explain briefly how the Kunze-Stein

construction extends from SL(2, IR) to more general semisimple Lie groups, with

profound consequences for representation theory. The results we discuss are con

tained in the series of papers by Kunze-Stein [20], [22], [33], [63], Stein [35], [48],

[70], and Knapp-Stein [43], [46], [50], [53], [58], [66], [73], [93], [97]. Let G be

a semisimple Lie group, and let urr be the unitary principal series representations

of G, or one of its degenerate variants. The urr all act on a common Hilbert space,

whose inner product we denote by (~, 1]). We needn't write down urr here, nor

even specify the parameters on which it depends. A finite group W, the Weyl

group, acts on the parameters Jr in such a way that the representations urr and

uwrr are unitarily equivalent for w E W. Thus there is an intertwining operator A(w, Jr) so that

(21) A(w, Jr)U;,rr = U; A(w, Jr) for g E G, w E W, and for all Jr.

If urr is irreducible (which happens for most Jr), then A(w, Jr) is uniquely de

termined by (21) up to multiplication by an arbitrary scalar a(w, Jr). The crucial

idea is as follows. If the A(w, Jr) are correctly normalized (by the correct choice

of a(w, Jr)), then A(w, Jr) continues analytically to complex parameter values Jr .

Moreover, for certain complex (w, Jr), the quadratic form

(22) (~, 1])) w,rr = «TRIVIALFACTOR)A(w, Jr)~, 1])

is positive definite.

In addition, the representation urr (defined for complex Jr by a trivial analytic

continuation) is unitary with respect to the inner product (22). Thus, starting with

the principal series, we have constructed a new series of unitary representations of

G. These new representations generalize the complementary series for SL (2, IR).

Applications of this basic construction are as follows.

(1) Starting with the unitary principal series, one obtains understanding of the previously discovered complementary series, and construction of new ones,

, :,*


e.g., on Sp(4, C). Thus, Stein exposed a gap in a supposedly complete list of complementary series representations of Sp(4, C) [GN]. See [33].

(2) Starting with a degenerate unitary principal series, Stein constructed new ir

reducible unitary representations of SL(n, C), in startling contradiction to

the standard, supposedly complete list [ON] of irreducible unitary represen

tations of that group. Much later, when the complete list of representations of

SL(n, C) was given correctly, the representations constructed by Stein played an important role.

(3) The analysis of intertwining operators required to carry out analytic continua

tion also determines which exceptional values of Jr lead to reducible principal

series representations. For example, such reducible principal series represen

tations exist already for SL(n, IR), again contradicting what was "known". See Knapp-Stein [53].

A very recent result of Sahi and Stein [139] also fits into the same philosophy.

In fact, Speh's representation can also be constructed by a more complicated

variant of the analytic continuation defining the complementary series. Speh's

representation plays an important role in the classification of the irreducible unitary representations of SL(2n, IR).

The main point of Stein's work in representation theory is thus to analyze the

intertwining operators A(w, Jr). In the simplest non-trivial case, A (w, Jr) is a

singular integral operator on a nilpotent group N. That is, A(w, Jr) has the form

(23) Tf(x) = 1K(xy-1)f(y)dy,

with K (y) smooth away from the identity and homogeneous of the critical degree with respect to "dilations"

(Ot)t>O: N ---+ N.

In (23), dy denotes Haar measure on N. We know from the classical case N = JR.l that (23) is a bounded operator only when the convolution kernel K (y) satisfies a

cancellation condition. Hence we assume fBI \8 K (y)dy = 0, where the Bi are 0

dilates (Bj = Oti (B)) of a fixed neighborhood of the identity in N.

It is crucial to show that such singular integrals are bounded on L 2 (N), generalizing the elementary L2-boundedness of the Hilbert transfonn.

COTLAR·STEIN LEMMA

In principle, L 2-boundedness of the translation-invariant operator (23) should be

read off from the representation theory of N. In practice, representation theory provides a necessary and sufficient condition for £2-boundedness that no one


,'.;....•.

16 CHAPTER 1 1tl SELECTED THEOREMS BY ELl STEIN 17

knows how to check. This fundamental analytic difficulty might have proved

fatal to the study of intertwining operators. Fortunately, Stein was working si

multaneously on a seemingly unrelated question, and made a discovery that saved

the day. Originally motivated by desire to get a simple proof of Calderon's the

orem on commutator integrals [Cal, Stein proved a simple, powerful lemma in

functional analysis. His contribution was to generalize to the critically important

non-commutative case the remarkable lemma of Cotlar [Co]. The Cotlar-Stein

lemma turned out to be the perfect tool to prove L 2-boundedness of singular in

tegrals on nilpotent groups. In fact, it quickly became a basic, standard tool in

analysis. We will now explain the Cotlar-Stein lemma, and give its amazingly sim

ple proof. Then we will return to its application to singular integrals on nilpotent

groups.

The Cotlar-Stein lemma deals with a sum T = Lv Tv of operators on a Hilbert

space. The idea is that if the Tv are almost orthogonal, like projections onto the

various coordinate axes, then the sum T will have norm no larger than maxv II Tv II. The precise statement is as follows.

Cotlar-Stein Lemma. Suppose T = L~l Tk is a sum oloperators on Hilbert space. Assume IIT/Tkll :s a(} - k) and IITjTtII :s a(j - k). Then IITII :s L~M Ja(j).

Proof 1IT11 = <IITT*II)!,so

II T11 2s :s II (TT*)' II = II LM

Th Tj:- .. Tj2s_,Tj: h···j2s=1

(24)

< LM

11 Th Tj: ... Tjz,_, Tj:, II· jl ... jz,=1

We can estimate the summand in two different ways.

Writing Tj, T; TjzH Tj: = (Tj, Tj:)(Tj) Tj:) ... (Th,_, TL), we get

(25) IIThTj: Tj2s_,Tj:1I ~ a(h - 12)a(13 - }4) ···a(j2s-1 - 12s)'

On the other hand, writing

Tj, Tj: ... Tjz'_l Tj:, = Tj, (Tj: Tj) )(Tj: Tjl ) ... (Tj:,_Jj2s-I) Tj:,.

we see that

(26) II Th Tj: ... Th-l Tj:, II 2

:s (mJx IITjll) a (12 - h)a(j4 - }s)··· a(12s-2 - hs-d·

Taking the geometric mean of (25), (26) and putting the result into (24), we

conclude that

1IT11 2r ~ . t (mJx II Tj ll) v'aV1 - h) /a(h - h)'" J 1I ...J2s=1

" ("'j'" UTjll . M) (~,ja(l) r' I 2.r-l

Thus, II Til :s (maxj II Tjll . M) 2> • (Ll ,Ja(l))"17 . Letting s ---+ 00, we obtain the conclusion of the Cotlar-Stein Lemma. •

To apply the Cotlar-Stein lemma to singular integral operators, take a partition

of unity I = Lv qJv(X) on N, so that each qJv is a dilate of a fixed CO' function

that vanishes in a neighborhood of the origin. Then T: I ---+ K * I may be

decomposed into a sum T = Lv Tv, with Tv: I ---+ (qJvK) * I. The hypotheses of the Cotlar-Stein lemma are verified trivially, and the boundedness of singular

integral operators follows. The L2-boundedness of singular integrals on nilpotent groups is the Knapp-Stein Theorem.

Almost immediately after this work, the Cotlar-Stein lemma became the stan

dard method to prove L2 boundedness of operators. Today one knows more, e.g.,

the T (l) theorem of David and Journe. Still it is fair to say that the Cotlar-Stein lemma remains the most important tool for L2-boundedness.

Singular integrals on nilpotent groups were soon applied by Stein in a context seemingly far from representation theory.

a-PROBLEMS

We prepare to discuss Stein's work on the a-problems of several complex variables

and related questions. Let us begin with the state of the subject before Stein's

contributions. Suppose we are given a domain D C en with smooth boundary. If we try to construct analytic functions on D with given singularities at the boundary, then we are led naturally to the following problems.

I. Given a (0, 1) form a = L~=I Ikdzk on D, find a function u on D that solves

au = a, where au = L~=I :;. ~Zk. Naturally, this is possible only if a satisfies the consistency condition oa = 0, i.e. at h = at Ik. Moreover, u is determined only modulo addition of an arbitrary analytic function on D.

To make u unique, we demand that u be orthogonal to analytic functions in L 2(D).


18 19 CHAPTER I

II. There is a simple analogue of the a-operator for functions defined only on the boundary aD. In local coordinates. we can easily find (n - 1) linearly independent complex vector fields L I ... L n - I of type (0,1) (i.e., L j = ajl ,~ +aj2 ,~ +.. .+ajn ,~ forsmooth,complex-valuedajk)whosereal

liZ I aZ2 uZn

and imaginary parts are all tangent to aD. The restriction u of an analytic function to aD clearly satisfies abu = 0, where in local coordinates abU = (Llu, L 2u, ...• Ln_IU), The boundary analogue of the a-problem (I) is the inhomogeneous ab-equation abU = a. Again, this is possible only if a

satisfies a consistency condition aba = 0, and we impose the side condition that U be orthogonal to analytic functions in L 2(aD).

Just as analytic functions ofone variable are related to harmonic functions, so the first-order systems (I) and (II) are related to second-order equations 0 and Db, the a-Neumann and Kohn Laplacians. Both fall outside the scope of standard elliptic theory. Even for the simplest domains D, they posed a fundamental challenge to workers in partial differential equations. More specifically, 0 is simply the Laplacian in the interior of D, but it is subject to non-elliptic boundary conditions. On the other hand, Db is a non-elliptic system of partial differential operators on aD, with no boundary conditions (since aD has no boundary). Modulo lower-order terms (which, however, are important), Db is the scalar operator £, = L~::(Xi + Y{), where Xk and Yk are the real and imaginary parts of the basic complex vector fields Lk. At a given point in aD, the X k and Yk are linearly independent, but they don't span the tangent space of aD. This poses the danger that £, will behave like a partial Laplacian such as 6' = .;; + :;2 acting on function u(x, y, z).

The equation 6'U = f is very bad. For instance, we can take U (x, y, z) to depend on z alone, so that N U = 0 with U arbitrarily rough. Fortunately, £, is more like the full Laplacian than like 6', because the Xk and Yk together with their commutators [Xb Yk] span the tangent space of aD for suitable D. Thus, £, is a well-behaved operator, thanks to the intervention of commutators of vector

fields. It was Kohn in the 1960's who proved the basic Coo regularity theorems for 0,

Db, a and ab on strongly pseudoconvex domains (the simplest case). His proofs

were based on subelliptic estimates such as (DbW, w) ~ cllwlI1E) - CllwlI 2 , and

brought to light the importance of commutators. Hormander proved a celebrated theorem on Coo regularity of operators,

N

L = LX; +xo, j=1

where Xo, XI, ... , XN are smooth, real vector fields which, together with their repeated commutators, span the tangent space at every point.


If we allow Xo to be a complex vector field, then we get a very hard problem that is not adequately understood to this day, except in very special cases.

Stein made a fundamental change in the study of the the a-problems by bringing in constructive methods. Today. thanks to the work of Stein with several collaborators, we know how to write down explicit solutions to the a-problems modulo negligible errors on strongly pseudoconvex domains. Starting from these explicit solutions, it is then possible to prove sharp regularity theorems. Thus, the aequations on strongly pseudoconvex domains are understood completely. It is a major open problem to achieve comparable understanding ofweakly pseudoconvex

domains. Now let us see how Stein and his co-workers were able to crack the strongly

pseudoconvex case: We begin with the work of Folland and Stein [67]. The simplest example of a strongly pseudoconvex domain is the unit ball. Just as the disc is equivalent to the half-plane, the ball is equivalent to the Siegel domain

DSiegel = {(z, w) E en-I x c I 1m W > IzI2}. Its boundary H = aDSiegel has an important symmetry group, including the following.

(a) Translations (z, w) f-----+ (z, w) . (z', w') == (z + z', W + Wi + 2iz . Z') for (Zl, Wi) E H;

(b) Dilations {)t: (z, w) f-----+ (tz, t 2w) for t > 0; (c) Rotations (z, w) f-----+ (Uz, w) for unitary (n - 1) x (n - 1) matrices U.

The multiplication law in (a) makes H into a nilpotent Lie group, the Heisenberg group. Translation-invariance of the Siegel domain allows us to pick the basic complex vector-fields L 1 ••• L n- I to be translation-invariant on H. After we make a suitable choice of metric, the operators £, and Db become translation- and rotation-invariant, and homogeneous with respect to the dilations {)t. Therefore, the solution I of Db W = a should have the form of a convolution W = K * a f,)Jl the Heisenberg group. The convolution kernel K is homogeneous with respect

. to the dilations {)t and invariant under rotations. Also, since K is a fundamental SOlution, it satisfies Db K = 0 away from the origin. This reduces to an elementary ODE after we take the dilation- and rotation-invariance into account. Hence one can easily find K explicitly and thus solve the Db-equation for the Siegel domain.

. To derive sharp regularity theorems for Db, we combine the explicit fundamental 8olution with the Knapp-Stein theorem on singular integrals on the Heisenberg group. For instance, ifDbW E L 2, then LjLkW, IjLkw, Ljlkw, and Ijlkw all belong to L 2• To see this, we write

DbW = a, W = K * a, LjLkW = (LjLkK) * a,

lKohn's work showed that ObW = a has a solution if we are in complex dimension> 2. In two complex dimensions, ObW = a has no solution for most a. We assume dimension> 2 here.


20 21 CHAPTER 1

and note that L j Lk K has the critical homogeneity and integral O. Thus L j LkK is a singular integral kernel in the sense of Knapp and Stein, and it follows that

ilL j Lk W II :5 C lIa II. For the first time, nilpotent Lie groups have entered into the study of a-problems.

e

Folland and Stein viewed their results on the Heisenberg group not as ends in themselves, but rather as a tool to understand general strongly pseudoconvex CR

manifolds. A CR-manifold M is a generalization of the boundary of a smooth domain D C en. For simplicity we will take M = aD here. The key idea is that

near any point W in a strongly pseudoconvex M, the CR structure for M is very nearly equivalent to that of the Heisenberg group H via a change of coordinates

w : M ~ H. More precisely, ew carries w to the origin, and it carries the CR-structure on M to a CR-structure on H that agrees with the usual one at the

origin. Therefore, if w = K * a is our known solution of Db w = a on the Heisenberg group, then it is natural to try

(27) w(z) = LK(ew(z»a(w)dw

as an approximate solution of Db w = a on M. (Since w and a are sections of bundles, one has to explain carefully what (27) really means.) If we apply Db to the w defined by (27), then we find that .

(28) DbW = a - fa,

where £ is a sort of Heisenberg version of (-L:l.)- ! . In particular, £ gains smoothness, so that (/ - £)-1 can be constructed modulo infinitely smoothing operators

by means of a Neumann series. Therefore (27) and (28) show that the fulJ solution

of Db W = a is given (modulo infinitely smoothing errors) by

(29) w(z) = L 1K(ew(z»(£ka)(w)dw, k M

from which one can deduce sharp estimates to understand completely 0; I on M. The process is analogous to the standard method of "freezing coefficients" to

solve variable-coefficient elliptic differential equations. Let us see how the sharp results are stated. As on the Heisenberg group, there are smooth, complex vector fields Lk that span the tangent vectors of type (0, 1) locally. Let X j be the real and imaginary parts of the L k • In terms of the Xj we define "non-Euclidean"

versions of standard geometric and analytic concepts. Thus, the non-Euclidean ball JB(z, p) may be defined as an ellipsoid with principal axes oflength '" p in the codimension 1 hyperplane spanned by the X j, and length'" p2 perpendicular to that hyperplane. In terms of JB(z, p), the non-Euclidean Lipschitz spaces r a (M)

SELECTED THEOREMS BY ELl STElN

are defined as the set of functions u for which lu(z) - u(w)1 < Cpa for w E

B(z, p). (Here, 0 < a < 1. There is a natural extension to all a > 0.) The non-Euclidean Sobolev spaces Sm,p(M) consist of all distributions u for which all

X · X· ... X}· u E LP(M) forO < s -< m.

11}2.

Then the sharp results on Db are as follows. IfDbW = a and a E Sm.p(M),

then w E Sm+2.p(M) for m ~ 0, 1 < p < 00, If DbW = a and a E ra(M),

then XjXkW E ra(M) for 0 < a < I (say). For additional sharp estimates, and for comparisons between the non-Euclidean and standard function spaces, we refer the reader to (67].

To prove their sharp results, Folland and Stein developed the theory of singular integral operators in a non-Euclidean context. The Cotlar-Stein lemma proves the crucial results on L2-boundedness of singular integrals. Additional difficulties

ariseJrom the non-commutativity of the Heisenberg group. In particular, standard singular integrals or pseudodifferential operators commute modulo lower-order errors, but non-Euclidean operators are far from commuting. This makes more difficult the passage from LP estimates to the Sobolev spaces Sm,p(M).

Before we continue with Stein's work on a, let me explain the remarkable paper of Rothschild-Stein {72]. It extends the Folland-Stein results and viewpoint to

general Hormander operators L = 1:.,7=1 XJ + Xo. Actually, {72] deals with systems whose second-order part is 1:., j XJ, but for simplicity we restrict attention here to L. In explaining the proofs, we simplify even further by supposing Xo = O.

The goal of the Rothschild-Stein paper is to use nilpotent groups to write down art explicit parametrix for L and prove sharp estimates for solutions of LU = f. This ambitious hope is seemingly dashed at once by elementary examples. For.', tak r 2 2·mstance, e J.- = X I + X 2 WIth

a a 2XI = -, X2 = X - on 1R.ax ay

. Then XI and [XI, X2] span the tangent space, yet L clearly cannot be approximated ";by translation-invariant operators on a nilpotent Lie group in the sense of Folland

~in. The trouble is that L changes character completely from one point to another. ;"way from the y-axis {x = O}, L is elliptic, so the only natural nilpotent group .~ can reasonably use is 1R2. On the y-axis, L degenerates, and evidently cannot i~approximated by a translation-invariant operator on 1R2. The problem is so ."'viously fatal, and its solution by Rothschild and Stein so simple and natural, that [72] must be regarded as a gem. Here is the idea:

Suppose we add an extra variable t and "lift" X I and X 2 in (30) to vector fields

(31) - a - a aX I =-, X2 =x-+- on 1R3

•ax ay at


22 23 CHAPTER 1

Then the Honnander operator '£ = xf + xi looks the same at every point of 1R3 , and may be readily understood in tenns of nilpotent groups as in Folland-Stein [67]. In particular, one can essentially write down a fundamental solution and prove sharp estimates for '£-1. On the other hand, '£ reduces to .c whe~ acting on functions u (x, y, t) that do not depend on t. Hence, sharp results on.cu = f

imply sharp results on .cu = f. Thus we have the Rothschild-Stein program: First, add new variables and lift

the given vector fields XI'" X N to new vector fields XI'" XN whose underlying structure does not vary from point to point. Next, approximate '£ = Lf XJ by a translation-invariant operator l = ~f YJ on a nilpotent Lie group N. Then analyze the fundamental so~ution of .c, and use it to write down an approximate fundamental solutionlor.c. From the approximate solution, derive sharp estimates for solutions of.cu = f. Finally, descend to the original equation .cu = f by restricting attention to functions u, f that do not depend on the extra variables.

To carry out the first part of their program, Rothschild and Stein prove the following result.

Theorem A. Let XI ... XN be smooth vector fields on a neighborhood ofthe ori

gin in JRn. Assume that the X j and their commutators [ [ [Xjt' X jJ, Xh] ... , X j,J

oforder up to r span the tangent space at the origin. Then we can find smooth vectorfields XI ... XN on a neighborhood iJ ofthe origin in JRn+m with the following

propertie~

(a) The X j and t,!:ir commutators up to order r are linearly independent at

each point of U, except for the linear relations that follow formally from the

antisrmmetry ofthe bracket and the Jacobi identity. _

(b) The X j and their commutators up to order r span the tangent space of U.

(c) Acti~g onfunctions on JRn+m that do not depend on the last m coordinates,

the X j reduce to the given X j'

Next we need a nilpotent Lie group N appropriate to the vector fields X I ... XN.

The natural one is the free nilpotent group NNr of step r on N generators. Its Lie algebra is generated by YI •.• YN whose Lie brackets of order higher than r vanish, but whose brackets of order S r are linearly independent, except for relations forced by antisymmetry of brackets and the Jacobi identity. We regard the Yj as translation-invariant vector fields on NNr' It is convenient to pick a basis {Ya}aEA for the Lie algebra ofNNr. consisting of YI ••. YN and some of their commutators.

On NNr we fonn the Honnander operator Z = Lf YJ. Then Zis trans~~tioninvariant and homogeneous under the natural dilations on NNr' Hence .c- I is given by convolution on NNr with a homogeneous kernel K (-) having a weak

SELECTED THEOREMS BY ELl STEIN

singularity at the origin. HypoeIIipticity of Zshows that K is smooth away from the origin. Thus we understand the equation Zu = f very well.

We want to use Z to approximate Zat each point y E iJ. To do so, we have to identify a neighborhood of y in iJ with a neighborhood of the origin in NNr' This has to be done just right, or else Z will fail to approximate Z. The idea is to use exponential coordinates on both iJ and NNr' Thus, if x = exp(I:aEA taY~ (identity) E NN" then we3se (t0aEA as coordinates for x. Similarly, let (Xa)aEA be the commutators of X I ... X N analogous to the Ya. and let y E iJ be given. Then given a nearby point x = exp(LaEA taXa)y E iJ, we use (fa )aEA as coordinates for x.

Now we can identify iJ with a neighborhood of the identity in NNr, simply by identifying points with the same coordinates. Denote the identification by

By: iJ - NNr. and note that By(Y) = identity. In view of the identification e y' the operators l and Zlive on the same space.

The next step is to see that they are approximately equal. To fonnulate this, we

need some bookkeeping on the nilpotent group NNr. Let {8t }t>o be the natural dilationsonNNr' Ifrp E C8"(NNr),thenwriterptforthefunctionx ~ 1p(8t x). When rp is fixed and t is large, then rpt is supported in a tiny neighborhood of the

, identity. Let D be a differential operator acting on functions on NNr. We say that '1) has "degree" at most k if for each rp E C8"(NNr) we have ID(lpt) I = O(tk) for large, positive t. According to this definition, YI , ... , YN have degree I while fYj, Yk ] has degree 2, and the degree of a(x) [Yj , Yk] depends on the behavior of a(x) near the identity. Now we can say in what sense '£ and Zare approximately

,'equal. The crucial result is as follows.

Theorem B. Under the map 8;1, the vector field Xj pulls back to Yj + Zy,j,

where Zy,j is a vector field on NNr of "degree" S O.

" Using Theorem B and the map 8 y • we can produce a parametrix for Zand prove Libat it works. In fact, we take

K(x, y) = K(eyx),

.... ~here K is the fundamental solution of Z. For fixed y, we want to know that

ZK(x, y) = 8y (x) + E(x, y),

Where 8y (') is the Dirac delta-function and E(x, y) has only a weak singularity at - NX = y. To prove this, we use e y to pull back to NNr. Recall that.c = LI X;

-.. N 2while.c = LI y • Hence by Theorem B, .c pulls back to an operator of the fonn .c --+ D y , with D y

J having "degree" at most l. Therefore (33) reduces to proving


24 25 CHAPTER 1

that

(34) (l + Vy)K(x) = Oid.(X) + [(x),

where [has only a weak singularity at the identity. Since £K(x) = Oid.(X), (34) means simply that VyK(x) has only a weak singularity at the identity. However, this is obvious from the smoothn~ss and homogeneity of K (x), and from the fact that Py has degree ~ 1. Thus, K(x, y) is an approximate fundamental solution

for £.. From the explicit fundamental solution for the lifted operator £., one can

"descend" to deal with the original Hormander operator £. in two different ways.

a. Prove sharp estimates for the lifted problem, then specialize to the case of functions that don't depend on the extra variables. _

b. Integrate out the extra variables from the fundamental solution for £., to obtain a fundamental solution for £..

Rothschild and Stein used the first approach. They succeeded in proving the

estimate

(35) IIXoullu(u) + IIXjXkullu(u) ~ Cp II (t X] + xo) ull + J I U(V)

Cpllullu(v) for 1 < P < 00 and U cc V.

This is the most natural and the sharpest estimate for Hormander operators. It was new even for p = 2. Rothschild and Stein also proved sharp estimates in spaces analogous to the r a and Sm,p of Folland-Stein [67], as well as in standard Lipschitz and Sobolev spaces. We omit the details, but we point out that commuting derivatives past a general Hormander operator here requires additional ideas.

Later, Nagel, Stein, and Wainger [119] returned to the second approach ("b" above) and were able to estimate the fundamental solution of a general Hormander operator. This work overcomes substantial problems.

In fact, once we descend from the lifted problem to the original equation, we again face the difficulty that Hormander operators cannot be modelled directly on nilpotent Lie groups. So it isn't even clear how to state a theorem on the fundamental solution of a Hormander operator. Nagel, Stein and Wainger [119] realized that a family of non-Euclidean "balls" Bc(x, p) associated to the Hormander operator £. plays the basic role. They defined the Bc(x, p) and proved their essential properties. In particular, they saw that the family of balls survives the projection from the lifted problem back to the original equation, even though the nilpotent Lie group structure is destroyed. Non-Euclidean balls had already played an important part in Folland-Stein [67]. However, it was simple in [67] to guess the correct family of


balls. For general Hormander operators £. the problem of defining and controlling non~Euclidean balls is much more subtle. Closely related results appear also in [FKP], [FS].

Let us look first at a nilpotent group such as NNr, with its family of dilations {or boo Then the correct family of non-Euclidean balls BNN, (x, p) is essentially dictated by translation and dilation-invariance, starting with a more or less arbitrary hannless "unit ball" BNN, (identity, 1). Recall that the fundamental solution for

Z = L~ YJ on NNr is given by a kernel K (x) homogeneous with respect to the 0,. Estimates that capture the size and smoothness of K(x) may be phrased

.' ~tirely in terms of the non-Euclidean balls BNN, (x, p). In fact, the basic estimate . '. is as follows.

C 2-m mpIy· y . ... y. K(x)1 <

JI J2 Jm - 1B (0 )vo NN, ,p

for x E BNN,(O, p) " BNN, (0, ~) and m ~ O. ,. Next we associate non-Euclidean balls to a general Hormander operator. For

!.,simplicity, take £. = L~ X] as in our discussion of Rothschild-Stein [72]. One definition of the balls Bc(x, p) involves a moving particle that starts at x and

"'ftavels along the integral curve of Xh for time t1. From its new position x' the particle then travels along the integral curve of Xh for time t2. Repeating the ptoeess finitely many times, we can move the particle from its initial position x to

i,final position y in a total time t = tl + ... + tm . The ball Be (x, p) consists of ~l ythat can be reached in this way in time t < p. For instance, if £. is elliptic,

Be (x, p) is essentially the ordinary (Euclidean) ball about x of radius p. If

take £. ~

= LIN YJ on NN,r, then the balls Bz<x, p) behave naturally under

translations and dilations; hence they are essentially the same as the BNN,,(X, p)

~aring in (36). Nagel-Wainger-Stein analyzed the relations betwee~Bz(x)... p),IIZ<x. p) and Be (x, p) for an arbitrary Hormander operator £.. (Here £. and £. are ~in our previous discussion of Rothschild-Stein.) This allowed them to integrate <, <It the extra variables in the fundamental solution of E, to derive the following

arp estimates from (36).

!eorem. Suppose XI'" X N and their repeated commutators span the tangent

.,e. A/so, suppose we are in dimension greater than 2. Then the solution of

tEf XJ)u = f is given by u(x) = J K(x, y)f(y)dy with

C 2-m mp for x E Bc(y, p) " Be (y, ~)(vol Bc(y, p»

andm ~ O.


27

.'

lUi

II 26 CHAPTER I

Here the X ji act either in the x- or the y-variable. 111

II Let us return from Honnander operators to the a-problems on strongly pseu

doconvex domains D C en. Greiner and Stein derived sharp estimates for the Neumann Laplacian Ow = a in their book [78]. This problem is hard, because

I two different families ofballs play an important role. On the one hand, the standard I

(Euclidean) balls arise here, because 0 is simply the Laplacian in the interior of 11I

D. On the other hand, non-Euclidean balls (as in Folland-Stein [67]) arise on aD,I[ because they are adapted to the non-elliptic boundary conditions for O. Thus, any

I understanding of 0 requires notions that are natural with respect to either family ofI 1

balls. A key notion is that of an allowable vector field on D. We say that a smooth I vector field X is allowable if its restriction to the boundary aD lies in the span r

II

of the complex vector fields LI '" L n- l • L 1 ••• Ln_ l • Here we have retained the notation of our earlier discussion ofa-problems. At an interior point, an allowable II vector field may point in any direction, but at a boundary point it must be in the natural codimension-one subspace of the tangent space of aD. Allowable vector fields are well-suited both to the Euclidean and the Heisenberg balls that control O. The sharp estimates of Greiner-Stein are as follows.

II

Theorem. Suppose Ow = a on a strictly pseudoconvex domain D C en. If

I a belongs to the Sobolev space Lf. then w belongs to Lf+1 (I < P < 00).

II Moreover, if X and Yare allowable vectorfields, then XY w belongs to Lf. Also. Lw belongs to Lf+l ifL is a smooth complex vector field oftype (0, 1). Similarly,I if a belongs to the Lipschitz space Lip(f.l) (0 < f.l < I), then the gradient of w

I III belongs to Lip(f.l) as well. Also the gradient of Lw belongs to Lip(f.l) if L is a

smooth complex vector field oftype (0, 1); and XYw belongs to Lip(f.l)for X and Y allowable vector fields. Iii

These results for allowable vector fields were new even for L 2 . We sketch the III

proof.II Suppose Ow = a. Ignoring the boundary conditions for a moment, we have II

/::"w = a in D, so

II11I1

(37) w = Ga + P.I.(w) illlIII where wis defined on aD, and G, P.I. denote the standard Green's operator and

I

Poisson integral, respectively. The trouble with (37) is that we know nothing II about wso far. The next step is to bring in the boundary condition for Ow = a.

According to Calderon's work on general boundary-value problems, (37) satisfies the a-Neumann boundary conditions if and only if

(38) Aw = (B(Ga)} Ian


for a certain differential operator B on D, and a certain pseudodifferential operator A on aD. Both A and B can be detennined explicitly from routine computation.

Greiner and Stein [78] derive sharp regularity theorems for the pseudodifferential equation Aw = g, and then apply those results to (38) in order to understand iii in terms of a. Once they know sharp regularity theorems for W, fonnula (37) gives the behavior of w.

Let us sketch how Greiner-Stein analyzed Aw = g. This is really a system of n pseudodifferential equations for n unknown functions (n = dim en). In a suitable frame, one component of the system decouples from the rest of the problem (modulo negligible errors) and leads to a trivial (elliptic) pseudodifferential

.equation. The non-trivial part of the problem is a first-order system of (n - I)

pseudodifferential operators for (n - I) unknowns, which we write as

O+w# = a#.

Here a' consists of the non-trivial components of (B(Ga)} IaD, w# is the unknown, and 0+ may be computed explicitly.

Greiner and Stein reduce (39) to the study of the Kohn-Laplacian Db. In fact, they produce a matrix 0_ of first-order pseudodifferential operators similar to 0+,

"and then show that 0_0+ = Db modulo negligible errors.2 Applying 0_ to (39) ' ..,yields

. (40) ObW# = O_a# + negligible.

;Prom Folland-Stein [67] one knows an explicit integral operator K that inverts Db ; modulo negligible errors. Therefore,

~'(41) w# = KO_a# + negligible.

~uations (37) and (41) express w in tenns ofa as a composition of various explicit Operators, inclUding: the Poisson integral; restriction to the boundary; 0_; K; G. Because the basic notion of allowable vector fields is well-behaved with respect to both the natural families of balls for Ow = a, one can follow the effect of each ,of these very different operators on the relevant function spaces without losing

~ormation. To carry this out is a big job. We refer the reader to [78] for the rest (of the story.

There have been important recent developments in the Stein program for several <lOmplex Variables. In particular, we refer the reader to Phong 's paper in this volume for a discussion of singular Radon transfonns; and to Nagel-Rosay-Stein-Wainger

2This procedure requires significant changes in two complex variables, since then Db isn't invertible.


29

__

I 1\

I

I'III

11

II II 11

Ii ~ ~ I~ 111,1

~ III(

'Ill..

III

~ 1

11.II1I

I1

1\1II1

1

1\11

'~IIII

28 CHAPTER I

[131], D.-C. Chang-Nagel-Stein [132], and [McNl, [Chr], [FK] for the solution of

the a-problems on weakly pseudoconvex domains of finite type in C2 •

Particularly in several complex variables are we able to see in retrospect the

fundamental interconnections among classical analysis, representation theory, and

partial differential equations, which Stein was the first to perceive.

I hope this article has conveyed to the reader the order of magnitude of Stein's

work. However, let me stress that it is only a selection, picking out results which

I could understand and easily explain. Stein has made deep contributions to many

other topics, e.g.,

Limits of sequences of operators

Extension of Littlewood-Paley Theory from the disc to ]Rn

Differentiability of functions on sets of positive measure

Fourier analysis on]RN when N -+ 00

Function theory on tube domains

Analysis of diffusion semigroups

Pseudodifferential calculus for subelliptic problems.

The list continues to grow.

Princeton University

BIBLIOGRAPHY OF E. M. STEIN

1. "Interpolation of linear operators." Trans. Amer. Math. Soc. 83 (1956), 482-492. 2. "Functions of exponential type." Ann. ofMath. 65 (1957), 582-592. 3. "Interpolation in polynomial classes and Markoff's inequality." Duke Math. J. 24

(1957),467-476. 4. "Note on singular integrals." Proc. Amer. Math. Soc. 8 (1957), 250-254. 5. (with G. Weiss) "On the interpolation of analytic families of operators action on HP

spaces." Tohoku Math. J. 9 (1957),318-339. 6. (with E.H. Ortrow) "A generalization of lemmas of Marcinkiewicz and Fine with

applications to singular integrals." Annuli Scula Normale Superiore Pisa 11 (1957), 117-135.

7. "A maximal function with applications to Fourier series." Ann. of Math. 68 (1958), 584--{j03.

8. (with G. Weiss) "Fractional integrals on n-dimensional Euclidean space." J. Math. Mech. 77 (1958), 503-514.

9. (with G. Weiss) "Interpolation of operators with change of measures." Trans. Amer. Math. Soc. 87 (1958), 159-172.

10. "Localization and summability of multiple Fourier series." Acta Math. 100 (1958), 93-147.

11. "On the functions of Linlewood-Paley, Lusin, Marcinkiewicz:' Trans. Amer. Math. Soc. 88 (1958), 43Q-466.


12. (with G. Weiss) "An extension of a theorem of Marcinkiewicz and some of its applications." J. Math. Mech. 8 (1959), 263-284.

13. (with G. Weiss) "On the theory of harmonic functions of several variables I, The theory of HP spaces." Acta Math. 103 (1960), 25-62.

14. (with R. A. Kunze) "Uniformly bounded representations and harmonic analysis of the 2 x 2 real unimodular group." Amer. J. Math. 82 (1960),1-62.

15. "'The characterization of functions arising as potentials." Bull. Amer. Math. Soc. 67(1961), 102-104; n, 68 (1962), 577-582.

16. "On some functions of Littlewood-Paley and Zygmund." Bull. Amer. Math. Soc. 67 (1961),99-101.

17. "On limits of sequences of operators." Ann. ofMath. 74 (1961), 140-170. 18. "On the theory of harmonic functions of several variables II. Behavior near the

boundary." Acta Math. 106 (1961),137-174. 19. "On certain exponential sums arising in multiple Fourier series." Ann. of Math. 73

(1961),87-109. 20. (with R. A. Kunze) "Analytic continuation of the principal series." Bull. Amer. Math.

Soc. 67 (1961),543-546. 21. "On the maximal ergodic theorem." Proc. Nat. Acad. Sci. 47 (1961), 1894-1897. 22. (with R. A. Kunze) "Uniformly bounded representations II. Analytic continuation of

the principal series of representations of the n x n complex unimodular groups." Amer. J. Math. 83 (1961), 723-786.

23. (with A. Zygmund) "Smoothness and differentiability of functions." Ann. Univ. Sci. . Budapest, Sectio Math., Ill-IV (1960-61),295-307.

24. "Conjugate harmonic functions in several variables." Proceedings of the International Congress of Mathematicians, Djursholm-Linden, Instut Mittag-Leffler (1963), 414420.

25. (with A. Zygmund) "On the differentiability of functions." Studia Math. 23 (1964), 248-283.

. -,·26, (with G. and M. Weiss) HP-classes of hoiomorphic functions in tube domains." Proc. Nat. Acad. Sci. 52 (1964), 1035-1039.

.~'21. (with B. Muckenhoupt) "Classical expansions and their relations to conjugate functions." Trans. Amer. Math. Soc. 118 (1965), 17-92.

!~8. "Note on the boundary ofhoiomorphic functions." Ann. ofMath. 82 (1965), 351-353. ~. (with S. Wainger) "Analytic properties of expansions and some variants of Parseval-

Plancherel formulas." Arkiv. Math., Band 537 (1965),553-567. <·;;30. (with A. Zygmund) "On the fractional differentiability of functions." London Math.

Soc. Proc. 34A (1965), 249-264. 31. "Classes H 2, multiplicateurs, et fonctions de Littlewood-Paley." Comptes Rendues

Acad. Sci. Paris 263 (1966), 716-719; 780-781; also 264 (1967), 107-108. ·32, (with R. Kunze) "Uniformly bounded representations III. Intertwining operators."

Amer. J. Math. 89 (1967), 385-442. 33. "Singular integrals, harmonic functions and differentiability properties of functions

of several variables." Proc. Symp. Pure Math. 10 (1967), 316-335. 34. "Analysis in matrix spaces and some new representations ofSL(N, C)." Ann. ofMath.

86 (1967), 461-490.

35. (with A. Zygmund) "Boundedness of translation invariant operators in Holder spaces and U spaces." Ann. ofMath. 85 (1967), 337-349.


30 31

e

CHAPTER 1

36. "Harmonic functions and Fatou's theorem." In Proceeding of the C.LM.E. Summer Course on Homogeneous Bounded Domains, Cremonese, 1968.

37. (with A. Koranyi) "Fatou's theorem for generalized ha1fplanes." Annali di Pisa 22 (1968),107-112.

38. (with G. Weiss) "Generalizations of the Cauchy-Riemann equations and representations of the rotation group." Amer. J. Math. 90.(1968), 163-196.

39. (with A. Grossman and G. Loupias) "An algebra of pseudodifferential operators and quantum mechanics in phase space." Ann. Inst. Fourier, Grenoble 18 (1968), 343-368.

40. (with N. 1. Weiss) "Convergence of Poisson integrals for bounded symmetric domains." Proc. Nat. Acad. Sci. 60 (1968),1160-1162.

41. "Note on the class L log L." Studia Math. 32 (1969),305-310. 42. (with A. W. Knapp) "Singular integrals and the principal series." Proc. Nat. Acad. Sci.

63 (1969), 281-284. 43. (with N. J. Weiss) "On the convergence of Poisson integrals." Trans. Amer. Math. Soc.

140 (1969), 35-54. 44. Singular integrals and differentiability properties offunctions. Princeton Mathemati

cal Series, 30. Princeton University Press, 1970. 45. (with A. W. Knapp) "The existence ofcomplementary series." In Problems inAnalysis.

Princeton University Press, 1970. 46. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of

Mathematics Studies, 103. Princeton University Press, 1970. 47. "Analytic continuation of group representations:'Adv. Math. 4 (1970),172-207. 48. "Boundary values of holomorphic functions." Bull. Amer. Math. Soc. 76 (1970), 1292

1296. 49. (with A. W. Knapp) "Singular integrals and the principal series II." Proc. Nat. Acad.

Sci. 66 (1970),13-17. 50. (with S. Wainger) "The estimating of an integral arising in multipier transformations."

Studia Math. 35 (1970), 101-104. 51. (with G. Weiss) Introduction to Fourier analysis on Euclidean spaces. Princeton

University Press, 1971. 52. (with A. Knapp) "Intertwining operators for semi-simple groups." Ann. of Math. 93

(1971),489-578. 53. (with C. Fefferman) "Some maximal inequalities." Amer. J. Math. 93 (1971), 107-115. 54. "LP boundedness of certain convolution operators." Bull. Amer. Math. Soc. 77 (1971),

404--405. 55. "Some problems in harmonic analysis suggested by symmetric spaces and semi

simple groups." Proceedings of the International Congress ofMathematicians, Paris: Gauthier-Villers 1 (1971),173-189.

56. "Boundary behavior of holomorphic functions of several complex variables." Princeton Mathematical Notes. Princeton University Press, 1972.

57. (with A. Knapp) Irreducibility theorems for the principal series. (Conference on Harmonic Analysis, Maryland) Lecture Notes in Mathematics, No. 266. Springer Verlag,

1972. 58. (with A. Koranyi)" H 2 spaces ofgeneralized half-planes." Studia Math. XLIV (1972),

379-388. 59. (with C. Fefferman) "HP spaces of several variables." Acta Math. 129 (1972), 137

193.


60. "Singular integrals and estimates for the Cauchy-Riemann equations:' Bull. Amer. Math. Soc. 79 (1973), 440-445.

61. "Singular integrals related to nilpotent groups and a-estimates." Proc. Symp. Pure Math. 26 (1973), 363-367.

62. (with R. Kunze) "Uniformly bounded representations IV. Analytic continuation of the principal series for complex classical groups of types Bn , Cn , Dn ." Adv. Math. 11 (1973), 1-71.

63. (with G. B. Folland) "Parametrices and estimates for the a complex on strongly b

pseudoconvex boundaries." Bull. Amer. Math. Soc. 80 (1974), 253-258. 64. (with J. L. Clerc) "LP multipliers for non-compact symmetric spaces:' Proc. Nat.

Acad. Sci. 7l (1974), 3911-3912. 65. (with A. Knapp) "Singular integrals and the principal series III." Proc. Nat. Acad. Sci.

7l (1974),4622--4624. 66. (with G. B. Folland) "Estimates for the ab complex and analysis on the Heisenberg

group." Comm. Pure and Appl. Math. 27 (1974),429-522. 67. "Singular integrals, old and new." In Colloquium Lectures of the 79th Summer Meeting

of the American Mathematical Society, August 18-22, 1975. American Mathematical Society, 1975.

68. "Necessary and sufficient conditions for the solvability of the Lewy equation." Proc. Nat. Acad. Sci. 72 (1975), 3287-3289.

69. "Singular integrals and the principal series IV." Proc. Nat. Acad. Sci. 72 (1975), 2459-2461.

.'70. "Singular integral operators and nilpotent groups." In Proceedings of the C.LM.E., Differential Operators on Manifolds. Edizioni, Cremonese, 1975: 148-206.

71. (with L. P. Rothschild) "Hypoelliptic differential operators and nilpotent groups." Acta Math. 137 (1976), 247-320.

}').. (with A. W. Knapp) "Intertwining operators for SL(n, r)." Studies in Math. Physics. E. Lieb, B. Simon and A. Wightman, eds. Princeton University Press, 1976: 239-267.

13.· (with S. Wainger) "Maximal functions associated to smooth curves." Proc. Nat. Acad. Sci. 73 (1976), 4295-4296.

74. "Maximal functions: Homogeneous curves." Proc. Nat. Acad. Sci. 73 (1976), 21762177.

15. "Maximal functions: Poisson integrals on symmetric spaces." Proc. Nat. Acad. Sci. 73 (1976), 2547-2549.

! ",76. "Maximal functions: Spherical means." Proc. Nat. Acad. Sci. 73 (1976), 2174-2175. 77. (with P. Greiner) "Estimates for the a-Neumann problem." Mathematical Notes 19.

Princeton University Press, 1977.

78. (with D. H. Phong) "Estimates for the Bergman and Szego projections." Duke Math. . J. 44 (1977),695-704.

','<79. (with N. Kerzman) "The Szego kernels in terms of Cauchy-Fontappie kernels." Duke Math. J. 45 (1978), 197-224.

, .. ·80. (with N. Kerzman) "The Cauchy kernels, the Szego kernel and the Riemann mapping function." Math. Ann. 236 (1978),85-93.

81. (with A. Nagel and S. Wainger) "Differentiation in lacunary direction." Proc. Nat. Acad. Sci. 73 (1978), 1060-1062.

82. (with A. Nagel) "A new class of pseudo-differential operators." Proc. Nat. Acad. Sci. 73 (1978), 582-585.


33

"

.

,

\~;

,

32 CHAPTER 1

83. (with N. Kerzman) 'The Szego kernel in terms of the Cauchy-Fontappie kernels:' In Proceedings of the Conference on Several Complex Variables, Cortona, 1977. 1978.

84. (with P. Greiner) "On the solvability ofsome differential operators ofthe type Db:' In Proceedings of the Conference on Several Complex Variables, Cortona, 1977. 1978.

85. (with S. Wainger) "Problems in harmonic analysis related to curvature." Bull. Amer. Math. Soc. 84 (1978), 1239-1295.

86. (with R. Grundy) "HP theory for the poly-disc." Proc. Nat. Acad. Sci. 76 (1979), 1026-1029. •

87. "Some problems in hannonic analysis." Proc. Symp. Pure and Appl. Math. 35 (1979), Part I, 3-20.

88. (with A. Nagel and S. Wainger) "Hilbert transforms and maximal functions related to variable curves." Proc. Symp. Pure and Appl. Math. 35 (1979), Part I., 95-98.

89. (with A. Nagel) "Some new classes of pseudo-differential operators." Proc. Symp. Pure and Appl. Math. 35 (1979), Part II, 159-170.

90. "A variant of the area integral." Bull. Sci. Math. 103 (1979),446--461. 91. (with A. Nagel) "Lectures on pseudo-differential operators~ Regularity theorems and

applications to non-elliptic problems." Mathematical Notes 24. Princeton University Press, 1979.

92. (with A. Knapp) "Intertwining operators for semi-simple groups II." Invent. Math. 60 (1980), 9-84.

93. "The differentiability of functions in JR"." Ann. ofMath. 113 (1981), 383-385. 94. "Compositions of pseudo-differential operators." In Proceedings of Joumees Equa

tions aux derivees partielles, Saint-Jean de Monts, Juin 1981, Societe Math. de France, 1\li Conference #5, l--<i.II

'II 95. (with A. Nagel and S. Wainger) "Boundary behavior of functions holomorphic in domains of finite type." Proc. Nat. Acad. Sci. 78 (1981), 6596--<1599.

96. (with A. Knapp) "Some new intertwining operators for semi-simple groups." In 1\ Non-commutative harmonic analysis on Lie groups, Colloq. Marseille-Luminy, 1981.

1 Lecture Notes in Mathematics, no. 880. Springer Verlag, 1981.

!11 97. (with M. H. Taibleson and G. Weiss) "Weak type estimates for maximal operators on certain HP classes." Rendiconti Circ. mat. Pelermo, Supp!. n.1 (1981), 81-97.

III 98. (with D. H. Phong) "Some further classes of pseudo-differential and singular integral operators arising in boundary value problems, I, Composition of operators." Amer. 1. Math. 104 (1982), 141-172.

1

Ilill 99. (with D. Geller) "Singular convolution operators on the Heisenberg group." Bull. II

1 Amer. Math. Soc. 6(1982),99-103.I 1

100. (with R. Fefferman) "Singular integrals in product spaces:' Adv. Math. 45 (1982), 117-143.

11 I I1 101. (with G. B. Folland) "Hardy spaces on homogeneous groups:' Mathematical Notes I' ~ 28. Princeton University Press, 1982. I

102. "The development ofsquare functions in the work of A. Zygmund." Bull. Amer. Math. ~I Soc. 7 (1982). 103. (with D. M. Oberlin) "Mapping properties of the Radon transform." indiana Univ.

III Math. J. 31 (1982), 641--<i50. III 104. "An example on the Heisenberg group related to the Lewy operator." Invent, Math. 69

I' (1982),209-216. . I,I,II1II "III"r:i


105. (with R. Fefferman, R. Gundy, and M. Silverstein) "Inequalities for ratios of functionals ofhannonic functions." Proc. Nat. Acad. Sci. 79 (1982), 7958-7960.

106. (with D. H. Phong) "Singular integrals with kernels ofmixed homogeneites." (Conference in Hannonic Analysis in honorofAntoni Zygmund, Chicago, 1981), W. Beckner, A. Calderon, R. Fefferman, P. Jones, eds. Wadsworth, 1983.

107. "Some results in hannonic analysis in JR", for n --+ 00." Bull. Amer. Math. Soc. 9 (1983),71-73.

. 108. "An HI function with non-summable Fourier expansion." In Proceedings ofthe Conference in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics, no. 992. Springer Verlag, 1983.

109. (with R. R. Coifman and Y. Meyer)"Un nouvel espace fonctionel adapte a l'etude des o¢rateurs definis pour des integrales singulieres:' In Proceedings of the Conference in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics, no. 992. Springer Verlag, 1983.

110. "Boundary behaviour ofhannonic functions on symmetric spaces: Maximal estimates for Poisson integrals." Invent. Math. 74 (1983),63-83.

111. (with J. O. Stromberg) "Behavior of maximal functions in JR" for large n." Arldv f Math. 21 (1983), 259-269.

112. (with D. H. Phong) "Singular integrals related to the Radon transform and boundary value problems." Proc. Nat. Acad. Sci. 80 (1983),7697-7701.

113. (with D. Geiler) "Estimates for singular convolution operators on the Heisenberg group." Math. Ann. 267 (1984), 1-15.

,,114. (with A. Nagel) "On certain maximal functions and approach regions." Adv. Math. 54 (1984),83-106.

'1'15. (with R. R. Coifman and Y. Meyer) "Some new function spaces and their applications to harmonic analysis." J.Funct. Anal. 62 (1985), 304-335.

, U6. 'Three variations on the theme of maximal functions." (Proceedings of the Seminar on Fourier Analysis, El Escorial, 1983.) Recent Progress in Fourier Analysis. I. Peral and J. L. Rubiode Francia, eds.

',117. Appendix to the paper "Unique continuation ... :' Ann. ofMath. 121 (1985),489-494. "US. (with A. Nagel and S. Wainger) "Balls and metrics defined by vector fields I: Basic

. properties." Acta Math. 155 (1985), 103-147.

tB>. (with C. Sogge) "Averages of functions over hypersurfaces." Invent. Math. 82 (1985), 543-556.

:tW. "Oscillatory integrals in Fourier analysis:' In Beijing lectures on Harmonic analysis. .... Annals of Mathematics Studies, 112. Princeton University Press, 1986.

,2I. (with D. H. Phong) "Hilbert integrals, singular integrals and Radon transforms II." Invent. Math. 86 (1986), 75-113.

122. (with F. Ricci) "Oscillatory singular integrals and hannonic analysis on nilpotent groups." Proc. Nat. Acad. Sci. 83 (1986), 1-3.

'123•. (with F. Ricci) "Homogeneous distributions on spaces of Hennitian matricies:' Jour. Reine Angw. Math. 368 (1986), 142-164.

]24. (with D. H. Phong) "Hilbert integrals, singular integrals and Radon transforms 1." Acta Math. 157 (1986), 99-157.

125. (with C. D. Sogge) "Averages over hypersurfaces: II." Invent. Math. 86 (1986), 233242.


34 CHAPTER 1 SELECTED THEOREMS BY ELI STEIN 35 126. (with M. Christ) "A remark on singularCalderon-Zygmund theory." Proc.Amer. Math.

Soc. 99, 1 (1987), 71-75. 127. "Problems in harmonic analysis related to curvature and oscillatory integrals." Proc.

Int. Congress of Math., Berkeley 1 (1987), 196-221. 128. (with F. Ricci) "Harmonic analysis on nilpotent groups and singular integrals I." J.

Funct. Anal. 73 (1987),179-194. 129. (with F. Ricci) "Harmonic analysis on nilpotent groups and singular integrals II." J.

Funct. Anal. 78 (1988),56-84. 130. (with A. Nagel,). P. Rosay and S. Wainger) "Estimates for the Bergman and Szego

kernels in certain weakly pseudo-convex domains." Bull. Amer. Math. Soc. 18 (1988), 55-59.

131. (with A. Nagel and D. C. Chang) "Estimates for the a-Neumann problem for pseudoconvex domains in C 2 of finite type." Proc. Nat, Acad. Sci. 85 (1988), 8771-8774.

132. (with A. Nagel, J. P. Rosay, and S. Wainger) "Estimates for the Bergman and Szego kernels in 1(:2." Ann. ofMath.l28 (1989),113-149.

133. (with D. H. Phong) "SingularRadon transfonns and oscillatory integrals." Duke Math. J. 58 (1989),347-369.

134. (with F. Ricci) "Harmonic analysis on nilpotent groups and singular integrals III." J. Funct. Anal. 86 (1989),360--389.

135. (with A. Nagel and F. Ricci) "Fundamental solutions and harmonic analysis on nilpotent groups." Bull. Amer. Math. Soc. 23 (1990), 139-143.

136. (with A. Nagel and F. Ricci) "Harmonic analysis and fundamental solutions on nilpotent Lie groups in Analysis and P.D.E." A collection of papers dedicated to Mischa Cotlar. Marcel Decker, 1990.

137. (with C. D. Sogge) "Averages over hypersurfaces, smoothness of generalized Radon transforms." J. d' Anal. Math. 54 (1990),165-188.

138. (with S. Sahi) "Analysis in matrix space and Speh's representations." Invent. Math. 101 (1990), 373-393.

139. (with S. Wainger) "Discrete analogues of singular Radon transforms." Bull. Amer. Math. Soc. 23 (1990),537-544.

140. (with D. H. Phong) "Radon transfonns and torsion." Duke Math. J. (fut. Math. Res. Notices) #4, (1991),44-60.

141. (with A. Seeger and C. Sogge) "Regularity properties of Fourier integral operators." Ann. ofMath. 134 (1991), 231-251.

142. (with J. Stein) "Stock price distributions with stochastic volatility: an analytic approach." Rev. Fin. Stud. 4 (1991), 727-752.

143. (with D. C. Chang and S. Krantz) "Hardy spaces and elliptic boundary value problems." In the Madison Symposium on Complex Analysis, Contemp. Math., 137 (1992), 119-131.

OTHER REFERENCES

[BGS] D. Burkholder, R. Gundy, and M. Silverstein. "A maximal function characterization ofthe class HP." Trans. Amer. Math. Soc. 157 (1971): 137-153.

[Cal A. P. Calderon. "Commutators of singular integral operators." Proc. Nat. Acad. Sci. 53 (1965): 1092-1099.

[Chr] M. Christ. "On the ab-equation and Szego projection on a CR manifold." In Proceedings, El Escorial Conference on Harmonic Analysis 1987. Lecture Notes in Mathematics, no. 1384. Springer Verlag, 1987.

[Co] M. Codar. "A unified theory of Hilbert transfonns and ergodic theory." Rev. Mat. Cuyana I (1955): 105-167.

[CZ] A. P. Calder6n and A. Zygmund. "On higher gradients of harmonic functions." Studia Math. 26 (1964): 211-226.

[EM] L. Ehrenpreis and F. Mautner. "Uniformly bounded representations of groups." Proc. Nat. Acad. Sci. 41 (1955): 231-233.

[FK] C. Fefferman and J. J. Kohn, "Estimates of kernels on three-dimensional CR manifolds." Rev. Mat. Iber. 4, no. 3 (1988): 355--405.

[FKP] C. Fefferman, J. J. Kohn, and D. Phong. "Subelliptic eigenvalue problems." In Proceedings, Conference in Honor ofAntoni Zygmund. Wadsworth, 1981.

[FS] C. Fefferman and A. Sanchez-Calle, "Fundamental solutions for second order subelliptic operators." Ann. ofMath. 124 (1986): 247-272.

[ON] I. M. Gelfand and M. A. Neurnark. Unitiire Darstellungen der Klassischen Gruppen. Akademie Verlag, 1957.

[H] 1. I. Hirschman, Jr."MuItiplier transformations I." Duke Math. J. 26 (1956): 222242;"Multiplier transformations II." Duke Math. J. 28 (1961): 45-56.

[MeN) J. McNeal, "Boundary behavior of the Bergman kernel function in 1(:2." Duke Math. J. 58 (1989): 499-512.

(2) A. Zygmund, Trigonometric Series. Cambridge University Press, 1959.


37

2

Geometric Inequalities

in Fourier Analysis

William Beckner*

1 INTRODUCTION

Geometric ideas occur in almost every aspect of Fourier analysis. Beginning with the symmetry structure of the domain and the product structure of the operator, geometric concepts control deep facts about the Fourier transform. The symmetry structure of a Riemannian manifold defines not only the natural objects of analysis for the domain such as the Laplace-Beltrami operator, Green's function, global transforms, and boundary operators, but also determines intrinsic ways to compare the "size" of such objects as measured by the classical function spaces. The geometric structure of a manifold is manifest in the character of analytic operator and variational inequalities. Such estimates are the building-blocks of "everyday analysis."

Convolution is a natural object viewed both as an averaging process for a translation-invariant measure and as the dual operator to the Fourier transform. Fractional integration is even more natural arising in the context of Green's functions and potential theory, restriction phenomena for the Fourier transform, intertwining operators for representations of the Lorentz groups and correlation functions in conformal field theory and statistical mechanics. A framework for the analysis of convolution inequalities on a manifold is developed here, especially in terms of (1) conformal invariance; (2) geometric symmetrization and equimeasurable rearrangement of functions; (3) complex symmetry structure on Lie groups; and (4) embedding of geometric and probabilistic information in exact constants

-This research was supported in part by the National Science Foundation.

GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS

for variational problems. Several themes from E. M. Stein's work have influenced this program, including his treatment ofRiesz transforms, spherical harmonics, and the Hardy-Littlewood-Sobolev theorem on fractional integration; his viewpoint on Lie groups and boundary manifolds including SL(2, JR.) and the Heisenberg group; and his emphasis on integral transforms and boundary behavior in several complex variables. Philosophically the roots for the overall approach go back to Hardy and Littlewood. In addition, a strong connection can be drawn to problems in physics, including the Bargmann-Fock analysis of the hydrogen atom, ideas of Bargmann, Fock, and Segal on quantum field theory, and Polyakov's quantum string theory. 1bree important calculations for sharp inequalities due to J. Moser, E. Onofri, and E. H. Lieb directly motivate much of this program. The interaction of ideas and methods from differential geometry, Fourier analysis, and quantum string theory has led to a richer understanding of the role of algebraic invariance and geometric structure in analysis on manifolds.

2 CLASSICAL INEQUALITIES

The basic convolution inequality on a unimodular Lie group is Young's inequality

(f * g)(x) = Lf(xy-l)g(y) dy

(1) Ilf * gllu(G) :s IIfl1u(G)lIgIILQ(G)

with lip + Ilq - 1 = llr, 1 :s p, q, r :s 00. Using product structure and radial symmetry, one can shOW that on the Euclidean space JR.n this inequality can be improved ([8],[16]) with the extremal result holding only for gaussian functions

(2) Ilf * gllu(JR") :s (A p Aq A r'YllflIu(JR")lIgllu(JR")

with

A p = [pl/p I p ,l/P'f/2

and primes always denoting dual exponents, 1I p + 1I pi = 1. It is relatively easy to see that this inequality extends to include weak Lorentz classes and in fact corresponds to the Hardy-Litllewood-Sobolev theorem for fractional integration ([68])

(3) II Ixl-A * fllu(lR") :s CIIfl1u(JR")

with A = nlq for I < q < 00 (r, p, q related as above). The function lxi-A is characteristic for the Lorentz class Lq,oo(JR.n) so this inequality has the equivalent


38 CHAPTER 2

fonn

(4) II f *gil U(JR") .s CIIg IILq•oo (JR") II fII U(IR")·

The linking step between these inequalities involves a symmetrization argument of Riesz-Sobolev type.

Riesz-Sobolev Lemma.

(5) I ( f(x)g(y)h(x - y) dx dy!.s ( j*(x)g*(y)h*(x - y) dx dy.k"x. _ k"x.

Here * denotes the equimeasurable radial decreasing rearrangement applied to

the modulus ofa function.

This technical result is central to the analysis of positive convolution operators and Sobolev inequalities ([70]), and is geometric in nature being equivalent to the Brunn-Minkowski inequality. The symmetry structure intrinsic to these inequalities includes: translation invariance, rotational symmetry, dilation invariance, and Euclidean product structure.

It is useful to take into account this product structure even though it does not provide good control of constants. Using the relation between arithmetic and geometric means, the one-dimensional Hardy-Littlewood-Sobolev inequality controls not only fractional integration on classical n-dimensional Riemannian manifolds but also on nilpotent Lie groups. Moreover, the Riesz-Sobolev symmetrization lemma can be applied on a nilpotent Lie group to remove the non-abelian structure and give the following extension for the inequalities that appear above.

Theorem 1. Let G be a nilpotent Lie group ofdimension n, homogeneous dimen

sion m with Ixl denoting the canonical distance on G. Thenfor IIp + l/q - I = I I r, I ,.s p, q, r ::::: 00

(6) IIf *gllu(G) ::::: (A p Aq A,,)nllfllu(G)lIgIILq(G)

A = [pl/p I p ,\/pl]I/2p

and for A = m Iq, I < q < 00

(7) IIf * gllu(G) ::::: CIIf11u(G)lIgIlLq,oo(G)

(8) Illxl-A * gllu(G) ::::: CIIf11u(G)'

No extremal/unctions exist for inequality (6) when the constant is less than one.

Interpolation arguments are the standard method used to prove weak Lorentz class inequalities, but dilation invariance and the Riesz-Sobolev lemma provide a

GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS 39

more elementary reduction to the one-dimensional fonn of Young's inequality on the real line. To illustrate this point consider the inequality

(9) I ( f(x)g(y)\x - yl-A dx dyl ::::: CIIf11u(IR")lIgIlU(lRn) 1M." xlRn

for A = n(llpI + l/q') < n and I < p, q < 00. Using symmetrization the problem of verifying this inequality is reduced to non-negative radial decreasing

functions which satisfy the estimates

Ixl n/ p f(x) ::::: CIIf11 p , Iyln/ p g(y) ::::: CIIgll q •

For the most part, the symbol C will denote a generic constant. Since the problem is one-variable, dilation invariance provides a fonnulation as a convolution inequality on the line. Set u(t) = Ixl n / p f(x), x = exp t and v(s) = Iyln/q g(y), y = exp s;

then the inequality above becomes

( u(t)v(s)1/I(t - s) ds dt ::::: CIIullu(lR)lI ii \lu(lR),JlRxlR

but under the conditions 1/1 E L 1 n Ln/ A•oo , lIull, ::::: CIIf11u(IR"), p ::::: r ::::: 00

and IIvll, ::::: ClIgIILq(IR"), q ::::: r ::::: 00. The estimate now follows immediately

from Young's inequality (1).

E. H. Lieb recognized ([48]) that when the map

f ~ lxi-A * f is a map from a space to its dual, then two additional types of symmetry are evident which can be used to calculate the best constant for the inequality

(10) I ( f(x)g(y)lx - yl-A dx dyj ::::: Cllfllu(IR")lIgllu(lR")JlR" xlR"

with A = 2n I pI and I < p < 2. There is a quadratic functional symmetry which makes the operator positive-definite self-adjoint so that one may take f = g. More importantly, the inequality is conformally invariant. Suppose < is a conformal transformation and let J denote the modulus of the Jacobian determinant for this

change of variables. Then under the transformation

f ~ lex) = f(u)J«, X)l/px ~ <x,

the functional inequality (10) is invariant. This application ofconformal invariance to convolution problems has a rich history in mathematical physics. It was used by Bargmann and Fock to give a group-theoretical analysis of the spectrum of the hydrogen atom and more recently by Onofri ([57]) to study the variation of the zeta-function detenninant of the Laplacian in two dimensions under conformal deformation. The critical issue here is the algebraic invariance of the metric. In

·:,~V',,,liil'.\\',.


41 40 CHAPTER '2

addition, conformal invariance implies that equivalent forms of inequality (10) exist on any domain conformally equivalent to the plane IRn, including the sphere sn and the two-sheeted hyperboloid lHIn •

Using the Riesz-Sobolev lemma twice, one can easily observe that extremal functions exist for inequality (l0). Symmetrization on IRn reduces the problem to

one variable. Then the dilation structure provides a one-dimensional inequality which can be put on the multiplicative group IR+. That is, letu(t) = Ixln/pf(x) ~

0, t = lxi, v(s) = Iyln/Pg(y) ~ 0, s = IYI and

Aifr(t) = tn-I [t + lit - 2~ . 1]r /

2 d~; now inequality (l0) becomes

1 dt ds (11) u(t)v(s)ljr(tls)- - :s CpliuIlU(IR+)lIvIlU(IR+).

~x~ t s

But the kemelljr is symmetric decreasing away from the origin t = 1 so one can apply the Riesz-Sobolev lemma a second time to insure that u and v are radial

decreasing functions on IR+ and as observed above they must also be uniformly

bounded. Now choose a sequence Un = Vn with lIunlip = 1 such that

dt dsf un (t)un(s)ljr (sit) - - --+ Cpt s

where Cp is the best constant in (11). Since the un's are decreasing functions,

one can use the Helly selection principle to choose a subsequence that converges almost everywhere to a function u E LP(IR+). By Fatou's lemma lIuli p :s 1. But

un(t) :s C[l + Ilnt!l-l/p = a(t)

and a(t)a(s)ljr(tls) E LI(IR+ x IR+) so applying Lebesgue's dominated convergence theorem gives

f dt ds dt dsfunk(t)Unk(s)ljr(tls) - - --+ u(t)u(s)ljr(tls) - - = Cpot s t s

Since C p is the best constant, II u II p = 1 and u must be an extremal function for inequality (11) which then provides an extremal function for the fractional integral inequality (10).

The different symmetries of the conformal structure determine the form of the extremal functions. This is clearly realized in terms of the conformal transformation (stereographic projection) from the plane IRn to the sphere sn where the

Jacobian determinant is proportionalto (1 + IxI2)-n. Since the sphere is a compact manifold, one expects constants to be extremal on that domain. This would then imply that on IRn the extremal functions are given by the function A(1 + IxI 2)-n/p

up to conformal automorphism.


To obtain this result, one makes use of the fact that the different kernel functions studied here are all strictly decreasing functions of a single variable. This means that whenever a symmetrization is effected, there must either be some positive increase in the value of the functional, or the unsymmetrized functions are translates of symmetric decreasing functions on the domain. The technical tool needed here is a variation of a symmetrization lemma of Baernstein and Taylor ([5],[9]). The

• first application of this style of argument for such functions is due to Lieb ([47]).

Symmetrization Lemma (after Baernstein and Taylor). Let M denote IRn, sn, or IHIn (real hyperbolic space) with symmetric decreasing rearrangement defined in terms ofgeodesic distance. Let K be a monotone decreasing function and d (x, y) denote the distance between the points x and yin M. Then

( f(x)K[d(x, y)]g(y) dx dy:s { !*(x)K[d(x, y)]g*(y) dx dy lMxM lMxM

where f. g are non-negative measurable functions with j*, g* denoting their respective equimeasurable geodesically decreasing rearrangements on M, dx denotes the measure invariant under the group action on the symmetric space M and the integrand on the left is in LI(M x M). If K is strictly decreasing, then the above inequality is strict unless f(x) = j*(rx) and g(x) = g*(rx) almost everywhere for r x a "translate" of x.

Using the conformal equivalence between IRn and Sn-{pole} given by

1 - Ixl 2X) 1 - \y12 2Y ) ~ = ( 1 + Ix12' 1 + Ixl2 . 1] = ( 1 + lyl2' 1 + IYl2

and setting

F(~) = (1 + Ix\2r/p f(x), G(1]) = (1 + \yI2r/Pg(y)

an equivalent fractional integral inequality to (l0) is obtained for the sphere sn.

(12) I ( F(~)G(1])I~ - 1]1-A d~ d1]1 :s BpIlFlIu(sn) IIGlIu(sn). 1sn xsn

Now consider an extremal function for the IRn fractional integral inequality (10)

which also has the inversion symmetry given by symmetrizing on the IR+ inequality

(11). If one now transforms this extremal function to the setting of the sphere, one finds that the inversion symmetry gives a function symmetric with respect to

hemispheres. But applying the Baernstein and Taylor lemma would produce a

positive increase to the left-hand side of inequality (12). This contradicts the fact that the function is extremal, so on the sphere this function must be radial decreasing


43

II 1IIIili

11,111

'II' I

III:

I

II 1

1'1""

1,1

~I :111

11

1

1"

I i

Illi

1: 1 1

Iii 11I I'

J

CHAPTER 242

away from some pole. But constants are the only measurable functions symmetric with respect to hemispheres and decreasing from a pole. Hence conformal factors given in terms of the Jacobian determinant are the only extremal functions for these conformally invariant fractional integral problems. The essential point of this argument is that symmetrization on JR+ picks out a specific extremal which "breaks the sn symmetry" unless the equivalent function on S" is constant almost

everywhere. In summary, the analysis of the fractional integral inequality as a dual-space

mapping depends on the invariance of the functional under the action of the conformal group and the strict monotonicity of the kernel (as a function of one variable).

In studying the variation of the zeta-function determinant of the Laplacian under conformal deformation of metric on S2, Onofri ([57)) had realized that this confor

mal group action was a natural technique for the solution of geometric variational problems. Carlen and Loss ([ 19)) later observed that the two techniques described

above, conformal action and geometric symmetrization, could be put together to solve the variational problem for the Hardy-Littlewood-Sobolev inequality (10) on

a single domain since in a generic sense a conformal transformation will break the geodesic symmetry. This was quite a useful remark since for some problems the

range of symmetrization techniques is more extensive in non-compact domains. A second example where the competing symmetries of different geometries determine the extremal functions for a variational problem occurs in the author's paper ([9)). There radial symmetry on JRn is played off against the infinite-dimensional

spherical symmetry of gaussian measure to provide an elementary proof of Nelson's inequality ([55)). That idea was an outgrowth ofearlier work with D. Jerison

on optimal information, gaussian symmetry, and a problem of H. Chernoff. The first immediate application of the sharp Hardy-Littlewood-Sobolev inequal

ity is that one obtains the classical sharp Soholev inequalities for L 2 control of the gradient and the Dirichlet form for harmonic extension from boundary

values a9],[1O)). Transforming these inequalities to the sphere and taking the infinite-dimensional limit also gives the logarithmic Soholev inequality of Gross for gaussian measure. Such arguments suggest that considerable geometric and

probabilistic information is contained in sharp fractional integral inequalities. It is also apparent that the most workable domain for developing the full consequences

of inequality (10) may be the n-dimensional sphere. In this setting spectral data can be visualized in the broadest sense. A good example to illustrate this point is

the spectrum of the hydrogen atom where the degeneracy was originally puzzling on the Euclidean side but readily transparent for the equivalent spectral problem

for the three-dimensional sphere. Using stereographic projection and the Funk-Hecke formula, the sharp Hardy

Littlewood-Sobolev inequality (10) gives a conformally invariant fractional


integral inequality on the sphere and an equivalent multiplier inequality expressed

in terms of spherical harmonics.

Theorem 2. For F, G E u(sn) with 1 :s p < 2 and A = 2n/p'

(13) I ( J

F(~)I~ -1/rAG(1/)d~d1/l:s BpllFllu(sn)IIGllu(sn) sn xS·

1 f(n)f( n-A )

- 11:- I-Ad -TA -2Bp- 5 1/ 1/- n A' sn f("2 )f(n - "2)

For F = L Yk and 1 :s p :s 2

(14) ~ Yk 1n IYkl 2d~ :s [llFllu(sn)]2

(~ ) ... (~ (15) Yk = A

(n - -) ... (n2

Equality is attained if 1 < p <

+ k - 1) f( ~ )f(? + k) A = n n .

- - + k - 1) f( - )f( - + k)2 p' p

2 only for functions of the form

F = G, F~(~) = All _I; . ~,-n/p, 11;1 < I.

The sharp value Bp here corresponds to

[ f(~)]-1+~C = nA/2 f(¥) f(n - ~) fen)

for inequality (10). The correspondence between these inequalities comes from considering the map from JRn to sn - (CO, 0, ... , -I)} with the "north pole"

corresponding to (0, 0, ... , 1):

c (2X I - IXI2) mn mnS=(U,S)= 2' 2 XEm.,uEm.,-l<s<1.1 + Ixi 1 + Ixl

The inverse map is x = u / (1 + s) and the change of measure is given by

d~ = n-n/2[fCn)/ f(n/2)](1 + IxI2)-n dx

where d~ is normalized surface measure on sn. Let x, Y E JRn with ~, 1/ the corresponding points on sn and set F(~) = (1 + IxI2)"/p f(x) and G(1/) = (l + Iy12)n/p g(y). The algebraic invariance of the metric is expressed by

1 2 2 1/2Ix - yl = 21~ - 1/1[(1 + x )(1 + y)] .

The equivalence between (13) and (14) follows from the fact that the kerne1l~ 171-). defines a positive-definite self-adjoint operator that commutes with rotations.


45 44 CHAPTER 2

Since such operators arise as intertwining operators for representations of the Lorentz groups, it is natural to see the relation with the Selberg point-pair product on symmetric spaces. The Funk-Heeke formula is used to find the multiplier action in terms of spherical harmonics that is used for (14).

Funk-Heeke Formula. For K an integralfunction on [-I, I] and Yk a spherical harmonic ofdegree k on sn, then

( K(~. 1/)Yk(1/) dry = )..kYk(~)Js'

, r(k + 1) )C(n-I)/2( )(1 2)!! -I d Cn

11 K( Ak = r(k + n _ 1) _I W k W - W 2 w

where C~(w) is the Gegenbauer polynomial defined by

(1 - 2wt + t 2)-V = L q(w)tk

and C = rr-12n - 2n nil )r( n21 ).n

Results in this paper are generally obtained as a priori inequalities for a smooth class of functions and then the full estimates follow by taking limits. Logarithmic integrals are in general indeterminate, but the form of the estimates occurring here will imply that all logarithmic integrals are well-defined.

It is particularly useful to study the inequalities in Theorem 2 under variation

of the parameters p and n, especially in terms of end-point information for p and asymptotic limits for n. A wealth of geometric and probabilistic information is contained in this fractional integral inequality including Nelson's hypercontractive estimates for gaussian measure, hypercontractive estimates for the Poisson and

heat semigroups on the sphere, logarithmic Sobolev inequalities, Moser-Trudinger inequalities in higher dimensions, Carleson-Chang inequalities, entropy estimates for logarithmic kernels and extremal properties for the zeta-function determinant of the conformal Laplacian and square of the Dirac operator in two and four dimensions under conformal deformation ofmetric. A nice feature of this structure

is that one often captures the intrinsic nature ofa problem by writing the inequality in a form where the operator norm is one. For end-point information, the idea is to find a value of the parameter where equality is attained in the functional inequality for a large class of functions and then to take a limit of difference quotients so that one obtains a "differentiated inequality" from the original inequality. Here

one can consider this process at p = 2 for general functions and at p = I for non-negative functions. In modem analysis this technique was utilized by L. Gross

in a striking manner to show that Nelson's hypercontractive semigroup estimates were equivalent to a logarithmic Sobolev inequality.


The first issue treated here is the asymptotic limit for large n. Observe that

k r( ~ )r(? + k) k

(16) (p - I) ~ r(?)r( ~ + k) ~ (p - 1) .

Now one has a choice of limits, both of which give Nelson's inequality for gaussian measure. If F is restricted to be a function of the polar angle, then the spherical harmonics are Gegenbauer polynomials which go over in an appropriate limit to

Hermite polynomials. The normalization used here is defined by

I tk00

exp(- - t2 + tx) = L ,Hk(X)2 k=O k.

corresponding to the one-dimensional gaussian measure

dJL(x) = (2rr)-1/2 exp(-x2j2) dx.

One could also rescale the inequality to be on a sphere of radius .jii and take the infinite-dimensional Poincare limit l to obtain an inequality on]Roo with gaus

sian measure ([9],[50]). For Iwl ~ I consider the operator defined on Hermite

polynomials kT",: Hk --+ w Hk

and its product extension to higher dimensions. T", may be represented as an

integral operator

(T",g)(x) = f T",(x, y)g(y) dJL(Y)

using the Mehler kernel

w2(x2 + y2) wxy }T",(x, y) = (1 - ( 2)-1/2 exp - 2(1 _ (2){ + I - w2

and in arbitrary dimension by the semigroup operator

ttNT", = e- , N = -f!,. + x . '\7, w = e-

where N corresponds to the "number operator" in Fock space.

Corollary 1 (Nelson's inequality). For d JL denoting the product gaussian measure

and real w with Iwl :::: ,.,;p=l for 1 :::: p :::: 2

(17) IIT",gIlL2(d/L) :::: IIgIlLP(d/L)'

IThough this limit is often attributed to Poincare, Dan Stroock has told me that it was used earlier

by Mehler.


47 46 CHAPTER 2

Using the left-hand side of (16), the analogous estimate is obtained for the Poisson semigroup (see [11]). For 0 < r < 1 the Poisson kernel on the sphere sn is defined by

I - r 2

Pr(~, 1J) = I~ _ r1Jln+1

with

(PrF)(~) = {isn Pr(~, 1J)F(1J) d1J.

The action of this kernel on spherical harmonics is given by

(PrF)(~) = L00

rkYk(~) k=O

ifF = LYk •

Corollary 2. For 0 < r < ...;'"ji"=l and 1 ~ P ~ 2

(18) IIPr FIIL2(s,) ~ IlFllu(S").

The most useful limit that can be applied in the context of Theorem 2 is the classical differentiation argument that extracts "end-point information" from the parameter range of the mapping. One starts with a functional inequality depending on a parameter P, say et>p(f) ~ 0 for which equality is attained for a class of functions f at the parameter value Po, i.e., et> Po (f) == O. Then one can differentiate the inequality at the value Po to produce a new functional inequality

(19) et>' 0 (f) = lim et> p(f) - et> Po (f) = lim et> p(f) ~ 0 p p~ Po P - Po p~ Po P - Po

if P ~ Po and the reverse inequality if p ~ Po. The inequalities of Theorem 2 provide "end-point information" at (L I, LOCi) and at L 2• The first case (L I, LOCi)

will be treated in Section 4 where this gives a sharp Moser-Trudinger inequality for the n-dimensional sphere and further geometric information about zeta-function determinants. In a rough sense, (L I, LOCi) is the "geometric end-point" while L 2 is the "probabilistic end-point" in that information is obtained which can be applied in an infinite-dimensional setting.

Inequality (14) becomes an equality for all functions when p = 2 so the differentiation argument given by (19) can be used for this end-point.

Theorem 3. For F E L2(sn) with J 1F12 d~ = 1 and F = L Yk,

(20) !IFI2InIFld~ ~~~k(n) 1.'Yk,2d~


k-I 1 ~k (n) = ~ L -n-- k ~ 1.

2 £=0 2: + e

Note that ~ I (n) = I, ~k (n) < k if k ~ 2 and asymptotically ~k (n) (n 12) In k for large k. The logarithmic character of this factor ~k (n) is not surprising if one thinks about the relation of this problem to fractional integration on the circle. Now using these estimates a more useful logarithmic inequality follows.

Tbeorem 4. For F E L 2(sn) with J 1F12 d~ = I and u denoting the harmonic extension of F to the interior ofthe unit ball in lRn+1 and en = 2rr(n+l)/21 r( nil)

(21) ! 1F121n1Fld~ ~ "ttkln IYk12d~

= ~ { IVul 2dx ~ ! { IVFI2d~. Cn i1xbl n isn

This logarithmic Sobolev inequality now gives by a standard argument ([9]) that both the Poisson and heat semigroups are hypercontractive on the n-dimensional sphere. This result was first realized by Weissler ([76]) in one dimension using a direct combinatorial argument and by Janson ([43]) in two dimensions using a weighted convolution of Bernoulli trials. The logarithmic Sobolev inequality for the heat semigroup was first obtained by Mueller and Weissler ([53]) and then by Bakry and Emery ([6]) using "iterated gradients" and the Bochner-LichnerowiczWeitzenbOck formula.

Theorem 5. The Poisson semigroup defines a contraction mappingfrom LP (sn) to Lq (sn) with 1 ~ p ~ q ~ 00 and r real ifand only if Ir\ ~ [(p -l)/(q - 1)]1/2:

(22) IIPr FIILq(s") ~ II Fllu(s')'

The possibility of proving such operator inequalities was first suggested to the author by E. M. Stein. Though the role of logarithmic Sobolev inequalities has been closely associated with arguments of Segal, Nelson, Federbush, and Gross in mathematical physics, an early application of a logarithmic inequality of this type relating entropy to smoothness occurs in the Calder6n-Zygmund theory of singular integrals ([18]).

3 MULTILINEAR CORRELATION INEQUALITIES

The interplay of Fourier analysis, differential geometry, and conformal field theory has marked several interesting directions in contemporary analysis. As one


48 CHAPTER 2

pushes to develop more fully the analysis and geometry of higher-dimensional spaces, both for the classical geometries and for Lie groups, the intrinsic operators that characterize basic geometric questions display greater complexity in terms of both multilinear and tensor structure. The natural resource to exploit here is the symmetry and algebraic invariance of the operators, especially in terms of such groups as the conformal group or the symmetric group. Conformal field theory and statistical mechanics constitute a useful laboratory producing exact model calculations that provide considerable insight for the development of this program.

A characteristic example that seems likely to be very rich in both analytic and geometric consequences is the conformally invariant multilinear fractional integral. This operator arises naturally in two different contexts: 1) as an invariant operatorvalued m-point correlation function in conformal field theory (see [26] for example) and 2) in terms of restriction phenomena for the Fourier transform ([23]). In addition, it is an intertwining operator for the Lorentz group, and the operator norms are related to higher-dimensional Selberg integrals. In his thesis, M. Christ showed that an end-point estimate for controlling the restriction of the Fourier transform of an LP (R") function to a curve in R" was guaranteed by a sharp fractional integral estimate. The foundation of this analysis rested on methods developed by Fefferman and Stein ([29]) and Carleson and SjOlin ([21]). A basic tool in the analysis of such estimates is the generalization of the Riesz-Sobolev rearrangement lemma by Brascamp, Lieb, and Luttinger ([17]).

Rearrangement Lemma (Brascamp, Lieb, and Luttinger).

(23) ILOX"'XRO Dfk(~aklxf) dXl" .dXml

~ { fI ft (t aklXf) dXI ... dXm JRox ... xRo k=1 f=1

where {akl} are real numbers and Xk E R". k = I, ... , m.

Using the framework established in Section 2, four equivalent theorems are set down here that give the conformally invariant multilinear Hardy-LittlewoodSobolev fractional integral inequality in different settings.

Theorem 6. Forfunctions h E Uk (R"), k = I, ... , Nand Pk > ], L .;; > 1 (24)

N NIf nh(xt> n IXi - Xjl-Y;j dXI ... , dXNI ~ Cp,y,".N nIlhIlUk(Ro) k=l 1<5) <j"",N k=1

49GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS

where Xk E R", P denotes {pd, y denotes {Yij}, n > Yij = Yji ~ 0,

t "Yik = 2n L Yij = n L I < (N - l)nL...J I' i<j Pki#k Pk

with Pk and pi dual exponents. The best constant Cp,y,".N is attained for the extremalfunctions hex) = A(l + !xI 2)-n/p, up to conformal automorphism.

Because this problem is conformally invariant, there is an equivalent inequality

on the sphere S".

Theorem7. Forfunctions Fk E U'(S"),k = 1, ... , Nand Pk > 1, L ...!... > 1Pk

N N

(25) If 0Fk(~k) DI~j - ~jl-Yij d~I" .d~NI ~ Bp,y.n,N 0IlFkllu,csO )

where ~k E S", d~k denotes normalized surface measure, n > Yij = Yji ~ 0,

"Yik = 2n L Yij < (N - l)nL...J '

i# Pk I

i<j

with Pk and pi dual exponents. The best constant Bp,y,",N is attained only for the extremal functions Fk(~) = A(l - I; . ~)-"/Pk where I; E jRn+1 with II; I < 1.

The two constants are related by the equation

(26)

Bp,y,n,N = r(n) ]L *Cp,y.n,N = fn - ~j I-. y.; d~1 ... d~N'(4;rr)n/2r(n/2) i<j I~i[

This formula makes the connection to the Selberg integral and the representation theory of the symmetric group readily apparent. By taking asymptotic limits of inequality (25) one obtains a multilinear version of Nelson's inequality which was first developed in the author's thesis from its equivalence to a generalized sharp Young's inequality. If the Yij are taken to be constant, then the one-dimensional form of inequality (24) (without sharp constants) was used by M. Christ ([23]) in obtaining optimal estimates for the restriction of the Fourier transform to curves

inRN. In addition to the flat and spherical geometric pictures for this inequality, the con

fonnal invariance of the problem provides a realization in terms of real hyperbolic geometry. Suppose JHIn denotes the unit two-sheeted hyperboloid in jRn+ I

JHIn = {p = (po, jJ) E jR X jRn: P5 - IjJI2 = 1}


51 5U CHAPTER 2

where JH[~ denotes the upper hyperboloid with Po ~ 1, and lHI~ denotes the lower hyperboloid with Po .:s -1. IHln is then a homogeneous space for the Lorentz group 0(1, n). Consider the map from JRn - {Ixl = I} to lHIn given by

2 1 + Ixl 2X)

P = ( 1 - Ix /2' 1 - Ix12

2 _ 11 - Ix l2111 - ly\21 11Ix - Y1 - 1pq2

where (only in the context of hyperbolic geometry) pq = POqo - pij and

dv(p) = 2150-- p2) dp = 280 + p2 - p6) dpo dp = 2nll - Ix12 1-ndx

is an 0 (l, n) invariant measure on lHIn • Then the multilinear Hardy-Littlewood-Sobolev inequality (24) has an equivalent fonnulation on the homogeneous manifold lHIn •

Theorem 8. For functions Gk E LPk(IHln), k 1, ... ,N and Pk > 1, "-1.>1L, Pk

f Ii Gk(qd IT Iqiqj - 11-Yij/2 dv(ql> .. . dV(qN)

k=1 i<j(27)

N

.:s Ep,y,n,N IT II GII Uk (JH[n)

k=1

where qk E lHIn , n > Yij = Yji ~ 0

LYik = 2~, L Yij < n(N - 1) i# Pk i<j

with Pk and p~ dual exponents. The best constant E p,y,n,N is attained for the extremalfunctions Gdq) = A[l + ij2]-n/(2p

;l up to conformal automorphism.

Inequality (27) results from using the stereographic projection map from the plane jRn to the two-sheeted hyperboloid JH[n described above, making the function change Gk(q) = 11 - IxI 2/n/Pk fk(x) in (24) and observing that

nL J,Ep,y,n,N = (../2) Pk Cp,y,n.N.

The relation between the parameters Y, n, and p is detennined by dilation invariance and the requirement for a confonnal structure. To check the upper bound on the size of y, consider each !k to be the characteristic function of a ball centered at the origin. Using the Euclidean product structure of the kernel, one could


show boundedness for multilinear fractional integrals by induction on dimension and an interpolation argument. A direct proof of Theorem 6 can be given by playing off the confonnal invariance which breaks the geodesic symmetry and symmetrization which improves the inequality by the Rearrangement Lemma. In fact, this lemma allows an equivalent fonnulation as a one-variable problem in terms of the dilation structure. Apply the lemma to (24) and let Yk = In IXk I with

hk(yd = exp(nyk/Pk)fk(Yk).

Theorem 9. For functions hk E LPk (lR), k = I, ... , Nand Pk > 1, L -1. > 1 Pk

yIf fI hk(Yd{O[2COSh(Yi - Yj) - 2 + l17i - 17jI2r ,j/2d17I ... d 17N} k=1 1<)

(28) dYI ... dYNI

N

.:s Ap.y,n,N n IlhkIlUk(IR)' k=1

where Yk E JR, 17k E sn-I, n > Yij = Yji ~ 0,

2n LYik = /' L Yij < n(N - 1). i# Pk i<j

The constants in Theorems 6 and 9 are related by

f(n/2)]L * Ap,y,n,N = [ 2:rcn/2 Cp,y.n,N.

Fractional integral estimates always have equivalent fonnulations in tenns of weak Lorentz spaces when rearrangement arguments are accessible, so Theorem 6

would, in fact, give sharp nonns for such functional inequalities. The interplay here between confonnal invariance and tensor structure provides

considerable insight into higher-order differential operators. Moreover, these theorems allow one to study confonnally invariant operators acting between non-dual LP spaces. In addition, algebraic invariance associated to the symmetric group is

central to understanding the geometric aspects of this work. The recognition that restriction phenomena would be closely associated to the geometric analysis of sharp constants arose in work with A. Carbery.

4 MOSER·TRUDINGER INEQUALITY

One of the most interesting aspects of the Hardy-Littlewood-Sobolev theorem is its relation to limiting Sobolev inequalities and geometric variational problems for


53 52 CHAPTER 2

confonnal defonnation. In view of the role played by sharp Sobolev constants in the arguments of Aubin, Obata, and Schoen for the Yamabe problem, this development is quite natural. But it was the impetus of quantum string theory and the focus on computing functional detenninants that emphasized the critical inequality of Moser-Trudinger-Onofri

(29) In { eF d~ S { F d~ + -41

( IVFI 2 d~.1s2 151 1s2 Here equality is attained only for functions of the fonn

F(~) = -21n II - ~ . ~I + c, I~I < l.

There are, in fact, two sharp constants here and two different geometric variational problems for confonnal defonnation: to characterize gaussian curvature and to extremize the functional detenninant of the Laplacian. The first constant is the factor one-fourth which was obtained by Moser using geometric symmetrization. Motivated by Polyakov's string theory models, Onofri detennined the sharp normalization constant using confonnal invariance. The effort to fully understand this inequality led to several incisive results, including: 1) a geometric derivation of (29) and its analogue for the circle by Osgood, Phillips, and Sarnak ([59]) in the context of computing functional detenninants on Riemann surfaces; 2) the Carleson-Chang theorem for the disc ([20]); 3) the explicit calculations of zetafunction determinants on four-manifolds by Branson and 0rsted ([15]); and 4) the study of the linearized Adams inequality on the four-sphere by Branson, Chang, and Yang ([14]).

Not only is inequality (29) and the corresponding Carleson-Chang inequality for non-negative functions with zero-boundary value on the unit disc in JR2

(30) In[~ { e2fdx]+[~ { 2f dx]-1 S 1+_1 { IVfl2dxeJr 11xI:'Oi Jr l!xl:'01 4Jr 11xI:'Oi

a consequence of the sharp Hardy-Littlewood-Sobolev inequality, but one can find by the same techniques a family of higher-dimensional Moser-Trudinger inequalities ofconsiderable geometric importance. A striking result is that under confonnal defonnation with fixed volume the zeta-function detenninant of the confonnal Laplacian is extremized by the standard metric up to confonnal automorphism (see [10],[14],[15]). The essential feature of the analysis is to combine conformal invariance and geometric symmetrization, while exploiting the paradigm that "fractional integration controls Sobolev estimates,"

Theorem 10. For a real-valued function F defined on the sphere sn with an expansion in spherical harmonics F = L~o Yk an exponential-class a priori

. GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS

inequality ofMoser-Trudinger type holds

(31) In ( eF(I;) d~ S ( F(~) d~ + 2- t q(n) f Iyd d~1s. 1s· 2n k=l S'

f'(n + k) 1q(n) = = -- k(k + 1) ... (k + n - 1).

f'(n)f'(k) r(n)

Using the spectral representationfor the Laplacian, this inequality can be written in terms of a positive-definite conformally invariant operator Pn(-b.) which is

(l differential operator for even n and a pseudo-differential operator ofboundary

type for odd n.

(32) In IS" eF(I;) d~ S Is. F(~) d~ + 2~! Is. F(PnF) d~ .-2 "'"2

Pn(-.1.) = n[-.1. +f.(n - 1 - f.)]n even i=O

.-3

1 2J1/2""2= -.1.+(~) n[-.1.+f.(n-I-f.)]nodd.[ 2 i=O

This inequality is invariant under the conformal transformation

F(~) -+ F(r~) + In J(r,~)

with r an element of the conformal group of sn and J the modulus of the corresponding Jacobian determinant. Equality in the above inequalities is attained

only for functions of the form

F~ (~) = -n In 11 - ~ . ~ I + C ~ E Sn

where ~ E Bn+1 = {x E JRn+l: Ixl < I}. Here d~ corresponds to normalized

. surface measure on sn.

There are three parts to the proof of the theorem: I) the sharp inequality, 2) confonnal invariance of the functional inequality, and 3) equality being attained only for confonnal factors, The first two parts are readily treated on the "differential operator side" using the differentiation argument discussed previously. Determining conditions for equality is more natural from the "fractional integral side" using a method that goes back to Sobolev for n-dimensional integrals. Invert the "differential operator" by a "fractional integral" and analyze an equivalent sharp inequality in the dual space setting. In fact, one could do the entire problem in tenns of fractional integrals, but the applications to geometric analysis suggest that one should understand as much of the problem as possible on the differential


5554 CHAPTER 2 GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS

operator side. For the exponential class the dual space setting is the Orlicz class Both Theorems 10 and 11 are consequences of Theorem 2 using end-point LIn L n L 1 since the exponential and the logarithm are complementary Young's differentiation arguments at p = 1. But a quick calculation of inequality (31) is functions. The dual space result is contained in the following theorem. obtained by inverting inequality (14) for q = pi ~ 2 and taking an appropriate

limit for q ~ 00. For G = L Yk Theorem 11. Suppose F and G are non-negative functions on the sphere sn with

J F d~ = JG d~ = 1. Then

(33) -n ( F(~) In I~ - 771 2G(n) d~ d77 ]snxsn

~ -n r In I~ - 771 2 d~ + { Fin F d~ + { Gin G d~h· hn h. and equality is attained only for functions ofthe form

F = G, Fdn = All - s . ~I-n, lsi < 1.

-n ( In I~ - 771 2 d~ = -n[ln 4 + 1{!(nj2) - 1{!(n)] 1{! = (In f)/.]s.

For F = G = I + L~1 Zk inequality (4) is equivalentto

n f: f(n)f(k) { Izdd~ ~ 2 ( GlnGd~(34) k=l f(n + k) ]sn ]sn

which by duality inverts inequality (3).

These two theorems are directly equivalent by duality, but either can be obtained

by a differentiation argument ([10]) applied to the Hardy-Littlewood-Sobolev in

equality (12) transferred to the sphere. They are the "end-point information" at

(L 1, L 00) for this inequality, just as the sharp logarithmic Sobolev inequality (20)

for the sphere is the "end-point information" at L 2 • Conformal invariance suggests

that the Green's function for the operator will playa critical role. This is already

implicit in the work of Carleson and Chang, though the equivalence of inequalities

(29) and (30) is not so apparent. Especially in conformal field theory, the Green's

function is often at the forefront of the analysis.2 The combination of these ideas

about the Green's function and nonlinear differential equations from analysis, dif

ferential geometry, quantum field theory, and statistical mechanics suggests that

much remains to be discovered about the implications of these results, especially

in terms of geometric structure.

2After the author proved these theorems, two preprints appeared that touched on similar themes. Using a mean-field calculation in statistical mechanics, M. Kiessling recognized the relation between the logarithmic potential and the Moser-Trudinger inequality in two dimensions. Subsequently E. Carlen and M. Loss developed a proof of the conditions for equality in much the same spirit as the argument given here.

(35) [IiGllu(s.)]2 ~ f:..!.- {IYkl2d~ = ( G*(TG) d~

k=O Yk ]s· ]s.

~ r(njq)r(njq/ + k) I 2 = LJ Ifkl d~ k=O r(njq/)r(njq + k) S·

where T is now a positive-definite self-adjoint operator. For a bounded real-valued

function F, set G = 1 + (1jq)F and apply the limit q ~ 00 to

i.11 + ~Flq d~

< [1 + ~ f F d~ + .2.. f f(njq)f(njq' + k) f IYkl 2 d~]q/2 - q S. q2 k=O f(njq')r(njq + k) S·

which immediately gives inequality (31). To rephrase this step as an end-point

differentiation argument, set e = 1j q and rewrite this inequality as

(36) [Ill + eFIiLl/8(sn)]2 ~ 1 + 2e f. F d~ + e 2 f. F(T F) d~. There is no loss of generality here since all inequalities are a priori and (36) is com

pletely equivalent to the original fractional integral inequality (13) on the sphere.

TheoperatorT dependsonebuteT is stable ate = 0, namely limo.....0 eT = ~ Pnn.

on functions orthogonal to constants where Pn is the operator occurring in Theo

rem 10. Now inequality (36) is an identity at e = °for all bounded F so that this

functional inequality has a positive derivative at e = 0, namely

(37) 2 f F d~ + ~ f F(PnF) d~ - 21n f e F d~ ~ O.

sn n! S. S·

But this is simply inequality (32). The condition on boundedness of F is easily removed by taking limits. Con

formal invariance can also be obtained by either a limiting argument from the

conformal invariance of (14), directly from the form of inequality (33), or by an

explicit analysis of the operators Pn (-b.). The fourth-order conformally invariant

operator

-b.(-b. + 2)

was observed by Paneitz ([60]) to be important in the interplay between the

conformal and gauge groups for Maxwell's equations.


56 CHAPTER 2 GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS 57

To see the confonnal invariance of inequality (37) directly from the calculation of the above limit process, observe that inequality (36) is invariant under the confonnal transfonnation

[1 +OF(~)] ~ [I +OF(r~)]J6(r,~)

where r is a confonnal automorphism of sn and J (r, ~) is the modulus of the corresponding Jacobian detennin~t. In the limit 0 ~ 0, this is equivalent to replacing F in inequality (36) by F(~) = F(r~) + In J (r, ~) so that

2 { Fd~ + ~ { F(PnF)d~ - 2In { eF d~Jsn n! Jsn Jsn

= 2 { F d~ + ~ ( F(PnF) d~ - 2ln { eFd~Jsn n! Jsn Jsn and in fact the fonn

2[ Fd~+~ (F(PnF)d~Jsn n! Jsn is invariant under the transfonnation F ~ F.

Since constant functions are extremal for inequality (37), confonnal invariance shows that the functions

F~(~) = -n In 11 - ~ . ~I + c for ~ E sn and ~ E lRn+l, I~ I < 1are also extremal. It remains to show thatthese confonnal factors are the only extremals. The classical method to obtain extremal functions is to apply symmetrization arguments and, for the case of a differential operator, to invert the operator and study a fractional integral in a dual setting. The idea goes back to Sobolev for the problem simply of showing boundedness and is exactly the step used to show that the multiplier inequality (35) is an equality only for functions ofthe fonn G~ (~) = All - ~ . ~ r n/ q with ~ a point in the open unit

1ball in lRn+ . Thus to detennine the extremal functions in Theorem 10, it suffices to detennine the extremal functions for Theorem 11.

The entropy inequality (33) can be directly obtained using an end-point differentiation argument at p = 1 from the fractional integral inequality on the sphere (13). For non-negative functions F and G having integral one

-n { In I~ - 1]12 d~ + { FIn F d~ + { GIn G d~hn hn hn

::: -n ( F(~) In I~ - 1]1 2G(1]) d~ d1].Jsn The logarithmic kernel is a monotone strictly decreasing function of the distance so one can apply the Symmetrization Lemma from section 2. The argument is usually

made for positive kernels, but the sign of the kernel plays no role as long as it is a monotone decreasing function of the distance between two points and the other functions are nonnegative. This symmetrization lemma shows that an extremal function for the inequalities in Theorem 11 must be radial decreasing with respect to distance on the sphere from some pole. On every domain which is confonnally equivalent to the sphere sn, one can find an equivalent inequality written in tenns of a strictly decreasing integral kernel. Moreover, one can find an equivalent one-dimensional inequality on IR+ which captures the "dilation character" of the inequality. The same symmetrization argument can be applied there (lR+ ~ lR with dilation on IR+ going over to the translation on lR) forcing the extremal function to be symmetric with respect to distance on lR+ which can then be traced back to its equivalent fonnulation on sn where this inversion symmetry means that the extremal function must be symmetric with respect to a hemisphere. Since constants are the only measurable function which are both radial decreasing away from a pole and symmetric with respect to a hemisphere, the only extremal functions for inequality (33) are the constants and the corresponding Jacobian factors for

confonnal transfonnations

G~(~) = All _ ~ . ~I-n

where ~ is a point in the interior of the unit ball in lRn+I and A is a positive nonnalization constant. Since (33) is equivalent to (31), this means that the only extremals for the n-dimensional Moser-Trudinger inequality are of the fonn

F~(~) = -n In 11 - ~ . ~I + c with I~ I < 1. This is the same method that was used above in section 2. The point is that symmetrization on lR+ picks out a specific extremal which "breaks the sn symmetry" unless the equivalent function on sn is constant almost everywhere. A second approach applies an observation of Carlen and Loss ([19]) that in a generic sense "confonnal transfonnations break symmetry" and allows the problem to be done on a single domain. Suppose one has an extremal function for inequality (33)

which is radial decreasing away from a pole. Perfonn a confonnal transfonnation

G(~) ~ G(r~)J(r,~) = G(~)

. where r is not simply a rotation; by the confonnal invariance of the inequality, this

will give a new extremal function which cannot be radial decreasing away from some pole unless the original extremal function was a confonnal factor. But then the symmetrization argument would force some increase in the integral

-n ( G(~) In I~ - 1]1 2G(1]) d~ d1]Jsnxsrr

so, in fact, the only extremals are confonnal factors.


5958 CHAPTER 2 GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS

The dual setting a priori inequalities used above are collected in the following theorem.

Theorem 12 (Entropy Inequalities). Suppose f and g are non-negative LIn L functions on JR.n with IRn f dx = IRn g dx = 1. Then (38)

-n { f(x)ln\x-y\2 g(y)dxdysCn + { flnfdx+ ( gIngdxJIRn x Rn JRn JIRn

Cn = n}n n + n[1/!(n) - 1/!(n/2)] - 2In[r(n)/ r(n/2)]

where 1/!(z) = 1i In r(z) and extremaIs are given up to conformal automorphism by

f(x) = g(x) = A(l + Ix/ 2)-n.

10Suppose u and v are non-negative LIn L functions on JR.+ with 1000 u(t) '¥

00 vet) '¥ = 1 and

q;(t) = -n 1.-1 In[t + 1ft - 2~ . 1)J d~.

Then

1dt ds dt dt(39) u(t)q;(t/s)v(s) - - S Dn + u In u - + v In v 1 1R+xR+ t s R+ t R+ t

Dn = {4n n/[r(n/2)f} Cn

and extremaIs are given up to translation on JR.+ by

u(t) = vet) = A(t + 1/0-n.

Since the kernel here has variable sign, one must check that the conditions on the functions guarantee that the inequalities are well-defined (that is, there is no cancellation of infinities). Inequalities (38) and (39) follow by straightforward differentiation arguments from the Hardy-Littlewood-Sobolev inequalities for smooth functions with compact support on the domains. Using the observation that the kernel is integrable over any compact neighborhood of the identity, then the inequality for smooth functions insures by a limiting argument that the integral of the positive part of the integrand over the product space must be finite. Hence, inequalities (28) and (29) are well-defined under the condition that the functions should be both in L 1 and L In L. It is useful to emphasize that the functional inequalities in Theorems 11 and 12 are directly equivalent for smooth functions with compact support using conformal transformations, but the class L In L n L 1 is not

preserved by these transformations. However, the class of extremal functions is

guaranteed to be preserved. As explained at the beginning of Section 2, fractional integral inequalities cor

respond to extending Young's inequality for convolution of functions on JR.n to include a weak Lorentz class. Inequality (38) has a corresponding interpretation here. Consider the class of measurable functions K on JRn such that

exp[K/2] E Ll,oo(JR.n)

which is equivalent to

K*(x) S -n In \x\2 + C

where K" denotes the equimeasurable radial decreasing rearrangement of K. The smallest value of C defines a size CK for this class. Since the sets (x: K (x) > a}

have finite measure, the following theorem is obtained using the Riesz-Sobolev

Lemma and Theorem 12.

Theorem 13. Let K satisfy the property exp[K /2J E LI,oo(JR.n) and f and g be non-negative LIn Lfunctions on JR.n with integral equal to one. Then

( f(x)CK * g)x) dx S CK + en + { fln f dx + ( gIn g dx(40) k kn kn Cn = n In n + n[1/!(n) - 1/!(n/2)] - 2In[r(n)/ r(n/2)J.

In addition, the Hardy-Littlewood-Sobolev inequality provides an extension of the classical sharp Sobolev inequalities in the context of dimensions one and two and more generally an interpolating result for the operators Pn ( - ~) of Theorem 10 in higher dimensions. These inequalities are not conformally invariant but do contain the higher-dimensional Moser-Trudinger inequality as a limiting case.

Theorem 14.

(41) ({ IFC~W d~)2/q S q - 2 { IVFI2 d~ + { 1F12 d~Jsn n Jsn Jsn

. for 2 S q < 00 if n = 1 or 2 and 2 S q S n~2 if n ~ 3;

(42) ({ 1F(;)lq d~)2/q S q - 21 IVul2 dx + ( 1F12 d~Jsn Cn lxl~l Jsn

for 2 S q < 00 if n = I and 2 S q S n~l if n ~ 2 where u is the harmonic extension of F to the interior of the unit ball in JR.n+l and en = 2n(n+l)/2/

r(Cn + 1)/2).


60 CHAPTER 2

Theorem 15. On sn for 2 ~ q < 00

(43) (r IFlq d~)z/q ~ q - 2 [ F[Pn(-ll)F]d~ + [ 1F12d~. 1 s n n! 1 s n 1sn

The proofs of these two theorems depend on convexity properties of the spectral data for the Hardy-Littlewood-Sobolev fractional integral on the sphere ([10)). An interesting geometric P.D.E. proofof the classical part of(42) was given by Escobar ([27],[28)) using the character of an Einstein metric.

Interest in this problem goes back to some remarks made by J. Moser at a lunch organized by Narasimhan at Chicago in the late 1970's and a talk given by S.

Y.A. Chang at Arcata in the summer of 1985. But the real impetus for seeing the possibility of these theorems came from the observation that a special case of (41),

1/2 (44)

( 12 1F14 d~ ) ~ 12 IV' FI

2 d~ + L1F12 d~, was a consequence of SL(2, JR) invariance for sharp Sobolev inequalities on the Heisenberg group. An unusual feature of these problems is that the algebraic invariance may not always be evident as with the Carleson-Chang inequality (30) or the Heisenberg inequality (44). But, in fact, this indicates a deep connection with geometric structure. The sharp form of the Carleson-Chang inequality and its equivalence with the Moser-Trudinger-Onofri inequality on the two-sphere is part of recent work by S.-Y.A. Chang and the author. It is now evident that there is a strong structural relation determined by conformal invariance and the Green's function which connects the Moser-Trudinger inequalities of Theorem 10, the work of Carleson and Chang ([20]), the Adams inequalities ([1]), and the entropy estimates of Theorems 11 and 12. This point will be discussed in detail in a forthcoming paper.

5 ZETA·FUNCTION DETERMINANTS

The interaction of ideas from differential geometry, representation theory, and string theory has focused attention on conformally invariant operators, their relation to conformal deformation and Einstein metrics, and especially the calculation of functional determinants. An elegant example coming out of Polyakov's string theory ([57],[58],[64]) was the computation that under conformal deformation of metric on the two-sphere ds z = e F dsJ

det(-3.)1 det(-ll) = e(-1/3)S(F)

S(F) = ~ 1 IV'FI 2d~ + 1F d~ - In 1eF d~

4 S2 S2 S2

GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS 61

so that the sharp Moser-Trudinger inequality (29) implies that the determinant of the Laplacian on SZ is maximized by the standard metric. Here, ~ denotes the Laplace-Beltrami operator for the deformed metric. But a classical result already occurs in this context as Widom pointed out that the one-dimensional MoserTrudinger inequality (31) is part of Szego's limit theorem for determinants of Toeplitz operators ([75],[35],[79]). Branson and 0rsted ([ 15]) calculated explicit formulas for functional determinants in four dimensions, including the conformal Laplacian and the square ofthe Dirac operator. Branson, Chang, and Yang ([14]) recently studied the general character of such operators under conformal deformation on a four-manifold and the relation to the linearized Adams exponential-class Sobolev inequality for second-order operators on S4. Using the Adams inequality, they were able to show the existence of extremals for the variational problem. S.-Y.A. Chang and the author together recognized that the sharp S4 inequality from Theorem 10 would determine the extremals for the determinant of the conformal Laplacian. Under conformal deformation of metric with fixed volume, this determinant is extremized by the standard metric up to conformal automorphism.

Theorem (Branson and 0rsted). Consider a conformal deformation ds z = eF / 2 dsJ on the sphere S4. Let A tf,:note either the conformal Laplacian or the

square of the Dirac operator while A denotes the same operator under conformal

deformation. Then

(45) det AI det A = e-{J,SI(F)-/hS2(F)

SI(F) = ~ ( F[-ll(-ll + 2)F] d~ + [ F d~ -In 1eF d~ 48 1s' 1S4 S4

Sz(F) = [ e-F/ 2(lle F /4 )z d~ - ~ { IV' Fl z d~. 1S4 4 1S4

Relating the values given in Branson and 0rsted ([ 15]) to the normalization used here, f31 = f3z = -180 for the conformal Laplacian and 7f31 = 22f3z = 77/180 for the square of the Dirac operator. Note that the notation here differs from that in ([14]) and ([l5]). SI (F) is non-negative by Theorem 1 and S2(F) is non-negative by the following lemma. Both functionals are conformally invariant and vanish only for functions of the form

F~(~) = -41n 11 - {' . ~I + C, I{'! < 1.

Lemma 1.

(46) 1e-ZW(lleW)2 d~ ~ 4 { lV'wlzd~. S' iS4


63 !IIII 62 CHAPTER 2

This lemma was proved by the author using "iterated gradients" and the BochnerLichnerowicz-Weitzenb6ck formula and by P. Yang using the sharp Sobolev inequality. Yang's method clearly demonstrates the conformal invariance of the functional and extends easily to include higher-order differential operators related to the Hardy-Littlewood-Sobolev inequality.

Lemma 2. Suppose T is a positive-definite linear operator with T 1 = 0 andfor someq > 2

[lifiILq(s")]2 S ~ [ f(Tf) d~ + [ Ifl 2 d~; cis" isn

then for f 2: 0

(47) c < -TfI + c II . - II f sq/(q-2) (sn)

From the Hardy-Littlewood-Sobolev inequality for f = I: Yk

00 f(!!. )f(.!!. + k) / [llfIILQ(s")]2 S ~ f(; W(~ + k) IYk l

2 d~

together with the lemma, one can obtain a family of conformally invariant inequalities for higher-order differential operators.

Theorem 16. On sn let a = n /2 - n / q for q > 2 and define

f(B + 4+ a)B = [ _ ~ + (n ~ 1 rr/2

D2a = 1 ' f(B + 2 - a)

then

2 f(!!. - a) [(48) [IIFllu(s")] S ~ + a) isn F(D2a F) d~T""'I

andfor f > 0

(49) fq + a) 1

S II 7D2a f II L("/2<r)(sn)'

Equality holds only for functions of the form

f(~) = All - ~. ~Ia-I, I~I < 1.

This inequality seems likely to be most useful in the cases where a is a positive integer and n/2a is an even integer (D2a is a differential operator). The


more explicit realization of the Hardy-Littlewood-Sobolev inequality in terms of intertwining operators for SO(n + I, 1) extends a suggestion ofT. Branson.3

6 LIE GROUPS

The principal theme of the program of geometric analysis outlined here is to understand the patterns of symmetry that exist for a problem. As the central questions in analysis involve increasingly and more fully higher-order differential operators, multilinear tensor analysis, non-homogeneous dilations, and surface singularities, "hard analysis" on Lie groups and symmetric spaces becomes more important. Techniques developed in this program have already made contributions in terms of Lie groups that arise naturally in real and complex analysis: namely SL(2, 1R), the Lorentz groups, the Heisenberg group, and quaternionic groups.

To illustrate this general approach, some results about SL(2, 1R) are described. Three intrinsic features of SL(2, 1R) are its natural action on the upper half-plane, the exponential growth ofgeodesic balls and the role of SL (2, 1R)/S0 (2) as the "dilation structure" of the Heisenberg group lHln • The hyperbolic structure extending the natural SL(2, 1R) action on a domain is at the heart of the inequalities described in the previous section. From representation theory Kunze and Stein recognized a surprising extension of Young's inequality for SL(2, 1R). This phenomenon reflected the exponential character of Haar measure on this group.

Kunze-Stein Phenomenon. For G = SL(2, 1R) and 1 S P < 2

(50) IIf * gllu(G) S Apllfllu(G)llgIIU(G)'

Stein extended this theorem to the complex classical groups and conjectured that it should hold for semi-simple Lie groups with finite center. In addition, for the case where f is bi-invariant, he gave a simple proof using Harish-Chandra's estimates for the decay of spherical functions. Cowling later gave a complete proof of the conjecture using representation theory. But symmetrization and the rearrangement lemma of Brascamp, Lieb, and Luttinger provide an immediate reduction for SL(2, 1R) to the case where f is bi-invariant.

Theorem 17. For G = SL(2, 1R)

(51) 1£ h(x)(f * g)(x) dxl s £h#(x)(f# * g#)(x) dx.

3A research announcement describing the results of Sections 4 and 5 appears in "Moser-Trudinger Inequality in Higher Dimensions;' Int. Math. Res. Not. (1991),83-91.


65 64 CHAPTER 2

# denotes an equimeasurable rearrangement of the absolute value of a function on G such that for a Cartan decomposition x = (p, k), the rearrangedfunction is radial decreasing in the p variable. Here p E M = SL(2, R)/SO(2) and k E K = SO(2).

This theorem allows the Kunze-Stein inequality (50) to be reduced to functions radial in the p variable and constant in the k variable, so, in fact, to bi-invariant functions. A similar proof works for SL(2, iC), the Lorentz groups, and any appropriate group ~ith abelian nilpotent part.

An interesting feature ofanalysis on the Heisenberg group and complex analysis in several variables is the lack of reflection symmetry. In part, this is simply a con

sequence of the two-parameter dilation structure given by M = SL (2, R) ISO (2).

Using the sharp Hardy-Littlewood-Sobolev inequality and analysis of fractional integration on the Heisenberg group, a one-parameter family of Sobolev inequalities is obtained for M. To set notation, let w = x + iy E M for y > 0; dm = y-2 dx dy is left-invariant measure on M, the gradient is D = yV, and the invariant distance function is

Iw - w'l d(w, w') = 2&

Theorem 18. For 1 S p < 2, lip + lip' = 1 and A = 2(n + 1)lp' for integral n

(52)11 F(w)G(w')Vt... [d(w, w')] dm(w) dm(w')1 S ApIlFIIU(M)IIGlluCM) MxM

Vt... (u) = (1 + u2)-"'/2F(A/2, A/2; n + 1; _1_ ), , 1 + u2

Extremalfunctions are given by [1 + d2(w, i)rCn+l)/p up to conformal automorphismsof M.

For q = 2(n + 1)ln (53)

1 [4 ]I/q {1 2 1 }1/2IlFlluCM) S.j1in: M IDFI dm + i (n - 2) M 1F12 dm

For n > 1, extremal functions are given by [1 + d2(w, i)rn/2 up to conformal automorphisms.

Further results for the Heisenberg group and nilpotent groups of type-H, F (iC) and SL(2, R) will be discussed in forthcoming work. Analysis of convolution

inequalities and complex symmetry structure is a very rich subject in Fourier


Analysis. The foundations for this program were set down by Hardy and Littlewood in the 1920's and in higher dimensions by Sobolev in the 1930's. Modem contributions have been made by many individuals mentioned in this article. The intersection of ideas from analysis, geometry, probability, and physics has made

this subject a useful part of contemporary mathematics.

Remarks. I have a special feeling for Eli Stein. His warmth and enthusiasm

that sparkle in his eyes-lit up the hallways of old Fine Hall the day I met him.

His lectures on Fourier Analysis that fall were filled with insight and marked the direction of my mathematical work. Both as a friend and as a mathematician, he

has given me much inspiration.

Acknowledgments. I want to thank the following individuals for their generous comments: Tom Branson, Tony Carbery, Alice Chang, Jose Escobar, Charlie

Fefferman, David Jerison, Carlos Kenig, Bent 0rsted, Aleksandr Pelczynski, D. H. Phong, Peter Samak, and Eli Stein. Portions of this work were completed

while visiting the University of Chicago.

University afTexas, Austin

REFERENCES

1. D. R. Adams. "A sharp inequality of J. Moser for higher order derivatives." Ann. of Math. 128 (1988), 385-398.

2. O. Alvarez. "Theory of strings with boundary." Nucl. Phys. B 216 (1983),125-184. 3. T. Aubin. "Meilleures constantes dans Ie theorem d'inclusion de Sobolev et un theorem

de Fredholm non lineaire pour la transformation conforme de la courbure scalaire." J. Punet. Anal. 32 (1979),14&-174.

4. T. Aubin. "Best constants in the Sobolev imbedding theorem: the Yamabe problem." In Seminar on Differential Geometry edited by S.-T. Yau, Princeton University Press, 1982.

5. A. Baernstein and B. A. Taylor. "Spherical rearrangements, subharmonic functions. and *-functions in n-space." Duke Math. J. 43 (1976), 245-268.

6. D. Bakry and M. Emery. "Hypercontractivite de semi-groupes de diffusion." C. R. Acad. Sci. Paris 299 (1984), 775-778.

7. M. Bander and C. Itzykson. "Group theory and the hydrogen atom." Rev. Mod. Phys. 38 (1966), 330-358.

8. W. Beckner. "Inequalities in Fourier analysis." Ann. ofMath. 102 (1975),159-182. 9. W. Beckner. "A generalized Poincare inequality for Gaussian measures." Proc. Amer.

Math. Soc. 105 (1989),397--400. 10. W. Beckner. "Moser-Trudinger inequality in higher dimensions." Int. Math. Res. Not.

(1991),83-91.


66 67 CHAPTER 2

11. W. Beckner. "Sobolev inequalities, the Poisson semigroup and analysis on the spheresn." Proc. Nat. Acad. Sci. 89 (1992), 4816-4819.

12. W. Beckner. "Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality." Ann. ofMath. 138 (1993), 213-242.

13. A. Beurling. Etudes sur un probleme de majoration. Thesis, Uppsala, 1933. 14. T. Branson, S.-Y. A. Chang, and P. Yang. "Estimates and extremals for zeta function

determinants on four-manifolds.." Comm. Math. Phys. 149 (1992), 241-262. 15. T. Branson and B. 0rsted. "Explicit functional determinants in four dimensions." Proc.

Amer. Math. Soc. 113 (1991), 671--684. 16. H. J. Brascamp and E. H. Lieb. "Best constants in Young's inequality, its converse, and

its generalization to more than three functions." J. Funct. Anal. 20 (1976), 151-173. 17. H. J. Brascamp, E. H. Lieb, and J. M. Luttinger. "A general rearrangement inequality

for multiple integrals." J. Funct. Anal. 17 (1974), 227-237. 18. A. P. Calderon M. Weiss, and A. Zygmund. "On the existence of singular integrals."

Proc. Symp. Pure Math. 10 (1967), 56--73. 19. E. A. Carlen and M. Loss. "Extremals of functionals with competing symmetries."

J. Funct. Anal. 88 (1990),437-456. 20. L. Carleson and S.-Y. A. Chang. "On the existence of an extremal function for an

inequality of J. Moser." Bull. Sci. Math. 110 (1986), 113-127. 21. L. Carleson and P. Sjolin. "Oscillatory integrals and a multiplier problem for the disc."

Studia Math. 44 (1972), 287-299. 22. S.-Y. A. Chang and D. E. Marshall. "On a sharp inequality concerning the Dirichlet

integral." Amer. J. Math. 107 (1985), 1015-1033. 23. M. Christ. "On the restriction of the Fourier transform to curves: endpoint results and

the degenerate case." Trans. Amer. Math. Soc. 287 (1985), 223-238. 24. E. D'Hoker and D. H. Phong. "On determinants of Laplacians on Riemann surfaces."

Comm. Math. Phys. 104 (1986),537-545. 25. E. D'Hoker and D. H. Phong. "The geometry of string perturbation theory." Rev. Mod.

Phys. 60 (1988), 917-1065. 26. Vi. S. Dotsenko. "Lectures on conformal field theory." Advanced Studies in Pure

Mathematics 16 (1988) "Conformal Field Theory and Solvable Lattice Models," 123-170.

27. J. F. Escobar. "Sharp constant in a Sobolev trace inequality." Indiana Math. J. 37 (1988), 687--698.

28. J. F. Escobar. "Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate." Comm. Pure Appl. Math. 43 (1990), 857-883.

29. C. Fefferman. "Inequalities for strongly singular convolution operators." Acta Math. 124 (1970), 9-36.

30. C. Fefferman. "Monge-Ampere equations, the Bergman kernel and geometry of pseudoconvex domains." Ann. ofMath. 103 (1976), 395-416.

31. C. Fefferman. "Parabolic invariant theory in complex analysis." Adv. Math. 31 (1979), 131-262.

32. C. Fefferman and C. R. Graham. "Conformal invariants" in "Elie Canan et les Mathematiques d'aujourd'hui." Asterisque (1985), 95-116.

33. G. B. Folland and E. M. Stein. "Estimates for the 8b complex and analysis on the Heisenberg group." Comm. Pure Appl. Math. 27 (1974),429-522.


34. G. B. Folland and E. M. Stein. Hardy Spaces on Homogeneous Groups. Princeton University Press, 1982.

35. U. Grenander and G. Szego. Toeplitz Forms and Their Applications. University of California Press, 1958.

36. L. Gross. "Logarithmic Sobolev inequalities." Amer. J. Math. 97 (1975),1061-1083. 37. G. H. Hardy, J. E. Littlewood, and G. polya.lnequalities. Cambridge University Press,

1934. 38. S. Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. Academic

Press, 1978. 39. S. Helgason. Groups and Geometric Analysis. Academic Press, 1984. 40. L. Hormander. "Oscillatory integrals and multipliers on F LP." Arkiv. Math. 11 (1973),

I-II. 41. C. Itzykson. "Remarks on boson commutation rules." Comm. Math. Phys. 4 (1967),

92-122. 42. H. P. Jakobsen and M. Vergne. "Wave and Dirac operators and representations of the

conformal group." J. Funet. Anal. 24 (1977),52-106. 43. S. Janson. "On hypercontractivity for multipliers on orthogonal polynomials." Arkiv.

Math. 21 (1983),97-110. 44. D. Jerison and J. M. Lee. "A subelliptic nonlinear eigenvalue problem on CR manifolds."

Contemp. Math. 27 (1984), 57--63. 45. D. Jerison and J. M. Lee. "Extremals for the Sobolev inequality on the Heisenberg

group and the CR Yamabe problem." J. Amer. Math. Soc. 1 (1988), 1-13. 46. S. Kobayashi and K. Nomizu. Foundations ofDifferential Geometry. John Wiley, 1969. 47. E. H. Lieb. "Existence and uniqueness of the minimizing solution of Choquard's

nonlinear equation." Stud. Appl. Math. 57 (1977), 93-105. 48. E. H. Lieb. "Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities."

Ann. ofMath. 118 (1983), 349-374. 49. P. L. Lions. "The concentration-compactness principle in the calculus of variations n.

The limit case." Rev. Mat.lber. 1 (1985),45-121. 50. H. P. McKean. "Geometry of differential space." Ann. Prob. 1 (1973), 197-206. 51. J. Moser. "A sharp form of an inequality by N. Trudinger." Indiana Math. J. 20 (1.971),

1077-1092. 52. J. Moser. On a nonlinear problem in differential geometry. In Dynamical Systems, edited

by M. M. Peixoto. Academic Press, 1973. 53. C. E. Mueller and F. B. Weissler. "Hypercontractivity for the heat semigroup for

ultraspherical polynomials and on the n-sphere." J. Funct. Anal. 48 (1982), 252-283. 54. A. Nagel and E. M. Stein. Lectures on Pseudo-differential Operators. Princeton

University Press, 1979. 55. E. Nelson. "The free Markoff field." J. Funet. Anal. 12 (1973), 211-227. 56. M. Obata. "The conjectures on conformal transformations of Riemannian manifolds."

J. Diff. Geom. 6 (1971), 247-258. 57. E. Onofrio "On the positivity of the effective action in a theory of random surfaces."

Comm. Math. Phys. 86 (1982), 321-326. 58. E. Onofri and M. Virasoro. "On a formulation ofPolyakov's string theory with regular

classical solutions." Nuc. Phys. B 201 (1982),159--175. 59. B. Osgood, R. Phillips, and P. Samak. "Extremals of determinants of Laplacians."

J. Funct. Anal. 80 (1988), 148-211.


68 CHAPTER 2

60. S. Paneitz. "A quartic confonnally covariant differential operator for arbitrary pseudoRiemannian manifolds." Unpublished preprint, 1983.

61. D. H. Phong. "Complex geometry and string theory." In Proceedings of the Geometry Festival. July 1990, edited by R. Greene and S.-T. Yau.

62. J. Polchinski. "Evaluation of the one loop string path integral," Comm. Math. Phys.l04 (1986),37-47.

63. G. P61ya and G. SzegO. Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, 1951.

64. A. M. Polyakov. "Quantum geometry of bosonic strings," Phys. Lett. B 103 (1981), 207-210.

65. R. Schoen. "Conformal deformation of a Riemannian metric to constant scalar curvature," J. Diff. Geom. 20 (1984), 479-495.

66. J. Schwinger. "Coulomb Green's function." J. Math. Phys. 5 (1964), 1606-1608. 67. A. Selberg. "Harmonic analysis and discontinuous groups in weakly symmetric Rie

mannian spaces with applications to Dirichlet series," J. Indian Math. Soc. 20 (1956), 47-87.

68. S. L. Sobolev. "On a theorem of functional analysis." Mat. Sb. (N.S.) 4 No. 46 (1938), 471-497.

69. E. M. Stein. "Analytic continuation of group representations." Adv. Math. 4 (1970), 172-207.

70. E. M. Stein. Singular Integrals and Differentiability Properties ofFunctions. Princeton University Press, 1970.

71. E. M. Stein. Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups. Actes Congres Intern. Math. (Nice, 1970), Tome 1, 173-189.

72. E. M. Stein. Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press, 1970.

73. E. M. Stein. Boundary Behavior of Holomorphic Functions of Several Complex Variables. Princeton University Press, 1972.

74. E. M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971.

75. G. Szego. On certain hermitian forms associated with the Fourier series ofa positive function. Festskrift Marcel Riesz (Lund, 1952),228-238.

76. G. Talenti. "Best constant in Sobolev inequality," Ann. Mat. Pura Appl. 110 (1976), 353-372.

77. N. Th. Varopoulos. "Sobolev inequalities on Lie groups and symmetric spaces," J. Funct. Anal. 86 (1989), 19-40.

78. F. B. Weissler. "Logarithmic Sobolev inequalities and hypercontractive estimates on the circle." J. Funct. Anal. 37 (1980), 218-234.

79. H. Widom. "On an inequality of Osgood, Phillips and Samak," Proc. Amer. Math. Soc. 102 (1988), 773-774.

80. A. Zygmund. Trigonometric Series. 2d ed. Cambridge University Press, 1959.

3

Representing Measures for

Holomorphic Functions

on Type 2 Wedges

Al Boggess and Alexander Nagel

1 INTRODUCTION

A striking fact about holomorphic function theory in several variables is that, under

Cnsuitable convexity hypotheses on a domain n c with n > I, if ZO E n is "close" to the boundary an, then there are representing measures for Zo whose

support on an is contained in a set which is concentrated near the orthogonal

projection of Zo onto the boundary. This is false for more general domains in Cn

and false for harmonic functions on domains in Rn•

Forasimpleexampleofthisphenomenon,letn = {(z, w); Re(w) > IzI2}. We

will denote the boundary of n by :E = {Re(w) = Iz12 } which can be identified

with the Heisenberg group. We wish to represent the value of a holomorphic

function F on n at the point (0, r) E n, r > 0, by integrating F against a

suitable measure on :E. To do this, we let <P E Cgo(C) be a radial function with

support in the unit disc and whose integral over C is one. Let

<Pr(w) = ~<p(2(W-r)) r r .

The function <Pr has support in the disc centered at r with radius § and the integral

of <Pr over C is one. The mean value property for holomorphic functions shows

that

F(O, r) = [ F(O, w)<Pr(w)dxdy ]WEC


71 70 CHAPTER 3

where we have written w = x + I y. The support of ,p, is contained in the square {1 ~ x ~ f, IYI ~ §}. Therefore, .jX is well defined on the support of ,p,.

Using the mean value property of F in the first variable, we obtain

2Jl

F(O, r) = - 1 1 F(.jXe iO , x + IY),p,(X + IY) d()dxdy.121l' WEe 0

Now, we change variables by letting x = t 2 and then we let z = te'B. We obtain

F(O, r) = !.. -I ( 1 F(z, Id + ly),p,(ld + ly)dJ..(z)dy1l' lZEC yER

where d)..,(z) denotes Lebesgue measure on the complex plane. The map (z, y) 1-+

(z, Izl2 + IY) for z E C and y E R is a parameterization for ~ and dJ..(z)dy is comparable to surface measure on ~, denoted by da. Therefore, we obtain

(1) F(O, r) = l F(S)K,(S)da(S)

where K,(·) is a smooth function. Note that the support of K,(S) is contained in a ball centered at the origin of radius Cy'r (where C is a uniform constant). Thus, K, is our desired local representing measure for holomorphic functions on the Heisenberg group (near the origin).

The same argument with inequalities replacing equalities implies that if u is a non-negative plurisubharmoriic function on n which is continuous up to ~, then

(2) u(O, r) ~ u(S)K,(S)da(l;).l The key idea in the above analysis is to slice n with analytic discs whose

boundary is contained in ~ (in the above case, the analytic disc is the map l; 1-+

(zl;, Id + iy) for l; in the unit disc in C). Then we average our holomorphic F (or subaverage our plurisubharmonic u) over the boundary of the discs. A similar slicing argument can be used for a more general convex or strictly pseudoconvex domain and a representation analogous to (I) and an estimate analogous to (2) can be derived. The case of weakly pseudoconvex domains of finite type presents additional difficulties since the boundary cannot always be locally convexified, but here, too, one can obtain local representing measures by imbedding suitable analytic discs (see [BDN]).

In this paper, we generalize these ideas to the case of "wedge domains" with an "edge" given by a CR submanifold M C en of real codimension greater than I. Our goal is to find local representing measures on the edge for points in the wedge near the edge. We restrict our attention to the case where the edge of the wedge has type 2, which is the analogue of the case of a domain with an edge given by a strictly pseudoconvex boundary (as discussed above). We have been

REPRESENTING MEASURES FOR HOLOMORPH1C FUNCTIONS

greatly influenced in our work by E. M. Stein's seminal observation that a strictly pseudoconvex boundary can be modeled at each point by a nilpotent Lie group, the Heisenberg group, and that the boundary behavior of holomorphic functions in strictly pseudoconvex domains is intimately connected with the approximating group structure on the boundary. In this paper, we carefully analyze a "model case" for our wedge domains where the edge is a nilpotent Lie group of step 2 (analogous to the Heisenberg group). This is done in section 4 of this paper. In section 2, we state our theorem on representing measures for more general wedge domains, and in section 3, we state an application to the boundary behavior of HP

functions defined on wedge domains. The details of the results given in sections 2 and 3 will appear in a laterpaper. The

results in section 2 for general wedges with type 2 edges are obtained by a threestage process which is again inspired by the work of Stein and his collaborators (see for example [FS] and [RS]): (i) we pass from the original object of study to a "free" object by adding appropriate variables; (ii) we solve the problem on the freed object by approximating it suitably by the model case; (iii) we return to the original object by integrating out the extra variables. The results of section 3 are

related to recent work of Rosay [R].

2 THE STATEMENT OF THE GENERAL THEOREM

In order to state our theorem, we first recall some basic facts about CR manifolds. For more details, see [B]. Let M be a Coo submanifold of Cn of real codimension d. For p EM, let TpM denote the real tangent space to M at p. Then the maximal complex subspace of TpM is T; = TpM n J(TpM), where J is the complex structure map given by multiplication by I = R. M is called aCR submanifold,

with CR dimension m if for all p EM, d imcT; M = m. M is called generic if for all p E M, TpM + J (TpM) = cn. If M is a generic CR submanifold of real codimension d and CR dimension m, it follows that n = m + d.

For p E M, let YpM denote the orthogonal complement to T; Min TpM, and let NpM be the orthogonal complement of TpM in TpCn (i.e., NpM is the normal

space of M at p). We have

TpM = T; EB YpM

Tpcn = TpM EB NpM

where all the direct sum decompositions are orthogonal. Let 1l'p: TpCn 1-+ NpM

be the orthogonal projection. Let H~·o(M) denote the subspace of the complexified tangent space to M at p spanned by tangent vectors of type (1,0). If Z E Hl.o(M), then the vector field *[Z, Z] is a real tangent vector field.


-- - - - ----

73 72

- --,.,-- - ~-.,.~::---------_._- _.::~- -----;:-------::: ;-- .-. - _. ~ ._""'-~ - ----:,.--.::

CHAPTER 3

Definition 2.1. The Levi form is the well defined quadratic mapping £p: H~·o(M) ~ NpM given by

1 £p(Z) = -1rp(J( - [Z, Z]p».

21

where Z E HI.o(M) is any vector field extension of Zp. The convex hull of the

image of £p is a cone in NpM and is denoted by r p'

Definition 2.2. Let M C C n be a generic CR submanifold. Then M is of type 2 at p E M ifand only-if r p has nonempty interior in NpM.

This definition is equivalent to the condition that the real and imaginary parts of the

tangential Cauchy-Riemann vector fields, together with their commutators, span the real tangent space to M at p.

For any subset K C NpM and any E > 0, let K, = {z E NpM I Z E K, and

Izl < E}. If Yl and Y2 are two cones in NpM, we say that YI is smaller than Y2 and

write YI < Y2 if YI n S is a compact subset of the interior of Y2 where S is the unit

sphere in NpM. We now describe the type of wedge domains that we will study.

Definition 2.3. Let M C Cn be a generic CR submanifold oftype 2. An open set Q C Cn is a domain with edge M if: (1) M C Q;

(2) For each p E M and for each cone Y < r p there exists an open set It) C M

containing p and an E > °such that

It) + Y, C Q.

Property 2) roughly states that near a point p E M, Q locally contains translates

of M in directions strictly interior to the cone r p'

The set It) + Y, is parameterized in a natural way by It) x Y,. After shrinking

E if necessary, we can consider the orthogonal projection IT of It) + Y, onto M so

that for z E It) + Y"

z - n(z) E Nrr(z)M.

We write

r(z) = Iz - n(z)1

where the absolute value denotes the length of a vector in Nrr(z)M.

We shall need to use the nonisotropic pseudometric and corresponding non

isotropic balls on M induced by the ambient complex structure on cn. We only

summarize the construction, which can be carried out for any CR submanifold M

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS

of finite type (see [NSW] for more details). Suppose L 1 = X I + 1 YI, ... , L m = X + 1Ym is a basis for HI.o(M) on an open set w eM.m

Definition 2.4. For pElt) and 8 > 0, let

R(P,,) ~ .j,Pj,Y E R(exp[t. jXj +PjYj +YT}P);

with tajl, l,Bjl < 8, lyl < 82 \,

where T is a fixed vector field transverse to L I, ... , L m .

There also exists a pseudo-metric D: M x M ~ R so that

B(p, 8) = {~ E M; D(p, {) < 8}.

These balls have Euclidean dimension 8 in the m complex tangent space directions

of M at p and so D behaves roughly Euclidean in these directions. In the d totally

real tangent directions, these balls have Euclidean dimension roughly 82 (since M has type 2) and so D behaves roughly like the square root of the Euclidean distance

in the totally real directions.

We can now state our main result on the existence of a local integral represen

tation formula for holomorphic and plurisubharmonic functions in domains with

type 2 edges.

Theorem 1. Let U C Cn be an open set and let M CUbe a generic CR

submanifold oftype 2. Let Q be a domain with edge M. Let p E M, let Y < r p, and let L I , .•. , L m be a basis for the space HI.o(M) near p. Then there exist a

neighborhood It) C M of p, a constant E > 0, a constant C < 00, and a Coo

function

K: {It) + y,} x M ~ [0,00)

with the following properties:

(1) For z E It) + Y, fixed, the function Kz({) = K(z, {) has compact support in B(IT(z), Cy'r(z».

(2) For every noncommuting polynomial peL, I) ofdegree k ::: °in the vector fields L I , ..• , L m , II, ... , I m , there is a constant Cp so that

IP(L, I)(Kz)({)1 ::: C pr(z)-k/2IB(IT(z), Jr(z»I-I.


75 74 CHAPTER 3

(3) If F is a function continuous on the closure ofthe set nand holomorphic on n, thenfor z E ill + y,

F(z) = LK(z, OF(O da(O

where da is the surface area measure on M.

(4) If u is a non-negative function which is continuous on the closure ofthe set n and plurisubharmonic on n, then for z E ill + y,

u(z) s LK(z, Ou(O da(O

where da is the surface area measure on M.

It is easy to see that the kernel for the Heisenberg group constructed in the introduction satisfies the properties listed in the above theorem. The function (z, y) f-+ ¢r(IZ!2 + iy) appearing in that kernel has support in the rectangle

{I zI s C y'r, IyI s Cr} for a suitable uniform constant C. This set is comparable to the projection of the nonisotropic ball B(O, C y'r) onto the tangent space of 1;

at the origin. In addition, from the definition of ¢r we obtain the estimate

C C l¢r(w)1 s r2 ~ IB(O, y'r)1

where I . Idenotes Lebesgue (surface) measure.

In section 4, we shall prove theorem 1 for a model case for wedge domains, which is a generalization of the Heisenberg group.

3 BOUNDARY BEHAVIOR OF H P FUNCTIONS

As an application, Theorem 1 can be used to study the boundary behavior of H P

functions defined on a wedge relative to a type 2 edge. We begin with the definition of this class of functions.

Definition 3.1. Let n c cn be a domain with edge M. Thenfor 0 0 and a neighborhood ill C M of q so that

sup1 W(~ + zW da(O < +00. ZEy, ~EW

If P = 00 we require that

sup W(z)1 < +00. ZEy,+W

Next we define certain nontangential approach regions in domains with edges.


Definition 3.2. Let n C Cn be a domain with edge M. Assume that M is a generic CR submanifold of type 2. Let qo E M; let y < r qO ,' and let a > O. By the definition ofa domain with an edge, there is a neighborhood ill of qo in M and an E > 0 so that ill + y, C n. Let q E ill. Then define

A(y, a, q) = {z E ill + y,; D(n(z), q)2 sa dist(z, M)}.

Recall that n is the projection from n to M which is defined at least near M, and D(p, q) is the nonisotropic distance between p and q. Also dist refers to the Euclidean distance. Since D is roughly Euclidean in the complex tangential directions. the approach regions A allow quadratic approach to M along these directions. Since D 2 is roughly Euclidean in the totally real directions. these regions allow only nontangential approach in these directions. Thus the regions A(y, a, q) are the analogues of admissible approach regions for strictly pseudoconvex domains.

It is easy to see that if n is a domain with edge M, and if F E Hl~c(n, M). then F has polynomial growth locally near M. To be precise, given p > 0, and given a point q E M and a compact cone y < r q, there is a real number s. a neighborhood ill C M of q and an E > 0 so that for every F E Hl~c (n, M) there exists a constant C so that

W(z)1 s C dist(z, M)-S

forz E ill+Y,. This polynomial growth in turn implies that every F E Hl~c(n, M) has a distributional limit F* along M in the sense that if rp E Cgo(M) then

lim [ F(~ + EZ)rp(O da(O = F*(rp). ,t-+O JM

Our main results deal with the existence of pointwise and dominated limits. rather than distributional limits, and we obtain a partial characterization of the boundary value distributions.

Theorem 2. Suppose M is a type 2 CR submanifold of en and suppose n is a

domain with edge M. Let f E Hl~(n, M), 0 0, then

lim fez) exists. Zt-+q

zEA(y,a,q)

This limit defines an element of Lt'oc(M). Conversely, if I S P S 00 and f E Lioc(M) is a CR distribution, then f has a unique holomorphic extension

F E Hl~(n, M)for some open set n with edge M.

The key ingredient of the proofof this theorem is the following maximal function estimate for plurisubharmonic functions which is derived from the estimates given


- - -----=--------_._--_._._------_._-_._----_._-- -;..; _. -~.=

CHAPTER 376

in Theorem 1 in section 2. Let u be non-negative and plurisubharmonic on the wedge Q and continuous up to M; then

sup u(z) ~ ey,aM(u)(q) ZEA(y,a,q)

where M denotes the maximal function associated to our family of nonisotropic balls and ey,a is a constant which depends only on the cone y and the aperture a. This maximal function estimate (with u = IF IpJ2) is used in the proofofTheorem 2 in much the same way as the analogous maximal function estimates are used in the classical analysis of the boundary values of H P functions on domains in the

complex plane.

4 THE MODEL CASE

In this section, we derive the reproducing kernel for Theorem 1 of section 2 for em2the following model case. We let Mm ~ denote the complex vector space of

m x m complex matrices, and let Hm denote the real vector subspace of m x m Hermitian matrices. For any r x s complex matrix A, let A* denote the conjugate transpose s x r matrix. For W E Mm, set

R(W) = ~ (W + W*);

1 I(W) = - (W - W*).

21

Then for W E Mm , R(W) and I(W) are Hermitian; W = R(W) + zI(W); and

M m = Hm EB I Hm. This exhibits Hmas a maximal totally real subspace of Mm. We shall view elements of em as m x I complex matrices, and so if

then z* = [ZI, ... ,Zm]'z= []

Define a quadratic form Qm : em H- Hm given by

Qm(Z) = zz* = Z2Z1

ZIZ]

Z2Z2

Z]Z2 ZIZm

Z2Zm

ZmZI ZmZ2 ZmZm

=~,=~~=""~.'- ~·-"--"'=-i~o;;'i~.i;.:r.tia:::i;;5;:~'~;.i,~:L'?'''''";;j-~';;w:~''ii;Nj~i,,~;,;r<i,;,;j .."..::_;,;..~K~~..:ci-~,i:;;;.;~iF:~~~~.;..;;_~~j.i;,:fu;~'''''_::_,ib;;jill"_:_.,,,.''''',i~;;;ji~;ijj:..·.~...''':.:.i....;;;;)'iia/#i''''~,;_~:w_~~~".'<~7F~;

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS 77

Definition 4.1. Define p: em x Mm H- Hm by setting

p(z. Z) = R(Z) - Qm(Z).

Then set

~m = {(Z; Z) E em x Mm; R(Z) = zz*} ,

= {(z; Z) E em x Mm; p(Z; Z) = O}

and

Q m = {(Z; Z) E em x Mm; R(Z) > zz*}

= {(z; Z) E em x Mm; p(z; Z) > O}

Here, p(z; Z) > 0 means that the matrix p(z; Z) is positive definite. It is easy to check that ~m is a generic CR submanifold of em x Mm ~ e m+m2

of type 2, and that Qm is a domain with edge ~m. We often identify ~m with em x Hm via the correspondence

em x Hm 3 (Z, Y) ++ (Z; Zz* + IY) E ~m.

A typical point in Q m is of the form (z; Z) where Z = zz* + X + I Y with X, Y E Hm , and X > O. There is a natural projection IT: Q m H- ~m given by

IT((Z; zz* + X + IY)) = (z; zz* + IY).

We are interested in the existence of analytic discs in Q m with boundary in ~m. Every analytic disc in em x Mm is a continuous map A = (Z; W): D H

em x Mm which is holomorphic on D. We write

ZI(S)] Z(O = : '

rZm(S')

where each Zj(.) is a (scalar) holomorphic function. W(.) is an m x m matrixvalued holomorphic function. Such an analytic disc maps the boundary of the unit disc D into ~m if and only if

R(W(e,II)) = Z(e'Il)Z(elll )*

for 0 ~ (j ~ 2n. We shall introduce the notation Ao(S) for the special analytic disc

Ao(S) = (Zo(S); WoeS))


-

78

,-.

CHAPTER 3

where r

{

{2

Zo({) = I

{m L

and r

1 0 0 ... 0

2{ 1 0 ... 0

2{2 2{ 1 ... 0Wo({) =

2{m-l 2{m-2 2{m-3 L

Note that - -2 -m-l

{ { ... {

-m-2 { 1 { ... {

-m-3R(Wo({» = I {2 { 1 ... {

{m-l {m-2 {m-3 L

Also, -2 -3 -m n {{ {{ ... {{ 2-2 2-3 2-m{2"f { { { { ... { { 3-2 3-3

{ { ... {3{Qm(ZO({» = {3"f { {

-2 -3{m"f {m{ {m{ ... {m"fm-

Thus when I{I = 1,

R(Wo({» = Qm(ZO({» = Zo({)Zo({)*

and so Ao maps the boundary of the unit disc to I:m • If ~ = (~l,"" ~m) E

em, then (R(Wo({»~,~) is a harmonic function of {, while (Qm(ZO({»~,~) 2= I L:j=l {j~j 1 is a subharmonic function of {. It is easy to check that

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS 79

(R(Wo(O» - Qm(ZO(O)))~,;) = 1;12 and hence it follows from the minimum principle for superharmonic functions that R(Wo({» - Qm(Z({» > 0 for I{I < 1. Therefore, the analytic disc Ao maps the open unit disc to Qm.

We now construct a local integral representation fonnula for Qm. Let F be holomorphic on Qm, and continuous on Qm' Let I denote the rn x rn identity matrix, so that (0; I) E Qm. Our first goal is to construct a representing measure for the point (0; l) E Qm with compact support and e oo density;

i.e., we wish to find a e oo function Ko with compact support, such that

F«O; l) = ( F({)Ko({) dAC{)JEm

where d"A is Lebesgue measure on I:m = em x Hm •

Since F is holomorphic on the subset {(O; W) E em x Mm; R(W) > OJ, we

can use the mean value property to conclude that

F(O; l) = ( F(O; I + w)q:>(w) dw,JMm

wher~ q:> is any radial function with compact support in a small neighborhood of

o E Mm and with total integral equal to one. Write w = H + I Y where both H, Y E Hm • For a suitable function q:> with compact support we have

F(O; I) = 11 F(O; (l + H) + IY)q:>(H, f) dH dY. Hm Hm

Note that the integrand has support in a small neighborhood of the origins of each

copy of Hm • Next, we make the change of variables

(l + H) = (l + X)2,

that is,

H = IJI(X) = 2X + X2

where X E Hm, and IJI: Hm ~ Hm. This mapping is a diffeomorphism from

some small neighborhood of the origin to a neighborhood of the origin. Hence

there is a smooth function 1/J with support near the origin of Hm x Hm so that

F(O; l) =11 F(O; (l + X)2 + IY)1/J(X, Y) dX dY. Hm Hm

Let U(rn) denote the group of rn x rn unitary matrices, and let dg denote Haar

measure on U(rn). Since Zo(O) = 0 and Wo(O) = I, for every g E U(rn) we have

F(O; (l + X)2 +If) = F«(l + X)gZo(O); (l + X)gWo(O)g*(l + X) + If)


- -- ------------

81

-_.. _--------,~..;.....~...~~,.~-'.,'~-_._--- ---_._.__ ..~"".,"""""'~

CHAPTER 380

and so F (0; I) is given by the integral

{ { ( F((l + X)gZo(O); JH", JH", JU(m)

(/ + X)gWO(O)g*(/ + X) + 1y)'I/J(X, Y)dgdX dY.

We replace I + X by the variable X E Hm • By averaging over the analytic disc

D 3 ~ 1-+ (XgZo(S); XgWo(Og*X + IY)

we see that F (0; I) is given by the integral

_1 { { { (2Jr: F(XgZo(e'o ); 2iT JH", JH", JU(m) J o

Xg Wo(e,o)g* X + I Y)'I/J(X - I, Y)d8 dg dX dY

Note that for any X, Y E Hm , g E U (m) and 8 E [0, 2iT], the point

W(X, Y, g, 8) = (XgZo(e'o); XgWo(e,o)g*X + IY)

belongs to :Em. We now study this mapping W: Hm x Hm x U (m) x [0, 2iT] 1-+

:Em. We have

Lemma 4.2. For each fixed go E U(m) and 80 E [0, 2iT], the mapping W has

maximal rank at the point (/,0, go, 80).

Proof. After we identify :Em with em x Hm , the mapping W becomes

W(X, Y, g, 8) = (XgZo(e'o), Y) E em x Hmo

Thus it suffices to show that the mapping

e'o

e2J O

(X, g, 8) 1-+ Xg I I E em

mlo e

has rank m at the point (/, go, 80)). This is easily shown by restricting X to diagonal matrices and g to diagonal multiples of go. •

Using this lemma, we can integrate out the extra variables in the integral formula for F (0; I) and obtain the following.

Lemma 4.3. There is a non-negative function Ko E ego(:Em ) such that:

-~-~_._--"".-_._~---~~------~- ~~---_..;;;;;.;...:;.=~--~.-~....",..._---------- --""_.. ..~~


(1) For everyfunction F continuous on Qm and holomorphic on Q,

F(O; /) = ( F(~)Ko(~) dA(~)Jr.",

= { ( F(z, Y)Ko(z, Y) dY dz; Jc'" JH",

(2) For every non-negative function u continuous on Q m and plurisubharmonic

on Q,

u(O; I):5 ( u(~)Ko(~) dA(~)Jr.",

= { ( u(z, Y)Ko(z, Y) dY dz. Jc'" In,,,

Our discussion so far has dealt with representing the value of a holomorphic function, but if the function is plurisubharmonic, all equalities are replaced by inequalities and estimate (2) can be established. The representation formula (1) and the estimate (2) are generalizations of (1) and (2) given in section 1 (with r = 1) for the Heisenberg group.

We can obtain a representation formula for other points in Qm by making use of the group structure and dilations on :Em. Let g be an element of U (m). Both :Em and Qm are invariant under the rescaling map

(z; Z) 1-+ (gz; g Zg*).

In addition for (zo; Zo) E :Em, the map

(z; Z) 1-+ (z + Zo; 2zz~ + Z + Zo)

also preserves :Em and Om. (This is the map induced by translation under the group structure of :Em.) Using lemma 4.3 and the translation map and rescaling map, we obtain the following representation fonnula for holomorphic functions at an arbitrary point in Qm' To state the result, note that an arbitrary point in Q m can be

: written as (zo; Zo) = (zo; zozo + Xo + I Yo) E Qm where Xo is positive definite 2~,(and therefore, Xo has a unique positive square root X0 1/ E Hm ).

Theorem 3. There is a non-negative, eoo function K : Qm x :Em 1-+ [0, 00) such

that if

(zo; Zo) = (zo; zoz~ + Xo + I Yo) E Qm

then:


82 CHAPTER 3

(1) For every function F continuous on Qm and holomorphic on Qm,

F(zo; 2 0) = f f F(z, Y)K(zo; 2 0), (z, Y») dY dz.lem lHm

(2) For every non-negative function u continuous on Qm and plurisubharmonic on Qm,

u(Zo; 20) :::: 1 [ u(z, Y)K(zo; 20), (z, Y») dY dz. em lHm

Moreover.

K(zo; Zoza + Xo + lYO), (z, Y»)

= det(Xo)-m-l Ko(X~I/2(Z - zo), X~1/2(y - Yo - 2zza)X~I/2).

If z = (Zo; 2 0) = (zo; Zoza + Xo + l Yo) is a point in Qm, then the quantity r(z) = IIXol1 is proportional to the distance from z to :Em. Moreover, in a fixed compact subcone, det(Xo) is proportional to IIXoli m • In view of the fact that Ko is a bounded function, the above formula implies the following estimate for K:

C IK(zo, z)1 :::: r(z)m2+m

The CR (complex) dimension of :Em is m and the real dimension of the totally real tangent space of :E (= dimR Hm ) is m2• Therefore, the right side of the above estimate is proportional to 1/IB(zo, Jr(z»l. This establishes the k = 0 part of the estimate on K given in theorem 1. The estimate for k ::: 1 can be obtained by differentiating the above expression for K by holomorphic tangent vector fields.

Texas A & M University University ofWisconsin, Madison

REFERENCES

[BDN] A. Boggess, R. Dwilewicz, and A. Nagel. "The hull ofholomorphy ofa nonisotropic ball in a real hypersurface of finite type." Trans. Amer. Math. Soc. 323 (1991), 209-232.

[FS] G. Folland and E. M. Stein. "Estimates for the "fh complex and analysis on the Heisenberg group." Comm. Pure Appl. Math. 27 (1974), 429-522.

[NSW] A. Nagel, E. M. Stein, and S. Wainger. "Balls and metrics defined by vector fields I: basic properties." Acta Math. 155 (1985), 103-147.

[R] J. Rosay. "On the radial maximal function and the Hardy-Littlewood maximal function in wedges." Preprint.

[RS] L. Rothschild and E. M. Stein. "Hypoelliptic differential operators and nilpotent groups." Acta Math. 137 (1976), 247-340.

4

Some New Estimates on

Oscillatory Integrals

Jean Bourgain

1 INTRODUCTION

This paper is mainly a summary of recent work of the author on the subject of the title, and most of its content relates to [B 1] and [B2]. The main objects underlying the problems discussed here are certain operators expressed by an oscillatory integral, roughly of the form

; (1.1) Tf(x) = ! eirp(x,y) f(y)dy

where rp will be a smooth function of x, y (this phase function rp will be more specified later). Our aim will be to understand the mapping properties of T, namely the pairs of (LPl, LP2)-spaces where T acts as a bounded operator, and to estimate its norm. This question is presently only very partially understood,

. even in the simplest cases. In this report, we specify the problem to the following , themes:

(i) Restriction and extension problems; (Ii) The Bochner-Riesz summation operators; (iii) The (more general) oscillatory integrals considered by Hormander [Hor].

We do not intend to give a survey of these topics but, instead, to comment on some progress made in [B 1], [B2]. We also want to mention the paper [Stl], which the reader may like to consult for more background and related material.

(i) The model cases (in the compact setting) are the operators

(1.2) f t---+ j I (restriction) Sd-l


85 84 CHAPTER 4

and

(1.3) J.t E M(Sd-l) t----+ fl (extension).

Here Sd-I = S the unit sphere {x E jRd I 'Lt xl = 1} in jRd with its invariant measure a = ad-I, and the measure J.t on S considered in (1.3) is assumed absolutely continuous with respect to a. The operators (1.2) and (1.3) are of course formally dual to each other.

The main relevant geometrical feature of the sphere is that it is a compact

smooth manifold with nonvanishing curvature. Considering locally such a

hypersurface as a graph, (1.3) may be expressed by an operator of the form

(1.1), namely

(1.4) Tf(x) = f ei(XtYt+"'+Xd-IYd-l+xd'/t(Y)) f(y)dy

Iwhere x E jRd, Y is taken in a neighborhood of 0 E Rd- , 1/1 is a smooth

function on that neighborhood satisfying

(1.5) det ( aa;2 1/1) i= 0,

and f is a function of the y-variable corresponding to the derivative ;; . (ii) The Bochner-Riesz multipliers are defined by the formula

[ m).(~) : ~1 _ 1~12»)' if I~I ~ 1 (1.6)

ifl~I>1

with A > O. We are interested in the corresponding Fourier multiplication

operator

(1.7) f t----+ (i. m).r, more precisely, when it acts as a bounded operator on LP(jRd). Thus the

operator is given by a convolution with kernel

i1xIe(1.8) m).(x) '" Ixl(d+I)/2+A'

One has obtained estimates using various methods, either dealing with the Fourier-multiplier definition (1.7) or the convolution operator

eilx-YI (1.9) Tf(x) = f f(y) Ix _ yl(d+I)/2+A dy.

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

Considering the form (1.9), restricting the variables x, y to disjoint balls at distance '" N > 1 and rescaling, one gets operators

(1.10) TN f(x) = f f(y)eiNlx-Yla(x, y)dy

where a is a smooth function on a neighborhood of (0, 0) in jRd x jRd vanishing

on a neighborhood of the diagonal. The approach of [C-S] and [Hor] consists of first passing from this jRd _jRd problem to an jRd-1 _jRd problem, by fixing

one coordinate of the y-variable, say Yd = 1, keeping x, (YI, ... , Yd-d in a neighborhood of 0 E jRd (resp. jRd-I). The phase function Ix - YI becomes

2 2 2] 1/2(1.11) [ (XI - Yl) + ... + (Xd-l - Yd-I) + (Xd - 1) .

Making an asymptotic expansion of (1.11), eliminating the purely x and y

terms and performing changes of variables in x and y separately, one is led

to a phase function of the form (1.4), but with the presence of non-linear x2terms 0 (Ix 12 1Y 1 ). This generalization of (1.4) gives precisely the operations

considered in [Hor]. (iii) In [Hor], one studies the behavior of operators

(1.12) TNf(x) = f eiN",(x'Y)a(x, y)f(y)dy

where a E CO'(jR2d-I) and ({J E coo (jR2d-l) is a real valued function

satisfying the conditions for (x, y) E supp a:

(1.13) rank a2({Jjaxay = d - I

2 (1.14) ~ ( a({J , t) = 0, 0 i= t E ]Rd ===} det ( a 2 (a({J ,t)) i= 0,

ay ax ay ax

condition (1.14) meaning that the map y 1-+ (*", t) has only non-degenerate critical points. After coordinate changes, ({J may be given the form

(1.15)

({J(x, y) = xlYI + ... + Xd-IYd-1 + xd(Ay, y} + O(lxIIYI(lxI2 + IYI2»

where A is a symmetric matrix with det A i= O. The number N > 1 is a parameter and one seeks uniform estimates of the form

(1.16) IITNfllq < C",N-d/qllfllr

for certain pairs (q, r). Those pairs are specified in [Hor] for d = 2 and the

higher-dimensional setting is proposed as a question. Thus the setting (iii) generalizes both (i) and (ii).


86 CHAPTER 4 SOME NEW EST/MATES ON OSCILLATORY INTEGRALS 87

It is conjectured that m.. defines a bounded Fourier multiplier on LP(~d) iff ). > 0 and

2d 2d(1.17)

d + 1 + 2), 0 is necessary because of C. Fefferman's solution ofthe ball multiplier problem [Fl]. It follows from (1. 8) that the range (1.17) is optimal.

It is conjectured that (1.2) maps boundedly from Lq' (~d) to p' (Sd-d or, more generally, (1.4) fulfills an inequality

(1.18) IITfIILq(IRd) :s Cllfllu(sd_Il

for 2d

(1.19) and d+l +~:S1. q > d - 1 (d - 1)q r

The condition (1.19) is the best possible one may expect (recall that o-d-l

IXI,Ll/2 for Ixl ~ 00).

One considers the same range (1.19) for the inequality (1.16) (proposed for d > 2 as a question in [HorD.

In dimension d = 2, problems (i), (ii), (iii) are essentially completely understood and the optimal results stated above are known to be valid. Thus the multipliers mA,). > 0 act boundedly (as a Fourier multiplier) on LP(~2) for p V p' < 4 and (1.16), (1.18) hold for (r, q) interpolated between the pairs (2, 6); (4,4) (not allowing (4,4». See [Stl] for a more complete bibliography. Apparently, the only results (preceding [BID in dimension d :s 3 were those derived from a purely "L 2-method." Thus (1.18) was shown for r = 2, q > d~1 by P. Tomas [T] and E. Stein (see [StID if r = 2, q ::: d~l' Also (1.16) is shown in [Stl] to be valid in this case. Following [F2], one may then obtain the boundedness of (1.7) under the condition (1.17), provided p* = p V p' ::: 2~~/).

The argument is based mainly on an L2-factorization of the LP-operator, invoking Parseval's theorem and the L 2-restriction result for spheres of [T] mentioned earlier.

The author obtained in [B 1] results beyond the range of the L 2-techniques. It is shown, for instance, that if IL E M(Sd-d, ~~ E U'O(S), then the Fourier transform fi is p-integrable for some p < 2~~11). Also the Bochner-Riesz conjecture is verified for certain p with p* < 2~~/). The approach makes essential use of certain new estimates on higher dimensional Besicovitch-type maximal operators, which will be described later. The role of such geometric objects became clear in [Fl], where the existence of the 2-dimensional Kakeya-Besicovitch set was

exploited to disprove the ball-multiplier conjecture. A systematic investigation of the geometric structures in order to approach the Fourier Analysis problems mentioned above was proposed in [F2]. The paper [Bl] contains the first results along these lines of investigation (for d > 2).

Surprisingly, for d ::: 3, inequality (1.16) may fail under the conditions (1.19) and hence this approach to the Bochner-Riesz multipliers for d > 2 is a bit too general. In fact, it turns out that if d > 2 is odd, (1.16) may only hold if q ::: 2~~/), even for r = 00. In [B2], we study the case d = 3, showing that for a "generic" phase function ({J one has (1.20)

3 < inf {q IllTNfll q < CN-3/qllflloo for N > I} < 4

II 11

2d 2(d + 1)

d - 1 d - 1

Here the relevant geometric structures are generalizations of the Besicovitch structures, in the sense that straight lines are replaced by curves r y' described by the equation

(1.21) B({J (x, y) = a(y).By

For such systems of curves, certain "compression phenomena" may occur, leading to the failure of (1.16) for r = 00 and certain q > }:!.l (with various degrees of strength).

Thus the non-linear x-terms in ({J given by (1.15) playa significant role if d > 2. Very recently the author observed that if d is even one may obtain (1.16) when

r = 00 for certain q < 2~~/) whenever ({J satisfies the conditions (1.13), (1.14) (this is not the case for d odd, as said previously). The main reason for this is a different behavior of the families of curves r y defined by (1.21), in the sense that they cannot be contained in a set A C ~d of dim A = di

1 (which is possible if d is odd). At the end of the paper, we give a sketch of the reason for this fact for even d. More precise statements and details will appear elsewhere.

We also give an application of the non-L 2 restriction results in the context of a.e. convergence of solutions of the 2-dimensional Schrodinger equation

I !J..u = iB,u (l.22)

u(x, 0) = f(x)

for t ~ 0, improving on Vega's result (for d = 2) (see [B3], [VD.


88 CHAPTER 4

2 ESTIMATES BASED ON L 2-METHODS

I~ was shown by Tomas [T] that for f E LP(~d), P < 2~:31), the restriction f lSd-I yields an L 2-function on the sphere. In [Stl], complex interpolation is used togettheendpointresuhp S 2~:31). Also (1.16) is proved for r = 2,q ~ 2~~/) (see [Stl], Th. 10). The aim of this section is to make some further comments on

those estimates, that will playa role later on (we will not be concerned here with the endpoint behavior). The discussion below will provide further information on

distribution of level sets.

Puning x = ~,Ix' I « N, we replace the operator TN defined by (1.12) by

(2.1) T~f(x') = f eiN<p(X'/N'Y)a(y)f(y)dy

where a is supported by a neighborhood of 0 in ~d-l .

Proposition 2.2. For 0 < ).. < 1

(2.3) let A). = {Ix I < N and IT~f(x)1 > )..}

denote the level sets. Assume Ilf 112 s I. Then thefollowing inequality holdsfor

R > 1:

(2.4) IA).I s ).. -2[R + L p-(d-l)/2 sup IA). n B(z, P)IJ. R<p<N Izl<N p dyadic

Here IAI denotes the measure of A and B(z, p) the ball centered at z of radius p.

Proof. Denote A = A). and consider a collection {Ba } of subsets of A such that

(2.5) diam Ba < R

(2.6) {disl[B•• BpJ > R for a =1= 13

(2.7) [UaBal '" IAI·

Let Xa stand for the indicator function of Ba. Denote T~ by T. From (2.7), one

may assume

Re L(Tf, Xa} > diAl

2(2.8) II~ T*Xall: < c)..2IAI .

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS 89

Expand and estimate the left member of (2.8) as

(2.9) L IIT*Xall~ + L(TT*Xa, XfJ} S 1IT*lIiz a a¥fJ (Rl

. L IAal

+ L c(a, f3)IAa l • IAfJl. a#

Here IIT*IILz denotes the norm of T* acting on L2-functions of norm ~ 1 sup(R)

ported by a ball of radius R; c(a, 13) = c(p), P = dist(Aa , AfJ) where c(p)

denotes a uniform bound on the kernel K(x, x') of TT* for Ix - x'i > p. K is

given by the oscillatory integral

(2.10) K(x, x') = f eiN[<P(x/N,y)-<p(x'/N'Y)]a(y)dy.

From the hypothesis on q; one has the estimate

C (2.11) IK(x, x')1 < Ix _ x'l(d-l)/2

(ct. [Stl], Prop. 6). Thus

c(p) < cp-(d-l)/2.(2.12)

The square of IIT*IILz may be evaluated by the L 2-norm of the operator with (R)

kernel of the form

(2.13) L(y, y') = f eiN[<p(x/N,y)-<p(x/N,Y')]bR(x - z)dx

where z E ~d, Izi < Nand bRdenotes a smooth function such that bR(x) = 1 if

Ixl < R, bR(x) = 0 if Ixl > 2R and fulfills the obvious derivative estimates. By

Schur's lemma, this L 2-norrn is bounded by

(2.14) s~p f IL(y, y')la(y') dy'.

Since IL(y, y')1 < cRd[1 + Rly - y'U-d, (2.14) is at most c.R. So (2.8), (2.9)

imply

(2.15) )..21A12 ;S RIAl + L p-(d-I)/2 L IAallAfJl R<p<N dist(Aa,A~)~p

from which (2.4) immediately follows. If we let in particular in (2.4) R '" ).. -4/(d-l), it follows that

IA).I ;S ).. -(2(d+l))/(d-l)(2.16)


91 90 CHAPTER 4

implying a bound on T acting between L 2 and LP'OO(lRd), p = 2~~/). The preceding shows, moreover, that if IA" I "'-' A- P , then also IA" n B I "'-' A- P for some ball B of radius A-4/(d-I). This statement is easily seen to be sharp. Denote, for

instance, JL to be the measure on Sd-I where ~. = IA~1/2 XA, A the e-cap centered at the north pole (0, ... ,0, 1). Thus IAI "'-' ed-I and fi(x) "'-' e(d-l)/2 = A in

the tube of essential shape f- 1 x ... X e- I, xe-2, of measure e-(d+l) = A-Po

d-I

The tube is contained in an e-2 = A-4/(d-l) ball. In [B4], it is shown that for

d = 3, (p = 2~~/).= 4) and ~~ = TATm XA, A a measurable subset of S2, the "extremal" case

(2.17) lIilIlL4(]R3) = 0(1)

may, essentially speaking, only occur for caps. More precisely, there is the

following estimate. •

Proposition 2.18. For 0 < 8 < 1 denote by C8 a collection of 8-caps < of S2 forming a covering ofbounded multiplicity. Let r denote the exponent ~ and L~

the space U «) where < is endowed with normalized measure. Then for A and JL

as above

4 11(2.19) lIilll4 ~ c{ L 84L II dJL }1/4

rJ;adi~ TEC, du L~

Thus, if (2.17) holds, one gets from (2.19) the following distributional property

for the set A:

L sup IA n <\(2.20) 0<8<.1 TEC, IAI4/51<11/5 = 0(1). 8 dyadIc

Hence for some cap < on S2

(2.21) IAI "'-' IA n <I "'-' 1<1

from which follows the previous claim.

Proposition 2.18 is shown by direct calculation of the L 4-norm. It seems of 2(d+I) )interest to have an analogous statement in arbitrary dimension (p = d-I .

3 BESICOVITCH-TYPE MAXIMAL OPERATORS

Recall that the classical Kakeya-Besicovitch set in 1R2 is a measurable set of zero

measure containing a line in every direction. As said before, the existence of


such sets was exploited in [FI] to disprove the ball multiplier conjecture, i.e., the

operator

(3.1) f f---+ (iIB(o.l)f

is unbounded on LP(lRd ), d > 1 (except p = 2 of course). Two-dimensional Besicovitch sets are necessarily of full Hausdorff dimension

(see [FaI] for more details on this and related facts). This result is unknown in

dimension d 2:: 3 (at least at the time of this writing). A positive answer to this would be implied by the Fourier Analysis conjectures described in the introduction,

more specifically, the restriction conjecture:

d(3.2) U(lR ) ---+ LI(Sd-I): f f---+ jls

is bounded for all p < }:I' In [B 1], new information on these Besicovitch structures was obtained, leading

to certain (sharp) maximal inequalities that will demonstrate their importance later

on. Next we summarize the main geometric measure theory results from [B 1]. Their

proof is of a more combinatorial nature. We will consider averages over tubes in IRd of unit length and width 8, with

direction ~ E Sd-I (see Fig. 1). Given a bounded measurable function on IRd ,

we define two maximal functions N, N* which we refer to as the Kakeya (resp. Nikodym) maximal function. These names are those appearing in [B 1]; in other

places they may have been used differently.

(e, 0) - tube

~

o Figure 1.


93

._~~_~":::"::?,":"~~~~~;::::';:':~~::-.:;"l'.'~'~:::~~""'::~'~~!-':<:::~':"'''::~''':',~:~

92 CHAPTER 4

(i) ft is defined on the sphere Sd-l, letting

(3.3) ft(~) = sup _11 1f(x)dx T rl T

where the supremum is taken over all (~, 8)-tubes (fixing ~ and considering different translations). Irl denotes the measure'" 8d- 1 of r.

We call N the Kakeya maximal function of f for 8 eccentricity. (ii) fr (the Nikodym maximalfunction) is defined on JRd,letting

(3.4) N*(x) = sup ~ 1f(y)dy T TIrl

where now the supremum is taken over all (~, 8)-tubes r centered at x (thus fixed center x and varying direction ~).

Considering the case of radial functions and the existence of Besicovitch sets, it seems natural to conjecture the following inequalities

1 )dIP-1+E(3.5) IIfnu(sd-J)« ( 8" IIf11u(lRd)

and, similarly,

I )dIP-l+E(3.6) II ft* II LP(lRd)« ( 8" II fII LP(lRd)

for I :::: p :::: d. These inequalities are valid for d = 2 and shown by Fourier Analysis methods.

In [B 1], the following result is obtained.

Proposition 3.7. For given dimension d > 2, (3.5), (3.6) hold provided

(3.8) p :::: Pd

where the exponent Pd is given by the recursive formula

(3.9) P2 = 2; Pd{2pd-l - I} = (d + 2)Pd-l - d.

Thus, in particular,

P2 = 2

7 P3 = "3

30 P4 =

11


155 PS = 49

etc.

Those numbers also yield lower bounds on Hausdorff dimension of Besicovitch

sets in corresponding dimension d. More generally, consider for a k-plane in JRd the Radon transform

(3.10) F(L) = i f(x)dx

and the corresponding maximal operator

(3.11) F*(L) = sup F(x + L) XElRd

defined on the Grassmannian G(d, k). From Proposition 3.7 one may derive the following corollary, for instance.

Proposition 3.12. (i) There is an a priori inequality

(3.13) IIF*IILP :::: cpllfll p G(4,2)

for P > 2 and assuming f supported by the unit ball of JR4. (ii) Let A be a measurable subset of JR4 offinite measure. Then for almost all

L E G(4, 2) each translate of L intersects A in a set offinite 2-measure and

SUPXElRd IA n (x + L)I < 00.

This result is in the spirit of the work of Oberlin and Stein [O-S]. Following [Fal], call a (d, k)-Besicovitch set a measure zero subset of JRd containing a translate of every k-plane. Thus Besicovitch's result affirms the existence of (2, I)-Besicovitch sets. It follows from Proposition 3.12 that there are no (4, 2)-Besicovitch sets. More generally, one shows in [BI]

Proposition 3.14. Assume (d, k) fulfill the condition

(3.15) d :::: 2k- 1 + k.

Then the (d, k)-property holds, i.e., there are no (d, k)-Besicovitch sets.

The proof relies on Proposition 3.7 and Fourier analysis techniques similar to the ones used in [O-S].


95 94 CHAPTER 4

4 ESTIMATES ON OSCILLATORY INTEGRALS

We will show in particular how the results of the last two sections may be applied to get new information on the higher-dimensional restriction problem.

Consider more generally q; of the form (1.15) and define for given N > 1

(4.1) 1/J(x, y) = Nq; (~ , y) .

Consider the oscillatory integral (2.1)

(4.2) -Ff(x) = TNf(x) = I ei!/t(x.y) f(y)dy

(f supported by a neighborhood of 0).

Fix 1 < R « N. Our purpose is to get a new distributional estimate on

T fIB(o.R) by estimating

2(d + I)(4.3) II T fIIU(B(O,R» for certain P<

d-I

This computation is summarized next. lConsider a partition of the y-domain (= neighborhood of 0 in IRd - ) in boxes

Qa of size JR centered at points Ya' Write for y E Qa

(4.4) 1/J(x, y) = 1/J(x, Ya) + (Vy1/J(x, Ya), Y - Ya} + O(ly - Ya1 2)

where the O(IY - Ya 1 2) = 0 ( *) will be dropped because of the assumption

Ixl < R. Define the operators

(4.5) Taf(x) = 1 ei ('Vy!/t (x ,Ya),y-Ya) f(y)dy Qa

and write

(4.6) Tf(x) = I: ei!/t(x. ya)Taf(x). a

The next point consists in introducing a new variable Z E B(O, -JR). Because of

the presence of Y - Ya, Iy - Yal < R.-1/ 2 in the phase function of Taf, one may,

roughly speaking, write for Ixl < R I / 2

(4.7) Tf(x + z) I: ei!/t(x+Z.Ya) Taf(x)rv

a

and estimate (4.3) as

(4.8)

IITfII~p(B(o.R» rv R-d /2 { {( II: ei!/t(x+Z.Ya) Taf(X)I

PdZ}dX.JB(O,R) JB(O,R'f

2 ) a


Define for fixed x

(4.9) 17(Z, y) = 1/J(x + Z, y) - 1/J(x, y)

and the function g = g (y) as

(4.10) glQa = ei!/t(x.Ya)Taf(x).

2Because in (4.9) Z is restrained to the ball B(O, RI / ), one may again roughly set 217(z, y) 17(z, Y + u) for lui of size R I / • Hence,

(4.11) I:rv

ei!/t(x+Z.Ya) Taf(x) = I: eiry(Z,Ya)g(Ya) '" R(d-I)/2 I eiry(z,Y)g(y)dy

a a

and we have to bound

(4.12) II I eiry(z,y) g(y)dy II U(B(O,R f »' 1 2

The exponent P will be taken in the range

2d 2(d + 1)(4.13) 2 < -- < P < PI == --

d-l- - d-l

to be specified later. At this point, we just use L 2-theory (see remarks at the end of this chapter) and

interpolate between the known L 2 - LPI result and the L 2 - L 2 bound

(4.14) III eiry(Z,Y)g(y)dyll ~ /4 1IgI12R ' L2(B(O.R1f 2 ))

observed in section 2 (see proof of Proposition 2.2). Hence, putting

1 1 - e e(4.15) --+-,

P 2 PI

it follows

(4.16) (4.12) ~ R(I-O)/4 . IIgl12

and thus from (4.11), (4.8) and the definition of the function g = gx

II Tfll P < R-d/2 . R«d-I)/2)p . R«(I-O)/4)p ·1 Ilg liP dx <LP(B(O,R» 'V x 2 'V

B(O,R)

(4.17) R-d/2+(p/4)(d-O) { (I: ITafI2)P/2JB(O,R) a

Next we will replace suitably

(4.18) ITa f1 2; Taf(x) = ~ ei('Vy!/t(x.Ya),Y) fa (y)dy


% CHAPTER 4

where

(4.19) Q = B(O, R- 1/ 2); fa(Y) = f(Ya + y) for Y E Q.

Clearly, one has

2 (4.20) ITafl 2 (x) = I fa~ 1 (Vy 1/l(X, Ya)) .

2Since supp fa C Q, one may write I fa 1 as an average of functions of the form

b( ~ ) where b is a standard bump function and the average is taken over points ~,with averaging weight R-(d-l)/21Ifall~.

At this point we make the following assumption on f

L oo(4.21) f E , If I s 1

improving on the L 2-hypothesis.

Since then Ilfall~ oS R-(d-l)/2, ITa fI 2(x) may be recaptured (as a convex combination) from functions of the form

(4.22) R-(d-I) . b (R- I/2(Vy 1/l(x, Ya) - ~a)) .

Thus for some choice of points {~a}, (4.17) is estimated by (4.23)

R-d/2+(p/4)(d-B)-(p/2)(d-I). { [L: b ( R- I/ 2(V 1/1 (x, Ya) - ~a) )J p/2dx. yJB(O,R) a

We estimate the l_power of the integrand in (4.23) by duality. Consider the p

following related geometric maximal operator

(4.24) M8g(y) = sup (o-(d-I) { Ig(X)ldX} ~ JII82Vy1{F(8-2x,Y)-~I<8}

assuming 8 » N- 1/ 2. Making a change of variable x = Rx', (4.23) has the upper

bound

pn (4.25) Rd/2+(p/4)(d-B)-(p/2)(d-l) ~ R-(d-I)/2(M -1/2g)(Ya)

[ R ]

for some g E L(p/2)' (B(O, 1)) of norm 1.

The sum in (4.25) may be replaced by the y-integral. Hence collecting previous estimates, it follows that if IfI S 1

< Rd/(2 p)+I/4(d-B)-(d-l)/21IM _ 11 1/ 2(4•26) IITfl1 LP(B(O,R» '" R 1/2 Up/2J' ->LI'

Finally, substituting the value of 0 given by (4.15), (4.26) yields

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS 97

Proposition 4.27, For T given by (4.2), R < N, one has the boundfor If I < 1

(428) II Tfll < R(3d+I)/(4p)-(3(d-I)/81IM _ 11 1/2 • LP(B(O,R)) '" R 1/2 L(p/2J' ->LI

where 2 S p S 2~~/) and the maximal operator M8 is given by (4.24).

It remains, of course, to find the estimates on M8, depending on the given phase

function f{J.

In the case of the restriction problem, f{J is linear in x and hence 1/1 = f{J in (4.1).

The operator M8 defined in (4.24) becomes

(4.29) M8g(y) = SUp15-(d-l) { Ig(x)ldx. ~ JIIVy",(x.Y)-~I<8}

Letting f{J be as in (1.4), Le.,

(4.30) f{J(x, y) = XIYI + ... + Xd-IYd-1 + xd1/l(y),

the gradient equations become

(4.31) Xi = -ai1/l(Y)Xd + ~i (i = I, ... , d - 1).

Because of (1.5), the map Y f-+ V 1/1 (y) yields a diffeomorphism on a neighborhood of 0 E lR.d - 1 and it is therefore easily seen that M8 corresponds to the Kakeya

maximal function from Section 3. Thus, applying Proposition 3.7, letting the exponent p from Proposition 4.27 be

p = 2p~ (Pd defined by (3.9)),

it follows that

1 ) (d/(pI2)')-1 (4.32) II M 8II Up/2>'->L' S IIM 811(p/2)' oS ( "8

We should observe at this point that Pd given by (3.9) satisfies Pd > di l , hence P < 2~~/). Substitution of (4.32) in (4.28) gives that

(4.33) II T fllu(B(o,R)) « R(d+I)/(4p)-(d-l)/8+<lIflloo.

We apply (4.33) in conjunction with Proposition 2.2. Observe that from (4.33) for

IfI < 1

(4.34) IA>. n B(O, p)1 « ).. -p p(d+I)/4-«d-I)/8)p+<,

which yields after substitution in (2.4)

(4.35) IA>.I oS )..-2 [R + ).. -p R-(d-3)/4-«d-l)/8)P] .


99 98 CHAPTER 4

Choosing R optimally, we get the following estimate on the measure of the level

set AA' assuming IfI < I:

(4.36) IAAj < A-2(2(d+l)+p(d+3»/(2(d+I)+P(d-I»-€, A <

where p = 2p~.

Observe that if we let p = PI = 2~~/), we get IA;l.1 < A-Pi-€. Hence,

since now p < 2~~JI), the exponent in (4.36) will improve. Thus, if we take

J.t E M(Sd_l) with *E LOO(a) rather than ~~ E L 2(a), better integrability properties for it are found. For instance (cf. [B4]),

Proposition 4.37. Let J.t be a measure carried by S2 with ~~ E L 00(S2)' Then*«it E U(lR.3) for p > 4). The same conclusion holds assuming ~~ E

LP(S2).

The second part of the statement follows from the Maurey-Nikishin factorization

theorem for general operators from L 00 to LP (p > 2) and considerations of

rotational invariance (thus using more specifically the symmetry properties of the

sphere).

Remark. In proving Proposition 4.37 by the above method, the use of the crude

inequality (4.17) is a weak point in the approach. In the case of the restriction

extension problem, the situation is the following: one considers a system of boxes

{Ba}approximating a spherical shell and functions fa' supp f: c Ba(see Fig. 2). The square function replacement (4.17) is the first step in the argument and consists

in comparing

withIlL fat !!(L Ifa I 2Y/2t·

In the 2-dimensional case, we let p = 4 and essentially obtain equivalence,

because of the geometric properties of the sum sets Ba + Bp (this family of

Figure 2.

Ba


sets is of bounded multiplicity). This estimate is more refined than invoking

the L 2-extension theorem. It is possible to improve a bit the exponent *in Proposition 4.37 by using Proposition 2.18 in developing a substitute for (4.17).

5 APPLICATION TO THE BOCHNER-RIESZ SUMMATION OPERATORS

In [B I], we used a variant of A. Cordoba's 2-dimensional argument (see [Co]),

breaking up the Fourier multiplier mA in rectangular pieces and applying the new

information in the restriction problem, together with an estimate on the Nikodym

maximal function described in section 3. We follow here a different approach,

namely the Carleson-Sjolin-Hormander reduction, evaluating (1.9) from estimates

on oscillatory integrals and, more precisely, again using Proposition 4.27. In fact,

this will give us a better result than proved in [BI], since it will establish the

conjecture provided again p* = p V pi 2: *for d = 3. Let b be a standard bump function localizing to a neighborhood of 0 in IRd.

Using a dyadic partitioning, estimate (1.9) by

(5.1) L 2-k«d+I)/2+J.) III f(y)eilx-ylb (2-k (x - y») dyll . k::::O LP(dx)

Fix N = 2k and rescale by putting x = N x', y = N y'. The LP -norm expression,

then becomes

(5.2) Nd/p+dIlSNgli p where g(y') = f(Ny')

and the operator SN is given by

(5.3) SNg(X) = I g(y)eiNlx-ylb(x - y)dy.

Thus, an estimate of the form

(5.4) IISNglip « N-d/P+€lIgli p

would yield from the preceding a bound on (5.1), provided p fulfills the condition

if, < A+ di I . Here p was assumed> 2 and this is thus (1.17). Again by general factorization theory and the rotational invariance, it suffices to get the L 00 - LP

inequality

(5.5) IISNflip :s N-d/P. IIf11oo.


101 100 CHAPTER 4

As proposed above, we perfonn the reduction to an lRd- 1 -lRd problem, freezing the Yd-variable and obtain the phase function (letting Yd = 1)

d-I ]I~ (5.6) rp(x, y) = L(Xj - Yj)2 + (1 - Xd)2

[ J=I

Here Y = (YI, ... , Yd-d and x are taking values in a neighborhood of O. Let 8 > N- I / 2 , M = 82N and

d-I ] 1/2 (5.7) T/(x, y) = ~(Xj - MYj)2 + (M - Xd)2

[ J=I

Rewriting the gradient equations

(5.8) VyT/(X, y) = M~

the regions involved in the definition (4.24) ofthe maximal function M~ are given by (5.9)

T = Ty.~ = {IXj - MYj + (1 _ ~~12)1/2 (Xd - M)I < 8 (l :'S j < d)}.

(Observe that necessarily I~ I = 0(1) if we want T to intersect a neighborhood of o[see Fig. 3].)

Thus the regions are neighborhoods of straight lines. Observe that if we fix a point i in a neighborhood of 0 and let Y

(YI, ... , Yd-I) vary, the tubes Ty.~y satisfying i E Ty,~y will describe all directions. By a straightforward adaptation of the proof of Proposition 5.6 in [B 1] on the Nikodym maximal function (to which M~ is obviously related, but with a

d·l lRMy

Figure 3.


missing y-dimension), one may obtain the same bounds

1 )d/p-I+e (5.10) IIM~gIlLP(lRd-I)« 8" IIgll p(

for p < Pd, Pd defined by (3.9). Substituting in (4.28) where T is given by (4.2), rp given by (5.6), yields the same estimate (4.34) on the measure A). n B(O, p) for the level set A). of T I, II I :'S 1. Hence (4.35), (4.36) are equally valid. Therefore,

if

(5.11) > 2 2(d + 1) + qed + 3) .

P 2(d + I) + qed - 1)' _

q -2 I

Pd

then

II! eiNrp(x,y) l(y)a(Y)dyt ;S N-d/Pllflloo.(5.12)

Integrating over variable Yd (taken here to be 1) leads to the required inequality (5.5). So finally we get

Proposition 5.13. m).(~) = (l - I~ 12)~ defines a bounded multiplier on LP (lRd )

if p* satisfies (1.l7) and also (5.11). In particular,for d = 3, one gets p* ::: ¥s as an extra assumption on p satisfying (1.17).

6 BEHAVIOR IN THE GENERAL CASE

It turns out that in dimension d ::: 3, Honnander's question on the estimates (1.16)

(6.1) II f e<N,,·.,) f (yla<x. Yldy II, "CN-d/, II f II,

has a negative answer, even if we let r = 00. In fact, there are examples of phase functions rp where the LOa - U estimate only holds for q ::: 2~~/) (thus in the L2-range), if we let d be odd.

Define rp(x, y) as follows (6.2)

rp(x, y) = xlYI + ... + xd-IYd-1 + 2Xd(YIY2 + Y3Y4 + ... + Yd-2Yd-d

2( 2 2 2 )+Xd YI + Y3 + '" + Yd-2 .

Take

I(y) = eiN(y;+yJ+"'+Y;_,)(6.3)


103 102 CHAPTER 4

so that the integral in (6.1) may be written as

f ei N {[(XdYI +Y2)2+(X, YI +X2Y2)]+" '+[(Xd Yd-2+Yd-1 )2+(Xd_2Yd_2+Xd_1 Yd-I)]J (6.4)

a(x, y)dy.

For 0 > 0, consider the set

(d-I)/2

(6.5) Q 8 = n [!X2j' Xd - X2j-11 < 0] n B(O, I) j=l

of measure

(6.6) IQ 1"v 8(d-I)/28

Letting 8 « ~,the expression between brackets in (6.4) can be replaced by (6.7)

[(XdYl +Y2)2 +X2(XdYl + Y2)] +... + [(XdYd-2 +Yd_d2 + Xd-l (XdYd-2 +Yd-dl

for x E Q8.

Putting Zl = XdYl + Y2, Z3 = XdY3 + Y4, .•. , Zd-2 = XdYd-2 + Yd-I. (6.4) essentially amounts to

(6.8) . n (1 eiN(Z]+XHIZj)a(Zj)dz j ).

j=I,3, ... ,d-2

The individual factors in (6.8) are of size "v N- l / 2, hence

(6.9) (6.8) N-(d-I)/4."v

Thus (6.1) admits the lower bound

(6.10) IQ 81 1/ q • N-(d-I)/4 N-(d-l)/(2q)-(d-I)/4"v

restricting x to Q8, using (6.6), (6.9). Inequality (6.1) may thus only hold if

d-I d-I d (6.11) -- + -- >

2q 4 - q

thus q > 2(d+I). - d-l

In particular, for d = 3, cp is given by

(6.12) cp(X, y) = XlYl + X2Y2 + 2X3YlY2 + xiY;

and the gradient equations VyCP = ~ are

Xl = -2Y2X3 - 2Ylxi + ~l (6.13)

{ X2 = -2YlX3 + ~2.


Put ~ = ~Y = (0, -2Y2). For this translation, one thus gets curves rye JR.3 contained in the 2-dimensional surface Xl = X2X3. This compression phenomenon is responsible for bad behavior of the maximal operators M8 defined by (4.24), in

qthe sense that for q :::: 2, IIM8 11 q--->q "v 8- 1/ .

The previous behavior is not generic, however. In fact, one shows in [B2] that,

if d = 3, in the generic case the best q satisfying (6.1) with r = 00 lies in the

open interval ] }!.l' 2~~/) [ (and depends on cp).

More generally, in the example (6.2), the gradient curves r yare translated in

the di l -dimensional surface

(6.14) X2j-l = X2j • Xd j = 1,2, ... , 2d-I

A compression in a set of di' -Hausdorff dimension seems impossible if d is even. We give a rough sketch of the argument for d = 4.

Proposition 6.15. Let cp be given by (1.15) (d = 4). For Y in a neighborhood of o E IRd - l • define

(6.16) r y ,8 = [IVycp(x, y) - ~yl < 8]

where the ~Y E jRd-1 are chosen arbitrarily. Consider the set

(6.17) Q 8 = U ry,8.

y

This set has8-entropy (in the metrical sense) > 8-«d+I)/2)-r for some L = L'I' > O.

The general idea of the proof is as follows. We will first observe that the entropy conclusion is true with L = O. Assuming then Q 8 of o-entropy '" O-(d+l)/:,

appearing as an "extremal" case, new geometrical information on the structure of

the r Y is gained, allowing us to see eventually that the o-entropy number of Q 8

needs to be ;(, 0-(d/2)-1. The argument produces the conclusion of 6.15, with a

L = L'I' which is numerical, provided bounds are imposed on sufficiently many derivatives of cpo To make matters a bit more concrete, take cp of the form

d-I I (6.18) cp(x, y) = LXiYi - 2:Xd(Y; + ... + Y~_I) + 0(lx121Y12+ IxIIYI3

)

i=l

so that Vy cp = ~ becomes

(6.19) Xi = YiXd + 0(lx!2IYI + IxIIYI2) + ~i (i=I, ... ,d-l)

describing the curve r y.


________ --- '0 __••'_._-----,--_._••••• _._--,,-- •• __• ,,_._•• ,,__'. _ __ __ __ _ .. __ _ _ _ •__" •• _0'_._" __ • '.'_0'__••••__ •••_.__, __ ._. ••••__"'_,,_. , 0'__• _

------- ------------- ------------------ ----- --------------------_. --- -- - - - ----- ------- -- - ------------ - - - -- -------- - - -- - ---------------------------

104 CHAPTER 4

(i) Ifone specifies a point i E r y and lets x' = x - i, (6.19) clearly becomes

(6.20) x; = YiX~ + 0 (lx'llyl(lil + Ix'i + Iyl)) .

From (6.20), Ix;J ;S IYllx~1 (1 ::: i ::: d - 1). Also, if we fix x~ away from 0, application of the implicit function theorem yields a diffeomorphic

correspondence between (x;, ... , x~_I) and y. For instance, Jy = x~ld + o(lx'l) and hence is invertible from the preceding (see Fig. 4).

(ii) Consider in the y-variable a net £ of 8-separated points. Thus 1£1 = card(£) '"" 8-(d=-l). Let £1 c £ such that the r y,8, y E £(, have a common

point. It follows easily from (i) that

(6.21) e8 ( U r y,8) .(; rII£d yEE:,

denoting by e8UJ) the 8-entropy numbers (= minimal number of 8-balls required for a covering of Q).

Hence, if we assume Q8 defined by (6.17) satisfies

(6.22) e8 (Q8) < 8-(d+I)/Z......, ,

it follows that

(6.23) 1£11 ;s r(d-l)/Z

for such family £1.

(iii) Fix separated values xJ, xJ of Xd such that the corresponding hyperplanes

H(J: Xd = x~ (a = I, 2) get intersected by all r y in points P; (see Fig. 5).

Consider the map y ~ (pi, pi) E (Q8 n HI) x (Q8 n Hz).

, Xd

ry

(d - I)-dimensional

Figure 4.

105SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

ad

-. 2

xd I

Xd I ---1

H2

• _ HI

Figure S.

It foIlows again from (i) that y may be derived from its image ina Lipschitz

way and therefore

(6.24) 8-(d-l) '"" e8({y}) ;s e8(Q8 n HI) . e8(Q8 n Hz).

Because of the free choice of the pair (Xt~' xJ), this forces

e8(QJ) > 8-(d+I)/Z(6.25) ......, .

Next, we will assume

(6.26) e8(Q8) '"" 8-(d+I)/Z

in order to derive more information on the structure of the curves r y , and

finally get a contradiction.

(iv) It follows from (iii), assuming (6.26), that a typical [Xd = Xd ]-hyperplane

H will satisfy

(6.27) e8(Q8 n H) '"" 8-(d-I)/Z.

Fixing H, there is thus a 8-separated system P of ~ 8-(d-I)/Z points P E H such that r y,8 n P =f. 0 for each y. Also, from (ii), each point PEP will belong to r y,8 for;S 8-(d-I)/Z values of y E £. This leads to a partition of

(a large subset) of £ in sets £p

(6.28) £ = U £p; I£pl '"" 8-(d-I)/Z

PEP

and for PEP a "bush" 13p of curves {r lYE £p}, where r has the y y

equation

(6.29) Vyrp(x, y) = Vyrp(P. y)


107

~---~~~~---::;;;::-~~....:;:==,...--==;;;,=---=-=---==--::--=--------=;=-~~-=-~. .---- -~~~-===~,-~==:-~---=-~~-----~==~~

106 CHAPTER 4

H p p'

Figure 6.

for y E [po The set Q.s is contained in a 8-neighborhood of UPEpB p (see Fig. 6).

(v) If we fix an individual point P, it follows from (6.28) that

e.s(B ) ~ 8- 18-(d-I)/2 = 8-(d+I)/2(6.30) p

and hence the 8-neighborhood B~ represents a "large" portion of Q.s.

(vi) Let 8 « E and fix an E-ball B(Q, E) in H. Then

IP n B(Q, E)I < (E ) (d-l)/2(6.31) '" "8

Indeed, let pi = P n B(Q, E), Ipil = K and [' UpEP'[p. Thus by (6.28)

1[11 '" K8-(d-I)/2.

Consider an E-net T in [' and let [' = UZEI[~' be a partitioning of[' in sets of diameter E. By construction, if y E [~', the curve r y will be in an IOEneighborhood of r z, since the basepoints in H are E-close and Iy - zl < E.

Therefore

(6.32) e.s(1J, r y ) ~e.s(~, r y ).'"

Also, the same considerations as in (iii) show that

(6.33) e.s(U r y ) ~ 8-11[~'11/2. YEf~'

------=-------==-r


Since I[~'I < (X )d-I, (6.26), (6.32), (6.33) imply that

-(d-1)/2

8-(d+I)/2 ~ 8-1L 1[~'11/2 ~ 8-1 i 1['1'" 8-1E-(d-I)/2K( )

ZEI

and hence (6.31) follows.

(vii) It follows from (v), thus (6.26) and (6.30), together with general measure theoretic considerations on set intersections, that

(6.34) e.s (~ n B~.) '" 8-(d+I)/2

for'" 8-(d-l) pairs (P, Pi) in the product P x P. From (vi), one can, moreover, assume that the points P, pi are far apart. If (6.34) holds, one gets additional information on the curves r y for y E [p, since a large subset of Bp admits a foliation in 2-dimensional surfaces S obtained by fixing some y' E [po and varying r y containing P and intersecting r y' (see Fig. 7). Based on this consideration, a straightforward construction permits us to replace Bp by a union of'" 8-(d-3)/2 surfaces S ESp,

(6.35) Bp = Us SESp

each S of dimension 2 generated by moving the curve r y described by (6.29) along some other curve, as indicated above.

P'

P Figure 7.


_';5II1;:II

108 CHAPTER 4 SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS 109

,'1'mI!<'"'9,;_"",fitl<'I;:Ioj/,,-':~",,;{'~""j""""~'--Y:"'r.1,r,'''4~{'''R'''';;-;;''1r,.·:-'T'''';;:~;;,' '1i"'~,,-:ri';·~"}"~"';·"'~';.j;:;J'Jl':·:"iS"'''*'=;,",",",77;:''''"' "~:'-":"'.ir,",'=g·~'GI;;;;:,-;:;'·i;;:P~,:",-:YO""'-'l'>',-'iP~::"i_'"-1·"N"":';1?'''=-'''''''f>'"'!iI';'if.''l'B)7''.'AI'=~,'''''':'Il:'"·",,t.''?~~'~=;1'J> Wi:!';;;j'!fl',:';;,9,,:,"-';i!:<IE;J';; ....··\Y..,:;.-6·' ,:;:"<:~:<I!"~~/\'W,,7,;:If'rr<"'7'~'

(viii) Fix a smooth 2-dimensional surface S away from H, a point i E S, and let Tj be the tangent plane of S at i. Consider the family of curves r y through i given by (6.20). By (6.20), the tangent vector"iy = ; = (tl, ... , td-I, 1)

of r y at i is described by the equations

(6.36) ti = Yi + O(ltIIYI(lil + lyl) i = 1, ... , d - 1.

Hence

(6.37) ti = Yi + O(lyl(lil + lyl)· .

Restricting t to lie in Tj , a smooth I-parameter family of directions is obtained, corresponding to a curve YI in the y-variable and a curve Y2 =

try n H; Y E yd of points in H. Consequently, one may find a neighborhood S' of i in S such that if the basepoint P of r y in H lies away from Y2, r y may only have a "transversal" intersection with S, assuming r y

intersects S'. (ix) From (6.34), (6.35), one may find a point PEP such that

(6.38) L e& (sn8~1) '" 0-(d+I)/2

SESp

for'" 8-(d-I)/2 points P' E P. From (6.33), there is some S E Sp

satisfying

2(6.39) e& (S n ~I) '" 0

for'" 8-(d-I)/2 points P' E P. Identifying S with a point neighborhood S'

as constructed in (viii), we get a curve Y in H such that the intersection of Sand r y (if any) is transversal, provided its basepoint pi lies away from

y. But from (vi), fixing 8 » 8, the number of points pi E Pat 8-distance

from Y is at most

8 )(d-I)/2(6.40) '" 8-1 8" = 8(d-3)/20-(d-I)/2

(

due to (6.31). Hence, from the preceding, letting 8 be small enough, a point P' E P

satisfying (6.34) and lying at 8 distance from Y is obtained (d = 4 > 3). Hence, a large subset of S lies on the r y (y E t'P' )-curves, intersecting, moreover, S in a transversal way (see Fig. 8).

Since the corresponding structure contains a diffeomorph of a neighborhood of 0 in]R3, this implies that

3(6.41) e&(Q&) ~ e& (~I) ;;: 0- ,

contradicting (6.26) for d = 4.

P'

FigureS.

In the general case, this last construction will yield a representation (6.35) where now the S are 3-dimensional. One repeats the transversality analysis (viii) and gets 2-dimensional exceptional sets. The estimate (6.35) becomes 8(d-5)/28-(d-I)/2, etc., so for d ~ 6, 4-dimensional structures are produced.

Continuing, one eventually gets

(6.42) e&(Q&) ;;: 0-[(d-I)/2j-2 = o-d/2-1 (d is even)

contradicting (6.26).

7 AN APPLICATION TO TWO·DIMENSIONAL SCHRODINGER OPERATORS

Based on the results from sections 2 and 4 of this paper, we sketch an alternative proof of the following result from [B3].

Proposition 7.1. Let u = u (x, t) be the solution ofthe Schrodinger equation with initial data f E S(]R3)

a (7.2) ~u = i - u; u(x, 0) = f(x).at


'---_._--.~~~-~ .. _~-~"'~~-_."-_ .._.""~--~_._-,_ .._--~ .. '..'----"-~~-~~_._---~"-",_.~.,_._ ..,...----~-'-~.'.'----'---,-~."--_.-.. ._, ._.__. - . _, _ .. .._. .._.' _ ....•.•.. - - ._..__ __,_.,__._._.,..__._,,_,_.,_., _. - - 0·'··----·----·-··--- _•.•._.. .._•...~ ,._, __._ __.,_. ..~ __ .~

110 CHAPTER 4

Then, assuming 1 in the Sobolev space H S (JR2),for some s > p where p = P2 <

t ' there is almost everywhere convergence

(7.3) lim u(x, t) = I(x) a.e. in x 1--->0

and a control on the maximal operators

(7.4) u*(x) = sup lu(x, t)l. 111>0

The new element is' the fact that the p-exponent is strictly less than ~. It is

known that for dimension d = I, the assumption 1 E H 1/4 (JR) ensures (7.3),

which is the optimal result (see [Cal, [D-K]). The statement of Proposition 7.1

with p = ~ was shown by Vega [V] in arbitrary dimension. Recall that the solution u of (7.2) is given by the Fourier integral

(7.5) u(x, t) = / j(~)ej«x,~>+11~12)d~.

The statement of Proposition 7.1 will result from an inequality of the form

(7.6) II sup I (j(~)ei«X'~>+II~e)d~111 ~ NP(/ Ij(~)12 d~)1/2 0<1<1 JI~I<N L2(B(0,l»

for some p < ~. Choose some q > 2 and estimate the left member of (7.6) by appropriate change

of variables (rescaling)

(7.7) N2- 2/qll sup I ( j(N~)ej«x'~>+llmd~111 . 0<I<N2 JI~I<1 L«dx)

Using standard considerations, one may then estimate it further by

(7.8) N 2(l-I/q) II (j(N~)ej«x'~>+t1~12)d~11 . JI~I<1 Lq(dx dt)

Consider the surface (~, 1~12) in JR3, restricting ~ to the unit disc. Since there is obviously curvature and smoothness, the restriction and extension theory applies

equally well here as it does for the sphere. Call (p, q) an admissible exponent pair

provided

(7.9) IIl1l1 q ~ ell ~~ t with p., a measure carried by the 2-sphere S2 or the restricted paraboloid P con

sidered above. The classical L2-restriction theorem states then that (2,4) is admissible, and in section 4 admissible pairs were obtained with q < .4.

111SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

Applying the (p, q) pair in (7.8), one gets a bound

P )l/P (/ P )l/P(7.10) N2(l-I/q) Ij(N~)1 d~ = N 2(1-1/p-l/q) Ij(~)1 d~ . (/

For p = 2, q = 4, that is inequality (7.6) with p = ~. It was shown in section 2 (see discussion following Proposition 2.18) that this

(2, 4)-estimate may be improved, unless the density "corresponds" to the indicator

function of a cap (as a rough statement). More precisely, if we assume j to be of

the form A XQ

(7.11) 1(0 = IQ1 1/ 2 (Q C B(O, N»

then an improvement will be obtained, unless for some square Q one has

(7.12) IQI "-' IQI '" IQ n QI·

Remark. j may always be broken up in level sets. The meaning of'" actually will allow factors of the form N° for some specific 8 and so yields an "improvement."

That this is the result of the reasoning below I leave to the reader to check.

So assume (7.12) holds. If IQI "-' N 2 , apply estimate (7.10) for an admissible

pair (p, q) with q < 4 gotten from [BI]. Since here

(7.13) (/ Ij(~)IP d~ yiP "-' N 2/p N- 1

we get an estimate of the form

(7.14) N l - 2/q

with I - ~ < 1q 2 • sIf IQI = N~, with N l = N 1- , thus Q C ~o + B(O, Nd, we proceed as

follows. Write in (7.7)

(7.15) ~ = ~~ + 1/; 1~12 = 1~~12 + 2{~~, 1/) + 11/12

h 1::' ~o I I !:!J..were 50 = N' 1/ < N'

It is clear from (7.15) that the parameter-values needed to recapture the supre

mum for t E [0, N 2] may be taken in a ~ -net and hence the passage to the

t-integral and (7.8) gives a saving factor ( ~ )l/q. Then continue with the (2, 4)

extension theorem to conclude also that case. If one pursues this argument a bit

more explicitly. it easily leads to (7.6) with an exponent p < ~.

Institut des Hautes Etudes Scientifiques. Bures-sur-Yvette. France


112 CHAPTER 4

REFERENCES

[Bl] J. Bourgain. "Besicovitch type maximal operators and applications to Fourier analysis." Geom. Funct. Anal. 1:2 (1991), 147-187.

[B2] ."LP-estimates for oscillatory integrals in several variables." Geom. Funct. Anal. 1:4 (1991), 321-374.

[B3] . "A remark on SchrOdinger operators." To appear in Israel J. Math. [B4] . On the restriction and multiplier problem in JR.3. Lecture Notes in

Mathematics, no. 1469. Springer Verlag, 1991. [Cal L. Carleson. "Some analytical problems related to statistical mechanics." In Eu

clidean Harmonio-Analysis. Lecture Notes in Mathematics, no. 779. Springer Verlag, 1979.

[Co] A. Cordoba. "A note on Bochner-Riesz operators." Duke Math. J. 46, N3 (1979), 565-572.

[CoS] L. Carleson and P. Sjolin. "Oscillatory integrals and a multiplier problem for the disc." Studia Math. 44 (1972), 287-299.

[D-K] B. Dahlberg and C. Kenig. A note on almost everywhere behavior of solutions to the Schrodinger equations. Lecture Notes in Mathematics, no. 908. Springer Verlag, 1982.

[Fal] K. J. Falconer. "The geometry of Fractal sets." Cambridge Tracts Math. 85. [Fl] C. Feffennan. "The multiplier problem for the ball." Ann. ofMath. 94 (1971), 330

336. [F2] . "A note on spherical summation multipliers." IsraelJ.Math. (1973),44-52. [F3] . "Inequalities for strongly singular convolution operators." Acta Math. 124

(1970),9-36. [Hor] L. Honnander. "Oscillatory integrals and multipliers on F LP." ArkivMath. II (1973),

1-11. [St] E. Stein. "Oscillatory integrals in Fourier Analysis." In Beijing Lectures in Harmonic

Analysis. E. M. Stein, ed. Annals of Mathematics Studies 112. Princeton University Press, 1987.

[T] P. Tomas. "A restriction theorem for the Fourier transfonn." Bull. Amer. Math. Soc. 81 (1975),477-478.

[V] L. Vega. "The SchrOdinger equation: pointwise convergence to the initial data." Proc. Amer. Math. Soc. 102 (1988), 875-878.

5

Dilations Associated

to Flat Curves in IRn

Anthony Carbery, James Vance, Stephen Wainger,* and David Watson

We would like to give a brief exposition of our recent work on Hilbert transforms and maximal functions along curves in Rn [CYWW). In particular, we wish to describe a theory of dilations that seems useful in this work. This is an outgrowth of joint work with Mike Christ that dealt with the case n = 2 [CCYWW). We would like to acknowledge the important contributions Mike made to this project

in its earlier stage. We let r (t) = (t, Y2 (t) ... , Yn (t» be a smooth curve in RR with r (0) = O.

For a function I in Cgo(Rn ), we define

(1) Hr I(x) = 1

I(x - rCt» -dt

, / _I t

and

l h

(2) Mr I(x) = sup -I I/(x - r(t»ldt.O<h~1 h 0

We are interested in the problem of obtaining estimates of the form

(3) IIHr IIIU(Rn) S A(p, r)II/IIU(Rn),

and

(4) IIMr fIIU(Rn) S A(p, r)II/IIU(Rn).

·Supported, in part, by a grant from the National Science Foundation.


114 CHAPTER 5

If the estimate (4) holds for some p < 00,

h

lim -h1 l f(x - f(tndt = f(x) a.e. h--+O 0

for every f locally in LP. Positive results for (3) and (4) have been known for a long time under an appro

priate "curvature hypothesis" on f. The following theorem was due to the efforts of Nagel, Riviere, Stein, and Wainger in the 1970s. See [SW].

Theorem of the 1970s. Suppose f(t) satisfies the following curvature condition:

(5) For small t, f(t) lies in the span of the vectors f'(O), f"(O) ... ,

fU)(O), ....

Then

IIHr fllu(RG) ~ A(p, f) II fIIU(RG) , I < P < 00,

and

IIMr fllu(RG) ~ A(p, Ollfllu(RG), 1 < P ~ 00.

Examples of curves satisfying the curvature hypothesis (5), are

kG )(6) f(t) = (t, t k2 ... , t

and

(7) f(t) = (t, tk2 + tk2H2 ... , t kG + tk"HG)

where the k j and i j are positive integers. Examples of curves not satisfying the curvature hypothesis are

(8) f(t) = (t, e-I/ltla2 ... , e-I/ItI"")

where a2 ... , an are positive integers. An important tool used in proving the Theorem of the 1970s was that of a

contracting group of linear transformations. A collection of non-singular linear transformations T),. on Rn defined for>.. > 0 is called a contracting group of linear transformations if

(9) T),.Tp, = T),.iL' for>.. > 0 and JL > 0,

(10) Tlx = x,

"'"~-::"~~~;';:';:~:"~~~~:;;k.-~~ii~~;~~~~~~~7~~;;;~~~~~;;;;~~~~~~;;;'~~~:<;:~·:;,;"",,~-.,,",,~~~~~~~;.~~.\.~;:i.;;'~~~~

DILATIONS ASSOCIATED TO FLAT CURVES IN JR" 115

for each x in Rn, and

(11) lim T),.x = 0, ),.--+0

for each x in Rn •

Note that the examples in (6) are "homogeneous" with respect to a dilation

group. That is

(12) f(>"t) = T),.f(t)

where T),. is the contracting dilation group defined by

T),.(XI,X2 ... ,Xn ) = (>"XI,>..k2x2 ... ,>..k"Xn).

The examples (7) do not satisfy (12), of course, but still for>.. positive and small and t small

f(At) rv T),.f(t).

(perhaps more importantly the set of curves

f),.(t) = T),.-Ir(At), >.. ~

forms a compact family of curves on [1, 2].) Groups of dilations played three roles in the Theorem of the 1970s. First the

dilations allowed one to "normalize" certain estimates. Furthermore, the homogeneity was used as an aid in bounding from below the absolute value of derivatives of ~ . r(t), for vectors ~ in Rn, so that the lemmas of Van Der Corput might be used. Finally, the dilations provided a variant of the Calder6n-Zygmund theory.

Let us first give an illustration of what we mean by "normalizing" estimates. It is not hard to see that the L 2-boundedness of H r is equivalent to the uniform boundedness of

dt m(~) = fl

exp(i~ . f(tn -, -I t

1 over vectors ~ in Rn • Let

dt mj(~) =. exp(i~ . f(tn -.

2-J~ltl~2'2-j t

One might expect the estimate for mj(~) to be very subtle depending on the curvature of f near the origin. In the examples f (t) = (t, tk2 ... , tk"), for instance, one might expect the behavior of m j (~) to depend in a complicated way on k2 ... ,kn • However, if f(t) is homogeneous with respect to a group T),., we


- --------- - -- - - -

116

......_.._-_...._.._.._-- ;======:..===c:==..-.,---- .. .. .. . -.- .. .. ,. -.- ... .... ___·,·_.. _·,·-_·_··'-:-.:.....:...:,~~ .~_w.'_..,...:..,::.._:~,;:~_.·_.~n~_ ..;:;:__~__"""~__.~ ~.~_._~_~ .~...._._.~~.,__._~,..~ .•~ ..~_,-"-':--

CHAPTER 5 DILATiONS ASSOCIATED TO FLAT CURVES IN JR" 117

can argue as follows: for an appropriate distance function peg). One shows HZ bounded on L 2 for

1 dt mj(~) = exp(ig . r(t))

2-j ::;ltl::;2.2- j t

i . dt = exp(ig . r(2-Jt))

1::;ltl::;2 t

i . dt = exp(lg. T2-jr(t))

1::;ltl::;2 t

i * dt = exp(iT2-j~ . r(t)) -. I::; It 1::;2 t

In the last integral, t stays in a compact interval not containing the origin, and we can expect the curvature of ret) to behave in a uniform way as it does in the examples. Thus the various types of behavior that might occur for m j (~)

(depending in the examples on k2 ... , kn) is expressed in terms of T2*-j~'

Next, the homogeneity of ret) implies that the curve 7/(t) = r(et) satisfies the differential equation

(13) 7/'(t) = A7/(t)

for an appropriate n x n matrix A (exp(A log t) = Tt). The differential equation (13) can be used to obtain estimates from below on derivatives of 7/(t) so that wellknown estimates of Van Der COIput can be used to obtain estimates on m j (g). See [SW]. As a matter of fact one obtains the estimate

(14) ImJC~)1 ::: CIIT2*_j(~)II-I/n,

which together with the trivial estimate

Imj(g)1 ::: CIIT2*-j(~)11

implies that meg) is bounded. It turns out that to obtain LP estimates for p # 2, one requires an appropriate

version of the Calderon-Zygmund theory. It is well known that there is a CalderonZygmund theory corresponding to a contracting dilation group. See [R]. In the 1970s LP estimates were obtained by using the complex interpolation of analytic families of operators due to Stein [S]. For example in studying the operator Hr , one introduced the analytic family of operators H; defined by

iffj(~) = mz(g)j(g),

where

mz(g) = pZ(g) f ItIZeiN(t) ~t

Re z ::: E, for some positive E. Further, one shows that for Re z < 0, HZ is convolution with a kernel satisfying the appropriate Calderon-Zygmund condition, and so is bounded in all LP. Stein's Theorem then implies that Hr is bounded in all LP, I < p < 00. More recent arguments inspired by work of Christ [C], and Duoandikoextea and Rubio de Francia [DR], employ a Paley-Littlewood argument. In order to employ the argument, one must show that for a smooth function ¢J with compact support vanishing near the origin, the multipliers

(15) mt (g) = Erj (t)¢J(T2*-j g)

(where rj(t) denotes the usual Rademacherfunctions) are bounded LP multipliers. One does this by checking that the inverse Fourier Transform of mt (g) satisfies a Calderon-Zygmund condition.

For a curve such as r (t) = (t, t k2 ... , t k"), the appropriate dilation group

T,l.(x) = (}..x!, }..k2X2 ... , }..k"Xn)

stares one in the face. However, it is not so clear how to choose dilations for a curve such as

ret) = (t, e-I/ltla, ... , e-I/ltla").

Perhaps a first guess might be to choose

T!(x) = (hi, Y2(}..)X2 ... , Yn(}..)Xn)

if ret) = (t, Y2(t) ... , Yn(t)) (p is for "provisional"). We shall see that this provisional guess has some problems, but nevertheless we can gain some insight into matters by considering T!. First of all, we must give up the group property (9). Moreover, it is not even true that r,l.(t) = T,l.-I reAt) forms a compact family on [1, 2]. (For example, el/,l.e-I/,l.t is not uniformly bounded for 1 ::: t ::: 2.) On the other hand, sums of derivatives of r,l. (t) . g do have uniform estimates from below of the type which are useful for the application of Van Der Corput's Lemmas.

We can see that instead of the group property (9), we might hope to have

(16) IIT -1T,1I ::: C(t/s)€, s::=: ts

for some positive E. This condition was essentially introduced by Riviere [R]. Something like (16) would seem natural if we want to bound a multiplier like mt (g) defined in (15), for by taking adjoints in (16) we can see that the inequality would imply that the supports of the functions ¢J(T2*_jg) would be essentially


118 CHAPTER 5

disjoint. Further, while we do not expect (12) to hold, at least we might expect

(17) f(t) = Tte

for some fixed vector e.

Also normalizing as in the homogeneous case, we see

m j (~) = [ . exp(i~ . f(t)) dt lZ-J :'Oltl:'OZ'Z-i t

1 . dt _ = exp(i~ . f(2-1t»I :'OI/I:'OZ t

1 . -I' dt = expI(~' TZ-iT _i f(2-1t)-.z1:'OItl:'OZ t

or

1 I . dt (18) mj(~) = exp(i. TZ*-i~' TZ-=if(Tlt))-.

1:'OItl:'OZ t

Thus if we let

(19) Uj(T/) = exp(iT/ . fj(t» dt ,1 1:'OItl:'OZ t

where

(20) fj(t) = Tz-=}reTjt ),

we seek estimates of the form

(21) IUj(T/)1 :::s CIIT/II-E

for some € > °uniformly in j. Let us tum now to the reason for discarding T/. First of all, the curve ret) =

(t, t log It J) is homogeneous and the dilation group G).. is given by the linear transformation

(22) T)"(Xl, xz) = ( A AIn IAI ~) (::) .

Thus, if we want our theory to include the homogeneous curve (t, y(t»

(t, tIn Itl), we can not define T).. by the formula

T)"(Xl, xz) = (Axl, y(A)XZ).

Note that the first column of the matrix in (22) is reA), and we ask how to interpret the non-zero entry, A, in the second column.

A quantity that had already played a role in the theory was the function

(23) h(t) = ty'(t) - y(t).

,;o~";=;';"-=~,;;:;;"':;;;"-;::;;;:;;:-"::;;:;;;;;;;;;;:=;;;;'7..:;;=~~~",-:;:::::~~-:::;C~~==T";;;:;""-::;:::::;:;;-~~-_-==:;;;;.;;;=,c;;~c..::;::.=." --=_-:;~.....~,~..:::;::,,;:;::;~;;;<:;;;;;;;;;:;;:;:;"~;;;;,,~--;::;:;;;;;;;;:-~~~:::;-,;:.~:;:~.~~,;:;;;;;:

119DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

(-h(t) is the y-intercept of the tangent line to ret) = (t, yet)) at f(t).)

For example we had the following Theorems:

Theorem A [NVWWI]. Let ret) = (t, yet)). Assume y(t) is convexfor t > 0, and that yet) is odd. Then a necessary and sufficient condition that Hr be bounded

in LZ(Rz) is thatfor some C > 1

(24) h(Ct) ~ 2h(t),

for t > 0.

Concerning the maximal function we have the following result.

Theorem B [NVWW2]. Suppose y(t) is convex and h(Ct) ~ 2h(t) for some

C > 1. Then Mr is bounded in LZ(Rz).

Now for the curve ret) = (t, t log Itl), h(t) = t. Thus for curves ret)

(t, y(t» in R Z, we might try

(25) h = (Y~A) h~A»)'

Moreover, the proof of Theorem B required the use of certain "balls," and those

balls can be obtained by applying T).. to one fixed ball. Of course in the case that

ret) = (t, t k ), we have been taking

T)..=(~ :k) which is different from the dilations in (25). However, in this case the two families

are equivalent in the following sense: Two families of transformations S).. and T).. are equivalent if there are balls BJ, Bz, and B3 such that

T)..B I C S)..Bz C T)..B3.

This means that the kernels treated by the Calder6n-Zygmund theory for T).. and

S).. are the same.

(Of course

S).. = (A 10: IAI ~) and

T).. = (~ A10~ IAI)


121 120 CHAPTER 5

are not equivalent.) There is another reason for discarding T!, which we shall come to when we discuss the n-dimensional situation. In any case, the following

theorem was obtained by using the dilations

AT= (Y~A) h~A»)' Theorem C [CCVWW]. Suppose r(t) = (t, y(t» with y(t) convex and that

there is an f > 0 such that

h'(t) 2: fh(t)(26) t .

Then

(27) IIMr fIIu(R2) :s A(P, r)llfIIU(R2), 1 < p :s 00.

If in addition r(t) is extended to negative t as an odd curve,

(28) IIHr fIIU(R2) :s A(p, r)lIfllu(R2), 1 < P < 00.

The condition (26) is stronger than (24). In fact (24) does not suffice for the

positive estimates of Theorem C if p < 2 for (27) and for p =1= 2 in (28). See

[CCVWW] and [C2].

The next questions are what convexity should mean in Rn and how TA in (25)

should be generalized from RZ to Rn, n 2: 3.

It turns out that there are different directions in which to proceed. One pos

sibility is to proceed inductively. If an estimate is true for a curve r(t) = (t, yz(t) ... , Yn(t» in Rn, then the same positive estimate holds for the curve

1r(t) = (t, Y2(t) ... , Yn-I (t» in Rn- • Thus one possible avenue is to add as

sumptions as n increases. This is the approach we shall take. The unfortunate

aspect of this approach is that the theory will not be GL(n, R) invariant. In other

words if r(t) satisfies the hypothesis and g is in GL(n, R) the curve gr(t) will

not necessarily satisfy the hypothesis (even though the estimates for Hr and Mr are equivalent to those for Hgr and Mgr ). It will be true that our theorems will

be invariant under the group of lower triangular matrices with positive diagonal

entries and 1 in the upper left-hand comer. An alternative approach which is

GL(n, R) invariant is developed in [CVWW]. In that theory the hypotheses are

more complicated. The basic difference is that in our theory the condition (16) is

proved, while in the GL(n, R) invariant theory (16) is assumed.

Since our theorem is inductive, as explained above, given a curve r(t)

(t, Y2 (t) ... , Yn (t» in Rn, it will be natural to consider also the curves

rr(t) = (t, Y2(t) ... , Yr(t» in R'

DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

for 2 :s r :s n. Our basic convexity assumption will be that for 2 :s r :s n,

r:(t) ) (29) Dr (t) = det : > 0

( ry)(t)

for t > O. We are mainly trying to understand what happens if Dn (t) vanishes

to infinite order at t = 0, so to simplify matters we might assume Dn(t) =1= 0 for

t =1= O. Now the inductive nature of our hypothesis leads to the assumption that

Dr(t) =1= Ofort =1= 0,2:s r:S n. Finally,byreplacingsomeyjbY-Yj,wearrive

at the condition Dr (t) > 0, t =1= 0, 2 :s r :s n. (A similar assumption occurs in

the theory of differential equations. See [H].)

There are also generalizations of h(t) for curves in Rn. Thus if r(t) =

(t, Y2(t) ... , Yn(t» and rz(t) = (t, Y2(t», we define hz(t) to be the h(t) for

rz(t). To define h3(t) consider the curve r3(t) = (t, Yz(t), Y3(t» in R3. We

consider the tangent line, L (t), to r3 at the point r3(t) in R3 • We let - r (t) be the

point of intersection of L(t) with the plane XI = O. Then under the hypothesis

(29) f'(t) is a convex curve in RZ• We let h3 be the "h" for r(t). We define hr for r :s n in an analogous manner. It turns out that the h j (t) can be expressed as

a ratio of determinants that is easy to calculate asymptotically in examples. As in

the case n = 2, the hj(t) were known to have had a role in the theory. This is expressed by the following theorem:

Theorem D. Suppose r(t) = (t, yz(t) ... , Yn(t» is convex in the sense of (29)

and that r (t) is odd. Then a necessary and sufficient condition that Hr be bounded in L z is that for some C > 1,

hj(Ct) 2: 2h j (t), t > O.

for 2 :s j :s n.

Thus it seems that the generalization of (25) should be

0 o yz(t) hz(t) 0 o

(30) TA = Y3(t) ? h3(t) o

Yn(t) ? ? hn(t)


123 122 CHAPTER 5

Looking again at (25) we note that the second column of r.. is obtained from the first column by applying to the first column a first-order differential operator. Namely, if

RW(t) = tZ( W;t))',

R (First column of T)J = 2nd column of TA• This suggests that we should try to find first-order differential operators R I ... , Rn-I such that

(31) R j (jth column of TA) = j + 1st column of TA•

It turns out that the convexity assumption (29) allows us to find unique Rj so that (30) and (31) are satisfied with the question marks filled in. If we denote by TA.i,j the i, jth element of the matrix for TA, we find

(32) TA,i,1 = Yi(t),

and

(33) TA,i,j+1 = R j+1 TA,i,j

where

f(A) )' h7(A)(34) (RJ(A) = hj(A) hi(A)'

It turns out then that the hypothesis that h j (Ct) > 2h j (t) implies

(35) HTs-I T,1I s C(~ r, s 2: t,

for some l: > O. Clearly also

(36) r(t) = T,e,

wh~u is the v~w m The next matter is somehow to relate the dilations to decay estimates of the

type (21). We found it convenient to prove a variant of the standard lemmas of Van Der Corput. We consider differential operators of the form

(37) MI!(t) = f'(t)

and

M·f(t»)'(38) Mj+I!(t) = aj(t) ( ~j(t) ,

where aj and {3j are smooth functions. Then we obtain the following theorem.


Theorem 1, Let J be a bounded interval on R. Assume

(39) aj(t) > 0 and {3j(t) > 0, 1 S j S n - 1

for tin J,

(40) Mn f (t) is single-signed for t in J,

(41) {31 (t) = 1 and al (t) S 1/l: for some positive l: and all tin J,

n

(42) L IMJ(t)1 2: l: for t in J, j=1

and, finally,

l Y

(43) {3j(t)dt 2: E(Y - x) max aj(t) x x9~y

. for x < y, x, y in J, and 1 S j S n - 1. Then for A 2: 1,

(44) I ( eiA/(tldtl < C(E n J) _1_.if -" AI / n

We apply the proposition with a j (t) = h j (t) and {3 j (t) = hi (t) where h j is the h j corresponding to the normalized curve

rj(t) = Tz-=Jr<Tjt).

It turns out that to verify the crucial hypothesis (42) it suffices to show

(45) II (T{)-I II s C(a)

for 1 S A Sa, where TA is the dilation matrix formed from r j' (This shows then the way the dilations are used to obtain the decay estimate (21).) Further, (45) can be shown to hold under the assumption hi (t) 2: E hjt(t

l .

The last point concerns the Paley-Littlewood decomposition. We choose a COO(Rn ) function¢(x) such that¢(~) = 1 for I~I S 1 and¢(~) = ofor I~I 2: 2, and we assume that we have a family of non-singular linear transformations Ak

satisfying the condition

(46) IIAk~IAkli Sa < 1.

(This condition was first introduced by Riviere [R].) We define

(47) mj(~) = ¢(Aj~) - ¢(Aj+l~)'

and SJ by

(48) fJ(~) = mj(~) . i(~).

We then obtain the following Littlewood-Paley estimate.


125 124 CHAPTER 5

Theorem 2. Iffor each integer k, IIAk~IAkll ~ IX < 1, then

(49) 1 < p < 00,II ~ ISi/(x) I' L:5 Cpll!II",( r and

(50) II ~Si/l :5 Cpll (DSj!JI't' II" 1 < p < 00

The key step in the proof of Theorem 2 is to show that for 0 ~ t ~ 1

(51) IIUr/IlLP ~ C(p)lIf11u, 1 < p < 00,

where

(52) Ur/(x) = L rj(t)Sj/(x), j

where again rj(t) are the standard Rademacher functions. To obtain the estimate (51) we need an appropriate variant of the Calder6n-Zygmund theory (which is not contained in the theory of spaces of homogeneous type of Coifman and Weiss [CW]).

We denote by B j, j E Z the "balls" of the theory. We assume for each j E Z

there is a "ball" B j' We assume that each B j is open, convex, balanced and bounded. We then have the following result.

Theorem 3 [CCVWW]. Assume

(i) UB j = Rn ;

(ii) nBj = {OJ; (iii) Bj+1 C B j .. and

(iv) there exists a K < oosothatlBjl ~ KIBj+i1. Let T be a convolution operator bounded on L 2(Rn), with convolution kernel

K(x). Assumefor each y E Bj (v) J;r¢2B

j IK(x - y) - K(x)ldx ~ A.

Thenfor fin CO'(W),

IITfIIU(R") ~ Cllfllu(R"jo 1 < P < 00.

The bound C depends only on p, n, A, the L2 operator norm of T, and the family of balls Bj, but not on f.

The above considerations lead to the following Theorem.


Main Theorem. Let r(t) = (t, Y2(t) ... , Yn(t» be convex in the sense of (29).

Iffor some E > 0,

I h/t). - 2h/t):::: E--, J - ... ,n, t

then

11M, fIIu ~ Cpllfllu, 1 0,0 < al < ... < an, it is easy to see that the hypotheses of the Main

Theorem are satisfied. We would like to add a remark pointing out a connection between our work

and the theory of uniform asymptotic stability of systems of ordinary differential equations. Suppose one considers the n x n system of differential equations

(53) y'(t) = B(t)y(t)

where y(t) is a vector in Rn and B(t) is an n x n matrix. Let F(t) be a fundamental matrix for the equation (53). That is, the columns of F(t) are solutions of the system and det F(t) =f. O. Then the system is uniformly asymptotically stable if

IIF(t)F- 1(s)1I ~ Ce-e(t-s), t:::: s,

for some positive E. See [CO]. Thus after a change of the time scale the adjoint of F (t) satisfies the estimate (16). In particular our results imply results on uniform asymptotic stability for systems in which the elements of the coefficient matrix are

unbounded. For example, the system

z' (t) = B(t)z(t)

with

B(t) = (_~ l+la (t») a(t) a(t)


~_~~~_ =:::_-=-~== =-c::-~-------=_-===-==-=-===-=-=..:;;;;;;;-~=_~::;;;-~=-===~,__ -== -=-==--=-==--

126 CHAPTER 5

is unifonnly asymptotically stable under the hypothesis 0 ::: a (t) ::: C, for some positive C.

University ofSussex, Brighton, United Kingtom

Wright State University

University ofWisconsin, Madison

Rutgers University, Camden

REFERENCES

[C] M. Christ. Personal communication. [C2] M. Christ. "Examples of singular maximal functions unbounded in LP."

Preprint. [CCVWW] A. Carbery, M. Christ, J. Vance, S. Wainger, and D. Watson. "Operators asso

ciated to flat plane curves: LP estimates via dilation methods." Duke Math. J. 59(1989),675-700.

[CO] W. A. Coppel. Dichotomies in Stability Theory. Lecture Notes in Mathematics, no. 629. Springer Verlag, 1978.

[CVWW] A. Carbery, J. Vance, S. Wainger, and D. Watson. "The Hilbert transform and maximal function along flat curves, dilations, and differential equations." To appear.

[CW] R. Coifman and G. Weiss. Analyse harmonique non commutative sur certains espaces homogenes. Lecture Notes in Mathematics, no. 242. Springer Verlag, 1971.

[DR] J. Duoandikoextea and 1. L. Rubio de Francia. "Maximal and singular integral operators via Fourier transform estimates." Invent. Math. 84 (1986), 541-561.

[H] P. Hartman. Ordinary Differential Equations. John Wiley, 1964. [NVWW1] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. "Hilbert transforms for

convex curves."Duke Math. J. 50 (1983), 735-744. [NVWW2] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. "Maximal functions for

convex curves." Duke Math. J. 52 (1985), 715-722. [NVWW3] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. "Hilbert transform for convex

curves in R" ." Amer. J. ofMath. 108 (1986) 485-504. [R] N. Riviere. "Singularintegrals and multiplier operators." Arkiv Math. 9 (1971),

243-278. [S] E. Stein. "Interpolation oflinear operators." Trans. Amer. Math. Soc. 83 (1956),

482-492. [SW] E. Stein and S. Wainger. "Problems in harmonic analysis related to curvature."

Bull. Amer. Math. Soc. 84 (1978), 1239-1295.

-----,,;=-=--~~"~"~~~~,=~~~-~~~~~--=~---~._~=~-----~-~~ -=--~ .~-~~=~~--=-~~.:;;~~~-~~~~~~=~-~

6

Nonexistence of Invariant

Analytic Hypoelliptic Differential

Operators on Nilpotent Groups

of Step Greater than Two

Michael Christ*

INTRODUCTION

The development of a large part of the theory of subelliptic partial differential equations has been guided by fundamental papers of Stein and collaborators. Among these are the explicit computation by Greiner and Stein [GS] of the Szego kernels for certain three-dimensional CR manifolds; the work of Folland and Stein [FS] which developed aspects of hannonic analysis on the Heisenberg group and established its connection with the aequation on strictly pseudoconvex domains; the proof by Rothschild and Stein [RS] of sharp regularity estimates for sums of squares of vector fields and their forging of the link between these operators and harmonic analysis on graded nilpotent Lie groups; and the estimation by Nagel, Stein, and Wainger [NSW] of the fundamental solutions of such operators in a very precise way. These works have pointed the way to a far-reaching domain of research centered on the confluence of partial differential equations, complex analysis in several variables, hannonic analysis on Euclidean spaces and on nilpotent groups, and real

variables.

'This research was supported by the Nalional Science Foundation and the Inslilut des Hautes Etudes

Scienlifiques.


129 128 CHAPTER 6

The present note serves to announce further progress in this direction. Detailed proofs will appear elsewhere [C5], [C6].

We are concerned with five interrelated themes:

• Analytic hypoellipticity of subelliptic partial differential operators, especially those which are left-invariant and homogeneous on some nilpotent Lie group.

• Hypoellipticity of certain PDE with complex coefficients, in the Coo sense. • Certain one-parameter families of irreducible representations of nilpotent Lie

groups and algebra~.

• Existence of eigenvalues for certain non-selfadjoint linear operators. • A scattering problem for ordinary differential equations.

Our investigations have been profoundly influenced by all four of the papers cited above.

Experts should not be misled by our title, in which at least three distortions are perpetrated in the interest of brevity.

1 REVIEW

To begin, we recall some facts concerning constant-coefficient, homogeneous differential operators on ]Rd. A differential operator L is said to be (COO) hypoelliptic in an open set Q if whenever Q' c Q is open, u is a distribution in [C<f(Q')]', and Lu E Coo(Q'), then u E Coo(Q'). L is said to be analytic hypoelliptic if Lu == 0 in Q' implies that u is real analytic in Q'. It is well known that for a homogeneous partial differential operator with constant coefficients in ]Rd, the following are equivalent:

(1) L is Coo hypoelliptic. (2) L is analytic hypoelliptic. (3) L is elliptic.

Moreover, if L has constant coefficients but is not necessarily homogeneous, then (2) and (3) remain equivalent.

For a certain very restricted class of nilpotent Lie groups of step 2, including the Heisenberg groups [M], these conditions (1) and (2) remain equivalent for homogeneous, left-invariant differential operators, and are equivalent to a modification of (3), the Rockland criterion; the latter and (1) remain equivalent for all graded nilpotent Lie groups.

We will be concerned in this chapter with the non-validity of the equivalence between (2) and the other conditions, for groups of step three or higher.

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS

In order to motivate later considerations, let us recall how it may be proved that a non-elliptic operator with constant coefficients, in ]Rd, admits solutions which are not analytic. Let L be such an operator. One approach is to note that if L were analytic hypoelliptic, then for any open sets Ql <E Q2 <E Q, there must exist C < 00 such that for every u E Coo (Q2) satisfying Lu == 0, for every multi-index

a,

(1.1) ~ull ~ C\+lallal!lIullL''''(~h)' II oxa

LOO(Q])

Assume that L has positive order n, and let P be its symbol. P is a polynomial, which may be considered to be defined on Cd, and then P(~) = 0 for certain ~ E Cd. If L is not elliptic, then a sequence {~j} C Cd may be chosen such that

P(~j) = 0,

I~j I ---+ 00,(1,2)

1£f(~j)1 ---+ 0 as j ---+ 00. I~jl

In order to violate the Cauchy estimates (1.1), set

/j(x) = exp(i(x, ~j)),

so that L/j == O. With a = a(j) a multi-index to be determined,

I aO;a Ii (0) 1 = I~jl while for any bounded neighborhood Q of 0,

Cl+1a1Ial!llliIlLoo(Q) ~ C1+1a11al! exp(CiI£f(~j)1)

with C\ = C\ (Q) fixed. If each ~j E ]Rd, then (1.1) is cl~arly contradicted by fixing any a satisfying a j i=- 0, and exploiting the condition that I~j I ---+ 00. In the general case, if the imaginary parts are nonzero, then exp(C\I£f(~j)l) will grow as j ---+ 00, so that both the left- and right-hand sides of (1.1) grow with j and the situation is more delicate. It can be shown that the third condition in (1.1)

still permits the choice of a sequence aU) such that (1.1) is violated for all finite C, as j ---+ 00; we shall not reproduce the details. However, for the sequel it is instructive to note that for such a construction to succeed in contradicting the Cauchy estimates, it is permitted that II /j II LOO(Q) grow as j ---+ 00, but is equally essential that it not grow too rapidly, in a suitable sense.


130 CHAPTER 6 ANALYTIC HYPOELLlPTlC DIFFERENTIAL OPERATORS 131

From the viewpoint of hannonic analysis, what we have done is to decompose the regular representation of lRd on L 2(lRd ) into its irreducible unitary components, which are parametrized by ~ E lRd ; to complexify so as to obtain non-unitary representations depending holomorphically on the parameter ~ E Cd; to associate to each representation n and differential operator L an operator n(L) on the Hilbert space 1fT( = L 2(lRO) = C; and to exploit those n for which n(L) fails to be injective. Ellipticity of L is determined solely by real ~, that is, by the unitary representations, but the nonunitary ones were vital in the analysis of analytic hypoellipticity. Note that.for homogeneous operators, the symbol is a homogeneous polynomial, so that if it is elliptic, then it is nonvanishing in a conic neighborhood of IRd in Cd, and hence no sequence satisfying (1.2) exists.

A similar situation occurs for homogeneous, left-invariant partial differential operators on graded nilpotent groups. The set of all irreducible unitary representations n may be described quite explicitly [K]. To each such L and each n is associated a differential operator n (L) acting on a dense subspace of1fT( = L 2(IRk)

for some ken). As conjectured by Rockland [R], and proved by Rothschild and Stein [RS] in the negative direction and by Helffer and Nourrigat [HNI], [HN2] in the positive direction, hypoellipticity of L in the Coo category is equivalent to the injectivity of n(L) (on its domain) in 1fT( for every irreducible unitary representation n. This is the modification of (3) mentioned above.

Thus it is natural to expect that the study of analytic hypoellipticity for such operators should be related to the properties of certain non-unitary representations. The subject was first approached from this point of view by Helffer [He]. It proved problematic to determine whether n(L) failed to be injective for some n, or not, and that is the problem to be addressed here.

We are able to treat only a very restricted class of groups, for which all the relevant representations satisfy 1fT( = L 2(lR1), the simplest case which does not arise for abelian groups. Then n(L) is an ordinary differential operator, about which much can be said by elementary means. One of our goals is to develop in detail the analysis of holomorphic one-complex-parameter families of such

representations. Some notation: "A ....., B" means that the ratio of these two positive quantities

is bounded above and below by positive constants. (z) = 1 + Izi. The natural logarithm is denoted In, the real part by!)i, and the imaginary part by:S. C'" denotes the class of real analytic functions.

2 THE MAIN RESULT

g will denote always a (finite-dimensional) nilpotent Lie algebra, and G a connected Lie group whose Lie algebra is g. Elements of g will be identified with

(real) left-invariant vector fields in a neighborhood of the identity element 0 E G via the exponential map.

Definition 2.1. (a) A Lie algebra g is said to be stratified ifit admits a vectorspace decomposition

asg = EBj~lgj,with[gj, gk] C gj+k!orallj, k E Z+,suchthatglgenerates

g as a Lie algebra.

(b) A stratified Lie algebra g is said to have k generators if dim(gd = k.

(c) A nilpotent, stratified Lie algebra g is said to be of step r if gr+l = {OJ but

gr =1= {OJ. (d) If g is stratified, then Un denotes the linear space ofall left-invariant differen

tial operators on G which may be represented as homogeneous polynomials

ofdegree n in the elements of gl.

Definition 2.2. For any formal polynomial P in two non-commuting variables,

with complex coefficients, Pdenotes the naturally associated polynomial in two

complex variables.

If g is stratified with two generators, and if a basis {X, f} for gl is fixed, then to each such polynomial, homogeneous of some degree n, is associated the element P(-iX, -if) of Un' Conversely, any element £, of Un may be so expressed; though P will not be uniquely determined, it is easily verified (by consideration of the abelian group whose Lie algebra is g/EBj~2gj) that P is uniquely determined by £'.

Definition 2.3. Let g be a stratified Lie algebra with two generators X, f E gl. (a) A homogeneouspolynomial P ofdegree n is said to be generic ifthe polynomial

z ~ P(z, I) in one complex variable has degree exactly n, and moreover has

n distinct complex roots.

(b) £, E Un is said to be generic if it equals P(-iX, -if) for some generic

polynomial P.

It may be verified that the property of being generic is independent of the choice of basis for gl, so long as the basis itself is suitably generic. Since it is determined by P, it is an intrinsic property of £, = P(-iX, -if), independent of the choice of the representing polynomial P.

Our principal result is:

Theorem 2.4. Let G be a Lie group whose Lie algebra g satisfies both:

• g is stratified and has two generators.


132 CHAPTER 6 ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS 133

• g is of step greater than or equal to three. Then any generic homogeneous, left-invariant differential operator .c on G fails to be analytic hypoelliptic.

Remarks. 1. The proof yields a stronger result in the scale of Gevrey classes. For some open

set Q and real number s ~ 1, GS (Q) denotes the class of all Coo functions

defined on Q such that, for each compact subset K C Q, there exists C < 00

such that for every multi-index a and every x E K,

laa f(x)1 ~ cl+lal(l + lal)slal.

For s = 1 this is simply the class of real-analytic functions, while for s > 1

there are strict inclusions CW C c Coo. Then our method establishes GS

existence of a solution to.cF == 0, in a neighborhood of 0, such that F fails to

belong to GS for any s < 3. 2. The case .c = X2 + y2 was treated in the union of the two papers of Helffer

[He] and of Pham The Lai and Robert [PRJ some time ago, so that Theorem 2.4

is no surprise. However, their argument depends on certain particular features of X2 + y2 which are absent for general .c.

3. The principal hypothesis is that g have step strictly greater than two. There

exists a class of nilpotent groups of step two, for which every left-invariant, homogeneous differential operator that is Coo hypoelliptic is automatically

analytic hypoelliptic [M]. In particular, this occurs for the Heisenberg groups

lHln •

4. The hypothesis that g have two generators may be weakened to dim (g2) = 1, with an appropriate redefinition of genericity. It would be interesting to decide

whether the result remains valid without any such hypothesis; this cannot be

done by our method as it stands.

S. The hypothesis that.c be generic is made to simplify certain technical aspects of the proof, and might conceivably be eliminated by a more elaborate argument

along the same general lines.

6. In order to simplify the exposition slightly we assume henceforth that the roots

of z f-+ P(z, 1) have distinct real parts.

3 OUTLINE OF THE PROOF

The first step is to decompose the regular representation of G on L 2(G) into irreducible components, and to focus attention on three particular families of com

ponents. For our purpose this amounts simply to separation of variables and may

be achieved in the following concrete fashion. Let n j denote the dimension of

gj, let N = L nj, and let Xjk be coordinates in ~nj for 1 ~ k ~ nj. In the coordinates given by the exponential map, G is identified with ~N, and X, Y take

the form a

LPjk(X)-,. k aXjkJ,

where the p jk are polynomials, homogeneous in a natural weighted sense. More

over, Pjk depends only on those variables Xii for which i < j. Thus the space of all functions on G which depend only on those variables x jk for which j ~ 3, is

invariant under X and y, hence under.c. It suffices to produce a function in this

space which is annihilated by .c in some open set, and which is not analytic. In the same way, we may restrict attention to functions independent of all variables

X3k with k ~ 2. Since gl and g2 have dimensions 2 and 1, respectively, our functions now depend

on four variables which will be called x, y, s, t. Y takes the form c1ax + C2 ay + l(x, y)as +p(x, y, s)a, where Ci are constants, lis linear, and p is a homogeneous

polynomial of degree two in :x, y plus a constant times s. X takes a similar form.

Mter composition with an appropriate polynomial diffeomorphism,

2X = ax and Y = ay - xas - x a,.

In effect, we are now working on a particular four-dimensional nilpotent Lie group

of step three.

We consider three families of irreducible representations of this group, or more concretely, the action of X, Y on three classes of functions F on ~4:

(I) F(x, y, s, t) = ei(~,(x,y)) with ~ E ~2,

(II) F(x, y, S, t) = f(x)eii.s with 0 -::j:. A E ~,

(III) F(x, y, S, t) = f(x)eiZYeirt with 0 -::j:. r E ~ and Z E iC.

The actions of X, Y on these classes offunctions generate representations of the Lie

algebra which they generate as (possibly unbounded) operators on iC l = L 2 (~o),

on L2(~1), and on L2(~1), respectively. X acts as ~[, -i d/dx and -i d/dx,

and Y acts as b -AS and (z - rx 2 ), respectively. The representations (I) factor

through g/EBj ::2gj, those of type (II) through g/EBj ::3gj, and those of type (III)

through g/EBj::4gj. To these representations of g correspond representations of G. In case (III),

when z E iC\~, these representations are not unitary, and indeed elements of G are represented as unbounded linear operators defined on a certain subspace of L2(~).

It is a general principle [G], [HN1], [R] that one should study in succession the

representations of type (I), then type (II), then type (III) in order to obtain increas

ingly more refined information about.c. For those oftype (I),.c acts as P(~l, ~2);


134 CHAPTER 6

if f>(~) = 0 for some 0 i- ~ E lR.2, then.c annihilates exp(ir(~, (x, y)}) for all r E ll~+. By the argument outlined in §1, it follows that .c is not analytic hypoelliptic, nor even Coo hypoelliptic. Therefore we may assume henceforth that Pis an elliptic polynomial.

The following is well known, and expresses the analogue of the last paragraph for the representations of types (II) and (III).

Lemma 3,1.

(i) If there exists· f E Loo(lR.), not identically vanishing, such that

P(-i 1:x: ,x)f == 0, or such that P(-i :f:x:, -x)f == 0, then there exists a function F, defined on an open subset of G, such that P(-iX, -iY)F == 0 but F is not Coo. Therefore.c = P(-iX, -iY) is not analytic hypoelliptic, nor even Coo hypoelliptic.

(ii) If there exist z E lR. and 0 i- f E L 00 (lR.) for which P (-i 1:x: ' (z - x 2»f == 0, then .c is again neither analytic hypoelliptic nor Coo hypoelliptic.

(iii) If there exist z E C and 0 i- f E L 00 (lR.) for which P(-i 1:x: ' (z - x 2»f == 0, then .c is not analytic hypoelliptic.

The proof is analogous to that for type (I). As in the argument outlined in § I , some growth restriction on the solution f is essential; in order to contradict the Cauchy estimates one considers the solutions of L

f (.1/3X )eir1fJ;;ye irt

where. is a positive parameter which will tend to 00, and where f, z are as in (iii) and are fixed. Thus the behavior of f (x) for large x comes into play. The condition that f be bounded may be relaxed in case (iii) to If(x)1 = O(exp(lxI3- e»for some e > 0, but it turns out that (if none of the roots of Pare purely imaginary) every solution will either decay rapidly to zero as Ix I -+ 00, or will grow like exp(8lx 13), that is, too rapidly to lead to the desired negative result. Thus it is irrelevant whether one seeks solutions in the Schwartz class, in L 2, in L 00, or in some class permitting moderate growth at infinity.

As far as the existence of such bounded solutions for representations of type (II) goes, only the sign of A is relevant, not its magnitude; this reflects the existence of a group of automorphic dilations of G. A bounded solution f for A = I leads by an appropriate dilation of the x variable to a bounded solution for any given A > O. Similarly, it is no loss to set. = + I in case (III).

Definition 3.2. .c is said to be nondegenerate if Pis elliptic and there exist no boundedfunctions in the nullspaces of P(-i 1:x: ' ±x).

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS 135

There are certain cases in which a very superficial analysis shows .c to be degenerate. Let

{Yj: I ::::: j ::::: n}

be the roots of Z f-+ P(z, I), recalling that Pis assumed to be elliptic.

Proposition 3.3. Assume.c = P(-iX, -iY) to be homogeneous, left-invariant

and generic on a nilpotent group whose Lie algebra is stratified with two gener

ators, and assume Pto be elliptic. Then.c is degenerate if any of the following

occur: (1) The degree of P is odd.

(2) Some root Yj is imaginary. (3) The degree n of P is even and the number ofindices j for which ffi(Yj) > 0

is not exactly n /2.

The idea of the proof will be, indicated later. There is the following consequence:

Corollary 3.4. Let.c = P (-iX, -iY) be a generic homogeneous, left-invariant differential operator on the Heisenberg group 1Hl1 ofdimension three. If the degree of P is odd, then .c fails to be Coo hypoelliptic.

We assume henceforth not only that P is elliptic, but also that none of these conditions are satisfied. These are by no means the only causes of degeneracy: the operators X2 + y 2 + ia[X, Y] on 1Hl1 fail to be Coo hypoelliptic for a discrete set of real values of the parameter a, while Pis independent of a. A fundamental issue in our analysis is the difficulty of detecting such more subtle degeneracies.

For our purpose the essential analysis takes place on the family of representations of type (III), which reflect the fact that the group is not of step 2. Let us consider a more general situation by letting m E (2,3,4, ... } be arbitrary and setting

d mL;; = P(-i -, (z - x -I ».

dx

Definition 3.5. z E C is said to be a nonlinear eigenvalue ofthe operator family {L;;: z E q If there exists f E L 00 (lR.), not identically vanishing, such that

L;;f = O.

In fact, the nonlinear eigenvalues are genuinely eigenvalues for a related problem; see [KeJ, [PRJ. The set of nonlinear eigenvalues has some structure.

Proposition 3.6. Assume that P is homogeneous and generic, and that Pis elliptic

and that none of the roots Yj are imaginary. Let m E (2,3,4, ... }. Then the set


137

-~---_.-._----------~~ -:-- .- ~, --".~-,~::;, ,.~- . ""!"~

136 CHAPTER 6

of nonlinear eigenvalues of (Lzl is equal to the set of all zeroes of an entire holomorphic function W, which satisfies

IW(z)1 = O(exp(C1zlm /(m-l))

for some C < 00. Moreover when m = 2, W(z) == Cl exp(c2z2) for some constants Cl, C2.

In §4 we shall explain how W is constructed. A class of examples in which it can be computed explicitly is

d d dm 1 m IP(-i -, (z - x - » = [- - + (z - x - )] 0 [- + (z - x m- 1)].

:.=_.

dx dx dx

Then

12 sW(z) = { i: e (zs-m- m) ds for m even

(3.7)

±l for m odd.

On the other hand, if m is even and

d d dm 1 m 1P(-i -, (z - x - » = [- + (z - x - )] 0 [-- + (z _ dx dx dx

then

W(z) == O.

It was observed in [CG] that the entire function j':x, exp(2(zs - m- I sm» ds

has order m/(m - 1). Thus at least three behaviors are possible: W can be

identically zero, a nonzero constant, or an entire holomorphic function of order

exactly m/(m - 1). In general, W will be of a transcendental nature, as illustrated already by (3.7), and will not be effectively computable.

On the technical level our crucial result is

Proposition 3.8. Fix an integer m ~ 3 and consider Lz = P(-i :fx ' (z - x m- I » where P is generic, homogeneous, and nondegenerate. Then there exists 8 > 0 such that

!W(z)! ~ 8 exp(8Izlm/(m-I) as IR+ 3 Z ~ 00.

Theorem 2.4 follows at once. If.c is degenerate, then analysis on the level of either type (I) or type (II) representations establishes the existence, on an open

subset of G, of a function F 1- Coo such that.cF == 0; hence .c is hypoelliptic

in neither the Coo nor the analytic categories. If.c is nondegenerate, then W is

mx - 1)],

ANALYTIC HYPOELLlPTIC DIFFERENTIAL OPERATORS

an entire holomorphic function of order exactly m/(m - 1), and since this order is not an integer, W has a nonempty, in fact infinite, set of zeroes. As discussed earlier, this implies that .c is not analytic hypoelliptic; if any of the zeroes happens

to be real, then it is not Coo hypoelliptic. A small extension of the same reasoning establishes

Theorem 3.9. In 1R3, with coordinates (x, y, t), set

x = ax and Y = ay - xm-1at •

Let P be a generic, homogeneous polynomial. Then for any m E {3, 4, 5, ...},

.c = P(-iX, -iY) is not analytic hypoelliptic.

Some additional work is required because the type (II) representations do not

arise directly as in the previous discussion, but only as a limit of the type (III) mrepresentations (with (z - x 2) replaced by (z - x - 1», in the fashion described

by Helffer and Nourrigat [HN3]. Therefore, in the degenerate case, their results

imply only the weaker conclusion that .c fails to be maximally hypoelliptic; see

[HN3] for the definition. In fact, just as in the group-invariant setting, .c turns out to be hypoelliptic in neither the Coo nor the analytic sense.

By two different arguments [PRJ, [HH], [C2], [C3] the case L = X2 + y 2 has

already been treated. Both used particular properties of this operator not present

in general.

4 THE WRONSKIAN W AND THE SCATTERING PROBLEM

Throughout this section it is assumed that P is nondegenerate and generic, and

for simplicity of exposition that none of the roots Yj are imaginary. The principal idea of the proof is to define W as the determinant of what can be thought of as a

scattering matrix. The ordinary differential equation L z has n linearly independent

solutions, and if r denotes the number of indices j for which ffi(Yj) > 0 then it happens (see below) that there exist exactly r linearly independent solutions which

remain bounded as x --+ +00, and exactly n - r independent solutions which

remain bounded as x --+ -00. This happens for every z E Co Now, the former solutions have an asymptotic behavior which may be understood rather precisely as

x --+ +00, and similarly for the latter as x ~ -00. Therefore the only possibility

for a globally bounded solution is that these n solutions should become linearly dependent, for some value of z.

The situation is mostly parallel for the level (II) representations, but certain

differences have their significance. The number of independent bounded solutions

for P(-i:fx, -2x), as x ~ +00, is again r, but is also r as x --+ -00. For


138 CHAPTER 6

P(-i Ix ' 2x), the number is n - r in both directions. Therefore if r > n/2, one has 2r > n solutions, which perforce must be linearly dependent. Hence there exists a bounded solution for P(-i Ix, -2x). The same goes for P(-i Ix, 2x)

if r < n /2. Taking the case where some Yj has vanishing real part into account, this argument establishes Proposition 3.3.

A precise formulation for the level (III) representations is as follows. Order the roots Yj so that ffi(Yd < ffi(Yj) whenever k < j. Denote

24>j.(x) = Yj(Zx - tx3) + Pj In(z - x )

where the Pj are certain scalars, and that branch of the logarithm function is chosen whose imaginary part is equal to i 7T: on the negative real axis.

Proposition 4.1. Suppose that P is nondegenerate and generic. Then for each j there exists Pj E C such thatfor each z E C, there exists afunction 1/IZ E COO(IR)

satisfying

L z1/lZ == 0

and

1/IZ(x) = e"'j(X)(1 + O(x- I )) as x ~ +00.

1/1"};. is uniquely determined, while in general each 1/IZ is uniquely determined

modulo linear combinations of {1/I;t: k > j}. There exists a constant CI < 00

such thatforall z E C,forallx ~ CdZ}I/2 and all 0 :s k < n,

dk k- (1/I+'(x) - y - I (z _x 2)k-I e<%>j(X») I < C(lxl + (z) 1/2)-II(z - x 2)k-Ie<%>j(x) I

1 dx k Z}} - •

The 1/1Z may be constructed so that each is an entire holomorphic function of z, modulo (jor each z) a linear combination of {1/I;t : k > j}.

There exist also solutions 1/1; possessing analogous properties as x ~ -00.

There is one formal change in the statement: 1/Iz} is unique modulo linear combinations of the more rapidly decaying solutions {1/Izk : k < j}, and the final clause of the proposition should be modified accordingly. Assume that the number of Yj

with positive real parts is n/2.

Definition 4.2.

1 dl d )

W(z) = determinant -l 1/1;(0): j :s n/2; -l 1/IZ(0): j > n/2;( dx dx

O:":l<n

ANALYTIC HYPOELLlPTIC DIFFERENTIAL OPERATORS 139

The last clause of the proposition implies that W is an entire holomorphic function. Straightforward estimates yield the upper bound

IW(z)1 = o(exp(C1d/2)).

Comment. The growth estimates for W are largely artifacts of its normalization, rather than an intrinsic feature of the representations under consideration. In the case m = 2, all representations with z E IR are mutually unitarily equivalent; an intertwining operator is induced by the change of variables x 1--+ x + z. Thus one might expect W to be constant, but in reality it is aGaussian.

To approach the lower bound of Proposition 3.8, in the case m = 3, assume z E 1R+ is large, let A denote a sufficiently large positive constant, and denote xt = ±ZI/2.

Proposition 4.3. For all sufficiently large z E 1R+ ,for all 0 :s e < n ,for all

x ~ xii + AZ- I/ 4 and all 1 :s j :s n,

d1

(4.4) -1/I+(x) = l-I(Z - x 2 )l-l e<%>j(x)(l + O(A-I)).dx1 Z} }

For Ix - x+1 < AZ- I / 4 o - ,

1 1

(4.5) "Z-l/4n-I I-1/I+'(x)d "-'" Z-l/4 I -dI n-I ,',+'(X+ + AZ- I / 4 ) I ~ dx1 ~ ~ dxlP~ 0 ' l=O l=O

uniformly in z.

In summary, one has excellent control of the 1/IZ for x - xii » Z-I/4, one loses much of this control for Ix - xii I :s Z-I/4, and one has no control for

4x :s xii - AZ- I / . Thus the situation is analogous to a scattering problem, in which the solutions 1/1Z undergo scattering by an obstacle which is localized where

4Ix - xiii :s AZ- I/ •

The same loss of control occurs for the solutions 1/1;, with xii replaced by xo' Hence the situation is more accurately viewed as one of scattering by two obstacles, widely separated for large z.

Normalize the 1/IZ by multiplying by scalars, depending on z, so that the new solutions t/JZthus obtained satisfy

1 "n-I -l/4 I_ d ,1,+ (x) I "-' 1 v Ix - x+1 < AZ- I / 4 ~ Z dx1 PZ} o - , l=O

uniformly in z E 1R+ provided z is sufficiently large.


__ __ IiiiliJ __ ~

140 CHAPTER 6

Define Ig~(y) = tz~(rl/2 + r- /

4 y).

Lemma4.6. As JR.+ '" Z ~ 00, the g~formanequicontinuousfamilyoffunctions

on every compact subset of R The same holds for all oftheir derivatives. For any

sequence r ofvalues of z tending to +00, if {g~ : z E r} converge uniformly on

compact subsets of lit then the limit is a solution g~,j of P(-i :x' -2x) which does not vanish identically. Moreover, g~,j (x) decays rapidly to 0 as x ~ +00,

for all j > n12.

There are similar functions gti, and possibly limits g~,j' to which an analogous statement applies.

Let us imagine for a moment that circumstances were simpler, that there was

only the single obstacle near xii, that the 1/tz} satisfied estimates similar to those 4for the 1/t~, for Ix - xiii ::s AZ- 1/ , and that the gti were redefined accordingly,

in terms of the behavior of 1/tti near xt. This is actually the situation, except for 2 m Ivery minor alterations, when (z - x ) is replaced by (z - x - ), with m even.

There are then two possibilities:

2(4.7) IW(z)1 ~ z-co exp( ~ L Iffi(Yj)lz3/ )

j

as JR.+ '" Z ~ 00, for some constant Co E JR.+; or as z ~ 00 through a sequence for which (4.7) fails for a sufficiently large but fixed Co, fixing a subsequence

for which the g~ and gti all converge to limits g~,j and g~,j' respectively, the solutions

(4.8) {g~.j: j ::s n12} U {g~.j: j > n12}

are linearly dependent. In the case (4.8) there exist scalars, not all equal to zero, such that

(4.9) L Cjg~,j == L Cjg~,j" j ~n/2 j>n/2

Defining

f = L Cjg~,j' j~n/2

f is a solution of P(-i fx ' -2x). It decays rapidly as x ~ -00 because each of the g~,j does (because ffi(Yj) < 0); it also decays rapidly as x ~ +00, because of the alternate representation afforded by (4.9). It follows from the estimates (4.4) at xii + AZ- 1/ 4 that the g~.j must be linearly independent, hence (granting the same for the g~.j in this hypothetical situation) that f does not vanish identically. Thus we would have Proposition 3.8.

141ANALYTIC HYPOELLJPTIC DIFFERENTIAL OPERATORS

2This scheme does not apply directly to P(-i :x' (z - x » because we know 1 4

essentially nothing about the 1/ttl in the large interval [xo+ AC I/4

, xii - AZ- / ].

To overcome this we introduce a third set of solutions 1/t~, whose behavior is well understood in exactly this interval. We introduce two additional determinants of

the same nature as W, one governing the scattering of the 1/tti into 1/t~j at Xo' the other governing scattering of 1/t~ into 1/t~ at xii. An inequality relating the three shows that if W is substantially smaller than the estimate in (4.7), then one of

the other two determinants must also be anomalously small. The argument of the preceding paragraph may then be implemented at either of xt, as appropriate. One obtains abounded solution of P(-i :x' =f2x). By means ofachangeofvariables,

this produces a bounded solution of P(-i :x' =fx).

5 RELATED QUESTIONS

In this section we consider in more detail some issues related to the operator

m£ = B; + (By - x -1Bt )2

in JR.3 , and the ordinary differential operators

d 2 mL = -- + (z _x - I )2.z dx 2

We assume always that m ~ 2 is even. It is useful to introduce

where z = rei/3.L/3" = L z

Fixm.

Definition 5.1. A ray JR.+ . ei/3 in C is said to be ofpolynomial growth ifthere exist

a E JR. and c > 0 such that for all sufficiently large r E JR.+,

(5.2) II L /3"fIIU(IR) ~ erG IIf11u(lR)

for all f in the domain of L /3,"

It is permitted that a be negative. As proved by Pham The Lai and Robert [PRJ,

whenever two rays of polynomial growth can be found such that 1131 - 1321 < 1C m;;; I , then there exists at least one nonlinear eigenvalue in the sector bounded by the two rays. When m is odd, any ray with 1C 12 < 13 < 31C 12 is a ray ofpolynomial growth, as is JR.+ (with a = (m - 2)/2(m - 1)); this yields sufficiently narrow

sectors to guarantee the existence of nonlinear eigenvalues, for odd m ~ 3. This is how the failure of £ to be analytic hypoelliptic was first proved for odd m. When m is even, JR.± are still rays of polynomial growth, with a = (m - 2)12(m - I),

but it is not apparent that there exist any other such rays.


143 142 CHAPTER 6

One may ask (we do so only for even m):

• Where are the nonlinear eigenvalues located? • Do there exist any rays of polynomial growth. besides JR±? • How does W behave off of the real axis? • What can be said about Coo hypoellipticity of operators

1Do = a; + (eiOay - xm - at )2,

which are sums of squares of non-real vector fields satisfying the condition of Hormander?

This last question is related to the others; to study Coo hypoellipticity of Do one should analyze the one-real-parameter family oflevel (III) representations in which Do corresponds to {Lo.r : r E 1R) acting on L2(JR).

Some partial answers may be obtained.

Proposition 5.3. For m ~ 4 and even, there are no nonlinear eigenvalues z satisfying I arg(z) I < I m~l or IJr - arg(z) I < I m~l .

This suggests that there may be a corresponding estimate which would affirm that every ray in these two sectors would be a ray of polynomial growth. but this hope is dashed by the next result.

Proposition 5.4. For any even integer m ~ 2, the only rays ofpolynomial growth

are JR±. More precisely,for any f3 E JR satisfying 0 < 1f31 < Jr, there exists 8 > 0 such that for every r E JR+ , there exists fr E L 2(JR) satisfying II fr II U = I but

II Lp.r fr IIU(IR) S C exp( _8rm /(m-l».

fr is constructed by the method of the preceding sections. Solutions 1/Ir± of Lp.r, which decay rapidly as x -+ ±oo, respectively, are constructed as before. Defining t;= by multiplying by scalars so that t;= are suitably normalized on the interval {Ix - xol S ).,r-(m-2)/2(m-l»), where Xo = (r cos(f3»I/(m-1) plays the

same role as xt in the previous analysis, and defining W to be the Wronskian determinant of t r± so that IWI S C uniformly in r. one finds

Proposition 5.5. The Wronskian of the normalized solutions satisfies

(5.6) \WI s C exp( _8rm /(m-l».

We emphasize that the normalizations are carried out so that the t;= are essentially bounded below by I near xo, and similarly for their derivatives. so (5.6)

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS

means that they are extremely close to being linearly dependent for large r. The functions fr required in Proposition 5.4 are obtained by pasting t;= together in the interval Ix - Xo I S ).,r-(m-2)/2(m-l) via a partition of unity. The functions

1/1;= extend to entire functions of x E C, their Wronskian is constant. and (5.6) follows by analyzing the equation obtained from Lp.r by shifting from x E IR to

x E IR . eia for a certain ex =F O. In a sense, the solutions 1/1;= constitute the only possible obstruction to the

inequality 5.2:

Proposition 5.7. For each even m ~ 2 there exists a ~ 0 such that for any f3, for any sufficiently large r E 1R+, there exists a subspace V of codimension one

in the domain of L p.r' such that

IILp.r fIIU(IR) ~ era II fIIu V f E V.

For m ~ 4, (J' is strictly positive. Proposition 5.4 plus standard arguments yield

Proposition 5.8. For any e such that eiO ¢ lR, De fails to be Coo hypoelliptic at

the origin.

This should be contrasted with the situation for operators which are sums of squares of real vector fields satisfying the Hormander condition. In that case. any small perturbation of the vector fields in the Ck topology, for suitably large k, produces an operator which is still hypoelliptic, even though the perturbation will

not in general be of lower order in any sense.

6 FINAL COMMENTS

For homogeneous. constant-coefficient differential operators in IRd, there is an

equivalence between ellipticity, Coo hypoellipticity. and analytic hypoellipticity. As presented. our results indicate that this equivalence, with ellipticity replaced, of course. by the Rockland condition that Jr(L) be injective for each irreducible unitary representation Jr ,breaks down for the class ofgroups in question. However, the analogy with the abelian case goes further, from an alternative point of view.

Consider again the case of JRd , and let L be homogeneous and elliptic. with constant coefficients. Then, although L will be analytic hypoelliptic. still the Cauchy estimates will be violated ifthe constant C1+lal is replaced by Ce • £Ial for

all sufficiently small £, no matter how large Ce is chosen to be. This amounts to the fact that while the symbol of L is free of zeroes in a conic neighborhood of

IRd C Cd, yet it still does have zeroes.


145 144 CHAPTER 6

Consider now the four-dimensional nilpotent group of step 3 which stood at the centerofour analysis, and let L be a (generic) homogeneous, left-invariant operator which satisfies the Rockland condition. Then we have proved the existence of certain non-unitary representations lr on which lr(L) is not injective in L 2 , an analogue of the fact that every nonconstant polynomial in one complex variable has at least one zero. This means not only that L is not analytic hypoelliptic, but that again, the constant Cl+lal appearing in the Gevrey estimates of order s = 3 for solutions of L cannot be replaced by C, . elal for arbitrarily small e.

It can be shown tha.t for any such operator on this group which satisfies the Rockland criterion, all solutions of Lu == 0 in an open set must belong to the Gevrey class G3 . Thus the situation is quite analogous to that for IRd , provided only that the class G 1 of analytic functions is replaced by G3•

Substantial progress on related problems has been achieved in the interval betwee1! preparation of this manuscript and its publication [C7],[C8]. More detailed information on the nonlinear eigenvalues has been obtained by Ching-Chau Yu in a UCLA PhD dissertation (in preparation).

University ofCalifornia, Los Angeles

REFERENCES

[CI] M. Christ. "On the aequation in weighted L 2 nonns in C I." J. Geom. Anal. 1 (1991),193-230.

[C2] . "Some non-ana1ytic-hypoelliptic sums of squares of vector fields." Bul/. Amer. Math. Soc. 116 (1992), 137-140.

[C3] . "Certain sums of squares of vector fields fail to be analytic hypoelliptic." Comm. Part. Dijf. Eq. 16 (1991), 1695-1707.

[C4] . "Analytic hypoellipticity breaks down for weakly pseudoconvex Reinhardt domains." Int. Math. Res. Not. 1 (1991), 31--40.

[C5] . "A family of degenerate differential operators." J. Geom. Anal. 3 (1993), 579-597.

[C6] . "Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem." Duke Math J. 72 (1993),595--639.

[C7] . "A necessary condition for analytic hypoellipticity." Math. Research Letters 1 (1994),241-248.

[C8] . "The Szego Projection Need Not Preserve Global Analyticity." Preprint. [CG] M. Christ and D. Geller. "Counterexamples to analytic hypoellipticity for domains

of finite type." Ann. ofMath. 235 (1992), 551-566. [CL] E. Coddington and N. Levinson. Theory of Ordinary Differential Equations.

McGraw-Hill, 1955. [FS] G. B. Folland and E. M. Stein. "Estimates for the ab complex and analysis on the

Heisenberg group." Comm. Pure Appl. Math. 27 (1974), 429-522.

ANALYTIC HYPOELLlPTIC DIFFERENTIAL OPERATORS

[G] D. Geller. Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability. Mathematical Notes 37. Princeton University Press, 1990.

[GS] P. C. Greiner and E. M. Stein. "On the solvability of some differential operators of type Db." Several Complex Variables, Proceedings of International Conferences, Cortona, Italy, 1976-77, Scuola Normale Superiore, Pisa, 197, 106-165.

[He] B. Helffer. "Conditions necessaires d'hypoanalyticite pour des opCrateurs invariants agauche homogenes sur un groupe nilpotent gradue." J. Diff. Eq. 44 (1982),

460--481. [HH] N. Hanges and A. A. Himonas. "Singular solutions for sums of squares of vector

fields." Comm. Part. DijJ. Eq. 16 (1991), 1503-1511. [HN1] B. Helffer and J. Nourrigat. "Hypoellipticite pour des groupes nilpotents de rang

3." Comm. Part. Diff. Eq. 3 (1978), 643-743. [HN2] . "Caracterisation des opCrateurs hypoelliptiques homogenes invariants a

gauche sur un groupe nilpotent gradue." Comm. Part. DijJ. Eq. 4 (1979), 899-958. [HN3] . Hypoellipticite Maximal pour des Operateurs Polynomes de Champs de

Vecteurs. Prog. Math. vol. 58. Birkhauser, 1985. [K] A. A. Kirillov. "Unitary representations of nilpotent Lie groups." Russian Math.

Surv. 17 (1962), 53-104. [Ke] M. V. Keldysh. "On the completeness of the eigenfunctions of classes of non

selfadjoint linear operators." Russian Math. Surv. 26 (1971),15-44. [M] G. Metivier. "Hypoellipticite analytique sur des groupes nilpotents de rang 2." Duke

Math. J. 47 (1980), 195-221. [NSW] A. Nagel, E. M. Stein, and S. Wainger. "Balls and metrics defined by vector fields

I: Basic properties." Acta Math. 155 (1985), 103-147. [PRJ Pharo The Lai and D. Robert. "Sur un probleme aux valeurs propres non lineaire."

Israel J. Math. 36 (1980), 169-186. [R] C. Rockland. "Hypoellipticity on the Heisenberg group-representation-theoretic

criteria." Trans. Amer. Math. Soc. 240 (1978),1-52. [RS] L. P. Rothschild and E. M. Stein. "Hypoelliptic differential operators and nilpotent

groups." Acta Math. 137 (1976), 247-320.


------.~----...~----=----=~==~--====--=.:;;~;~==~<~~~=-:=;:;:=~===~=.~"§.t:~~~g;j::"'E~,,~~':£~~~~~~~_==_~.::..'"~~~~~~~~-=--='55i''''''''., ..~~~::==~~~~=~:=,;;;;

7

Operateurs Bilineaires

et Renormalisation

R. R. Coifman, S. Dobyinsky, et Y Meyer

1 INTRODUCTION

Une fois renonnalisees, certaines expressions bilineaires se comportent mieux que prevu et la fonne precisee du lemme du div-curl nous pennettra d'illustrer cette remarque. Cette fonne precisee, conjecturee par P. L. Lions et demontree dans [2], utilise l'espace 'HI (JRn) de E. Stein et G. Weiss. Rappelons que l'espace de Hardy 1{1(JRn) est I'ensemble des fonctions I, appartenant aLI (JRn) , dont les n transfonnees de Riesz R I (f), ... , Rn (f) appartiennent egalement aL I (IRn ). La

nonne de I dans 'HI (lR.n ) est la somme 11/111 + IIRI(f)lh + ... + IIRn (f)lll et, muni de cette nonne, 'HI (lR.n ) est un espace de Banach, compose de fonctions d'integrale nulle. Comme I'ont montre C. Feffennan et E. Stein [4], Ie dual de 'H I (JRn) est l'espace B MOde John et Nirenberg. La fonne precisee du lemme du div-curl est l'enonce suivant:

Theoreme 1. Soient E(x) = (EI (x), ... , En(x» et B(x) = (B I (x), ... ,

Bn(x», X E IRn, deux champs de vecteurs verifiant les proprietes suivantes:

(1.1) Ej(x) E L 2(JRn), Bj(x) E L 2(IRn) pour 1 S j S n

(1.2) div E(x) = 0, curl B(x) = O.

Alors E(x)· B(x) = EI(X)B1(x) + ... + En (x)Bn(x) E 1{1(lRn).

Les derivees qui interviennent dans (1.2) sont prises au sens des distributions. Chacun des tennes Ej(x)Bj(x) du produit scalaire E(x) . B(x) appartient a

OPERATEURS B/LlNEAlRES ET RENORMALlSATlON 147

LI(lRn), sans appartenir a1-{l(lR.n). Si la somme de ces n tennes appartient a 'HI (lR.n), ce fait remarquable est dfi ala presence de cancellations, provenant des hypotheses (1.2). Ces cancellations seront analysees en appliquant un algorithme de renonnalisation achacun des produits Ej(x)Bj(x). Un algorithme de renormalisation pennet d'ecrire Ie produit ponctuel uv entre deux fonctions u (x) et v (x)

sous la fonne

(1.3) uv = P(u, v) + R(u, v)

ou l'operateur bilineaire P, definissant l'algorithme, satisfait les conditions

suivantes:

(1.4) P(u, v) = P(v, u)

(1.5) P(u, v) = 0 chaque fois que la fonction u(x) est une constante

et ou I'operateur bilineaire R possMe les proprietes suivantes:

(1.6) u E L 2 et vEL2 impliquent R(u, v) E 'HI

si u j, j ~ 1, et vj, j ~ I, sont deux suites bomees

(1.7) en nonne L 2 et si Uj --->.. u (j -+ +00), Vj --->.. V (j -+ +00),

alors R(uj, Vj) --->.. U(u, v) (j -+ +00).

Nous avons note Ii --->.. I (j -+ +00) la convergence au sens des distributions: JIi (x)rp(x) dx -+ JI(x)rp(x) dx pour toute fonction de test rp. Dans la conclusion de (1.7), on peut supposer que rp(x) appartienne aI'espace VM 0 qui est la fenneture, pour Ia nonne BMO, de I'espace vectoriel des fonctions continues et nulles aI'infini.

Signalons enfin que 1{1 est Ie dual de V MO. Voici la signification des deux tennes de la decomposition (1.3). Le tenne

P(u, v) est la partie principale du produit uv. Si au lieu d'appartenir aL 2 (JRn),

les deux fonctions u et v appartenaient aLP(lR.n) et si p < 2, alors Ie produit uv

n'aurait plus de sens, en tant que distribution, et it conviendrait de lui soustraire certaines quantites infinies pour retrouver un terme suffisamment oscillant pour etre une distribution temperee. Ces quantites infinies sont presentes dans Ie tenne P(u, v). Le tenne R(u, v) appartiendra al'espace de Hardy 'H p / 2 lorsque u et v

appartiennent aLP et que pest inferieur a2. Le terme R(u, v) est donc Ie produit

uv, renormalise par soustraction de P(u, v).

Cette renonnalisation doit etre la plus simple possible: nous ne voulons soustraire que Ie strict minimum et, la plupart du temps, ne rien soustraire du tout. Si par exemple la fonction u (x) est une constante, Ie produit entre cette constante et une fonction arbitraire vex) n'a pas besoin d'etre renonnalise et P(u, v) sera donc


149 148 CHAPTER 7

nul, comme l'indique la condition (1.5). Cette meme condition (1.5) nous perrnettra de verifier (section 8) que Ie produit entre une fonction u(x) qui est reguliere et asupport compact et une fonction arbitraire vEL2 conduit aune correction P(u, v) faible, au sens que l'operateur qui av associe P(u, v) est compact.

L'operation de renorrnalisation doit perrnettre de multiplier deux distributions temp6rees arbitraires. Elle s'apparente au paraproduit, tel que J. M. Bony l'a defini. La condition (1.7) signifie qu'en se limitant ades suites bornees dans L 2,

la renorrnalisation est compatible avec la convergence au sens des distributions. Signalons enfin qu'il existe plusieurs solutions au probleme de la renorrnalisa

tion du produit. Nous avons choisi celIe, presentee dans la section 3, qui conduit aux calculs les plus simples.

Revenant au theoreme 1, on a automatiquement et sans tenir compte de (1.2),

(1.8) Ej(x)Bj(x) - P(Ej , Bj)(x) E HI (lR") , 1::: j ::: n.

Demontrer Ie tMoreme 1 revient donc a prouver que la somme S(x) = P(EI, BI) + ... + P(En, Bn ) appartient aussi aHI lorsque les conditions (1.1)

et (1.2) sont satisfaites. En fait, on a beaucoup mieux et cette somme S(x) appartient ai'espace de Besov

homogene iJ~' I qui sera defini soigneusement dans la section suivante et qui est un sous-espace "rudimentaire" de HI (lRn

). L' enonce du theoreme 1 prete done a confusion puisque Ie rOle joue par l' espace de Hardy HI (lRn

) y est relativement

super/lCiel. alors que l' espace de Besov iJ?·1 est au CfPur du probleme.

Cette meme analyse (1.3) nous foumira une demonstration particulierement elegante du lemme "classique" du diY-curl, tel que F. Murat et L. Tartar 1'ont enonce. II s' agira de montrer que, sous les hypotheses (1.2), Ie produit scalaire E· B

possede egalement la propriete de continuite faible decrite dans (1.7). Dans la section suivante sont rappeles quelques resultats generaux concernant les op6rateurs bilineaires et l'espace de Besov homogene iJ?·I. Nous pourrons alors construire toute une collection d'op6rateurs P ayant les proprietes (1.4) a(1.7). A I'interieur de cette collection, nous choisirons un exemple explicite, se reliant aux "paraproduits" de J. M. Bony et conduisant aune demonstration particulierement simple du theoreme 1 et du lemme de Murat et Tartar. Nous conclurons en indiquant (sans demonstration) les liens qui existent entre la renorrnalisation et les series d'ondelettes.

2 LA DEFINITION GENERALE DES OPERATEURS BILINEAIRES P

Pour la commodite du lecteur, nous rappelons quelques enonces figurant dans [1] ou [5].

OPERATEURS BlLINEA/RES ET RENORMALISATION

Les symboles bilineaires servant aconstruire les operateurs bilineaires sont des fonctions r(~, TJ) definies sur lRn x lRn prive de (0,0) et verifiant les conditions

usuelles

(2.1) la;affr(~, TJ)I ::: Ca,p(I~1 + ITJ\)-laI-iPI

ou a = (ai, ... ,an) est un multi-indice de longueur lal = al + ... + an et ou

a (a )a l (a)~ a~ = a~1 . .. a~n .

En particulier, les conditions (2.1) sont satisfaites si r(~, TJ) est indefiniment derivable dans lRn x lRn prive de (0, 0) et si r (A~, ATJ) = r (~, TJ) pour tout A > O.

On associe aun symbole bilineaire r (~, TJ) l' operateur bilineaire

(2.2) T(j, g)(x) = (2nT 2n II ei(Hry)·xr(~, TJ)j(~)g(TJ) d~ dTJ

OU j, g sont les transforrnees de Fourier de f et g. Les conditions (2.1) impliquent que T, qui est evidemment defini si f et g

appartiennent ala classe S (lRn ) de Schwartz, se prolonge en un operateur (encore note T), defini sur L2(IRn) X L 2(lRn ), avaleurs dans L I (IRn). Un cas particulier

evident est Ie symbole r = 1 conduisant a T (j, g) = f g. La continuite de T: L 2 X L 2 ~ L I sous les hypotheses (2.1) est donc une generalisation de l'inegalite de Holder. Cela amene aconjecturer que T se prolonge en un operateur

lineaire continu de LP x Lq avaleurs dans L' si 1 ::: r < 00, 1 < p ::: 00,

1 < q < 00 et i + ~ = ~. Ceci est vrai (valeurs limites comprises) et la

demonstration se trouve dans [5]. Une fois pour toutes, les conditions (2.1) seront supposees satisfaites dans les

enonces qui suivent (lemmes 1, 2, et 3). Dans ces conditions, la fonction r sera

appelee Ie symbole bilineaire de l'op6rateur bilineaire T.

Lemme 1. Une condition necessaire et suffisante pour que l' operateur bilineaire

T: L 2 X L2 ~ L I, defmi par (2.2) ait la propriete suivante

(2.3) T(j, g) E H1(IRn) lorsque f E L2 et g E L2

est que son symbole bilineaire r (~, TJ) verifie

(2.4) r(~, -~) = 0 pour tout ~ =1= O.

Les conditions (2.3) et (2.4) sont aussi equivalentes a

(2.5) ( T(j, g)(x) dx = 0 pour toute f E L2 et toute g E L2 JJR.

I


151 150 CHAPTER 7

ou encore acette meme condition (2.5) forsque f et g appartiennent a fa classe de Schwartz.

Ce lemme foumit une demonstration (fausse) du theoreme 1. Pour verifier que E . B appartient a'HI, il suffirait donc de montrer que la condition (2.5) est satisfaite. Or on a B = VU (x) puisque curl B(x) = O. II vient E (x) . B(x) = div[U(x)E(x)] puisque div E(x) = O. Finalement J E(x) . B(x) dx = O. Cette demonstration est insuffisante parce que Ie produit scalaire E . B n'est pas un des op6rateurs bilineaires auxquels s'applique Ie lemme I.

La preuve du lemme I consiste aevaluer l'integrale I = J T(f, g) . u(x) dx lorsque u (x) appartient a vM 0 et que II u II B M 0 ::: 1. Mais on a I = JL(f, u)gdx oil I'operateur bilineaire Lest defini par Ie symbole A(~, lJ) = r(~, -~ - lJ). On a alors IIL(f, u)1I2 ::: ClJfll2l1ullBMo en appliquant Ie theoreme 34 du chapitre VI de [I].

Le lemme suivant n' intervient pas dans la demonstration du theoreme 1. II nous apprendra que tous les choix de l'operateur T que nous ferons sont equivalents.

Lemme 2. En conservant les notations du lemme 1, une condition suffisante pour que T (f, g) appartiennearespace de Besov homogene B?' I (Rn), pour tout couple (f, g) de deuxfonctions de L2 (Rn), est que l' on ait

(2.6) r(~, -~) = 0 pour tout ~ =I 0

et

(2.7) r(t 0) = 0 = r(O, lJ) pour ~ i- 0, lJ i- O.

Avant de passer ala definition de B?,I, un demier resultat doit etre mentionne.

Lemme 3. En conservant les notations du lemme 1, la condition (2.4) implique fa propriete de continuite faible suivante:

si Ii et gj sont deux suites bornees dans L2(lRn) (2.8)

et si Ii ->. f et gj ->. g, alors T(/j, gj) ->. T(f, g).

Le lecteur trouvera dans I'appendice les preuves tres simples de ces deux lemmes.

Rappelons maintenant la definition de I'espace de Besov homogene B?·I .

OPERATEURS B1LINEA1RES ET RENORMALISATION

nOn designe par 1/I(x) une fonction de la classe de Schwartz S(lR ) dont la

transforrnee de Fourier est nulle si I~ I ::: 2/3 ou si I~ I ~ 8/3 et verifie, si ~ =I 0,

o < c ::: L00

1~(2-j~)I.(2.9) -00

jOn designera alors par 1/1j (x) la fonction 2nj 1/1 (2 x) et par /:!" j l'operateur de

convolution avec 1/1j' Vne fonction f ELI (Rn ) appartient aB?' I si et seulement si

L00

lI/:!"j(f)111 < 00(2.10) -00

et cette condition ne depend pas du choix des operateurs /:!" j' Vne seconde definition equivalente s'obtient en decomposant f(x) dans une

base orthonorrnee d'ondelettes 1/IA' A E A, de regularite r ~ l. Alors f E B?,I si et seu1ement si

(2.11) f(x) = L UA1/IA(X) et L luAI1I1/IAlii < 00.

AEA AEA

Selon la terrninologie de G. Weiss et de ses collaborateurs, (2.11) s'appelle une

decomposition atomique en "atomes speciaux." Les operateurs bilineaires P que nous utiliserons dans la preuve du theoreme 1

seront definis par des symboles bilineaires 1f(~, lJ) verifiant les conditions

suivantes:

(2.12) 1f(~, lJ) = 1f(lJ,~)

(2.13) 1f(~, 0) = 1f(0, lJ) = 0 si ~ i- 0 et lJ i- 0

(2.14) -~) si ~ i- O.1f(~, = I Compte tenu des lemrnes I et 3, ces proprietes entrainent les conditions (1.4) a

(1.7). En outre, si deux symboles bilineaires 1f1 et 1f2 possedent ces proprietes, alors

1f3 = 1f1 - 1f2 verifiera les conditions suffisantes du lemme 2. eela entrafne que, modulo B?,I, tous les choix de P que nous ferons seront

equivalents.

3 RENORMALISATION ET PARAPRODUITS

Le but de cette section est de donner une demonstration complete du theoreme 1 en utilisant un choix de l'operateur P qui se reliera naturellement ala theorie du

paraproduit de J. M. Bony.


153 152 CHAPTER 7

Pour definir P on utilise la decomposition classique de l'espace de Fourier jRn \ {OJ en couronnes dyadiques ainsi que l'analyse de Littlewood-Paley-Stein associee a cette decomposition.

On designe pour cela par 1{! (x) une fonction a valeurs reelles, appartenant a la classe S(jRn) de Schwartz et dont la transformee de Fourier verifie les deux conditions

(3.1) ;j,(~) = 0 si I~I ~ 2/3 ou I~I ~ 8/3

et

00(3.2) L 1;j,(rj~)12 = 1 pour tout ~ ¥ o.

-00

Compte tenu de (3.1), la condition (3.2) se reduit a

2 4(3.3) 1;j,(~)12 + 1;j,(2~)12 = 1 si _ < lei < -.3 - ., - 3

On pose ensuite 1{!j(x) = 2nj 1{!(2j x), j E Z, et ron designe par!i. j 1'0¢rateur de convolution avec 1{!j. La condition (3.2) signifie que

00 (3.4) I = L !i.j!i.j

-00

ou !i. j est l'adjoint de !i. j'

On pose finalement, si / et g appartiennent a L 2(jRn),

00 (3.5) P(j, g) = L(!i.J!)(!i.jg).

-00

II est trivial de verifier que P (j, g) appartient a L I(jRn) puisque l'on a

1I(!i.J!)(!i. j g)1I1 ~ 1I!i.J!11211!i.j gIl2 = aj{3j

etqueL~00lajl2 = IIf11~'L~001{3jI2 = IIgll~,gracea(3.2). Onadonc

(3.6) IIP(j. g)1I1 ~ 1If112I1g!l2.

Le symbole bilineaire de P est Jr(~, 11) = L~oo ;j, (2- j ~);j, (2- j 11). II verifie evidemment (2.12). La condition (2.13) provient de (3.1) tandis que (2.14) decoule de (3.2) et de ;j, (-~) = ;j, (~).

Pour demontrer Ie theoreme 1, on ecrit chaque produit E1(x)BI(x), ... , En(x)Bn(x) sous la forme P(Ej, Bj ) + R(Ej , Bj ). On a R(Ej , Bj ) E HI et l'amelioration du theoreme 1 que nous avons en vue est Ie fait que

. 0 I (3.7) Sex) = P(EI, B» + ... + peen, Bn) E BI '

OPERATEURS BILlNEAlRES ET RENORMALlSATlON

Ona 00

Sex) = L(!i.jE). (!i.jB) -00

et nous allons, en designant par II . II la norme dans l'espace de Besov iJ?,I,

demontrer Ie theoreme suivant:

Theoreme 2. En consefllant les notations precedentes et les hypotheses du

theoreme 1, on a

(3.8) 1I(!i. j E) . (!i. j B)1I ~ C(n)lI!i. j EII211!i. j BII2

ou C (n) ne depend que de la dimension n.

II resulteevidemment de (3.8) que Sex) E iJ?,I. Eneffet. on a L~oo lI!i. j EII~ = II E II~ et L~oo 1I!i. j B II~ = II B II~ et cela entraine

00 L II (!i.jE) . (!i. j B)1I ~ C(n)IIEII2I1BII2· -00

Pour etablir (3.8), nous utiliserons Ie lemme suivant:

Lemme 4. Soit lex) une /onction de/mie sur lRn, localement integrable, et

possedant les deux proprietes suivantes:

1 n(3.9) V/= (-a f , ... ,-/a) EL(lR) aXI aXn

(3.10) / = div F ou F = (Flo ... , Fn ) ELI (lRn ).

Alors / (x) appartient a iJ?,1 (lRn) et l' on a

(3.11) II/II ~ C(n)[II V/IIII1 F lllfl2.

On a pose IIF[1t = 11F11i1 + ... + IlFnlll et IIV/1I1 est definie de meme. La preuve elementaire du lemme 4 est renvoyee a la section suivante.

Pour demontrer Ie theoreme 2, nous appliquerons Ie lemme 4 a la fonction

hex) = (!i.jE) . (!i.jB). La transformee de Fourier de h est nulle hors de la

boule I~ I ~ ¥2j ce qui entraine

16 . 16 . (3.12) IIV/jll l ~ 32JllfJIII ~ 32JII!i.jEII211!i.jBII2.

grace au lemme de S. Bernstein dont nous rappelons l'enonce.


155 154 CHAPTER 7

Lemme 5. Si la transformee de Fourier f(~) d'unefonction f E LP(lRn), I < p ~ 00, est nu/le hors de la boule 1~1 ~ R, alors on a

(3.13) IIV flip :s Rllfll p

et si, en revanche, fest nu/le sur la boule I~I < R, l'inegalite inverse de (3.13) est verifiee au sens suivant:

(3.14) Rllfll p ~ C(n) IIV' flip

ou C(n) ne depend qne de la dimension n.

On observe ensuite que curl B(x) = °signifie B(x) = Vu(x) ou la fonction (scalaire) u (x) est unique, modulo les fonctions constantes. Alors (3.14), appliquee Ii la fonction I::J.ju, s'ecrit

(3.15) lII::J. j (u)1I2 ~ CTjllI::J.jBI12.

L'hypothese div E (x) = °entraine

(3.16) fj(x) = div Fj(x)

ou Fj(x) = (I::J.ju)(I::J.jE) et IIFjl1I ~ C2- j llI::J. j BII211I::J. j EII2Le lemme 4 entraine donc Ie theoreme 2.

4 LA PREUVE DU LEMME 4

On observe tout d'abord que, pour tout a > 0, les fonctions f(x) et an f(ax) ont la meme norme dans iJ?·I. En choisissant convenablement a > 0, on peut se

ramener au cas ou II Vfill = IIF III . Pour demontrer Ie lemme 4, il suffit alors de verifier que, pour tout j ~ -I,

on a

(4.1) II I::J. j(f)111 ~ C2 j IIFIII

tandis que, pour tout j ~ 0, on a

(4.2) III::J. j (f) II I ~ CTjIlVfII1'

Ces deux estimations resultent des deux assertions du lemme de Bernstein. Pour etablir(4.1), on observe que f = div FetqueI::J.j(f) = div(I::J.j(F». Onapplique alors (3.13) Ii chacune des composantes du vecteur I::J. j (F) et 1'on obtient (4.1).

Pour verifier (4.2), on utilise l'inegalite inverse (3.14) que l'on applique aI::J. j (f). Ceci tennine la preuve du lemme 4.

OPERATEURS BILlNEAIRES ET RENORMALISATION

5 LE LIEN AVEC LE PARAPRODUIT DE J. M. BONY

Pour definir Ie paraproduit, on part d'une fonction radiale, lp(x), appartenant Ii la classe de Schwartz S(]Rn), avaleurs reelles et dont la transfonnee de Fourier fjJ(~)

verifie les conditions

(5.1) fjJ(~) = ° si I~I ~ 4/3

et

(5.2) fjJ(~) = I si I~I ~ 2/3.

On pose alors lpj(x) = 2nj lp(2 jx) et l'on designe par Sj l'operateur de convolution avec lpj. On definit 1/I(x) par 1/I(x) = 2nlp(2x) - lp(x), on pose

1/1j (x) = 2nj 1/1 (2 jx) et l'operateur de convolution avec 1/1jest Dj = Sj+ I - Sj. Soient f et g deux fonctions arbitraires, appartenant aL2(]Rn). On definit alors

PI (f, g) par

(5.3) PI(f, g) = L L (Djf)(Djg). lj-kl::::2

Alors on a l'identite remarquable

(5.4) fg = PI(f, g) + R;(f, g) + R~(f, g)

ou 00

R;(f,g) = L(Sj_2!)(Djg) -00

et 00

R~(f, g) = L(Sj-2g)(Dd) = R; (g, f) -00

sont les paraproduits definis par J. M. Bony. Le symbole bilineaire Jrl (~, 1/) de PI vaut

00 00

1- L fjJ(T(j-2)O~(Tj1/) - L fjJ(T(j-2)1/)~(Tj~) -00 -00

et verifie les conditions (2.12), (2.13), et (2.14). Cela signifie que l'operateur P que nous avons utilise dans la section 3 peut etre

remplace par PI et que l'erreur commise appartiendra a iJ?' I .

6 LE LEMME DE MURAT ET TARTAR

Rappelons tout d'abord l'enonce classique du lemme du diY-curl [2].


------------------

156

..------.~.=~~----.---- ~~~-.~~.-~.~ ..~-==~= ......."""_"'"~~"""'-''''~''''~=~'T'"~','''..:-:."="''~''''.:;=_'''~':..~.z'"~~~~~""'='''''.~'~~=:''='-''._-=7=~-:.7"'''".:..;.:.~_=;~,~ __=_.~.z...==:_~'''":..·~~._ ,,_.=;;;~.;~,,~~~.-_-"="=.=~"'''''':.'''~:=,~..:;::.:...:;~=_""~_-~

CHAPTER 7

Soient E(m) et B(m), mEN, deux suites de champs de vecteurs verifiant

(6.1) IIE(m)1I2 ~ Co, IIB(m)112 ~ Co

ainsi que

(6.2) E(m) -----" E, B(m) -----" B (m ~ +(0).

Supposons, d'autre part, que div E(m) = 0 et curl B(m) = O. Alors on a

Lemme 6. Sous les hypotheses precedentes,

(6.3) lim f E(m) . B(m)(x)u(x) dx = f E(x) . B(x)u(x) dx m~oo

pour toute/onction de test u(x), continue et nulle a /'in/mi.

II suffit evidemment de demontrer (6.3) lorsque u(x) est une fonction de classe C I , a support compact.

On utilise pour cela l'optSrateur P de la section 3. La propriete de continuite faible est automatiquement satisfaite pour les restes R(E(m), B(m), grace a (1.7), et il suffit donc de considerer p(E(m), B(m).

On pose

It) = f (lljE(m) . (lljB(m)u(x) dx

et I'on doit verifier que

00 00

(6.4) lim "I~m) = " I. m~+oo~ J ~ J

-00 -00

ou

I j = f(lljE) . (lljB)u(x) dx.

Pour cela, il suffit d'etablir l'existence de deux constantes C, et C2 telles que

(6.5) IIt)j ~ C,Tj si j::: 0

(6.6) IIjm) I ~ C22nj si j ~ O.

La condition (6.2) implique que, pour tout j fixe, on a

~n ~ IJ~)=~ m~+oo

et (6.4) resultera de (6.5), (6.6), et (6.7). Pour etablir (6.6), on ecrit

(6.8) IljE(m)(x) = f l/tj(X - t)E(m)(t) dt

157OPERATEURS BILlNEAIRES ET RENORMALlSATJON

et I'on a

Co2nj /2II ll j E(m)lIoo ~ 1I1/1jIl2I1 E (m)1I2 ~ .

Cela permet de majorer IIt)1 par Cfj2nj lIu III. Pour demontrer (6.5), on observe que

IljE(m) . IljB(m) = div[(llju(m), (lljE(m)](6.9)

et I'on a donc, apres integration par parties,

(6.10) Ijm) = - f (llju(m))(lljE(m) . Vu) dx.

On majore finalement cette integrale par IIllju(m)11211 IljE(m) 11211Vu1l 00 • Or,

comme nous l'avons deja note,

II ll ju(m) 112 ~ CITj et IIlljE(m)1I2 ~ Co.

Les proprietes (6.5) et (6.6) etant demontrees, il reste a examiner (6.7). Or la convergence faible des E(m) implique la convergence simple des fonctions

Il j E(m) (x). On a meme convergence unifonne sur tout compact. Puisque u(x)

est a support compact, (6.7) en resulte. Le lemme de Murat et Tartar est donc demontre.

7 LA RENORMALISATION PAR LES ONDELETTES

Considerons une base orthonormee, pour l'instant arbitraire, 1/IA(X), A E A, de L 2 (]Rn). Soit u(x) une fonction de L 2(]Rn). On a alors u(x) = LAEA aA1/IA(x) et il en decoule que

2(7.1) lu(x)12 = L laAI 11/1A(X)12 + w(x) AEA

OU w(x) E LI(]Rn) et f w(x) dx = O. On peut aussi considerer deux fonctions arbitraires u(x) et v(x) de L 2 (]Rn) , les

decomposer dans la base orthononnee 1/IA' A E A, et ecrire

(7.2) u(x)v(x) = L a APAI1/IA(X)!2 + w(x) AEA

OU, la encore, w(x) ELI (]Rn) et f w(x) dx = O. En generalla fonction w(x) n'a pas d'autre propriete. Si I'on considere l'ex

emple du sysreme trigonometrique (2Jr)-1/2eikx sur l'intervalle [-Jr, Jr] au lieu

de (1/IA) sur ]Rn, alors w(x) = u(x)v(x) - c ou c est une constante et w(x) est donc n'importe quelle fonction de L I [-Jr, Jr] d'integrale nulle.


_ ~•••~ __c ,_._. _ ,,_ ___" __,,_ •• _'"''-''~;;''' ""-'ii£"'" ,""""'-'.',:""'''i'.i'''","''''''''''''.'''''<=~.'~ """_=,,," .-""'."-","","~" =-,",,'_C' __C'~~'>;"~~~~==-====--"='~:~':=-";'-~~~~d"1..~~:::"~~~~~"~¥;:;:·~~j;."'-~~"'-.2,OE"s,;:""","?_"'·;i."""",~'"''''''''''~ .-~._'~-".='.=='-'--'-"-~"------'---""-~'-'-

159158 CHAPTER 7 OPERATEURS B1LlNEAlRES ET RENORMALlSATJON

Soit maintenant VrA' A E A, une base orthononnee d'ondelettes reelles. On se reportera a[5] pour y trouver une description detaillee de ces bases. Excluons Ie cas particulier du systeme de Haar et supposons que la regularite r des fonetions VrA ne soit pas nulle (r ::: 1 dans les notations de [5]).

Alors la fonction w(x) precedente appartient aHI des que u et v appartiennent aL2(lR.n ).

On se limite dans l'enonce qui suit a des fonctions avaleurs reelles et, si u et v sont deux fonetions de L 2(]Rn), on pose

(u, v) = f u(x)v(x) dx.

Alors on a [3].

Theoreme 3. Soit VrA' A E A, une base orthonormee de L2(W), d' ondelettes de

regularite r ::: 1. Si u et v appartiennent aL 2 (]Rn), posons

P2(u, v) = ~)u, VrA)(V, VrA)IVrA(X)1 2. AEII

Alors eet operateur bilineaire P2 possede les propriites (1.4), (1.5), (1.6) et (1.7). . 0 I 2 2

De plus P2(U, v) - PI (u, v) E B I ' si u E L et vEL.

L'operateur P2 fournit donc la renormalisation la plus simple possible du produit

UV, c'est-a-dire celIe ou 1'0n soustrait Ie moins de tennes possible. Cet operateur peut etre utilise pour demontrer la forme precisee du theoreme 1. Cependant eette

demonstration, que 1'0n trouvera dans [3], est techniquement plus compliquee que celIe que nous venons de presenter. Les diffieultes techniques viennent du fait que 1'0perateur P2 ne verifie pas la fonnule de Leibniz qui est satisfaite par Ie produit usuel et par les operateurs P et PI. En d'autres tennes, P2 n'appartient pas ala

classe generale des operateurs bilineaires definis par (2.2).

8 APPENDICE (PREUVES DES LEMMES 2 ET 3)

Reprenons, pour demontrer Ie lernme 2, les notations de la section 5. On a done

00

T(u, v) = L[T(Sj+lu, Sj+lv) - T(Sju, Sjv)] -00

00 00 00

= LT(Dju, Sjv) + LT(Sju, Djv) + L T(Dju, Djv). -00 -00 -00

Nous allons, en designant par II . lila norme de l'espace de Banach n?'\ etablir

que

(8.1) L00

IIT(D j u, Sjv)1I :s CIIull21lvII2 -00

et qu'it en est de meme pour les deux autres series. Les trois preuves etant semblables, nous nous limiterons a (8.1). Obser

vons que T(Dju, Sjv) = T/u, v) ou Ie symbole bilineaire rj(~, 17) de Tj est

Vt(2- j ~)fP(2- j 17)r(~, 17). Sur Ie support de rj, on a necessairement 1171 :s ~ 2j et

12j :s I~ 1:s ~ 2j . Cela impIique

2 . . (8.2) '3 2J :s I~ 1+ 1171 :s 4.2J

si bien que l'on a

(8.3) lara~rj(~, 17)1 :s Ca.tJr(\al+ltJl)j.

Les symboles r/~, 17) s'ecrivent donc rj (~, 17) = (}j (2- j ~, 2- j 17) ou les fonc

tions ()j (~, 17) sont portees par I~ 1+ 1171 :s 4 et verifient, ainsi que toutes leurs

derivees, des estimations uniformes en j. Par hypothese r s'annule si ~ = 0 ou si 17 = 0 ou si ~ + 17 = O. 11 en est de

meme de rj et de (). On peut done ecrire

n n

. (}j(~' 17) = L L(~k + 17k)17I(}?,I)(~, 17) I 1

ou les symboles (}tl)(~, 17) sont portes par I~I + 1171 :s 4 et verifient, ainsi que

leurs derivees, des estimations uniformes en j.

Finalement on a n n

(8.4) rj(~' 17) = 4- j L L(~k + 17k)17I(}tl)(rj~. rj17). 1 1

Pour tenniner cette analyse de rj(~' 17), on tient encore compte du support de

rj(~, 17) pour ecrire rj(~' 17) = rj(~, 17)q(2-j~)p(2-j17) oil p et q sont deux fonctions de la classe V(lR.n ) de Schwartz et ou q est nul au voisinage de O.

On a, en revenant al'operateur T j ,

(8.5) Tj(u. v) = 2- j div Lj(uj, Vj)

OU jUj(~) = u(~)q(2-j~), Vj(17) = r 17IP(rj 17)v(17)

et ou l'operateur L j est defini par Ie symbole (}tl)(2-j~, 2- j 17). On a done,

unifonnement en j, IlLj(f, g)lh :s CIIf112IlgI12'


161 160 CHAPTER 7

La preuve du lemme 2 se termine en appliquant Ie lemme 4, comme nous I'avons fait dans la section 3. n vient

(8.6) IITj(u, v) II S Cllujll211vjll2

et I'on concluten observant que L lIujlj~ S Cllull~ etque L IIvjll~ S CIIvll~. Venons-en ala preuve du lemme 3. EIle repose sur 1'0bservation suivante:

Lemme 7. Soit R un nombre positij et K(x, y), x E JRIl, Y E JRIl, un noyau possMant les propriet~s suivantes:

(8.7) K(x, y) = 0 si Ix - yl > R

pour tout Z E JRIl, on designe par B(z) la boule de JRIl X JRIl de centre (2, z) et de rayon 2R et l' on demande que

(8.8) [( IK(x, Y)1 2dx dy tende vers 0 quand Izl ---+ +00.JJB(Z)

Alors l' operateur T defini par Ie noyau K est compact.

Pour demontrer ce lemme, on pose, pour tout m 2: I, Km(x, y) = K(x, y) si Ixj s met Iyl S met Km(x, y) = 0 sinon. Alors I'operateur Tm definit par ce noyau Km est un operateur de Hilbert-Schmidt. A ce titre Tm est compact. Par aiIleurs la nonne de T - Tm tend vers 0, grace ala condition (8.8), quand m tend vers l'infini. Donc T est compact.

Revenons au lemme 3. n s'agit de verifier que

(8.9) ,lim (T(fj, gj), u) = (T(f, g), u) J-++OO

pour une classe convenable de fonctions de test u. Nous supposerons, parexemple, que uappartient Ii la classe de Schwartz V(JR").

Alors (T(f, g), u) = (L(f, u), g) et Ie lemme 3 sera demontre si nous verifions que 1'0perateur defini par f Ho L(f, u) est compact.

En passant aux transfonnees de Fourier, Ie noyau K(~, 1]) de cet operateur est u(-~ - 1])r(~, 1]). Ce noyau verifie evidemment (8.7). Puisque r(~, -~) = 0, on a, grace Ii (2.1), K (~, 1]) = O((l~ I+ l17f)-l) quand I~ I + 1171 tend vers l'infini, ce qui entraine (8.8).

Nous concluons cet appendice en reliant les algorithmes que nous avons utilises dans la section 3 au celebre theoreme T(l) de David et Journe [5]. La demonstration de ce theoreme repose sur l'existence, pour toute fonction b(x) E BMo(JRIl) , d'un op6rateur de Calder6n-Zygmund Tb tel que Tb(l) = b, (n(l) = 0 et qui soit borne sur L2 (JRIl). Cette demiere condition fait partie de la definition des op6rateurs de Calder6n-Zygmund que ron trouvera dans [I] ou [5].

OptRATEURS BILlNtAlRES ET RENORMAUSATION

On a designe par (L Ie transpose de 1'0p6rateur L; Ie noyau-distribution de (L est

L(y, x) si celui de Lest L(x, y). Considerons alors, en revenant aux notations de la section 3, 1'0perateur Sb

defini, si b E BMO(JR"), par 00

(8.10) Sb(f) = L(~jb)(~j/). -00

Cet op6rateur est borne sur L2 (JRIl) lorsque b appartient Ii 8Mo(JRIl) , c'est un

op6rateur de Calder6n-Zygmund et ron a Sb(l) = O. On a JJR' Sb(f)(X) dx = L~oo J(~jb)(~j/) dx = J bf dx puisque L~oo ~j~j = I. Cela signifie que (Sb(l) = bet que Sb est Ie transpose de l'op6rateur Tb que ron cherche Ii construire lorsqu'on demontre Ie theoreme T(I).

Lorsque b(x) est une fonction de classe C 1, Ii support compact, on a lI~j(b)lIoo S C2- j si j 2: 0 et lI~j(b)lIoo S C'2"j si j S O. En fait C' = IIVrlloollulll. En outre, chaque tenne (~jb)(~j/) qui figure dans (8.10)

est un op6rateur compact de L2 dans L2• 11 en resuIte que Sb: L2 ---+ L 2 est compact. Puisque la nonne IISbll de Sb: L 2 ---+ L 2 est majoree par CIIbIl BMo, I'op6rateur Sb est encore compact lorsque la fonction b(x) appartient Ii VMO.

Comme nous l'avions annonce dans I'introduction, Ie produit uv entre une fonction reguliere u et une fonction arbitraire vEL2 (JRIl) n'a pas besoin d'etre renonnalise. Cette affinnation ne peut etre prise au sens strict car si P(u, v) etait nul chaque fois que u appartient Ii la classe de Schwartz V(JRIl) et que v appartient Ii L2(JRIl), alors P(u, v) serait, par densite, nul si u et v appartiennent Ii L2(JRIl).

Mais la regularite de u inftue sur l'importance de la renormalisation et Ie sens precis Ii donner Ii cette assertion est que 1'0p6rateur qui av associe P(u, v) est

compact si u appartient Ii VMO.

Yale University University ofParis, Dauphine University ofParis, Dauphine

REFERENCES

[1] R. Coifman et Y. Meyer. Au dela des operateurs pseudo-dijferentiels. Asterisque 57, 2eme edition, Societe Mathematique de France (1978).

[2] R. Coifman, P. L. Lions Y. Meyer, et S. Semmes. Compacite par compensation et espaces de Hardy. CRAS Paris, tome 309, serie I (1989), 945-949.

[3] S. Dobyinsky. These du CEREMADE. [4] C. Fefferman and E. Stein. HP spaces of several variables. Acta Math. 129 (1972),

137-193. [5] Y. Meyer. Ondelettes et operateurs. Tomes I, net III. Hermann, 1990.


163

8

Numerical Harmonic Analysis

R. R. Coifman, Y Meyer, and V Wickerhauser

INTRODUCTION

The purpose of this chapter is to describe recent developments involving the numerical implementation of methods from classical harmonic analysis in signal processing and computational P.D.E.

As an example, Littlewood-Paley theory, in which a function or a Fourier multiplier is analyzed by partitioning the frequency space in dyadic blocks, has recently been translated into a powerful numerical tool through expansions in orthonormal wavelet bases. (See [1], [2].)

In this numerical setting one sees a general Calder6n-Zygmund operator or IJID.O. as given by an "almost" diagonal matrix having a simple analysis and being implementable by fast numerical algorithms (Le., algorithms of complexity eN log N, N = number of discretization points). Pseudo-differential calculus is translated into an efficient numerical calculus in which smoothing operators are represented by "small" matrices of low numerical rank (see [1]) permitting its use in explicit calculations of solutions to P.O.£. In particular, we can obtain a fast algorithm for the numerical computation of the Green's function for a variable coefficient Laplacian (with smooth coefficients).

In this exercise of translation of methods and ideas from harmonic analysis into fast computational algorithms, one soon realizes that the ability to implement efficiently an integral operator applied to a function is equivalent to a good understanding of the interaction between geometry of the underlying space and cancellation properties of the operator. In the particular case ofCalder6n-Zygmund operators we see an efficient computational algorithm as being a translation of the method of proof of the T(l) theorem of David and Joume. For the case of fractional integrals and operators of potential theory, the need to come up with

NUMERICAL HARMONIC ANALYSIS

efficient computations has led V. Rokhlin to the independent discovery of various versions of Calder6n-Zygmund theory as embodied in his multipole algorithms.

As it turns out, in this case, the question of fast computation is more elementary than boundedness on L 2 or other spaces. It leads directly to issues of geometry of interactions and cancellations.

This interaction between harmonic analysis and a number of concrete problems in applications, such as signal processing and computations, has opened a number of new fundamental questions in analysis.

Our goal is to describe some of these problems on a few simple examples. We start with a fundamental question of signal processing, the question of compression of a signal. Stated simply, given a function (or more precisely, a vector which is a sampled function) one would like to represent the function with as few parameters as possible (here a representation is always assumed to have a given fixed precision). Such a representation could be given in terms of expansion coefficients, Fourier, Taylor, etc., or by stating that the function solves an equation which is easy to describe (say by giving coefficients of a differential equation). The ability to represent a function simply with few parameters is not only desirable in applications for storage purposes, it is also a test of our understanding of the structure of the function and its numerical complexity. Traditionally, the first attempt to represent a signal (or a function not described analytically) would be to expand the signal in a Fourier series, or in terms of some other orthogonal (or non orthogonal expansion). This leads to a variety of problems familiar to all analysts. Assume that a smooth function is supported on a number of disjoint intervals. It is "clear" that separate Fourier expansions restricted to these intervals will be much more "efficient" than a single expansion on the union. The actual answers are not so obvious since some intervals could be close to each other and the term efficient has not been defined. We see that we are confronted with the issue of selecting an optimal expansion inside a class of possible expansions. This leads naturally to the concept of a library of orthonormal bases, as well as to precise definitions of efficiency of an expansion.

DEFINITION OF MODULATED WAVE FORM LIBRARIES

We start by observing that it is impossible to construct an orthogonal basis ikxby localizing smoothly e . This is clear for the case of two adjacent win

dows WI (x), W2(X) since the requirement of orthogonality between WI (x)eikx and W2 (x )eijx implies that

f WI (x)eikxw2(x)ei(k-j)xdx = 0


165 164 CHAPTER 8

which implies WI (x )W2(X) == °(if it is supported in an interval of length smaller than 2:rr).

Recently Daubechies, Jaffard, and Journe, as well as Malvar, observed that by taking equal windows and sines or cosines orthogonality can be maintained. It was observed in [3], [5] that the windows can be chosen to different sizes enabling adaptive constructions. (See Figures 5, 6.)

We start by defining this library of trigonometric waveforms. These are localized sine transforms associated to covering by intervals of R (more generally, of a manifold).

WeconsideracoverR = U~ooI;, 1= [Cli, Cl;+d,Cl; < Cl;+), writei; = Cl;+lCl; = 11;\ and let p;(x) be a window function supported in [Cl; - ii-II2, Cl;+l + i;+1/2] such that

L00

p7(x) = 1 -00

and

pt(x) = I - pt(2a;+l - x) for x near Cl;+l

then the functions

2 :rr Sik(X) = r,:;-;;-:- p;(x) sin[(2k + 1) - (x - Cl;)]'v2ii 2l;

form an orthonormal basis of L2(R) subordinate to the partition p;. The collection of such bases forms a library of orthonormal bases.

It is easy to check that if HI, denotes the space of functions spanned by S;,k,

k = 0, 1,2, ... then H li + Hli+' is spanned by the functions

2 :rr P(x) sin[(2k + 1) (x - Cl;)]

J2(i; + ii+l) 2(i; + ii+d

where

p 2 = p7(x) + P7+1 (x)

is a "window" function covering the interval I; U /;+1. This fundamental identity permits the useful implementation of the adapted window algorithm described in Figure l. (Other possible libraries can be constructed. The space of frequencies can be decomposed into pairs of symmetric windows around the origin, on which a smooth partition of unity is constructed. Higher dimensional libraries can also be easily constructed-as well as libraries on manifolds-leading to new and direct analysis methods for linear transformations.)

-1

-1

o


-1

-1

Figure 1. Local trigonometric waveforms.

RELATION TO WAVELETS-WAVELET PACKETS

WeconsiderthefrequencylineRsplitasR+ = (0, oo)unionR- = (-00,0). On

L 2(0, 00) we introduce a window function p(~) such that L~-oo p2(2-k~) = I

and p(~) is supported in (3/4,3). Clearly we can view p(2-k~) as a window function above the interval (2k , 2k+l) and observe that

k

[1 (~ - 2 )]sin (j + 2'):rr ~ p(Tk~) = Sk,j

form an orthonormal basis of L2(R+). Similarly,

k I (~ - 2 )]Ck.j = cos (j + 2'):rr ~ p(Tk~)

[

gives another basis. If we define Sk,j as an odd extension to R of Sk,j and Ck,j

as an even extension, we find Sk,jJ..Ck',jl permitting us to write Ck,j ± iSk,j = e±;j1r~/2k 'if;(~ /2 j ) where 'if;(~) = e;1r/~ p(~) is the Fourier transform of the base

wavelet \11 (see [4]). We therefore see that wavelet analysis corresponds to windowing frequency

space in "octave" windows (2k, 2k+1).


167

=~_='~=_~c ._._~~~~~~:~~

166 CHAPTER 8

15

-1

5

1

~o 1

-1 1 -1

I ·.1

Figure 2. Local trigonometric wavefonns.

A natural extension, therefore, is provided by allowing all dyadic windows in frequency space and adapted window choice. This sort of analysis is "equivalent" to wavelet packet analysis.

The wavelet packet analysis algorithms permit us to perform an adapted Fourier windowing directly in the time domain by successive filtering of a function into different regions in frequency. The dual version of the window selection provides an adapted subband coding algorithm.

This new library of orthonormal bases constructed in the time domain is called the wavelet packet library. This library contains the wavelet basis, Walsh functions, and smooth versions of Walsh functions called wavelet packets. (See Figure 7.)

We'll use the notation and terminology of [5], whose results we shall assume. We are given an exact quadrature mirror filter h (n) satisfying the conditions of

Theorem (3.6) in [5], p. 964, Le.,

L h(n - 2k)h(n - 2£) = lh,e. Lh(n) = h. n n

We let gk = hl- k(-ll and define the operations F; on £2(Z) into "£2(2Z)"

(1.0) Fo{sd(i) = 2 L Sk hk-2i


Figure 3a. Wavelet packet library.

Fdsd(i) = 2 L Skgk-2j·

The map F(Sk) = FO(Sk) EB F1(Sk) E £2(2Z) EB £2(2Z) is orthogonal and

(1.1) F;Fo + FtFI = I.

Figure 3b. Schematic Description.


---- -

169

__ ...,._u _ __".,_,._. ... ~._, .'_._,- _. ..'_. -". "~·~jiy.'jf~~~"'~5..7atj~'AW;~f'...""·:;"',n·7't'~;''-._~;;,:,,,,,":,:~n~!;;l>,~''__;~!:;~\1;"':{i,"f,~'r"":·:'-~'"'·"·"'·\'~~,1·'(>:"o.;·'·';'ii';.'; _·"" ..1"",·.·"''''>·.;,... ,.":'=,'',·,:·,'-,,,...·.~·.. _.~",''.,,·""~""",,,,"",", ..-~.".~,.~,.,~,=.,;,"~

168 CHAPTER 8 NUMERICAL HARMONIC ANALYSIS

We now define the following sequence of functions. Proposition. Any collection ofindices (i, n) such that the intervals [21n, 21n + 1) form a disjoint cover of [0, (0) gives rise to an orthonormal basis of L2.1

W2n(X) = hL:hkWn(2x - k) (1.2)

{ (These intervals correspond to the partition of frequency space alluded to in §1.)W2n+I(X) = hLgkWn(2X - k). Motivated by ideas from signal processing and communication theory we were

Clearly the function Wo(x) can be identified with the scaling function f{J in D and led to measure the "distance" between a basis and a function in terms of the WI with the basic wavelet 1/1. Shannon entropy of the expansion. More generally, let H be a Hilbert space.

Let us define mo(~) = h L hke-ik~ and Let v E H, Ilvll = 1 and assume

- 1 ml (~) = _ei~ mo(~ + Jr) = - L gkeik~. H = ffi L Hi

v'2 an orthogonal direct sum. We define

Remark. The quadrature mirror condition on the operation F (Fo, FI) is 2equivalent to the unitarity of the matrix e (v, {Hi}) = - L IIvill 2enllvdl2

M _ [mo(~) ml(~)] as a measure of distance between v and the orthogonal decomposition.

- mo(~ + Jr) ml(~ + Jr) . e2 is characterized by the Shannon equation which is a version of Pythagoras'

theorem.

Taking the Fourier transform of (1.2) when n = 0 we get Let

H = ffi (L Hi) ffi (L H j )Wo(~) = mo(~ /2) Wo(~ /2)

= H+ ffi H_.i.e., 00

Hi and H j give orthogonal decompositions H+ = L Hi, H_ = L H j • ThenWo(~) = nmo(~ /2 j

)

j=1 e2 (v; {Hi, H j }) = e2 (v, {H+, H_}) and

A A 3 2 2 WI(~) = ml(~/2)Wo(~/2) = ml(~/2)mo(~/4)mo(~/2 ) ... + IIv+1I e (11::11 ,{Hi})

More generally, the relations (1.2) are equivalent to 2 2 ( v_ )

00 +lIv-lIe IIv_II,{Hj }.

(1.3) Wn (~) = nmej (~/2j) j=1 This is Shannon's equation for entropy (ifwe interpret II PH+ v 11 2 as the "probability"

,",00 0-1 of v to be in the subspace H+, as in quantum mechanics). and n = L.j=1 e j 2J (ej = 0 or 1).

This equation enables us to search for a smallest entropy space decomposition The functions Wn(x - k) form an orthonormal basis of L2 (R 1). We define a of a given vector. library of wavelet packets to be the collection offunctions of the form Wn (21 X - k)

In fact, for the example of the first library restricted to covering by dyadic where e, k E Z, n E N. Here, each element of the library is determined by a intervals we can start by calculating the entropy of an expansion relative to a local scaling parameter i, a localization parameter k and an oscillation parameter n. trigonometric basis for intervals of length one, then compare the entropy of an (The function Wn (21x - k) is roughly centered at 2-1k, has support of size ~ 2-1

adjacent pair of intervals to the entropy of an expansion on their union. Pick the and oscillates ~ n times.) We have the following simple characterization of subsets forming orthonormal

I We can Ibink of Ibis cover as an even covering of frequency space by windows roughly localized bases. over Ibe corresponding intervalso


171 170 CHAPTER 8

'I,

~ : :;'1

~J..~__,.,.,.,.,.,...,.,..."..__. .~ _ ... "'''' ''''''''''''~'''''"''''''''''''''''''''''''''''''__''''·N'''''''''''''''''''·''W._''''_-:-''''''''''''·''~''

!t""-----~-..~'~lMfYI~-......,..............--~-

~'AA:I: ~~lm,,"'X;;';KiXo':r~.W<'#~~;:~-'~"f:i'i~i·_~-~~'~"" ·~·,%:···.~~~i-;::;.x.;olAA~~N*'~~"l<W~'''~l;$}'~'~~<>;;

Figure 4. Wavelet coefficients of a function.

expansion of minimal entropy and continue until a minimum entropy expansion is achieved. (See Figure 4.)

Of course, while entropy is a good measure of concentration or efficiency of an expansion, various other information cost functions are possible, permitting discrimination and choice among various expansions.


::.,'"? ,§'-:.":'..:<', ~::>., '~,.:~' .

......,k,··,·,,·;,·,·, •.•.,.·,,·;··,·,,·...·,.·;·,,·. ,,,.,,,.,,,·· __ .·v·.,·,·.,·····

----~--..........- .............~.....--.~~~,~~"I,.,.,~~

-i==:~:~~~;;:=~:;~~~:::~":'~~';:=:~=;=,;:~~;;;-;:;:~~;: :::::~;;:~~:~;:~~:~:?@~:~-$J

Figure 5. A best-basis wavelet packet analysis. This analysis corresponds to selection of windows in frequency space, to minimize the entropy of the expansion.

We illustrate these points, as well as the effect of various analysis methods, in figures 5-7 in which the vertical axis represents the frequency axis and the horizontal is the time (or space) axis. The signals have 512 samples (and are wrapped around). Each rectangular box in this phase space corresponds to a coefficient obtained by correlating the signal with an element of the wavelet packet


NUMERICAL HARMONIC ANALYSIS 173172 CHAPTER 8

L/;;::,.:~:,:.. ~ .. .•.:J.i.~. .....•.>::~~:~:~;,.:\\y.:~.t&·~-::::;.\· .. -·~·~ ·"~/·;;:~:t.~):~,:,, ,.\>;,;<

Q>.::::::::.;'-;:,::. :"};':",::;:--:,:/::'/~ ::;:,<:,;~

,:;:~:~:J;~~f.;.~,;~.:~.}:~ ........".-.. " .

........... "" ...... '''...- ........ ........',1' ,,, / : / • .-"."

. :~c :;,.·~~,::G:;;:;;; .:::.. ,:::: ;,::;,;:;;... :,;, :~, .;" ;;:;:":.. ;:,;;,~ ..'''''.''.

:,. g~ ~

~ -~ :'<

~ ::: .~ -J ; : ~ :'.'

t -;

;~

~

~~

t: f

,I ~~;:.~:.:~~:".::.~. ,

<:. ':?:,::t\x:"; «< :';:-;"-:'; ~ ~.':~., ';:'(::"< :, :-:',:';...... :~~: ::~:'~~': f

.... .. .. i!~~

:··t F

: ~ f., 'j

.........---.....,.,...--..--.~~-.-JM~fJ'~~~[-. W/tJh\\/i~'rl~~1 ~:'s;~ if;- ~~

it~

,>; _____._-. _. --..."'.__..~_, ."'O"""",,..<.............__,y...i<..._.<.,..,'....,·.~.'ij

Figure 6. A two-windows expansion with no adaptation. Figure 7. A best-level expansion in which a fixed window size is chosen to minimize entropy.

library whose time support lies "below" the box and whose frequency support is in the projection of the box on the vertical axis. Each box has area 512 pixels (Le., vious candidates as well as functions which can be well approximated locally by a cover of the discrete phase plane has 512 elements). trigonometric polynomials of short length. Various obvious definitions come to

The compression rate can be computed as the ratio of the visible gray area to mind. At the moment it would seem that experiment will provide a better guide. the total area of the box (Le., the relative number of visible boxes). The procedure for signal analysis described above is very similar to the usual

Of course we can try to characterize classes of functions which are well com methods of studying Fourier multipliers in which we break the multiplier by an pressible, i.e., for which we can estimate the number of coefficients needed for appropriate partition of unity to simpler components whose spatial localization representing the function with a prescribed accuracy. Smooth functions are ob- and structure are easier to understand. We can describe a similar procedure for


174 CHAPTER 8

integral operators

T(f) = k(x, y)f(y)dyLwhich are not necessarily convolutions.

Our goal is to implement a fast discrete version of the operator. A procedure that is equivalent to PI QI decompositions of Calder6n-Zygmund

operators (see [1]) can be obtained by trying to compress the k(x, y) viewed as an image, i.e., k(x, y) represents light intensity at pixel (xy). Here again the analysis consists in finding an 'optimal windowed expansion for k(x, y) (or k(t 1])) by selecting that combination of windows most efficient in capturing the kernel.

Since the kernel is represented as a sum of products of functions of x and y it is easy to convert an efficient two-dimensional representation into a corresponding efficient computation. Observe also that each box selected represents an interaction between two windows on the line.

It can be proved that for k(x, y) a single or double layer potential for Helmholtz on a curve or surface (with bounded curvature), this procedure leads to an order pN log N algorithm, where p is the number of decimals desired, and N is the number of discretization points (~ number of wave lengths on the surface).

For more general curves or surfaces one has to develop specific, highly oscillatory analogues to multipoles. (Smooth bases are useless.) This has been done by Rokhlin with a resulting description of local oscillatory interactions.

Yale University

University ofParis, Dauphine

Washington University, St. Louis

REFERENCES

Software for adapted waveform analysis is available by anonymous ftpfromceres.math.yale. edu, Internet address 130.132.23.22. [1] R. Coifrnan. Adapted multiresolution analysis. computation, signal processing and

operator theory. ICM 90 (Kyoto). [2] R. Coifrnan, Y. Meyer, and V. Wickerhauser. Wavelet Analysis and Signal Processing.

Proceedings of the Conference on Wavelets, Lowell, Mass. 1991. [3] R. Coifman and Y. Meyer. Remarques sur l'analyse de Fourier afenerre, serle I C. R.

Acad. Sci. Paris 312 (1991), 259-261. [4] Y. Meyer. Principes d'incertidudes, bases hilbertiennes, et algebres d'Operateurs.

Seminaire Bourbaki, 1985-86. Asterisque, 662. [5] I. Daubechies. Orthonormal bases ofcompactly supported wavelets. Comm. Pure Appl.

Math. XLI (1988), 909-996. [6] E. Laeng. Une base orthonormale de L2(R), dont les elements sont bien localises dans

l' espace de phase et leurs supports adaptes a toute partition symetrique de l' espace desfrequences. serle I C. R. Acad. Sci. Paris 311 (1990), 677--680.

9

Some Topics from Harmonic Analysis

and Partial Differential Equations

Robert A. Fefferman

It is a pleasure to celebrate, together with many colleagues, the occasion of the sixtieth birthday of Elias M. Stein. To me, Professor Stein is not only an extremely great mathematician, but he is also my teacher and friend. He gives to each of his students his time and his energy very generously. Most of all, he provides us with such an inspiring example in the way that he does mathematics. I think this is the greatest gift that a teacher can give to a student, and I shall never forget this gift.

lit this article, I would like to discuss some of the mathematical issues which

have interested me and have guided much of my work. They are in large part taken directly from E. M. Stein's celebrated book, Singular Integrals andDifferentiability

Properties ofFunctions [27] and will be motivated by the statement of three "model

problems." These can be stated as follows:

Problem 1. If Q ~ jRn is a "nice" region, and 1 < p < 00, define TQ by

jfJ(~) = XQ(~)?<~)'

When is it true that IITQfllu(lRn) .::: Cp ,nllfllU(lRn ) where Cp,n is a constant

depending only on p and n, but not on f?

Problem 2. Understand "Marcinkiewicz multipliers." By a Marcinkiewicz multiplier m(~, 1]), ~ E jRn, 1] E JRm, we shall mean that m(~, 1]) satisfies all the same

estimates that ml (~)m2(1]) does, where ml and m2 are in some appropriate set of "classical" mUltipliers (such as Honnander multipliers). We shall not assume that m(~, 1]) takes the special fonn ml (~)m2(1]). This problem was first investigated by


177

~~{~"a-~~~_~~~~~~~~~~?i'.~"'~";"'='--~-~'''=~~~~~-'~;j~¥.'!2!\1j,'if~':_'::'~''.;;~-

176 CHAPTER 9

Marcinkiewicz (see Stein [27]) who obtained LP estimates for these multipliers. when I a} ~ -llfIILJog+ L(IR"xlRm )

a

and these estimates are not available by the (now standard) methods presented in [I] for the LP theory. In this article we shall be interested in the recent work in harmonic analysis which allows us to obtain finer estimates on these operators near L l ,

Problem 3. It is a very important fact from the classical theory ([27]) that if B ~ IR" is the unit ball and we are given a function f(x) defined on S"-I which belongs to LP(da) (l < p < 00. and da is the surface area measure). then there exists a unique harmonic function u on B taking on the values f (x) as non-tangential boundary values and such that u* E LP(da), where u* denotes the non-tangential maximal function of u. Our third problem is to extend this classical theorem about harmonic functions to the solutions of a class of elliptic operators with bounded measurable coefficients. The solution to this problem provides at once a generalization of the theory of harmonic functions on Lipschitz domains. and also some interesting harmonic analysis involving classical weights and the Helson-Szego theorem.

Let us begin to consider each of these problems in some detail. Starting with the first problem, we quote two results:

Theorem A (Charles Fefferman [14]). If S is a "nice region" in 1R2 whose bound

ary has a point ofnon-zero curvature then XS is not an LP (1R2) multiplier for any

p =f=. 2:

Theorem B (A. Cordoba-R. Fefferman [8]). If S is a region whose bound

ary consists of line segments, then the boundedness properties of the multiplier

Xs are equivalent to the boundedness properties of a certain maximal operator

corresponding to S.

In Theorem B above, the maximal operator corresponding to the multiplier XS

is very easily described. Suppose that B = {Bj }j=1,2.... denotes the sequence of directions of the normal vectors to the sides of the (infinite) polygonal region S.

SOME TOPICS FROM HARMONIC ANALYSIS

Let M9 denote the maximal operator defined by

M9!(x) = sup (I ) l lf (y)! dy xER m R R

RERn

where n9 is the family of all rectangles in 1R2 whose side lengths are arbitrary, but whose orientations are restricted to belong to one of the directions Bj E B. Then the behavior of the multiplier operator corresponding to Xs on LP is equivalent to the boundedness properties of M9 on the space L (p/2j'.

For this reason in order to answer Problem 1. it is sufficient to gain an understanding of the operators M9 • To do this. we shall discuss a systematic approach which can often be applied to determine the action on (Orlicz) LP classes of various maximal operators. We emphasize that just as in the case of M9 , the maximal operators we deal with are defined by a supremum over averages with respect to a family of sets whose geometry may be considerably more complicated than in the

classical case of balls or cubes. Our method proceeds as follows. Suppose we are given a family of sets n in

IR" and we wish to analyze the operator M defined by

Mf(x) = sup _1_ ( If(y)ldy XER meR) JR RE'R

and obtain (for instance) the estimate

C mIx E Q IM(f)(x) > a} ~ -lIf11LO>(Q)

a

(where Q is the unit cube and L denotes an Orlicz class corresponding to the convex increasing function <1». Then we shall describe a method which often makes it possible to prove a lemma along the following lines:

Given a family of sets {R j }, Ri E n, Ri ~ Q. there exists a subfamily Rj such that (a) II L X'R;lIv"(Q) ~ C; and (b) m(U Ri ) ~ cm(U Rj ) for some pair of constants c, C > O.

Here L 1jI is the Orlicz class which is dual to the class L which occurs in the estimates on M which we are trying to prove. That this covering lemma is sufficient to prove the weak type estimate of M on L above is rather trivial. In fact,settingEa = {x E QIM(f)(x) > a}, for each x E Ea.chooseanRx En

so that m(k,) JR !!(y)ldy > a. Now apply the covering lemma to the collection x

{Rx }xEE to get {R j } satisfying (a) and (b) of the lemma. It follows that a

m(Ea ) ~ m (U RX) ~ ~ m (U Rj ) < ~ 1

Lm(Rd jXEE.


-----"-------------

179

~)iiiiIiI--------------- --------·----------··i5iii:-------------;;iiiiiiirlzi------~---ilii~ ,- ~.;,.,·~~~!iT~i.ii~~u,:;,.·"iW'":';;:n·_·~-~··~·-~-iiiii!ii~_~~~-~~~~~iiWiiiiiiW;-----

178 CHAPTER 9

::s -1 L -1 1_ If(y)ldy = -1 1If I LXii, dy c i Cl R; CCl Q

1 1 ~ -. -lIf11L4>(Q)II"XR:IIL"'(Q)c a ~ ,

C 1 ~ - . -llfIIL4>(Q)'

c Cl

and this is the estimate on m(Ea ) we require.

A similar argument shows immediately that if we desire to prove that M is of weak type (p, p) for some p with 1 < p < 00, then it is enough to prove a covering lemma 'exactly as above (where M is mapping into weak L I ) except that we must replace (a) by the estimate

IILXIi,IIU'(Rn)::S Cm (URi)'!P' To show how to prove covering lemmas for various types of sets, we present

some concrete examples. The first of these involves proving the sharp covering lemma for rectangles in IRn with sides parallel to the coordinate axes. This covering lemma is exactly what is needed to prove the celebrated lessen-MarcinkiewiczZygmund Strong Maximal Theorem [22]. m{x E Q IMn(f)(x) > a} ~

~ IIfIlL(log+ L)n-l (Q) when Mn is the strong maximal operator in IRn defined by

Mnf(x) = sup _1 [If(y)ldy, XER IRI JR

the sup being taken over all rectangles in IRn with sides parallel to the axes. We have the following:

Covering Lemma for the Strong Maximal Function [7]. Suppose {Rdi=I.2, .. , is a given sequence of rectangles with sides parallel to the axes, wit.!! Ri ~ Q, where Q is the unit cube in IRn. Then we may select a subsequence {Ri}from the Ri so that

(a) II L X~ lIexp(L),/n-1 ::s Cn; and

(b) m(U Ri) ::: cnm(U Ri) and where cn, Cn are positive constants depending

only on the dimension n.

Proof We order the rectangles Ri so that their Xn side length decreases as n

increases. Select the Ri according to the following rule: Set R = R I. AssumingI

we have chosen R I, R2' ... , Rk-I, go down the list of Ri following Rk-I, until we first have a rectangle R such that

m (R n [g Ri]) / meR) < ~.

SOME TOPICS FROM HARMONtC ANALYStS

This rectangle R then becomes Rk • In this way we continue to obtain the selected rectangles Ri , i = 1,2,3, .... We now claim that the Ri have properties (a) and (b) as stated above.

To prove estimate (a) we use duality, and fix rp ::: 0 such that II rpllL(1og+ L)n-I (Q) = 1 ~d estimate JL Xli, rp dx. To do this, set Ei = Ri - Uj <j R j , the disjoint part of R j • Then

f LXli,rpdx = L (rpdx ::s 2 Lm(Ei)~ (rpdx. j i hi j m(Rj ) hi

(Recall that the selection process assumes that m(Ej ) ::: ! meR;), and we have used this in our last inequality.) It is also clear that for each x E Ej ,

----L- h rpdx ::s Mn(rp)(x) so that m(R;) ,

L m(Ej ) [~ (rp dX] ::s [ Mn(rp)(x) dx. j meR,) hi JQ

This estimate is not useful, because, first, if rp E L(log+ Ly-l, then J Mn(f)dxQ

may diverge, and second, we are trying to prove a result concerning the operator Mn • Both of these difficulties can be avoided by using the geometry of the rectangles R In the argument that follows, the Ri will be assumed to be dyadic. The j •

general case requires only minor modifications which a~ best left to the reader. The key point is that since the Xn side lengths of the R j are decreasing, if we slice these rectangles with a hlperplane perpendicular to the X n axis to obtain n - I-dimensional rectangles T;, then

1 < -m (R,)2 n Im. ( R, n [~RJ])

implies

m.-l (T, n [Wi]) < ~ m._l (T,)

(The reason for this is thatmn_1 (T; n[U j<i Tj ]) is independent of the X n coordinate of the hyperplane that is used to slice Rj as long as i!.,passes through R!.:) The exact same duality argument, only applied to the slices Tj rather than the R j, produces the estimate

f LXT,:rpdxI ... dXn-l ::s 2 f Mn-1(rp)dxl ... dXn-l,


-'-.,,----_--- w -,- ••• _ ••••,: : ,_"- " __ • __ ••••,-::,,:'- :.=::::';:':::::-===--=':::':::::::=-':""--=:::::,:::=:::::::='=:"'-=:::'::::==':.'=:==--==_':'=:::=:==;::5=-~"'-.-._' - '::_ __~._ ...::'. - .,.. - ,._.

180 CHAPTER 9 SOME TOPICS FROM HARMONIC ANALYSIS 181

and integrating this in the Xn variable yields oriented in anyone of a sequence of "lacunary directions".) Define

f L X'R;rp dx S 2 f Mn- I(rp) dx (dx is n dimensional measure here).

According to the theory of the strong maximal function Mn - I (which could be

assumed by induction), if IIrpIlL(log+ L)"-'(Q) = 1, then 1Q Mn-I(rp) dx S Cn and so this proves part (a) of the covering lemma.

To prove part (b), suppose R is one of the Ri which have not been selected. Then by the definitiol} of the selection criterion it follows that

- ImeR n [ URil) > 2meR)

where, in U iii, the union is taken only over those iii which precede R on the list of rectangles. Now, slice the rectangle R as above with a hyperplane perpendicular to the Xn direction, calling slices of R an,.,? iii, T and T;, respectively. Then just as above, since the Xn side length of each Ri preceding R exceeds the Xn side length of R, we have

Umn-I (T n [ Ii)) > 21

mn-I (T).

This clearly implies that on R we have

Mn-I(Xu'R;) > 2' or equivalently that

1 URi ~ {Mn-I(Xu'R;) > 2}'

By the theory of the n - I-dimensional strong maximal operator, Mn - I , this gives

m (U Ri) SCm (U iii)

which is part (b) of our covering lemma. • The moral of the story in the first example of our technique for proving covering

lemmas is that by using the geometry of the relevant family of sets involved, we have controlled the maximal operator at hand (in this case Mn ) by a simpler operator (Mn-I). This method will be employed in the nextexample ofthe lacunary directions maximal operator Mlac defined for functions in the plane.

Suppose that Rlac denotes the family of all rectangles in ]R2 whose longest side makes an angle of2-k with the positive x-direction, for some positive integer k, but whose side lengths are arbitrary. (In other words R 1ac is made up of all rectangles

Mlac(f)(x) = sup (I ) rIf(y)ldy. XER m R JR

RE'R,oc

Then the first estimates for this operator were obtained using the method described above for the strong maximal operator. They were obtained by J. O. Stromberg [28], and involved weak type estimates on Mlac(f), for f belonging to a certain Orlicz class near L 2

• We shall present here an extension of the results in [28] where we obtain weak type estimates on Mlac(f) for f E L 2 (A. Cordoba-R. Fefferman [9)), and where the argument is considerably simpler than that in [28].

We wish to show that m{Mlac(f) > a} S ~ Ilflliz(R.z) for a > O. This will be done by proving that given a sequence of rectangles Ri E 'lilac, there exists a subsequence iii satisfying

(1) II LXi lIiz S Cm(U iii) (2) m(U Ri ) ::: cm(U Ri ),

where c and C are positive constants. In order to do this, assume that the Ri are arranged so that their longest side length is decreasing. We select the iii according to a slightly different rule from that given by rectangles with sides parallel to the axes: Choose iiI = R I and assume iii, ... , iik- I have already been chosen. We continue down the list of Ri until we first come upon a rectangle R so that

k-I 1 (*) L meR n iii) S - meR).

i=1 2

We then set R = iik • Now, we claim that the {iii} satisfy properties (1) and (2). To prove (1), we write

II LX'R;lIiz = i LX'R;XRj = 2 Lm(iii n iij ) + Lm(iii). IRZ

i,j i<j i

From the selection rule (*), we see that for each fixed j,

j-I

2 L m(iii n iij ) S m(iij )

i=1

so that

2 L m(iii n iij ) S L m(iij ) S 2m(U iij ).

i<j j


183 .... 182 CHAPTER 9

This gives

II L Xli; IIl2 S 4m(U Rj )

and establishes (1).

To prove (2) we must use the geometry of the rectangles involved, in a way quite analogous to the previous example. Let R be a rectangle which is one of the R j

that was not selected. Let S denote the smallest rectangle with sides parallel to the axes such that S contains R. Finally suppose Sdenotes the concentric double of S. Then, since R was not selected,

meR) _ L meR n Rj ) > ~ R before R - 2'

i

Very simple geometry shows that

meR n Ri ) m(S n Ri )<C A

meR) - m(S)

for all Ri whose longest side length exceeds that of R, and whose orientation is different from that of R. Now, there are two cases:

1 Case 1. meR) _L meR n Rj) ::: 1/4

_ R; before R

R, oriented differently from R

1Case 2. L meR n Ri ) ::: 1/4.

meR) _ R; before R

R, and R have same orientation

In Case 1, from our geometric observation, it follows that

1 ~ 1L ------:x- m S n R· > m(S) i ( ,) - 4C'

so that on R,

M2(L XR;) ::: 1/4C.

In Case 2,

M1 (_L, XR;) > ~ R;E'R.J


where M1 denotes the two-dimensional strong maximal operator with respect to·

a pair of ~es oriented along the sides of R and where R) denotes the collection

of those R j oriented in the same way as R. It follows that

URi ~ {M2(LXR;) > 4~ }uulM1 (_L,XR;):::~} ] R; E'R.J

so that

m(URi)SCII~XR;112 +~mIM1(_L,XR;):::~} all, L2 ] R,E'R.J

2 2 ]

S C L XR; + L L XR; . [ 11." L j ~,~ L'

Again, by the selection rule of the R j , and our estimates to show (1),

II L Xli; IIi2 S Cm(U Ri )

and for each j,

2 S Cm (_U, Ri) ._L Xli; R; E'R.J L2 R,E'R.J

Summing this on j gives

L II L Xli;lIi2 S Cm(U Ri ),

j R; E'R.J all i

and this proves (2) and the desired covering lemma. We should point out that in [26], Nagel, Stein, and Wainger were able to extend

our results on L 2 for I < p < 00, and their methods are entirely different. It would be quite interesting to obtain, by a covering lemma argument, the boundedness of the M1ac on the full range of LP spaces.

The reader will notice that in the preceding two examples of covering lemmas, the rule for selecting rectangles in the sparse subcollection changed. In the case of the strong maximal operator, Mn , we chose a rectangle R if and only if

U- 1meR n [ RiD S -meR).

i<k 2


-.::_ __ •. :=',,", ." .....::",,"'........""_= __ . ;;;,;;; . .~.";; . ..__ _" ..::-.._.,.;;r"~~"=:::!~"'~£:"'~'~~~-''7",",~,~~"? .."",,,, '~O''''''=''''''""",''~''''·''"":.f",,,,,:",,- ,- ...·"'·~~~'"',,~""'.~"' ..~_'''i'';''"''_ =,;:,~",~""<:"",,,,-,,,,,,,,.~,,,,.,,,, ..,,,,,:,,..,,,,,,,="""""=",'--",,,==,.,,,"""~_.'

185184 CHAPTER 9 SOME TOPICS FROM HARMONIC ANALYSIS

In the second example of Mlac, a rectangle was selected if and only if and iii, S and Si, respectively. Then, because the z side lengths of the iii before

" - I~ m(R n Ri) ~ - m(R), i<k 2

which is a s~ghtly stronger control of the overlap of R with the preceding selected rectangles Ri than in the case of Mn above. In our last example, the reader will notice a further strengthening of the selection rule.

Consider, in JR.3, the collection of all rectangles R z with sides parallel to the coordinate axes, contl!ined in the unit cube, whose side lengths are of the fonn s, t, and 1/I(s, t) (in the x, y, and z directions, respectively) where 1/1 is increasing in each of the s and t variables separately. Then, A. Zygmund conjectured many years ago that if we set

Mzf(p) = sup _1_ ( IfI pERERz m(R) JR

then we get the estimate

C m{p E QIMzf(p) > a} ~ -lIf11L(log+L)(Q)'

a

In 1978, A. Cordoba [6], using the methods we have described, proved this conjecture using a beautiful argument which we reproduce here.

In fact, assuming thatJ Ri } is a sequence of rectangles in R z we prove that there exists a subcollection {R;} so that

(1) II L Xi, lIexp(L) ~ C, and (2) m(U Ri ) ::: cm(U Ri ).

To do this we order the Ri so that the z side lengths of the Ri are de~reasing.

(We shall also assume in what follows that the Ri are dyadic.) Then set Rl = Rh

and, assuming that iii, i < k have been chosen, choose the first rectangle R on the list such that

I ([ Ek-

1X'::"] 1 m(R) JR e ,=1 Ri - I dx ~ 2:'

It is a simple matter to verify that eE,X;; - 1 E L1(Q) and lI/aIli X;; - Illv(Q) ~

Cm (U iii) for some constant C. We need to verify now that m (U Ri) < Cm(U iii). To do this, suppose R is an unselected rectangle so that

I i EX'::" I-- [e Ri - I] > m(R) R 2 '

where the sum only extends over those iii before R on the list. Slice R and iii with a plane perpendicular to the z axis, and call the two-dimensional slice of R

R exceed the z side length of R, we have

1 EX"::' 11-- [e 5i - 1] > m2(S) S 2

where again the sum only extends over those i for which Ri comes before R on our list of Ri • It is at this point that we make use of the fact that all of the rectangles in question (in JR.3) have side lengths of the form s, t, and 1/1 (s, t). It follows from this that each of the Si in the sum ~ove has eitherjts x side length greater than the x side length of S (call such an Si of type I) or Si has its y side length greater

than the y side of S (type II). Then

_1_ ([/t:;, _ 1] = _1_ ([e(EIt:;,+EIlt:;,) - 1] m2(S) Js m2(S) Js

(here Ll indicates that we sum over only the Si of type I,and similarly for Ll/)

_1 ( [e EI t:;, _ 1] [eEll t:;, - 1] m2(S) Js

1 1 E/X"::' 1 1 EllX"::'+ __ (e 5i - 1) + -- (e 51 - I). m2(S) s m2(S) s

Since the sum above exceeds 1/2, at least one of the terms must exceed 1/6.

In case

-- - Ell X"::' 1 1/6,1 1[eEI X"::'5i 1] [ e 5i - ] dxdy > m2(S) s

since eEl t:;, - 1 is a function of the y variable only considered as a function on S, and eEllt:;, - 1 is a function of x only, we have (writing S = J x K)

1 ({ [ EI X"::' ] [ Ell X"::' ]m2(S) JJs e 5i - 1 e 5i - I dxdy

1 ( [ E, X"::' ] 1 f [Ell X"::'] 1 = mj(K) JK e 51 - 1 dy ml(J) e 5i - 1 dx > (5

Thus in this case at least one of the inequalities

I ([ EI X"::'] 1 ml(K) JK e 5, - 1 dy > J6

or 1 Ell X"::' 1 1ml(J)1 J[e 5 -

I ]dX> J6 is valid.


187 186 CHAPTER 9

In case

m2:S) 1[eE1XS; -I]dxdy > ~. it follows that

---I 1[eEO:-::-]- -,Sj I dy > 1 ml(K) K 6

and a similar analysis holds in the remaining case where

---I 1[eElJX"::']-Sj 1 dxdy > -I m2(S) S 6

Putting all of this together yields

URj ~ {Mx (eEX;; - I) > 1/6} U{My (lX;; - I) > 1/6}. (Here Mx and My denote one-dimensional maximal functions in the x and y

directions,respectively.) Using the estimate IIlX;; - IIIL'(~) ::: Cm(U Ri ) and

the weak type L1 estimates for Mx and My, we see that m(U Ri ) ::: C(U Ri )

finishing the proof of the covering lemma.

As the reader can see, it is far from routine to apply the covering lemma method to get infonnation about a particular maximal operator. Nevertheless, this method

has had a number of deep applications, and can be expected to have more in

the future. Before moving on to consider the second "model problem" it should also be pointed out that in order to do the operator theory of operators which are

invariant with respect to certain actions ofl~n (certain groups ofdilations, rotations, etc.) it is essential to understand the geometry of the relevant sets associated to

these operators and for this, covering lemmas must be proved. It is not enough

just to know the sharp estimates for the associated maximal operator. This will be illustrated quite clearly in the following part of this article dealing with the analysis of singular integrals on product spaces.

Now, let us consider the second problem, namely obtaining sharp estimates for

Marcinkiewicz-type operators. We shall begin by explaining the precise meaning

of this class of operators, but the reader should be aware that the details of the definition could be altered considerably without making essential changes in their basic theory.

Recall that a function m (~), ~ E ~n is called a "Honnander multiplier" provided all the dilates m(8~), 8 > 0 satisfy, unifonnly in 0, estimates of the fonn

Im(~) I ::: C, ~ E ~n

and

II Da [m(~)({l(I~ I)] 11L2(lR") ::: C


for all multiindices a with lal ::: [~] + 1; here ((l(x) is any function COO(R1)

which satisfies ({l(x) = I for all x E [1,2] and ({l(x) = 0 for all x f/:. [~, 4]. It is a fundamental fact from the theory of singular integrals (see Stein [27]) that

the multiplier operators T corresponding to the Honnander class are bounded on m

LP(~n), I < p < 00, and are of weak type (l,I). Now, to motivate the definition

of the Marcinkiewicz multipliers, the reader should consider the estimates satisfied,

on ~n X ~m, by multipliers of the fonn ml (~)m2('7), ~ E ~n, '7 E ~m, where ml

and m2 are Honnander multipliers on ~n and ~m , respectively. These estimates

on a function m defined on ~n X ~m are as follows:

The functions m(81~, 02'7) satisfy (unifonnly in 81 and 82) the estimates

Im(~, '7)1 ::: C, for ~ E ~n, '7 E ~m

IIDnm(~. '7)({l(I~I)]IIL2(lR") ::: C

n for each '7 E ~m whenever laI ::: [ "2 ] + I

IID~[m(~, '7)({l(I'7I)]IIL2(1Rm) ::: C

m for each fixed ~ E ~n whenever IIlI ::: [ "2 ] + I

and

IIDf D~[m(~, '7)({l(I~I)({l(I'7I)]IIL2(lR"xlRm)

n m for all a, Il such that lal ::: ["2 ]+ I and IIlI ::: [ "2 ]+ 1.

Functions m (~, '7) satisfying these estimates are called here "Marcinkiewicz-type

multipliers," and their corresponding multiplier operators, the "Marcinkiewicz

operators." The problem we investigate now is that of obtaining the weak type inequality

C m{(x, y) E Q IITm(f)(x, y)1 > a} ::: -llfIILIog+ L(Q)

a

for all a > 0, where Q denotes the unit cube of ~n x ~m, and Tm is the

Marcinkiewicz operator. This estimate is easily derived by iteration in case m(~, '7) takes the special

fonn ml (~)m2('7). However the general case is much more difficult, and we shall explain why this is the case.

One of the best ways to obtain the LP theory (I ~~__ __~~_~ _


where g and gr are the classical Littlewood-Paley functions (Stein [27]). The This theorem forms the basis for the definition of product Hardy spaces: f E

quantity A > I in the inequality depends on how much smoothness the multiplier H P(IRn x 'f?m) if and only if f* E U (IRn x 'f?m) or equivalently S(f) E U ('f?n x

has. If the multiplier is minimally smooth ([ I ]+ I derivatives in L 2 away from 0 'f?m) for p > O. We define II!11HP(lRnxlRm) = II f*1I LP(lRnxlRm). There are certain

and (0) then A is not much bigger than I. Unfortunately gr is not bounded on obvious examples of functions in product H P, namely tensor products of H P

U(lRn) unless AP > 2, i.e., in this case gr is not of weak type (1,1). Thus functions of the x and y variables, respectively:

when we are interested in the action of our multipliers near L 1 we must resort where It E HP(jRn), !z E HP('f?m).f(x, y) = It(x)!z(y)to something other than (#). The other way to handle T on LP(lRn ) is to realize that T is essentially a classical Calderon-Zygmund operator. One then uses the In particular, when p :s I if II = aj and !z = a2, where the aj are classical HP

decomposition into "good" and "bad" parts, or the atomic decomposition of the atoms, then a (x, y) = al (x )a2 (y) is an H P(jRn X 'f?m) function which serves as a

classical Hardy spaces HP(lRn) in order to obtain LP boundedness and the weak model for what we shall call an H P('f?n x 'f?m) "rectangle atom." (The necessity

type estimates on L1(lRn). Our point here is that the classical Calderon-Zygmund of inserting the word "rectangle" will be clarified shortly.) Clearly these a(x, y)

decomposition and atomic decomposition cannot be used to analyze the more have the following properties. complicated Marcinkiewicz operators. m

(1) a(x, y) is supported on the rectangle R = I x J where I ~ 'f?n and J C IRThe approach we take will be to study different spaces HP('f?n x IRm ) (we write

are cubes; HP('f?n+m) to denote the Charles Fefferman-E. M. Stein classical HP space, and (2) II a(x, Y)XU dx = 0 for all multiindices a with lal :s Np(n) and for each

HP('f?n x 'f?m) to denote our non-classical Hardy spaces) adapted to the study of y E J, where Np(n) = [n( i - 1)];

the operators at hand. We construct the appropriate theory of BMO('f?n x 'f?m) and (3) I a(x, y)yfJdy = 0 for all multiindices Pwith IPI :s Np(m) and for each

the atomic decomposition of H P('f?n x 'f?m), and study the interpolation properties J x E I, where Np(m) = [m(i -l)];andfinally,

of these spaces. Once all this is well understood, we will be in a good position to 1 1

solve our second problem. (4) lIaIlU(R) :s m(RP-p.

Let us begin with the space HP(lRn x 'f?m), for p :s l. Suppose that f is a Such a function a(x, y) on 'f?n X IRm is called an HP(lRn x IRm ) rectangle atom.

tempered distribution on IRn x IRm• Let ~I (x) E CO'('f?n), fP2(y) E CO'(lRm) with For a while, it was thought that these rectangle atoms would span H P(IR

n x IR

m)

I~j = l,i = 1,2. Set~ol,02(x,y) = ~1(x/81)~(y/82)81n82m,for81,82 > O. in a way that would immediately reduce the harmonic analysis of product spaces We define the product non-tangential maximal function f* (x, y) by to the consideration of the action of the relevant operators on these very simple

atoms. The conjecture was that every f E HP(jRn x IRm) could be written as

J*(x, y) = sup If * ~OI,02(X' y)1 01,02>0

f = LAkakIx-.:cl<o, IY-yl<82 k

where ak are rectangle atoms of HP(lRn x IRm) and Ak are scalars satisfying and the product Lusin area integral S(f)(x, y) by Lk \AkI P :s CllfII~p(lRnXIR~)'

2 i~ 2 d"Xd8 1dyd82 In 1974, in [31], Lennart Carleson disproved this conjecture, suggesting that -S (f)(x, y) = If * lh,02(x, Y)I ~n+1 ~m+l f(.:c)xf(y) 01 02 the harmonic analysis of product spaces had to proceed along lines very different

from the ones of the classical theory. The fact that rectangle atoms did not properly where 1/1 (x, y) = 1/11 (x) 1/12 (y) and 1/11 E CO' (lRn), 1/12 E C8" ('f?m), I 1/Ii = 0 and span H P (IRn x IRm ) also suggested that the product theory could not be nearly as 1/1; are suitably non-trivial. Then we have the following fundamental theorem. simple and satisfactory in its final form as the classical one. We now know that

both of these suggestions are wrong, and to see this, the reader need only consider

Theorem of Gundy-Stein [20]. For all p > 0, S(f) E U('f?n x IRm) ifand only the following:

if f* E U('f?n x jRm). In this case the ratio IIS(f)IILP / 1If*IILP is bounded above and below by strictly positive finite constants depending only on p, n. and Theorem [15]. Let T be an L 2(lRn x IRm

) bounded linear operator. Suppose.

m. further. that whenever a is a rectangle atom of HP(lRn x IRm) (0 O. CRy

(Here Ry denotes the concentric dilate of R by thefactor y.) Then T is a bounded operatorfrom HPCWl,.n x ffi.m) to LP(Wl,.n x ffi.m).

The meaning of this is that, if T is to map H P boundedly to LP, then it is obviously necessary t1}at

(t) [ IT(aW dxdy ~ C for all rectangle atoms a. JR"xRm

Carleson's counterexample shows that (t) is not sufficient for T to be bounded between H Pand LP. Nevertheless, if we trivially strengthen the assumption above to say that not only is the integral (t) bounded, but the contribution to the integral from integrating over that part of space which is away from the support of the atom vanishes as we look farther and farther from this support, then this stronger condition is sufficient. Thus, it is not important that rectangle atoms fail to span HP(ffi.n x ffi.m). By examining the action of an operator on these atoms, we may

still conclude its boundedness. Below, we shall give the main elements in the proof of this result.

The first thing we must discuss is the correct notion of an H P (Wl,.n X ffi.m) atom.

This atomic decomposition can be found in S. Y. Chang-R. Fefferman [4]. We set the following notation: If Q s:; IRn x ffi.m is an open set of finite measure, then the maximal dyadic subrectangles of Q will be denoted by M(Q). If R is a rectangle, then Rdenotes its concentric double, and if Q is an open set, then Q*

denotes {Ms(Xn) > 2"~m J where Ms is the strong maximal function in ffi.n x ffi.m

defined by a supremum of averages over rectangles (products of cubes in ffi.n and ffi.m, respectively).

Let 0 < p ~ 1 and suppose that Q s:; ffi.n X ffi.m is an open set of finite measure. A function a(x, y) on ffi.n X ffi.m is called an HP(~n x ffi.m) atom (with associated open set Q) if and only if:

(1) a is supported in Q*; 1 I

(2) Iia lIu(R" xR'") ~ m(Q)2 - P ;

(3) a can be decomposed as a = LReM(Q) CXR, where each CXR has the following properties: (i) CXR is supported in R; (ii) Jcx R(X, y)x" dx = 0 for each y E ffi.m and for all multi-indices v such

that Ivl ~ (n( ~ - 1)] = Np(n) and JCXR(X, y)y" dy = 0 for each x E ffi.n and for all v such that Ivi ~ Np(m) = [me ~ - 1)];

.".-=,~

191 SOME TOPICS FROM HARMONIC ANALYSIS

(iii) [LREM(R) lIaRII~]'!2 ~ m(Q) 4- *. Observe that (i) and (ii) describe aR as a rectangle atom except without any

normalizing condition on liaR 112. This is, of course, compensated for. n m

The atomic decomposition in [4] then states that every f E H P (lR x lR ) (here

o < p ~ 1) can be decomposed as

f = LAkak h

where Ak are scalars, ak are HP(ffi.n x ffi.m) atoms, and

L !AkIP ~ CIIfll~P(R"xRm)" k

The second ingredient in the proof of our theorem is a geometric lemma due to J. L. Joume [23], which he used in order to obtain estimates on operators such as product commutators. We set the following notation in what follows: Let Q s:; IRn x ffi.m be an open set of finite measure. M 1(Q) will denote the collection of all dyadic subrectangles of Q which are maximal in the x coordinate. A similar definition applies to M2(Q). Let Q be a cube. We denote by Qy the y fold concentric dilate of Q. Then, given R, a dyadic subrectangle of~, R = I x J (l s:; ffi.n, J s:; ffi.m are cubes), we define Yl(R) = sup{t ~ lilt x J <; Q*}.

Then we have:

Journe's Lemma. If Q s:; ffi.n X ffi.rn is an open set offmite measure, and /) > 0,

then

L m(R)Yl(R)-s ~ Csm(Q). ReM2(n)

Of course, if Q is an open set in lR1, then the sum of the lengths of the maximal dyadic subintervals of Q equals the measure of Q. Although for Q open in ffi.2 it is not true that the sum of the areas of the maximal dyadic subrectangles of Q is finite (or less than Cm(Q», it is true if in the sum, we multiply the area of each maximal dyadic subrectangle by a suitable small constant depending (roughly speaking) on how much we may expand the rectangle without leaving Q (more precisely Q*).

This is the meaning of Journe's result. Now let us prove the theorem. Let T be L 2(ffi.n X lRm

) bounded and linear, and

suppose that

f- IT(aW dxdy ~ Cy-8

CRy


~~_ _~,__'!'!l' .~if'",,,,~,,",!:.l"!~~_:,,-.,.= ...,~..,.,,,",",,, ....~-,,-,,,~~~~~~~~~,;;:"",,!~,,!!;,1'?';:'~":;"o,,'" .,,,,,,,,,;,,~,,,.,~,,

192 193SOME TOPICS FROM HARMONIC ANALYSISCHAPTER 9

for all Y ~ 2 whenever a is an HP('R,.n x JRm) rectangle atom supported in R. We dominated by claim that for f E HP(IRn x 'R,.m), p ::: 1, we have

L m(R)I-p/zllaRllfYI-8(R) + L m(R)1-p/zllaRlliYz-8(R) REM(n) REM(n)II T fIIU(lll.nxlRm ) ::: CpIlfIlHP(lll.nxlRm ).

= (A) + (B).By the atomic decomposition this will follow once we prove that

We estimate (A) by HOlder's inequality: IIT(a)IIU(ill.nXlRm) ::: Cp

p/Z ( ) I/(Z/p)'whenever a is an HP(lRn x JRm) atom. So, suppose a is such an atom with (A) = L lIaRII~ L m(R)(I-p/Z)(z/p)' y;8(Z/P)' (R) associated open set Q. Then

( )REM(n) REM(n)

P/ZlnXlll.mIT(aW dxdy = L" IT(aW dxdy + In" IT(aW dxdy.

::: m(Q) i -I (

L lIaRII~ )Clearly ReM(n)

::: CP by 10ume's lemma. { IT(aW dxdy ::: ({ IT(a)IZdXdY)P/Z m(Q**)I/(*Yh.. ~x~

In order to estimate (B) we proceed similarly. .::: m(Q) i -lm(Q**)I-p/Z ::: Cpo

p/Z ( ) I/(Z/p)' 8As for !co." IT(a)iP dxdy, we write a = LReM(n) aR as in the definition of an (B) ::: L IjaR II~ L m(R)(I-p/Z)(2/p)' yz- /1 <fi)

( )atom. Then we proceed as follows: Suppose R E M(Q), R = I x J. Let f REM(R) ReM(n)

denote the largest dyadic cube containing I so that f x J S; Q*. Then, let Jdenote I/(Z/p)'

the largest dyadic cube containing J so that f x J ~ Q**. We then estimate ::: m(Q)p/Z-1 L m(R)Yz-8

/1 (R) P [ ]

REM(n)

! IT(aW dxdy =! IT ( L aR) I dxdyco." co." REM(n) Then

811L m(R)Yz-8 /1 (R) = L yz- (S) L meR).<! L IT(aR)iP dxdy

ReM(n) SEM,(n') REM(n)co." REM(n) R=S

< L (:x.:x. IT(aRW dxdy Clearly any two dyadic R E M(Q) such that Ii = S must be disjoint, so REM(n) J(lXJ)

L m(R)::: mrS),< L j:x. \T(aR)iP dxdy REM(n)

REM(n) C(l) xlRm

R=S

+ ( :x.. IT(aRW dxdy. and so llll.nxc(J)

11 811L y;8 (S) L m(R)::: L yz- (S)m(S) ::: Cm(Q*)We now use the fact that aRI(lIaRllzm(R)(\/P)-(I/Z») is an HP(JRn x JRm) rectSEM,(n') ReM(n) SEM,(n')

angle atom, so that by the hypothesis of our theorem, the right-hand side above is R=S


===--;c--------------- -----_~, "=-'='~~~-==__=_=_"'"c==._;=_--="~='"====_== -----~.«-~-,-.--~---~.-.---.-----------------=----------~-------:==, .........,."'''~---


by Joume's lemma. This proves that singular integrals T, that

(B) ~ Cm(QV/2-'m(Q*)'/(2/ p)' ~ C'.

This completes the proof of the boundedness of T from H P(Rn x ]Rm) to LP (]Rn x ]Rm).

Once we understand that in order to check the boundedness of an operator on

product H P, it is enough to check its action on rectangle atoms, it is not difficult

to settle the question of the boundedness properties of Marcinkiewicz operators

near L'. This proof of the weak type estimate

C m{(x, y) E Q IITf(x, y)1 > o:} ~ -llfIIL(log+ L)(Q)

0:

(Q denotes the unit cube of]Rn x ]Rm) where'T is a Marcinkiewicz operator

proceeds much in the same spirit as the argument given in detail above for H P•

In fact, it involves an atomic decomposition of L(log+ L)(Q) much like that of HP(]Rn x ]Rm). For the details, the reader should consult [19].

Again, we wish to stress that it is not the technical details that are impor

tant here, but rather the philosophy that although the function spaces in the

product theory are inherently more complicated than in the classical case, the

operator theory is not. This philosophy has some very concrete applications be

sides the one we have been discussing. As another example, we should mention

the weighted inequalities for product singular integrals, which can be found in

[15] and [16]. It is quite non-trivial to prove weighted norm inequalities for

these singular integrals, because there are no known analogues of the classical

Burkholder-Gundy good ,\ inequalities in this setting. One studies this problem

by determining rather precisely the way that the strong maximal operator Ms

controls the singular integral. In what follows below we shall be a little more

precise. To begin with, let us recall that a weight w (x, y) defined on]Rn x ]Rm is said to

belong to product AP (written AP(]Rn x ]Rm» if and only if for all rectangles R

(products of cubes in]Rn and in Rm, respectively) we have

(m;R) 1w(x, y) dXdY) (m(IR) Lw(x, y)- P~' dxdyr-' ~ c.

This is equivalent to w satisfying a uniform classical AP condition in the x and

y variables separately fixing the other variable. Then the way we prove weighted

inequalities for general product singular integrals (not necessarily convolutions)

in [15] and [16] is by studying a variant of the C. Fefferman-Stein sharp function,

f# in the product setting. Thus, if we wish to prove, for very general classes of

{ IT(f)(x, YWw(x, y)dxdy ~ C ( If(x, YWw(x, y)dxdyk.x. ~x. for I < p < 00 when w E AP(]Rn x ]Rm), then we need to have an analogue of

the fact, from classical singular integrals, that

(##) (Tj)#(x) ~ C[M(fHe)(x)]'/l+e.

Here T is a Calderon-Zygmund singular integral, M is the Hardy-Littlewood

maximal operator, and

f#(x) = sup (I ) ( If(y) - fQldy, XEQ m Q JQ

(where fQ = mlQ) J f and the sup is taken over all cubes Q ~ ]Rn containingQ

x) is the C. Fefferman-Stein sharp function. In the classical theory, the value

of the inequality (##) lies in the fact that f# E U(Rn) implies f E U(]Rn).

Unfortunately, as we would expect, this is not true in product domains. To be

more specific, suppose we define the mean oscillation of a function f (x, y) on

]Rn x ]Rm over the rectangle R by

I 21 )1/2oscR(f) = inf -- If(x, y) - fl(X) - 12(y)1 dxdy(fJJ1 meR) R

where the inf is taken over all pairs of functions fl and 12 depending only on the

x and y variables respectively. If we then define the (product) sharp function of

f, f#(x, y) by

f#(x, y) = sup oscR(f) (x,y)ER

where the sup is taken over all rectangles containing (x, y), then it is simply not true that f# E LP(]Rn x ]Rm) implies that f E LP(]Rn x ]Rm). In order to use

the product sharp function as a useful tool to prove weighted inequalities, we must

transfer the definition to the context of operators, rather than functions.

Intuitively, we would like to define the sharp operator S of an operator T by

Sf(x, y) = sup OSCR T(f). (X,y)ER

As we stated above, the boundedness of Son LP(]Rn x ]Rm) does not imply the

boundedness of T on this space. However, suppose, in addition, we require that

whenever the support of f is far away from the rectangle R, i.e., if supp(f) ~ CRy then for some 8 > 0, and y ~ I

Sf(x, y) ~ yS oscR(Tj) when (x, y) E R.


197 196 CHAPTER 9

Then we say that S is a sharp operator for T. It is shown in [15] that if p ::: 2 then if S is bounded on LP(jRn x jRm). then so is T. Also, if T is a product singular integral, then M s (f2)1/2 is a sharp operator for T. This precise expression of

the control of product singular integrals by the strong maximal operator can then be used in order to prove the product version ofthe Hunt-Muckenhoupt-Wheeden weighted inequalities for singular integrals, above. It is another graphic illustration

that in product domains, one can use the philosophy mentioned above in order to

bring things into a very simple and satisfactory form quite like that of the classical

theory. Our last problem concerns elliptic differential equations, but the mathematical

issues involved have quite a lot in common with model problems 1 and 2. In fact,

we will see that just as in the first two problems, the theory of maximal functions, singular integrals, and especially the topic of weighted inequalities will play a

central role.

We shall begin to discuss the third problem by formulating it precisely. We consider elliptic operators in the unit ball B of jRn in divergence form. This means

that Lu = div(AVu) where A(x), x E B, is a symmetric n x n matrix valued

function satisfying

AI~12 ~ ajj(x)~i~j ~ A-ll~12, A(x) = (ajj(x»

for all ~ E jRn and some fixed A > O. We will assume, in general, no regularity

of the coefficients aij (x) except for measurability. The question we investigate is the solvability of the LP Dirichlet problem. To define this properly, we require

several definitions. First, we say that a function u E H/oc(B) (Hl~(B) denotes those functions in B which are, together with their distributional first derivatives,

locally in L 2 ) is a solution to Lu = 0 provided a(u, ({I) = 0 for all ({I E CO'(B),

and where a(u, v) = fB aij(x)oiU(X)Ojv(x) dx. We denote by r(x) the right circular non-tangential cone with fixed aperture having vertex at x E 0B. Then,

if u is- a function in B, we denote its non-tangential maximal function by u*(x)

where

u*(x) = sup lu(z)I, for x E oB. ZE['(X)

Then, by the LP Dirichlet problem for L in B we mean the following. Let there

be given a function f E LP(da) (da is the surface area measure on aB). Find a

solution u(z), Z E B to Lu == 0 in B, satisfying

lim Z-"+X

u(z) = f(x) for x E oB, ZE['(X)

and also the estimate

lIu* IILP(du) ~ CII fIILP(du)'


The question we ask is: Given an elliptic operator L, when does there exist a solution u to the problem above given any f E LP(da)? Here p is a fixed exponent satisfying 1 < p < 00. The estimate on the non-tangential maximal

function is included to insure uniqueness of solutions to the problem. Before giving a summary of the known results concerning this question, we

shall discuss some general background information on the solutions to the elliptic

equations we are considering. It is a celebrated fact from the classical theory that

positive solutions of these equations satisfy the Harnack principle (Moser [25)). This means that if K S; B is a compact set, and u(z) > 0 is a solution of Lu = 0 in B, then there is a constant C depending only on K and A such that

sup u(z) ~ C inf u(z). ZEK ZEK

Another beautiful result from the classical theory of these equations is that

the classical Dirichlet problem is always solvable for any elliptic L (Linman

Stampacchia-Weinberger [24)): Given any continuous function f (x) on 0B, there exists a unique function u (z)

continuous on Ii so that

Lu == 0 in Band u(x) = f(x) for all x E aB.

There is also the maximum principle for solutions to Lu = 0 in B: If u is

continuous on B and satisfies Lu = 0 in B then

max u(z) = max u(x). ZEB xEaB

By the previous results it is possible to define the harmonic measure wi associated

to the operator L with respect to the point z in a manner exactly analogous to the familiar case of the Laplace operator: Suppose uf, f E C(aB) denotes the unique

solution to Lu = 0 in B with uf(x) = f(x), x E aB. Then by the maximum principle, the map f --+ uf(Z) is a positive linear functional on C(aB). Hence,

according to the Riesz Representation Theorem, there exists a unique probability

measure wi so that

uf(z) = r f dwi, f E C(oB). JaB

This measure dwi is the harmonic measure. It is a trivial consequence of the

Harnack principle that if Zl, Z2 E B, then the ratio wi' (E)jwi2 (E) is bounded above and below by a pair of finite, strictly positive constants for all Borel sets

E S; 0B. Thus, the exact choice of z E B makes virtually no difference when

considering the properties of harmonic measure. In fact, if z = 0 we will write dw~ often simply as dwl, (or even dw if L is understood). Now the nature of the


199 198 CHAPTER 9

harmonic measure dWL is of critical importance. To be more exact about this, we recall the classical weights from the theory of singular integrals and maximal functions.

We say that a measure J-L on aB is in the class BP(da) if and only if J-L == k da where

(_1 {kPda)IIP < C(_1 (kda) a(l1) ill - a(l1) ill

for all surface balls l1. S; aB. A measure J-L == k da is said to belong to the Muckenhoupt AP class if and only

if

_1 f kda) (_I (k- p~, da)P-l < C( a(l1) II a(l1) ill

for all surface balls l1 s; aB. Then the key point for our purposes here is the following:

The LP (da) Dirichlet problem is solvable for L if and only if the harmonic measure dWL belongs to BP' (da) where.!.

P + 1,

P == 1.

Unfortunately, it is not true that dWL is always in some class BP(da) for an arbitrary elliptic operator L. In fact in Caffarelli, Fabes, and Kenig [I] an example of an elliptic operator L is given whose harmonic measure dWL is singular with respect to da. Thus, the nature of the solvability of the LP Dirichlet problem depends on the operator we are dealing with. Considerable progress has been made on this issue. In what follows, we shall give a quick overview of some of the highlights of the theory.

One of the motivations for considering the class of elliptic operators with bounded measurable coefficients is to achieve a better understanding of the harmonic functions on Lipschitz domains.

One strategy for solving the Dirichlet problem for the Laplacian on a Lipschitz domain is to change variables in such a way that a harmonic function on the Lipschitz domain is pulled back via the change of variables to a solution, in the upper half space, of an elliptic operator with bounded measurable coefficients. (The coefficients of this operator come by taking the derivative of the defining function of the Lipschitz domain; hence, they are merely in L00.) The coefficients of the operator gotten in this way are independent of the Xn coordinate in ffi.+.. It is a well-known result of B. Dahlberg [10] that the LP Dirichlet problem is solvable for the Laplacian in a Lipschitz domain when p 2: 2. Corresponding to this (stated for the unit ball rather than the upper half space) is a basic positive result for the theory we are now considering:


Theorem of Jerison and Kenig [21]. Suppose Lu == div(A'V'u), is an elliptic operator whose coefficients A(x) are bounded measurable functions in B ~ ffi.n.

Suppose further that A(x) is independent ofthe radifll variable, i.e., A(x) == A(y) whenever x/lxl == y/lyl. Then the harmonic measure dWL E B

2(da).

Thus, if the coefficients of the operator are "radially independent" then the LP Dirichlet problem is solvable when p 2: 2, and this can be thought of as an

extension of Dahlberg's result on Lipschitz domains. A bit later, E. Fabes, D. Jerison, and C. Kenig [13] proved the following result:

Suppose Lu == div(A'V'u) is elliptic, and that A(x) is continuous on B. Let w(o) == sup Ix-Yl:oo IA(x) - A(y)1 denote the modulus of continuity of A on B.

x,YER Assume that

11 w(o)2 -- do < 00.

o 0

Then WL E BP(da) for every p < 00.

This result holds only for continuous coefficients, and, in particular, does not cover the operators which arise via a change of variables from the Laplacian on Lipschitz domains. In order to remedy this, B. Dahlberg proved a perturbation theorem which applied equally well to the case of discontinuous coefficients. In order to state his result, we recall a definition. Suppose that J-L is a positive measure on B. Then we say that J-L is a Carleson measure provided, for any x E aB and

r > 0, J-L(B(x; r) n B) < C.

r n- 1

We say that J-L is a Carleson measure of vanishing trace provided

J-L(B(x; r) n B) . . 1 --+ 0 umformly 10 x as r --+ O.

rn-

Theorem of Dahlberg [11]. Suppose Lo and LI are elliptic operators with coefficients Ai (x) bounded and measurable in B. For each Z E B, set a (z) == SUPYEB(z; '-;1'1) IAo(Y) - AI (y)l. Suppose that the measure a2 (z) o~~) is a Carleson measure of vanishing trace (here o(z) == 1 - Izl). Then WL o E BP(da) implies

thatwL, E BP(da)forany I < p < 00.

In Dahlberg's theorem, we have an easily-checked criterion that tells us that the coefficients of two operators are close enough near the boundary so that the solvability of the LP Dirichlet problem for one of the operators guarantees its solvability for the other operator. This result depends in a crucial way on the smallness of the Carleson measure on small balls. If the vanishing trace assumption


-------- -'-"" -"--" --"- -

201

=,::",~-==--~--=-~~~--~:=~~""-~ ~"',,="-"::::~~----=-~=-=~~='::""'~=::-~'~-==--~~':-----'''~::-_:::~~:-~--===.:::".::-----==--=-====:=:-=-

200 CHAPTER 9

is removed, then simple counterexamples show that the conclusion of the theorem is no longer valid. Nevertheless, in the absence of the vanishing trace condition, Dahlberg made the following conjecture:

Dahlberg's Conjecture. Assume that Lo and L [ are elliptic in B with bounded measurable coefficients Ao and A I, respectively. As above, set

a(z) = sup IAo(Y) - Al (y)1 yEB(z; ¥)

and assume only that ~2 ¥ is a Carleson measure. Then if WLo E BPO(da) (for

some Po > 1) then it follows that WL, E BPI (da) for some PI > 1.

This says that when we no longer assume smallness of the relevant Carleson measure, then what is preserved is the solvability of the LP Dirichlet problem in

some range of P not necessarily the same for both operators. In what follows we wish to sketch the proof of this conjecture, which is given

in [18] (R. Fefferman, C. Kenig, and J. Pipher). Let us begin by mentioning the method of proof of Dahlberg in his vanishing

trace theorem.

Assume that Lo and L I are elliptic operators, and that a2 ¥ is a Carleson measure of vanishing trace. Dahlberg introduces the one-parameter family L t = (1 - t)Lo + tLI, and sets Q(t) equal to the BP(da) norm of the harmonic measure

Wt of Lt. Then the argument proceeds by showing that for some large N we have \Q'(t)1 ::: CQ(t)N whereC ~ OastheCariesonnormofthemeasurea2 ¥ ~ O.

Since we are assuming that a2 ¥ has vanishing trace, we may essentially assume that C in the differential inequality above is as small as we like; hence, Q{l) is controlled by Q(O), and the proof will be finished. What remains is the proof of the differential inequality, or more precisely, something essentially equivalent to

it. Fixt E [O,l],andconsidertheBP(da)constantofwt = kt da,name1y Q(t).

By definition, there exists a surface ball on aB, ~,so that

Q(t) ::: 2(_1_ [kf da)IIP/ _1_ [ ktda. a(~) i", a(M i",

Suppose that ~ has radius r, and has center Xo E aBo Take A = B n B(xo; 2r). Then it is a simple consequence of the so-called "comparison theorem" (see Caffarelli, Fabes, Mortola and Salsa [2]) that if we restrict all our operators to the

region A, then the resulting harmonic measure is equivalent on ~ to that of the operator considered on all of B. (Equivalent in this context means that the ratio of the two measures applied to any Borel set in ~ is bounded above and below by

strictly positive constants.) We shall abuse notation by henceforth denoting all the


harmonic measures of Lt on A by Wt. Then this rescales everything so that

c ::: Wt(~) ::: I,

where c depends only on the ellipticity constant of L t (see [2]). This means that

to estimate Q{t) it suffices to estimate (using duality)

i fktda

under the assumption that II f II u' ( ~) = 1 and f is a non-negative function on a(tq

aA supported on ~.

The next step in the proof is that if rp ::: 0 is a smooth bump function with compact support in A, which is suitably normalized, then by Harnack's principle,

i fkt da ::: Co i Utrp dz

where LtUt = 0 and Ut laA = f. It is therefore enough to estimate fA Utrp dz, and

it is actually this quantity whose t derivative we shall bound. In fact, letting a dot stand for differentiation in t, we see that

i L,(Ut)h, dz == 0 V t where Lth, = rp, htlaA = 0,

so that

0= i Lt(Ut)htdz = i Lte(Ut)htdz + i L,(Ut)htdz + i Lt(Ut)ht dz.

Now the last of these three terms is 0, since LtUt = 0, and integration by parts in

the second term transforms it into

i Ut,pdz.

Integrating by parts in the first term gives the identity

i Ut,p dz = i sij(z)aiUt(z)ajht(z) dz

where (sij) = S and s = Al - Ao We must then prove that if s(z) = maXi,j ISij(z)\, then

i s\VUtllVhtldz ::: CQ(t)N.

It is quite easy to show that

iA[ s\VutllVhtl dz ::: C1 St U)(X) H(k,) da xE'"


203 202 CHAPTER 9

where S, denotes the Lusin area integral, i.e.,

S;(f)(x) = r IVutl Z8Z-

ndz Jr(X)

and where H is a classical Calder6n-Zygmund singular integral. Now to finish the proof, Dahlberg uses HOlder's inequality to arrive at

Ii U,qidzl s CIIS,jIlU'(dcr)II H (k/)IIU(dcr,Il)'

By the boundedness of singular integrals on LP(da), we have

IIH(kr)IIU(dcr,ll) S CIIk/ ll u (dcr,ll) S a(ll)llp.

As for II S,j lI u '(dcr) , according to a theorem of Dahlberg, Jerison, and Kenig [12] the area integral S, is bounded on all LP (dw,) for I < P < 00 for all elliptic

operators. We need to estimate here the quantity II S,j II u' (dcr) , not IIS,jllu(dw,). However, by a good ).. argument one can show [12] that the operator S, is also bounded on all LP(dlL), where IL is any measure satisfying an Aoo condition with

respectto w,, Thus II S,j II u' (dcr) sell f II U' (dcr) where the constant C is dominated by a constant depending only on P times the norm II d a II A 00 (dw,) and this last quanti ty turns out to be less than C Q(t)N for a large power N. This competes the proof of the differential inequality and Dahlberg's theorem.

Now, let us pass to the proof of the conjecture, where we only assume that the measure aZ ¥ is a Carleson measure in B, with no smallness assumption. We assume that the LP Dirichlet problem is solvable for the operator Lo when

P = Po, and then seek to show that the LP Dirichlet problem can be solved for

L I when P = PI, and where PI might be much larger than Po. In terms of the harmonic measure Wi, i = 0, I of the operators L;, this means wo E BPO(da)

implies WI E BPI (da) for some I < Po, PI < 00. It will be important to think of this in terms of the so-called A 00 class. This class of weights can be defined in a great many different ways (see Coifman-C. Fefferman [5]). Here we recall that

AOO(da) = U BP(da), p>1

so our conjecture assumes that Wo E AOO(da), and we now show that WI E

AOO(da) which proves it.

To do this we must first single out an important special case: Assume rather

than aZ ¥ a Carleson measure, the stronger condition

11 dt (A) aZ((1 - t)x) - S C for all x E aBo

o t


Then we show under assumption (A) that WI E AOO(da) (see R. Fefferman [17]). To do this we use a novel characterization of AOO(da): Observe that W = k da E

A 00 (da) if and only if it satisfies

IIkllLlog+ L(Il' E....)O'l6.) S Cllk ll v (Il', E....)• 0(6)

and we let L , = (l - t)Lo + tL\ as before, but here we set

ilk/ilL log+ L(Il; E....) Q(t) = sup Ilk II 0(6) for t E [0, 1].

Il / V (Il; O~~) )

The idea is then to show that because Q(t) is now defined in terms of an Orlicz

class near L I, we actually have

IQ(t)1 s CQ(t)

and this controls Q(l) in terms of Q(O) without any smallness of C. We do this

as follows: We show that the quantity Q(t) can be bounded by fll fk, da for

some choice of II and a non-negative f supported in II satisfying II f IIBMO + IIfllv(..!!!!-) S 1. Then an analysis similar to Dahlberg's shows that

(J"(~)

IQ(t)1 Sci s(z)IVu/(z)IIVh/(z)1 dz

where

L/u/ = 0 in A, u/laA = f and

L/h/ = cp in A, u/laA = O.

We estimate fA sIVu/IIVh/ldz by breaking this up into two parts A = AI U Az where we write II = B(xo; r) naB for some Xo E aB and r > 0, and then define Al by AI = B(xo; 3/2r) n B and Az = A - AI. The main estimate is the term

fA slVutllVh/ldz which we carry out below. I

{ sIVu/IIVh/1 dz JAr

S Ca { ~ 8I -

n(z)s(z)IVu/(z)IIVh,(z)1 dZ) da(x).(1lXEIl fa(x)

Now we estimate

1 81-n(z)s(z)IVu/(z)IIVh/(z)! dz fa (x)

ill ~ S 8I -n(z)s(z)IVu/(z)IIVh/(z)1 dz

o B«(I-p)x;ap) P


---

204

~--

CHAPTER 9

I ( )IPS C ( pl-naa((l - p)x) ( lVur(z)12 dz10 1B((I-p)x;ap)

1/2 dp

IVhr(z)1 2 dz ( )l«(l-p)x;ap) p

(here aa(Z) = sUPYEB(z;a6(z» 1£(z)1)

I _ ( ) 1/2 S C ( pl-naa((l - p)x) IVur(z)1 2 dz1

10 B«(l-p)x;ap)

(*) p-lpn/2p2-nWr(ilx,p) dp p

where ilx,p is the surface ball centered at x of radius p, and we have used Cacciopoli's inequality as well as the estimate

hr((l - p)x; ap) S Cap2-nwr(ilx,p).

Then

1 ( ) 1/2(*) s ( aa((l - p)x) ( IVur(Z)12c52-n dz pl-nwr(ilx,p) dp .

10 1B«(l-p)x;ap) p

Thus

{ £IVur(z)IIVhr(z)1 dz S ({ aa(Z)Szwr(ilz) d:lA 1 llA1 c5

1/2 where Sz = ( ffB(z;a6(Z» lVur(y)1 2c52- n(y) dy ) and ilz = il( Vi; c5(z)). The last integral above is easily seen, by Fubini's theorem and the Cauchy-Schwarz inequality, to be bounded by

C ( S(f)(x)dwr(x).laA

Therefore our claim is reduced to showing that

( S(f)(x)dwr(z) S CQ(t),laA

We proceed as follows:

1/2 ( 1/2lA S(f)(x)dwr(x) S [lA S2(f)dwr(z) S C lA f2 dWr)J

",.~~,.... _~~~":;~~-';;":~;;::'';'=:~~ :;:.-:~,';~~~;';~~~::::::~~.~.-;:,-::~~<, '-;';.~.-;;,~ ;.:~·::~':,".~:::.,~;:;:;;,~,..:--;;:~,.;=~_.~';"'-;;,"~.;;~:;.";':'7o,~-;:-.:"'-:::"·~;;';;;;"';;:-':::;;;;::;:;":7~~~~;:~~:~~';'~~~~;~~:';:;:;';;;;'~~~~~~~;~~~~:7.".;~;;: ..;;::~~-..;.~.;;.-..;;,.-;,c_=~

205SOME TOPICS FROM HARMONTC ANALYSTS

Let h,w, = w)!1) It. f dWr. Then

1/2 ( )1/2 ( )1/2lA f2 dWr S i If - h,w,1 2 dWr + i fl,w dWr( )

"c [(w)D.> i If - fA~12dW,r + IfA.~I]. We observe that

h,w, = W)il) i fkr da S W)il) (i If - hlkrda + h) where ft. = rr/t.) It. fda S 1. Also

1 { 1 1 ( wr(il) It. If - hlkr da = [w,(t.)] a(il) It. If - hlkrda

rr(!1)

< Cllfll IIkrllLlog+ L(drr/rr(t.» - BMO(drr) It. kr da/a(il)

S CQ(t).

The term

1 ( 2 ) 1/2

( wr(il) 1t. If - ft.,w, 1 dWr

is dominated by a constant depending only on the doubling constant of Wr (which, by [2] depends only on the eIlipticity constant of L r ) times the quan

tity IIfIlBMO(dwl)' Now, we claim IIfIIBMo(dwl) is bounded by CQ(t). To see this we estimate: For ilo ~ il

I 1 11 1-(A) If - holdwr = -(A)a(ilo)-(A) If - ft.olkr da Wr uo t.o Wr uo a uo t.o

Cllfll IIkrllLlog+ L(drr/rr(t.o»S BMO(drr) [ ]

WI (t.o) rr(t.o)

S CQ(t).

This completes the proof of the conjecture under the more stringent hypothesis lthati a2«(l- t)x)~ E LOO(da(x)). In ordertoreduce the general case where o

a2 ¥ is a Carleson measure to this one we proceed as follows:

Choose a surface ball il = il(xo; r) ~ aB and a subset F ~ il with ~f~~ > 1 - ~. We will show that WI E AOO(da) by proving that ::f~~ > £ for some

£ > O.


206 207 CHAPTER 9

Observe that

~ r (r a2((l _ t)x) dt) da(x) ~ C a (Ll) JXEt. Jo t

by the Carleson condition we are assuming. This means that on a set E such that aCE) ~ 4a (.1.) we have

21,a ((1 - t)x) -dt

~ c' = 2C for each x E E.o t

Define r~(x) to be the right circular cone with vertex at x E aB, aperture ex (suitably small) and truncated at height r. Set R = U r~ (x). Then define the

xEE intermediate operator Mu = div(AVu) by

Ao(z) ifz fI. R A(z) =

( AI(z) ifz E R

A simple geometric argument shows that if

B(z) = IA(z) - Ao(z)l, z E B and if

a(z) = sup B(y) yEB(z;a8(z»

then for ex sufficiently small, we have

11 dt Q2((l - t)x) - ~ C' for all x E aBo o t

Thus, by the special case treated above, the harmonic measure of the operator M, WM E AOO(da). Now the main question is "What is the relationship between WM

and WL, ?" This is answered by a lemma from [12].

Lemma. Suppose that.1. ~ aB is a surface ball of radius r. Let R = UXES r~(x) for some set S ~ .1.. Then there exists a constant C > 0 and e > 0 so that for all subsets E ~ S and operators M and L I whose coefficients agree on R we have

WL j (E) [WM(E) JII-->C -WL, (.1.) - WM (.1.)

Applying this lemma in our context gives

WI (F) wI(F n E) C [WM(F n E)]II II-- ~ > > > C11WI (~) WI (.1.) - WM (.1.)


if 11 > 0 is so small that by the A00 property of WM,

a(H) 1. . wM(H) -- > - 1ffiphes -- > 11 for all H ~ ~. a (.1.) 4 WM (~)

Thus, under the assumption that :~~j > 1 - 1510' we have proven that ::~~j > C1111 and this shows that WI E AOO(da) completing the proof of Dahlberg's conjecture.

Finally, we would like to discuss a beautiful connection between the Dahlberg

Conjecture and the Helson-Szego Theorem from harmonic analysis.

It is a simple matter to construct, given any doubling measure JL = f dx on JEt I,

an elliptic operator L on JEt~, whose harmonic measure WL is JL. The difference (using the notation discussed above) between the coefficients of this L and the

identity matrix a(x, t) for (x, t) E JEtt is essentially given by

a(x, t) = if * Vrt(x)! f * CPt (x) .

Here Vr, cP E Co (JEtI), with JVr dx = 0 and cP ~ 0 with JcP = 1. It is a special case of Dahlberg's Conjecture that if

If * Vrt(x)\2 dxdt

(f * CPt (x))2 -t

is a Carleson measure in JEtt, then f E AOO(dx). Actually this condition gives a complete characterization of Aoo(JEtn) for any dimension n.

Theorem [18]. Let JL = f dx be a positive measure on JEtn which satisfies the doubling condition JL(B(x; 2r)) ~ CJL(B(x; r)) for all x E JEtn, and r > O. Then f E A00 (JEtn) ifand only if

f * Vrt(X) /2 dxdt -- is a Carleson measure on JEt~+l.

I f * CPt (x) t

It is interesting to note that the assumption that the measure f dx is doubling is crucial, and the theorem is incorrect if this assumption is deleted.

Now how is this related to the Helson-Szego Theorem? The latter states:

Helson-Szego Theorem. A measure JL = f dx on JEtI is such that the Hilbert transform is bounded on L2 (d JL) ifand only if

log f = b l + H(b2) where bi E LOO (JEt1), i = 1,2, and IIb2 1100 < 77:/2.


209 208 CHAPTER 9

Taking into account the celebrated Hunt-Muckenhoupt-Wheeden Theorem on weighted norm inequalities for the Hilbert transform, as well as the C. Fefferman

Stein Decomposition ofBMO, we see that except for the restriction II b2 11 "" < Jr/2, the He1son-Szego Theorem would say that / E A2(~1) if and only if log / E

BMO(lR1). In a sense this says that log / E BMO is almost enough to guarantee

that / is a weight in A2 , and it tells us exactly the difference between these two

conditions. What we have in our theorem above is an analogue of this for A00

weights in all dimensions. To see this, let us write that log / E BMO using

C. Fefferman's characterization of BMO in terms of Carleson measures:

log / E BMO(1R1) if and only if dxdt

j(1og f) * VrtI 2 - t

is a Carleson measure in 1R~. Suppose Vr = qi. Then this last condition can be

written

2 2 OlfJr 1 dxdt I a 1 dxdtt (log f) * - -- = t - [(log j) *IfJr] -- is a Carleson measure. ax t ax tI

Now suppose we commute the taking of the logarithm with the process of

averaging (convolving with IfJr). We get (the inequivalent) condition

2 a [2 dxdt I / * Vrt 1 dxdt .t - [log(f * IfJr)] -- = -- -- IS a Carleson measure. I ax t / * 1fJ, t

But this is exactly the condition which we proved in [18] is equivalentto / E A"". This tells us that our Carleson condition, shown to characterize in all dimensions the

space A00 (IRn ) is nothing more than a commutation away from log / E BMO(lRn ).

In this way we have a version of the Helson-Szego Theorem for the space A"". Not only does the Dahlberg Conjecture imply a special case of our character

ization of A00, but the characterization tells us that the Dahlberg conjecture is

completely sharp. Thus, for instance, if we start with the Laplace operator and run

through all elliptic operators whose coefficients satisfy

2 dz. C a 8" IS a arleson measure,

then the class of harmonic measures which results is precisely the class A"". The

conclusion of Dahlberg, given his assumption, is therefore absolutely sharp. Sim

ilarly, the equivalence of the A 00 condition with the Carleson condition shows that

under any assumption weaker than that made by Dahlberg, we cannot conclude

that the resulting operator will have its harmonic measure in A"" (da).

University a/Chicago


REFERENCES

[1] L. Caffarelli, E. Fabes, and C. Kenig. "Completely Singular Elliptic Harmonic

Measures." Indiana Univ. Math. J. 30 (1981). [2] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa. "Boundary Behavior of Non-Negative

Solutions ofElliptic Operators in Divergence Fonn." Indiana Univ. Math. J. 30 (1981). [3] L. Carleson. "A Counterexample for Measures Bounded on HP for the Bidisc." Mittag

Leffler Report No.7 (1974). [4] S. Y. Chang and R. Feffennan. "A Continuous Version ofthe Duality of HI and BMO

on the Bi-Disc." Ann. ofMath. 112 No.2 (1980). [5] R. Coifman and C. Fefferman. "Weighted Nonn Inequalities for Maximal Functions

and Singular Integrals." Studia Math. 51 (1974). [6] A. Cordoba. "Maximal Functions, Covering Lemmas, and Fourier Multipliers." Proc.

Symp. Pure Math. 35, Part I. Amer. Math. Soc., 1978. [7] A. Cordoba and R. Feffennan. "A Geometric Proof of the Strong Maximal Theorem."

Ann. ofMath. 102 (1975). [8] A. Cordoba and R. Feffennan. "On the Equivalence Between the Boundedness of

Certain Classes of Maximal and Multiplier Operators in Fourier Analysis." Proc. Nat.

Acad. Sci. 74 No.2 (1977). [9] A. Cordoba and R. Feffennan. "On Differentiation of Integrals." Proc. Nat. Acad. Sci.

74 No.6 (1977). [10] B. Dahlberg. "On Estimates ofHarmonic Measure." Arch. Rat. Mech. Anal. 65 (1977). [11] B. Dahlberg. "On the Absolute Continuity of Elliptic Measures." Amer. J. Math. 108

(1986). [12] B. Dahlberg, D. Jerison, and C. Kenig. "Area Integral Estimates for Elliptic Differential

Operators with Non-Smooth Coefficients." Arkiv. Math. 22 (1984). [13] E. Fabes, D. Jerison, and C. Kenig. "Necessary and Sufficient Conditions for Absolute

Continuity of Elliptic Harmonic Measure." Ann. ofMath. 119 (1984). [14] C. Feffennan. "The Multiplier Problem for the Ball." Ann. of Math. 94 (1971). [15] R. Feffennan. "Calderon-Zygmund Theory for Product Domains: H P Spaces." Proc.

Nat. Acad. Sci. 83 (1986). [16] R. Feffennan....AP Weights and Singular Integrals." Amer. J. Math. 110 (1988). [17] R. Feffennan. "A Criterion for the Absolute Continuity of the Harmonic Measure

Associated with an Elliptic Operator." J. Amer. Math. Soc. 2 (1989). [18] R. Feffennan, C. Kenig, and J. Pipher. "The Theory of Weights and the Dirichlet

Problem for Elliptic Equations." Ann. ofMath. 134 (1991). [19] R. Feffennan and K. C. Lin. "A Sharp Marcinkiewicz Multiplier Theorem." To appear

in Ann. Fourier Inst. Grenoble. [20] R. Gundy and E. M. Stein. "HP-Theory for the Polydisk." Proc. Nat. Acad. Sci. 76

(1979). [21] D. Jerison and C. Kenig. "The Dirichlet Problem in Nonsmooth Domains." Ann. of

Math. 113 (1981). [22] B. Jessen, J. Marcinkiewicz, and A. Zygmund. "A Note on Differentiability ofMultiple

Integrals." Fund. Math. 25 (1935). [23] J. L. Joume. "Calderon-Zygmund Operators on Product Spaces." Rev. Mat. Iber. 1

(1985).


210 CHAPTER 9

[24] W. Littman, G. Stampacchia, and H. Weinberger. "RegularPoints for Elliptic Equations With Discontinuous Coefficients." Ann. Scuola Norm. Sup. Pisa 17 (1963).

[25J J. Moser. "On Harnack's Theorem for Elliptic Differential Equations." Comm. Pure Appl. Math. 14 (1961).

[26] A. Nagel, E. M. Stein, and S. Wainger. "Differentiation in Lacunary Directions." Proc. Nat. Acad. Sci. 75 (1978).

[27J E. M. Stein. Singular Integrals andDifferentiability Properties ofFunctions. Princeton University Press, 1970.

(28] 1. Stromberg. "Weak Estimates on Maximal Functions with Rectangles in Certain Directions." Arkiv. Math. 15 (1977).

---C;~';;~""~~,",,:::;-~~~;~~=;;;:W~_:~'~~':~"~~,~_...;.;.;.;;,;:,~~~~~~:.: __ ~,._.-:-_._.:~~~", _:,~~~_~~.~~~~_ .._:::-:::::~,:"~-=-_"~-;;;;;~--:.:;::'~:;:,;;;:~~;;;;=

10

Function Spaces on Spaces

of Homogeneous Type

Yongsheng Han and Guido Weiss

1 INTRODUCTION

There are many topological vector spaces that arise naturally in Analysis. In the theory of partial differential equations and in harmonic analysis these spaces consist of functions, or, more generally, distributions defined on the n-dimensional real Euclidean space ]R". Some of the best known examples are the Lebesgue spaces LP(]R"), I S p S 00, the Lipschitz spaces A a , 0 < CL < 1, the Hardy spaces H P, and the Sobolev spaces L~, CL > 0, I S p < 00. These are examples of a scale of spaces that are naturally defined by a method known . as Littlewood-Paley-Stein Theory. Our purpose is to describe how this theory can be extended so that this scale of spaces can be introduced in the general setting of spaces of homogeneous type. There are, in fact, two scales: the Besov spaces and the Triebel-Lizorkin spaces. The former includes the Lipschitz spaces and the latter includes the Hardy, Lebesgue, and Sobolev spaces.

We shall not present here a detailed description of the Littlewood-Paley theory in its various classical settings. We refer the reader to [FJW) for a presentation of this subject that is appropriate for our purposes. Another account of this theory that, in addition, explains E. M. Stein's considerable contribution to this subject can be found in [eWl). In order to explain our program, however, we do have to present those aspects of the classical Littlewood-Paley theory associated with ]R"

that are needed for the extension of the spaces in question in the setting of spaces of homogeneous type.


213 212 CHAPTER 10

We begin with a version of the Calder6n reproducing formula. It is clear that we can construct a function ({J E S satisfying

Supp~ C {~: 1/2 ~ I~I ~ 2},(1.1)

I~(~)I ~ c > 0 if 3/5 ~ I~I ~ 5/3.

The result that we call the Calderon reproducing fonnula then asserts that we can find two such functions ({J and 1/1 in S such that

(1.2) f = L 1/1k * iPko *f, kEZ

where 1/Ik(x) = 2kn 1/l(2kx), ((Jk(x) = 2kn ({J(2kx) and iP(x) ~ ({J(-x) (see [PI]). The convergence of the series can be taken in several senses: for example, in

L2 and, much more generally, S'/P (that is, the space of tempered distributions modulo the polynomials). For each dyadic cube Q of side length l(Q) = 2-k ,

Q = Qkj = {x E IRn : 2-k

ji ~ Xi ~ rk(jj + I), i = 1,2, ... ,n} , where k E Z and j = (jl, h. "', jn) E zn, let

({JQ = 2kn /2({J(2kx - j)

and, similarly,

1/1 Q = 2kn/21/1(2kx - j).

Then it is not hard to show, once (1.2) is established, that

(1.3) f = L (j, ({JQ}1/I Q' QEQ

where {g, h} = g(ii) when g E S' and h E S (this is the ordinary L 2 inner

product when g E L 2), and Q denotes the collection of all dyadic cubes. The relatively simple argument establishing (1.3) is an extension of one that gives us the Shannon Sampling Theorem and can be found in [FJW].

The function ({J and the coefficients (j, ({J Q) can be used to characterize the spaces that are of interest to us. For the moment we shall restrict ourselves to

the homogeneous versions of the Besov and Triebel-Lizorkin spaces. For a E IR, p "f: 00,0 < p, q ~ 00, the Besov space i3;.q consists of all elements f in S'I'P such that

q

(1.4) IIf11n;,· = [L (2ka ll({Jk * flip rJ 1

/ < 00. kEl

FUNcnONSPACES ON SPACES OF HOMOGENEOUS TYPE

The Triebel-Lizorkin space P;,q consists of all elements f in S'I'P such that

(1.5) IIf11!;" ~ ~[fu (2"1~, < co.* fI)'rl The "dot" denotes the fact that we are considering the homogeneous version of

these spaces. It is most important to show that these definitions are independent of the choice

of the function ({J E S satisfying (1.1). This independence follows from the Calderon reproducing fonnula. To see that (1.4) defines the Besov space i3;.q independently of the choice of ({J, it suffices to show that there exists a constant C

such that

qJl/q [ qJ1/q

(1.6) [L (2ka lllh * flip) ~ C L (2ka ll({Jk * flip)

kEZ kEZ

whenever 11 is another function satisfying (1.1). For simplicity we show this when -1 < a < 1 and 1 < p, q < 00. By interchanging the roles of ({J and iP in (1.2),

however, we can find 1/1 satisfying (1.1) such that

(1.7) 11k * f = 11k * L 1/1 i * ({Ji * f. iEZ

Thus,

lI11k * flip ~ L II 11k * 1/I i * ({Ji * flip ~ L lI11k * 1/Iilllll({Ji * flip· iEZ iEZ

An easy calculation shows that

2-k V 2-i

(1.8) 1(11 * .1, )(x)1 < crlk-il ~__...,.. 'Ik 'Yi - (2-k V 2-i + Ixj)n+l'

where a vb = max{a, b}. This estimate clearly implies

1k illI11k * 1/I i lll ~ cr

and, consequently, the left side of (1.6) does not exceed

c{ L(L 2karlk-illl({Ji * fIIp)q},/q kEl iEZ

= C{L(L {2iall({Ji * flip [2(k-i la-1k-i l]'/q} kel iEl

{2(k-i)a-1k-i l} I/q,) q } I/q


215 214 CHAPTER 10

S C{L{L2iaqllrpi * fIIP(k-Oa-1k-i l } kEZ lEZ

{L 2(k-i)a-lk-il }q/q,} I/q

iEZ

I/qiaqS C L 2 IIrpi * fII~ L 2(k-Oa-lk-i l

{ }iEZ kEZ

i }I/q

= C L ziaqllrPt * fII~ , fEZ

where (1/q) + (l/q') = 1. HOlder's inequality was used to establish the first inequality, and C denotes a constant (depending on a and q) that varies appropriately.

Similarly, the finiteness of the expression (1.5) that defines the norms for the Triebel-Lizorkin spaces is independent of the choice of the test function rp satisfying (1.1). Again, this can be shown with the aid of the Calderon reproducing formula: By (1.7) and (1.8) we have the pointwise estimate

l17k * f(x)1 S C L r1k-iIM(rpi * f)(x), iEZ

where M is the Hardy-Littlewood maximal function operator. Thus,

111~C2'"'",. fl)' rt "C III~(~ 2'"T'HMC_,' fl)'j 11'11,

But, applying first HOlder's inequality and the assumption -1 < a < 1 and then the Fefferman-Stein vector valued maximal function inequality [FS] (valid here since 1 < p, q < 00), we see that the last expression is dominated by

c III~(2'" MC_, • f))f'll, "C III ~(2'" 1_, •f1lf'll, The equivalence of the norms defined in terms of 17 and rp clearly follows from these inequalities.

Let us now return to equality (1.3). We have stated that the coefficients sQ = (f, rp Q} can be used to characterize the spaces we have introduced. Toward this end we introduce the spaces b~,q and i;,q of sequences S = {sQ} indexed by the dyadic cubes Q and satisfying

I a I JP)q/P}I/qIlsllb;,q ==

{ L

( L

[ IQIP -;; - ilsQI < 00

kEl. i(Q)=2-'

FUNCTION SPACES ON SPACES OF HOMOGENEOUS TYPE

and

< 00.II s llj;,q == 11(~[IQI-;;-tISQIXQry/q P

The following theorem gives us this characterization (see [FJW]):

Theorem (1.9). Suppose a E R 0 < p, q S 00, and the functions rp and 1/1 are as in (1.1) and (1.2). Given f E S' /P let S~ = (f, rp Q} ,for each dyadic

cube Q (so that equality (1.3) holds). Then f E B;,q ifand only if the sequence

S = {sQ} E b~,q;and f E P;,q if and only if the sequence s = {sQ} E i;,q,

provided p < 00. When this is the case we have the norm equivalences

II s II jU,q ;:::: II fII F"q and lis II ba ,q ;:::: II fII Ba ,., Jp p p p

There are several reasons why these spaces are "natural" and why Theorem (1.9) is useful. The space L 2 is rather special in many ways; for example, it is endowed with an inner product with respect to which the Fourier transform is a unitary operator, and the bounded convolution operators on it are characterized by the fact that their kernels have bounded Fourier transforms. These properties can be used to give rather simple proofs of the boundedness of many operators on L2 • While these properties are no longer true for LP, P # 2, several of these boundedness results still hold when 1 < p < 00, but they fail to be true for either L I or LOa. There are, however, appropriate replacements for these two spaces on which such boundedness properties are valid. In the one dimensional case, the Hilbert transform provides an example of this situation, where the Hardy space H I replaces L I and BMO replaces LOa. The scale of spaces P;,q includes the spaces LP for 1 < p < 00 (p~,2 is LP in this case) and it also includes HI

(= p~,2) and BMO (= p~2). In addition to this the Hilbert space techniques that simplify these problems in the case p = 2 have somewhat successful extensions to the scale of spaces we are describing. This is a consequence of Theorem (1.9) and the "almost orthogonal" expansion (1.3). In a well defined sense the operators of interest correspond to matricial operators on the sequence spaces b~,q and i;,q

that are "almost diagonal" (see [FJW] and [T] where many of these features of the Triebel-Lizorkin and Besov spaces are discussed).

As stated at the beginning, our purpose is to describe how one can define TriebelLizorkin and Besov spaces in the setting of spaces of homogeneous type. We are now in a position to be more specific about our goal. In this general setting we do not have an underlying group that gives us a convolution which can be used in the definitions of the norms given in (1.4) and (1.5). The lack of a convolution structure is also an obstacle for obtaining the reproducing formula (1.2). Nevertheless, the


217 216 CHAPTER 10

original motivation for the introduction of spaces of homogeneous type was to introduce a most general setting in which the ideas of hannonic analysis can be extended. Moreover, many important convolution operators, such as the CalderonZygmund Singular Integral Operators, have been extended to operators defined by more general kernels, such as the Calderon-Zygmund Operators (see [CM] and [J]). Thus, it is natural to try to extend more general operators and their properties

when the underlying space is of homogeneous type. Such extensions can be found in the literature (see [DJS], for example); however, these results do not include the

scale of spaces that we are considering. Very recently considerable progress has

been made in this dire~tion. In this paper we describe this effort. We shall not

prove all the results we announce; however, we will give complete references of

the work now in progress. This paper should be considered to be a companion to

[HS], where the details that are not contained here are given. On the other hand, much of the motivation for the material developed in [HS] is presented here.

2 MOTIVATION FOR A CALDERON REPRODUCING FORMULA ON SPACES OF HOMOGENEOUS TYPE

A quasi-metric on a set X is a function p: X x X ---+ JR+ = {r E JR: r ::: O} satisfying

(a) p(x, y) = 0 if and only if x = y, (b) p(x, y) = p(y, x) for all x, y E X,

(c) p(x, y) ~ K[p(x, z) + p(z, y)] for some K < 00 and all x, y, and z in X.

That is, a quasi-metric satisfies the properties of a metric except that the triangle

inequality is replaced by the more general condition (c) (clearly K must be at least

1 if X contains more than one point). A space of homogeneous type (which we will denote as an HT space) consists of a measure space X that, in addition to a

measure J-L, is endowed with a quasi-metric p; moreover, J-L and p are related in

the following way:

(i) J-L(B(x, r)) < 00 for all x E X and r > 0, where B(x, r) = (y E

X: p(x, y) < r}, (ii) there exists a positive constant c such that for all x E X and r > 0

J-L(B(x, 2r)) ~ cJ-L(B(x, r)).

If we want to introduce norms that are analogous to the ones in (1.4) and (1.5),

we must find appropriate substitutes for the convolution operators f ---+ ({Jk * f. Moreover, in order to develop the theory of the spaces defined by these new norms along the lines we indicated above, we also need a version of the Calderon

FUNcnONSPACES ON SPACES OF HOMOGENEOUS TYPE

reproducing formula that has the same basic features as the one involved in equality (1.2), that does not depend on a convolution. We claim that this can be done in JRn by using certain families of Calder6n-Zygmund operators. We begin by describing

a version of this program that can be found in [HITW]. In this context the operators we consider are linear and map the space of test

functions V(JRn) continuously into the space of distributions V'(JRn). By the

Schwartz kernel theorem each such operator T is associated with a distribution

K E V' (JRn x JRn) such that

(Te, 1J) = (K, 1J 0 e),

for all e, 1J E V. K is called a Calder6n-Zygmund kernel if its restriction to the set of all (x, y) E JRn x JRn such that x i- y is a continuous function K (x, y) that

satisfies: 1

(2.1) IK(x, y)1 ~ c for all x i- y,Ix - yin

Iy - y'l'(2.2) IK(x, y) - K(x, y')1 ~ c I . ,_ whenever21y - Y'l ~ Ix - YI,

x-y

Ix - xii' (2.3) lK(x, y) - K(x' , y)1 ~ c I ., _ whenever21x - xii ~ Ix - YI,

x-y

for some constant c > 0 and E E (0, 1]. In this case T is called a Calder6n

Zygmund operator and we write T E CZ 0 or, if we wish to emphasize the

"Lipschitz condition of order E," we write T E CZO(E). If T E CZO, e E V,

and x ¢ Supp e, then

Te(x) = { K(x, y)(j(y)dy.JR'

The family of operators that can be used to give a more general Calderon repro

ducing formula consists of such integral operators, where the above equality holds

without the support restriction. We call such a family, {SklkEZ, an approximation

to the identity. It consists of a sequence of integral operators with kernels Sk(X, y),

k E Z, satisfying the following conditions:

2-kf

(2.4) ISk(X, y)1 ~ c (2-k + Ix _ yl)n+, , for all x, y E JRn;

2-k' [IX - xii ]'

(2.5) ISk(x, y) - Sk(X' , y)1 :::: c (2-k + Ix _ yl)n+' 2-k + Ix - yl

for Ix - xii ~ ~ (2-k + Ix - yl);

2-kf Iy - y'I ]'(2.6) ISk(x, y) - Sk(X, Y')I :::: c (2-k + Ix _ YI)n+, [2-k + Ix yl


-----------

219 218 CHAPTER 10

for Iy - y'l ::s ! (2-k + Ix - yi);

(2.7) { Sk(X, y)dy = 1 for all x E JRn and k E Z;JIRn

(2.8) { Sk(X, y)dx = 1 for all y E JRn and k E Z.JRn

Christ and Ioume [CI] considered families such as these that satisfied conditions (2.4) and (2.5). When the last three properties are added one can use (Ski to obtain norms that are equival<:nt to the ones given in (1.5) and (1.6). More precisely, the following result is shown in [HJTW]:

Theorem (2.9). Suppose that {Sk}. k E Z, is an approximation to the identity and Dk = Sk - Sk-l· Thenfor -E < ex < E and I ::s p, q ::s 00,

l/q (2.10) IlfIlB;'. ~ L(2ka IlDdllp)q[ ]

kEZ

and

(2.11) 11111;.;' ~ II[fu(Z""Ddll'J'l The role played by a more general Calderon reproducing formula in establishing

these equivalences is the following. Suppose we want to show that

l/q (2.12) IIf11B;'. ::s e L(2kaIlDdllp)q .{ }

kEZ

We can do this by making use of an idea due to R. R. Coifman. He observed that it follows from the properties (2.4)-(2.7) that

(2.13) 1= LDk. kEZ

From this equality (which is easily established when, say, the operators involved are applied to test functions), we obtain

(2.14) I = {L Dk} {L Dt} = L DkDt + L DkDe == RN + TN, kEZ eEZ Ik-lI>N Ik-el~N

where N is a fixed positive integer. In [DIS] it is shown that the operators TN are bounded on L 2 (Rn

) and converge to I, as N -+ 00, in the strong operator topology. Moreover, it is shown in [HITW] that

I/q (2.15) IITNfIIB;. ::s e L(2kaIlDdllp)q .{ }

kEZ

._--,_...__.-.-_._--------,_._....__._


The operators RN , on the other hand, belong to CZO(E') for 0 < E' < E and their kernels RN (x, y) satisfy the following conditions: There exist e, .5 > 0 such

that

(i) IRN(X, y)1 ::s erN8 1x _ yl-n;

IRN(x, y) - RN(x', y)1 ::s erN8 1x - x'I€'lx _ yl-(n+€')

, 1 (ii) for Ix - x I ::s 2: Ix - YI;

jRN(x, y) - RN(x, y')1 ::s erN8 1Y - y'j€'lx _ yl-(n+€')

(iii) for Iy - y'l ::s 2:I

'x - YI;

moreover, these operators satisfy the following version of the weak boundedness

property

(iv) I(RNf, g)1 :s crN8 tn(lIf11oo + tliVflloo)(llglioo + tllVgll oo )

for all f, g E V having support in a cube Q whose diameter is t > 0, as well as

the David-Iourne condition

(v) RN I = R~ I = 0 modulo constants.

We can then apply the "Tl theorem" for Besov and Triebel-Lizorkin spaces that is proved in [FTW], [T], or [HITW] to conclude that

(2.16) IIRNfIIBu,. < erN8 I1fIlBu,., p - p

(2.17) IIRNfllpu,. :s e2-N8 I1f11pu,., p p

when -ex < E < ex and I ::s p, q ::s 00. These estimates applied to TNl

(/ - RN) -I = L~=o R~ show that this operator is bounded on iJ~·q and on i;·q for these parameters ex, p, and q. We thus obtain another Calderon-type

reproducing formula:

(2.18) f = TN1TNf = TN 1 LDfDd,

kEZ

where Df = LIiI~N Dk+j (so that TN = Llk-e\~N DeDk = LkEZ Df Dk). We can now give a simple proof of (2.12): Since TN is bounded on iJ~·q ,

I/q 1IIfIlB;. = II TN TN fIIB;'. ':S ell TN fIIB;·. ::s e L(2ka II Ddllp)q ] ,[

kEZ

where the last inequality follows from (2.15).


221 220

CHAPTER 10

The inequality in the opposite direction to the one in (2.12),

(2.19) { L(2ka

llDk/ll p )q }

l/q

::5 cllfllB~,q, kE'Z

is easier to establish. In fact, it can be done by exactly the same argument used for proving (1.6) by replacing the operator g --+ 1]k * g with the operator g --+ Dkg.

All that is needed is to observe that inequality (l.8) remains valid if 1]k * 1/1i is replaced by Dk 1/1i" This, then, gives us (2.10). Completely analogous estimates give us (2. I I). We are now ready to extend these ideas and results to HT spaces.

3 THE CALDERON REPRODUCING FORMULA ON SPACES OF HOMOGENEOUS TYPE

When working on HT spaces it is often quite useful to introduce the measure distance m(x, y), equal to the measure of the smallest ball containing x and y. Without loss of great generality we can assume that m (x, y) is the quasi-metric metric p (x, y) and that it satisfies a Lipschitz smoothness condition

!p(x, y) - p(x' , y)1 ::5 cp(x, x,)p[p(x, y) + p(x' , y)]1-tl

(see [CW2] and [MS]). For example, replacing the usual Euclidean distance in n

lR by its nth power gives us such a measure distance which does not change the topology of the underlying space. The estimates we shall present require a smoothness property of this type (though much of what we present is also true when the HT space is discrete) and their fonn and derivations are simpler if we are dealing with a measure distance.

There are many ways of introducing approximations to the identity in HT spaces. A construction by Coifman gives us a "continuous approximate identity"

Prf(x) = l Pr(X, y)f(y)dJ-t(y),

where Pr(x, y) = 0 for p(x, y) > cr, Pr(x, y) = Prey, x), IPr(x, y) Pr(x, y')1 ::5 p(y, y')p jrl+fJ , 0 ::5 Pr(x, y) ::5 ~, and f Pr(x, y)dJ-t(y) = 1. It can be easily shown that lim,....o Pr f = fin L 2(X) (but we shall be interested in other types of convergence, which will be discussed in a moment). Let the "variation" of Pr, Dr, be defined by either Dr f = (Pr - Pr/2) f or by Dr f = r ~ P f.rA discrete version is then obtained by letting r = 2-k , k E Z. Then, as in (2.13), we have

(3.1) f= LDk/. kE'Z


Our task, then, is to carry out the program described in the previous two sections. We want to establish an analog of the Calderon reproducing formulas described in §1 and in §2. Moreover, we have to show how this analog can be used to introduce a class of Besov and Triebel-Lizorkin spaces associated with HT spaces. Guided by the material in §2 the definition the spaces iJ;,q and p;.q should be, simply,

that these are the spaces of all f such that

l/q

(3.2) IIfllB~q == [ L(2ka llDk/ll p )q ] < 00 kE'Z

and

(3.3) 11/11;;·' =~ [~(2kaIDdl)' rl < 00.

We must, however, give meaning to these inequalities. In a general HT space we do not have a theory of distributions; thus, it is not clear what we mean by f. Moreover, we do not have a class of test functions available. Finally, and most importantly, we must show that the definition of these spaces iJ;·q and P;,q we have just given in terms of the norms (3.2) and (3.3) is independent of the choice of the family {Dk}. In this section we give the properties of the class of operator families {Dk} that can be used in these definitions and describe in detail the program that we are following. As stated before, the technical details are contained in the companion paper [HS].

We begin by introducing the classes of test functions that are needed for our purposes. All this is in strong analogy with the theory of Hardy spaces associated with HT spaces that we developed in [CW2]. There the test functions were elements of Lipschitz spaces that turned out to be the duals of the Hardy spaces. In the present situation the test functions are "localized smooth molecules." We say that a function f on X is a strong smooth molecule of type (f3, y), centered at Xo EX, and having width d > 0 if and only if

dY (i) !f(x)1 ::5 c (d + p(x, XO»l+y ,

(ii) P(x, x I)}P dY

If(x) - f(x ' )! ::5 c { d + p(x, xo) [d + p(x, xo)]l+y ,

d + p(x, xo)for p (x, x') ::5 2K

(iii) l f(x)dJL(x) = O.


223FUNCTION SPACES ON SPACES OF HOMOGENEOUS TYPE222 CHAPTER 10

Theorem (3.5). Suppose {Sd, k E Z. is an approximation to the identity on a Denote by !JJ1(P,Y) (xo, d) the collection of all such molecules and let the "norm" space of homogeneoul type and Dk = Sk - Sk-l' Then there exist families ofof such an f be the infimum of all constants c for which (i) and (ii) hold. If we operators {Dd and {Dd. k E Z, such that choose a fixed Xo EX, then the space !JJ1(P, y) (xo, d) is our space of test functions.

The theory of Calderon-Zygmund operators has been extended to HT spaces by f = :LDkDd several authors. (For a good account of this see the monograph by Christ [Ch].) In [HS] it is shown that they are bounded on the spaces !JJ1(P,Y), where the indices f3 and y are adapted to the degree of smoothness of the kernels of these operators.

The analog of (2.18) can be derived from the following result in [HS]:

Theorem (3.4). Suppose {Sd, k E Z, is an approximation to the identity on an HT

space and Dk = Sk - Sk-I. Set TN = LkEZ Df Db where Df = Llil:SN Dk+ j

and N is a positive integer. Thenfor N large enough, TN l is a Calder6n-Zygmund

operator, TN! (1) = (TN!)*(1) = 0 and TN! has the weak boundedness property.

In the setting of JRn this theorem follows from results of David and Journ6 [DJ] and techniques described to us by Tchamitchian that involve aspects of the theory of wavelets that are not available to us for the HT space case. The method used in [HS] parallels the approach we described in §2 (see the estimates (i), (ii), (iii), (iv), and (v) that follow inequality (2.15)). More precisely, Theorem (3.4) is obtained

from the following estimates on the operators RN = L\ll>N LkEZ De+kDk: if o < € ' < € then there exists a constant c such that for all m = 1, 2, 3, ...,

(i) IR~(x, y)1 ::: cmTN~mp(x, y)-l;

(ii) IR~(x, y) - R~(x', y)1 ::: cmTN~mp(x, X')f'p(X, y)-O+f')

p(x, y)for p(x, x') ::: ~;

(iii) IR~(x, y) - R~(x, y')1 ::: cmTN~mp(y, y,/p(x, y)-(l+f')

p(x, y)for p (y, y')::: 2K

(iv) I(R~f, g} I ::: cmT N8m /l-(B(xo, r»

for all f and g having support within B(xo, r), with Lipschitz norms not exceeding r-T/ and L oo norms bounded by 1;

v R~(l) = (R~)*(1) = 0,

where R~ (x, y) is the kernel of RM, fJ > 0 and 0 < TJ < 1.

These estimates and the fact that TN 1 = L~=o R'N are the basic tools needed to prove Theorem (3.4). We can now announce a version of the Calderon reproducing formula for HT spaces:

kEZ

and

f = LDJjd, kEZ

where these series converge in the topologies of !JJ1(P,y). Moreover, the kernels

Dk (x, y) of these operators satisfy

_ 2-kf'

(i) IDk(x, y)1 ::: c (2-k + p(x, y»l+f' ;

f' k ' _ - [ p(x x') ] 2- f

(ii) IDk(x, y) - Dk(x', y)1 ::: c 2-k + ~(x, y) (2-k + p(x, y))l+f'

') < 2-k + p(x, y)fior p (x, x - 2K

for all y E X and k E Z(iii) l Dk(X, y)d/l-(x) = 0

The kernel}, Dk(X, y) satisfy the same conditions except that in (ii) Dk(x', y) is

replaced by Dk(X, y') and p(x, x') is replaced by p(y, y').

These are the basic definitions and results that are needed to carry out the known program on JRn to the new setting of HT spaces. As mentioned above, the details are carried out in [HS]. There one can also find the atomic and molecular de

compositions of these general Besov and Triebel-Lizorkin spaces. The theory of interpolation of operators acting on these spaces is also extended to this general setting. We hope that this exposition furnishes sufficient motivation to make it easier to go through the rather complicated technical details that are needed for

the thorough presentation contained in [HS].

University ofWindsor, Windsor, Ontario

Washington University, St. Louis

REFERENCES

[Ch) M. Christ. Lectures on singular integral operators. CBMS Regional Conference

Series, Number 77 (1991), 1-132.


----------_.. ._-

224

,.~~~~;;;-~-~-_._.__

CHAPTER 10

[CW] M. Christ and J.-L. Journe. "A boundedness criterion for generalized CalderonZygmund operators." Acta Math. 159 (1987),51-80.

[C] R. R. Coifman. "Multiresolution analysis in non-homogeneous media." In Wavelets. Time-Frequency Methods and Phase Space. Combes, Grossman, and Tchamitchian, eds. Springer Verlag, 1989.

[CM] R. R. Coifman and Y. Meyer. "Au dela des operateurs pseudo-differentiels." Asterisque 57 (1978), 1-185.

[CWI] R. R. Coifman and Guido Weiss. "Littlewood-Paley and Multiplier Theory, by Edwards and Gaudry." Bull. Amer. Math. Soc. 84, Number 2 (1978), 242-50.

[CW2] R. R. Coifman and Guido Weiss. "Extensions of Hardy spaces and their use in analysis." Bull. Amer. Math. Soc. 83 (1977), 569--645.

[DJS] G. David, J.-L. Journe, and S. Semmes. "Calderon-Zygmund operators, paraaccretive functions and interpolation." Rev. Mat. Iber. I (1985), I-56.

[FJ] M. Frazier and B. Jawerth. "A discrete transform and decomposition ofdistribution spaces." J. Funct. Anal. 93 (1990), 34-170.

[FJW] M. Frazier, B. Jawerth, and Guido Weiss. Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series, Number 79 (1991), 1-132.

[FS] C. Fefferman and E. M. Stein. "HP spaces of several variables." Acta Math. 129 (1972),137-193.

[HITW] Y.-S. Han, B. Jawerth, M. Taib1eson, and Guido Weiss. "Littlewood-Paley theory and €-families of operators." Colloq. Math. LX/LXI (1990), 321-59.

[HS] Y.-S. Han. and E. T. Sawyer. "Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces." Memoirs of the AMS lIO (July 1994), 1-126.

[J] J.-L. Journe. Calder6n-Zygmund operators, pseudo-differential operators and the Cauchy integral ofCalder6n. Lecture Notes in Mathematics, no. 994. Springer Verlag, 1983.

[MS] R. Macias and C. Segovia. "Lipschitz functions on spaces of homogeneous type." Adv. Math. 33 (1979), 257-70.

[T] R. Torres. "Boundedness results for operators with singular kernels on distribution spaces." Mem. Arner. Math. Soc. 442 (1991),1-172.

11

The First Nodal· Set

of a Convex Domain

David Jerison*

INTRODUCTION

Our pwpose in this paper is to prove that the first nodal set of a long, thin convex domain touches the boundary. Let

B(z, r) = {y E Rn: Iy - zl < r}

inradius(Q) = max{r: B(z, r) C Q for some z}.

Theorem. Let Q be a convex, open subset of Rn. There is a dimensional constant

C such that if diameter(Q)/ inradius(Q) ~ C, then the nodal setfor the second

Dirichlet eigenfunctionfor Q touches the boundary. In other words, if /).u = -AU in Q, u = 0 on aQ, and A is the second eigenvalue for the Dirichlet problem in Q,

then A = {z E Q: u(z) = O} satisfies A n aQ =I- 0. In the case n = 2, An aQ

consists of exactly two points.

This theorem was announced in [J] in the case ofplanar domains. A few months later, A. Melas [M] proved the result for all smooth convex planar domains, not just ones with large eccentricity. (He also showed by a limiting argument that even if the domain is not smooth, the nodal line touches the boundary in at least one point.) Our method of proof is much more crude, but it has some advantages. First of all, it applies in higher dimensions, as we show here. Secondly, it applies

'This research was supported in part by a Presidential Young Investigator Award and National Science Foundation grants DMS-9106507 at the Massachusetts Institute of Technology and DMS9100383 at the Institute for Advanced Study.


~~~~;;;.;=.;.=-==~=,~.;~~.=-v ·•

226 CHAPTER 11

directly to all convex domains, with no smoothness hypothesis. Finally, it is more quantitative than the one given by Melas, in that it gives some indication as to where the nodal set is. We hope to return to this in the future.

L. E. Payne made the conjecture that the first nodal line touches the boundary for any planar domain [P]. S.-T. Yau made the conjecture for all convex planar domains [Y]. The best previous results on the first nodal line are as follows. L. E. Payne [PI] proved that the nodal line touches the boundary of a convex domain in the plane which is symmetric under a reflection. This result was extended by C.-S. Lin [L] to the case of conveJLdomains which are symmetric with respect to a rational rotation instead of a reflection.

A basic notion underlying our approach is that eigenfunctions behave like harmonic functions. This was motivated in part by results of [KP]. Another result that motivated this work is the well-known lower bound for the first eigenvalue in terms of the inradius due to W. K. Hayman [H]. We would like to thank Carlos Kenig for pointing out the generalized maximum principle (Proposition I). The use of this maximum principle simplified our original proof which was based on comparisons with harmonic functions using a Neumann series. We would also like to thank Charles Fefferman for pointing out an error in Remark 2 of the original manuscript.

OUTLINE OF THE PROOF

F. John's theorem [dG, p. 139] says that any convex domain is comparable to a rectangle. More precisely, there is a dimensional constant C1 such that Q can be rotated and translated so that

(1) {Z E Rn : IZkl :s L k} c Q C {Z ERn: IZkl :s C1Ld.

We may assume without loss of generality that the sidelengths Lk are increasing with k. After a dilation we may assume that L I = I. Let n = p + q. We will consistently split the variables into Z = (x, y) E RP x Rq. The theorem follows from

Claim. For any M there exists N (depending on M and n) such that if L P :s M and L p+ I ~ N. then the nodal set for Q touches the boundary.

We prove by induction that the claim implies the theorem. Let N I be the value given in the claim for M = I. If L 2 ~ N I, then the nodal set touches. If not, then since L2 < NI we can choose N2 according to the proposition corresponding to M = N I • If L3 ~ N2, then the nodal set touches. If not, then we may continue in this fashion until finally the nodal set touches the boundary provided L n ~ Nn- I .

_,•.;;.;;-~.;";:~,;;;;;;,,,;,~.~.;;;=~;~,;;.;.;.:..;.;;...~-,;.~~~;~~~~~;;~~,.;:;,;;.._ __=.'._~_,~=·~~ •.__ ~<

THE FIRST NODAL SET OF A CONVEX DOMAIN

Because Nn-] is a dimensional constant and because Ln = LnlL] is comparable to the ratio of the diameter to the inradius of Q. the theorem is proved.

We will prove the claim by contradiction. Assume that the nodal set does

not touch the boundary. Denote Q- = {z E Q: u(z) < O} and Q+ = {z E Q: u(z) > OJ. Without loss of generality we may assume that Q- is the component that is relatively compact in Q. Denote u_(z) = -u(z) for z E Q_ and u_(z) = 0 elsewhere. Denote u+(z) = u(z) for z E Q+

and u+(z) = 0 elsewhere. Denote Q(y) = {x E RP: (x, y) E Q}, Q_(y) = {x E RP: (x. y) E Q_}, and similarly for Q+(y). For any domain DIet )\.j (D) denote the first eigenvalue of D, and let ).dD) denote the second

eigenvalue. Let AQ = inf A](Q(y)).

yERq

The proof goes as follows:

1. When N is large, A2(Q) is very close to AO (Lemma 1). 2. It follows from Step 1 that for "most" points y in a large cube Q', Al (Q- (y))

is very close to Al (Q (y)) and AQ (Lemma 4). 3. It follows from Step 2 that Q_(y) is "most" of Q(y), that is, the Lebesgue

measure, m(Q+(y)), of Q+(y) is very small (Lemma 3). In other words, Q+

is very thin in some places. 4. By the Hopf lemma (Lemma 5) and Carleson lemma (Proposition 3) max u_

dominates max u+ on the thin portion of Q+. (See (12).) 5. u+(z) decays exponentially as z approaches the center of the thin portion of

Q+. On the other hand, because we have assumed that Q- is relatively compact in Q, Q_(y) is relatively compact in Q(y). In particular, Q+(y) is nonempty, and, using the Hopflemma again, max u+ on the central part of the thin portion of Q+ dominates max u_ nearby. Thus we have come around in a circle, and max u_ on a certain subset is bounded by a factor times itself. The exponential decay (the factor 2-w for some c > 0) dominates all of the earlier bounds (which grow at most as fast as powers of N) so that for large N the factor is

less than 1, and we obtain a contradiction.

The reader may also wish to consult the paper [J] for the very similar argument in the case n = 2. The main differences between the two-dimensional argument and the one in higher dimensions are these. In R2

, Step 3 is trivial because the cross-sections Q (y) are one-dimensional. In Rn. n ~ 3, Step 4 requires an extra technical lemma (Lemma 6) to show that very thin filaments of Q- do not prevent the control of max u+ by max u_ via the Hopf lemma and Carleson lemma.

The rest of the paper consists of one long section giving the details of the argument for n ~ 2. Then in the final section we indicate the modifications

o~~_".-~;,_c_-",;,.;;;.",~=._.,~~ .. ,,~~.~;-";;;","·~;;~,,"~ .. ~<-;,.,.;-., _.~.~"=.~~~,,,_~~," __ ",,_._~'~-'''~-'-'-'-''---'''---.'~'-'~"O='~"~~_,-~~~~._~="~,~-~.",-.'~~;~""~.~"-~~~~'"--'

227


228 CHAPTER 11

needed to prove that when n = 2 the nodal line touches the boundary in two points.

Throughout the proof we will use the convention that the constants c > 0, and C < 00 are either dimensional constants or depend on dimension and M and may vary from line to line. Lower case c will be used for lower bound constants and upper case C for upper bounds. The numbered constants c\, C2, CI, etc., do not change from line to line, which should help the reader to follow their interdependence.

THE PROOF

Lemma 1. There is a constant C2 depending only on n such that the second eigenvalue A = A2(0) satisfies

Ao S A SAo + C2N-2/ 3 .

Proof It is easy to see that Al (0) 2: Ao. In fact, if v is the first eigenfunction for o then

{ IVvl2 2: { ( IVyv(x, y)12dxdyIn JRq In(y)

2: Ao { ( Iv(x, y)!2dxdyJRq In(y)

2 = Ao l v •

Moreover, A 2: Al (0), so the lower bound follows.

Choose D = O(y) so that AI(D) SAo. By (1), (0, y') E 0 provided IY~I S N, k = I, ... , q. Therefore, by convexity, «(1 - N-2/3)x, (1 _ N-2/3)y +

N-2/3 y') E 0 for all xED. In other words, D' x Q c 0 where D' = (1 _ 2

N- /3)DandQ = (1-N-2/3)y+{y': IY~I S N I/3},acubeofsidelength2N I /3. Separation of variables shows that A2(D' x Q) = Ao(1 _ N-2/3)-2 + rr2(q +

3)(2N1/3)-2. (1) also implies that at least one cross section O(y) contains a cube of sidelength 2 so thatAo S rr 2p/4,adimensionalconstant. ThusA2(D' x Q) S Ao +C2N-2/3 for some dimensional constant C2. The min-max principle implies that A == A2(0) S A2(D' x Q), so the lemma is proved. •

Let us recall the generalized maximum principle for solutions to eigenfunction equations. (See [PW] for a more detailed discussion.)

ProPOSition 1 (Generalized Maximum Principle). Let h be a bounded function. Let u and w satisfy li.u + hu = 0 and Ii.w + h w S 0 in a bounded domain D.

THE FIRST NODAL SET OF A CONVEX DOMAIN 229

Suppose that u and ware continuous in D. Finally, suppose that u > 0 in D and w > 0 in D. Then

max u/w S max u/w.D aD

To prove this, observe that v = u/w satisfies the inequality li.v + 2V log w .

Vv 2: O. The inequality then follows from the usual maximum principle. (See [PW].)

Remark 1. Let B = B(x, r), a ball of radius r S 1. The boundary problem li.u + u = 0 in B and u = f on aB can be solved for every continuous function

f and

u(x) = cn(r) a(;B) iB fda,

and cn(r) S 2. (Here, and elsewhere s will denote surface measure.)

Proof The existence of a solution follows from the generalized maximum principle and the existence of a solution to the equation that is positive in the closed ball B. Such a solution exists for r S 1, and even for r less than the first zero of an appropriate Bessel function. We will need a few facts about Bessel functions later on anyway, so let us write down the solution in detail. Denote

l

Fk(r) = i cos(sr)(1 - s2l-I/2ds.

The relationship of Fk with the Bessel functions is 1k (r) = Ckrk Fk (r) for some constant Ck ([SW] pp. 153-154). If k = (n - 2)/2, then Bessel's equation implies (Ii. + I)Fk (lxl) = O. Because 1/2 < cos(t) S 1 for 0 S t S 1, we have Fk(O) S 2Fk(r) for all r, 0 S r S 1. In particular, Fk(r) > 0 for 0 S r S 1. Now, we can confirm the formula for u(x). By symmetry we see that u(x) is some constant multiple cn(r) of the average of f over the boundary. The constant can be evaluated using the radial solution Fk(!Z - xl) for k = (n - 2)/2:

cn(r) = Fk(O)/ Fk(r) S 2. • Proposition 2 (Harnack Inequality). There is a dimensional constant C3such that

if 0 < r < 1/(max h)I/2, li.u + hu = 0 and u > 0 in B(x, 2r), then

max u S C3 min u. B(x.r) B(x,r)

(If h S 0, then the inequality holds for all r > 0.)


230 231 CHAPTER 1 I

Proposition 2 can be proved by using the Bessel function w(z) = F(n-2)/2(lz xl) and the Harnack inequality due to Serrin [Se] for the function v = ulw. (See [PW].)

In Lemma 3 below we will use Propositions 1 and 2 for a function h which is a constant plus the characteristic function ofa ball. However in all other cases we will

have h = A, a constant. In that case there is another way to prove Propositions 1 and 2, namely, to deduce them from their counterparts for parabolic equations using the observation that if (~ + A)U = 0, then the function e-Atu satisfies the

heat equation (at - ~)(e-At u) = O. (For other applications of this idea, see [KP].)

The parabolic analogues of Propositions 1 and 2 are well known. (See [PW].)

Proposition 3 (Carleson lemma). Let ¢: Rn-

1 -+ R satisfy the Lipschitz condition W¢I s M and ¢(O) = O. Denote D = {x E Rn

: Xn > ¢(Xl, ... , xn-d. Let 0 < r S 1/0.anduisapositivesolutionto(~+A)u = OinB(0,2r)nD such that u vanishes on B(O, 2r) n aD. Let z = (0, r). There is a constant C4

depending only on M and n such that

max u S C4u(z). B(O,r)nD

The parabolic version ofthe Carleson lemma was proved by Salsa [S], and Proposition 3 is an immediate consequence, as in the case ofconstant h for Propositions 1 and 2.

Proposition 3 was proved by Carleson [C] and Hunt and Wheeden [HW] in

the case of harmonic functions. (In that case, A = 0, and the upper bound on

r is superfluous.) There are many subsequent versions. An examination of the

proofs in [C] and [CFMS] shows that the result only depends on the scale-invariant

Harnack inequality (Proposition 2) and a uniform HOlder continuity for solutions

at the boundary, which is obtained by a majorization of solutions by an explicit supersolution (barrier) on a cone, that can be constructed using Bessel functions.

We will not carry out the proof, since another one is available. But we will need the barriers in annuli given by the next lemma.

Lemma2. ForanyE > OandO Ofor p S r S 2p, (b) (~ + l)w.(lxl) S Ofor p S Ixl s 2p (x ERn), (c) w.(r) = 1 + Efor r = 2p, (d) As E tends to zero, w. tends uniformly to afunction Wo such that

awowo(p) = 0, wo(2p) = 1 and a;:- (p) sCIp

for some dimensional constant C.


Proof Forn ~ 3, define

" [lOg riP]w.(r) = (rI2p)l- 2 log 2 + E •

Then ~w.(lxl) = -( ~ - 1)2 Ixl -2 w.(lxl) and the properties listed are easy to 2check. In particular, w~ (p) = 2n/ -

1I p log 2.

For n = 2, define

w r = Fo(r) [lOg rip E] . •( ) Fo(2p) log 2 +

Then (~+ l)w.(lxl) = F~(lxl)/lxlFo(2p) log 2. This last expression is negative

for Ix I < 1 because

F~(r) = -11

s(sinrs)(l - s2)-1/2ds sO

for 0 S r S 7r since the integrand is positive. Once again, the other properties

are easy to check. For example, because Fo is decreasing on the interval under

consideration,

2W' (p) = Fo(p) < Fo(O) < o Fo(2p)p log 2 - pFo(/2) log 2 - P •

Lemma 3. Let D be an open convex subset of RP such that B(O, 1) CDC B(O, M). There exist constants Cl and K depending only on M and p such that for any open subset V of D,

Al(D\V) ~ Al(D) + Clm(V)K.

Proof Suppose that w is the first eigenfunction for D, with f-L = Al (D). Nor

malize w so that max w = 1. Let B = B(O, 1/2). Consider a dilation of the

function w. of Lemma 2 in dimension p so that it satisfies (~ + A)W. S O. This changes the bound on the derivative in part (d) by only a bounded amount because

Ais bounded above by a dimensional constant. Now translate w. so that the inner circle is tangent to aD and contained in the complement of D. This is possible

because D is convex. The generalized maximum principle for the intersection of

D with the annulus implies that w S w., where W. is the translated and dilated w•. Taking the limit, we find that w S WOo Thus, there is a dimensional constant C such that w(x) S C dist(x, aD). It follows that w attains its maximum, 1, at a

point whose distance to aD is greater than l/C. It then follows from Harnack's

inequality that there is a constant C2 > 0 depending only on M such that

(2) min w ~ C2. B

Now consider the first eigenfunction 1/1 for the operator Lu = -~u + XBU.

Thus L1/I = f-Ll1/1 in D and 1/1 = 0 on aD. Normalize 1/1 so that max 1/1 = 1. Since


_,_:~'":::'::::~:-=--=~=-~~:-:::'::::::::::=""-:::':=:::=::=::::::-:::"-==::='::=.==='.:::,:::::.:==-=~~..:::':::::~::~.:::::'::::::::::::_-:=-=::~':;~~=-' '--::::-;;::-'::::-::.::::"- --,----~~--~~'--,- ~---" -,--~ ..~:':_:::.:~v:_:::--_:"":;. _.-.:::::.":"~' ~_'.C:::,-::=,-==-=,,==~-"=,--===

232 CHAPTER 11

w. in the proof above is only required to be a supersolution, the proof applies to J/r, so that (2) holds with J/r in place of w. In particular, there is a constant C3 > 0 depending only on M for which

Is J/rz ~ C3 LJ/rz.

Therefore,

LIVJ/rlz + i J/rz ~ (JL + C3) LJ/rz.

Hence, JLI ~ (JL + C3) and we have

Z ~ Z(3) LIVviz+ i v (JL + C3) Lv

for any function v that vanishes on aD.

The function w is superhannonic. We can bound from below the rate of vanishing of w at the boundary of D by comparing it with the rate of vanishing of a positive hannonic function that vanishes on the boundary of a cone. The hannonic function vanishes at most as fast as a power of the distance to the vertex, and the power is bounded above in terms of a lower bound on the aperture of the cone, that is, in terms of M. So we deduce from (2) that there is K I depending only on Mforwhichw(x) ~ cdist(x, aD)K,. It follows that

m(D n {x ED: w(x) < t}) :::: Ctli,

for some constants C and 8 > 0 depending on M. Choose t by the equation m(V) = 3Ctli . Then

m(V n {x ED: w(x) > t}) ~ 2Ctli .

Choose an open subset U of V n {x ED: w(x) > t} with a smooth' boundary and such that m(U) ~ Ctli. Since Al (D\ V) ~ Al (D\U) we will confine our attention to D I = D \ U. The boundary of D I consists of a smooth part, au, and a convex part, aD. In particular, it is a Lipschitz domain (without any control on the Lipschitz constant). Thus we can solve

t:.¢ = 0 in D I ; ¢ = t on au, ¢ = 0 on aD.

Denote by v the first eigenfunction in D" with v > 0 and t:.v = -vv in D I . Because D I is a Lipschitz domain the following calculation based on Green's formula is valid. (Here, a/an denotes the outer normal derivative of aD

I • Note

that av/an :::: 0.)

(v - JL) { vw = ( v(t:.w) - (t:.v)w = - { w av da lDI lDI laD, an

-':"_'::=':':::':::_:.':::':.":':::"_':-':':"'-:'::'::::::::::=~'-'~::~-:::".'-:::::~:::,:::='::"'-:::::.~~-~~.::::::,,::".;:':';'-:-._'=~:':-.'::7:~::::::::-'-::::'''-


av 1 (a¢ av)~ -t -da = - v - ¢ - da au an aD, an an1

= { (t:.¢)v - (t:.v)¢ = v ( v¢. lDI lD,

In order to obtain a lower bound for the last integral, consider any point x E B.

By Fubini's theorem there is a dilate D' == r(D - x) + x of D, with respect to x as the origin, for which

a(aD' n U) m(U)>-

a(aD') - m(D)

for some r, 0 < r < 1. On the other hand, the A oo estimate for hannonic measure in Lipschitz domains due to Dahlberg [D,CF] implies that the hannonic measure w for D' at x satisfies

S (a(s) )K2w( ) ~ c a(aD')

for some constants c > 0 and Kz depending on M, and for every Borel subset S of aD'. But if S = aD' n U, then by the maximum principle, ¢(x) ~ tw(S).

Therefore, K3min ¢ ~ C4t

B

with K3 = I + Kz8. Extend v by zero outside D I • It follows that

(4) { v¢ ~ C4tK3 { v. lDI lB

Define a by the equation

{ v = a { v. lB lD I

We consider two cases.

Case 1. a < c for a constant c, depending on M, to be specified later.

Let us first verify that there are dimensional constants A > 0 and C5 such that

(5) i fZ :::: C5E i IV fl z + C5EA (i IfI)Z-

for all E > O. Let p > 2. HOlder's inequality and the fact that A I A2 < A:+a + Ai+a for positive numbers AI, Az,anda = (p - 2)/p > oyields

i gZ :::: (i gp) I/(p-I) (i g) (p-Z)/(p-l)


234

;''!'. !,!;f~ PEB_ .... '"""._-:~-.t";'\.;~..".!'!'1':;~~~w.~~:;t~~~~~~,.~~i~~~\WF..;~~~~~~£~~~:r,n~",~....,t,=.~~~"''';;''~J;~~~~~~~~\~'f{3i~';'~.~~~~~~;

CHAPTER I I

S E (l gP) 2!p + E-P!(p-2) (1 g) 2

for any non-negative function g. Recall the Sobolev estimate

(1 If - alPy!P S C1IV fl 2

where a = mlB) 1B f, the average value of f, and C is a dimensional constant. Applying this inequality and the previous one for g = If - a I, we have

•. 2 21If - al S CE 1IV fl 2 + E-P!(p-2) (1 If - a l)

2

S CE 1IV fl 2 + 4E-P!(p-2) (1 If I) Therefore,

21f2 S 21If - al2 + a

S 2CE i IV fl 2 + 4E-P!(p-2) (1 If I) 2 + 2m(B) (i If I) 2,

and (5) is proved.

Now, we apply (5) and then (3) to the function v extended by 0 outside D I to get

2l, IVvl2 ~ (l - CSE) l, IVvl2 - CsE-

A (1 v) 2 + i v2~ (l - CSE)(JL + C3) II v - CsE-A (i v) 2

~ (I - C")(I" + c,) In, v' - c,,-'o (10, v)'

We can choose E depending only on M so that (l - CSE)(JL + C3) > JL + C3/2.

CE-

Then there is a constant C depending only on M such that if () < c then JL +C3/2 _ A

() > JL + C3/4. It follows that Al (D\ V) ~ v > JL + C3/4 and the lemma is proved in this case.

2. () ~ c. Then 1B v ~ C 1D vw and consequently 1

v - JL ~ CC4vtK3 ~ clm(V)K3!J,


for some constant CI depending only on M. This concludes the proofofLemma 3.•

Remark 2. Let Q be a cube in Rq of sidelength s. Let E C Q. There are

dimensional constants C6 and K (q) such that

~ f(y)2dy S C6 (:~~;) K(q) (s2 ~ IV f(y)1 2dy + 1f(y)2 dY) .

Proof. Without loss of generality we may assume that s = 1. We will prove by induction on q that for 0 < r S I,

~ IV f(y)1 2dy + r 1f(y)2dy ~ c(q)(rm(E»K(q) ~ f(y)2dy.

The remark is the special case r = 1. For q = 1 we let I = [0, 1]. It is easy to show that

f2 S 5 /U')2 + 5 min f2, / I I I

Hence,

U')2 + r ( f2 ~ rm(E) /U')2 + rm(E) min f2 ~ rm(E) / f2. / I JE I I 5 I

For the induction step, let Q = Q' x I where Q' is the unit cube in Rq-I. Let

E(x') = {tEl: (x', t) E E} and F = {x' E Q': m(E(x'» > m(E)/IO}. Then

m(E) = 1 m(E(x'»dx' + ( m(E(x'»dx' S ~m(E) + m(F). Q'\F JF 10

Thus m(F) > m(E)/2. If V' denotes the gradient in the first q - 1 variables,

then using the inequality for q = 1 and the induction hypothesis we have

2 2 2~ IV fl + rXEf = ~ IV' fl

2+1(/ lar!(x', t)1 dt + r ( f(x', t)2dt ) dx' Q' I JE(X 1

)

~ ~ IV'fl2 + ~, ~m(E(X'» / f(x', t)2dtdx'

~ ~ IV' fl 2 + ~, ~ mi~) XF(X') 1f(x', t)2dtdx'

: f (10, IV' f(x', 1)1' + '~~E) XF(x')f(x', t)2dX ) dt ~


~~~~====~.~~=====~T=-~=====~~=~==~~=~=~~~~~'~-==TT -,_. ~.,-

236 CHAPTER 11

1[ (E) ]K(q-O~ c(q-l) r~m(F) f(x',t)2dx'dt!I Q' 50

2 ]K(q-I) ~ c(q - 1) r m(:; fa f2,

[ l

which proves the remark. • We can now show that ~L is very wide (in the x variable) for many values of y.

Lemma 4. Let ex = l/lOK(q). There is a constant C depending only on M and n such that for any N 2: C, there exists a cube Q in Rq satisfying (a) m(Q) = Na ;

(b) m(E n Q) ~ N-a. where E = {y E Rq : Al (~L(y» ~ AO + N-1/3}; (c) If B denotes the unit cube with the same center as Q, then

N [1 u_(x, y)2dxdy ~ 1 1 u_(x, y)2dxdy ly€B X€a(y) y€Q XEn(y)

(d) If Q' denotes the cube concentric with Q ofhalfthe diameter, thenfor y E Q' the eccentricity of Q(y) is controlled: the inradius is bounded below by a dimensional constant and the diameter is bounded by a constant depending onM.

Proof Choose a tiling of Rq by cubes Q of measure N a , that is, a family F of cubes that overlap only on their boundaries and whose union is all of Rq. Let s

denote the sidelength of Q. Thus s = Na/q. Define E(Q) by

IVyU_(x, y)12dxdy = E(Q) [[ u_(x, y)2dxdy.11 lQhMQ aM

We consider three subfamilies of F.

F] = (Q E F: E(Q) ~ 5C2N-2/3 and (b)failsfor Q\,

F2 = {Q E F: €(Q) > 5C2N-2/3},

F3 = {Q E F: (c) fails for Q}.

N2aFor QinFl wehavem(Q)/m(EnQ) ~ . Fixx,anddefinef(Y) = u_(x, y) if (x, y) belongs to Q_ and zero otherwise. If we apply Remark 2 with E replaced by E n Q and integrate in x we find

2N I[[ u_(x, y)2dxdy ~ C6s /51[ IVyu_(x, y)1 2 +1Qlo(y) Q In(y)

237 THE FIRST NODAL SET OF A CONVEX DOMAIN

+ C6 N1 /51 [ u_(x, y)2dxdy EnQ In(y)

~ C6S2E(Q)Nl/5j [ u_(x, y)2dxdy Q In(y)

+ C6 NI /5 [ [ u_(x, y)2dxdy lEnQ In(y)

For N sufficiently large, C6s2 N 1/5E(Q) < ~,and, hence,

(6) 1r u_(x, y)2dxdy ~ 2C6NI/5 r 1 u_(x, y)2dxdy. Q In(y) 1EnQ n(y)

On the other hand, making use of (6) and bounds in terms of Vx, we have

2 2

1r u2 = (1 - t) r 1 u2 + t r 1 u + r 1 u Qla(y) 1Q\E O(y) 1Q\E n(y) 1EnQ n(y)

5~ (1 - t) 1 1 IVxu_1 2 + (1 + 2tC6N I/ ) f r u~

AO Q\E n(y) 1EnQ In(y)

(1 + 2tC6N 1/5) < (1 - u_1 2 +t) 1 1 IVx ),,0 + N-l/3- AO Q\E Q(y)

r r IVxu_12

lEnQ In(y)

(1 - t) (1 + 2tC6Nl/5») 11 I 2 ~ max --, 1/3 Vxu-I .

( Ao ),,0 + N- Q n(y)

Lett = lOC2 AoI N- 2/ 3 • Then, noting that Ao 2: 1/M 2, we find that for sufficiently

large N, - .

2 (Ao + 5C2N-2/3) 11 u~ ~ 1r IVxu_1(7) Q n(y) Qlo(y)

for all Q in Fl. Next, we clearly have

23(8) (Ao + 5C2N- 2/ ) 1r u~:: r r IV(x,y)u_1Qla(y) lQ lQ(y)

for all Q in F2. For Q in F3 we use Remark 2 again but with E replaced by B:

u2 = (1 - t) 11 u + t 11 uQ Q(y) Q Q(y) Q Q(y)11 2 2


239 238 CHAPTER II

< (1 - t) {{ lV'xu_IZ + tC6sZNI/1O ({ lV'yu_I Z Ao 1Q In(y) 1Q In(y)

+ tC6NI/LO ({ u~ lB In(y)

< max (1 , - t I tC6SZN I / LO) 11 IV'~,~ u_I Z

11.0 Q n(y)

+tC6N-9/LO [[ U~. lQ In(y)

Therefore, if t = N- I/ z and N is sufficiently large,

(l - tC6N-9/1O) [[ u~ S 1 - t [[ IV'(x,y)u_l z. lQ In(y) AO lQ In(y)

Forlarge N, (l - t)/(l - C6tN-9/LO) < 1 - lOCzAol N- z/3, and, consequently,

(8) holds for all Q in :F3 as well.

Now if every cube in :F is in one of the three families, then we can sum over all cubes and obtain

/3(Ao + 5CzN-z

) Lu~ s LlV'u_l z = A Lu~

which contradicts Lemma 1. So there must be a cube that does not belong to any

of the three families, in other words, a cube for which (a), (b), and (c) of Lemma 4 hold.

Finally, let us deduce (d) from (b). The theorem of F. John (1) implies that the

diameter of Q (y) is bounded by a constant times M for any y. AO is bounded

above by a dimensional constant. Therefore there is a dimensional constant cp

such that every domain contained in a rectangle with side of length less than cp

has first eigenvalue greater than AO + 1. By the John theorem applied to the convex

set Q (y), there is a dimensional constant c' such that if the inradius of Q (y) is

less than c', then Q (y) is contained in a rectangle with a side smaller than cp,

so that AI(Q(y)) > AO + 1. Now suppose that there is a point yO in Q' such

that the inradius is smaller than c'. Then AI(Q_(yo)) ~ Al(Q(yO)) > AO + 1.

In particular, yO belongs to E. The set of all y for which the inradius is larger

than c' is convex. Therefore, there is an entire halfspace containing yO for which

the inradius is smaller than c'. But this half-space intersects Q in a set of large

measure, and all of its points belong to E. This contradicts (b). (The proof of (d)

can be simplified using a theorem of E. Lieb that Al (Q (y» is a convex function of y.)


Let B and Q be given by Lemma 4. Denote

E Z - ({ uZ

1B In_(y)

Let Q" be the cube with the same center as Q' but with half the diameter. Define

R' = {(x, y) E Q: y E Q'} and R" = {(x, y) E Q: y E Q"}. We claim that

(9) "1~xu- S CgNI/zE.

To prove (9), let S' = (x, y) E R'. By Lemma 4(c) and Fubini's theorem there is

a dimensional constant C such that for some r, I/2JI S r S 1/..II,

-- u_da < CNI/ZE.1 1 a(aB(S', r» aB(~,r)

Consider the function 1/1 that solves

(~+ A)1/I = 0 in B(S', r); 1/I(z) = u_(z) + E for z E aB(S', r).

The maximum principle implies that u_ S 1/1 in B(S', r) n Q_, and by Remark I,

u_(S) S 1/I(S) s 2 [ (u_ + E)da S CgNI/Z E. • a(aB(S', r» laB(~,r)

LemmaS. Suppose that 0 < r < I/4JI, B(S', r) C Q+ and aB(S', r) nA =f:- 0.

Then

u(O S C9 max u_. B(~ ,2r)

Proof Let S = maxB({,2r) u_. Let w€(z) = wf(JI(z - S')) where w€ is the

function given in Lemma 2 with p = rJI < 1/4. Then (~ + A)w€ = 0 in B(S', 2r)\B(S', r), and W€ is positive in the closed annulus. Therefore, by the

maximum principle for the region Q_ n B(S', 2r)\B(S', r), u_(z) S Sw€(z).

Taking the limit as E tends to 0, we have u_(z) S SWo(z). Let 11 belong to

aB(S', r) n A. Harnack's inequality implies that there is a dimensional constant

C such that

u(S) S C min u. B(VI2J

The function u is superharmonic in the ball B(S', r). Therefore, u is larger than a Z ndimensional constanttimes the harmonicfunction G(z) = u(O(lx/ rl - - 1) on

B(S', r)\B(S', r/2). (When n = 2, we put G(z) = log Iz/rl.) Since both G and

u vanish at 11, it follows that IV'G(I1)1 S CjV'u(I1)I, and hence.

u(S) S CrlV'u(I1)I·


241 240

.,.".."'."-""""""'-......"', ....._-' ..'" - ~ -~- ·,.,.....,"-at--·~~·- ~~~';~~~.....r-""'===-~~~~~~~:::"~=:::::~-~~."."~~~~<d""""'_"--'-f"'''_;.'''''''''''L :- '!<:~~~.c=_=""'="-,~~_~~_

CHAPTER 11

In particular, VU(TJ) i= 0, so that, near TJ, A is a smooth hypersurface tangent to aB(~, r). Because bothu and iVovanish at 1/, it follows that IVu(TJ)1 s ISVwo(1])I. Thus, the upper bound from Lemma 2(d) implies u(O S C9S, as desired. •

Remark3. Suppose that °< v S 1and (~+A)V = Oin D. LetO < r S 1/..rA. If ~ E D, m(B(~, r) n D) S m(B(~, r»/4, and v = °on B(~, r) n aD, then v(O ::: 1/2.

Proof By Fubini's theorem, there exists s, °< s ::: r, such that a(aB(~, s) n D) ::: a(aB(~, s»/4. For any E ~ 0, let v. satisfy (~ + A)V. = °in B(~, s).

v. = v + E on aB(~, s) n D and v. = Eon aB(e s)\D. By the generalized maximum principle in D n B(~, s), v ::: v. provided E ~ 0. Passing to the limit as E tends to 0, we find v S vo. Remark 1 implies

. 2 da < -.1v(O s vo(O :::: a(aB(~, s) aB({,s) Vo - 2

I

• Lemma 6. Denote Y = maxw u_. Choose Zo E aQ such that Izo - ~ I = r = dist(~, aQ). There is a constant CIO depending on M such that if ~ belongs to R', B(~, lOr) n Q c R', and

(10) u(O > ClOy

then B(zo, 2r) n A = 0.

Proof If B (~, r) is not contained in n+, then it intersects A. Choose s < r such that B(~, s) is contained in Q+ and aB(~, s) intersects A. Then by Lemma 5 and Harnack's inequality the opposite inequality to (10) holds. So (10) implies B(~, r) C n+.

Denote Bo = B(zo, 4r) and

Bk = B(zo, rk) with rk = 2(1 + rk)r.

Thus the intersection of Bk for all k is B(zo, 2r), and rk+1 = rk - 2-kr. Denote the convex hull of z with B(~, r) by fez). We claim that

10) sup u_ ::: C1l2-Zk- y Bk

and

(il) An fez) c B(z, (n-Zkr) forall z E Bk n aQ,

Where 0 > 0 will be specified later. We prove the claim by induction. For

k = 0, 1, ... 5 part (i) is true by the definition of Y, with a suitable choice of CII.

let B be the ball of largest radius in fez) centered on the axis of fez) such that

THE FJRST NODAL SET OF A CONVEX DOMAlN

Be Q+,butaB intersects A. ItfollowsfromLemma5thatu(zz) S C9 Y.where Zz is the center of B. If we assume that part (ii) fails for some k S 5. then the radius of B is greater than 2-200r. On the other hand. by following the cone as it expands, this ball can be connected by a chain of balls which overlap in a large fraction of their volume to the ball B(~, r). The number of such balls is at most

a dimensional constant times log l/e. Therefore. the Harnack inequality applied log l/e times shows that u(O ::: Ae-Au(zz) for some dimensional constant A.

To assure that (ii) is true for k = 0, 1, ... 5 we assume CIO ~ AO-A.

Next, for the induction step, suppose that (i) and (ii) hold for some k ~ 5.

Suppose that Zl E Q_ n B(zo, rk - 10· 2-Zkr). We now show that

(11) m(B(ZI, r 2k r) n Q_) < C 120m(B(ZI, r 2k r»,

by showing that every z E B(ZI, 2-Zk r) n Q_ is within e2-2k r of an. The ray

from ~ to z meets an at a point Z3 and because IZ3 - ~ I ::: 5r and the convex set n contains the ball B(~, r), the angle of the ray with a plane tangent to an at Zz is at least 1/5. It follows that Z3 E Bk n aQ and by part (ii) of the induction hypothesis, z E B(Z2, OrZkr). In particular, z is within erZkr of aQ. Furthermore. the fact that B(~, r) is contained in n implies that the Lipschitz constant of an n Bk is

less than 5. so that (11) holds for some dimensional constant C12.

Ifwe choosee so that CIZO < 1/4. we can apply Remark 3 repeatedly to obtain for j = 1,2, ...

u_(z) ::: rjy for all z E B(zo, rk - (10 + j)r2kr).

In particular. if 10 + j = 2k we have

Zku_(z) ::: 210r y for all z E Bk+ l ,

which is the induction step for part (i). Now consider any z E Bk+1 n an. If there is a point of fez) n A at a distance greater than e2-Zk- Zr from z, then there is a ball B in f(z) of radius at least a dimensional constant times er2k- 2r which

is tangent to A and such that the convex hull of B with B(~, r) is contained in n+. By Lemma 5, the value of u at the center of B is less than C92102-zk Y.

Moreover, B can be connected by a chain of overlapping balls to the ball B(C r) as above. The number of balls in the chain is less than a dimensional constant times log(2zk+ZIe). Therefore, by Harnack's inequality,

u(~) ::: A(2-2k- zO) -AC92 IOr Zk Y.

Choosing CIO sufficiently large. we have acontradiction. This proves the induction step for part (ii).

Now, take the intersection over all k to obtain B(zo, 2r) n A = 0, which proves Lemma 6. •


243 CHAPTER II

We can now deduce that if R" denotes the subset of Q that projects onto a cube Q" concentric with Q' with half the diameter, then

(12) max u+ :s C l3 max u_. R" R'

For any ~ in Q+ n R", either u(O :s ClOy and we are done, or else the opposite inequality holds and the hypothesis ofLemma 6 is satisfied. With Zo as in Lemma 6, consider the ball B(zo, s) disjoint from A but such that aB(zo, s) intersects A. Thus s ~ 2r, but because of the estimates ofLemma 4, s must be smaller than some

negative power of N. In particular, s < I/../).. Consider a point Z4 E B(zo, s) at a distance, say s / M from a(Q n B(zo, s». By Lemma 6 and Harnack's inequality, U(Z4) :s CCIOY, for some constant C depending on M. Now by Proposition 3, uCO :s CC4ClO Y, which proves (12).

Note that Q+ n R' c {(x, y): y E En Q'} U (x, y): x E Q+(y) and y E Q'\E}. Lemma 4 implies m«((x, y): y E En Q'j) :s (CIM)P N-1/1O.

Lemma 4(d) and Lemma 3 imply that for y E Q'\E,

AI(Q(y» + clm(Q+(y»K :s Al(Q_(y» < AO + N-1/ 3.

It follows that m(Q+(y» :s C;I/KN- J/3K. Let Z5 E Q project onto the center of

the unit cube B given by Lemma 4. All the points of B(Z5, T), where T = C5Na/q,

project onto Q". Choose a dimensional constant C7 < minO/../)., 1/1O}. The estimates above for the measure of E and for Q+(y) when y is in the complement of E show that if N is sufficiently large, then for any z E B(Z5, T),

I m(B(z, C7) n Q+) :s 4" m(B(z, C7).

Therefore, we can apply Remark 3 repeatedly to conclude that for j = I, 2, ...

u+(z) :s 2-j

maxu+ forall Z E B(Z5, T - j C7)'RO

In particular,

Tu+(z) :s r max u+ for all z E B(Z5, T /2).RO

Now take any point z E Q- that projects onto B ofLemma 4. Because we assumed that Q- is relatively compact in Q, there is an open ball centered at z contained in Q- whose boundary intersects A. By the same reasoning as in Lemma 5 (with the roles of u+ and u_ reversed), we find that

u_(z) :s C92-T max u+. R'

Integrating and using (9) and (12), we have

E :s C142-T max u+ :s CI4 Cl3 2-T CSN 1/ 2E. R'


If N is sufficiently large, the 2-T term dominates, and this inequality cannot hold unless E = O. But if E vanishes, then u vanishes on an open subset of Q, a contradiction. This concludes the proof that the nodal set touches the boundary.

THE CASE n = 2

We would now like to show that in R 2 the nodal line must touch the boundary in exactly two points. The Courant nodal domain theorem says that the sets Q+ and Q_ are connected. Therefore, A cannot intersect aQ in more than two points. In

the very last stage of the proof above we used the fact that any ball in Q_ can be expanded until it touches A. In the case n = 2, we prove a variant of Lemma 4

which will imply that Q+ must touch both sides of aQ somewhere in the region that projects onto Q.

We will make a slightly different normalization of the domain Q for notational convenience. We claim that we can assume that

Q c (x, y) E R 2 : 0 < x < I},

the diameter of Q is N and the inradius is at least I - N- I • In fact, suppose first

• that the inradius is 1/2. Choose a disk D of radius 1/2 in Q. If aD touches aQ in two points on the same diameter of D, then Q is contained between the two tangent

lines perpendicular to the diameter, which are a distance I apart. If no such pair of points exists, then aD n aQ contains at least three points and the tangent lines at those points fonn a triangle containing Q. The longest side of the triangle has length at least N and it is easy to check that the altitude perpendicular to that side has length at most I + N -I. Thus, n is contained in a rectangle of length N parallel to the longest side and width I +N-I • If we dilate by the factor (1 +N- I ) -I and

rotate so that the rectangle is horizontal, we have our normalization. As in the case of Lemma I, an easy consequence of this normalization is

Lemma I'. rr 2 :s A :s rr 2 + C2 N-2/3.

Lemma 4'. Let Q be a convex subset of R2 with the normalization above. There

is a constant C IS such that If N ~ CIS, then there exists an interval J of length comparable to N 1/ 4 such that

(a) the rectangle I x J C n_,

where I = (x: N-l/8 < x < I - N- I / s). Moreover, if Yo denotes the center of

J ,the square

QI = {(x, y): Ix - 1/21 :s 1/8, Iy - Yol :s 1/8}


w. --j"- .=----

244 CHAPTERlI

satisfies

(b) 1U2 :::: N 2 { U2.~L JQ1

The differences between Lemma 4' and Lemma 4 are that we show here that an entire rectangle is contained in ~L, instead of showing that all but a subset of small measure of a rectangle is contained in ~L and also that we can control the integral of u2 on a large set by the integral on a "unit-sized" set Q I whose double is contained in ~L. .

Suppose that 11. intersects aQ in at most one point. Then, since aQ_ is a simple closed curve that encloses Rio either for every y in J there exists x such that o < x < N- I/8 and (x, y) E 11. or else for every y in J there exists x such that 1-N- I/8 < X < 1and (x, y) E 11.. In either case, one can use the same argument as above to control u+ by u_. Then one finds exponential decay of u+ along a thin strip of Q+ either in 0 < x < N- I/ 8 or 1 - N- I/8 < X < 1. Then one can

control u_ on Q I in terms of u+ in either one of the strips by translating a disk of radius 118 around a point of Q I to the left (decreasing x) or to the right (increasing x) until its boundary touches 11.. This yields the same type of contradiction as above, and concludes the proof that the nodal line touches in two points.

It remains to prove Lemma 4'.

Lemma 7. Suppose that 0 < f < 1/2 and (f, 0) ¢ Q_. There exists '1. I'll < f

such that if E = {y E R: '1 < Y < '1 + f/8}, then

rr 2 {{ u~dxdy:::: (l - f) ({ IVu_1 2dxdy. JE JO(y) JE JO(y)

The proof makes use of the inequality

(13) rr2iL j(s)2ds :::: L2iL j'(s)2ds,

for all functions j such that j(O) = j(L) = O. (The minimizer is j(s) sin(rrsI L).) Also,

(14) rr2iL j(s)2ds :::: 4L2iL !,(s)2ds,

forallfunctionsjsuchthatj(O) =0. (Theminimizerisj(s) = sin(rrsI2L).) Because Q_ is simply--connected, there is a continuous curve in the complement

ofQ_ that joins (f, 0) to the real axis. Denote V = {(x, y): Ix - fl < f/4, Iyl <

f/8}. Let y denote the component of the curve from (f, 0) to the first time that

","''--=-'-'~f~~",,~-io.,",_\'......i;;~-.~~'j;i,~~~;;;;'~.d~AA;~~-i';;i;;;~:ij:;::..;;;.,''f,,~~~~.:;.,.--i;~aiit."Y~c;;;r-¢;;i;-""_,,,,~,;i:i~""'~:'~~;.;i;.:~•.:i.i'~~:~T';';;~~~~~,~';,;:o,

245THE FIRST NODAL SET OF A CONVEX DOMAIN

the curve exits V. Let

and '11 = max y.'1 = min y (X,y)EY(X,y)Ey

If '1 :::: y :::: '11 then there exists x such that Ix - fl < f/4 and u_(x, y) = O.

Therefore,

2 i l rr2i1 u_(s, y)2ds = rr2ix u_(a, y)2ds + rr u_(a, y)2ds

:::: x2ix (aIU_(S, y»2ds + (l - X)2 i l

(aIU_(S, y»2ds

1

:::: (l - 3f /4)21 (a1u-(s, y»2ds,

wherealu-(x, y)isthepartialderivativeofu- with respectto x. If'1I-'1::: f/8, then we are done. If not, then y exits V on one of its vertical sides. Suppose that the side is the one for which x = 3f14. (The case in which y exits V on the

side x = 5f14 is similar.) We divide the rectangle R = {(x, y): '1 < Y < '1 + f/8,0 < x < l}intofourparts. Denoteh = '1 -'11 +f/8, P = (f -h, '1+ f / 8), PI = (f, '11), and P2 = (f - 2h, '1 + f 18). Let SI be the segment from P to P[, and let S2 be the segment from P to P2. The region RI is defined as the u,nion of

horizontal segments with left endpoint on SI and right endpoint on the line x = 1 with the shortest vertical segments with upper endpoint on SI and lower endpoint

on y. R2 is union of horizontal segments with right endpoint on the line x = 0 and left endpoint on S2 with the shortest vertical segments with upper endpoint on S2 and lower endpoint on y. R3 is the union of the shortest horizontal line segments with right endpoints (1, y) for all '1 < y < '11 and left endpoint on y. Finally, R4 is the rest of the rectangle R. It is easy to see that R4 is a disjoint union of vertical

line segments at least one of whose endpoints belongs to y. The region RI is foliated by curves that start at x = 1, move horizontally until

they reach S[, and then drop vertically to y. These curves have length at most

1 - 5f18. We claim that this implies that

(15) rr 2 { u~:::: (1 - 5f/8)2 { IVu_12 •

JRI JRI In fact, we can make an area preserving change of variable of the form

(s, t) s+t:::O

F(s,t)= { (-t,s+2t) s+t::::o


247 246 CHAPTER 11

Note that

s+t>Oaas v(F(s, t» = !alv(F(s, t)) (hv(F(s, t» s+t<O

if v is defined on a set U and v vanishes on au. For any numbers a < b let W = un {F(s, t): a < t < band s E R}. Let

L = max length(W n {F(s, t): s E RD, a:9:::,b

the length of the longest "right angle" curve in W. Using (3) and the change of variable F we find

Jr2 { v2 :::; L21 ~v(F(s, t))2dsdt :::; L2 ( lV'vI2.Jw F-l(W) as Jw The segment SI has slope -1, so a translate of this change of variables yields (5). The region R2 is also foliated by right angles with comer on S2. These curves have length at most 9E /8, so that by the same argument as for (15),

Jr2 { u~:::; (9E/8)2 ( lV'u_12.JRz J~

The region R3 is foliated by horizontal line segments of length at most 1 - 3E/4, so that by (3),

Jr2 { u~:::; (1 - 3E/4)2 ( lV'u_12.JR3 JR3

Finally, R4 is foliated by vertical line segments of length at most E/4, so that by (4),

Jr2 { u~ :::; 4(E/4)2 ( lV'u_12.JR JR44

Now, since all of the coefficients on the right-hand integrals over R1, R2 , R3 , and R4 are less than 1 - E, Lemma 7 is proved.

We also need the inequality

(16) Jr2 11j(s)2ds :::; -711 j'(s)2ds + 1~ j(s)2ds, . 0 8 0 3/S

for all functions j for which j(O) = jO) = O. (The essential feature here is that 7/8 < 1, so that there is an improvement on the eigenvalue for the Dirichlet problem on the interval if one adds the extra integral on the right-hand side.) Inequality (16) can be proved by direct computation of the lowest eigenfunction; the calculation is left to the reader.

[f

""""'''''''''''''''''''......... ._--_._._


We can now proceed with the proof of Lemma 4' in much the same way as we 4proved Lemma 4. Let J be an interval of R of length N 1/ • Define E(J) by

l 1 E(J) f i u_(x, y)2dxdy = f 1 (a2U-(X, y»2dxdy.

If E(J) > 1/2C6NI/2, then since

1Jr2 f 1 u~dxdy :::; f 1\alu_)2dXdY,

we have

2(17) (Jr2 + 1 1/2) 1t u~dxdy :::; 1t lV'u_1 dxdy.

2C6N JJo JJo If dJ) :::; 1/2C6NI/2 and (a) fails for an interval l' with the same center as J, but of length N 1/4 - 1, then by Lemma 7 with E = N-1/ S, there is an interval E C J of length N-I/s /8 for which

1 SJr2111 u~dxdy :::; (I - N- 1

/ ) 11 lV'u_1 2dxdy.'

:~ Remark 2 implies'.

1j(y)2dy :::; C6 {N 1/21J'(y)2dy + 8N3

/ s1j(y)2} ,

and, hence,

1 1 2 3III u~dxdy :::; C6! N / 11 (a2U_)2dxdy + 8N /

s 111

u~dxdy 1 1

1 2 3 = C6! E(J)N / 11 (u_)2dxdy + 8N /

s 111

u~dxdy I· It follows that

1 3 s111

u~dxdy :::; 16c6 N / 11 u~dxdy. Now

t u~dxdy = (1 - t)1 t u~ + tit u~ + { t u~ 1E Jo J\E Jo J\E Jo JE Jo

< I ~ tit(aIU_)2 + (l + 16C6N 3/ St) 1t u~ Jr J\E Jo EJo

1

< ---=;- 1 1(aIU_)2 + Jr-2(1 + 16C6N 3/ St).I Jr

t J\E 0


-- --- ~- -"' ~~~

249

-.--,-.----..--.-.------.--.-"-.-'-~ •• --,.;.~~,~0iG'~"""':,;;;~~,,~~~~"'>:F~~,~,,~ ~;';';~Y_"""''''';;;'~~...~~~;,,~~~ .. ;"',~;;.' ....~~~_"'!""' ..._----'----------------- ----'--.-------.-.---•.."---.------.-----.,,'-•.-.-•. .--.--.---------•.:--...

248 CHAPTER 11

8. (1 - N- 1/ ) Ie i1

lV'u_1 2•

Choose t = N-5/ 8• Then for large N,

(1 + 16C6N3/8t)(1 - N- 1/ 8 ) S 1 - N- 1/ 8/2 S 1 _ N-5/ 8 ,

and, hence,

(18) rr2111 u~ S (1 - N-5/ 8) 111 lV'u_12. Finally, consider the case in which E(J) S 1/2C6N 1/ 2 and (b) fails for J. Then the interval E = [Yo - 1/8, Yo + 1/8] C J, and, by (16),

rr 2 { t u~dxdy S (1 - 1/8) { t Uh u_)2dxdy + { t/8 u~.

1E 10 1E 10 1E 13/8 The same argument as the one leading to (18) yields

rr 2 j t u~ s (1 - N-1/ 2) j t lV'u_1 2 + 1{5/8 u~. J 10 J 10 E13/8

And since (b) fails,

(19) rr2111 u~ (1 - N-1) 111 lV'u_12+ N-2 1u~.S /

2

Now if Lemma 4' were false there would be a disjoint covering of the projection onto the y-axis of n by N 3/

4 intervals J satisfying one of (17), (18), or (19). Taking the sum, we obtain

{ {N 3/ {rr2 1n u~ s (1 - N-5

/ 8) 1n lV'u-f + N2

4

1 u~. n Therefore,

5 4 8(rr2 - N- / )(1 - N-5

/ )1u~ s 1,V'u- ,2 = A1u~,

which contradicts Lemma I' when N is sufficiently large.

Institute for Advanced Study Massachusetts Institute ofTechnology

REFERENCES

[C] L. Carleson. "On existence of boundary values for hannonic functions in several variables." Arkiv Math. 4 (1961),393-399.


[CF] R. R. Coifman and C. Feffennan. "Weighted nonn inequalities for maximal functions and singular integrals." Studia Math. 51 (1974),214-250.

[CFMS] L. A. Caffarelli, E. B. Fabes, S. Mortola, S. Salsa. "Boundary behavior ofnonnegative solutions of elliptic operators in divergence form." Indiana Univ. J. Math. 30 (1981), 621-640.

[0] B. E. J. Dahlberg. "Estimates for hannonic measure." Arch. Rat. Mech. Anal. 65 (1977),275-283.

[dG] M. deGuzman. Differentiation ofIntegrals in Rn. Lecture Notes in Mathematics, no. 481. Springer Verlag, 1975.

[H] W. K. Hayman. "Some bounds for principal frequency." Appl. Anal. 17 (1978), 247-254.

[HW] R. Hunt and R. Wheeden. "On the boundary values ofhannonic functions." Trans. Amer. Math. Soc. 132 (1968), 307-322.

[J] D. Jerison. "The first nodal line of a convex planar domain." Int. Math. Res. Not. 1 (1991),1-5.

[KP] C. E. Kenig and J. Pipher. "The h-path distribution of the lifetime of conditioned Brownian motion for non-smooth domains." Prob. Theo. Rei. Fields 82 (1989), 615-624.

[L] C.-S. Lin. "On the second eigenfunction of the Laplacian in R2." Comm. Math. Phys.ll1 (1987),161-166.

[M] A. Melas. "On the nodal line of the second eigenfunction of the Laplacian in R 2."

J. Difj. Geom. To appear. [P] L. E. Payne. "lsoperimetric inequalities and their applications." S./AM. Rev. 9

(1967),453-488 (Conjecture 5). [PI] L. E. Payne. "On two conjectures in the fixed membrane eigenvalue problem." J.

Appl. Math. and Phys. (lAMP) 24 (1973), 721-729. [PW] M. H. Protterand H. F. Weinberger. Maximum Principles in Differential Equations.

Springer Verlag, 1984. [S] S. Salsa. "Some properties of nonnegative solutions of parabolic differential

operators." Ann. Mat. PuraAppl.128 (1981),193-206. [Se] J. B. Serrin. "On the Harnack inequality for linear elliptic equations." J. d'Anal.

Math. 4 (1954-56), 292-308. [SW] E. M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces.

Princeton University Press, 1971. [y] S.-T. Yau. "Problem Section." Seminar on Differential Geometry, S.-T. Yau, ed.

Annals of Mathematics Studies 102. Princeton University Press, 1982.


-- - -251

~ ... 12

On Removable Sets for

Sobolev Spaces in the Plane

Peter W Jones*

1 INTRODUCTION

Let K be a compact subset of C = R 2 and let K C denote its complement. We say K E H R, K is holomorphically removable, if whenever F: C -+ C is a homeomorphism and F is holomorphic off K, then F is a Mobius transformation. By composing with a Mobius transform, we may assume F(oo) = 00. The contribution of this paper is to show that a large class of sets are HR. Our motivation for these results is that these sets occur naturally (e.g., as certain Julia sets) in dynamical systems, and the property of being H R plays an important role in the Douady-Hubbard description of their structure. (See [4].)

To prove that the sets in question are H R we establish what may be a stronger result. A compact set K is said to be removable for W I •2 if every f which is continuous on R 2 and in the Sobolev space WI,2(KC) (one derivative in L 2 on KC)

I 2is also in W • (R2

). It is a fact that if K is removable for W1,2, K is HR. We do not know the answer to the following question:

If K is H R, is K removable for W I ,2?

To prove the fact we first show that the two dimensional Lebesgue measure of K, IKI,iszero.lfnot,letFn = 7<1,"- * (ein(x+Y)XKY))· Thenlim -+ IlFnIlLoo(R2) = 0n oo and Fn is continuous. Since laFnl = Xn, lIaFnllu(R2) = IKI 1/ 2. On the other hand, Fn (z) -+ 0 for z f/. K, so L4 bounds on convolution with ~ when combined with Holder's inequality show IIF~IIL2(K') -+ O. (See [11], Chapter 1.) Taking a sum of functions like the Fn , we obtain a globally continuous F E

'This research was supported by the National Science Foundation under grant DMS-89l6968.

'1" .. _ ~~-ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

.. •. W' ,2 (K'), F ¢ W ',2 (R2). Now us;ng the fact that IK I ~ 0, we deduce K E HR.

\, Take a homeomorphism F with F'(oo) = 1. Thenf(z) = F(z)-z E W 1•2(K C )

4a" i1; because integrating IF'1 2 gives the area of the image. Now f E W I ,2(R2) and c~ af = 0 except on a set of measure zero implies (Weyl's lemma) f is holomorphic.

Therefore, F(z) = z + a. We recall some elementary facts concerning HR. If IK I > 0, it follows from

the "measurable Riemann mapping theorem" (see [1]) that there is a nontrivial quasiconformal mapping F which is holomorphic off K. (Thus K ¢. HR.) If K has Hausdorff dimension less than 1, Dim(K) < 1, the fact that K E H R follows from the Cauchy integral formula (Painleve's theorem). Similarly, if K is a rectifiable curve, Morera's theorem implies K E HR. Kaufman [7] has produced examples of curves where Dim(K) = 1 but K ¢. HR. The "difficult" case is the one that occurs in conformal dynamics: K is connected and has some "fractal" properties. (The case of "pure" Cantor-type sets is easy; they are HR.

By a pure Cantor set, we mean, e.g., one arising from a Cantor construction with a constant ratio of disection, or the Julia set for Z2 + c where c is not in the Mandelbrot set.) We also point out that the case where K is a quasicircle seems to be folklore-again, K E HR. That the property of being H R is related to quasiconformal mappings is seen from the following:

Remark. K is H R if and only if whenever F is a homeomorphism of C which is M quasiconformal on K C , F is globally quasiconformal (and hence M quasiconformal). (See [8], page 200.)

To prove the remark, first assume that K is HR. By the measurable Riemann mapping theorem there is a globally quasiconformal mapping G such that G 0 F

is holomorphic off K. Since G 0 F is a Mobius transformation and IK I = 0, F is globally (M) quasiconformal. For the other direction, standard LP estimates (see [1]) show that necessarily IK I = O. IfF is a homeomorphism which is analytic off K, F is globally quasiconformal and hence (I K I = 0) a Mobius transformation.

Let n be a domain on the Riemann sphere and let Zo E n. Then n is a John domain (with center zo) if there is s > 0 such that for all Zl E n there is an arc yen which connects zo to Zl and has the property that

d(z) ~ sd(z, Zt), Z E y.

Hered(z, Zl) is the chordal distance from z to Zl and d(z) is the chordal distance of z to an. We call such an arc ya John arc. In this paper we will choose coordinates so that Zo = 00, and this allows us to replace d(z), d(z, Zl) by the corresponding Euclidean distances. The property of being a John domain is preserved under globally quasiconformal mappings. If n is a simply connected John domain, it is

1..

i1I

"


·~=~;;';;;;::::;;~~==--='--=-"-=:;~~';':"'~~~~;;;;;;;;;;;';:;;;:~=~=;'=':::"-:: '--,-'-:_" --.:i.o~~-:;:,~~:;.~,,:,~,z;;~~,,:"=::=::,= ....7.i'-~~""""''<i~oi.I~~~;;:';:;':';';;:~·:i;i-~~0·~~",-~~·.6':ii .."".'~;;,-'...-.;:;.i,.<-i,,,..i:",:;,~·...::>',,,=c,,,,,,-=,,,.=,,,,,,,,,=~,,

252 CHAPTER 12

easy to show that the are y may be taken to be the hyperbolic geodesic from Zo to 00. (See [9] for an exposition of properties of John domains.) The main result of this paper is

Theorem 1. If n is a John domain and K = an, then K is removable for w1.2 . JI,

Notice that the hypothesis demands that K = an, but says nothing about the other components of C\K. This is because the hypothesis will be seen to force some geometry on those.other components. (For example, the interior of a cardioid is a John domain while the exterior is not. The parabolic basin for Z2 + i is also a John domain. while the basin for oo-the exterior domain-is not.) It is of some philosophical interest to note the similarities between Theorem 1 and the results of [6] on extension problems for Sobolev spaces.

Since the John condition is quasiconformally invariant, we obtain directly (see also "Remark")

Corollary 1. If n is a John domain and K = an, any global homeomorphism

which is quasiconjormal off K is globally quasiconjormal (with the same constant

ofquasiconformality).

We say that a polynomial P(z) is subhyperbolic on its Julia set J if there is a metric A(z)ldzl such thatA(z) - Lj Iz - zjl-a j is Coo for some numbers aj < 1, and P(z) is hyperbolic on J in the metric A. In other words there are c, e > 0 such that for all n ~ 1.

d A(Z)-IA(Pn(z))l dz Pn(z)1 ~ c(l + et.

Here Pn(z) = Po··· 0 P (z) is the nth iterate of P. (This definition may be a bit restrictive, but it is all we will need for this paper.) The following question is open:

If J is the Julia set for a polynomial. is J E H R?

It is proven in [3] that whenever a polynomial P (z) is subhyperbolic on its Julia set J, then Aoo , the basin of attraction at 00 for P, is a John domain. Since J = aAoo ,

we obtain

Corollary 2. If P(z) is subhyperbolic on its Julia set J, then J E HR.

The corollary answers a question of A. Douady and J. Hubbard and was the starting point of this investigation. Douady posed the question to the author for the particular (subhyperbolic) case where P(z) = Z2 + c has the (Misiurewicz)

...,,,':=.;'""'=-."...-.......,:...-~"'04,· ...";"",.~'",=-=a·"""""'~·==,,~""."":;:.:,·::--"'=;;;:=~'"..;m;~"••;"''''''''';...'''''''''''''''''''''''=''"'''"''''';,.,'".;:;;,;: ";,,,••;,.~'""

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE 253

property that the origin is preperiodic but not periodic (e.g., Z2 + i). This case is not fundamentally different for the general case of subhyperoblic polynomials. An amusing feature of our proof is that the Julia set for a Misiurewicz point (from the family Z2 + c) is actually easier to deal with than those arising from the hyperbolic case. (When K C = n, our argument is a bit simpler. The arguments of sections 5

and 6 are not needed.) The proof of Theorem 1 starts by proving it in the case where n is simply

connected on t, Le., K is connected. The general case then follows from

Theorem 2. If n is an (e) John domain, there is a (c(e» John domain n' with

n' simply connected and

an Can'.

While the proof of Theorem 2 is perhaps not immediately obvious, it turns out

to follow from a simple construction with planar graphs. Section 2 contains background material, and sections 3-7 are devoted to the

proof of Theorem 1. The idea is to redefine F near K so that it is Coo near K and so that the Sobolev norm does not change much. Theorem 2 is proven in section 8.

~{

2 BACKGROUND MATERIAL

Let F E W I ,2(KC ) be continuous on C. An easy argument with the Dirichlet principle shows that to prove F E W1,2(R2) it is sufficient to treat the case we

now assume, where F is harmonic near K. We also assume the reader is familiar with elementary properties of logarithmic capacity, which we denote by Cap(o). See, e.g., [10] for the first two of the next three lemmata. Let f: D --+ C be univalent, f(O) = 0, 1'(0) = 1. Then f has a Fatou extension to T = aD and this extension is always defined except on a set of capacity (and hence Lebesgue measure) zero. In our applications, all image domains will have locally connected boundaries, and, hence, f will be continuous on D. The following results are due to Beurling. The values of c below are various universal constants.

Lemma 2.1. If E C T,

Cap(f(E» ~ cCap(E)2.

Lemma 2.2. Let go = f({reio : 0 ~ r < I}). Then If £0 denotes arclength,

Cap({ej(l : £(g(l) > A}) ~ CA-1/2


255 254 CHAPTER 12

Lemma 2.3. Suppose H is harmonic andcontinuous in D, and (IV'HI2 = IH I2+ x2IH y I ),

2Ii IV'HI dxdy = 1.

Then

Cap({eiO : IH(eiil

) - H(O)I ~ An :::: ce-rrJ.2

•

This last lemma can be found on page 30 of [2]. We next require some elementary geometric facts about simply connected John domains. For the next result see [5].

Lemma 2.4. If g is a Poincare geodesic from 00 to Zo E aQ where Q is an (e)

John domain, then g is an arc ofa K (e) quasicircle.

Suppose now Q is a bounded (e) John domain and suppose the John center Zo satisfies d(zo) = 1, where

d(z) = distance(z, aQ).

Then diameter(Q) '" 1. Let f: D -+ Q, f(O) = Zo be any choice of Riemann mapping, and define for E C aQ,

Cap(E, zo, Q) == Cap({eiil : f(e jil

) E ED.

Lemma 2.5. For any Borel set E C aQ,

Cap(E, zo, Q) '" Cap(E).

In the last line we mean that A '" B if there is a constant M = M (e) such that

M-1AM :::: B :::: MA 1/ M •

Proof Let G(z) = G(z, zo) be Green's function for Q with pole at zoo Then it follows from the John condition and the Koebe ! theorem that

G(z) ~ cd(z)a, a = aCe),

whenever Iz - zol ~ ~. Suppose now that Zj E E, Zj = f(~j), j = 1,2. Fix a point ~3 E D such that

(l - 1~31) '" 1~1 - ~31 '" 1~2 - ~31 '" 1~1 - ~21

and let Z3 = f(~3)' Then by the John condition

IZI - z21 :::: Cd(Z3),

t ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

while by our last estimate,

d(Z3) :::: CG(Z3)I/a '" CI~1 - ~211/a.j~ ''Ii

:,'1~l

l!~ ,,·v In other words,

I~l - ~21 ~ CIZI - z2l a

,

and it follows from the definition of logarithmic capacity that

Cap(E, zo, Q) ~ M-1 Cap(E)M.

The other direction of the lemma follows from Lemma 2.1. • Lemma 2.6. Suppose Q j are (e) John domains with centers Zj, j = 1,2, and

suppose d(ZI), d(Z2) '" 1. Suppose also that F is harmonic on Q 1 U Q2 and

continuous on f.!l U f.!2. Then if E C aQI n aQ2 satisfies

Cap(E) ~ 8 > 0,

there are geodesics gj C Q j from Zj to aQ j such that gl and g2 terminate at the

same point ~ E aQI n aQ2, and

2W(n - F(zj)1 :::: A(e, ll)(fl'V'FI dxdy)I/2, =j 1,2. }

Proof Let

E j = {z E aQ j : IF(z) - F(zj)1 ~ A(fllV'FI 2dxdy)I/2}. }

If A is large enough, Lemmata 2.3 and 2.5 show Cap(E I U E2) < 8. Then

E\(E1 U E2) =f:. rp, so we may select ~ from that set. •

3 QUASICIRCLES

We now give a quick outline of our proof for the case where K is a quasicircle. This represents the only idea of the paper. The rest of the sections contain only technical arguments which make the same philosophy work for the general case.

Let Q+ and Q_ denote, respectively, the unbounded and bounded components of C\K. Fix two points Z± E Q± satisfying

8(z+) '" 8(z_) '" Iz+ - z-I '" 8,

and build domains V± C Q± which are bounded by quasicircles and such that av+ n av_ is a subarc of K with diameter'" 8. The points z± are made to be


256 257 CHAPTER 12

the "centers" of D±. Then by Lemma 2.6 there is A such that

IF(z+) - F(z_)1 ::: A( rr IVFl2dxdy) 1/2.JJv+uv_

Standard smoothing techniques now show there is F E W1•2(R2) such that F = F outside of K8 = (z: d(z) ::: 8}, F is Coo near K, and

rr IVFI2dxdy ::: c rr IVFl2dxdy.JJK~ JJKe,

Sending 8 to zero we see that F E WI,2(R2) and

fl2lVF,2dXdy = fie IVFl2dxdy.

If F is M quasiconforrnal on K C , Lemma 2.2 and an argult:lent similar to the one

above show that F is globally quasiconforrnal. The point of this vague remark is

that, whatever argument we use, it should show that F being M quasiconforrnal

on K C implies F is globally quasiconforrnal. (See the "Remark" in Section 1.)

4 SOME GEOMETRY

In this section we construct certain domains related to a point Xo E K and a scale

r. Since the John condition is scale invariant, we may assume Xo = 0 and r = 1. We will add to K certain curves to obtain a new set K so that, in a certain sense, C\K looks like a union of quasidisks of diameter about 1 (near K).

Let I: D* = {lzi > I} ~ n be univalent with 1(00) = 00. Since K = an is locally connected, I is continuous up to T. Select angles 0 = 00 < 01 < (h < ... < ON = 2Jr so that

illI/(e ) - l(eiOj )! ::: 1, (}j::: () ::: OJ+l,

and

illj I (eilljI/(e ) - +J )1 :::: ~. Now fix M :::: 1 and let rj < 1 be the largest value of r so that

llj I(eillj )I = M.I/(ri )

Setting L j = {reiOj , rj ::: r < I} we see that

(4.1) distance(Lj,L t ):::: c(l-rj), j ;:f=k,

for otherwise the John condition would be violated for the corresponding geodesics in n.

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

Lemma4.1. 11 - rjl "-' 11 - rj+i1 "-' distance(L j , Lj+I).

Proof. We show that l(}j+1 - (}jl ::: C(I - rj). The proof that /(}j+1 - (}jl < C(I - rj+l) is the same. The lemma will then follow from (4.1). Let I {eill : (}j ::: () ::: (}j + Jr}. By symmetry

* Iw(~j, I, .D ) = 2

where ~j == rjeiOj . Here w(z, E, V) denotes the harmonic measure at z of E C

aVin V. By Beurling's so-called! theorem [10], if we set I j = {eill : (}j ::: () :::

(}j+1 }, I

w(~j, I j , D) = w(f(~j), l(lj), n) ::: CM- 2,

because diameter (f(lj)) ::: 1 and distance (f(~j), l(lj)) :::: M - 1. Thus

* Iw(r. 1\1· D ) > '>J' J' - 4

t' if M is large enough, and the lemma follows from simple estimates on the Poisson ~' kernel. •

II: 'i Let V j be the domain bounded by T, L j , Lj+l, and the line segment [~j, ~j+l]

and let 'ij = Rjei"'j where Rj - 1 = ! min(rj - 1, rj+1 - 1), and flJj = ! «(}j + (}j+d. Then since we are assuming diameter (K) » 1, each V j looks like a quadrilateral (in D* with one side on T) with bounded geometry.

Lemma 4.2. nj = I(Vj ) is an (8') John domain with John center Zj = I('ij ).

Proof. Let ~ E V j and let L = [~, 'ij ] be the line segment from ~ to'ij . Then

L C V j and if ( E L, distance(~', aVj ) :::: cl~' - n (This follows from

the elementary geometry of V j .) Now if L' is the geodesic from ~ to 00 in D*, L' = {R~: R :::: I}, p(~', L') ::: C for all ( E L, where p is the hyperbolic

metric on D*. The lemma now follows from the John property on the arc I(L')

and the distortion theorem for I. The details are left to the reader.

Lemma 4.2 is actually a special case of the following fact:

If I: D ~ n, 1(0) = zo, and n is an (8) John domain with John center Zoo and if V c D is a (8) John domain with John center the origin, then I (V) is an

7/(8,8) John domain with John center Zoo

We leave a proof of this statement as an exercise for the reader. •

At this point we remark that Qj = interior of Qj is a 8(8) quasicircle ifC\K = n. (This is. e.g., the case for the Julia set corresponding to Z2 + i.) A most

unfortunate complication is that this statement is easily seen to be false if C\K is

(II·


259 258

_-'C''',"_--',_' _">'-- - _.__••_. , _ __'.~---~'"-

CHAPTER 12

allowed to have bounded components. This necessitates the technical construction of our next section. The reader interested only in the case where n = C\K may skip to Section 7, noticing that Proposition 6.1 has already been proven for quasicircles.

5 SOME ADDITIONAL CURVES

We now add some additional curves to K. Let OJ be a bounded component of C\K. Then by the defiqition of the domains nk, each ank intersects aOj in either

a connected set or the empty set. Let us for the moment reorder the n k so that

ani, anN are exactly those domains such that ann n aOj consists of more than

one point (and hence an arc). Let 0 > 0 be a small constant to be fixed later

and fix a Riemann mapping h: D --+ OJ so that II, ... , IN are intervals with h(ln) = ann n aoj . By selecting h(O) to lie very close to ani n aOj we may

assume £(/1) ~ 2;rr. Let TI be the tent-shaped region bounded by TVI and two straight lines in D which intersect TVI at angle O. The TI is a "thin sliver." Define

UI = UI = D\TI so that aUI n T = II. For n ~ 2 let L~, L~ be the two lines which start at the endpoints of In, go into D, and make angle = 0 with TVn' Let

ifiIn = {(l - o-I£(/n»e : 0 ~ 0 ~ 2;rr}, and let Un be the domain bounded by the four arcs In, L~, L~, In. Then Un almost fills up a rectangle with length (along T)= 0-2£(ln) and width (in the direction orthogonal to T) = o-I£(/n). Then by elementary estimates on the Poisson kernel,

(5.1) {w(z, I", D) ~ CIO} C Un C (W(z, In, D) > C20}.

By reordering we may now assume that

£(h) ~ £(h) ~ ... ~ £(/N).

AnN N A •

Define Un = Un \ Uk=1 Uk SO that Un=1 Un = Un=1 Un. Recall that a domam U is called an M Lipschitz domain if there is Zo E U and R > 0 such that

au = {zo + Rr(O)eifi : 0

(l + M)-I ~ reO) ~ 1

where

and

~ 0 ~

for all

2;rr}

0

Ir(O) - r(O')j ~ MJO - Oil.

Lemma 5.1. Un is a M(o) Lipschitz domain, I ~ n ~ N. Furthermore, if { E DnaUn,thereisf/J E [0, 2;rr]suchthatthelinesegmentDn{{+reifi : r ~ O}

"."_---

ON REMOVABLE SETS FOR SOBOLEV SPACES 1N THE PLANE

lies in Un and has endpoint on In whenever

210 - f/JI ~ co •

The proof of the lemma is an exercise in elementary geometry. Now let OJ = /j(Un). If we consider any nk, we have for each OJ, such that aOj n ank is an

arc, obtained a domain oj C OJ (sometimes oj = OJ) with the property that

aOj n aOj C ank. Leth = {OJ: aOj n ank is an arc} andledik = interior

of closure of nk U UFt OJ.

Lemma 5.2. ank is an 7/(e, 0) quasicircle.

Proo! Let yf = OJ n aOj. Then ank = ank U U Ft yf. We first :Iaim nk is an 7/(e, 0) John domain. It is only necessary to find for every Zo E ank an arc y C n k which has endpoints zo and Zk such that

distance(z, and ~ 7/lz - zol, z E y.

If Zo E ank this is clear by Lemma 3.3. We therefore assume Zo E yf for some j. By Lemmata 2.2, 2.3, and 5.1 there are angles f/J-I < f/Jo < f/JI such that

Jf/Jf - f/Jm I "J 02, £ ::j:. m, such that

ifiD n U;I(ZO) + re , r > O} C Uj ,

whenever f/J-I ~ 0 ~ f/JI, and such that

£(f'm) == £U/Uj-I(zo) + rei'Pm : r > OJ) ~ Cd(zo)·

Furthermore, Lemma 5.1 allows us to assume that f'm is a Jordan arc and if z E r f

and Iz - zol ~ ! d(zo),

(5.2) distance(z, r m) ~ cd(zo) , £::j:. m.

Now let Om be the endpoint of r m on ank and let y_I (resp. yd be the John geodesic from {-I (resp. {d in n k to Zk (the John center of nd. Then the curve

y = r -I uri U Y_I U YI surrounds {o and by the John condition on nk,

(5.3) distance({o, y) ~ cd(zo).

(Notice here that we are implicitly using the fact that z E oj implies d(z) ~ C. This, in tum, follows from (5.l) and either Lemma 2.2 or 2.3.) Notice also that the interior of Y must lie entirely in nk.

Let Yo be the John geodesic in nk from {o to Zk. We claim that the John condition

for n k holds on ro U yo. First suppose that ZEro and Iz - zol ~ ! d(zo). Then by the distortion theorem for h,

distance(z, and ~ distance(z, y) ~ clz - zol.

I


260 CHAPTER 12 ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE 261

Now by inequality (5.2),

d(z, aQk) ~ d(z, y) ~ elz - zol

whenever z E f o and Iz - zol ~ 1d(zo). (Here we have used the John property

on nk to obtain distance (z. Y-l U Yl) ~ elz - zol·) We must finally check the John condition on Yo. Ifz E YO and lz - {ol s: ed(zo),

the inequality on distance (z, aOk ) follows from (5.3) and the fact that Izo - {ol :::: Cd(zo). If z E Yo and Iz - {ol > ed(zo), the inequality on distance(z, aQk) follows from the John condition distance(z, ank) ~ elz - {ol. We have thus established that Ok is a- John domain.

We now claim that Gk = C\(Od is a John domain. We note that by the definition of Qb aOk = aGk. Now fix a point zo E Gk.

Case A. Zo E (Qj) for some j. (Then j I- k.) First draw the John geodesic in n j

from Zo to Zj. We then draw the geodesic (in the Poincare metric ofn) from Zj to 00. By the construction of the domains n j (Lemma 3.2) this is a John geodesic. The union of these two geodesics provides the arc joining Zo to 00.

Case B. Zo E n\ Uj Qj. Let Y be the Poincare geodesic in n from Zo to 00.

Then by Lemma 3.2, y is a John geodesic in Gk •

Case C. Zo ¢. n u Uj fl. j . Then Zo E Ojo for some jo. Let

A = sup diameter n j, j

so that A ~ I. If d (zo) ~ 2A there is a half line y (to 00 from zo) which is a John geodesic in Gk. Ifd(zo) < 2A there is a hyperbolic geodesic (which is also a John

arc in Gk) Yl from ZO to Zl E Ojo,f where £ I- k, d(z) ~ I on Yl, and £(y) s: C. (This follows from the definition of the domains OJ,k') By Case A there is a John geodesic Y2 from Zl to 00 in Gk. The curve Y = Yl U Y2 is the required JolIn arc.

The proof is now completed by first observing that a simply connected domain D with aD locally connected, D = Interior(V), and C\V connected is bounded by a Jordan curve, and then invoking the following fact (see [9]):

A Jordan domain D is bounded by a quasicircle if and only if D and C\T> are

John domains. •

6 AN ESTIMATE ON CAPACITY

We now seek to imitate the proofgiven in Section 3. What is required is an estimate implying that IF(z) - F(zj)! is not too large on aQj, except for a set of small

capacity. While nj is a quasidisk, Qj n K C is not necessarily connected. This ~ means we cannot simply apply Lemma 2.3. We state our result as a proposition; its proof will be broken into several steps. The result we state is far from optimal, but it is all we need. Let Qk be the domain obtained by adding to Qkthe set

U{z E OJ: p(z, aOj,k) < I}, j

where p is the hyperbolic metric on OJ. The domains Qk then satisfy

L~. s: C.

(. ~roposition 6.1. Suppose H is continuous on the closure of Qk and harmonic on

Qk \K. Then if

r~ IVHI2dxdy = 1.J.!?ik\K

we have the estimate

Cap({z E aQk: IH(z) - H(Zk)1 > A}) = 0(1)

as A ~ 00.

t Proof. Let EI = {z E aQk n ank: IH(z) - H(Zk)1 > A} and let E2 = {z E :y3 ~::~ aQk\ank: IH(z)-H(Zk)1 > ),,). ThenbyLemmata2.3and2.5,Cap(E I ) = 0(1)~K

I i,!I,

as )" ~ 00, so it is sufficient to show Cap(E2) = 0(1) as)" ~ 00.

It Step 1. Construction of Some Special Points. Let {Zn) be a collection of points

in aOk \ank satisfyingI

IZn - zml ~ 4:1

d(zn), V n, m

f and

I1

i~f lz - znl s: "2 d(z), V Z E ank\ank.

We will now form for each Zn a point z~ E nk. For an arbitrary point z E

aoj,k\ank we let K z = {{ E ank: Iz - {I S 2d(z)) so that Cap(Kz) ~ cd(z). I,t By the John condition and Lemma 2.5, there is a point z* E nk such that d(z) ~ ~"

d(z*) ~ Iz - z*1 and there is a set Kz C K z such that

(6.1) Cap(Kz, z*, n), Cap(Kz, z, OJ) ~ C.

Denote by f a Riemann mapping from nk to D with f (Zk) = 0, where Zk is the "center" of nk' We can move z· slightly so that f(z*) has the form

Ie ~

(6.2) f(z*) = (l-Tf )exp{im2-f n}


263 262 CHAPTER 12

for some.e, mEN. By this method we produce from our collection {Zn} a new collection {z~}. Notice that it is possible that z~ = Z;, even if n =1= m, but then d(zn) '" d(zm) '" IZn - zml.

Step 2. Another Geometric Construction. Let {Zn} be the collection of points in Step l. Let {In} be a collection of subarcs of ank\ank such that Un In = ank \ank, In n 1m = ¢ when n =1= m, diameter(In) '" d(zn), and Iz - Zn I .::: 4d(Zn) for zEIn. We also define I; to be the arc

f-1({(l- rf)exp{i(m + t)rfJr}: O::s t .::: I}).

See (6.2) for notation. Then I; has diameter'" d(zn) and if Z E I: the hyperbolic distance from Z to z~ (in nk) is bounded by C.

With the notation of (6.1) we also denote by I n the subarc of T

{e illI n = : m2-f Jr < () ::s (m + 1)rfJr}

and we denote by Qn the "square"

Qn = {re iO : (l-r i ).::: r'::: l,ill E I n}.

We now use the standard terminology that an arc Jm is maximal in a subcollection

F of {In} if Jm E F and It E F, i =1= m, imply either Ji n Jm = ¢ or Jf C Jm. Notice (by the John condition) that if Ji C Jm ,

(6.3) IZf - z~ I ::s Cd(zm).

Finally, if:f is a subcollection of {In} and E = U d In we denote by E* the1n set

E* = U I; J. EJ'"

where

F = {In: In E j: and In is maximal}.

Step 3. A Capacitary Estimate. Let E and E* be sets as in the previous paragraph.

Lemma 6.2. Cap(E*) 2: c Cap(E).

Proof Let J.L be a probability measure on E satisfying

f log _1-dJ.L(O .::: y, Z E C. Iz - ~I

We relabel the intervals In E F so that F = {II. J2, ...} and d(ZI) 2: d(Z2) 2: ... 2: d(zn) 2: d(Zn+l 2: .. '. Define

En = {z E E: Iz - znl .::: Cd(zn) and Iz - zml > Cd(zm), m < n},


SO that by (6.3).

E= UEn .

n

Notice that the sets En are pairwise disjoint.

Now define a probability measure J.L * by setting J.L * to have uniform distribution on the center half! I; of I;, J.L*(l;\ 4I;) = 0, and

J.L*(l;) = J.L(En)·

By the construction of I; and 4I;, 1 I

dist( 2" I;, 2" I~) 2: cd(zn), Vn =1= m.

Let Z E E and z' E 4In where n satisfies z* E In. Then

log I dJ.L (0 = +f *! 1 1

Iz' - ~ I (Iz'-{I~Ad(z)l (Iz'-{I>Ad(z))

1 ::s c + log -- dJ.L(O

1 {lz'-{I~Ad(z)} Iz - ~I

I+ c + log , dJ.L(O (lz'-{I>Ad(z)} Iz - ~I

.::: 2c + y,

and Lemma 6.2 is established. • Step 4. Proofofthe Proposition. By Lemmata 2.3 and 2.5 and by estimate (6.1),

IH(z) - H(z*)1 ::s 1.

Now let 'D = nk \ UF Qn, where the Qn are as defined in Step 2 and Qn f- I (Qn)' Then by Lemma4.1, 'D is an (£') John domain. We define our collection

F to be {In : 3z E In, IH(z) - H (zk)1 2: A}. Then ifIn E -G IH(O - H (zk)1 2: A - c for all ~ E In. (This is why we slightly enlarge nk to Qk .) By our previous estimate,

IH(z) - H(Zk)1 2: A 2c on I;, for any In E F. Setting as before

E = Uln

F

and E* = U I*m'


265

iiiijjj,"C'jiii> - -- - 'I 'I:lIlW ,.. ~,._- - '~-

264 CHAPTER 12

we have Cap(E*) ~ c Cap(E). Now since

2f1 IVHI dxdy ~ 1,

it follows from Lemmata 2.3 and 2.5 that

Cap(E*) = 0(1) as A -+ 00.

This completes the proof of Proposition 6.1. • 7 PROOF OF THEOREM 1

Let Q j and Qk be two domains satisfying

distance(Q j, Qk) ~ 1.

It is an exercise to find domains Qjl' .. " Q jN where h = i, iN = k, and

oQj~ n aQj~+l is an arc of diameter ~ c, and N ~ C. (Use the fact that Q

is a John domain and each Q j is an TJ quasicircle, i.e., Lemma 5.2.) Then by

Proposition 6.1,

N-I

IF(Zj) - F(Zk)1 ~ L IF(Zj~) - F(zj~+)1 m=1

N ( ) 1/2

~ C ~ fkj~\K IVFI 2 dxdy

1/2

< c' IV FI 2dxdy)- (flzEKC: Iz-zjl.:5C)

We also notice by (6.1) that if Z E Qk and d(z) ~ 1,

IP 2

IF(Z) - F(zdl ~ C fkk\K IVFI dxdy( )

Putting our last two estimates together we see there is F E C OO (R2) such that

F(z) = F(z) when d(z) ~ 1 and

rr IV FI 2dxdy ~ C rr IV Fl 2dxdy.JJlzEKc: Id(z)I.:5l1 JJIZEKc : Id(z)I.:5Cj

Here we are using the fact that, by the construction of the Qt, {z E KC : d(z) ~

I} C Uk Qk. Since the John condition is dilation invariant, we may now build a


sequence Fn E C OO (R2) with Fn(z) = F(z) when d(z) ~ ~ and

ff IV Fnl2dxdy ~ C rr IVFl2dxdy.JJIZEKc: d(z).:5 ~ } JlrzEKc: d(z).:5 ;}

SincelKI = O,itfollowsthat F E W1.2(R2)and

2 2fL2 IV FI dxdy = fic IVFl dxdy.

8 PROOF OF THEOREM 2

Let Q be an (e) John domain with compact boundary K of diameter one, let {Q j}

denote the Whitney decomposition of Q into dyadic squares [11], and let Zj be the

center of Qj. Let A = A(e) be a large constant and define Tn = {Zj: A-n ~

d(zj) ~ A}. It is an exercise with the John condition to construct a connected

graph Go such that every edge in Go is of the form [z j, zd where aQ j n 0Qk =f. ¢, and where the vertices Vo of Go satisfy

To C Vo C {Zj: 1 ~ d(zj) ~ A 2 }.

We also build Go so that

(8.1) If d(Zj) ~ 1 and Zj ¢. Vo then Qj is in the unbounded component of

C\ UZkEFo Qk. It is now an exercise (with induction) to construct connected graphs G withn

the following properties:

(8.2) Every edge in Gn is ofthe form [Zj, zd for some Zj, Zk E Tn+1 U Vo where

oQj n OQk =f. ¢. (8.3) Every Zj E Tn is in Vn, the vertices of Gn.

(8.4) If p(Zj, zd is the graph distance on Gn,

inf p(Zj, zd ~ C, Zk E Gn .ZkEF. (8.5) Gn C Vn+l •

Notice that we have chosen Go to be connected. Let ZO E Vobe an extreme

point of the (planar set) convex hull (Go). We may assume by induction that ~. each Gn is actually a directed graph in the following sense. Each edge [Zj, zd 'J

is directed in the sense that (perhaps switching i and k)

(8.6) p(Zj, zo) = p(zt, zo) + 1.

Such an edge is an outgoing edge from Zj. It is not hard to see that we may choose the Gn so that

(8.7) Each Zj =f. Zo has exactly one outgoing edge.


267

.__. ..-"'-"'-"'-"'.:"'....,,...,...,,.,,"""""'".

266 CHAPTER 12

Lemma 8.1. The graph G n is simply connected, i.e., it contains no loops.

Proof Suppose, to the contrary, that there is a loop in Gn . Let Zj be a vertex in the loop maximizing p(Zj. zo). Then Zj has two outgoing edges (by (8.6» and this contradicts (8.7).

Let G = limn Gn be the limiting graph, so that G is simply connected. It is clear that K U G is connected. Notice by (8.3) that (8.8) For every Zj E G there is an arc y C G from Zj to zo which satisfies the s'

John condition in Q.

In other words, G is a John graph. For a Whitney square Qj with Zj E G let {.c~} denote all the edges of G with

one endpoint being Zj' Define

11 = {z E aQj : distance(z, .c~) < 8 diam(Qj)},

where 8 is a small constant, and put

Sj = aQj\ U11, j f:. o. k

For the special point Zo E G we select a Whitney square Qe such that Ze if. G, aQo n aQe f:. ¢, and we put

So = aQo\(lf U U If), k

Q = Q\ U Sj' •ZjEG

Lemma 8.2. Qis simply connected.

Proof Let Q+ = U{Q n Qj : Zj if. G}, Q_ = U{Q n Qj : Zj E G} so that

~ ~ ~ 0 Q = Q+ U Q_ U Ie .

By condition (8.1), Q+ is simply connected (in C), so it is only necessary to check that ~'L is simply connected.

We first verify that Q_ is connected. Let Z E Qj n Q_ and let y be an arc in G connecting Zj to Zoo Then y' = [z, Zj] U y is an arc in Q_ which connects Z to

Zoo Now suppose that y is a loop in Q_ that is not homologous to zero. It is then

an elementary exercise to homotopy y to y', a loop in G that is not homologous to zero. This contradicts Lemma 8.1.

It is clear from the construction of Q that aQ c aQ. To verify that Q is a John domain we must look at two cases.


Case 1. Z E Q+ n Qj. There is arc y from Zj to some Zk if. G such that length(y) :::: C, d(z) ::: 1 on y, and Zk is not in the convex hull of aQ. By selecting a suitable ray R from Zk to 00 we then see that

[Z,Zj]UyUR

is the required John arc.

Case 2. Z E Q_ U If. Let y be a John arc from Ze (the center of the special Whitney square Qe adjacent to zo) to 00. Then if Z E Qj and y' eGis the John arc from Zj to Zo guaranteed by condition (8.7), we see that

[z, Zj] U y' U [zo, zd U y

is the required John arc. • Yale University

REFERENCES

[l] L. V. Ahlfors. Lectures on Quasiconformal Mappings. Wadsworth, 1966; 1987. [2] . Conformal Invariants. McGraw-Hill, 1973. [3] L. Carleson and P. W. Jones. "On coefficient problems for univalent functions and

conformal dimension." Duke Math J. 66 (1992), 169-206. [4] A. Douady and J. H. Hubbard. "Etude dynarnique des polynomes complexes, I, II."

Publ. Math. d'Orsay, 84-102. [5] F. W. Gehring and K. Hag. "Quasi-hyperbolic geodesics in John domains." Math.

Scand. 65 (1989), 75-92. [6] P. W. Jones. "Quasiconformal mappings and extendibility of functions in Sobolev

spaces." Acta Math. 147 (1981), 71-88. [7] R. Kaufman. "Fourier-Stieltjes coefficients and continuation of functions." Ann. Acad.

Sci. Fenn. 9, 27-31. [8] O. Lehto and K. 1. Virtanen. Quasiconformal Mappings in the Plane. Springer Verlag,

1973. [9] R. Nlikki and J. Viiisiilii. "John Disks." Exp. Math. 9 (1991),3-43.

[10] Ch. Pornmerenke. Univalent Functions. Gottingen, 1975. [11] E. M. Stein. Singular Integrals andDifferentiability Properties ofFunctions. Princeton

University Press, 1970.


=====-----

13

Oscillatory Integrals and

Non-Linear Dispersive Equations

Carlos E. Kenig*

In this chapter I will describe a collection of results, obtained jointly with G. Ponce and L. Vega, on well posedness and non-linear scattering, with data in Sobolev

spaces, for solutions to a variety of non-linear dispersive equations. I hope that I will be able to convey, in describing these results, and the meth

ods used in their proofs, how many of the ideas pioneered and developed by E. M. Stein, such as complex interpolation, the application of oscillatory integrals to the study of restriction theorems for the Fourier transform, and the application of the Kolmogorov-Seliverstov-Plessner method to study maximal operators in which curvature is present, can be used in a natural way in the study of non-linear

dispersive partial differential equations. The results thus obtained are sharp in many instances, and do not seem to be attainable by any other approach.

When considering an initial value problem (LV.P.) of the form

au - + A(u) = 0at

{ ult=o = Uo

where A is a linear or non-linear operator and Uo E X, a Banach space, we say that the problem is globally well posed in X if given Uo EX, there exists a unique u E C«-oo, +(0); X) n V'O«-oo, +(0); X) satisfying ~~ + A(u) = 0, with

u(0) = Uo, and such that the mapping

Uo t-+ u E C«-00, +(0); X) n L00«-00, +(0); X)

"This research was supported, in part. by the National Science Foundation.

---~-~~

OSCILLATORY INTEGRALS AND NON·LlNEAR DISPERSIVE EQUATIONS 269

is continuous. If this is true when the time interval (-00, +(0) is replaced by (-T, T), T = T(uo) > 0, we say that the problem is locally well-posed.

From now on, X = HS(JRn) = (f E L 2(JRn): (-,-~)'/2 f E L 2(JRn)}, or X = HS(JRn) = {f E Lfoc(JRn): (_M s /2 f E L 2 (JRn)}. To illustrate these

concepts, we present two families of well studied examples.

Example 1 (Generalized Burger's equation).

au k au - +u - =0 k~

(1) at ax{

ult=o = uo·

Then (see [58]), (1) is locally well posed in HS(JR), s > 3/2, and is not locally well posed in HS (JR), S :::: 3/2, for any k ~ 1.

Example 2 (Semilinear Schrodinger equation in JR x JR).

au a2u - + i -2 + ilulku = 0, k > 0 (2) at ax

{ ult=o = uO·

Then (see [58], [21] and [62]), for 0 < k < 4, (2) is globally well posed in L2(JR),

while for 4 :::: k, (2) is locally well posed in H S (JR), S ~ Sk = k;4, and globally well posed for small data in HSk (JR). Analogous results hold in JRn; there 4 is replaced by 4/n, and k;4 by k~k4 .

Example 1 is treated by using energy estimates. Example 2 is treated using the extensions due to R. Strichartz ([57]) of the Stein-Tomas ([61]) restriction theorem for the Fourier transform.

In our work, the general strategy when studying a non-linear dispersive problem is to use oscillatory integral estimates to obtain sharp inequalities for the homogeneous and inhomogeneous associated linear problems. Many times, these are estimates that involve fractional derivatives and are measured in mixed norms, where the time variable norms are taken first. In order to apply those estimates to our non-linear problems, we develop mixed norm (and weighted norm) Leibniz and chain rules for fractional derivatives. Using our linear estimates, we then construct suitable space-time Banach spaces, in which we solve our non-linear problem by using the contraction mapping principle.

In the remainder of this chapter, I will describe the most salient results thus obtained. I will then illustrate the proofs by sketching the arguments used in one


270 CHAPTER 13

particular case. This case will clearly show the impact of the ideas of E. M. Stein

on the subject at hand. We start out with our results on the generalized Korteweg-de Vries equation:

3au a u k au-+-+u-=O k>l (3) at ax3 ax '

{ ult=o = Uo

For k = 1, this equatiop. was derived by Korteweg-de Vries as a model for long waves propagating in a channel. Subsequently, the cases k = 1, 2 have been found to be relevant in a number of different physical systems. Moreover, these equations have been studied because of their relation to inverse scattering theory

and to algebraic geometry.

Theorem 4. (i) For k = 1, (3) is locally well posed in HS(JR), s > 3/4, with time interval of

length T = T(lIuoIIH')' (ii) Fork = 2,(3)islocallywellposedinW(JR),s::: 1/4,T = T(ll uoIIH'/4). (iii) For k = 3, (3) is locally well posed in HS(JR), s ::: 1/12, T = T(lI uoIlH'/12). (iv) For k = 1,2,3, if s ::: 1, then we have well posedness On any time interval

[-T, T], and the solution belongs to V·O(JR; H[s](JR». (v) For k ::: 4, (3) is locally well posed in W (JR), s ::: Sk = (k - 4)12k. (vi) For k ::: 4, (3) is globally well posed for small data in JiSk (JR). It is also

globally well posed in HS(JR), s ::: sk,for data small in JiSk(JR).

(i) was first obtained in [33], (iv) for k = 2, 3 was obtained in [31], while for k = 1, it was obtained in [33]. The remaining results are in [36].

We have also obtained a sharp non-linear scattering result.

Theorem 5. Let k ::: 4, and assume that lIuollli'k(IR) ~ Sk, Sk the "smallness" prescribed by Theorem 4(vi). Then, there exist unique wt E JiSt(JR) such that

(6) lim Ilu(t) - W(t)wtIIH"k = 0,t---+±oo

where u(t) is the solution to (3) given in Theorem 4(vi), and W(t)wo = w(t) is

the solution to the associated linear problem

3aw a w -+-=03at ax(7)

{ wlt==o = Wo

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERS1VE EQUATIONS 271

Remarks on Theorems 4 and 5. (iv) in Theorem 4 follows from (i), (ii), (iii) because of the conservation laws

1+00

2 +00 ( au )h(u) = -00 u (x, t) dx; h(u) = (_)2 - Ckuk+2 (x, t) dx.

/ -00 ax

Previously, ([4], [5], [50], and [27]) the energy method showed local well posedness in H S (JR), s > 3/2, just as in example 2. Moreover, using this result and energy estimates for the second derivative, it was shown in [5] and [50] that there is global well posedness in H S (JR), s ::: 2 (with small HI norm for k ::: 4).

In [59] and [28], using the conservation laws /z and h global weak solutions with HI data were constructed (for small HI data if k ::: 4). Moreover, in [28], taking advantage of the term ~:~ in (3), a "local smoothing effect" was discovered (which is not true for solutions of (1». Thus, it was proven that there exists a weak solution u, with data Uo E HI (JR) (small data if k ::: 4) such that

[ lR a2u

(8) 1-2 u(x, t)1 2 dxdt ~ C(R, lIuollHI) Ixl<R -R ax

and that, (for 1 ~ k ~ 3) if Uo E L 2(JR) , there exists a weak solution such that

2 (9) { r Iau (x, t)1 dxdt ~ C(R, lIuollu).

J1xl<R LR ax

In [23], it was shown that, for k ::: 2, the weak solution with HI data is unique. Finally, in [22], and using the methods of [32], [33], and [35], results for k ::: 4, weaker than those in Theorem 4(v), are announced.

Note that the results in Theorem 4 are very similar to the ones in example 2. The novelty here is that the non-linear term includes derivatives, complicating the analysis.

The results in Theorem 4 are sharp (in a suitable sense) for k ::: 4. To explain this, and to clarify Theorem 5, let us fix our attention on the case k = 4. A perfect balance between the dispersive effect and the non-linearity in (3) is represented by the existence of solitary wave solutions

Uc,k(X, t) = ¢c,k(X - ct), c > 0

where

(k + 2)c ( k )} 1/4(10) ¢c,k(X) = 2 sech2 2.,JCx{

A simple computation shows that, for k ::: 4, and Sk = k;4,

lI¢c,kllli'k = at.


272 CHAPTER 13

where ak > 0 is independent of c, and if s i= Sk. then

lI¢c,k II !i' ~ 0 as c ~ either 0 or +00,

Returning to the case k = 4, the solution that we construct to (3) for lI uoliu ~ £0

has the property that

1-00+00 1+00 6 2-00 ID I/ U(X, t) 1 dxdt < 00.

2This result fails for powers k i= 4 as can be seen from Uc,k. and also forlarge L (lR) nonn as can be seen from u 1,4- Similarly, the scattering result of Theorem 5 cannot hold in the L 2 nonn for k i= 4, or for the L 2 nonn k = 4 and large data.

Next, we tum our attention to the results that we have obtained for the hierarchy

of the generalized KdV equations. Thus, we consider

I2 1 2j

au a j+ u (au a u)at + ax2j+l + Q u, ax"'" ax2j = 0 (11)

ult=o = Uo

where j is a positive integer, x, t E JR and

Q: JR2j+1 -----+ JR

is a polynomial having no constant or linear tenns, i.e.,

(12) Q(z) = Q(zJ, Z2, ... , Z2j+l) = Lp

aa za , with k::: 2. lal?k

Theorem 13 ([37]). (i) There exist m, So E N, and 8 > 0 such that, for any s ::: so, and any

Uo E HS(JR) n L 2(lxl mdx), of norm less than 8 in that space, (11) is well

posed on a time interval (-T, T), T = T(Q; lI uoII w onL2(1xl'" dx»)' (ii) If k in (12) ::: 3, we can take m = 0 in (i). (iii) If Q does not depend on Z2j+l, the results in (i) and (ii) hold without any

smallness assumption on uo. (iv) The results in (ii) are global in time ifone of the following two hypotheses is

verified: (a) k ::: 4j + 3 in (12), or (b) k ::: 5 and Q(z) = Q(Zi, ... , z2j+d with l ::: (2j - 1)/4.

l _~!§\~~~ _~ __ I_4!@.~~ 5liJ&!lS2!5 2&~ ~ __• __ _ _.

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS 273

Remarks on Theorem 13. (11) generalizes the KdV hierarchy introduced in [41]. In [18], it was shown that the eigenvalues of the time independent SchrOdinger operator

d 2

L(g) = dx2 - q(x)

remains unchanged when the potential q(.) = u(·, t) evolves according to the

KdV equation ~ + ~ + u ~; = O. This discovery was the starting point of the inverse scattering method. In [41], it was shown that the same principle holds for the sequence (KdV hierarchy)

au (14) - + [B j ; L(u)] = 0,

at

(with [A; M] = AM - MA), and Bj denotes the skew-symmetric operator

d2j+l j-l ( dU+l dU+1 )

Bj = etj dx2j+1 + L bji dX2f+l + dxU+1 bji , i=O

and the coefficients bji = bji(u) are chosen so that the differential operator [B j ; L(u)] has order zero. Equations of the type considered in (13) also appear as higher order models in water waves, in elastic media with microstructure ([42]), and in recent developments in physics ([67]).

It is interesting to note that the classical approaches-such as energy estimates, abstract semi-group theory, or space-time estimates-used to study other evolution equations cannot be applied to (13) except for very particular fonns of the polynomial Q. Previously, existence results for the KdV hierarchy (14) were obtained in [48], and uniqueness for (14) was shown in [47]. Both sets of results depended heavily on the Hamiltonian fonn of (14). In [45], for some fifth order equations of the type appearing in (11) both in Hamiltonian and non-Hamiltonian fonn, well posedness results were obtained.

Next, we tum to higher dimensional results. We have been able to establish local well posedness results for non-linear Schrodinger equations, with first order derivatives in the non-linearity. Thus, we consider

Iau - = if),u + P(u, Vxu, U, Vxu) t E JR, x E JRn

(15) at

ult=o = Uo

where u = u(x, t) is a complex valued function, and

P: C2n+2 -----+ C


274 CHAPTER 13

is a polynomial, PCl.) = P(ZI, ... ,Z2n+2) = Ld::S:lal::S:p aaza, and d ::: 2. (ytfe always assume that there exists aao' laao I =I 0, laol = d.)

Theorem 16 ([35]).

(i) Assume that d = 2. Then, there exists 0 = o(P) > 0 such that,jor any Uo E HS(IRn) n H 2n+3 (IRn: Ixl2n+2 dx), with s ::: So = 3n + 4 + 1/2. and

IluoIlH'onH2n+3<1xlz.+zdx) = 00::: o,then(15)islocallywellposedon(-T, T), T = T(oo).

(ii) Assume that d :::- 3. Then there exists 0 = o(P) > 0 such that,for any Uo E W(IRn), with s ::: So = n + 2 + 1/2 and lIuollH'o ::: 0, then (15) is locally well posed on (-T, T), T = T(lIuoll H'o),

Remarks on Theorem 16. As mentioned in the discussion of Example 2, the

semilinear case P = j(lul)u (with j a real valued function) has been studied extensively. The general initial value problem (15) had been treated before mainly

when one could either use energy estimates or deal with analytic data and analytic solutions. Energy estimates for (15) can be established when, using integration by parts, one can show that

L [ (~)ap(U' V'xu,"ii, V'xU) (~)a u dxl ::: Cs(I + lIull~,)lIull~" lal<s JR. ax ax

for any u E W(JRn), with s > 1 + 1, p = pep) E Z+. The above estimate

can be guaranteed only if P exhibits an appropriate symmetry. For example, n = I, P = aa (Iulku), k E Z+ (see [63], [64]), n ::: 1 and a.E!!. P, a au P, j = x aXj aXj

I, ... , n are real valued functions. In this case the local well posedness in H S (JRn), s > 1 + 1 follows from the argument used for quasilinear symmetric hyperbolic systems ([26]). Indeed, for those P's the same proof works if one removes the

Laplacian term from the equation in (IS). Also, when n = 1, P = aa (luI 2u), in x

[30] the inverse scattering method is applied.

The approach that deals with analytic data uses analytic function techniques to overcome the loss of derivatives introduced by the non-linearity ([25]).

Our method works without any special assumptions on P. It works for real or complex valued functions, or for systems of equations. The same method applies to non-linearities given by smooth functions F (u, V'xu, "ii, V'xU) with Taylor

expansion at the origin having no constant or linear terms. We have not attempted to obtain the best results provided by our method, since, in any case, it is not clear

that they would be optimal.


Nevertheless, in cases where both the energy estimate and our method apply, our method seems to give better results. An instance of this is our study of the initial value problem for the generalized Benjamin-Ooo equations

au k au a - +u- ax Dxu = 0, t, x E JR, k E z+ (17) at ax

{ ult=o = Uo

where Dx = (- aa;2 )1/2. In the case k = I this equation was deduced in [3] and

[43] as a model in internal wave theory. The generalized Benjamin-Ono equation presents the interesting fact that the dispersive effect is described by a non-local

operator and is weaker than the one exhibited by the Korteweg-de Vries equations.

The energy method proves local well posedness in H S ,s > 3/2. Our result here is

Theorem 18 ([34]).

(i) Let k::: 2 and s be such that

S>1 k=2

Is > 5/6 k = 3

s ::: 3/4 k ::: 4.

There exists 0 = o(k) > Osuchthat,joranyuo E HS(IR),lIuolIs::: 0,(17) is locally well posed on (-T, T), T = T(lIuolls, k).

(ii) For k ::: 4, s ::: 1 there is global well posedness,jor small initial data. (iii) For k ::: 4, Uo E HI (JR), II Uo 111 ::: 0, if u is the solution oj (17) there exist

unique Wo± E HI (JR) such that

lim Ilu(t) - V(t)WO±IIHl = 0,t-->±oo

where V(t)wo is the solution oj

Iav a ---Dv=Oat ax x

vlt=o = Wo

Remarks on Theorem 18. The previou~ly known results for (17) were: For k = 1 or 2, Uo E HS(JR), s = 0 or 1/2, there exists a weak solution u with u E

Lfoe (JR; ~': 1/2) (local smoothing effect) ([20]). For k = I, the same is true for 1

Uo E H (JR) ([20], [60]). For k = I, (17) is globally well posed in H 3/ 2(JR)


276 CHAPTER 13

([44]). For k > 1, (17) is locally well posed in HS(lR), S > 3/2 ([1], [24]). Note that none of these results requires a restriction on the size of the initial data.

The results of Theorem 18 extend also to the initial value problem

au a2u au au - + i- + P1(u, u) - + P2(U, Ii) - = 0 at ax2 ax ax(19)

{ u!t"'O = Uo

where : C2 --+ C, j 1,2 are polynomials, with Pj(ZI, Z2) = Pj

'" afJL..a+fJ~2 ·ajafJzl Z2'

Next, in order to illustrate our method, we will sketch a proof of Theorem 4(v),

case k = 8. Thus, we will prove

Theorem 20. There exists 8 > 0 such that for any Uo E H 1/4(R) with

IID /4 uoIlU(lR) < 8, there exists a unique strong solution of '

3au a u 8 au-+-+u-=O at ax3 ax

{ ult",o = Uo

satisfying

1/4 au u E C(R; H1/4(R» n L OO(R; H1

/4(R»; liD -11L''''L2 < +00;ax x t

lIullL I2 LOO < +00; sup II Diyu II L4 LOO < +00. , x yE]R x t

Moreover, the map Uo t-+ u(t)from {uo E H1/4 (1R): lIuolilil/4 < 8} into

4B = {f E C«-oo, +00); H 1/ (JR»: IIf11B =

max["fIlLoo(]Roli L/4 (]R)); IID'/4 af IILoo L2 ; IIfIIL12 Loo ; sup II Diy fIIL 4 LOO ]. ax I Y tA: t.J .J

< +oo}

is Lipschitz.

In order to prove Theorem 20, we introduce the quantities

4f31(f)= sup IID 1/ f11u

-00<1<+00


1/4 af f32(f) = liD -IIL oo L 2

ax tx

f33(f) = IIf11L:2L~

f34(f) = sup II Diy fIIL4 Loo , I

Y

and for a > 0, we let Ba = {f E B: II fliB S a}. We have the following two claims:

Claim 21. If Uo E H1/4 (R), and we let fo = W(t)uo (Le., fo =

r~:: eiW+X~)uo(~) d~ is the solution of (7», then

(22) IIfoliB S CIIuoII li1/4.

Claim 23. If for Uo E H1/4(IR), v E Ba we define

av u(t) = 4>uo(v)(t) = W(t)uo - W(t - t')(v8 - )(t') dt'io

t

ax

then there exists 8 > 0, a = a(8) such that, if lI uoII lill' < 8, then 4>(B) C B,

and 4> is a contraction.

Notice that once 23 is established, Theorem 20 follows from the contraction mapping principle and Duhamel's formula.

We start out with a verification of (22). These are inequalities for oscillatory integrals, which are versions of the "local smoothing effect" of Kato ([28]), the maximal function estimates of L. Carleson and others ([7], [38]), and the L 2

restriction theorem of Tomas-Stein ([61]) and Strichartz ([57]). Note first that the inequality

f31 (fo) s lI uoII li l /4

is an immediate consequence of the fact that W (t) is a unitary group on L2 (JR).

Lemma 24. Let W(t)wo be defined as in (7). Then

aw (t)woll s Cllwollu. ax LOOL211 x I

Proof -Ix W(t)wo = Jei(t~3+X~)(i~)wo(~) d~. Now, fix x, and perform the

change of variables 1) = ~3. Plancherel's theorem in t and a new change of variables shows that, for each fixed x,

12

+00 I-a W(t)wo 1 dt = cllwolli2' -00 ax

The lemma follows from this. •


279 278 CHAPTER 13

Note that this is, for the solution of (7), a refinement of the estimate (9) obtained by Kato for solutions of the Korteweg-de Vries equation. Local smoothing effects for linear dispersive equations have attracted a lot of attention recently ([13], [14], [66], [51]). This sharp form of it was first observed in [65].

Corollary 25. fj2(W(t)Uo) :::: clluoll/i'/4.

Lemma 26. For Uo E ifl/4 we have the estimate

IIW(t)uoIlLr I2L

xoo :::: CII uoll/iI/4.

This is a version of the Strichartz type inequalities ([57]) for the Airy equation

(7). Its connection with the restriction theorem for the curve r = (;, ;3) is that, since W(t)uo = JeiW+x~luo(;)d;, the solution operator to (7) is the dual of

the operator restriction of the Fourier transform to r (i.e., the extension operator)

applied to the density uo(nd; viewed as a measure on r. Lemma 26 says that

if J 1;11/2Iuo(;)12d; < +00, the extension operator yields a function in LJ2LC:.

By duality this is equivalent to a restriction estimate. The main tool in the proof

of Lemma 26 is an estimate for oscillatory integrals.

Lemma 27. Let I!(x) = f+oo ei(le+X~).!!:1. Then a -00 1~18 • ,

I Cs 1 (28) for - < (l < 1IIs(x)1 :::: Ixl H 2

I Cs 1 (29) for --<(l<l.IIs(x)1 :::: Itl(l-S)/3 2

The proof of Lemma 27 is an application of the version of Van der Corput's lemma found in E. M. Stein's article ([54]). See [31] and [36] for the details.

Once we have Lemma 27, the proof of Lemma 26 follows the lines of the proof

of the Tomas-Stein ([61]) L 2restriction theorem (see [36]). In fact, 26 is equivalent

to the estimate

(30) IID-I/4W(t)uoIlL:2L~ :::: CIIuo1lL2.

We claim that (30) is in tum equivalent to

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

This is because, by duality, (30) is equivalent to

4(32) U.:OO D-'/ W(t)0(-, t)dtL"Clio II L:"" L) .

In tum, the left-hand side of (32) squared equals

1 4f (f D- / W(t)g(-, t) dt) (f n- I/4W(t')g(-, t') dt')dX

2= ff g(x, t) (f D-1/ W(t - t')g(-, t')dt') dxdt.

Hence, (32) and (31) are equivalent. In order to prove (31), we apply Lemma 27, estimate (29), (l = ~, to obtain

00

111:00

D-'I'W(t - t')o(-, t')dtl" " It _ :'1'/6110<-' t')IIL)dt'.c1:Fractional integration in t now finishes the proof.

Corollary 33. fj3(W(t)Uo) :::: CII uoll/il/4.

Lemma 34. For Uo E if 1/4 we have the estimate

IIW(t)uoIIL4x L

I oo :::: CII uoll/il/4.

Lemma 34 is a maximal function inequality for the solution of (7). It, of course, implies that limq,o W(t)uo = uo a.e., for Uo E ifl/4. This kind of a.e. convergence problem was first studied by L. Carleson ([7]), who showed that, for the

linear Schrodinger equation on JR, Uo E ifl/4 suffices for a.e. convergence. Later

on, in [15] it was observed that the exponent 1/4 is sharp, and in [38] the maximal

inequality in Lemma 34, for the case of the Schrodinger operator is proved. It was E. M. Stein who first noticed that inequalities as those in 34 are simple con

sequences of the Kolmogorov-Seliverstov-Plessner method. Lemma 34 was first proven in [65]. To prove 34, note that, arguing as in the proof of Lemma 26, it suffices to show that

(31) /+ 00

D-I/2W(t - t')g(-, t')dt'll :::: CIIgIlL~2flIL~' (35) 1 D- /

2 /+OO

- :::: CIIgIlL~/3L:'W(t t')g(-, t')dt'll11 -00 L12L~ II -00 L~L~


I

~",~_."":""",,:._-~---,.,,,~~~,",,"""~~;'~~~~~' ..,_._.~-_.':""~ .._ _~t--~:--~ _.7_ --- - - ~',-..-:-...... ,---;-~~ .• ,::'..,=---'_.. ,. ._~-- ;.~~~!:!.._".--,- -.~~'--'- -- -- .--.---- - _.::- ....~: ._._--=--_.. .~~%:~ ,_., ..•..__ _.•.• ._ _ "_..• _0'.'_ .,...... •• " ·•.m ••" ••••• ,.,•• _.__ •.•",.,. ".

280 CHAPTER 13 OSCILLATORY INTEGRALS AND NON·LlNEAR DISPERSIVE EQUATIONS 281 To show (35), we use Lemma 27, estimate (28), with 0 = 1/2, to obtain

00

II D-'{2 1: W (I - I')g(-, "ldl'L "c Ix :'/2 * Ilg(-, I) lit:, I

and (35) follows by fractional integration in x.

Corollary 36. ,84(W(t)Uo) ~ CIIuoIlIi'/4.

This finishes the proof of (22). In order to prove (23) we need a non-linear estimate, which is contained in the following:

Lemma 37. If v E Ea , then

4 8 9 (38) l:oo IID I

/ (V :: )IL dt ~ Ca •

Taking (38) for granted, in order to establish 23, we see that, by Minkowski's inequality

1+00 av

lIu(t)IIB ~ IIfoliB + IIW(t - t')(v8 - ) II Bdt' -00 ax

1+00 av4~ Co + C IID I / (v8 - )lIudt'

-00 ax

by (22). Using (38) we see that

lIuliB ~ Co + Ca9 •

Using the following variant of (38)

(39) +00 II D 1/4 ( VI8 -aVI - v28 -aV2 ) II dt ~ ClIvI - v211Ba8 , -00 ax ax L;1

where Vi E Ea , we see that

IIUI - u211B ~ ClIvI - v211Ba8,

and hence 23 follows. We will now indicate the proof of (38). What is needed here are vector valued

versions of the Leibniz and chain rules for fractional derivatives. Previous results in this direction are due to [56], [29], [9], and [58]. Our results are a consequence of the ideas of Coifman-Meyer ([10]) and Bony ([6]), and the vector valued inequalities of C. Fefferman-Stein ([16]), Benedek-Calderon-Panzone ([2]), and Rubio de Francia, Ruiz, and Torrea ([46]). To deal with the endpoint results, we also need

the HP theory as developed by C. Fefferman and E. Stein ([17]), as well as the "tent spaces" of Coifman-Meyer-Stein ([12]). The results needed here are:

Lemma 40. Let F be of class C I, F(O) = 0. Let a E (0, n, p, q, PI, P2, q2 E

O,oo),ql E O,oo]besuchthat ~ = *+ -};. ~ = t +~. Then

(41) liD" F(f)IILPLq ~ CIIF' (f)IILP' Lql liD" fII LP2 Lq2, x t x I x t

Also, if r > 1, h E L~:'(lR), then

(42) liD" F(f)hll p ~ CIIF'(j)lIooIID"(j)M(hrp)l/rPllp,

where M denotes the Hardy-Littlewood maximal operator.

Lemma 43. Let a E (0, 1), aI, a2 E [0, a] with a = al + a2. Let

P. PI, P2, q, ql, q2 E (1,00) be such that'! = ..!. + ..!., 1 = .1 + .1 P PI P2 q q, q2

Then

(i) IID"(jg) - fD"g - gD" fIIL"L q ~ CIID"'fIILP'Lq, II D"2g II LP2 Lq2.

MoreoverJor al = 0, the valu; ~I = 00 is allow~i x r

(ii) IID"(fg) - fD"g - gD" flip ::: CIIglloolID" flip.

Let us now explain why (38) should hold. By the Leibniz rule,

DI/4(v8 av) '=' v8D I/4 av + D I/4(v 8) av . ax ax ax

For the first term, note that

av avI 4 2 4IIv8 D / -IIL2 ~ IIvll~oo IIv D I / -IIL2

ax x x ax x

and so

1/2+00 av +00 av 1/28 I 4 I 4 2~ loo IIv D / ax ilL; ~ loo IIVII1~ (ff Iv2

D / ax 1 dXdt)( )

~ ,83(v)6 ,84 (v)2 ,82(V).

For the second term, by the chain rule, it "equals" v7D I/ 4 (V) i;. Stein's complex interpolation theorem ([52]) now shows that

IIDI/4(v)IILaL~ ~ C,82(vl,84(V)HI x I

avII ax IIL~L~ ~ C,82(V)~,84(V)I-~,


-_. - ~~_.- "._.~;:. <-- ...: ....""''''-''''''.•''''. ~._- ,_:.._;=:.••; •. --;.:.=-~r.::~~;(~T~"V~".:=-""'"-i~-'-'---'';'~~;;:'

282 CHAPTER /3

where

1 (J (1 - (J) 1 (J (I-B) 1 1-=-+-- - = B(1 + -) + (1 - B)Oa 00 4 :a=2+ 00 ' 4 4

1 '7 (1-'7) 1 TJ (1 - '7) 1-=-+- - = - + 1 = '7(1 + -) + (1 - '7)0./-L 00 4 v 2 00' 4

(It is in this step that Diy f in the definition of fJ4 is needed.) Then,

av av IIv?D1

/4 (v) -IIL2 ::: CIIvll~oollvD1/4(v)-IIL2ax' , ax '

. One then proceeds as before, and then has to estimate

4(ff !vD 1/ (V) :: f dXdt) 1/2

This is accomplished by means of Holder's inequality and the above estimates. Finally, the error terms are handled by using (40) and (43) appropriately. The details are given in [36]. (39) is proven in a similar manner, and this concludes our proof.

University ofChicago

REFERENCES

[1] L. Abdelouha~, J. L. Bona, M. Felland, and J. C. Saut. "Nonlocal models fornonlinear dispersive waves." Physica D 40 (1989), 360-392.

[2] A. Benedek, A. P. Calderon, and R. Panzone. "Convolution operators on Banach space valued functions." Proc. Nat. Acad. Sci. 48 (1962),356-365.

[3] T. B. Benjamin. "Internal waves of permanent form in fluids of great depth." J. Fluid Meeh. 29 (1967),559-592.

[4] J. L. Bona and R. Scott. "Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces." Duke Math. J. 43 (1976), 87-99.

[5] J. L. Bona and R. Smith. "The initial value problem for the Korteweg-de Vries equation." Roy. Soc. London, Ser A 278 (1978), 555--601.

[6] J. M. Bony. "Calcul symbolique et propagation des singularires pour les equations aux derivees partielles non lineaires," Ann. Sci. E. N. S. 114 (1981), 209-246.

[7] L. Carleson. "Some analytical problems related to statistical mechanics, Euclidean harmonic analysis," Lecture Notes in Mathematics, no. 779. Springer Verlag, 1979.

[8] T. Cazenave and F. B. Weissler. "The Cauchy problem for the critical nonlinear SchrOdinger equation in HS," Nonlinear Anal. TMA 14 (1990), 807-836.

[9] F. M. Christ and M. I. Weinstein. "Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation." To appear in J. Funct. Anal.

[10] R. R. Coifman and Y. Meyer. "Au dela des operateurs pseudodifferentielles." Asterisque 57 (1973).


[II] . "Nonlinear harmonic analysis, operator theory and P.D.E," In Beijing Lectures in Harmonic Analysis, edited by E. M. Stein. Princeton University Press, 1986.

[12] R. R. Coifman, Y. Meyer, and E. M. Stein. "Some new function spaces and their applications to harmonic analysis," J. Funet. Anal. 62 (1985), 304-335.

[13] P. Constantin and J. C. Saut. "Local smoothing properties of dispersive equations," J. Arner. Math. Soc. 1 (1988),413-446.

[14] W. Craig, T. Kappeler, and W. A. Strauss. "Gain of regularity for equations of KdV type," Preprint.

[15] B. Dahlberg and C. E. Kenig. "A note on the almost everywhere behavior of solutions to the SchrOdinger equation, Harmonic Analysis," Lecture Notes in Mathematics, no. 908. Springer Verlag, 1982.

[16] C. Fefferman, and E. M. Stein. "Some maximal inequalities." Amer. J. Math. 13 (1971), 107-115.

[17] . "HP spaces of several variables," Acta Math. 129 (1972),137-193.

[18] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. "A method for solving the Korteweg-de Vries equation," Phys. Rev. Letters 19 (1967), 1095-1097.

[19] . "The Korteweg-de Vries equation and generalizations. VI. Method for exact solutions," Comm. Pure Appl. Math. 27 (1974), 97-133.

[20] J. Ginibre, and G. Velo. "Smoothing properties and existence of solutions for the generalized Benjamin-Ono equations." Preprint.

[21] . "Scattering theory in the energy space for a class of nonlinear SchrOdinger equations," J. Math. Pure Appl. 64 (1985), 363-401.

[22] . "Smoothing properties and retarded estimates for some dispersive evolution equations," Preprint.

[23] J. Ginibre and Y. Tsutsumi. "Uniqueness for the generalized Korteweg-de Vries equations," SIAM J. Math. Anal. 20 (1989),1388-1425.

[24] N. Hayashi. "Global existence of small analytic solutions to nonlinear Schrodinger equations," Duke Math. J. 62 (1991), 575-592.

[25] R. J. Iorio. "On the Cauchy problem for the Benjamin-Ono equation." Comm. Part. Diff. Eq. 11 (1986), 1031-1081.

[26] T. Kato. "Quasilinear equations of evolution, with applications to partial differential equations," Lecture Notes in Mathematics, no. 448. Springer Verlag, 1975.

[27] . "On the Korteweg-de Vries equation," Manuscripta Math. 19 (1979), 89-99. [28] ~. "On the Cauchy problem for the (generalized) Korteweg-de Vries equation."

Adv. Math. Suppl. Stud. Appl. Math. 8 (1983), 93-128. [29] T. Kato and G. Ponce. "Commutator estimates and the Euler and Navier-Stokes

equations," Comm. Pure Appl. Math. 41 (1988), 891-907. [30] D. J. Kaup and A. C. Newell. "An exact solution for a derivative non-linear Schrodinger

equation," J. Math. Phys. 19 (1978), 798-801. [31] C. E. Kenig, G. Ponce, and L. Vega. "On the (generalized) Korteweg-de Vries

equation," Duke Math. 1. 59 (1989), 585-610. [32] . "Oscillatory integrals and regularity of dispersive equations," Indiana Univ.

Math. J. 491991,33-69. [33] . "Well-posedness of the initial value problem for the Korteweg-de Vries," J.

Amer. Math. Soc. 4 (1991), 323-347.


~'k < :.._ __ .::: __~_:~":" _ .m_,:::'~:~;._ ~;._ ..:=~~-"",.~",,~ .,.,. ,~~£~~=="="",,,~~~~~~_~~,,;'e:.,,"g:~~~;;'~~

284 CHAPTER 13

[34] . "On the generalized Benjamin-Ono equation." To appear, Trans. Amer. Math. Soc.

[35] . "Small solutions to nonlinear SchrOdinger equations." To appear, Ann. Inst. Poincare, Analyse Non-Lineaire.

[36] . "Well posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle." To appear, Comm. Pure Appl. Math.

[37] . "On the hierarchy of the generalized KdV equations." To appear, Proceedings. Nato Lyon Workshop on Singular Limits ofDispersive Waves.

[38] C. E. Kenig and A. Ruiz. "A strong type (2, 2) estimate for the maximal function associated to the SchrOdinger equation." Trans. Amer. Math. Soc. 230 (1983), 239-246.

[39] S. Kichenassamy and P. 1. Olver. "Existence and non-existence of solitary waves solutions to higher order model evolution equations." Preprint.

[40] D. J. Korteweg and G. de Vries. "On the change offonn oflong waves advancing in a rectangular canal, and on a new type oflong stationary waves." Philos. Mag. S (1895), 422-443.

[41] P. D. Lax. "Integrals of nonlinear equations of evolution and solitary waves." Comm. Pure Appl. Math. 21 (1968),467-490.

[42] P. L. Olver. "Hamiltonian and non-Hamiltonian models for water waves." Lecture Notes in Physics, no. 195. Springer Verlag, 1984.

[43] H. Ono. "Algebraic solitary waves in stratified fluids." J. Phys. Soc. Japan 39 (1975), 1082-1091.

[44] G. Ponce. "On the global well posedness of the Benjamin-Ono equation." Diff. and Int. Eqns. 4 (1991), 527-542.

[45] ."Lax pairs and higher order models for water waves." To appearJ. Diff. Eqns. [46] J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea. "Calder6n-Zygmund theory for

operator-valued kernels." Adv. Math. 62 (1988), 7-48. [47] J. C. Saut. "Sur quelques generalisations de l'equations de Korteweg-de Vries, II." J.

Diff. Eqs. 33 (1974), 320-335. [48] . "Sur quelques generalisations de l'equations de Korteweg-de Vries." J. Math.

Pures Appl. S8 (1979), 21~1.

[49] J.-C. Saut and R. Temam. "Remarks on the Korteweg-de Vries equation." Israel J. Math. 24 (1976), 78-87.

[50] M. Schwarz, Jr. "The initial value problem for the sequence of generalized Kortewegde Vries equation." Adv. Math. S4 (1984),22-56.

[51] P. Sj61in. "Regularity of solutions to the SchOdinger equations." Duke Math. J. 5S (1987),699-715.

[52] E. M. Stein. "Interpolation of linear operators." Trans. Amer. Math. Soc. 83 (1956), 482-492.

[53] . Singular integrals and differentiability properties of functions. Princeton University Press, 1970.

[54] . "Oscillatory integrals in Fourier analysis." In Beijing Lectures in Harmonic Analysis, edited by E. M. Stein. Princeton University Press, 1986.

[55] E. M. Stein and G. Weiss. Introduction to Fourier Analysis in Euclidean Spaces. Princeton University Press, 1971.

[56] R. S. Strichartz. "Multipliers in fractional Sobolev spaces." J. Math. Mech. 16 (1967), 1031-1060.

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS 28S

[57] . "Restriction ofFourier transforms to quadratic surface and decay ofsolutions of wave equations." Duke Math. J. 44 (1977), 705-714.

[58] M. E. Taylor. "Pseudo-differential operators and nonlinear P.D.E." Preprint. [59] R. Temam. "Sur un probleme non lineaire." J. Math. Pures Appl. 48 (1969),159-172. [60] M. M. Tom. "Smoothing properties of some weak solutions of the Benjamin-Ono

equation." Diff. and Int. Eqns. 3 (1990), 683~94.

[61] P. Tomas. "A restriction theorem for the Fourier transform." Bull. Amer. Math. Soc. 81 (1975),477-478.

[62] Y. Tsutsumi. ..L 2-s01utions for nonlinear SchrOdinger equations and nonlinear group." Funkcialaj Ekvacioj 30 (1987), 115-125.

[63] M. Tsutsumi, and I. Fukuda. "On solutiQns of the derivative nonlinear SchrOdinger equation. Existence and Uniqueness Theorem." Funkcialaj Ekvacioj 23 (1980), 259277.

[64] . "On solutions of the derivative nonlinear SchrOdinger equation, II." Funkcialaj Ekvacioj 24 (1981),85-94.

[65] L. Vega. Doctoral Thesis, Universidad Autonoma de Madrid, 1987. [66] . "The SchrOdinger equation: pointwise convergence to the initial data." Proc.

Amer. Math. Soc. 102 (1988),874-878. [67] E. Witten. "Two dimensional gravity and intersection theory on moduli space."

Preprint.


287

14

Singular Integrals and

Fourier Integral Operators

D. H. Phong*

I INTRODUCTION

The theory of singular integrals and pseudo-differential and Fourier integral operators [17], [39], [43], [45] constitutes one of the most impressive achievements in

classical analysis of the last few decades, both in its wide ramifications and in the

coherence and unity of its structure. The operators falling outside of the scope of

this classical theory have usually surfaced in widely disparate areas of analysis and geometry, often to the extent that their mutual existence was hardly known. They required methods devised individually, and with more limited success in many

cases. However, they have been appearing increasingly frequently in recent years, and remarkably, many common threads have begun to emerge.

In this chapter we shall focus on two themes which seem to unify many of the

very disparate operators not covered by the classical theory of Fourier integral

operators. In the first theme the Lagrangian manifold may fail to be locally the

graph of a canonical transformation. This was the case for diffraction theory [19], [20], [41], [42], where the Lagrangian is a Whitney fold. More systematically, it is

generically the case whenever we deal with Radon transforms along submanifolds of codimension higher than 1. Thus Radon transforms and maximal functions along families of curves [34], [25], [40] and X-ray transforms [10], [12] are prime

examples, as are analysis and deformation theory on manifolds of geodesics [10],

[14], [44]. In the second theme, the densities of the Fourier integral operators

'This research was supported, in part, by the National Science Foundation, under grant DMS-9004062.

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

are allowed singularities of an either fractional or Calder6n-Zygmund type. This again seems dictated by rather general situations. A phenomenon probably typical of subelliptic problems is the following. The main term of the Green's function for the a-Neumann on the Siegel upper half-space [29], [32], [3] is smooth outside of the diagonal of the Heisenberg group. Yet it satisfies no better estimates than a ker

nel supported on the distribution of maximal complex hyperplanes with additional Calder6n-Zygmund singularities, and must be treated as a singular Radon trans

form [31], [32]. Another case of a singular density is the fundamental solution of

a hyperbolic partial differential equation [21], [15]. Its wave front set is contained

in the union of the cotangent space at the origin with the flow out of its intersection

with the characteristic variety of the equation. This occurrence of two intersecting Lagrangians is shared by singular Radon transforms, with the Lagrangians given

in this case by the diagonal and the normal bundle of the distribution of hyper

planes. Recognition of this common feature has already borne fruit, as it has led

to a microlocal proof of the boundedness of singular Radon transforms [11]. It is

very intriguing that the two themes we have just described seem to be themselves intimately related. In fact, composition of the Fourier integral operators singular

in the first sense leads under different guises to Fourier integral operators singular

in the second sense. Examples of such occurrences are given in Sections 11.3 and IV below.

It can be hoped that a theory with composition calculus and Sobolev and LP - Lq bounds can be constructed which encompasses these degenerate Fourier integral operators and singular Radon transforms. Here we shall discuss encouraging

progress in a number of directions [34], [35]: first, in a class of operators modeling

Fourier integral operators with arbitrarily high order of degeneracy, and whose

bounds can be formulated in terms of the stratification of the Lagrangian; second, in the LP - Lq bounds for Radon transforms with folding canonical relations and

their relation with singular Radon transforms; third, in establishing bounds even

in some cases when a normal form for the Lagrangian may not be available; and finally, in the understanding of the role of the Monge-Ampere determinant in the

localization procedures required for the Sobolev estimates.

This paper is organized as follows. In Section II we provide an introduction to several problems in Fourier analysis and integral geometry leading to the operators

considered here. In Section III we describe bounds for the model Fourier integral operators. The key ingredients in our approach are a refined method of stationary

phase with uniform bounds and sharp estimates for the distances between the roots of an algebraic equation. In Section IV we present some aspects of Radon

transforms on curves in two dimensions, with special emphasis on the emergence

of the Monge-Ampere determinant in the hard analysis, and of singular densities in the composition.


288 CHAPTER 14

This paper describes joint work with E. M. Stein, some of which stretches back to the days when I was one of his graduate students. It has been an exceptional privilege to carry on a research program with him continuously since that time. On this occasion of his sixtieth birthday, I would like to express my immense gratitude to him, and also my pride and happiness at participating in an enterprise which, if I were to judge from the boundless energy and enthusiasm he brought to it, must have been especially close to his heart.

U BASIC EXAMPLES

In this section we provide a briefdiscussion of several areas of analysis and integral geometry leading generically to singular Fourier integral operators. The singularities emerge under many different guises, either in the symbols or in the projections of the Lagrangians, and it is remarkable that they all seem to be intimately related.

1 Radon Transforms along Curves

We consider first the distribution of curves (Mp = P + y(t), P ERn} in Rn given by translates of a fixed curve y(t ), and the corresponding integral transform

(1) (Tf)(P) == r+oo

f(P + y(t»X(t)dt == r fJ-oo JMp

where X is a fixed Co cut-off function, == I and supported in a small enough neighborhood of O. The question is to determine geometric conditions on the curve y(t) which insure the optimal regularity properties for T. As a convolution operator the multiplier of T is given by

r+oo

(2) m(A) = J-oo ei{A,y(I») X(t)dt.

The most natural geometric condition, namely that y(t) have non-zero torsion at t = 0, i.e., that the n vectors y(O), ... , y(n)(o) be linearly independent, is also the one guaranteeing uniform bounds in Afor the phase function in (2)

(3) I(A, y}(O) I + ... + !(A, y(n)(O)}1 ~ ciAI.

The van der Corput lemma implies Im(A)1 S CIAI- I/n, and we obtain the sharp estimate

Theorem 1. If y(t) has non-vanishing torsion at 0, then the operator T of(1) is boundedfrom H(s)(Rn) to H(s+l/n)(Rn).

~~<'t!~ijtlt&""i!lit\\i&t<it"""',}J!~~:, ..ii'iMiil4tiiii;Jtiiiij "l~~ ---:&f:i:it:>£i6:,~

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS 289

This simple example already falls outside the scope of the classical theory of Fourier integral operators. Indeed it exploits an n-th order derivative condition (torsion) rather than the second order conditions of Fourier integral operators. More precisely the kernel of (1) is the Dirac 8 measure along the defining relation

(4) C = {(P, Q) E Rn x Rn; Q E M p }.

Its wavefront is contained in the Lagrangian manifold N*(C) C T*(Rn) x T*(Rn). The calculus of Fourier integral operators applies and gives L 2 estimates when the projections JrL and JrR from N*(C) on the left and right factors

/N'(C)~ (5)

T*(Rn) T*(Rn

)

have invertible differentials. By convention the P variables are on the left and the Q variables a~e on the right. Now an element (P,~; Q, 1/) is in N*(C) when

Q = P + y(t), ~ + 1/ = 0, and

(6) (1/, y) = O.

An element (8P, 8~; 8Q, 81/) is then in T(P.~;Q''1)(N*(C» when

(7) 8Q = 8P + y8u, (81/, y) + (1/, ji)8u = 0, 8~ + 81/ = 0

for some 8u in R. In particular the kernel of dJrL at (P,~; Q, 1/) is non-trivial exactly at points where

(8) (1/, ji) = O.

For n > 2 there are at least n - 2 directions 1/ satisfying both (6) and (8), and the Lagrangian N*(C) fails there to be a local graph.

What are the optimal estimates when the condition that N*(C) be a canonical graph breaks down? A classic theorem of Hormander [16], [17] says that when the kernels of dJrL,dJrR have dimension at most k, then the corresponding Fourier integral operator satisfies bounds which lose -k/2 derivatives compared to when the Lagrangian is a local graph. In the formalism of Fourier integral operators, the Radon transform along curves (1) is of class I- 1/ 2(M, M, N*(C». The fact that it is of order -1/2 can be seen by choosing n - 1 defining functions <1>1 (P, Q) = ... = <1>n-l (P, Q) = 0 for (4) and representing the Dirac measure along (4) as a Fourier integral distribution with oscillating variables AI, ... , An-I. phase


291 290 CHAPTER 14


L~:: Aj<l>j(P, Q), and symbol I (12) (7], y) = (7], ji) = ... = (7]. y(n») = O}

f n-I n-I

8c<P. Q) = exp(i I: Aj<l>j(P, Q» ndAj. j=1 j=l

The order of the operator is then

I order(symbol) + (2" #oscillating variables)

I . b(9) - - - (#P vana les + #Q variables) = -1/24

and would have been the amount of smoothing, had the projections JrL and JrR been!lli' I locally invertible. In this case the kernels of dJrL and dJrR are one-dimensional,

and the sharp estimate in the absence of any additional information as given by 1~lnl

Hormander's theorem is then boundedness on H(s). Theorem I shows that this can be improved by natural and generic assumptions on the geometry of JrLand JrR.

The next issue we address is whether the torsion condition, natural as it is for the family of translates of a curve, can be extended to general families of curves. This is the case, as the torsion condition for distributions of translates of a curve arises naturally from the stratification structure of the singular varieties of JrL and JrR. Let

:~~ (10) I: = {(P,~. Q. 7]) E N*(C) where JrL. JrR are not locally 1 - I}

be the singular variety of dJrL and dJrR. The key geometric information is the position of the kernels of dJrL, dJrR relative to the tangent space to I:, viewed as subspaces of T(P.~;Q.'1)(N*(C». We discuss first, say, the left projection JrL, and make the generic assumption that the kernel of dJrL is one-dimensional and the determinant of dJrL vanishes only of first order along I:. The variety I: is then smooth, and

Kerd(JrLIl;) = KerdJrL n TI:.

In particular, the transversality of Ker dJrL and T(p,~; Q,'1)(N*(C» is equivalent to the local injectivity of JrL when restricted to I:. This leads us to define successively

(11) I:2 = I:,I I!I

I:1 = {(P,~; Q. 7]) E I:~-I; KerdJrL(P,~; Q. 7]) C T(P,~;Q,'1)I:~-I}.

In the language of singularity theory, I:L = 0 means for example that the projection JrL is a (Whitney) fold. For the distribution of translates of y(t) it is readily seen that

2I:in - ) = {(P,~; Q, 7]); Q = P + yet),

I

I', I . i~1

and the condition that yhave torsion is equivalent to I:£-2 be empty. Finally we come to the issue of possible asymmetry between the left and right

projections. To define I: there is no need to distinguish between them, because one of them is singular if and only if both are. This is a consequence of the fact that N*(C) is Lagrangian and can be seen as follows. First observe that

dJrR(Ker dJrd and I mage(dJrR) are orthogonal complements with respect to the symplectic form on T*(R). In fact 88, 8n are in these spaces respectively when (0,88) and (8'11, 8n) are in T(N*(C» for some 8'11. Since N*(C) is Lagrangian we have

0= w;"(C)(O, 88); (8'11, 8n» = Wp(Rn)(8?, 8n).

This implies that dJrR(Ker dJrd and Ker dJrR have the same dimension, and thus that Ker dJrL and Ker dJrR have the same dimension since dJrR restricted to Ker dJrL is manifestly one-to-one. For translation invariant distributions of curves, there is symmetry between left and right projections to all orders of stratification. In general, however, we also need to consider the stratification varieties I:~ of JrR,

as well as the mixed singular varieties defined as follows. Let W = (WI, ... , wd denote a sequence of k indices Wi which can be either an L or a R index. Then

(13) I:k = I:k-IW1W2"'Wk - W2'''Wk

n {(P,~; Q, 7]); KerdJrw, (P,~; Q. 7]) C T(P,~;Q,'1)I:~~'Wk}'

This leads us to the extension of the notion of torsion for a single curve that we were looking for. In [34] a (not necessarily translation invariant) family of curves was said to have left (respectively, right) torsion if the variety I:£-2 (respectively, I:~-2) is empty. More generally it will be said to have non-vanishing "w-torsion" at (P, Q) if for some k and w of length k

(14) I:~ n N(p,Qj(C) = 0.

The simplest family of curves in Rn incorporating an asymmetric behavior is given by the parametrization

M(x,f) = ley, s); s= t + sex. y)}

(15) C = {(x';; y, s); (y, s) E M(x,i)}

Iwith x. yin R, and S, t, Sex, y) vectors in Rn- . The Lagrangian becomes (16)

.. -+, - -+, -+ n-l}N (C) = {«x. (A, Sx(x, y)}), (t, A); (y, (A, Sy(x. y)}), (s, -A»; A E R .


293 292 CHAPTER 14

In particular it can be parametrized by (x, y, t, X), and its tangent space by (8x, 8y, 8t, 8A). Linearizing the defining equations for N*(C), we see that the singular variety 1:; and the kernels of drrL, drrR at 1:; are given by

1:; = N*(C) n {(A, S;y(x, y)) = O}

KerdrrL = (8x = 8t = 8A = 0, 8y E R)}

(17) KerdrrR = (8x E R, 8t = 8A = 8y = O)}.

To each W we associate now a set I w of indices (a, fJ) E N2 by the following inductive process reflecting that of (13)

IwtWJ.···W! = IWJ. •••wk U {(a + I, fJ); (a, fJ) E IWJ. wk}, if WI = R

(18) IwlWJ. •••wk = IWJ.· ..wk U {(a, fJ + I); (a, fJ) E IWJ. wk}, if WI = L.

Then we have (19)

1:;~tWJ."'Wk = {(x, y, t, A); (A, a:+la:+IS(x, y») = 0, (a, fJ) E IwtWJ. ...wk}.

We can finally formulate the first main goal of a theory of Fourier integral operators with degenerate Lagrangian manifolds: to establish optimal estimates, or, more precisely, identify the nature of TT* and T*T, for each possible stratification structure of the projections rrL and rrR, as described by the subvarieties 1:;~. In Section III we shall provide a systematic study of a class of models which shows how the L 2 estimates and smoothing properties are indeed dictated by the stratification of the Lagrangian.

2 Green's Functions for Subelliptic Problems and Singular Radon Transforms

It is perhaps a surprising fact that Radon transforms arise in Green's functions for subelliptic problems. Indeed they are normally associated rather with propagation of singularities phenomena inherent to hyperbolic problems. More precisely the singular support of the Green's function of a subelliptic problem is contained in the diagonal, whereas the singular support of Radon transforms is usually given by a manifold of higher dimension.

Let Hn = en x R be the Heisenberg group, with the group operation (z, t)(z', t') = (z + z', t + t' + 2Imz . Z'). The vector fields

a a - a a a (20) z· = - +iz·- z· = - -iz·- T = -i-

J a J a 'J a- J a ' aZj t Zj t t

SlNGULAR lNTEGRALS AND FOUR1ER 1NTEGRAL OPERATORS

are left invariant and form a basis for the Lie algebra of Hn. The Laplacian is given by

n1 2 IL:(21) Ii. = - T - - (Z·Z· + Z·Z·) - (n - 2)T2 2. JJ

- -JJ '

J=I

and our task is to study the Green's function for the following operator

(22) 0+ = li. 1/ 2 + -

I T.

.j2

This problem is closely related to the a-Neumann problem on the domain U =

H X R+. Let (z, t, p) denote points in U, and set

I a2

(23) D=---+Ii..2 ap2

Then the a-Neumann problem is the following boundary value problem

a a (24) Ou = f, (-+i-)ul=o=Oat ap p

where fez, t, p) is a given Co function on iI. More precisely U can be identi

fied with the Siegel upper half space {(z, Zn+I); ImZn+1 > Iz1 2} by (z, t, p) ~

(z, Zn+1 = t + i(p + IzI2». The Levi metric is the metric which makes j - d j . - I d n+1 - 21/ 2 "n -·d· '2-1/2d ' W - z, J - ... , n an W - - L..,j=1 ZJ ZJ - I Zn+l mto

an orthonormal basis of (I,O)-forms. The corresponding Laplacian on (O,I)-forms

Lj~: UjWj is diagonal. The boundary condition on Un+1 is the Dirichlet condition Un+II p=o = 0, while the Laplacian and boundary conditions on the remaining components Uj, j = I, ... , n are given by (24).

The L 2[7], U and Schauder estimates [8], [13] for both problems (22) and (24) are by now well-known. Here we would like to discuss instead explicit formulas for their Green's functions (more accurately, their parametrices) and bounds for the corresponding classes of integral operators. Translation invariance shows that the Green's function for 0:;:1 is a convolution operator T on the Heisenberg group

(25) (T¢)(z, t) = II Gb«Z, t)(w, s)-I)¢(w, s)dwds.

On the other hand, the parametrix for the boundary value problem (24) must be of the form

(Nf)(z, t, p) = 100 II Go«z, t)(w, S)-I; Ip - J-Ll)f(w, s; J-L)dwdsdp

(26) + 100 II G«z, t)(w, S)-l; p + J-L)f(w, s; J-L)dwdsdp.


SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS 295294 CHAPTER 14

The first term on the right hand side of (26) is pseudolocal and is the same as for, say, the Dirichlet problem. The second term is the key one insuring the a-Neumann condition. Such terms are called Hilbert integral operators. There is a remarkable relation [30J linking directly the Green's function of 0+ to that of (24)

(27) 0+141 = -(N(¢ ® 8(JL))Ip=o.

It is now well known [29], [32J, [3J that G(z, t; p) is given by an asymptotic expansion in kernels of the form Ek(Z, t; p)H/(z, t; p), where Ek(Z, t; p) and H/(z, t; p) are homogeneous of order -k and -1 with respect to the two different

notions of dilations

(28) Ek()..z, At; )..p) = rkEk(Z, t, p), H/()..z, )..2t ; )..2 p ) = ).. -/ H/(z, t; p)

respectively. Similar notions of homogeneity are introduced for kernels which are functions of (z, t) alone. The sharp estimates for the a-Neumann problem and for 0:;' are given by bounds on ZjZkN(f), Z/ZkN(f), Z/ZkN(f), and Z j O:;I¢, ZjO:;'¢. The theory of singular integrals with homogeneous kernels on the Heisenberg group has been developed in [8J. The above expressions lead, however, to convolutions on the Heisenberg group and Hilbert integral operators with products of kernels of two different homogeneities. The following theorems provide bounds for such operators [30J, [31J, [32J:

Theorem 2. Let Gb(z, t) = Ek(Z, t)H/(z, t) be a kernel ofmixed homogeneity,

and let T be the corresponding convolution operator (25). Then

• If k + 1 < 2n + 2, k < 2n or k + (1/2) < 2n + 1,1 < 2, Tis boundedfrom

Lfoc(un) into itselffor 1 ::: p ::: 00;

• If k + 1 = 2n + 2, k < 2n, and the kernel Ek(Z, O)H/(z, t) has mean value 0

on the parabolic (z, t)-sphere Izl4 + t 2 = 1, then T is bounded on LP(Un)for

1 < P < 00;

• If k + (1/2) = 2n + 1,1 < 2, and the kernel Ek(Z, t)H1(z, 0) has mean value 0

on theeuclidian (z, t)-sphere Id +t2 = 1, then T is again bounded on U(Un)

for 1 < p < 00.

Theorem 3. Let G(z, t; p) = Ek(Z, t; p)HI(z, t; p) be a kernel ofmixed homo

geneity, and let H be the corresponding Hilbert integral operator defined by the

second terms on the right hand side of(26). Then

• If k + 1 < 2n + 4, k < 2n or k + (1/2) < 2n + 2, 1 < 4, H is bounded from

L{oc(un x R+) into itselffor 1 ::: p ::: 00;

• If k + 1 = 2n + 4, k < 2n, or k + (1/2) = 2n + 2, 1 < 4, then H is bounded

on LP(Un x R+)for 1 < P < 00;

• If k + 1 = 2n + 4, k = 2n, and the kernel Ek(Z, t; p)H/(z, t; p) has mean

value 0 on the z-sphere Izi = 1 for any (t, p), i.e.,

(29) Ek(Z, t; p)H/(z, t; p)dz = 01 £<lzl<8

for any E, 8,0 < E < 8, then H is bounded on LP(Un x R+)for 1 < P < 00.

The conditions k + 1 < 2n + 2 and k + (1/2) < 2n + 1 of Theorem 2 define a region in the (k, 1) plane, inside of which the kernel is locally integrable. On the edges k + 1 = 2n + 2, k < 2n, away from the intersection point k = 2n, 1 = 2, the kernel behaves as the parabolically homogeneous kernel Ek(Z, O)H/(z, t)

which has singularities along the plane z = O. These singularities are, however, integrable in view of the condition k < 2n, and the condition that the mean value on the (z, t)-sphere of Ek(z, O)H{(z, t) is 0 leads to LP bounds just as in the case of

smooth kernels homogeneous of degree 2n + 2. The case of k + (1/2) = 2n + 1, 1 < 2 is similar, with the kernel Ek(Z, t)H{(O, t) homogeneous of the (euclidian) critical degree 2n + 1 and having integrable singularities on the (euclidian) (z, 1)

sphere. The estimates for ZjO+'¢, ZjO:;! fall in these cases. The case k = 2n,

1 = 2 is not needed here, but may be of independent interest. It is likely that it can be treated by the method of singular Radon transforms used for Theorem 3, to be described next, but this has not been established.

We tum now to Hilbert integral operators. Inside the region k + 1 < 2n + 4, k + (1/2) < 2n + 2 the kernel Edz, t; p)H/(z, t; p) is locally integrable, and the first statement of Theorem 3 follows at once. On each of the edges k + 1 = 2n + 4, k < 2n and k + (l/2) = 2n + 2, 1 < 4 the kernel behaves again as a homogeneous kernel with integrable singularities on the sphere, the homogeneities being parabolic and euclidian, respectively, in their critical degrees 2n + 4 and 2n + 2. Such kernels lead to operators bounded on LP(Un x R+), by arguments similar to the one-dimensional case of K (x, y) = (x + y) -Ion functions on U(R+). Unlike in the case of 0:;', sharp bounds for Nf lead to Hilbert integral operators at the doubly critical degree of homogeneity k = 2n, 1 = 4, requiring singular Radon transforms on the Heisenberg group.

The key geometric ingredient of singular Radon transforms on un is a distribution of hypersurfaces M p passing through each point P E un. At P = (0, 0). Mp is the maximal complex space M p = {Q = (w, 0); WEen}. For general P = (z, t) it is the image of M(o,o) under the group multiplication by (z, t)

(30) M(z.t) = {Q = (z + w, t + 2/mz . w)}.

Let Lo(w)dw be a density on the hyperplane M(o,o), which we can transport to a density dJLp(Q) on each M(z.t) again by group multiplication. Typically Lo(w)


297 296 CHAPTER 14

is a homogeneous kernel on en, and, in particular, singular at w = O. The singular Radon transform associated to the distribution of hyperplanes M p with their densities L p is given by averaging over each M p with respect to the density dlLp(Q)

(31) R¢(P) = { ¢(Q)dlLp(Q) = { ¢(z + w, t + 2lmz . iiJ)Lo(w)dw.lMp len When Lo(w) is a Calderon-Zygmund kernel on en the operator R extends as a bounded operator on LP

(32) IIR¢IIU(H") :5 CII¢IIU(Hn), 1 < P < 00.

The key observation is that Hilbert integral operators are effectively averages of

singular Radon transforms. For fixed s, p, IL we can construct the singular Radon

transform Rs,p,1J. with density Lo(w) = K(w, s, P + IL) and the function ¢S,IJ. on the Heisenberg group defined by ¢S,IJ.(z, t) = f(z, t - s, IL). Then

100 1+00(33) Hf(z, t, p) = (Rs,p,IJ.¢S)(Z' t)dsdlL.

o -00

Now the densities K (w, s, P+IL) are smooth in w, but their bounds as CalderonZygmund kernels are only O(s2 + p2 + 1L2)-1. Thus (32) implies

IIRs,p,1J.1I :5 C(s2 + p2 + 1L2)-1.

It follows that

100 1+00 I II(Hf)(·", p)llU(Hn) :5 C ( 2 + 2 + ? II¢sllu(H")ds)dlL

o -00 s P IL

100 I

(34) < C --lIf(-' " 1L)IIU(Hn)dlL· o p + IL

The LP boundedness of the one-dimensional Hilbert integral operator gives the

desired result, establishing Theorem 3.

In [9] the estimate (32) was proved using the group Fourier transform on the Heisenberg group. The natural setup for singular Radon transforms is, however,

in integral geometry, and the following general theorem holds [31], [32]:

Theorem 4. Let M p be a smooth distribution of hypersurjaces passing through

P in a Coo compact manifold M, and let dlLp(Q) be a density on each M p with

a Calder6n-Zygmund singularity at Q = P. Define the corresponding singular

Radon transform and maximal operator by

(R¢)(P) = { ¢(Q)dlLp(Q)lMp


(35) (M¢)(P) = sup~ , ~/ ~ ( ¢(Q)dlLp(Q)0"

lB(P,~)

where we have chosen a metric on M and denoted by B(P, 8) the geodesic ball

centered at P and of radius 8. Assume that the distribution ofhypersurjaces M p

has non-vanishing rotational curvature along the diagonal, i.e., if <f>(P, Q) = 0, 'VP,Q<f> #- 0 is a defining equation for the relation manifold C = {(P, Q); Q E

M p }. then

<f>(P, Q) 'Vp<f>(P, Q) ) (36) det(]) == #- O.

( 'VQ<f>(P, Q) 'Vp'VQ<f>(P, Q)

Then the operators R. M are bounded on LP(M)for 1 < p < 00.

It should be pointed out that (36) is exactly the Monge-Ampere determinant

] (<f» introduced by Fefferman [6] in his study of the Bergman kernel and parabolic

invariants. It will also playa major role in the study of Lagrangians with folds, as

discussed in Section IV below. It is instructive to study models for the singular Radon transforms (35) in their

own right. If we consider the family of hypersurfaces in an+1 given by M(x,t) = {(x + y, s = t + S(x, y)} with x, yEan, tEa, and the density dlL(x,t)(y, s)

which is the image under y --+ (x + y, t + S(x, y» of a fixed homogeneous density K (y)dy on an, then the singular Radon transform of (35) can be viewed

as a vector-valued pseudo-differential operator with symbol

(37) (a()..)1jr)(x) = f ei'-S(x,y) K(x - y)1jr(y)dy.

Since rotational curvature is a second derivative condition, we assume that the

phase S(x, y) is a quadratic form. We also setA = 1. The analogue of Theorem 4 is [32]:

Theorem 5.· Let the rank of the quadratic form S(x, y) be k, and let -IL be the

degree of homogeneity of the kernel K (x). When IL = n we also assume that

K(x) has mean value 0 on the unit sphere. Assume also that k > O. otherwise

(37) reduces to a familiar fractional integral. Then the operator a is bounded on

LP(an)for 11/2 -11pi < (IL -n +k)/(2k), 1 < p < 00, andfor 11/2 -11pi :5 ILI(2n) when k = n.

More recently progress has been made in several different directions. First it has

been shown in [27] that the singular Radon transforms associated to the distribution

ofhyperplanes (30) in the Heisenberg group and a density Lo(w )dw homogeneous of degree 0 are smoothing of order -n on Sobolev spaces. This is a generalization


299

~. m_" _"C"'-" .,_~ ._••_"•• - • - .",n" __"._-_ • i:C:-"-_C·"'=-=-=--"'""--""'-"'"="='=""""'·_·.~.'-C--"-.""~ -...,:""_":-=--="'-=:~-c

298 CHAPTER 14

of the similar estimate for smooth densities which would follow from the classical theory of Fourier integral operators. Although we may expect a similar result to hold in the more general setup of Theorem 4, this has not been established to date. Second, Theorem 5 suggests that the condition that the rotational curvature form be non-degenerate can be weakened considerably if we restrict our attention to LP bounds for singular Radon transforms with Calder6n-Zygmund singularities or maximal functions. In the context of nilpotent groups, such improvements can be found in [4], [23], [36]. This has now been done in full generality by Christ, Nagel, Stein, and Wa~nger [5], who showed that the LP bounds hold as long as the rotational curvature form does not vanish to infinite order. At the other extreme, when the densities are smooth, singular Radon transforms become Fourier integral operators, and non-vanishing rotational curvature is essential to insure the optimal bounds since it reduces to the condition that the Lagrangian be a local graph. It may be interesting to determine exactly how much rotational curvature is needed for a fractional singular Radon transform.

3 Integral Geometry

In a remarkable parallel development, operators sharing many of the features of the previous examples arise in integral geometry, particularly in the problems of inverting the X-ray transform and constructing Lorentz metrics, all of whose light rays are closed. More precisely, they are certain Fourier integral operators X;: whose Lagrangian exhibits an asymmetric type of singularity as discussed in Section 1.1, and whose composition X}X;: is a Fourier integral operator with a singular density, analogous to the singular Radon transforms of Section 11.2. This important phenomenon was discovered independently by Guillemin [14] and Greenleaf and Uhlmann [10], [12], and we discuss it briefly here.

The main example is the X-ray transform on the family of light rays in R3• Let a light ray 1 in R3 be any line making an angle of Jr/4 radians with the X3 axis, and let F be the 3-dimensional family of all light rays. The corresponding X-ray transform is the operator sending a function I E CO(R3) to the function X;:I on F, defined by averaging I on each 1 E F. If we parametrize a light ray 1 by its intersection with the (Xl, X2) plane, and the angle B that its projection on the (XI, X2) plane makes with the x-axis, then

(38) (X;:f)(xJ, x2,B) = i: I(x] +sCOSB,X2 +ssinB,s)X(s)ds

where we have restricted the support of I to a fixed compact subset K of R3, and chosen a cut-off function X (s) which is even, and == I in a neighborhood of K.


As usual we define C = {(l, Y); Y E I}, and the Lagrangian of X;: is

N*(C) = (((XI,X2,B), S);

((YI = Xl + s cos B, Y2 = X2 + s sin B, Y3 = s), \11)}

S = (-J.LI, -J.L2, Y3(J.Lt sin B - J.L2 cos B))

(39) \11 = (J.L I, J.L 2, - (J.L I cos B + J.L2 sin B)).

It follows that N* (C) can be parametrized by (XI, X2, B, Y3, J.LI, J.L2) and its tangent space by infinitesimal variations (hI, 8x2, 8B, 8Y3, 8J.LI, 8J.L2). In this instance JrR

is the projection on the Y, \11 variable, so that its kernel requires 8J.L] = 8J.L2 = 8Y3 = 0,8xI = Y3sinB8B,8x2 = -Y3cosB8B, and (-J.LIsinB + J.L2cosB)88 = O. Thus Ker dJrR is non-trivial exactly on the variety

(40) b = N*(C) U {-J.LIsinB + J.L2cosB = O}

and is generated by a non-vanishing 8B. In particular it is one-dimensional and transversal to b, and the Lagrangian is a fold with respect to rrR. On the other hand, it is easy to see that dJrL is singular along b, but its kernel is one-dimensional and generated rather by a non-vanishing 8Y3. This implies that Ker dJrL is tangent to b, which is an asymmetric behavior. In view of later generalizations, it is helpful to note that the description of \11 in (39) is equivalent to its being perpendicular to the line 1. Furthermore, the condition (40) defining b just means that \11 is also orthogonal to the vector e2 = (-s i nB, cosB, 0) which, together with 1, generates the tangent plane to the light cone. Thus b is actually characterized by requiring that \11 be perpendicular to the light cone. The statement about X}X;: can be verified by an explicit formula which we give below.

The family of light rays is only a special case of a rather general setup, as shown by Greenleaf and Uhlmann. Let £, be the family of all lines in Rn. We shall see shortly that it is a smooth manifold of dimension 2n - 2. Given a function I E Co(Rn), its X-ray transform is the function on £, defined by

(Xf)(l) 1 E £'.= ii, Recapturing I from XI is an overdetermined problem. Gelfand raised the problem of finding sets of uniqueness for X, i.e., find n-dimensional families F of lines 1 for which

(41) X;:I = XII;:

suffices to characterize I. This problem was solved in the case of complex lines in Cn , the sets of uniqueness being the ones satisfying the following cone condition.

For each X let r x be the cone generated by the lines passing through x. Then r x


301 300 CHAPTER 14

is tangent to r y along the line joining x and y for any y in r x' Greenleaf and Uhlmann addressed the analogous problem for real lines, namely to reconstruct explicitly as much of f as possible from XF f when F satisfies the cone condition. Although not generic, this condition is satisfied by many families of lines, such as the family of light rays discussed above, and the family of all lines passing through a given curve. It turns out to imply that XF is a Fourier integral operator with a fold on one side and a blow-down on the other. Furthermore X;'XF is again essentially a singular Radon transform. We provide now some details in the simplified setting of families of lines in R3•

Asymmetric Behavior of the Lagrangian of XF The first task is to obtain an explicit parametrization of the Lagrangian of XF. A

line 1can be described by a reference point P and a unit vector y. Locally we may view then the space of all lines C as a submanifold in T(R3 ). As a consequence T*(C) can be identified with the restriction to T(C) of T*(T(R3 )), viewed as linear functionals on T(T(R3

)). Similarly T*(F) is the restriction to T(F) of T*(T(R3», and in particular

~*(F) = (T;(R3) x T;(R 3»IT/(F)'

More explicitly the defining relation for XF is

C = {(I, Q); Q E l}

(42) = {«P. Y). Q); Eijk(Q - p)jyk = O}

where Eijk is the antisymmetric symbol. In the above formalism the Lagrangian N*(C) becomes (43)

~ . k . k . k N*(C) = {«P, y). (-EijkA,1 y ). -EijkA,1 (Q - P) )); (Q. EijkA,1 Y »I1i(F)}'

At this point we need a more concrete characterization of Tt(C) and Tt(F). Let s be the parametrization of 1 by arc length, starting from P, and let eo == y, e,. e2

be an orthonormal basis of vectors in R3• The line 1can be deformed to another line by

2

(44) P + sy ~ (P + sy) + L(aiS + bi)ei i='

where ai, bi are any constants. This means that the vector fields along 1given by

(45) X(S) = ei, or sei. i = 1,2

can be viewed as a basis for Tt(C). They are called Jacobi fields. We note that this same discussion is valid in Rn, in which case there would be n - 1 vectors ei


orthogonal to y, and C would be (2n - 2)-dimensional, as asserted earlier. When 1 is viewed as a pair (P, y) E T(R 3 ), the deformation (44) is expressed as

(46) (P, y) ~ (P + X(O). Y+ X)

so that a tangent vector in Tt(C) is identified with the pair

(47) (X(O), X) E Tp(M) x Tp(M).

Since Tt(F) is a subspace of Tt(C), it can also be described by Jacobi fields as in (45) and (47), and the covectors (Eijk») Yk. Eijk») (Q - p)k) appearing in (43) pair off naturally with vectors in (46).

The next task is to introduce a basis 4>" 4>2, 4>3 of I-forms on ~*(F), in which the covectors of (43) take a succinct form. Here we exploit the cone condition satisfied by the family F. Now F is a codimension 1 submanifold of C, so its normal space is a one-dimensional subspace of T* (C). An important ingredient of the geometry of C is that it is a symplectic manifold, with symplectic form

2 2 2

w.c<L(ai s + bi)eit L(CjS + dj)ej) = L (bicj - aidj). i=' j=' i,j='

This allows identifying T (C) with T* (C) via VI ~ w~J VI. In particular the normal space to F corresponds to a Jacobi field X4 (s) along l. The cone condition asserts that X4(S) is proportional to a fixed vector as S varies. If we consider the cone r Q generated by the lines in F passing through Q

(48) r Q == UQEl,lEFl

this means that the cones with vertices along 1all share this vector as normal vector. We choose the vector e, used earlier in the construction of Jacobi fields to be this common normal, and the vector e2 is determined accordingly. We note that e2 is tangent to rQ. It is now not difficult to obtain a basis for Tt(F). It consists of

(49) X,(s) = (as + b)e" X2(S) = e2, X3(S) = se2.

The constant coefficients a, b are specified up to a constant multiple by the requirement that w.c<X" X4) = O. We can now define 4>i, i = 1,2,3 to be the dual basis to Xi, i = 1,2.3.

Returning to the expression (43) for N*(C). we let Q = P + sy and write

j -k j Q k) - ~ ~ ~ »(50) ( EijkA y ,EijkA ( - P) == (A,e, + e2. s(A,e, + A2e2 •

In terms of the (P, y) parametrization, the Jacobi fields X" X2, X3 of (49) become (be,. ae,), (e2, 0), and (0. e2). respectively. Evaluating their pairings with (50)


._..,,""==-----'----==,.,-'-"""=_""""~~"_=, __._._. ._=~"""'_=",.,.~,=__"=~==.",..__","_.,,~,_~ .....o...;."'_.~ ..,---'" "";;";';~""'':'';_:::'',"--~~"'~'';-'"'_~.','_.';"_ .".""".",.__;,;,=",;""""",~~~~,,",,.,...;..-~.,--; ."'o_•.~. ",,,,,,..,_ ''"''''''''''-'''"'''''-''''''''~''"""-;-,,,-,,,,,,",,_.",_,,_-=",,-''C"=_~.'''~~-'-~=_~''''''--''O''''""""""= .•._ ,:~~~~,·:,7;;:'::;;;;;;"':;C':;;."':'

302 CHAPTER 14

gives

(51) N*(C) = {«(P, y), 8 == -[(2as + b)AI1 + Az<l>z + sAz<l>3)];

(Q, Aiel + Azez))}.

Finally we can evaluate drrL and drrR. First we note that N* (C) can be parametrized by (P, y) E F, s, and AI, AZ. Let the variations of the parameters be

3

(52) L 8cxiXi. 8s. HI, Hz. i=l

The corresponding infinitesimal variations drrLCT(N*(C))) are given by

3

8P = L8a;X;(0) ;=1

3

8y = L8aiX i=1

(53) 88 = 8S(2aAI 1 + Az<l>3) + 8AI(2as + b) I + 8Az(z + s<l>3)'

For (50) to give an element in Ker(drrd we need that 8a; = 0, i = I, 2, 3. Furthermore for Q generic, 2as + b is different from 0, and Ker(drrd is nontrivial (in fact, one-dimensional) at exactly the subvariety of N*(C) defined by AZ = 0

(54) ~ = {(((P, y), 8 = (2as + b)Al <1>1); (Q. \II = AIel))}.

Since T(~) = T(N*(C» n {8Az = OJ. and Ker(drrd is generated by (52) with 8ai = Hz = 0, it follows that Ker(drrd is tangent to ~. On the other hand we show now that if the cones r Q satisfy a generic curvature condition, the kernel of drrR is transversal to~. In fact. the infinitesimal variations drrLCN*(C» under (52) are

3

8Q = L 8aiXi(S) + 8seo i=1

z z (55) 8\11 = L Aj8ei + L 8Ajej.

i,j=l j=1

For (55) to vanish, we must have 8s = 8al = 0 and 8az + s8a3 = O. This means that we need evaluate 8ei only when the line I is deformed infinitesimally along the tangent plane to r Q. keeping the point Q fixed. At ~, AZ = 0 and we need only evaluate 8e\. Now the variation of el as we vary I along ez is proportional to ez, with the coefficient of proportionality given by the geodesic curvature of

303SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

the curve of intersection between r Q and the unit sphere. If we assume that the cones r Q have non-vanishing geodesic curvature, it follows that the elements (52) corresponding to Ker(drrR) must have 8Az i- O. This means that the kernel of drrR is transversal to the singular variety ~. This is the asymmetric behavior of the Lagrangian N*(C) that we wanted to establish,

X:FXF and Singular Radon Transforms Formally the operator X:FXF is given by

(56) (X:FXF!)(Q) = { !(P)d/LQ(P)JrQ

where d/LQ is a density on the cone r Q (48), which is singular at Q. For example, when F is the family of light rays in R3 we have

{Zrr (57) (X}g)(YI, Yz, Y3) = X(Y3) J g(YI - Y3COS(}, YI - Y3sin(}. (})d(}

o

and the operator X:FXF can be written explicitly as

(58) (X:FXF)(f)(Xl, Xz, X3)

= {{ !(XI - U, Xz - V. X3 ± Juz + VZ)X(z ± Juz + vZ) dudv . ZJJ ..Juz+ v

Thus X:FXF is ofthe form (35), with the difference that the submanifolds passing through each Q are cones with vertices at Q and thus singular at Q, and the densities d/LQ are of fractional degree rather than critical degree. Also as the dimension n of the ambient space increases, r Q remains of dimension 2 instead of staying a hypersurface. From the microlocal viewpoint

(59) WF'(X:FXF) C ~T'(R3) U N*(9)'

where~T'(R3) C T*(R 3) x T*(R3) is the diagonal and 9 = {(Q. P); P E r Q} is the defining relation of X:FXF. Operators whose wave fronts are contained in a pair of intersecting Lagrangians have appeared in a number of other contexts, and have been extensively studied [20], [15], [I]. Nevertheless a calculus powerful enough to yield Sobolev estimates is only available in certain special cases, notably when the Lagrangian N*(9)' is a flow-out [I]. A crucial observation of Greenleaf and Uhlmann is that this turns out to be the case for N* (9)' when the family F satisfies the cone condition. For families of lines in R3 we can illustrate directly the underlying geometry as follows.

For each Q in Rn, we parametrize the unit vectors generating the cone r Q by

Y(Q. (}). The orthonormal frame eo. eJ, ez introduced earlier is now given by

~ ~ ~

eo = y, ez = ooY/llooYIl. el = ez x eo.


=,=~-=-=,.====, ===="=·~""'~='===;O~="=='-=--=:;="='="'='===='="'='_"~"=='m ~_. ..e·.••,.~... .

304 CHAPTER 14

The dual cone to r Q is the cone r Qof all covectors orthogonal to r Q at some line in rQ. It is generated by the vectors el. We set r* = uQrQ. Then

N*(Q) = {«Q, ~), (P = Q - sy(Q, 0), ~lIel»}

(60) 1/i = ~i - saxiyk~k'

Under the condition that each r Q have geodesic curvature, the closure N*(Q)'

extends smoothly to t. P (R3), and

(61) N*(Q)' n t.P (R3) = t. r •.

Next we derive two key consequences of the cone condition. The first is that 1/

in (60) is actually parallel to ~, and the second is a simple characterization of the

symplectic complement of T (f*). In fact, the cone condition says that the tangent planes to rP and rQ are the same along the P, Q line, Le.,

y(Q - sy(Q, 0), 0) = y(Q, 0)

(62) (aoY)(Q - sy(Q, 0), 0) x (aoy)(Q, 0) = O.

Differentiating the first equation with respect to s shows that

ia .(63) y xiyJ = 0

so that axi yk~k is orthogonal to y. Differentiating the first equation with respect to oand using the second one shows next that (aOyi)(Q, O)axiY(Q - sy(Q, 0), 0) is parallel to aoy(Q, 0) for any s i= O. Letting s tend to 0 gives

(64) (aoyi)(Q, O)axiy(Q, 8)llaoy(Q, 0).

Since ~ in (60) is parallel to y x aOY, it follows at once that

(1/, y) = (1/, aoY) = 0

and 1/ is parallel to~, as claimed. Next T(Q,~)(r*) is generated by infinitesimals (8Q, 81/) ofthe form

(65) 81/ = -s(y x a~oY + aQy x aoy)8Q - (y x aoy)8s - s(y x a~oY)80.

Thus w«a, fJ), 8Q, 81/» = 0 implies that a is proportional to y, while

fJi = ElpqaxiypaOyqyl.

In view of (63), (64) fJ is orthogonal to both y and aOY, and hence must be parallel to y x aOY.

We can identify now the bicharacteristic through (Q, 1/). By the curvature condition on each cone, the covector 1/ corresponds to a unique ray y(Q, w).

Points (P, ~) are on the bicharacteristic when P flows along y(Q, w), and ~

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS 305

along 1/. This identifies the Lagrangian N*(Q) as the flow-out of r*, which is the

desired statement. The asymmetric behavior of the Lagrangian of XF can lead to very different

operators for X}XF and XF X}. Again returning to the example of light rays in

R 3 and the operator (38) we find

2 2 {{ ( u + v )

(66) (XFX}g)(l) = JJ g(x + u, y + v, 0 -1/I)X 2(u cosO + v sinO)

dudv

lu cos 8 + v sin 01 2where 1/1 is the angle defined by cos 1/1 = 2uv(u2 +V )-I, sin 1/1 = (v2

- u2)(u2 + v2) -I. We note that the kernel of the integral opemtor (66) has singularities along

a line, but its support is narrow enough to make it integrable. Operators of type XFX} are central to the problem of deforming the Minkowski metric on R4 to

metrics all of whose light rays are closed. We refer to [14] where a detailed microlocal analysis for them is provided. It would be interesting to explore these operators also from the local viewpoint, with an aim towards LP - Lq estimates.

III MODELS OF DEGENERATE FOURIER INTEGRAL OPERATORS

A full theory of Fourier integral operators encompassing the stratification (11),

(13) of the Lagrangian and singular densities is still far ahead at this point. it is

important first to construct and study simple models which reflect the high order of degeneracy of the projections ]fL and ]fRo Returning to the family of curves

(15) in R n , the corresponding Radon transforms can be expressed as

Rj(x,t) =1 j MexJ)

n-I

= ! ei()...i-S)[! ei()",S(x,y») X(y)!(y, )..)dy] 1] d)..j(67)

where j (y, s) is a Coo function with support in a fixed compact set, ! (y, )..) is the

partial Fourier transform of j with respect to s, and X (y) is a cutoff function which is identically 1 on the support of j. As in (37) the operator (67) can be viewed

as a pseudo-differential operator whose symbol a()..) is the operator from Coo functions of y to Coo functions of x given by the expression between brackets. For

each fixed)" this is an oscillatory integral with phase S)..(x, y) = ()../I)..j, Sex, y»).


~ ~ ~

307 _____......... 7"""'7' ..,

306 CHAPTER 14

Non-vanishing right torsion at (x, y) means that for any A

n-Z (68) L la:ays).(x, y)1 = 0 ===? a;-Iays).(x, y) 'I- O.

01=1

Similarly the condition that any of the varieties E~-z be empty translates into the non-vanishing of an n - th derivative of S).(x, y) when certain derivatives of order :s n - I vanish. Fixing A we are led then to the following models for degenerate Fourier integral operators

(69) - (T¢»(x) = f eiAS(x,y)X(y)¢>(y)dy

where Sex, y) is a homogeneous polynomial in (x, y) of degree n

n-I (70) Sex, y) = L ai_Ixi yn-i.

i=1

We assume that at least one of the coefficients ai, i = 0, ... ,n - 1 is not O. Evidently the norm IITII is unaffected by the inclusion of terms proportional to either x n or yn, so we can ignore them. The models (70) generalize oscillatory integrals with quadratic phases which are models for Fourier integral operators associated to canonical graphs [18], as well as the well-known Airy operator

+00

(71) (Ai¢»(x) = -00 ei ).(x-y)3 X (y)¢> (y)dy /

which is the prototype of Fourier integral operators where both rrL and rrR are Whitney folds [19], [41]. It is remarkable that no systematic study of the operators (69) has been undertaken to date, and we shall address this issue here [35].

1 Low degrees of degeneracy n = 2, 3, 4

For low degree of degeneracy we have a complete understanding for the L Z bounds for (69)

Theorem 6. For each n ::; 4 the following bounds hold and are sharp

(72) 1IT11 ::; CIAI- I/n

except when Sex, y) isproportional either xyn-I or xn-Iy, and n is strictly greater

than 2. In these cases T satisfies the weaker estimate

(73) II TIl ::; C1AI- 1/(Z(n-l)).

Proof The case n = 2 is trivial, since the operator T reduces to the Fourier transform (acting on X¢» after a rescaling. For n = 3 either Sex, y) = aoxyZ +

SINGULAR lNTEGRALS AND FOUR1ER lNTEGRAL OPERATORS

alxZy with aoal 'I- 0, or Sex, y) is proportional to either xyz or xZy. In the first case, the operator T is equivalent to a similar operator with phase

Z Za aSex, y) = «_0 )1/3x + (_I )1/3 y)3.

3al 3ao

After a linear change of variables, T reduces to a convolution operator whose multiplier is readily seen to be uniformly bounded by IA\-1/3. In the second case

assume, say, that sex, y) =xl. The kernel K).(x, y) of TT* is then given by

IK).(x, y)1 = e').(X-y)z2 x(z)ZdzI I '" irr I'/Z(74) If ·

A(X - y)

which shows that 1IT11 = 0(\AI- ' /4), as stated in (73). We tum next to n = 4.

3When S(x, y) is proportional to either xy3 or x y, the kernellK). (x, y)1 of either TT* or T*T can be evaluated to be proportional to I(A)-1/3 (x - y)-I/3I. It follows

that II TIl ::; 0(IAI- ' /6), which is the estimate (73) for n = 4. We now show that

the better estimate II TIl ::; 0(IAI- ' /4 ) holds for any other phase function which is

Za homogeneous polynomial of degree 4. If Sex, y) = x y 2, the kernel K).(x, y)

of TT* is IK).(x, y)1 '" IA-1/ Z(X - y)-1/2(x + y)-I/zi. Thus TT* is bounded on L 2(R) with norm O(IA\-I/Z) and IITII = 0(IAI-I/4

) as was to be shown.

Otherwise we may assume that the phase Sex, y) is given by

(75) Sex, y) = xl + alx2l + a2x3 y

with at least one of the coefficients ai, az different from O. The kernel K).(x, y)

is given by

(76)K).(x, y) = f ei).(S(x,z)-S(y,z»X2(z)dz

= ei).(x-y)[?7 0I;(X+y)3- t OI\OIz(x+y)(xz+xy+yz)]

x f ei).(x-y)(z3+[az(xz+xy+i)- ta;(x+y)Z]z)Xz(z + ~ (x + y))dz3 .

The second integral in (76) is of the form

+00

I (A) = -00 ei).(Qz- t z3) X (z)dz(77) /

with A(X - y) replaced by -A/3. An explicit analysis, say, using Airy functions,

yields

(78) II(A)1 ::; C min{IAI- I/ 3, (IA\IQI'/z)-,/z}

(79) If A < 0, then II(A)\ ::; C([).IIQi)-1

where C is a constant independent of A and A.


~ -

309

-:"'iiiO'-

308 CHAPTER 14

In particular, we obtain the following bound for the kernel K).. (x, y)

I(80) IK)..(x, y)1 ~ C/A(X - y)I-I/2/a2 (X 2 + xy + l) - 3a~(x + y)2/-1/4.

The right hand side of (80) is homogeneous of degree -1. Lemma 1 below asserts that the corresponding kernel is bounded on L 2(R) with norm 0(IAI- /2) as long

'as it has at most a finite number of singularities 8i on the circle x 2 + y2 = 1, near which it is bounded by

/Arctan(y/x) - 8d- I+Si

for some ~i > 0 in general, and ~i > 1/2 if the singularity 8i is on one of the x, yaxes. These conditions can only be violated if the quadratic form in (80) has double roots at either x = 0, y = 0, or x = y. It is readily seen that the first two possibilities cannot occur, while the third one occurs exactly when 9a2 - 4a~ = O. In particular, we have proved the desired estimate for all phase functions of the form (75), except when

S(x, y) = xy3 + 3ax 2y2 + 4a 2x 3y

for some a =f. O. In this case, however, the phase in (76) becomes A(X - Y)[Z3 + a2(x - y)2Z]. Since a2(x - y)2 ::: 0 we can apply (79) and obtain

(81) /K)..(x, y)1 ~ Cmin{lA(x - y)I-I/3, IAI-I/x - Yr 3 }.

As a consequence,

(82) i: IK)..(x, y)/dx = r IK)..(x, y)ldx At-YI<I)..I-I/4

+ r IK)..(x, y)ldx J1X-YI> 1)..\-1/4

~ C/AI- ' /31 Ix - yl-I/Jdx

Ix-yl<I)..I-1/ 4

+ C/AI- I r Ix - yl-Jdx J\X-YI> 1)..\-1/4

~ C/Ar l/ 2.

Similarly, the L I norm in y of K)..(x, y) is 0(IAr l / 2). Thus, we still have IITII ~ O(lAI- ' /

4) in this case, and the proof of Theorem 6 is complete.

General Order of Degeneracy nIi In the formalism of Fourier integral operators, the non-vanishing of a coefficient

I!I ai in S(x, y) in (70) corresponds to the condition that some variety I;~-2 be empty.

" 1~1!llj. il~!

-.,--- ~-_.-----


Theorem 6 suggests that the sharp decrease O(IAI-I/n) for the operator norm II Til holds when at least two varieties I;~-2, I;:;-2 with suitable w and Wi are empty, or when n is even and I;~-2 is empty when w contains as many L as R indices. Strong evidence for these phenomena is provided by the following theorem which extends Theorem 6 to all orders n of degeneracy [35]:

Theorem 7. Let S(x, y) be as in (70), T the corresponding operator (69). Then T extendsasa bounded operator on L2(R) with norm II Til ~ C/AI-I/n when

• an-2 =f. O. n is even, and la(n/2)-" + + lao I > 0; • an-2 =f. 0, n is odd, and la(n-I)/2-d + + laol > 0; • n is even, a(n/2)-1 =f. 0, and ai = 0 either for all i ::: n/2 or for all i ~

(n/2) - 2.

Evidently the roles of x and y can be interchanged, and an analogous statement holds starting from the assumption ao =f. O. Perhaps the most encouraging feature of Theorem 7 is that sharp bounds depend only on the non-vanishing of individual coefficients in the phase function S(x, y), instead of, say, the rank and signature of some quadratic form. This is certainly a necessary condition for any eventual generalization to a full-fledged theory of Fourier integral operators based on stratification of the Lagrangian by the singular varieties of 1rL and 1rR.

The key estimates required for the proof of Theorem 7 are contained in the following Theorem 8, which can be viewed as a composition theorem for these models of degenerate Fourier integral operators [35]:

Theorem 8. Assume that one of the sets ofconditions in Theorem 7 holds. Then the kernel K)..(x, y) of the operator TT* satisfies the following bounds: (a) (83) IK)..(x, y)1 ~ C/A(X _ y)I-'/(n-')

(b) There exists a finite number ofpairs (ai, bi ), i ~ N, at + b; = l, such that

(84) IK).. (x , y)1 ~ C/AI-2/ n- v LN

laix + biy\-IHi(!xl + Iyl)-Si- un

i=1

for some v ::: 0, and (x, y) outside ofa conic neighborhood ofthe line x = y.

Here the exponents ~i are strictlypositive, and strictly greater than 1/2 ifeither aj or bi vanishes;

(c) In the conic neighborhood of x == y omitted above. either an estimate of the form (84) holds, i.e.,

(85) IK)..(x, y)1 ~ C\AI-2/n- ulx _ YI-I+S(lxl + Iyl)-S-vn II


311

I

310 CHAPTER 14

for some /) > 0, or we have an estimate of the form

I"i (86) IKJ..(x, y)1 .::s CI)..I-v- 2/nlx _ yl-I-vn

with v ~ O.

I, Theorem 7 follows from Theorem 8, according to the following lemma:

Lemma 1. Let K)..(x, y) be any kernel on R x R satisfying either the conditions 1:111 (84) and (85), or the conditions (83) and (86). Then the corresponding operator

is bounded on L 2(R) with norm O(I)..I-2/n).

When the kernel satisfies (84) and (85), Lemma 1 is an easy generalization of the standard Hilbert integral lemma which establishes the L 2(R+) boundedness of the operator with kernel (x + y) -I. When it satisfies (83) and (86), the desired

!IIIIII estimate can be established as in (82) assuming (81).

We discuss now the main ideas in the proof of Theorem 8. The kernel K)..(x, y)

of T T* can be expressed as an oscillatory integral

K).. (x, y) = Jei)..(x-y)P(x.y.z) X (Z)2dz

(87) P(x, y, z) == (x - y)-I (S(x, z) - S(y, z». In principle, the asymptotics in ).. of such integrals are given by the method of

stationary phase or the van der Corput lemma, except that here it is crucial to keep track of the dependence on x, y. For example, the estimate I)..QI/21-1/ 2 in (78)

and (79) can be worse than the estimate)..-1/3 if IQI is small enough. What is needed here is a version of the method of stationary phase which keeps track of the dependence on external parameters. Since the coefficient IQ11/ 2 in (78) can be interpreted as twice the distance separating the critical points ± Q1/2, we are led to the following theorem which may be of intrinsic interest.

Let K).. be an oscillatory integral of the form

(88) K).. = lb ei)..P(z) X(z)dz

where P(z) is a C2 phase function on R, X(z) is a C l function which is taken to be compactly supported if either one or both of the bounds a and b are infinite. Assume that

N

(89) IP'(z)j ~ A IT Iz - Sjl j=1


where A is a positive constant. The Sj 's are called "critical points," and are allowed to be complex. If Sj has an imaginary part, then its conjugate also occurs in (89).

We set

(90) Sj == aj + ib j , j = 1, ... , N;

P"(z) changes sign only a finite number M of times. By a cluster L around a given critical point Sk we designate any subset L of the

sj's which contains Sk. U will denote L \ {sd and lUI will denote the number of critical points in U. For each integer m ~ 2 and each k, k = 1, ... , N, we define KJ..(k, m) to be

m(91) KJ..(k, m) == minu,lUl=m-2(1A)..1 IT lak - SjD- I/

j¢L

when Sk is real, and

mK)..(k, m) == min{minU.IUI=m-3(1A)..! IT lak - SjD- I/ ,

j¢L

mminu,lul=m-2(IA)..llbkl IT lak - SjD- I/ ,

j¢L

m(92) minu.IUI=m_I(IA)..llbkI2 IT lak - SjD- l/ }

j¢L

when Sk = ak + ibk has a non-vanishing imaginary part. Then

Theorem 9. The oscillatory integral (88) can be estimated by

(93) IK)..I .::s CM.Nmaxl~k~Nminm~2[K)..(k, m)]

where the constant CM.N depends only on M, N, and the CI norm of x.

In Theorem 9 we can evidently choose to deal only with real critical points, simply by dropping the imaginary part bj of Sj from the bounds (89), (91), and (92). This will lead to slower decays for K)... The proof of Theorem 8 actually requires the full Theorem 9 with complex critical points. To illustrate the estimate (93), we take A = 1, Sj = aj to be real, and choose min (91) to be 2. Then Theorem 9 implies the following bound for K)..:

IK).. I .::s CM,NI)..r l/ 2 L

N

IT lak - ajl-I/2. k=1 j#

This is the estimate predicted by the method of stationary phase, with the key improvement that the constants are uniform in all phases with bounded number of critical points and bounded number of intervals of monotonicity for P' (z).


313

~~ _efts• .,...-- .~....._ .119"",___,.iiilliii ~tti\'1RI~ti"=---·-

312 CHAPTER 14

Returning to the kernel K)..(x, y) of TT* (87), we replace).. by )..(x - y),

the phase is given by P(x, y, z), and thus the critical points ~j of (90) are the (possibly complex) roots ~j{x, y), j = 1, ... , n - 2 of the polynomial in z(x

y)-l(S~(x, z) - S~(y, z». They are homogeneous functions of (x, y) of order 1, so we may assume that (x, y) is on the circle x 2 + y2 = 1. For each fixed phase Sex, y) there are at most a finite number of points Arctan(yIx) = Ok. k = I, ... , N on the circle at which two or more of the roots Sj coincide. Theorem 6 shows that estimates for the kernel K)..(x, y) reduce to estimates for l~i(X, y)

Sj(x, y)1 as (x, y) i:" close to the line of slope Ok' Exploiting the factthat these roots are algebraic functions, we can show that under the hypotheses of the Theorem not more than (n 12) - I when n is even, and (n - 1)12 when n is odd can get too close together in a precise sense. This combined with Theorem 9 leads to the bounds in Theorem 8 for K)..(x, y), and Theorem 7 follows.

IV THE MONGE.AMPERE DETERMINANT AND COVARIANCE

The models we discussed in Section III correspond to Fourier integral operators with "constant coefficients," For example the symbol a()..) of (67) does not depend on 1. The task ofconstructing a full theory defined on smooth manifolds even when these models are already well understood turns out to be surprisingly difficult. The only reasonably general results are for n = 3, that is, when at least one of the projections JrL or JrR is a fold. There it is known that

• Fourier integral operators whose Lagrangians are folds with respect to both JrL and JrR can be microlocally conjugated to the Airy operator [19], [20] and are smoothing of order -1/3;

• If JrL is a fold and JrR is a blow-down, then the FlO is smoothing of order -1/4 [11], [12];

• Sobolev and LP - Lq estimates are known for Radon transforms along general variable distributions of curves in any 2-dimensional manifold [34].

The methods for the first two classes of results are microlocal in nature. Here we shall discuss the local approach of [34], which makes the role of the MongeAmpere equation (36) transparent and also confirms the intimate relation between degenerate Fourier integral operators and singular Radon transforms.

Let M be a Coo two-dimensional manifold, and let M p be a curve passing through P for each point Pin M. If dJ-Lp(Q) is a Coo density on Mp supported in a small neighborhood of P we set

Tf(P) = iMp f(Q)dJ-Lp(Q).(94) [

SfNGULAR fNTEGRALS AND FOURfER INTEGRAL OPERATORS

Let itL, itR be the projections from the Lagrangian manifold N* (C) on the left and

right factors as in (5). Then [34]

Theorem 10. • If either itL or itR is a fold along the singular variety ~, then T is bounded

from H(s)(M) to H(s+1/4) (M); • Ifboth JrL and itR are folds, then T is boundedfrom

(a) H(s)(M) to H(s+I/3)(M); (b) LP(M) to U(M)for (lip, l/q) in the intersection of the closed triangle

with vertices at (0,0), (1, 0, (2/3, 113) and the closed half-space of the

equation

I 1 I(95) 4 ~ o.

p q

As mentioned above, Part (a) of the second statement is the well-known estimate of Melrose and Taylor [19], [21] in diffraction theory. The estimate in the first statement has been obtained by Greenleaf and Uhlmann [12] for any dimension, under the additional hypothesis that the other projection is completely degenerate,

or, more precisely, a blow-down. We discuss in some detail two ingredients of the proof. We focus on the case

when both itL and itR are folds since its treatment also contains many of the ingredients required for the case when only one projection is assumed to be a fold. It suffices to prove the theorem for M = R2 and f is supported in a fixed small neighborhood of the origin. We can choose coordinates (x, t) on R 2 so that locally

the curve M(r ,r) is the graph of a function

(96) M(x,r) = ley, s); s = t - sex, t, y)}

with

(97) Sex, t, x) == 0, S(O, t, y) == o.

The operator T can be written as

(98) (Tf)(x, t) ei )..(s-r+s(x,r,Y»1/J(y)f(y, s)dydsd)" = fff where 1/J(y) is a suitable cutoff function in y. The standard almost-orthogonality argument reduces the Sobolev bounds to showing that the localized operators in )..

(99) (Tmf)(x, t) = fff ei)..(s-r+S(x,r,y» X(Tm)..)1/J(y)f(y, s)dydsd)"


-~"----- --

314

----'-'--'-=-, ------------------"-------------------------_._----------_._------

CHAPTER 14 SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS 315 satisfy the unifonn estimates

(100) II Tm II S crm / 3 •

Let J (x, t, y) be the Monge-Ampere detenninant (36) with cI> (x, t; y, s) = s _ t+S(x,t.y)

(101) J(x. t. y) = (1 - S;)S:y - S~S:~'

The condition that JrL (respectively JrR) be a fold along its singular variety translates into the fact that the zero set of J (x. t. y) can be parametrized as the graph of a smooth function of y = x (x, t) (respectively x = y(t, y». In particular the map x ~ x(x, t) is invertible for each t if both projections are folds. We introduce a dyadic decomposition Tm = :L;:o Ii,m away from the zero set of J (x, t. y)

(102) (Ii,mf)(x, t)

= III ei,\(s-t+S(x,t,y))x (2-mA)X (i J(x, t, y»1{t(y)f(y, s)dydsdA.

The Sobolev bounds for T are consequences of the following two bounds for Ii,m

IlIi,mll S cr l

(103) IlIi,mll S qrmi)I/2.

The index m is fixed from now on, so we drop it from our notation. Now the fact that both JrL and JrR are folds implies that the support of the kernel of Ii in each variable is of length "-' 2-1 when the other variable is fixed. The first estimate follows at once. We note that we would obtain only the weaker estimate IlIiII S C2-1

/2 if only one projection is assumed to be a fold. The key to the

second estimate is a further localization

(104) (Ii,j/)(x, t) = x (ci(x(x, t) - Xj»

x III ei,\(s-t+S(x,t,Y))1{t(rmA)X(i J(x, t, y»f(y, s)dydsdA

where cis a large constant, and x j are points regularly spaced at a distance (c) -12-1•

Since (x, t) ~ (x(x, t), t) is a smooth change of variables, we may define

(105) ci;,j/)(x, t) == <1i,j/)(x, t).

The second estimate for Ii can be verified to follow from a similar estimate for t,m' In tenns of (y, s) and (x, t) the operator t,m is of the same fonn as (104) but with a different phase function

(106) S(x, t, y) == S(x(i, t), t, y).

Lemma 2. In a neighborhood ofthe origin, the phase function S(x, t, y) satisfies

(107) IS:I = 0(1), IS:~I = 0(1).

Lemma 3. Let S(x, t, y) be a phase function satisfying the condition (l07),

together with fold conditions on both sides. Then

• In a sufficiently small neighborhood ofthe form It - s I < clx - YI, the equation

(108) S~(x, t, z) - S~(y, s, z) = 0

admits a unique solution Zc (x, t, y, s);

• Let y = x(x, t) denote the zeroes of the Monge-Ampere determinant

corresponding to S(x, t, y), and set

(109) X(X, t, y, s) = S(x, t, i(x, t» - S(y, s, i(x, t».

Then in the region where Iz - i(X, 1)1 S 2-1 , It - s - X(x, t, y, s)1 >

c2-l lx - yl we have

(110) It - s - X(x, t, y, s)1 "-' It - s - S(x, t, z) + S(y, s, z)1

while in the region

(111) It - s - X(x, t, y, s)1 s 2- l lx - yl

the critical point zc can be approximated by the zero of the Monge-Ampere

determinant

(112) Izc(x, t, y, s) - i(x, t)1 s C(lx - yl + (c)-lrl).

Lemma 4. Let j (x, t, y) be the Monge-Ampere determinant determined by the

phase function <il(x, t; y, s) == s - t + S(x, t, y). Then we have the following

transformation law

- ax (113) J(x, t, y) = (ax )J(x, t, y).

We sketch the proof of the second statement in Lemma 3, as it is one of the key steps in the argument. First let x(x, t) be any function of (x, t) and define X as in (109). We note that X can be expanded in tenns of (x - y) and t - s - X with smooth coefficients

(114) X == (x - Y)P(x, t, y, s) + (t - s - X)a(x, t, y, s).

In fact, a Taylor expansion gives X == (x - y)f3o + (t - s)ao, and laol = 0(1)

since IS:I = 0(1). The equation (114) follows with

(115) P= f3o(1 - ao)-I, a = ao(1 - ao)-l.


317

"-"-'-'ii_-~-'-- ",irliIr- :iti:~ -_~_~ - '-iiiiii"j"'''"iliii'iiiiiiiiii"i" -.·s-... 'Dif;;-- n ;f~~~~"'-""'-~"g_-ina;;i'i!fi""''U'iii~;:tF30-'':'.~~;;';;":-''':''&:i!'~"''~;ii;l£''-;i!ii'~,i@~~~~~';i'·"-·w,";"B'~S"~iia:;j'i;j""i:<:b'~~%jik - 'iIiiill;~:i·;!l"'l.i~;ilfll!l.:-'~

316 CHAPTER 14

Next we rewrite the equation for the critical points in more explicit fonn. Setting S(.x, t, z) - S(y, s, z) == (x - y)y + (t - s)8 we find

Sex, t, z) - S(y, s, z) = (x - y)W(x, t, y, s, z)

- t-s-X W(x, t, y, s, z) = [y + P8] + _ (8 + (8)

x-y

(116) azW(x, t, y, s, z) = 0 {::::::} z = zc(x, t, y, s).

We can obtain approximations of Zc up to errors bounded by the right hand side

of (112) by adding to the equation az II' = 0 any tenn bounded by 0 (Ix - yl) or O(C-12-1). In particular in azw we can drop (t - s - X)(x - y)-1(8 + (8),

replace y by x, and s by t. The approximating equation is

(117) az(Y + ~8) = 0

with up to negligible tenns

- f30(118) Y = S~(x, t, z), 8 = S;(x, t, z), P=

1 - ao

Finally, we derive explicit fonnulas for ao, Po, up to similar negligible tenns

(119) ao = S;(x, t, x(x, t», Po = S~(x, t, x(x, t).

Altogether the equation (116) for zc(x, t, y, s) becomes

(120) S;/x, t, zc)(l - S;(x, t, x(x, t))) + S~(x, t, x(x, t))S;~(x, t, zc) = o.

Thus for the critical point zc to coincide with x(x, t), we must choose x(x, t) to

be the zero of the Monge-Ampere detenninant! This establishes the key estimate (112) in Lemma 3.

We return to the proof of L 2 bounds for 1;, j' Forming 1;,j 1;~j' we obtain an operator whose kernel is an oscillatory integral in z and in ).. with phase given by

)"(S(x, t, z) - S(y, s, z»

and a cutoff given by

(121) X (ci(x - xj»X(c21(y - xj»X(2i ](i, t, z»X(2l ](y, s, z».

The main contribution to the oscillatory integral comes from its critical point zc(x, t, y, s). Now Lemma 4 implies in particular that the zero set of j is given by y = X. In view of Lemma 3 we have then

(122) Izc(x, t, y, s) - xl :s CClx - yl + (c)-ITI )


in the region defined by (11l). Since Ix - yl :s (c)-12-l on the support of the kernel of tt(123) Iz - zc(x, t, y, s)\ "" Iz - xl "" T l

and we obtain the following lower bound for the phase variation

(124) IS~(x, t, z) - S~(y, s, z)1 "" Ix - YITI .

This allows to exploit integration by parts with respect to z in the region (111).

Outside of this region, the desired estimates follow from a simpler integration by parts in )" using (11 0).

We would like to point out an interesting feature of the above arguments. For

classical Fourier integral operators we can choose coordinate systems so that the non-vanishing of the Monge-Ampere determinant in small neighborhoods is equiv

alent to the non-vanishing of the Hessian in x, y of Sex, t, y). This is done from

the outset and the Monge-Ampere determinant itself no longer enters the analysis [32], [33]. In the above argument on the other hand it is crucial to introduce dyadic

partitions away from the zeroes of the Monge-Ampere determinant itself, rather than from the zeroes of say the Hessian, since these approximate better the critical

points Zc in (111). Now the critical points Zc are zeroes of a linear expression in

S, while the zeroes of J are of course zeroes of a non-linear expression in S. The appearance of this non-linearity is due to the fact that the approximation (112) does

not hold everywhere, but only in a region determined by the cutoff itself. It is very intriguing that the localization procedures required by linear analysis inevitably

introduce some non-linearity and lead us exactly to a covariant expression.

We conclude by sketching the relation between the Radon transfonns of (94) and singular Radon transforms. For the sake of simplicity we shall assume that

the family Mp is semi translation-invariant in the sense that the defining function

Sex, t, y) of (96) is independent of t. The operators T and TOT can then be

expressed as

(Tf)(x, t) = I fey, t - sex, y»dy

(125) (r+Tf)(x, t) = II fey, t + sex, z) - S(y, z»dzdy.

Fixing (x, t) we make the change of variables (y, z) ~ (u, v)

u=x-y

(126) v = S(y, z) - Sex, z) - (S(y, zc(x, y» - sex, zc(x, y)))

where zc(x, y) is the critical point S~(x, zc) - S~(y, zc) = O. Observe that

IJacobianl = IS~(x, z) - S~(y, z)1


319 318

CHAPTER i4

""' Ix - yllz - Zc(X, Y)I

(127) Ivl ""' Ix - Yllz - Zc(X, y)e

SO that the operator T* T corresponds to the case s = 112 ofthe following operators

(128) (Us f)(x, t)

= II f(x - u,t - v + S(x, Zc(x, x - u» - S(x - U'Zc(X,X - u)))

dudp

lulslvlS

This relation reduces the LP - L q bounds for T to L 2 bounds for singular Radon

transforms. In fact, the estimates along the segment (11 p) - (llq) - (1/4) ~ 0

of Theorem 10 follow from the boundedness of T*T from L4/3 (R2) to L4(R2). By complex interpolation this is a consequence of the endpoint bounds

• Us is bounded from LIto L 00 for Re s = 0;

• Us is bounded from L 2 to L 2 for Re s = 1.

The first statement is trivial. As for the second, the operator Us can be factorized as Us = LsMs where

I dv(129) (Msf)(x.1) = f(x, t - v)-

Ivl s

(Lsf)(x, t) = I f(x - u, t + sex, Zc(X, x - U»

du - S(x - U Z (x x - u») , c , luis'

The boundedness of M s on L 2 is the standard one-dimensional theorem on singular

integrals. As for L s we recognize that it is a singular Radon transform with phase function

(130) cI>(x, y) == Sex, zc(x, y» - S(y, zc(x, y».

In general, this phase function is degenerate. However we have

Lemma 5. Let z(x) be the zero ofthe Monge-Ampere determinant S~yCx, z(x» = O. Then

(131) III 1 Sill ( -( (dz(y) )2 0(1 )cl>XYY = 4 xyy y, z y» ----;ty" + x - YI .

This shows that for two sided folds IcI>~~y I ~ c > O. Thus the Hessian of the

phase (130) does not vanish to infinite order, and the refined boundedness Christ-

SiNGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

Nagel-Stein-Wainger theorem [5] (see also the discussion at the end of Section II. 2)

for singular Radon transforms applies, giving the desired estimate.

Columbia University

REFERENCES

[I] J. Antoniono and G. Uhlmann. "A functional calculus for a class of pseudo-differential operators with singular symbols." Proc. Symp. Pure Math. 43 (1985) 5-16.

[2] V. Arnold, A. Varchenko, and S. Goussein-Zade. "Singularites des applications differentiables." Editions Mir, 1986.

[3] M. Beals, C. Feffennan, and R. Grossman. "Strongly pseudo-convex domains." Bull. Amer. Math. Soc. 8 (1983).125-322.

[4] M. Christ. "Hilbert transfonns along curves I." Ann. ofMath. 122 (1985),575-596. [5] M. Christ, A. Nagel, E. M. Stein, and S. Wainger. "Singular and maximal Radon

transfonns." In preparation. [6] C. Feffennan. "Monge-Ampere equations, the Bergman kernel, and geometry of

pseudo convex domains." Ann. ofMath. 103 (1976), 395-416. [7] G. Folland and J. J. Kohn. "The Neumann problem for the Cauchy-Riemann complex."

Annals of Mathematics Studies 75, Princeton University Press, 1972. [8] G. Folland and E. M. Stein. "Estimates for the ab-complex and analysis on the

Heisenberg group." Comm. Pure Appl. Math. 27 (1974), 429-522. [9] D. Geller and E. M. Stein. "Estimates for singular convolution operators on the

Heisenberg group." Math. Ann. 267 (1984), 1-15. [10] A. Greenleafand G. Uhlmann. "Non-local inversion fonnulas for the X-ray transform."

Duke Math. J. 58 (1989), 205-240. [11] . "Estimates for singular Radon transforms and pseudo-differential operators

with singular symbols." J. Funct. Anal. 89 (1990),202-232. [12] . "Composition of some singular Fourier integral operators and estimates for

the X-ray transform I." Ann./nst. Fourier 40 (1990), 11,1991 preprint. [13] P. Greiner and E. M. Stein. "A parametrix for the a-Neumann problem." In Estimates

of the Neumann Problem, edited by P. C. Greiner. Mathematical Notes 19. Princeton University Press, 1977.

[14] V. Guillemin. "Cosmology in (2 + I) dimensions, cyclic models, and defonnations of M2•1." Annals of Mathematics Studies 121. Princeton University Press, 1989.

[15] V. Guillemin and G. Uhlmann. "Oscillatory integrals with singular symbols." Duke Math. J. 48 (1981),251-267.

[16] L. Honnander. "Fourier integral operators I." Acta Math. 127 (1971), 79-183. [17] . The Analysis ofLinear Partial Differential Operators /-N. Springer Verlag,

1985. [18] A. Melin. "Lower bounds for pseudo-differential operators." Arkiv Math. 9 (1971),

117-140. [19] R. Melrose. "Equivalence of glancing hypersurfaces." /nv. Math. 37 (1976),165-191. [20] R. Melrose and M. Taylor. "Near peak scattering and the correct Kirchhoff

approximation for a convex obstacle." Ailv. Math. 55 (1985), 242-315. [21] R. Melrose and G. Uhlmann. "Lagrangian intersection and the Cauchy problem."

Comm. Pure Appl. Math. 32 (1979), 483-519.


~II---===-="-==;======:::-"~--

I, CHAPTER 14

[22] G. Mockenhaupt, A. Seeger, and C. Sogge. "Wave front sets, local smoothing, and Bourgain's circular maximal theorem." Ann. ofMath. 136 (1992), 207-218.

[23] D. Milller. "Singular kemels supported by homogeneous subrnanifolds." J. Reine Angew. Math. 356 (1985), 90-118.

[24] D. Muller and F. Ricci. "Analysis of second order differential operators on the Heisenberg group I." Inv. Math. 101 (1990),545-582.

[25] A. Nagel, N. Riviere, and S. Wainger. "Hilbert transforms associated with plane curves." Bull. Amer. Math. Soc. 223 (1974), 235-252.

[26] D. Oberlin. "Convolution estimates for some measures on curves." Proc. Amer. Math. Soc. 99 (1987), 56-Q0.

[27] Y. Pan. Ph.D. dis~., Princeton University, 1989.

[28] Y. Pan and C. Sogge. "Oscillatory integrals associated to folding canonical relations." Colloq. Math. 60 (1990), 413--419.

[29] D. H. Phong. "On integral representations for the Neumann operator." Proc. Nat. Acad. Sci. 76 (1979),1554-1558.

[30] D. H. Phong and E. M. Stein. "Some further classes of pseudo-differential and singular integral operators arising in boundary value problems I." Amer. J. Math. 104 (1982), 141-172.

[31] ---. "Singular integrals related to the Radon transform and boundary value problems." Proc. Nat. Acad. Sci. 80 (1983),7697-7701.

[32] --_. "Hilbert integrals, singular integrals, and Radon transforms I and II." Acta Math. 157 (1986), 99-157 and Inv. Math. 86 (1986), 75-113.

[33] ---. "Singular Radon transforms." Duke Math. J. 58 (1989), 349-369. [34] --_. "Radon transforms and torsion." Int. Math. Res. Notices 4 (1991), 49-60. [35] ---. "Oscillatory integrals with polynomial phases." Inv. Math. 110 (1992),39

62.

[36] F. Ricci and E. M. Stein. "Harmonic Analysis on nilpotent groups and singular integrals, I. Oscillatory integrals." J. Funct. Anal. 73 (1987),179--194; "II. Singular kemels supported on submanifolds." J. Funct. Anal. 78 (1988), 56-94.

[37] C. Sogge and E. M. Stein. "Averages of functions over hypersurfaces: smoothness of generalized Radon transforms." J. Analyse Math. 54 (1990), 165-188.

[38] E. M. Stein. "Maximal functions: spherical means." Proc. Nat. Acad. Sci. 73 (1976). [39] --_. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory

Integrals. Princeton Mathematical Series 43. Princeton University Press, 1993. [40] E. M. Stein and S. Wainger. "Problems in harmonic analysis related to curvature."

Bull. Amer. Math. Soc. 84 (1978), 1239-1295.

[41] M. Taylor. "Propagation, reflection, and diffraction of singularities of solutions to wave equations." Bull. Amer. Math. Soc. 84 (1978), 589-611.

[42J --_. "Fefferman-Phong inequalities in diffraction theory." Proc. Symp. Pure Math. 43 (1985), 261-300.

[43J --_. Pseudo-differential operators. Princeton Mathematical Series 34. Princeton University Press, 1981.

[44] A. Thompson. "A singular Radon transform in Rp3." Comm. Part. D!ff. Eq. 14 (1989), 1461-1470.

[45] F. Treves. Introduction to pseudo-differential and Fourier integral operators. Plenum, 1980.

,_ ___"_.':::.,,, .:'-::',",::_::::,":::," ~·'·'_·, ..'""r,~,_ ,,,,,,.'U~~-'=:::""'" V"~".::,_ ._:'::'::~:'~=~'M""" "'~~_'· .•T.~_~.'.'_.'~".'_~·' "~_~~" ':'"" ,._ :-~~_" ..•.""""_,,,.. ...'" =.":~~':-'''~~~:'::=''''d'''''''.,

15

Counterexamples with

Harmonic Gradients in }R3

Thomas H. Wolff"

INTRODUCTION

Many results about hannonic functions in 1R2 are proved using complex analysis explicitly or implicitly, so it is natural that some of them should fail in higher dimensions, or should at least need to be reformulated. The purpose of this paper is to work out three such counterexamples. Let a > 0 be small enough.

Theorem 1. There is a nonconstant harmonic function on the upper half space 1R~ C 1R3 which is Cl+a up to the boundary, and whose gradient vanishes on a boundary set ofpositive measure.

Theorem 2. There is a harmonic function u on 1R~ such that

(0.1) sup [ lV'u(x, tWdx\dx2 < 00, 1>0 JIR 2

and such that V'u does not have nontangentiallimits almost everywhere.

Theorem 3. There is a bounded domain in 1R3 whose harmonic measure puts no mass on any set with Hausdorff dimension less than 2 + a.

These results extend easily to IRn , n ::: 4. What they have in common, in addition to being counterexamples to the obvious generalizations of two-dimensional

'This research was supported by the National Science Foundation, under grant DMS 87-03456.


323COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN JR.3322 CHAPTER 15

inner functions on the ball in en and by Lewis [14] and myself [31] to constructtheorems, is that the two-dimensional results may all be proved using the subharmonicity of log lV'ul when u is a harmonic function on a domain in 1R2. For p-harmonic functions without almost everywhere boundary values. The method

itself goes back to Men'shov-the Men'shov correction theorem and the fact that Theorems 1 and 2 this is well known, e.g., if u: jR~ -+ jR and V'u vanishes on every measurable function is a.e. the sum of a trigonometric series. Consider (0.3). a boundary set E with positive measure, then log lV'ul is -00 on E and, being One shows first that there is no true obstruction to (0.3) such as the subharmonicity subharmonic, is identically -00. I will now give a brief discussion of Theorem 3, in the 2-dimensional case. This comes down to proving that if e is a unit vector in which motivated this paper. jR3, then there is a harmonic gradient V'u with suitable decay at 00 such that In [12], Jones and I showed that harmonic measure for any domain in jR2 puts

full mass on some set with Hausdorff dimension I. This completed a line of (0.4) { log Ie + V'uldx\dx2 < O.work due to Oksendal [19], [20] (who also made the conjecture), Kaufman-Wu JR2

[13], Carleson [6], and Makarov [15]. In particular, Makarov had proved the Once (0.4) is known, one sets up a recursive construction as follows. Let V' g\

same result for simply connected domains in [15, Theorem 3]. It is sharp for be any smooth harmonic gradient. On a small enough scale V'g\ is approximately

any smooth domain, so the apparent generalization to jRn would be that harmonic constant, so by adding on dilates of the functions V' u one obtains V' g2 with

measure always lives on a set with dimension n - 1. The relevance of subharmonicity of log lV'ul for Oksendal's problem is due { log!V'g2/dx\dx2 < { loglV'gIidx 1dx2 -I], I] > ofixed.

to Carleson [6] who used the following argument. Consider a scale invariant JlR2 JR2 boundary such as the von Koch snowflake. Harmonic measure at a fixed interior One then changes to a still smaller scale and repeats the procedure. (0.2) is point is then mutually absolutely continuous to an invariant measure for the scaling treated the same except that the modifications are made in the domain and the transformation on the boundary. In ergodic theory it is known that the dimension initial step is to find a perturbation, n, of the upper half-space with the property of such an invariant measure and its entropy are equivalent quantities. Computing that • the entropy turns out to be a matter of evaluating { IV' goo Ilog IV' goolda < 0,

Jan . 11(0.2) hm - lV'gnllog lV'gnlda, where goo (Green's function with pole at 00) satisfies goo(x) = 0 when x E ann--+oo nann and lim..--.oo goo(x) - X3 = O.

where nn is the nth stage in the construction of the snowflake and gn is the Green's Theorems I, 2, and 3 leave open a number of questions having to do with the function, and a is surface measure. Subharmonicity of log IV' gn I away from the size of the number a. Consider first Theorem 2. Stein and Weiss [25] found pole implies the integrals in (0.2) are bounded below in terms of an upper bound for the analogue of subharmonicity of log lV'ul in higher dimensions, namely that diam nn' So (0.2) is certainly nonnegative, which is what is needed to conclude IV'u !(n-2l!(n-\) is subharmonic if u is harmonic on a domain in jRn. They introduced

athat harmonic measure is at most I-dimensional. the space of functions satisfying (0.1) (now called the Stein-Weiss H space) as This "argument is reversible, so the main step in proving Theorem 3 is to find a a generalization of H p theory to jRn, and proved that functions in H a have a.e.

snowflake-type construction in jR3 for which (0.2) is negative. Theorems 1 and 2 boundary values if a ::: :=i. One can ask whether Theorem 2 can be pushed are similar. Instead of needing to construct a domain and its Green's function, we up to their critical exponent. The main obstacle to doing this by our method is have a fixed domain (the upper half-space) and we need a sequence of harmonic that one would need the analogue of (0.4), Le., for each p < :=i and unit vector functions Un with suitable normalization, such that e E jRn a harmonic gradient V' u on jR~ with rapid decay at 00 such that

(0.3) ( log lV'unldx\dx2 -+ -00. (0.5) { (Ie + V'ui P - I)dx\ ... dxn-l < O.JQOl JRn-1 Here Q(I) is the unit square in jR2 and jR2 is the boundary of the upper half-space. Such functions probably exist, although my argument for (0.4), which consists

The constructions for (0.2) and (0.3) are done by the same method. This method of explicitly writing down u and computing the relevant integral, does not show how to construct them. Similar remarks apply for questions like Theorem 2 where seems to be the canonical one for counterexamples involving boundary behavior;

e.g., it has been used by Aleksandrov, Hakim-Sibony, and Low (see [I]) to construct Riesz transforms are replaced by double Riesz transforms, etc., as in [5].


324 325 CHAPTER 15

One can also ask whether C 2 or Coo can replace Cl+a in Theorem 1. This has to do with the rate at which (0.3) can diverge and could be a delicate question. A C2 counterexample would lead (by taking one derivative) to a harmonic function u with u = ddu = 0 on a boundary set of positive measure, which is how the "'3 problem has sometimes been stated.

Regarding Theorem 3, Bourgain [3] showed that any harmonic measure in lRn

puts full mass on some set with dimension n - En where En > 0 depends only on n. It would be interesting if one could compute the best value of En. Both Bourgain's argument and the argument of this paper are too crude for that.

I want to mention some previous work related to Theorems I and 2. Uchiyama [26], [27], [28] gave a new method for studying singular integrals on LP, p ::::

I. His method does not use subharmonicity at all and applies to more general situations than harmonic gradients, but a special case of the result of [28] is another proof that Stein-Weiss H P functions have boundary values for p > 1 - E. In [27], he considered the small p case and gave counterexamples like Theorem 2 for related problems on discrete martingales. The proof of Theorems I and 2 uses some of the ideas of his method for the positive results in [26]; in particular, the functions satisfying (0.4) play the same role as his Lemma 2.3.

The question of whether there are uniqueness theorems for boundary sets of positive measure in higher dimensions was posed by BeTS. The previous work that has been done on it appears to be concerned mainly with local properties near one boundary point; cf. Mergelyan [18] and for the strongest known result, Rao [21].

Acknowledgments. I am grateful to J. Goodman for verifying Lemma 1.2 numerically. This work was done while I was at the Courant Institute.

Added September, 1991

Almost all current research in Euclidean harmonic analysis owes something to the work of E. M. Stein and this paper is quite clearly not an exception. I dedicate it to him with appreciation and respect.

Since it was first circulated in October 1987 there has been some further progress which I will now summarize. A. B. Aleksandrov and P. Kargaev [2] have given an alternate, less computational approach to the existence of the building block functions (0.4), (0.5). For given n 2': 3, let lR~ = {x E lRn : X > O} andn

denote by SH(lR~) the harmonic gradients Vu: lR~ ~ lRn vanishing at infinity, smooth up to the boundary and with boundary values in the Schwarz space. Then Aleksandrov and Kargaev show that for any p < n~2 and unit vector e E lRn

there is a harmonic gradient Vu E SH(lRn) such that (0.5) holds. They also give several additional applications of this construction including among others the

it-~"'~.7~;<>:Mh;;{~#.i~~i.i.'i~~i..;i>;~~~.$';'~~ii£<;~i".,;w~~';.,i;i~-,,=,,",;;,M;'~-iWI*1~.i'~imi~.~~.M:iQ~~';;W[~

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN ]R3

following: (1) given any continuous function f: lRn-

1 ~ lRn , n 2': 3, there is a harmonic function u on lR~Cl up to the boundary and such that Vu coincides with f on lRn outside a set with arbitrarily small measure; (2) for p < n~2, any LP function f: lRn - 1 ~ lRn coincides with the a.e. boundary values of some harmonic gradient Vu belonging to the Stein-Weiss HP space; (3) for p < n~l

there is a function in the Stein-Weiss H P space whose a.e. boundary values belong to a ray {e: r > OJ, for some e E lRn

•

J. Bourgain and the author [3] used the construction of Aleksandrov and Kargaev

to show also that there are harmonic functions u on lR~ C1 up to the boundary and such that u and ddu vanish on a common boundary set of positive measure. This

"'. answers one of the questions mentioned above. W. S. Wang [30] has proved further related results, in particular, that examples analogous to those of Bourgain and the author exist on any domain with a Cl+f boundary.

On the other hand the question ofhigher smoothness in Theorem I remains open and no further bounds have been proved on the dimension of harmonic measure. It is also still unknown whethe~(0.5) can be pushed all the way up to the Stein-Weiss

critical exponent :::::~ . I have made some changes in the preceding introduction mainly reflecting the

fact that in 1987 I was unaware of relevant previous work-Men'shov's classical work as well as [17], [21]. I have also corrected some misprints and minor inaccuracies in the subsequent sections but have not attempted a serious revision. It should therefore be pointed out that several arguments can be done more simply, and in particular, the appendix is now definitely unnecessary because of the work

of Aleksandrov and ~argaev.

I would like to take this opportunity to present the following observations which show that functions proving (0.5) will tend to depend strongly on p as p ~ :::::i. First we have the following "rigidity" result for the Stein-Weiss subharmonicity inequality.

Proposition A. Suppose u is harmonic on a domain Q C lRn• n 2': 3, and IVu I :=~

is also harmonic. Then u is either affine, or ofthe form blx - ej2-n + cfor some

b, c E lRand e E lRn.

Proof Since u is real analytic we may shrink Q and may therefore assume that Vu never vanishes. Ifwe examine the proofof the Stein-Weiss inequality [24], [25] we see that strict inequality must hold at a given point unless Vu is an eigenvector of the Hessian Hu and, furthermore, all eigenvalues other than the one corresponding to Vu are equal. Then in the present situation we conclude that there is a function A: Q ~ lR, such that (i) Hu(x)(Vu(x)) = A(X)VU(x)


327 326

___"'-"".'-~~'=""c:~_","_-=,=:'=-.:-:o",.- _ _. -- - ':;,..~i;,;;;;;;:-=•.- -~._._._. .,.-- ~ - ""'-.,; ..",-'" -_~i""'- ..··~_m'·'..":·,·, .'- -i<..."",-_ _X·~';"ii"d'ci'''':iiF;i;.i'C'''''''~~"",,"':,~.'''';'''''"'':.'·:''';·''''';;;';'·'''''''''''~'~~~''''~·'''~",;,<,;';~0>if;i;.O;;"';;""'''''~''''''''';;''''''-'''''''''''''~''''',,"~" '''''''''~~~:''''''-=-''''''"''''''''''''''''''''''''' ~

CHAPTER i5

(ii) Hu(x) r (VU(X)).1 = - n~l A(X) . id.

where id is the (n - 1) x (n - 1) identity matrix. Now (i) implies that IVul is constant on level sets of u, while (ii) implies that the level sets are umbilical (all principal Curvatures at any given point are equal). By [23] Theorem 26, the level sets are contained in planes or spheres. The proposition now follows by applying the uniqueness statement in the Cauchy-Kowalewski theorem on one of the level sets. •

A somewhat deer.er application of the same general principle is given in Theorem 2 of [8]. We now apply Proposition A to prove

Proposition B. Suppose that n ~ 3, e is a unit vector in IRn, {Vud c SH(IR~){pkl --+ :=i and JlRn-1 (Ie + VUkl Pk - l)dx < O. Then the only possible

L :=~ + L I limit point o/the sequence {Vud is zero.

Proof. Let p = :=i. We have

PLn-I (Ie + VUkl - l)dx = Ln_, (IV(x. e + UkW - l)dx

~ { xnt.(IV(x. e + ukW)dx.JRn +

The Laplacian is taken in the pointwise sense. It is well defined except at critical points and therefore a.e. The inequality is easily justified b1. a limiting argument starting from the smooth subhannonic functions (IV(x·tl~k!Jph€2)PI2 - 1 and using Fatou's lemma.

Suppose now that (Vud converges in LP + L I • Then the maximal function estimates in [25] imply {VUk} also converges unifonnly on compacts of IR~ and the limit function is a harmonic gradient Vu. Then using the LP + L I convergence and the fact that lie + Vulq - 11 ;S min(IVul, IVuIP) for q ~ p, we get

lim { (Ie + VUklPk - 1) = { (Ie + Vul P - I)k-+oo JRn~1 JlRn-1

= lim { (Ie + VUklP - 1).k-+oo JRn-1

In particular, limk-+oo J Ie + VUklP - 1) ~ O. On the other hand t.(IV(x . e + UkW) --+ t.(IV(x . e + u)IP) on IR~ except at critical points of x . e + u, and therefore a.e. By Fatou's lemma

( xnt.(IV(x· e + uW) ~ lim inf ( xnt.(IV(x. e + UkW)JIR~ k-+oo JR~

COUNTEREXAMPLES WiTH HARMONiC GRADIENTS IN ]R3

~ lim inf { (Ie + VUklP - 1)k-+oo JRn-1

~ O.

It follows that t.(IV(x . e + u) IP) must vanish identically, so Proposition A applies to x . e + u. Since Vu cannot be bounded below at infinity, the only possibility is that x . e + U = x . e + const and the proposition is proved. •

We remark that convergence to zero can take place only in a fairly weak sense, if at all. Variational calculations like those in Section 1 below imply that {Vu n }

cannot converge to zero in LOa.

1 BOUNDARY PROBLEMS FOR RIESZ SYSTEMS

We work in 1R3 and often write points of 1R3 as x = (x, X3) with x E 1R2 . The

upper half space is IR~ = ~ E 1R3 : X3 > 0). We write IR~ for its closure and

identify 1R2 with IR~ \IR~. Squares in 1R2 with sides parallel to the axes are denoted

Q, and Q(N) = [-! N, !N] x [-! N, !N]. The standard basis is el, e2, e3.

A hannonic gradient is a function /: IR~ --+ 1R3 which is the gradient of some harmonic function u, Qr else the boundary values of such an /. Both / and U

will always be assumed to vanish at 00. If X is a function space on 1R2, then

X H = {f: 1R2 --+ 1R3 : / is a harmonic gradient and the components of /

belong to Xl. Thus SH is the harmonic gradients whose components belong to the Schwarz space S, etc.

Before doing the constructions we summarize some properties ofSchwarz space harmonic gradients.

(i) Let /: 1R2 --+ 1R3 be a harmonic gradient such that h E S. Then / E SH

if and only if JXU hdx = 0 for all multiindices a = (ai, (2) of length ~ O. This follows by taking Fourier transforms: 11 = -i~I1~I-1 13 and 12 =-i~21~ I-I 13 are smooth at zero if 13 vanishes to infinite order there.

(ii) (a) SH is dense in L~, 1 < P < 00.

(b) For any k E Z+ there is e E Z+ making the following true: if / is a

harmonic gradient such that h is a smooth function with compact support and with JXU / (x )dx = 0 when la I ~ e, then for any E > 0, there is g E SH such that sUPxER2(1 + Ixl)kl/(x) - g(x)1 < E.

I believe these facts to be well known but will include a proof. The proof in my 1987 manuscript was incorrect; I thank W. S. Wang, who very tactfully pointed this out.

Let 1/1 be a Schwarz function on 1R2 such that ;j, has compact support and;j, = 1 in a neighborhood of O. Let 1/Ir (x) = t-21/1 (t- I x). To prove


I

328 CHAPTER 15

(a) it suffices to approximate functions I E L~ such that h E S, since such functions are dense in L~ by boundedness of the Riesz transfonns. However, if h E S, then by (i) and the fact that.(fJ = 1 near 0, we will have

1-1/11 * I E SH. Moreover 1/11 * I --+ 0 as t --+ 00 by Young's inequality since lEU for 1 < q < p.

To prove (b) note first that the moment assumption implies /I (x) I ;S (1 + Ixl)-C£+2). It therefore suffices to show that if f XU I(x)dx = O(lal :::: l)

and I/(x)1 ;S (l + /xi)-C£+2) then sUPxEIR2(1 + Ixl)kl1/l, * l(x)1 --+ 0 as t --+ 00. This is done as follows: Fix x and t ~ 1. On the one hand, since 1/IES

2 Ix - yl )-C£+2)11/11 * l(x)1 ;S t- (l + lyi)-CH2) 1 + -t- dxf (

Ix I ) -C£+2)< ( 1+ - . '" t

On the other hand let p be the degree l - 2 Taylor polynomial of 1/1 at 'i-; then

2/1/11 * l(x)1 = t- f I(Y)(1/I(X ~ Y) - p(X ~ Y))dY )1

;S t-2 f I/(Y)II f If-Idy

< t-CHI )'" . Taking l = 2k we obtain

11/11 * l(x)l:::: ((1 + 1~lrC2k+2)t-C2k+I)Y/2

:::: t- I / 2(t + Ixl)-k,

which suffices.

(iii) If g E SH then g is rapidly decreasing on the upper half space, i.e., Yk3Ck : Ig(x)1 :::: Ck(l + Ixl)-k, x E lR~.

One expands the Poisson kernel PJ as a Taylor series at 0 with remainder, scales to obtain an expansion of PI for t large, and uses (i).

(iv) Hannonic gradients g E SH such that g =f. 0 on Q(2) are dense in SH.

Choose finitely many ¢I ... ¢n E SH such that (¢j(x)} spans lR3 for each x E Q(2). This can be done by compactness since (x: ¢j(x) do not span}

is closed and clearly SH(X) spans for each fixed x. If I E SH is given, then let 0 be small and consider

II + EO j ¢jl-2doJ ... dOndxldx2.1 [Q(2) J~ ~<8j <~

~il;'t·'W m ' !nC:fiHI!.6C"iEll"~' '. rrh~fiij.hll~Mi T f~!Ii!illi

COUNTEREXAMPLES WlTH HARMON1C GRAD1ENTS 1N 1R3 329

The inner integral is finite for each fixed x since the zero set of f (x) + Eo j ¢ j (x) has codimension 3, and is bounded by a compactness argument on the minors of the matrices (¢j(x)}. On the other hand, if f + EOj¢j has a

zero for a certain 01, ... , On then fQ(2) II + EOj¢jl-2dxJdx2 = 00. If this happened for all choices of the o's, Fubini's theorem would be contradicted.

The first step in proving Theorems 1 and 2 is to distinguish the 3-dimensional

case from the 2-dimensional case by means of the following fact.

Lemma 1.1. For each unit vector e E lR3 there is q E SH such that Iri2 log Ie + q(x)ldx < O. We can choose q so that e + q never vanishes on lR2

.

It is natural to try to prove Lemma 1.1 by perturbing off the case q = 0, and

we now look where this leads. Fix q E SH and let I(t) = fIR2 log Ie + tqldx. By calculations which are justified (for t small) by the rapid decay at 00,

d/ = [ (e + tq, q) dx dt JlR2 Ie + tq 12

d2I = [ Iql21e + tql2 - 2(q, e + tq)2 dx. dt 2 JIR2 le+tql4

Taking t = 0,

dI I = [ (e,q)dx = 0, dt 1=0 JlR2 2

d I I (lql2 - 2(e, q)2)dx. -2 = i dt 1=0 ]R2

If e = e3 (or -e3) it follows, e.g., using the Fourier transfonn that ~~t 11=0 = 0 for any q E SH. But if e has a nonzero tangential component, then for similar

reasons ~ 11=0 > 0 for any q E SH, i.e., 0 is a local minimum.

ProofofLemma 1.1 when e = ±e3 We can assume e = e3. Differentiating once more,

3

d I I 2)i ( 3-3 = 8q3 - 6q31ql dx dt 1=0 lR2

= 2 [ (qj - 3q3(q~ + qi»)dx.JlR2

Temporarily drop the requirement that q E SH and try q V¢, ¢(x) -Ix + e31- 1• Then

xf + xiq3 = Ix + e31-3• qf + qi = Ix + e316


331 330 CHAPTER 15

2L2 (qj - 3q3(qr + q;) )dx = 2100

(r 2 + 0-9/ - 3r2(r 2 + 0-9

/ 2)rdr

2Jr = -- < O.

35

Approximating this q in L 3 nonn by one in SH we obtain q E SH with ~ /1=0 <

O. Then for small positive t, fn~.2 log Ie + tq I < 0 and e + tq has no zeros.

ProofofLemma 1.1 when e has nonzero tangential component We can assume e = (a, 0, .JI=(2),- 0 < a ~ 1. Again we temporarily drop the requirement that q E S. To emphasize the +el-direction it is natural to consider q = Y' / where in polar coordinates on 1R2

, /(XI, X2, 0) = ¢J(r) cosO. Next observe that if ¢J(r) = -ar for r < I, then (e + q)1 = (e + qh = 0 when r < 1. Having defined ¢J this way for r < I we decide to extend it to 1R2 keeping the L 2 nonn of q as small as possible. That leads to the choice q = a Y'g, where g is the hannonic function with boundary values

-r cosO, ifr I

We then need to show

def 1 rLemma 1.2. 1(a) = 2JT JlR2(log Ie + ql - (e, q})dx < O/or each a E (0, I].

The integral is proved absolutely convergent at 00 by expanding the logarithm to second order and using that q E L 2. It is easily seen to be absolutely convergent locally, e.g., this follows from fonnulas (A 1) and (A2) and statements A3.1 and A3.5 in the appendix to this paper.

Lemma 1.2 could in principle have been proved by computer work and error analysis, but I decided to treat it as a calculus exercise instead. This argument is messy and is therefore postponed to the appendix. It turns out that ifone uses polar coordinates the O-integral can be evaluated explicitly in tenns of certain elliptic integrals (cf. Lemma AI.I and the subsequent remark). Jonathan Goodman offered to integrate the resulting function of r numerically, and found that 1 is decreasing in a with 1 -+ 0 as a -+ 0 and 1(1) ::::::: -0.28, so the reader can believe the lemma without going through the technicalities in the last three sections of the appendix.

We now show how to obtain Lemma 1.1 from Lemma 1.2. If e = (a, 0, ~), andq isas in Lemma I, thenq E L2 and (e + qh is everywhere nonnegative. So by first convolving q with a strictly positive approximate identity and then approximating the resulting harmonic gradient by one in the Schwarz

3COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R

space, we obtain qn E SH such that qn -+ q in L~ and e + qn never vanishes. By

(i), Jqn = O. Also

log Ie + qnl - (e, qn) ~ C1qnI 2 •

Sinceqn -+ qinL2andlogle+ql-(e,q} E L1,wecanpasstoasubsequence

and assume that

log Ie + qnl - (e, qn) - (log Ie + ql - (e, q}) ~ /,

with / ELI. By Fatou's lemma,

lim sup { (log Ie + qn I - (e, qn) )dx ~ { (log Ie + ql - {e, q} )dx < O. n->oo JlR2 JlR2

Since J(e, qn) = ONn will satisfy the conditions of Lemma I when n is large

enough. •

Lemma 1.1 leads immediately to it refinement ofitselfwhere instead ofa constant

vector e we consider a constant vector plus a small error.

Lemma 1.3. There are finitely many harmonic gradients qj E SH and numbers K > 0, P > 0,77 > 0 making the/ollowing true. 1/ N is sufficiently large, then/or each v E 1R3 there is j such that if g: Q(N) -+ 1R3 and IIg - vll oo < KN-2Ivl, then I

plgl ~ Ig + Ivlqjl ~ p-llgl on Q(N),

1 log Ig + Ivlqjl ~ -11 + 1 log Igl· Q(N) Q(N)

Proof. Lemma 1.1 and a compactness argument show there are finitely many qj and numbers p, 11 such that if e is any unit vector then for some j,

2p ~ Ie + qjl ~ (2p)-1

(1.1) ( log Ie + qj I ~ -311·JlR2

Replacing 377 by 277, the integral in (1.1) may be taken over Q(N) provided N is large enough. When v is a unit vector, the lemma follows by easy estimates with absolutely convergent integrals. The general case follows by properties of

the logarithm function. •

We now set up a recursive procedure with the basic step provided by Lemma 1.3,

which will prove Theorem 1 and the following result:


333 332 CHAPTER 15

Lemma 1.4. Suppose p > 0 is small enough, and let k = kp < 00 be large

enough. Then for given E > 0 there is a harmonic gradient g which is continu-3

ous on lR+ and satisfies: Ig(x)1 < Elx\-k when Ixl > 1, flR2g(X)XadXldx2 = o for all a = (aI, a2) with lal ~ k. SUPt fJR2 Ig(X. tWdxldx2 ~ C, UJR2Ig(X,OWdxldx2)I/p < E, and I{x E lR2 : Ig(X. a)1 ::: C-I}I ::: c- I fora

certain a = a€ > O.

Here C is independent of E. The last two properties are the significant ones. Theorem 2 follows easily from Lemma 1.4 as we explain at the end of the section.

Choose three large -enough constants A, B. N in that order. They depend on

properties of the functions q j and will be kept fixed throughout the proofs. Next

choose an initial harmonic gradient gl E SH with IgII :f= 0 on Q(2), and then a large number A and, finally, a small number 8 > 0 with 8-1 E Z. Constants will

initially be independent of A, B, N. gl, A. 8.

Let Gn be the grid on Q (1) consisting of squares with side 8n. For each Q E Gn,

let aQ be its center point. For each j, let

qf(x) = qj(N8-n(x - aQ))

with qj as in Lemma 1.3.

At stage n (n ::: 2) we will have constructed a harmonic gradient gn-I E SH

obeying the following estimates when x E Q(2), Y E Q(2) x [0, 1].

(1.2) Ign-I (Y)! ~ A max(l, r(n-I)lx - yJ)lgn-l(x)l.

If Ix - YI < 8n- l , then

(1.3) Ign-I(Y) - gn-l(x)1 ~ BNr(n-I)lgn_l(x)llx - YI.

We will also have a subset An-I C Gn- I, called nonstopped squares at stage n - 1. Take gl as above and take A1 to be those squares in G1 which do not touch the boundary of Q(l). Then (1.2), (1.3) are satisfied when n = 2, provided 8 is small enough.

We now define An and gn' A square Q E Gn belongs to An, provided every Q' E Gn- 1 which contains or touches Q belongs to An-I and, furthennore,

(1.4) mgx 10g(lgn-i1/lgt!) ~ - ¥N-2n + A(n log n)I/2,

with 1/ as in Lemma 1.3. And

gn = gn-I + L IVQlqf<vQ) QEA.

where vQ = gn-I (aQ) and j(vQ) is the index provided by Lemma 1.3. We now make some estimates and show in particular that (1.2), (1.3) are again satisfied.

COUNTEREXAMPLES WfTH HARMON1C GRADfENTS fN JR3

Claim. For any d ::: 4on and Y E Q(2) x [0, 1],

(1.5) L IVQllq~vQ/Y)1 ~ CAN-482n lgn_ 1CY)lr l max(rl , 8-(n-I») QEA.

ly-aQI~d

(1.6) L IVQIIVqf<vQ)(Y)1 ~ CAN-482nlgn_1CY)lr2 max(r l . r(n-I»)

QEA. ly-aQI~d

Proofof(1.5) Sinceqj E SH we have Iqf<vQ)(Y)1 ~ CN-484nly_aQI-4. Using

this fact and property (1.2) of gn-], the sum in (1.5) is

4 4n~ CA L max(l, r(n-I)IY - aQDlgn-ICY)lly - aQI-4N- 8QEA.

ly-aQI~d

~ CAN-484nlgn_lCY)I( L Iy - aQI-4 + L r(n-I)IY - aQI-3).

a·-1>ly-aQI>d ly-aQI~m8Jl(d.a·-I)

Since the sums are comparable to integrals, the expression in brackets is

1

ICr2nd-2, ifd < 8n

~ C8-2nr(n-l)d-1• ifd::: 8n -1

and (1.5) follows. For (1.6), use IVqf<vQ) (Y)I ~ CN-48-4n IY - aQI-5 and argue

the same way. •

Let

gn(Y) = gn-l(Y) + L IVQlqf<vQ)(Y)' y¢Q QEA.

Taking d = 48n and using (1.5),

4(1.7) 18n(Y) - gn-l(y)1 ~ CAN- Ign_Iwl.

when Y E Q(2) x [0, 1]. We make N large enough that C AN-4 < ~ KN-2 with C as in (1.7) and K as in Lemma 1.3. Now restrict temporarily to x = x E JR.2. If x E Q E An then

jgn-I(X) - gn-l(aQ)1 ~ BN8Ign_l(aQ)1


----------.------~".-----"-_--.-.-- -,;:",,,,,,-,-,--=---.---.-:......!,.;-'----------,...~'";;--.,','i=_~..=,'~~~~,;;;;:~,.-------------.",- ..-;;----;:;..-- -----------··;;;----;;;-·,-·-----:"",""";,·';::,;....r~-----,----------,-.-- .."

334 CHAPTER 15

by (1.3),andwe take 8 small enough that BN8 < ~KN-2. Then x E Q E An implies

Ign(x) - gn-l(aQ)1 .:s [gn(x) - gn-l(x)1 + Ign-I(x) - gn-l(aQ)1

.:s 3I KN-2 (Ign-l(x)1 + Ign-l(aQ)1)

.:s ~KN-2(2[gn_l(aQ)1 + Ign-I(x) - gn-l(aQ)J)

2< KN- Ign_l(aQ)I.

Therefore, Lemma 1.3 applies to g = gn and qj = qj~vQ)' after scaling by N8-n,

and we get p-llgn I ::: Ign I ::: plgn I on U(Q: Q E An). Using that gn = gn on Q(2)\ U {Q: Q E An}, (1.7) implies

(1.8) 2p-I Ign-Ii ::: Ign I ::: 2I plgn-II,

on Q(2). Moreover if Q E An then

2n~ log Ign I .:s -T/8 N-2 + ~ log Ign I

2n.:s -T/8 N-2 + ~ log [gn-II + CAN-682n

by (1.7). So

2(1.9) ~ log Ignl .:s - ~ T/N- 8

2n + ~ log Ign-Ii,

provided N is large enough.

Next we prove (1.2) for gn' Observe that (1.3) for gn-I implies

1(1.10) Ign-l(y)1 .:s (I + BN8 /2)/gn_l(x)1 if Ix - yl < 8n- I /2.

Now ify E Q E An (the case where y ¢ U{Q: Q E An} is slightly easier),

Ign(Y) - gn-l(y)1 .:s Ign(Y) - gn(y)1 + Ign(Y) - gn-l(y)1

.:s C1gn-l(aQ)1 + Ign(Y) - gn-l(y)1

.:s C(I + BN81/2)lgn_dy)1 + CAN-4 Ign_ICy)I,

using (1.7), (UD), so if N is large and 8 small,

Ign(y)1 .:s C(lgn-ICY)I + Ign-I(Y)I)

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3 335

and by (1.8),

Ign(y)1 < Co Ign-l(y)1 + Ign-l(y)1 .

Ign(x)1 - Ign-l(x)1

If Ix - yl > 8n- I /2, (1.2) for gn now follows from (1.2) for gn-I provided 8 is small enough. If Ix - yl .:s 8n- I /2, (1.2) for gn follows from (1.10) by choosing

8 small enough, provided A > 2Co.

To prove (1.3) for gn, write

Ign(x) - gn(y)1 .:s Ign-I(x) - gn-l(y)1 + I(gn - gn-I)(X) - (gn - gn-I)(y) I·

The first term is .:s BN8-(n-1) Ix - yllgn-l(x)l. To bound the second suppose

first that x and y belong to the same Q E An. Then.. . J(gn - gn-d(x) - (gn - gn-d(y)1 .:s Ign-l(aQ)llq~vQ)(x) - q~vQ)(Y)1

+I(gn - gn-I)(X) - (gn - gn-I)(y)l·

The first term is .:s C1gn_l(aQ)IN8-nlx - yl where C depends on Lipschitz

bounds for the qj' The second term is .:s CAN-4 Ix - yl8-n sUPQ Ign-d by (1.6)

(with d = ~ 8n) at the points of a straight line path from x to y and the mean value

theorem. By (1.2) for gn-I,

I(gn - gn-I)(X) - (gn - gn-d(y)1 .:s CAN8-nlx - yllgn-l(x)1

if N is large. If neither x nor y belongs to any Q E An' the same estimate is valid and slightly easier. For general x, y with Ix - y I < 8n , draw a straight line from

x to y and then from y to y. That gives

l(gn-gn-l)(x)-(gn-gn-I)(y)l.:s CANo-nlx-yl max Ign-l(tX+(I-t)y)I, tE[O,I)

and the right side is .:s CA2N8-nlx - Yllgn_l(x))lby(1.2). So

Ign(x) - gn(y)1 .:s (BNo-(n-l) + CA2N8-n)lx - yllgn-l(x)1

.:s 2p-I(BNo-(n-l) + CA2No-n)lx - yllgn(x)I,

which proves (1.3) for gn provided B > 2Cp-1 A 2 and 8 < ~ pB- I (B

2Cp-1 A 2).

That means that the construction makes sense, and now we show that it con

verges. From here on, A, B, N are fixed. Constants can depend on them. Let

E = nn U{Q: Q E An}. Let r: Q(2) --+ Z+ U (00) be the stopping time in the

construction: rex) = 00 if x E E and otherwise rex) = min(n: x ¢ Q E An).

We use the usual notation gr(x) = gr(x)(x).


337 336 CHAPTER 15

-3 Lemma 1.5. The gk converge uniformly on lR+ to a harmonic gradient g. This g is Holder continuous with g = 0 on E and C-llg~ I s Igl s C1g~ I on Q(2)\E. If a > 0, E > 0, k < 00 are given then the following estimates hold if 8 is small enough: Ig(x) - gl(x)1 < Elxl-k when Ixl > 1 and Ig(x) - gl(x)1 < E when x E Q(2) x [a, 1].

Proof. Fix x E Q(2)\E. If k 2: r(x) + 2 and Q E Ak then Ix - aQ) 2: 8~(x)+1

by the rule about touching squares in the definition of Ak. By (1.5), with d = 8~+1 2: 8k - 1,

Igk(X) - gk-l(X)1 s C1gk_l(x)18k-~.

Calculus then shows Igk-l (x) I S 21gHI (x)j for k 2: r + 2 if 8 is small. By (1.8),

Igk - gk-Ji s C8k-~lg~1 whenk 2: r + 1. This implies C-llg~1 S Igkl s Clg~1 whenk 2: r,andthat{gk(x)} converges uniformly on {x E Q(2): r(x) s n}for

any given n. However if k S r(x) then the stopping rule (1.4) implies Igk(X)1 s Ce-ak (here a and C dependongl and),,), so it follows that {gd converge uniformly

on Q(2), and clearly the limit satisfies g = 0 on E and C-llg~I s Igj s C1g~ I

on Q(2)\E. To estimate when Ixl > 1note the preceding argument gives a bound

IIgj IIL""(Q(2» S M where M = M(gl, ),,) is independent of 8 for 8 small enough. Using that qj E SH and that there are at most 8-2j squares in A j,

(1.11) Igj(x) - gj-l(x)1 S Ckr 2j (r j lxl)-kM,

for any given k if Ixl > 1. This implies uniform convergence on 1R2\Q(2) (hence

on IR~) and also the estimate Ig - gJi < Elxl-k when Ixl > 1. The estimate

Ig - gJi < E on Q(2) x [a, 1] also follows in this way; replace Ixl by a on

the right side of (1.11). We prove the Holder continuity on Q(2) only since on

1R2\Q(2) it follows easily from (1.6). In this argument, constants depend on all

parameters including 8. By the stopping rule (1.4) we know Igki S c,l for

some '7 < 1, when k < r, and therefore Igi S C'7~. Also (1.3) together with

boundedness of the gk implies IVgk I s C8-k. If j 2: r (x) + 2, then using (1.6) and the rule about touching squares, IV(gj - gj_l)(x) I s C8j-2~(x). Taking

k = r(x) + 1 and summing a series gives IVgl S C8-~, i.e., IVgl S C1gl-W when g ::j:. 0, where w = (log 8) (log '7)-1. This estimate implies g is Holder with exponent (1 + w)-I. •

Lemma 1.6. Fix t E (0, 1]. If t > 8, then Ig(X, t)1 S C(lgl (x, 01 + Igl (x)l)for all x E Q(2). If t < 8, then Ig(X, t)1 S C1gkWI where k is the largest number with 8k 2: t.

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3

Proof. Wedothet < 8 case, leaving the other to the reader. By (1.2), !gk(X, t)1 s C1gk(x)l. By (1.5), (1.8), if j 2: k + 1 then

Igj(X, t) - gj-I (X, t)1 S C1gj-l (X)I82j -k-l-max(k+1.j-l)

j k 1S C1gj-l (X)18 -

28 )j-k-lS C ( P Igk(X)I,

and the result follows if 8 < t p. • We still have to show why g can be taken to have the special properties in

Theorem 1 and Lemma 1.4. Basically, this is because of (1.9). Let r 1\ k

min(r, k), and if Q is·a square, ~ a function, let Q(<p) = IQI- 1 fQ <p.

Lemma 1.7. On Q(2), log Igkl Slog IgJi - t N-2'7r 1\ k + hk where hk satisfies

1 ephk _ < C1eClnp2 xEQ(2) ; ~ I\k=n

for all n E Z+, p E (0,00).

Proof. Let <Pn = log Ignl - log Ign-Ji. By (1.3), log Ignl and, therefore, <Pn are Lipschitz with Lipschitz norm S C8-n. Also Q(<Pn) S - tN-2'7 for each Q E

An by (1.9). Using a partition of unity, write <Pn = t/Jn + ~n where ~n S - t N-2'7 onU{Q: Q E An}andQ(t/Jn) = OforeachQ E An,andlit/JnIlLip(l) S C8-n.

For n 2: 2 define fn: U{Q: Q E An} -+ IR by fn = L:J=1 t/Jj. The t/Jj are weakly dependent and we can estimate fn by a standard argument from elementary

probability. (See the proof of Khinchin's inequality in the appendix of [24].) Fix cp2Q E An. If Q(t/J) = 0 then Q(e PVr ) S e where C depends on 1It/J1l00' Since

t/Jn + fn~1 - Q(fn-d has mean zero on Q and Lipschitz norm S C8n, hence

sup norm S C,

Q(eP!n) = ePQUn-l) Q(eP(Vrn+fn-1-Q(!n-l»)

S epQUn-lleCp2

S ecp2 Q(eP!n-,)

{ eP!n < ecp2 { eP!n-1 ]u/Q: QEAnl ]u/Q; QEAnl

< ecp2 { eP!n-l. ]U(Q: QEAn-d


339 338 CHAPTERi5

Iterating n - 1 times,

(1.12) eP!, ~ eC(n-l)p2.1U[Q: QEA.}

Now we have, using Lemma 1.5,

log Igkl ~ log IgrAkl + C2

= log Igil + L ¢>j + C2 2::oJ~rAk

~ log Igil - 2"1 N-

2 Tl(. 1\ k - 1) + frAk + C2.

The set {x: • 1\ k = nl is contained in U{Q: Q E An-d, so the lemma

follows from (1.12) if we take hk = frAk + 4 N-2Tl + C2. •

ProofofTheorem 1 Fix gl E SH with gl # 0 on Q(2). We claim that lEI> 0 if ).. is large and 0 small. For n 2: 2, let Fn be the union of all squares such that

Q C Q' E An-I and (1.4) fails on Q at stage n. Then. 1\ n = non Fn. Also, if Q is one of the squares comprising Fn then hn- 1 2: )..(n log n)1/2 somewhere on

Q by the fact that (1.4) fails. So hn- I 2: )..(n log n)1/2 - C everywhere on Q by

(1.3), and then by (1.8), hn 2: )..(n log n)1/2 - Cion Q. SO hn 2: 4 )..(n log n)1/2

on Fn provided)" > 2CI(2 log 2) -1/2 . Lemma 1.7 implies

IF, I < C eC2p2n- 4Ap(n log n)I/2n _ 2 ,

for any p. Taking p = (4CI )-I)..n-1/2 (1ogn)1/2 gives

-_IA2IF,n I ~ CIn 16CI •

Accordingly 1; IFn I may be made arbitrarily small by taking).. large. If x E

Q (1) \ E, then x belongs either to the double of one of the squares used to define

one of the Fn , or to the double of one of the squares in G 1/A h so

IQI\EI ~ 4(LlFnl +40), n:;:2

and the proof is finished. • ProofofLemma 1.4 We need p < (2C I) -I N-2Tl with C I as in Lemma 1.7. Let

k be large enough and fix E > O. We need gl to satisfy Igil < lO(4Ixl)-k when

Ixl > !, JJR2 xagt(x) = 0 when lal ~ k,lgll ~ C and I{x E Q(1): IgI(x) > C- III > C- 1• C as well as the constants appearing below may depend on k, but

not on lO. Such a gl is easily constructed using "atoms": let e be large and fix a

smoothharmonicgradient¢>with¢>Jsupportedinlxl < ! andJ xa¢>dx = owhen

COUNTEREXAMPLES WiTH HARMONIC GRADIENTS IN]R3

lal ~ e, and with I{x: 1¢>3(X)1 > Clill > Cll. This is simply the existence of compact support functions with many vanishing moments. Let 1/1 be a Schwarz

space approximation to ¢> obtained by convolution as in (ii). We now modify 1/1 so as to obtain the first mentioned property of gl. For a given E, let t be small enough

and hex) = L 1/1(t- I x - z) where the sum is over lattice points z E 71} with z Iz I < (8t) -I. The rapid decay of 1/1 implies that Ih I ~ C, and (1 + Ix I)k 11/13 (x) I may be made arbitrarily small when Ixl > ~ as described in (ii). So there is no

significant cancellation among the different functions 1/13 (t -I X - z) and it follows

thatl{x: Ih(x)l2: 4Cll}12: 4Cll. Iflxl2: !,thenlt-Ix-zl > 4t-I lx lfor all z, so Ih(x)1 ~ C3t-2( 4 t-Ilx 1)-(k+2) which is < E(4Ixl)-k if t is small enough.

To obtain gl, approximate h by·a harmonic gradient without zeros on Q(2) using

(iv). • Let no be a large enough integer. By taking ).. very large we can guarantee

that if .(x) ~ no then x E Q(2)\Q(1) or x belongs to the double of one of

the squares in G I\Al. In either case, Ixl 2: !. Let a > 0 be such that I{x E

Q(1): IgI(X, a)1 > C-I}I > C- I and make 0 smallenoughthatthe last statement

of Lemma 1.5 is valid with this a. If g is as produced by the construction, then

I{x: Ig(x,a)1 > 4C-lll > 4C-I. Moreover,

IglP f Ig/PJ IglP = L f IgIIP- p + L IgII P pQ(2) n~no r(x)=n Igil n>no r(x)=n Igil

< lOP L IglP + C L IglPf f - n~no r(x)=n IgIIP n>no r(x)=n IgIIP

By Lemma 1.7,

IglPfr(x)=n IgIIP ~ Cle-pn~/2N2 eCIP2n

= Cle- fJn ,

where fi > 0 since p < Tl/2CIN2. SO JQ(2) IglP < CEP if no is large enough. If o < t ~ I, then choosing k as in Lemma 1.6,

J Ig(X, t)IP ~ C + C J Igk(X)IPQ(2) Q(2)

< C + C J Igk(X)IP - Q(2) IgI(x)IP

fJn~ C + C + C L en

~ C.


341 340 CHAPTER 15

The next to last line here follows from Lemma 1.7 by conditioning on 'r /\ k. Estimates when Ix I > I follow from Lemma 1.5 if 0 is small enough, and the

vanishing moment condition is built into the construction. •

ProofofTheorem 2 This is a very standard argument; probably it is well known that Lemma 1.4 implies Theorem 2.

It g is a harmonic gradient, then we call the function whose gradient is g the

primitive of g and denote it Ug' The example will have the following additional properties which willpe used in extending to higher dimensions: u g is continuous

-3 on ~+ and C I for large Ix I, and for suitable l, fIR2 x a Ug (x) = 0 when laI ~ l and g and Ug vanish at 00 faster than Ix I-t .

Fix k large enough. Let gE be the function in Lemma 1.4. Note that ME ~ lIug,lloo and IUg,1 ~ CElxl l - k forlarge lxi, and

(L2 IgE(X, tWdxldX2) lip < E unless t E ('rE, 2),

with 'rE > O. Let

gE,N(X) = L gE(Nx - m) m

where m runs over all points (2z, 0) with Z E ~} and Izi ~ ! N.

Consider Lm gE(Nx - m) where we now restrict to points m as above with

INx - ml > 1. In view of the estimate IgE(X)1 ~ Elxl-k when Ixl > 1, this sum will be ~ CE for all x and in fact ~ CEd-(k-2), where d = d(x) = minm(INx - ml: INx - ml > 1). It now follows that

(1.13) Ifx: IgE,N(X, ~)I > C;I}I > C;I.

with UE = U as in Lemma 1.4. Also,

f !gE,N(X, tWdxl dx2 ~ L f !gE(Nx - m, NtWdx l dx2 m

which is ~ C for all t and ~ CEP if t fj. (~, ~).

Let UE,N be the primitive of gE,N. Since UEvanishes rapidly at 00, an argument like the one for (1.12) bounds lIu E,N 1100; due to the different scaling we obtain

lIuE,Nlloo ~ CMdN,

Now let {Ej}'f and fNj}'f converge rapidly enough to 0 and 00 respectively. Specifically, let B be a large constant. We require

Ej < B-j.

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

The intervals ( :;;, N2 ) are disjoint ] ]

'"L.J ME·(1.14) __I < 00. Nj

These are obtained recursively by choosing Ej < B-j and then N j > jmax(2 ME]' 2Nj_1 'rE~~I)' Let g = Lj gE],Nr Property (1.14) implies that the

series converges unifonnly on compact sets in lR.~ and the limit has a primitive

which extends continuously to R~. For any t,

f Ig(X,!W ~ L f IgEj,Nj(X, tW

~ C,

Isince at most one tenn in the sum is ~ B-jp. Moreover,ift fj. Uj('rEj N j- , 2Nj l),

then

f Ig(X, tW < L B-jp,

which can be made arbitrarily small by taking B large. In particular, we can guarantee that for such t

1 1 I{x: Ig(X,t)1 > '3 C;I}1 < '3C31,

with C3 as in (1.13). In the same way, we guarantee that for t E ('rEjNjl, 2Nj l),

1 1 Ifx: Ig(X,t) -gE],Nj(X,t)1 > '3C31}1 < '3C31,

and, therefore,

2 2 Hx: Ig(X, NjIUE)1 > '3C31}1 > '3 C31 .

It follows that there is a set of x with measure at least ~ C3 1 on which

lim SUPHO Ig(X, t)1 > ~ C;I and lim infHo Ig(X, t)1 < ~ C;I, so Theorem 2 is proved. From the construction, U g has as many zero moments as we like, and it is

easily checked that it is C I for large Ixl and that g and ug die faster than a given power of lxi-I. •

The extension to higher dimensions is achieved by adding dummy variables. For Theorem 1 this may be done in a trivial way, but for Theorem 2 an argument

is required. Denote variables in i:(n ~ 4) by z = (x, y, t), x E lR.2, Y E 3lR.n-3, t ~ O. Let y,: lR.n- ~ lR. be a smooth function with y,(y) = 1 when

IYI ~ 1, y,(y) = 0 when Iyl ~ 2, and with sufficiently many zero moments.


343 342

CHAPTER 15

Let u = ug where g is as in Theorem 2, and let F be the harmonic extension of 1{r(y)u(x, 0) to ~~. By the reflection principle, F(x, y, t) - u(x, t) extends smoothly across the set{(x, y, 0): Iy) S I} C ~n-l, so VF doesn't have radial limits almost everywhere. We will show that VF is in the Stein-Weiss H P space. Since ¢(y)u(x, 0) is C 1 for large Ixl + IYI and rapidly vanishing at 00 and has lots of zero moments, it is enough to show

sup r IV F(x, y, tWdxdy < 00 0</<1 Jmax(lxl,IYI)<R

for fixed R. Fix Yo and write

1{r(y)u(x,O) = q(y)u(x, 0) + (1{r(y) - q(y»u(x, 0)

where q(y) = 1{r(yo) + (V1{r(yo), y - Yo). The function Gyo(x, y, t) q(y)u(x, t) is harmonic (compute its Laplacian) and

1 IVGYo(x, Yo, tWdx S ~3 7rR3I1ull~IV1{r(yoW Ixl<R

+ 11{r(yoW 1 lu(x, tWdx S C Ixl<R

uniformly in Yo· Let Hyo(x, y, t) be the harmonic extension of (1{r(y)

q(y»u(x, 0). Since 1(1{r(y) - q(y»u(x, 0)1 s Cjy - YOl2 locally and S k

CjYllxl- at 00, the formUla for the derivatives of the Poisson kemel applies and shows that IV'Hyo(x, Yo, t)1 is bounded. We conclude

r IVF(x,y,tWdxdy Jrnax(lxl,IYI)<R

"1(1[IVG,(x, y, IW + IVB,(x, y, IW]dX) dy

sf Cdy

< 00.

2 DIMENSION OF HARMONIC MEASURES

The purpose of this section is to prove Theorem 3. We will do the main construction in somewhat greater generality than is needed for this since it doesn't require any extra effort and tends to show that there is no reason at all for a harmonic measure in ~3 to be 2-dimensional. As explained in the introduction, ideas of Carleson [6]


show that for scale invariant boundaries the dimension of the harmonic measure should be determined by the quantity

(2.1) lim ~ r IVgnlloglVgnlda n-->oo n Jan'

where nn is a natural smooth (enough) approximation to our domain n and gn

is its Green's function with fixed pole p. (We normalize Green's function to be

positive and satisfy IVg I = ~~ where (J is surface measure and w is harmonic measure at p.) In particular, in ~3 the sign of (2.1) determines whether w is > 2or < 2-dimensional. In contrast to the planar case considered in [6], the sign of (2.1) may be chosen freely by adjusting the parameters of our construction.

The situation is similar to section I in,that once we know how to "improve a constant" in a suitable se'hse, we can set up an iterative procedure to construct fractill boundaries with control over (2.1). Now the role of a constant is played by the upper half space. Suppose ¢: ~2 ~ lR is a Lipschitz function with supp ¢ C Q(l). Let U = {x E ~3: X3 > ¢(X)}. Then there is a unique harmonic function goo: U ~ ~+ ("Green's function with pole at 00") such that goo vanishes on au and limx--> 00 (goo (x) - X3) = O. One shows that for x E lR2,

IVg(x)1 = I + O(lxl-3) as Ixl ~ 00. This is done as follows: letwbeharmonic measure for the upper half-space of a disc centered at the origin and containing supp ¢. Then using the maximum principle,

X3 - Cw S goo S X3 + Cw, x E U n lR~

for suitable C. And IVwl is O(/xl-3) at 00 when x E ~2 by the Poisson integral formula, so the statement follows from l'Hopital's rule. The integral

I(¢) = r IVgoollog IVgooldaJau

is then easily (given [7]) seen to be absolutely convergent. At the end of the section we will find piecewise linear functions ¢ with support in Q (l) and arbitrarily small Lipschitz norm, such that I(¢) has either sign.

For now, fix any piecewise linear ¢ with supp ¢ C Q(l). Next fix b > 0 and small. If Q is a square (Le., a square in some plane in ~3) with center point aQ

and given unit normal e, define "pyramids" PQ and PQ by

PQ = cch(Q U (aQ + bf(Q)e})

PQ = int cch(Q U (aQ - bf(Q)e})

where cch denotes the closed convex hull. Let N be a large number. In particular, N should be large enough that (x : X3 =

N-1¢(Nx)} is well inside PQ (1) U PQ (1). Constants will be independent of N


344 CHAPTER 15

provided N is large enough. Define (with e = -e3)

A = {x E PQ(l) U PQ(l) : X3 > N-I¢>(Nx)}

a = {x E lR3: x E Q(l), X3 = N-I¢>(NX)}

Then a· C PQ(I) U PQ(1). a is a polygon with lots of faces. Fix a Whitney decomposition W ofeach face into squares Q with side g-k, k = 1, 2, .... Choose a distinguished edge for each square. These choices may be made in an arbitrary way, but once made, they are kept fixed. The following is easy to see if N is large enough.

Lemma 2.1. There are at most Cgk squares Q E W with f(Q) = g-k. Andfor each k ::: 1,

U{Q E W: f(Q) ::s g-k} c U{D(aQ, Cg-k ): Q E W, f(Q) = g-k}.

Suppose next that n is a domain and Q c an is a square with P n n = 0, PQ c n, and with a distinguished edge y. Let e be the normalinto PQ.

Q Form a new

domain n as follows: Let T be the affine map with T(Q(l)) = Q, T(O) = aQ,

T(-e3) = e, and T({~} x [-~'~] x {OJ) = y. Let A Q = T(A) and aQ = T(a). Then n n (PQ U PQ) = A Q and n\(PQ U PQ) = n\(PQ U PQ).

We call this construction "adding a blip to n along Q." Transferring W by T,

ann (PQU PQ) = aQhas a natural decomposition into squares with distinguished edges together with a set of a-finite length.

The domains we want are obtained by iterating this procedure. Let no be the unit cube. If nn-I has been constructed, then its boundary will be given as ann-I = En-I U (UQ.,_, Q) where En-I has a-finite length and each Q E Q,,_I

is a square with a distinguished edge and, with PQ n nn-I = 0, P C nn-I, and,Q

moreover, if QI, Q2 E Q,,-1' then intPQ, n intPQ2 = 0 and PQ, n PQ2 = 0. To form nn, add a blip along each Q E Q,,_I'

Thenann = En-IU(UQEG aQ)isdecomposedasEnU(UQEG Q')where ---II-I ---n-I

the Q' are obtained from the preceding decomposition of the aQ

• If Q E Q,,_I' Q' E Q" and Q' C aQ then we say Q' is directly descended from Q.

If N is large enough then it is clear that for Q' directly descended from Q, PQ' U PQ, C PQ U PQ and in fact by properties of the Whitney decomposition, dist(PQ, U PQ" (PQ U PQ)C) ::: C-If(Q'). Moreover, PQ, C A Q, PQ, C A Q,

and if Q'I' Q; are directly descended from the same Q then PQ, n PQ' = 0, - - , 2

PQ; n PQ; = 0. So the induction hypothesis is satisfied and the construction can continue.

Let n = limn->oo nn. Also let G = U:I Q". If Q, Q' E G we say Q' is descended from Q in n stages if there are Qo = Q', QIo ... , Qn = Q with Qj

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN R3 345

directly descended from Qj+1 for j = 0, ... , n - 1. If Q' is descended from Q

we write Q' -< Q. We will write m(U, Y, z) for the harmonic measure of the set Y (Le., Y n aU),

relative to the domain U, evaluated at z E U. Let w= wen, " 0).

Main Lemma. Suppose ¢> has sufficiently small Lipschitz norm and I(¢» =f:. O. Then if N is large enough there is a number d such that

10gw(D(x, r))lim = d for a.e. (dw) x E an r->O log r

where D(x, r) is the intersection of an with a ball of radius r, centered at x.

Thus w puts full mass on some set with dimension d and no mass on any set with

dimension less than d. If I(¢» < 0, then d > 2, and if I(¢» > 0, then d < 2.

We will prove only the I(¢» < 0 case explicitly but we have set up the lemmas so that the I(¢» > 0 case follows by changing the notation.

Certain estimates of harmonic measure will have to be made. Fortunately the whole construction takes place within the class of NTA (nontangentially accessible) domains of Jerison and Kenig [9] and most of the necessary estimates are proved in [9]. Let n be a bounded domain. If diam n = 1, then n is an NTA domain with NTA constant A if it has the following properties:

• For each x E an, r < A -I, there are points Ar(x) E n, Br(x) E n c

with IAr(x) - xl ::s Ar, IBr(x) - xl ::s Ar, and dist(Ar(x), an) ::: A-Ir,

dist(Br(x), an) ::: A-Ir. Existence of the points Ar(x) (resp. Br(x)) is called the interior (exterior) corkscrew condition.

• If x, yEn then there is a path y from x to y with length(y) ::s A Ix - y Iand dist(y(t), an) ::: A-I min(ly(t) - xl. Iy(t) - yl). This is called the Harnack chain condition.

In general n is an NTA domain with NTA constant A if a dilation of it with diameter 1 is such. This definition is easily seen to be equivalent to (say) the one in [11].

If n is an NTA domain, t > 0, then we say n is Lipschitz on scale t with Lipschitz constant M if D(x, t) n an is the graph of a function with Lipschitz norm ::s M for each x E an.

Now suppose r c G is closed under the descent relation, i.e., Q E r and Q -< Q' imply Q' E r. Then we can form a domain n r by adding blips only along squares in r: n~ = no, n~+1 is obtained from n~ by adding blips along squares Q Ern Q"-it is clear by induction that such squares are contained in

an~ and satisfy PQ c n~, PQ n n~ = 0--and n r = limn->oo nr'


___________._.._.__ .~ _ ~: ~.-- .•~~ ""- .-"0_ .._-_.

347 346

CHAPTER 15

Lemma 2.2. n r is an NTA domain with NTA constant independent of Nand r. If r contains no squares with length < t then nr is Lipschitz on scale C- 1t with C and the Lipschitz constant independent of N, t, and r.

Proof We can assume r is finite. Squares denoted Q, Q', Q, etc., will be understood to belong to r.

For each Q, define VQ = AQ\ U {PQ': Q' directly descended from Q} and WQ = (PQ U PQ)\(A Q U U{PQ,: Q' directly descended from Q}). Then VQ and WQ are Lipschitz domains provided b is small enough, and V C n ,

Q r WQ n n r = 0. Moreover, VQ n VQ, = 0 if Q :f:. Q', and n r = S U (U V

Q)

where S = no\ U {PQ : Q E f1J n r} is a "trivial" set. For each Q, let bQ

beQ

a point of VQ with dist(bQ, aVQ) ~ diam VQ ~ i(Q). There are three things to check in order to know that n r is an NTA.

Exterior corkscrew condition. Fix x E anr , 0 < r < 1. We will ignore the

easy case where x E ~ E f1J \r. Jiggling x slightly to avoid edges, it follows that x E aQ for some Q E r. Let Qx be a smallest such square. Let Q be a smallest square such that Qx -< Q and i(Q) 2: r. Since x E PQ U P it

Q follows that dist(x, WQ) S Cr. Let z be a closest point to x in awQ. Since W

Qis Lipschitz with diameter 2: r there is Br(x) E WQ with IBr(x) - zl S Cr

anddist(Br(x). aWQ) 2: C-1r. Then IBr(x) - xl S Cr anddist(Br(x), nr) 2: dist(Br(x), aWQ) 2: C-1r.

Interior corkscrew condition. Choose Qx and Q as before, let z be a closest point toxinaVQ,andletAr(x)satisfYIAr(x)-zl S Cranddist(Ar(x), Vo)2: C-1r.

Harnack chain condition. It is here that we use that 1I¢IILip(1) is sufficiently small. We use this assumption only to assert that domains (2.2) below are Lipschitz and therefore NTA.

We show first that if Q, Qjil, i = I, 2, j = I, 2, are squares with Q(1) directly descended from Q and Qj2) directly descended from Qjl) then }

2

(2.2) VQ U VQ(I) U VQ(2) and VQ U U(VQ(I) U VQ

(2))

J J J Jj=1

are LipSChitz domains. Consider, for example, the first domain. It is obtained from PQ by adding a blip, adding two more blips, then deleting certain pyramids all of

whose bases make a small angle with the plane containing Q (provided 1I¢IILip(1) is small). resulting (if b is small) in a figure whose boundary is aPQ \ Q together with a Lipschitz graph over Q.

-,


Suppose x E anr\S, Qx is the square with x E VQ,' and Qx -< Q.

Claim. There is a path Y from x to bQ with length(y) S Ci(Q) and dist(y(t), anr) 2: C-1Iy(t) - xl·

Proof. Let Qx = Qo -< QI -< ... -< Qn = Q with Qj directly descended from

Qj+l. Using the Lipschitz domains VQo and VQjul U VQj we can find paths Yo

from x to bQo and Yj from bQj_ to bQj (j = I, .... n) such that 1

length(Yo) S f:i(Qo),

dist(Yo(t), anr) 2: c-I min(IYo(t) - xl. IYo(t) - bQol)

length(Yj) S Ci(Qj),

dist(Yj(t), anr) 2: c- I min(IYj(t) - bQj-L I, IYj(t) - bQj I).

Now IbQj -xl S diam(PQj U PQ) S C dist(bQj' aVQj ) S C dist(bQj' anr) for any j. It follows that Iy - x I S Iy - bQj I+ C dist(bQj • anr ) for all Y E n r , and, therefore, that dist(Yj(t), anr) 2: C-1IYj(t) - xl. Let Y be Yo followed by YI followed by ... followed by Yn. Then dist(y(t), anr ) 2: C-1Iy(t) - xl and length(y) .::: C Lj i(Qj) .::: Ci(Q), since i(Qj) S ki(Qj+l). _

Now fix x, y E n r . We assume neither of them belongs to S. So there are

Qx,Qywithx E VQ"y E VQy.LetQbetheminimalsquaresuchthatQx ~ Q

and Qy ~ Q.

Case 1. Each of Qx, Q y is either equal to Q or descended from Q in at most two

stages.

Then we simply use the Lipschitz domains (2.2).

Case 2. One of Qx or Qy is descended from Q in more than two stages but not

the other.

We suppose Qx is descended from Q in more than two stages. Let Q satisfy Qx -< Q and Q is descended from Q in exactly two stages. Then Ix - bQI S C Ix - y I. To see this, let Q' satisfy Q -< Q' -< Q. Then Ix - y I 2: dist(x, (PQ' U

PQn 2: dist(PQ U PQ' (PQ' U pQ,n 2: C-1i(Q) 2: C-11x - bQI, proving the assertion. Now connect x to bQ by a path Yl as in the claim. The length of this path is S C1x - y I. Connect bQ to y by a path Y2 with length(Y2) S C1bQ - y I and dist(Y2(t), anr) 2: c-I min(IY2(t) - bQI, 1Y2(t) - yl) using the Lipschitz

domains (2.2) associated to Q. Since IbQ - yl .::: Ix - yl + IbQ - xl .::: C1x - yl,


348 CHAPTER 15

we have dist(Y2(t), aQr ) :::: C- I min(IYj(t) - YI, !Yj(t) - xl) and length(Yj) :s C/x - YI. Now let Y be YI followed by Y2.

Case 3. Both Qx and Qy are descended from Q in more than two stages.

Choose Q~I), Q~I) so that Q~l), Q~l) are descended from Q in exactly two stages

and Qx -< Q~I), Qy -< Q~I) Let bx = bQ~I), by = bQ~l). It follows as before that Ix - bxl :s Clx - YI, IY - byl :s C1x - yi. Use the claim to connect x to bx by YI and Y to by by Y3, and (2.2) to connect bx to by by Y2. Then YI followed by Y2 followed by Y3 is the desired path.

To prove the last statement of the lemma, let:EQ = A Q6.PQ (6. =

symmetric difference)-the change in Qr made in adding the blip along Q. Note

dist(:E Q , (PQUPQy):::: C-If(Q).lfx E aQr,letQ E rbeminimalsubjectto D(x, C-It) n:E Q =j:. 0. Then D(x, C-It) C PQ U PQ. By minimality it follows that D(x, C-It) n aQ r = D(x, C-It) n aQ, which is a Lipschitz graph. •

We now give some general estimates for hannonic measures. We will assume

that Q C IRn is NTA and a basepoint pEn has been fixed; a lower bound for (diam n)-I dist(p, aQ) is incorporated into the NTA constant A. All constants

C, Ct, etc., depend only on A. The basic result proved in [9], Lemma 4.10, and

Theorem 7.9 is the "rate theorem": if u and v are positive harmonic functions on

n, x E an, r > 0 and u and v vanish continuously on D(x, r) n aQ, then on

D(x, ~), ; :s C ~i1;g;;. Here D(x, r) denotes a ball of radius r, centered at x. Moreover; extends to a HOlder continuous function on n n D(x, ~) and satisfies the following estimate when Y, z E an n D(x, 0 (2.3) Iu(y) - u(z) I < C( Iy - ZI)/3 u(Ar(x)) , f3 > O.

v(y) v(z) - r v(Ar(x))

Other results that will be used, such as the doubling property of harmonic measure, are in Section 4 of [9]. We also need some further estimates which are not in [9] but follow easily from the techniques used there.

Lemma 2.3. There is Ct > n - 2 such that if ° < r < R, x E Q then w(D(x, r)) .::: C( f? )aw(D(x, R)).

Proof. If we required only Ct > 0 this would follow from the doubling property,

as does a corresponding inequality in the opposite direction. To get Ct > n - 2, normalize Q so that diam Q = l. By "localization" ([9], Lemma 4.11; see also

[lID we can assume R = l. Lemma 4.1 of [9] applied to the Green's function g of Q with pole at p on the domain n/D(O, C-I) gives g(Ar(x)) :s Cr/3 with f3 > O. Lemma 4.8 of [9] then implies w(D(x, r)).::: Cr/3+n - 2 = Cra. •

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3 349

Lemma 2.4. Let Q 1 and Q2 be two NTA's with base points PI. P2, and Wj = w(Qj, " Pj). Suppose E C aQI n aQ2 satisfies D(x, Cilr) n aQI C

E, diam E :s rand dist(E, aQI6.aQ2) :::: Cilr. Suppose Y C E. Then ~ < C W7.(E) ",,(Y) - "'1(E) .

Proof. The assumptions imply that E is contained in the union of a bounded

number of discs of radius iCilr whose doubles do not intersect aQI6.aQ2. The

statement now follows from the rate theorem, Lemma 4.8 of [9], and the doubling

property. •

• Lemma 2.5. Let Q I and Q2 be two NTA's with a common base point p. Suppose

theHausdorjfdistancebetweenaQlandaQ2is:s Cit, and that x E aQ I naQ2 with D(x, t) n ani = D(x, t) n aQ2. Then w2(D(x, t)) ~ wI(D(x, t)) where

the constants depend on A and CI.

Proof. We will show w2(D(x, t)) :s CWI(D(x, t)). By the doubling property

we can assume D(x, 2t) n aQ I = D(x, 2t) n aQ2. Let D = D(x, t), and let D j = D(x, 2j t), j = 0, 1,2, .... We have

(2.4) w2(D) - WI (D):s r W(Q2, D, {)dwi (0lnlnan1

To see (2.4), regard the function w(Q2 , D, z) as extended to Q I U Q2 by setting it to zero on QI \Q2. Then W(Q2, D, z) - W(QI, D, z) is subhannonic on Q I and

zero on aQ I\Q2 and (2.4) follows.

Let k~ be the kernel function d"'C;:~"{) • If { E (Dj +1\Dj ) n Q2 n aQI, then, estimating k~ by [9], Lemma 4.14, we have

w(n2 , D, {) :s w2(D) max k~(Y)YEDnan2

w2(D) (dist({, aQ2))a<C ., Ct>O - W2(D j+ 1) 2f t

< cr ja w2(D) - w2(Dj +l)

by the assumption about the Hausdorff distance. So,

W2(D) - wI(D) :s C L wI(Dj+1 n Q2) r jaW2(D). . w2(Dj+d

f

We make two estimates on WI (Dj+1 n Q2). By [9], Lemma 4.2 we know

w(n2, Dj+2' {) :::: C-I if { E Dj+1 n ani n Q2. It follows by the maximum principle and doubling property that

WI (Dj+1 n Q2) :s CW2(Dj+2) :s CW2(Dj+I).


350 CHAPTER 15

On the other hand WI (Dj+l) ~ Cj+IWz.(Dj+d :~~i by the doubling property. So for any jo < 00,

Wz.(D) - wI(D) ~ C LTjotWz.(D) + C L C j2- jotwI(D). j>jo j::;jo

The lemma follows by taking jo large enough that C Lj> jo 2- jot < 1. •

Suppose next that n is also Lipschitz on scale t. Then ([7]; see also [10])

harmonic measure (with basepoint p) is absolutely continuous to surface measure and the density ~~ is·the normal derivative of Green's function. Moreover IVgl satisfies an A2 condition with respect to w on discs of radius ~ M t for any

M < 00, i.e.,

(2.5) rn~1 1 IVgl 2da < c( w(D(x, r)))2r,,-ID(x,r)

if r ~ Mt, where C depends on the NTA constant A, the Lipschitz constant and M, Actually only the case t = diam n is stated in [7], [10] but the localized

version follows using Lemma 4.11 of [9].

Lemma 2.6. Let n be NTA and Lipschitz on scale t. Let x E an, r ~ Mt, and let).., > 0 satisfy

-I w(D(x, r)) w(D(x, r))C <).., < CI .I rn-I - - r,,-I

Then

( Ilog()..,-IIVgl)l IVgl da~C. JD(X,r)nan w(D(x, r))

Proof. This is a formal consequence of (2.5). One can quote the fact that the logarithm of an A2 weight is BMO, or prove it using Jensen's inequality and boundedness of the function x log_ x. •

The next lemma is one of the main steps in the proof. We are assuming 1(4)) < 0, i.e., adding a blip to a half-space decreases the value of f IVgoollog IVgool, and we want to conclude that adding a blip to an NTA-domain decreases the value of f IVg I log IVg I. This will follow in a standard, if slightly complicated, way using the rate theorem to compare with the half-space case, provided N is large enough that the blip can be regarded as far from the nonflat part of the boundary.

Lemma 2.7. Suppose n is NTA and Lipschitz on scale t. Let Q c an be a square with f(Q) ~ Mt, and with PQ n n = 0, PQ c n. Let nbe the result ofadding

====-~-:-~-=..-=-="::=-~~::..-==-==-~-=~,:::-::_.=::::::::::.~=-~-=-=~,~-=="=-"==~=:='--..__.'---- -_.._-_..__.._--------

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN JR.3 351

a blip along Q, and let g and g be the Green'sfunctions with pole at p, wand W

the harmonic measures at p. Then

2{_ IVg\log I'Vglda ~ ( IVgllog IVglda - TJN- w(Q)

Jan Jan

provided N > No is large enough. No and TJ depend on the Lipschitz and NTA

constants and on M, and of course on 4>.

Proof. We will use [9] as a reference for estimates although often only the smooth

or Lipschitz case is needed. We scale so that Q = Q(l) and PQ C JR~. Let I

q = !be3 and let goo be the Green's function of {x: X3 > N- 4>(Nx)} with pole

at 00. Ilt is easy to see that w(D(O, t)) ~ w(D(O, t)) for all t > N- . (If we

want, we can use Lemma 2.5 and the doubling property for this.) Likewise

fO(o,l) IVgoo Ida ~ t2 for all t > N-I.

Alsog(q) ~ w(Q(1))by[9],Lemma4.8,soby[9],Lemma4.4,g ~ w(Q(l))

on PQ(l). By the reflection principle and Hopf lemma,

1 IVgl ~ w(Q(l)) on Q( -)

2 1

(2.6) IVg(x) - Vg(y)1 ~ Cw(Q(l))\x - yl if x, y E Q( 2:)'

Similarly goo (q) ~ 1. Let m = (Cd N)e3 where CI is a constant which is

> 2 II 4> II Lip(l), so that dist(m, an) ~ ~' Theng(m)-g(m) ~ CNw((D(O, ~)) by [9], Lemma 4.8. So by [9] Lemma 4.4, we have g - g ~ Cw(D(O, *))gm on n\D(O, i) where gm is the Green's function ofn with pole atm. So IV(g - g) I ~ Cw(D(O, *))\Vgm! on an;D(O, ~) and by [9] Lemma 4.14, it follows that (for

some fJ > 0)

IV(g - g)1 ~ Cw(D(O, ~ ))(Nlzl)-.8w (D(O, !zl))-IIVg\

on n\D(O, i). Using the preceding bounds,

(2.7) IV(g - g)(z)1 ~ CN-2w(Q(1))w(D(O, Izl))-I(Nlzl)-.8jVg(z)l.

If in addition Z E Q(!), this implies by the formula preceding (2.6) that

(2.8) IV(g - g)(z)1 ~ C(Nlzl)-(2+.8)w(Q(l)).

In particular, IVg(z)! ~ w(Q(l)) when Z E Q(!) and Nizi is large enough.

Likewise, comparing goo with X3, we get

\Vgoo(z) - e31 ~ C(N!zl)-(2+.8),


353 352 CHAPTER 15

if z E ]R2 and Nizi is large enough (actually, we could take f3 = I here). By the rate theorem,

lV'g(z)1 '" w(Q(I» ~ w(Q(I». lV'goo(z)! "" woo(Q(l»

Moreover, by (2.3) and L'Hopital's rule,

(2.9) I lV'g(Y)1 - lV'g(z)1 I ::: Cjy - zltlw(Q(I» /V'gCl')(Y)/ lV'goo(z)1

if Y, z E graph t/J with y, Z E Q(!). Now let A = lV'g(O)I and fix a 2 and (for notational convenience) a > (l - a)(2 + f3).

Claim.

f_ A-IIV'gllog(A-IIV'gJ)da - f A-IIV'gllog(A-IIV'gJ)da JannD(O,N-.) JannD(o.N-.)

::: -7/N-2

for N large enough.

To prove this, fix z E ]R2 with Izi N-ot. Let)'. IV$(z) I Then A ~ IV$oo(z)I •

w(Q(l» and

I)'. - AI ::: IlV'g(O) I - lV'g(z)11 + IIV'g(z)1 - lV'g(z)l/

lV'g(z)1 I IV'- ()I II + lV'goo(z)1 goo z

::: C(N-ot + N-(l-ot)(2+tl) + N-O-ot)(2+tl»w(Q(l»

(2.10) I)'. - AI ::: CN-(I-ot)(2+tl )w(Q(l»

where we used (2.6), (2.8). Also if Ix I < N-ot , X EOn,

--IIIV'g(X)1 _ lV'g(z)j I lV'goo(x) I !)'.-IWg(X)1 - IV'goo (x) II = A IV'goo (x) I IV'goo(z) I

(2.11) ::: N-ottllV'goo(x)/

3COUNTEREXAMPLES W1TH HARMON1C GRAD1ENTS 1N 1R

using (2.9) and)'. ~ w(Q(l). The expression in the claim is I + II + III + IV

where

1= f_ [A-IIV'gllog(A-IIV'gl) - )'.-IIV'gllog()'.-IIV'gJ)]da JannD(O,N-Q)

II = f_ [)'.-llV'gllog()'.-IIV'gJ) -1V'goollogIVgool]da JannD(O,N-.)

III = f_ . lV'goollog lV'goolda JannD(o.N-·>.

IV = - f A-IIV'gllog(A-IIV'gJ)da. JannD(o.N-.)

If N is large enough, then expression III is ::: -7/N-2 by a change of scale and the fact that I(t/J) < O. We will show the others are error terms.

111:::A-Illog(A)'.-I)1 f_ lV'glda+I)'.-I-A-II J annD(O.N-·)

f_ lV'glllog()'.-IIV'gJ)lda. J annD(O,N-·)

The first integral is ::: CN-2a w(Q(l». Lemma 2.6 applies to the second integral showing that it is ::: Cw(D(O, N-ot » ::: CN-2a w(Q(l». Since (2.10) implies the constants in front of the integrals are ::: C N-(l-ot)(2+/l)w( Q(l»-I, we

get

III ::: CN-(l-ot)(2+tl)-2ot ::: CN-2-tl(l-ot).

Expression II is

1 j,-IIV'-'

::: j,-llV'gllog _ g da aflnD(O.N-:-.) 1 (,V'goo,)1

+ f_ 1)'.-IIV'gl-lV'goollllog/V'goollda J annD(O,N-·)

::: Cj,-I N-ottl f _ Wglda + N-ot/l f _ lV'goolllog lV'goollda. JannD(O,N-.) JannD(O,N-.)

Lemma 2.6 also holds for the Green's function at 00 of a Lipschitz domain, so

the second integral is ::: CN-2ot . Thus

1111 ::: CN-ot(2+tl).

Finally, IIVI ::: C N- 3ot by (2.6); the claim follows.



357 356 CHAPTER i5

The doubling propeny implies

[ J'V'gkjlog IVgklda.:s [ J'V'l-lllog J'V'gk-llda - P3,JoQ! J00,1-1

and the lemma follows by iterating on k. •

ForeachQ E G,letfQ = ann(PQUPQ). ThenQ' -< QimpliesfQ, C f Q and the f Q are otherwise disjoint.

Lemma 2.9. If 71 > - 0 is small enough, then for each sufficiently large k there

are squares QI '" Q. with f(Qj) = 8-k, such that EJ=I w(fQ) :::: 71 and w(rQ .) < .

log l(Qj12 _ -TJk for all J.

Proof. Fix 71 > 0 sufficiently small. We first find such squares with Wk- I (Q j)

replacing w (fQ). Let Q I ... Q, be all squares of length 8-k, with Q I ... Q.

being thosefor which log w;~~~~f) ::::: -TJk. Let D j = D(aQj' C8-k). By (2.13),

there are pairwise disjoint regions Rj with Qj C Rj C D j such that (Jnk- I =

Uj Rj . Take)., = w;(Q~~f) and apply Lemma 2.6 on D j , restricting the integral to R j' This gives

, Wk-I(D-) LWk-I(Rj ) log ;::::: CLWk-I(Dj ) j=1 f(Qj) j

+ 4: i Ivl-Illog IVl-llda. J 1

By Lemma 2.8 and the doubling property

, wk-I(Q-) "Wk-I(R·) log J < -pk - C < -£>-ok. ~ J f(Q .)2 - - f"'<' J=I J

If EJ=I Wk- I (Qj) .:s 71, then using Wk- I (R j) .:s cd-I(Qj) and Wk- I (Qj) :::: C-k , we get

-CTJk(log C - log 64) - TJk ::::: -f>2k,

which is a contradiction if 71 is small enough.

Now, for each j E {I, ... , s} let n Qj = nr where f = {Q E G: Q i Qj}.

Since nQj is obtained from nk- I by adding blips along squares of side ::::: 8-k ,

the Hausdorff distance from (Jn Qj to ank- I is ::::: C8-k and Lemma 2.5 shows w(nQj , Qj, 0) ~ Wk-l(Qj)' A symmetry argument (or [9], Lemma 4.2, and the doubling property) and the maximum principle imply w(fQ) ~ w(nQj • Qj, 0).

So w(fQ) ~ wk- I (Qj). The lemma follows. •

COUNTEREXAMPLES WiTH HARMONiC GRADiENTS iN]R3

We now follow [6] and derive the main lemma from Lemma 2.9 using elementary results in ergodic theory. Fix a square I; E fb. The shift T on infinitely many symbols acts on f I;' Namely, for each Q E G I with Q -< I;. the restriction of T to f Q is the conformal affine map which takes Q to I; and preserves distinguished edges. This defines T except on a set with a-finite length and therefore a.e.

(dw). Letting ~ be the partition {rQ: Q E Gd, the nth partition V~:~ T-j~ is {rQ: Q E ~}. For x E fE, let Q,,(x) satisfy x E fQ.(x) and Q,,(x) E ~.

D~fine a,,(x) = f(Q,,(x»2 and h,,(x) = w(fQ.(x)'

Lemma 2.10. w is mutually absolutely continuous to a certain invariant measure

/-L for T. T is ergodic with respect to /-L and ~ is a generating partition. The limits

lim,,~oo .!. log ---L-() and lim,,~oo .!. log h (I ) exist a.e. / (d/-L) and are constant. n an .t n" .t

Proof. Let /-L(")(y) = ~ E'j:~ w(T-jy). Let /-L be a weak· limit point of{/-L(")}.

Then /-L is an invariant measure [22]. Let E, = {x E f E : dist(x, B) < E} where B is the boundary of the square I;. Then

(2.14) If Y n E, = 0, then C;IW(Y) .:s /-L(Y) .:s c.<.V(Y).

(2.15) /-L (,,) (E,) ::::: CEil with f3 > O. independently of n.

For let Y be as in (2.14). Fix Q E ~. Let S be the affine map with S r f Q = T". By Lemma 2.4 with nl = n, PI = 0, n2 = Sn,P2 = S(O), E = fE\E"

we have

(2.16) C;lw(Y)w(fQ) ::::: <.V(T-"Y n f Q ) ::::: C,<.V (Y)<.V (fQ).

. So C;lw(Y) ::::: <.V (T-" Y) ::::: C,<.V(Y) and (2.14) follows. To do (2.15) fix again Q E ~. Then T-" E, n fQ can be covered with CE-1 discs ofradiuSEf(Q). By

Lemma 2.3 and the doubling property,

w(T-"E, n fQ) ::::: CEIl<.V(fQ).

with f3 > O. (2.15) follows. By (2.14) /-L and w are equivalent on each fE\E, and since (2.15) shows

lim,~O/-L(E,) = lim,~o<.V(E,) = O,theyareequivalent. Given (2.16),thefact that T is ergodic and ~ is a generating partition follows by a standard argument as in [6]. Next, define T' by T'(x) = f(Q)-1 if x E f Q and Q E G I.

Claim. I/(log T') 0 T"I/L'(dw) ::::: C independently ofn.

Fix Q E ~. The set {x E f Q: (log T') 0 T" (x) > ).,} is identical with U{rQ': Q' directly descended from Q and f(Q') < e->"f(Q)}. By Lemma 2.1,


- "

359 358 CHAPTER 15

the latter set can be covered by CeA discs of radius e-A• By Lemma 2.3,

w({x E r Q : (log T') 0 Tn(x) > AD S Ce-IlAw(rQ),

with fJ > O. The claim follows. From the claim we conclude that log T' E

L1(dJJ,). Since ~ log t = ~ L~:~ log T' 0 Tj, the ergodic theorem shows limn-+ oo 1. log...!... exists a.e. dJJ, and is constant. Now fix QI E G . By the n ~ _ I doubling property, there is a constant r such that

w(T-nrQ, nrQ )::=: C£(QI)'w(fQ),

for all Q E fin. Itfonows thatJJ,(rQ,) ::=: C£(Qdr and, sincelogT' E L1(dJJ,), that

1L JJ,(fQI) log -- < 00. Q, EG JJ,(rQ,)

-I

Since; is a generating partition, this implies that T has finite entropy with respect to JJ,. Let hn(x) = JJ,(Qn(x)). Then limn-+ oo ~ log t exists a.e. (dJJ,) and is constant by the Shannon-McMillan-Breiman theorem (the version we need for infinite partitions with finite entropy is in [16]). By (2.14), limn->oo 1. log -hi

n •exists and is constant. •

Proof of Main Lemma Note that any squares Q E G with £(Q) = 8-k must belong to fin for some n S k. By Lemma 2.9, we can choose E so that

I . hn(x) }) TJw({x E rI;: - rnflog -- S -TJ ::=:k nsk an (x) 6

for a sequence of k tending to 00. It follows that limn->oo 1. log !!..a. < 0 and n fIn

therefore that

I I !!..1mn-+ 00, log hn 1, Ii og II.(2.17) hm --=1+, 1 fIn

=pn-+oo log an hmn-+ oo Ii log an

exists a.e. and is> l. Suppose x is a point where (2.17) and limn->oo ~ log an (x)

exist. For r > 0, let n be the largest number such that £(Qn(x)) > r. By the doubling property

10gw(D(x, r)) 2 log hn(x) 0((1 I )-1)log r ::=: logan+l(x) + og an+l(x)

> 2 log hn(x) + 0(1) - log an (x)

O' l' log fIn±! (x) I Sas r -+ , Since 1mn -+ 00 IOgfI.(X) = . 0

liminf 10gw(D(x, r)) > 2fJ > 2. r-+O log r

0.__ "",. ~

~ ~_" ~ =~"-'" ._~._.~- ,~- .. ..~


Likewise, , 10gw(D(x,r))

I1m sup S 2fJ r->O log r

and the proof is complete. • We still have to construct suitable variations tP. First we get rid of the piecewise

linearity requirement.

Lemma 2.11. Suppose {tPn} is a sequence of Lipschitz functions on 1R2 with common compact support and tPn -+ tP in Lipschitz norm, Then l(tPn) -+ l(tP),

Proof. Let gn, g be Green's function at 00 for Qn = (x: X3 > tPn(X")} and A = (x: X3 > tPCX)}. WecanassumesupptPn C Q(1).

There is no difficulty at 00 since lV'gn(x)1 = I + 0(lxr3) uniformly in n as x E 1R2 -+ 00 and similarly with IV'g I. It follows that it is enough to prove

(2.18) [ lV'gCX, tPCX)) - V'gnCX, tPnCX))12dxldx2 -+ 0 as n -+ 00.JQ(2)

In proving (2.18) we may assume tPn ::=: tP· To see that, let En = IItPn - tPlioo. Let 1{fn satisfy 1{fn = tPn on Q(3) and -En S 1{fn - tPn S 0 outside Q(3) with 1{fn - tPn = -En outside Q(4). Let 8n be Green's function at 00 of X3 > 1{fn ei),

i.e.,8n = 0 when X3 = 1{fnCi) and 8n - (X3 + En) -+ 0 as x -+ 00. The rate theorem applied to 8n - gn and gn implies that lV'gn - V'8nl S 8(En)lV'gnl on (x E aQn : X E Q(2)}, where 8(E) depends only on E and a bound for IItPn IlLip(I),

and 8(E) -+ 0 as E -+ O. So

(2.19) [ 1V'gn CX, tPnCX)) - V'8nCX, tPnCX))12dxldx2 -+ 0,JQ(2)

as n -+ 00. We can replace Q(2) by Q(8) in (2.19) by standard theorems on continuous dependence of solutions of the Dirichlet problem, since both tPn and 1{fn are smooth outside Q(1).

Ifwe now replace Q(1) by Q(4) andtPn by 1{fn + En we have a reduction of(2.18) to the case tPn ::=: tP. This case will follow from the regularity for the Dirichlet problem on Lipschitz domains due to Verchota [29]. That theorem implies

[ lV'g(x, tP(x)) - V'g(x, tPn(x))l2dxldx2 -+ 0JQ(2)

as n -+ 00. In particular, by the Lip(1) convergence,

[ /V'Tg(X, tPn(X))12dxldx2 -+ 0,JQ(2)


360 CHAPTER 15

where VT is differentiation along ann' Using regularity for the Dirichlet problem again, this time on the domain nn, we obtain

( IVgn(x, l/J,,(x» - Vg(x, l/Jn(x))l2dx, dx2 lQ(2)

~ C ( IVT(gn - g)1 2dx, dx2laQ. ---+0 as n ---+ 00

and the lemma follows. • Now fix l/J E Ccf(JR.2). Let Q t = {x E JR.3: X3 > tl/J(X)}. Let get, x)

(t E JR., x E nt ) be the Green's function of Q t with pole at 00. Then get, x) is smooth in t and x jointly. Denote I(tl/J) by I(t). We claim that I vanishes precisely to third order at zero for suitable l/J, as in the first case of Lemma 1.1.

Lemma 2.12. 2 3

dl I d 1I d 1 1[( d~ )3 - 2d~ ]-d = -d2 = 0 and -3 = - - 31Vl/JI - dx,dx2t t=O t t=O dt]R2 dX3 dX3

Here ~ is the harmonic extension into the upper halfspace and V is the ]R.2 gradient.

Proof We also let t::. be the ]R.2 Laplacian. Let g', gil, gill be the first three tderivatives of g with x fixed. Then g'(t, .), g"(t, .), glll(t, .) are harmonic on Qt>

vanish at 00, and vanish on ant outside the supportofl/J. Moreover, g!(0, .) = _~

("Hadamard variational formula"); to prove it on JR2, which suffices, differentiate the equation g (t, x, t l/J (X» = 0 with respect to t.

We do the calculations using coordinates (t, u) E JR x R~ and the map (t, u) ---+

(t, x); with x E Q t defined by x = (ii, U3 + tl/J(ii». We write get, u) for get, Ii, U3 + tl/J(ii», n(t, Ii, 0) for the normal into Q t at (Ii, tl/J(ii», etc. Then

d d2

(2.20) -I da = 0, -d21 da = IVl/J1 2du,dU2dt t=O t t=O

2 3 ( dn)/ (d n)1 - 2 (d n )1(2.21) dt' e3 t=O = 0, dt2 ' e3 t=O = -IVl/JI, dt3 ,e3 t=O = 0

dg dg - = g

I + l/J (ii) dt dU3

d dg d dg d ( dg ) dg' d 2g- - = - - = - g I +l/J(ii)- = - +l/J(ii)

dt dU3 dU3 dt dU3 dU3 dU3 du~ , 2 2

d dg d d g d d ( _ dg ) - -- = - - = - - g

I + l/J(u)dt2 dU3 dU3 dt2 dU3 dt dU3

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN ]R3 361

d2 3d ( dg' d dg ) dg" g' + l/J2 d g= - gil + l/J - + l/J - - = + 2l/J du~ d 3 'dU3 dU3 dU3 dt dU3 u3

11d3 dg dg"l d2g 2 d 3g' 3 d4g - -- = - +3l/J- +3l/J - +l/J -. dt3 dU3 dU3 du~ du~ duj

When t = 0, g is linear and the last term in each of the last three expressions

drops out. If U3 = 0, then IVgl = n3)~' so when U3 = t = 0 (2.21) gives ~

dlVgl dg' ----;j"( = d U3 '

d21VgI dg" d 2g' - 2 (2.22) -d2 = -d + 2l/J-d2 + IVl/J1 ,

t U3 u3

dg llld31VgI d 2g" 2 d3g' dg' - 2 -- = - + 3l/J- + 3l/J - + 3-IVl/J1 .

dt 3 dU3 du~ du~ dU3

Also, for t arbitrary but U3 = 0 and Ii outside the support of l/J we have

dlVgl dg' dt = dU3'

d 21VgI dg"(2.23)

dt 2 dU3 '

dg llld 3/Vgl

~ = dU3

In computing It> Itt, 1m we can differentiate freely under the integral sign. For there is no problem locally since all functions are smooth, while if Ii is outside

the support of l/J, then the first three t-derivatives of IVgllog IVglda are sums of terms each of which (by (2.23» involves either log IVgl, d , ~d/I or <!.L-11/E.8- , •I

d"3 U3 d"3

Since g', gil, gill and g(x) - X3 are harmonic functions vanishing at 00 and with

boundary values of compact support, their u3-derivatives die like lu ,-3 as u ---+ 00

along JR2, uniformly in any finite t-interval. For g - X3 this was proved at the

beginning of the section and the same proof works in general. So we have uniform integrability.

Keeping in mind that IVgl = I when t = 0, and using (2.20), (2.22),

dl I 1 dg' - = -dUldu2 dt t=O ]R2 dU3

2 2d 1 1 dg" d g' dg' )2 2](2.24)- = - + 2l/J- + ( dU3 + IVl/J1 du,du2,dt2 Lo ]R2 [ dU3 du~

2 3 3 1 d I dg' d g' 2 d g' (dgl )3~ = 91Vl/J12 --.£ +6l/J- - +3l/J - -

(2.2S)dt 3 It=0 ]R2 [ dU3 dU3 du~ du~ dU3


362 CHAPTER 15

d2g" dg' dg" dg'" J +34>-- +3- - + - dUldu2.

du~ dU3 dU3 dU3

Now we use that if I : jR~ ~ jR is harmonic and vanishes at 00 then fJR2 ~

0, f JR2( -%;)2 = fJR2 IV112• Therefore, It/I=o = O. Also, the first term on the right

side of (2.24) is zero, and, using g' = -¢, 2 2d 1 d ¢ d¢ 21-2/ = [-24>-2 + (-d ) + IV4>12Jdu1du2dt 1=0 JR2 dU3 u3

2= LJ24>£:,.4> + (:~ r+ IV4>1 Jdu 1du 2

2 = LJ(:~ r -IV4>1 Jdu 1du2

=0.

The last term in (2.25) is zero. The preceding two terms are

1 d2g" d¢ dg"3 [4>-- - - -JdU 1du2

JR2 du~ dU3 dU3

which is zero by Green's identity for the harmonic functions ¢ and ~. The remaining terms in (2.25) are

2 31[ - 2d¢ d¢ d ¢ 2d ¢ ( d¢ )3J-9IV4>1 - + 64>- - - 34> - + - dUldu2 JR2 dU3 dU3 du~ duj dU3

2 d¢ d¢ - 2- d¢ ( d¢ )3J= 1[ - - - 6¢-£:"4> + 34> £:,.- + - dUldu2-9IV4>j JR2 dU3 dU3 dU3 dU3

1[ - 2d¢ d¢ - 2 2- d¢ ( d¢ )3J= -3IV4>/ - - 3-£:''(4)) + 34> £:,.- + - dUldu2. JR2 dU3 dU3 dU3 dU3

The middle terms vanish by Green's identity for £:,. since 4> has compact support.•

Remark. Similar calculations may be done with any initial domain in place of the upper half space and for most domains one does not have to compute so many derivatives since curvature terms enter into the first variation. However, this is useless for proving Theorem 3. The curvature effects scale out as N ~ 00.

3 The expression f [(5#; ) - 31 V4> 12 5#; Jalso came up in the proof of Lemma 1.1

and was shown to be negative if 4> = 4>0 = -Ix + e31- 1. Approximating 4>0 in the W ,3 Sobolev norm by a Co function t/t, and then approximating tt/t with

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3 363

t > 0 small in Lipschitz norm by a piecewise linear function, we have a proof of Theorem 3.

Taking t < 0 instead of t > 0 we obtain a domain whose harmonic measure is less than two-dimensional. Domains homeomorphic to a ball with this property (infact, with I-dimensional harmonic measure) may be found much more easily by other means (a closely related result is in [32]), but it may be new that there is a scale invariant and/or NTA example. It is also amusing to note the following: the {froof of the main lemma works just as well if Q is replaced by the complement of Q, except that 1(4)) must be replaced by f IVgllog IV'gl where g is the Green's function at 00 on the domain below the graph of 4>. It follows that if Q is the domain just used to prove Theorem 3, then harmonic measure for rr is < 2dimensional. It is probably possible to make similar examples where the interior and exterior harmonic measures are both > 2-, or both < 2-dimensional. This

wotild involve working with a function 4> as in Lemma 2.12 for which ~ 11=0 = 0 and ~;! 11=0 has the appropriate sign. The calculations for the fourth derivative are rather complicated and 1 have not worked this out. Another remark is that some of the technicalities in the proof of the main lemma could be avoided by working in a special situation where it is possible to use a finite partition. However, one would then have to compute sgn 1(4)) for special 4>, which (unless it were done numerically) would appear to be a difficult problem.

Finally, we show how to generalize to jRn, n ::: 4. An unbounded domain with dim w ::: n - 1 + E may be obtained by crossing the domain in Theorem 3 with jRn-3. To get a bounded domain, intersect domains Ul ... Un-2 of this type with different choices of the dummy variables. If U is the resulting domain and E c au with dim E < n - 1 + E, then

w(U, E) ~ L w(U, En aUj) ~ L w(Uj, En aUi) = O. i i

APPENDIX: PROOF OF LEMMA 1.2

This argument uses a lot of high school math; the details of some of the calculations will be omitted. The last part is a proof-by-cases and a hand calculator is used to evaluate logarithms and the like. The reader who wants to check through it all will need a calculator and scratch pad.

While 1 was working out the proof, it often seemed to be a huge waste of time. It would be desirable to have a conceptual proof of Lemma 1.1.

Added September, 1991

See the remarks after the introduction, and reference [2]. 1


364

-="-= __ -=.-------~---~":'S~.=:;::::::~~~~~~1'..::;:~"":"--"'~~,~-~-::::2'~~'?_::::=:=..':":"".-~~~":~~::::~~~::;~~':~;:;:.;_:.:~,~s~.s'£~:~''::,',,·~"~.:-~"fZ!"''':2.ti~''d::tJ:::::: 2j!"-"-::..""~~~:T;zg;?:;::_~"~~'·~';li:..':<-:J!2;:(.':''Oc''S0''~~'~t~~£:..c~~:?:'

CHAPTER 15

A.l The setup

In this section we reduce the problem to estimating an explicit integral. We will use cylindrical coordinates r, &, z as well as rectangular coordinates x, y, z on ~~.

Lemma A.l.l.

(a) 1::1< (e, q (r, &») g~ = 0for all r.

(b) Let 1/I(r) = ~ 1::1< (J4-rL~;:~n<m\1/2 drjJ. Thenfor r < 1,

r . d&1-1< log Ie + q(r, &)1 21T

2 2= !log( ~ (~+ Jl - a - a 1/1(r)2)) , 1/I(r) S ~ a

a 1/1 (r) 2log- 1/I(r) :::: Jl - a

2 a

(c) Let a = a 2( 4- + 1/1 (r)2), b = 2a~1/I(r), c = 1 _ ~ + ~. r r r

Thenfor r > 1,

11< d& 1 log Ie + q(r, &)1- S log( - (.JC + .J(a + c)2 - b2)).

-1< 21T 2

Remarks.

(1) It is easy to check that 4ac - b2; therefore, (a + c)2 - b2, and also 1/I(r), are :::: 0 for all r.

(2) It will be clear from the argument for (c) that we could also obtain a formula

for the integral appearing there. This formula is too messy to be useful in the

proof, but was used by J. Goodman to evaluate lea) numerically.

Proof Recall e = (a, 0, ~), g is the harmonic function on IR~ with boundary values

if r < 1g(r, &) = { -r cos &,

-r -I cos &, if r > 1

. (d d) d d2 Whand q = aVg. We wnte VT = dx' dy ,t!,.T = fiX! 2 + d y2' en z = 0 we

have

(-I, 0) if r 1 r r

3 36S

and t!,.Tg = 0 when r 7'= 1. Thus as distributions on 1R2 = aR.~,

2

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN R

d g dg dg I I- = -t!,.Tg = -(- + -)d& = -2 cos &d& . dz2 dr+ dr_ r=1 r=1

If (x, y, z) E R.~ then by the Poisson integral formula

d2g 1 r 2z cos rjJdrjJ

dz2 = - 21T 1-1< ((x - COSrjJ)2 + (y - sinrjJ)2+ Z2)3/2'i

When z = 0, the fundamental theorem of calculus gives

dg 1 11< cos rjJdrjJ(A.2) - =

dz 1T -1< ((x - cosrjJ)2 + (y - sinrjJ)2)1/2

= 1/I(r) cos&,

with 1/1 as in the lemma. Part (a) of the lemma is now obvious. Also

Ie + ql2 = 1 + a21Vgl2 + 2(a2gx + a~gz)

= {(~ + a1/l(r) cos &)2, r < 1

a cos2 & + b cos & + C, r > 1.

We record the formula

11< d& (A.3) log Icos& - pl- = log Ip + JP2=l1 -log2, pEe,

-1< 21T .

where the sign of the radical is chosen so that IP + JP2=l1 :::: 1. When

PI , P2 E IR we obtain

IpI! s Ip211< d& \ log /P21 - log 2,

1 loglp2·cos&-pJ!- = jr---1< 21T 10g(lpII + Vpr - pD - log 2, Ipil :::: Ip21·

Part (b) of the lemma follows from this. For (c) we use

Lemma A.l.2. Suppose pet) = at2 + bt + C is nonnegative on the real axis.

Then 1< d& 1

log p(cos&) - s 210g( - (.JC + J(a + c)2 - b2)). / -1< 21T 2

Proof Let D = 4ac - b2 :::: O. Write p(t) = aCt - {)(t - I), { = fa (-b + i..{ij). Then

r d&1-1< log p(cos &) 21T = log a - 2 log 2 + 2 log I{ + ~I,


367 366

CHAPTER 15

with the sign of the radical as in (A.3). Now

I~ + ~12 = :. + '!'J(a + c)2 - b2 + .!.re-J"E, a a .a

2 2where E = 4c - 2b + 4ac + 2ib...{ij and../E is taken to have nonnegative real part. Estimating er../E by ..;rET, we obtain

Is + ~12 s :. + ~ /(a + c)2 - b2 + ~ .JC-t(a + c)2 _ b2 a a a

.!. (.JC + j(a + c)2 _ b2)2, a

and the lemma follows. • Part (c) of Lemma A.U is immediate from this. •

A.2 Simplifying the formulas

We want to get rid of the roots and logarithms. Define lin(a) = 1,<1 log Ie + q(r, 8)1 ~~ rdr, lout(a) = 1,>1 log Ie + q(r, 8)1 ~~ rdr. Also define

f 1/f2rdr, x = fd = 1/f2r dr, Q = f rdr. r<1 r<1

y,~CX-1 (I_cx2)'/2 r<1 y,~a-I(I_cx2)'/2

ZLemma A.2.t. lin S ~ 10g(1 - a ) - ! 1~:2 x + i (i - Q) log[ :(Y=-~~].

ZLemma A.2.2. loul S k(1 - a ) log( :~~ ) + V where V satisfies the following two estimates. Let

2 2 1 )r(r) = min 1/f(r)z, [1/f(r)z + 2' (2a Z - I + a Z( 2' + 4'))]+ .( a r r

Then

z zI f""( a )V(r) S _a 1 - Z Z r(r)rdr,4 I (1 + a )r

ZV(r) S 12-3/\4a z(1 - a ))-1/4a Z11f""( I + r1 )-I/Zr(r)rdr

Z


ProofofLemma A.2.1 It is understood that r < I in all fonnulas.

1 d8 log Ie + ql- rdr

y,~~/cx 27r

Z= { log[ ~ (~ + /1 - a - a Z1/fZ)]rdr1y,~JI-cx2/cx 2

~ = { ( ~ 10g(1 - a Z) + 10g[.!. (I + II -~ 1/fZ)])rdr1y,~JI-cx2/cx 2 2 V I - a

Z Q Z 1 I (/ a z)S - 10g(1 - a ) - - I - I - --z 1/f rdr 2 y,~JI-cx2/cx 2 1 - a

Q 1 a Z

< -log(l-az) - - --x - 2 4 I - a Z '

where we used 10g(1 - t) S -t, 1 - JT'=t 2: ! t. Also

1 d8 a1/f1logle+ql-rdr = logl(-)Irdry,>~/cx 27r y,>JI-cx2/cx 2

Z1 ( 1 a 1/fz rdr1 (- - - Q) log -)- 2 2 y,>~/cx 4 i - Q

Z1 ( I ) [ a (d - x) ]s 2: 2: - Q log 4(! _ Q)

by Jensen's inequality. • Proof of Lemma A.2.2 Here it is understood that r > 1. We use the following

fact.

"Lemma A.2.3. Let e E [-1, I] and let y = min( i, 2- I /z3-3/ 4 (1 - eZ)-1/4).

Then (1 + 2et + t Z)I/4 S I + Y min(t, (t + 2e)+)for t E [0,00).

Proof Let f(t) = (1 + 2et + t Z)I/4 - 1. Then f(O) = f( -2e) = 0, and f < ° on (0, -2e) if e < 0, so it is enough to show I'(t) s y when t > max(O, -2e).

We have I'(t) = i (t + e)(1 + 2et + t Z)-3/4, f"(t) = (- !tZ - ~t + i %eZ)(1 + 2et + tZ)-7/4, so the critical points of I' are -e ± J2 - 2ez.

If e > J2/3, then I' is decreasing on [0,00); hence I' S 1'(0) = ~. If°S e < J2/3, then I' has a maximum at -e + J2 - 2ez; its value there is 2- I / z3-3/ 4 (1 - eZ)-1/4. One computes ~ S Y for all e while 2- I/ z3-3/ 4 (1

eZ)-1/4 = y when e < J2/3. The e < °case follows the same way: if


----- -----,,-----~--_._-~--=-----=="'"'==-,==-----=---_._---..,....--,_._ ~._-====---==-=,-=---=====--==--===-,==--==.===----===--===--========.._--='--"._-.;, .'-.'- - .._·_'<=....·==~~~-=-=~~-~~=~~~-~~7-

368 CHAPTER 15 COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3 369 e :::: -,J213, then f' is decreasing on [-2e, 00) and if -.../2/3 < e < 0 then f' We set V = 1100

II(r)rdr and make two estimates corresponding to y :::: 4and has a maximum at -e + ../2 - 2e2. • y ~ 2-1/ 23-3/ 4 (1 - e2)-1/4. Using y :::: 4, we get

2 2I J 2a aNow, working from A.Ll (c), let II(r) :::: - I + -2 + 4" min(t, (t + 2e)+)4 r r

a 21{!2 2 2 Z1 ( 2a a )-I/Z2a -I +a2(~.+ ;.) E [-1,1), = - I + - + -4

r(r)a z t = 2a2 a2 ' e = 2a2 ~ 4 r Z rI+-;r+~ I + "72 + r4

and the first stated estimate for V follows using and define y relative t~ this e. Then

22az a Z)-I/Z ( a

2 )-1 a ( 1+ -z + 4" ~ I + 2" ~ 1-.. ~,~ , r > 1.

r r r~ (...;c + 1(a + c)2 - b2) The other estimate comes from

22 I / 2az a1 (J 2

2a 2

a 2- / 2a

2 a ) II(r) ~ "2 T I

/2r 3/\1 - e2)-I/\.; 1+ ?2 + r min(t, (t + 2e)+)= - 1- - + - + 1+ - + -1t 2 + 2et + I 42 r r 4 r 2 r 4

2az a Z )-I/Z2 2 2 = 12-3/\1 - eZ)-1/4 I + - + - r(r)a z

(1 (J 2a a - / 2a2

a ) r2 r4<- 1--+-+ 1+-+- 2 r 2 r 4 r 2 r 4

because (1 - eZ)(1 + ~ + ~)2 = 4a2(1 - a 2)(1 + -4- )z.r r r •2 2 y J 2a a+ - 1+ -2 + 4" min(t, (t + 2e)+)2 r r

A.3 Estimates for 't/J where we used A.2.3. So, since log x ~ x - I,

We give some formulas for 1{! and then some estimates.

2 2 (A3.1) 1{!(r- l ) = r1{!(r)f dO [ 1 J 2a (i2 I / 2a2 a ]log Ie + q I - ~ - 1 - - 2 + - + - I + - + - - 1

2Jr 2 r r 4 2 r 2 r 4 001(A3.2) 1{!(r)2r dr = 22 2 y J 2a a+ - 1+ -2 + 4" mm(t, (t + 2e)+)2 r r

Proof On the one hand 1g; ;~ rdr = 411{!(r)zrdr, and on the other hand = I(r) + II(r). 1g; ;~ rdr = 1 IVr gl2

;; rdr = 101rdr + 1100 r- 3dr = 1. •

Now .."

2 t t l/2dt2 4 2 4 (A3.3) if r < I.a - a ( a a - a ] 1{!(r) = ;r 10 (1 _ t)I/2(1 _ r2t)I/2I(r) = ~ [ (I - ~:) 1+ + 1+-

2 )J1+ -1(r2 - a 2)2 r 2 (r2 + a 2)2

1 a2 - a4 Proof The function ( (!+r 2)z':'r(l +z2) ) I/Z has a single valued branch on the unit <--

- 2 r 4 - a 4 disc slit from 0 to r. Accordingly,

since .Jf+X ~ 1 + 4x. It follows that ( Ii Z dZ)1{!(r) = e - -Jr Izl=1 (1 + r2 - r(z + z))I/Z i z

f oo 1 I + a 22I(r)rdr ~ - (1 - a ) Iog(--2)' =e ( - 1 ( )1/2dz)I Z

I 8 I - a Jri Izl=1 (I + r 2 )z - r(7z + ])


6!I,~iQ<\~l'- ,('If .-·-.)boljo>~t,.*'''''_'lfn~-'''''''''''''''--'''''''~''''·~,'''''''''L''''__>~·_'·"'~_·." ~

370 CHAPTER 15

X=e(--21'( )1/2dx ) ni 0 (1 + r 2)x - r(x2 + 1) ,

and now let x = r1. • 1/I(r) . . . [0 I)(A3.4) -- IS increasing on , .

r This follows from A3.3. Note it implies 1/1 is also increasing.

(A3.5) 1/I(r)::: rand(withx, QasinA2)x ~ Q2.

The first statement follows from A3.4 since 1/1' (0) = 1. The second statement follows from the first.

2 7 d .(A3.6) - < d < - (as In A2).3 - - 8

2Proof· Let v(r) = tp. Then d = fol v r 3dr and by A3.l, A3.2, 2 - d =

fd v2rdr. v2 is increasing, so

11 2 3 I 2 4 21 1 2 4 dv2II I v (r - 2r )dr = - (r - r)v - - (r - r ) - dr

o 2 0 0 2 dr

::s O.

It follows that d ::: ~. Also,

t v2(r - r 3)dr::: t (r - r 3)dr = _,10 10 4

by A3.5, and this implies d ::s ~.

J•

OO I (A3.7) (1 + 2" )-1/21/1 (r)2r dr ::: 2(.y2 - 1)(2 - d).

1 r

Proof· Let P = 2(.,,12 - I). Let v be as in A3.6. Let I(r) = (1 + r 2)1/2 24pr - l. Then 1(0) = 1(1) = 0 and 1 > 0 on (0, I) (this follows because 1

is concave down at the critical point r = (p-2 - l) 1/2) so

2

11(r) 11 11 dv~ 2 - pr v(r)2dr = Iv2 - I-dr o 1 + r 0 0 dr

< O.

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN Ii3 371

(A3.8) For r < l, drd

((1 + r 2 )1/21/1(r» ::s (1 - r 2)-I(1 + 4r2 - r 4)1/2.

Proof. Lett = 6.1+, Then (1 + r 2)1/2,'r(r) = ! foo-00 cosIJd9 ~'I':n; (I-t coslJ)I/2 1(1) •

dl 1 /:n; cos2 9d9 di = 21T -:n; (1 - 1 cos 9)3/2

::s A(t)1/2B (t)1/2,

where

1 jJr cos29 A(t) = -2 ---d9

n -:n; l-tcos9

I cos 9 d9jJr 2 B(t) =

2n -Jr (1 - t cos 9)2

may be evaluated using residues to be

(1 + r 2 )2A(t) = t-2((1 - t 2)-1/2 - l) = -'---

2(1 - r2 )

(l + r2 )2 4B(t) = 1-2(1 + (I - t 2)-3/2 - 2(1 - t2)-1/2) = (1 +4r2 - r ).2(1 - r 2 )'

The statement follows from these formulas and the fact that -ddt = 2(1(1 -~21 ., +, ) • (A3.9)

1/I(r) ::s (I + r 2)-1/2(log --I + log(l + r) - arctan r) when r < 1.

I - r

Proof. A3.8 implies

:r ((1 + r 2)1/21/1 (r» ::s 0 - r 2)-I«(I + r 2)2 + 2(r2 - r 4»1/2

r2 r 4

)< (I - r 2)-I(I + r2 ) 1 + (- (1 + r 2)2

1 1 1---+----- I - r 1 + r 1 + r 2 • •

By A3.I, ftJ (1 + ~ )-1/21/1(r)2r dr = fdo + r 2)-1/2rv (r)2dr, and the d1/l 4 -1/2(1 _ r 2)-1(A3.10) < -r when r < l.statement follows by A3.I, A3.2 as above. • dr - n


372 373

.~.___ _~,"'-_._._ .~...__ __-- - '" __... _ .n__--.'- -,__ - .__....., .... __ .,____ - _

CHAPTER 15

l 2Proof Let f(r) = r / (1 - r 2) ~. We claim f(r) .:s Hm SUPHI f(t). This will follow if f(r) > f(r 2). From A3.3,

21 1 r l / 2(1 - r2)t l

/ 2 r - - dt1( ) - 7Z' 0 (1 - 1)1/2(1 - r 2t)3/2

221 1 2 r(1 - r 4 )t l

/r - dtI( ) - -;; 0 (1 - t)I/2(1 - r 4t)3/2

_ 211

¢J(r, u)r l /2(1 - r 2)u l /2 2 _ - - du (u - t),

'7Z' 0 (1 - U)I/2(1 - r 2u)3/2

with

2u 3/ 2r l / 2(1 + r2)

¢J(r, u) = (1 + u)I/2(1 + r 2u)3/2 .

One computes ~ > 0, so ¢J(r, u) < ¢(r, 1) = ( ~ )1/2 < I, and the claim follows. To evaluate the limit, write

d1/l 117< (cos(}-r)cos()- - - d(}dr - 7Z' -7< «(1 - r)2 + 2r(1 - cos (}»3/2 .

We show this is -: (1 - r)-I + lower order as r ---+ 1. The denominator is ~ max«1 - r)3, (}3), so we can ignore O«(}2) terms in the numerator and O«(}4) terms in the denominator. This leads to

d1/l 1 17< 1 - r - = - 2 2 3/2 d(} + lower order terms dr 7Z' -7< «1 - r) + r(} )

00 r 1/2(} )= 1 1 (1 + t 2)-3/2dt + lower order terms (t =

7Z'(1 - r) -00 1 - r 2

= + lower order terms, 7Z'(1 - r)

and A3.10 follows. • (A3.1l) 1 r1/l(r)2dr .:s ~ (1 - s2)(1/I(s)2 + 1(s)1{t(s) + K(s»

s<T<1 2

16 l-s'/2 _ 128 (l+s'/2)-2_ * where 1(s) = Jr J=Sr' K(s) - Jr2 l- 2

s

Remark. 1 and K are decreasing on (0, I). For K, this may be seen by letting t = s 1/2_then ~ = (1 + t 2) 0+1)3 and calculus shows that (1+1)3 is

128K (s) 3+1 ' 3+1 increasing.


P~oof j} r1{t(r)2dr = ~ (1 - s2)1{t(s)2 + 21{t(s) j}(1{t(r) - 1{t(s»rdr + .r. (1{t(r) - 1{t(s»2r dr and

2 [I (1{t(r) _ 1{t(s»rdr = 2 [I [T 1/I'(t)dt rdr

= [1(1 _ t2)1/I'(1)dt

I 4 [ -1/2dt< - t

- 7Z' s

1 =1(s)"2(1-s2)

[ I [I [I 1(1/I(r) - 1{t(s»2r dr = - (I - max(t, u)2)1/I'(t)1/I'(u)dt du s s s 2

= [I [I 1{t'(u)1/I'(t)(1 _ t2)du dt

.:s 216 [I [I t- I/2U- I / 2(1 - U2)-ldu dt 7Z' s s

/1= 264 (1 + V)-I(1 + V2)-ldv 7Z' .fi

128 /1.:s -2 (1 + v)-3dv 7Z' .fi

= K(s)"2(1-S2). 1 •

A.4 Final calculations

The argument will proceed roughly as follows. Suppose a function I, involving

logarithms or whatever, can be shown by sections A2, A3 to be an upper bound

for 1 (a) on the interval [0.2, 0.27). Suppose 1 is increasing. If 1 (0.27) < 0,

then Lemma 1.2 is proved when 0.2 .:s a :::: 0.27. The fact that 1(0.27) < 0 will

be proved by evaluating 1(0.27) on a calculator. With V as in A2.2, we can use r :::: 1/12 and A3.1, A3.2, A3.7 to get

1 1 + 2a2 V < min(-a2(2- d) 2(21/2_1)12-3/4(4a2(1-a2»-1/4a2(2-d»)

- 4 1 + a 2 ' •

Another frequently used fact will be that all Taylor coefficients at 0 of ¢ (a) = 2i (l - a 2) log( ~~:~) + ~ 10g(1 - a 2) = i (1 - a 2) log(l + a ) + i (1 +


~ - - ~ ""'., _.~~ ---.-----~ ""'-.:.~''.;,;;~ -,-~T~~~~~'~.;.~,~,'_'-.;.,-;,',_.._,_"__...""3i,"".~"'·,[,; ..;,.""'......__",. __....,_,.,"'-"""::;_~_,.._ .."A'i!~....""'"""_. """""",",,,"---""''-'~''''-~~

374 CHAPTER 15 COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3 375

a 2) log(l - a 2) are nonpositive, and in particular (look at the first nonzero Use the same bound for Jin and maximize over Q for fixed d. The worst Q is 2coefficient) q, (a) :::: - i a 4 • 4 - 4e(~~a') leading to Jin :::: ~ 10g(l - a ) + ge(~~a2)' The second bound for

V leads to 2Case 1. 0.85 :::: a :::: I.

1 2 ( 1 + a 2 ) 1 2 a

2d

J :::: - (l - a ) log --2 + -4 log(l - a ) + -8-1- 2)8 1 - a e( - a

+ 2(21/ 2 _ 1)12-3/ 4 (2 _ d)(4a2(l _ a2))-1/4a2 I 2 ( I + a 2

) I 2 ( I + 2a2

)1 I < - (I - a ) log -- + - a 2 - d au - 8 I - a 2 4 I + a 2

, 1 ( 1 + a 2

) 1 2 a 2d< - (l - a 2) log -- + - log(l - a ) + --- 2 - 8 1 - a 2 4Q - 2 I ( I ) ( a d )1· < - log(l - a ) + - - - Q log

m - 2 2 2 4( 4 - Q) + 0.153a2(2 - d)

where we have overestimated in Lemma A2.1 using x :::: O. Adding the right using that 2(21/ 2 - 1)12-3/ 4 (4a2(l - a 2))-1/4 is increasing and a calculator for

sides, then maximizing over d E (0, (0) gives 2the value when a 2 = 0.85. The bound is increasing in d when a :::: i, so by

A3.6 we can take d = 7/8 gettingI 2 ( I + a 2

) I 2 Q 2J:::: - (I - a ) log -- + - a + - log(l - a )8 1 - a 2 2 2 2 2

I 2 1 + a 1 2 7 a 2 J :::: "8 (l - a ) log -1---a-2 + 4" log(l - a ) + -64-e -1---a-2 + 0.173a .1 I 1 + a 2

(A,4) + 2: (2: - Q) log[ ,",_11 I ,",_.7J· The calculator shows this is negative, as follows: Evaluate 64e(;-a2) + 0.173 at

This is linear in Q so we need only check the endpoints, 0 or 4. a~ = 0.85, ai = 0.82, ar = 0.79, obtaining values :::: 0.442 = f3,:::: 0.397 = f2, 2:::: 0.365 = fl. Let aJ = 0.75. By calculator, k(l - a ) log( :~~~ ) + ~ log(l

Q -- !. 2' a2)+fja2 is:::: -0.01 < owhen a = aj_1 (j = 1,2,3). The statement follows

2 using monotonicity, as with (A5) in Case I.1 ( 1 + a ) 1 I 2(A,S) (AA) = - (l - a2) log --2 + - log(l - a2) + _a . 8 I-a 4 2

2Case 3. 0.5 :::: a :::: 0.75.Q =0:

1 2 ( 1 + a 2 ) 1 2 1 [ 1 + a 2

] SubcaseA. Q:::: ~. Lets satisfy !/J(s) = ..!l-a2

• Thens :::: 2- 1/2 andinA3.11,(A.6) (AA) = "8 (l - a ) log I _ a2 + 2: a + 4" log 2e(l + 2a2) . a

J(s):::: J(2- 1/2):::: 1.7,K(s):::: K(2- 1/2):::: 1.2. InthenotationofA2, (A.5){a 2 is decreasing in a by the remark about negative coefficients. The

2 calculator shows (A.5) < 0 when a 2 = 0.85, so it remains to consider (A.6). (

l-a ~ )(1 )d - x:::: ~ + 1.7 a + 1.2 2: - Q Letting x = 1 - a 2 , we get

1 ell 1 3(2 - x) :::: 2( 1:2a2

+ 1)(~ _Q).(A.6) = - (log - - 2x + -2 x log(2 - x) + - x log - + log[ 2 ]).4 3 2 x (3 - 2x)

Using this to bound the last term in A2.1 and A3.5 for the second term, we getBy concavity, if a > 0 then 4flog f :::: 4a log ~ + 4(t - a) (log ~ - 1) =

2Q 2 1 a 1 ( 1 )~ flog -!e + ~ a. Taking a = 2 log ~ and evaluating log -!e on the calculator, J. < - 10g(1 - a ) - - -- - - - - Q log 2

In - 2 64 1 - a 2 2 2we obtain ~ flog f :::: 0.32f + log ~; hence, using log x :::: x - I, 2

1 2 1 a 1 ( 1) 2- log(l - a ) - - -- - - - - Q log(2(l - a ))1(1 )x(A.6) :::: 4" -2x + 2: x log 2 + 0.32x + 6 _ 4x :::: -0.2x. 4 64 1 - a 2 2 2

1 2 I a 2 log 2 2 2 (A.7) < -log(l-a) - - -- + --(2a -1)Case 2. 0.75 :::: a :::: 0.85. - 4 64 1 - a 2 4


=",===.=."~.,,,= --""""""""'''''-~''''-''''''-''''''''~~~'''''~'='''''~'•..=''''.~-''''''''='.-,~=.====~=..,........."'-~~""-...,." ~""""",.",,,,,,,",,,~=-,,,=,-=-==--,,,==,,,"=,==,,,, ..'~ "".-- - •..' --"--''';';;'~~.;;;:~~~~~j.''=''''''''''';;'-~'''';;;'''=:ii''~=:;;;;;:.&i~;;;;';..•;,; "'liI'i'-~ '~',~':'~ .... ··--'ri'·*'~:J"'=:i"·_-i~!'. IF- ':'-i~r.'·M '..'" {'4i!~ ·'···"'::,r' 'ii'5if"'"'i~: .~-.,<, :'ijj'" - '~'7 'Jm!~I~wi1-:#it;:P&ii;'ril&-"it"i_TaE!iij;j;"jj;;? -.-.• --•• - '" ·--ilj;··!¥~··'jtr1.~-ijt·'ii'''~~;i;

376 COUNTEREXAMPLES WITH HARMONIC GRAD1ENTS IN IR3 377 2

CHAPTER 15

where the last line follows by concavity of the logarithm function on the interval I 2 ( I + a ) I 1 ( 7 ) - (1 - a ) log --2 + - log(4a2(1 - a 2»+ - log - + 0.16a2[!' I]. Also 8 I - a 8 8 32

I 2 ( 1 + a 2 ) I 2 2 21 2 ( I + a 2

) ::s - (1 - a ) log -- - - (2a - I) - 0.18 + 0.100IoU[::S g(1-a )10g l-a2 +2(J2-I)12-3/4(2-d)(4a2(1_a2»-1/4a2. 8 I - a 2 8

2By A3.6, 2 - d ::s ~. By calculator ~ (J2 - 1)12-3/4 (4a2(1_ a 2»-1/4 ::s ~ using log x ::s x - I and evaluating klog -lz. The function k(1 - a ) log( :~:~ ) when a 2 = 0.75, so is concave down in a 2 and so has a unique maximum. Checking values at a 2 =

2 • 0.55,0.56,0.57, we see that the maximum occurs between 0.55 and 0.57. At theI 2 ( 1 + a ) 3 2], t < - (I - a ) log -- + - a maximum0.43 < l-a2 < 0.45, I +a2 < 1.57, so the value is < 0.073 < 0.08,

and ou - 8 - 1 - a2 16

2 I 2 ( I + a ) I 2 I a 2 3 2I < - (I - a ) log -- + - 10g(1 - a ) - - -- + - a I I 2 2 2I < - - - - (2a - I) + 0.16a- 8 I - a 2 4 64 I - a 2 16 - 10 8

log 2 + (2a2 - 1) < O.

4

3 4 I 2 3 2 2 log 2 < - - a - - a + - a + (2a - 1)- Case 4. 0.36 ::s a 2 ::s 0.5. - 8 32 16 4

<0 2If a = 0.36, then ~ (J2 - 1)12-3/ 4 (4a2 (1 - a 2»-1/4 ::s 0.175, so using

where the last line is easily checked since the preceding line is a quadratic 2 - d ::s ~' we have as before polynomial in a 2•

2I 2 I + a 2lout ::s - (I - a ) log --2 + 0.175a .

8 I-aSubcase B. Q ::s !. Estimating lin as in case 1, then rearranging some terms, we find that 2If a ::s !, then v'~-a2 ::: I, and A3.9 shows 1{f(2-1/2) < I, so if 1{f(s) >

I 2 I I ( a 2d )],. < - log( 1 - a ) + - (- - Q) log . v'~-a2 then s ::: 2-1/2 . So we can bound lin as in the argument leading to (A.7),

In - 4 2 2 4(1 - a2)O _ Q) the only difference being at the last step where log(2(1 - a 2»is now positive so

This is increasing in Q since d ::s i, Q ::s !' a 2 .::S ~' so the worst case is Q = !. We obtain

I I a 2 Ii < -1 log( 1 - a 2) + -I log(a--

2d) j. < - 10g(1 - a 2

) - - - In - 4 8 1- a2 In - 4 64 I - a 2

2 23 4 I a 2I 2 ( I + a ) I a 2) + I (a2d)I ::s - 10g(1 - a ) log -- + - log(l - - log __ I < --a - - -- +0.175a.28 I - a 4 8 I - a 2 8 64 I - a 2

2+2(2 1/ - 1)12-3/ 4 (2 - d)(4a2(1 _ a 2»-1/4a 2. lfa2

::: 0.4, this implies I ::s (- ~ (0.4) - (64)~O.6) + 0.175)a2 < O.

2 2 2Using that (4a 2(1 - a »-1/4 is largest at a = ~ (and a calculator for the If 0.36 ::s a ::s 0.4. we modify as follows: If a 2 ::s 0.4, then v',;;a2 ::: 1.22, 2coefficient of d in the last term when a = ~) this is seen to be increasing in d and if r = 0.8, then 1{f(r) < 1.19 by A3.9. So if a 2 < 0.4, then Q > (O.~)2 =

ford < 1 so evaluating 2(2 1/ 2 - 1)12-3/4(2 _ 1)(4.1. !)-1/4 < 0.16 we 0.32, ! Q2 ::: 0.0256, and we can improve the second term in the bound for lin - 8' , 8 4 4 - , get in obtaining

2I 2 I + a 2 I 2 I [ 7a2 ] 3 4 a 2I ::s - (I - a ) log --2 + - 10g(1 - a ) + - log 2 + 0.100 2 I < --a - 0.0256-- + 0 175a8 I - a 4 8 8(1 - a ) 8 I - a 2 •


378 CHAPTER 15

3 0.0256 ) 2 5 ( - 8" (0.36) - 0.64 + 0.175 a

=0.

Now we make estimates that will be used for all the remaining cases. 2We always assume a 5 0.36. Then .J~-a2 2: 1. A3.9 shows 1fr(0.83) 5

1.30 < 1and in A3.11, 1(0.83) 5 ~ and K(0.83) 5 l. It follows that if a 2 5 0.36, then

2 I - a 3 ~ ) ( I )

d-x~ ( --+- +1 --Q. a 2 2 a 2

l-a2

3 ~ d C a2A A2 h . I'We set Aa = --;:;r- + "2 -a- + I an a = ~' .1 t en Imp les

I ( I) a2

(A.S) ],. - -I 10g(1 - a 2) < - - - Q log C - -I --x In 4 - 2 2 a 4 I _ a2 '

where we know x 2: max(Q2, d - Aa ( ~ - Q)).

Claim. ~ (~ - Q) log Ca - ~ 1~:2 Q2 is increasing in Q on [0, ~].

Proof The critical point is Q = QcritZ(a2) = - (l:~2) log Ca' Let t = ~' ...., l-a2

- log( 1 + 1 t+ 1 t 2 )

Then QcritZ = / 4 This is decreasing in t as long as ~ + i t +4 •

2 2~ t < 1, as will be the case since a < 0.36. So QcritZ 2: Qcrid0.36) > 4. •

So in using (A.8) to bound lin we can restrict to the case where Q2 5 d

Aa ( 4- Q). Writing this as a quadraticfor 4- Q it implies 4- Q 5 ~-JI' so. 1(1lin - -I log(l - a 2) 5 max - - - Q) log Ca

4 0::; ~ -Q::;(d- ~ )/(A.-I) 2 2

- ~~ (d - A(~ - Q))a4 I - a 2 2

1 < max ( - - Q) (~ log Ca + Ca)

0::; ~ -Q::;(d- ~ )/(A.-I) 2

2I a-4 --d

I - a 2 '

and now we have the following:

LemmaA4.1. I II 2 la2d la2

(a) If "2 log Ca + Ca < 0, then lin - 4 og(l - a ) 5 - 4 l-a2 5 - (; l-a2 '

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN ]R3 379

(b) If 4log Ca + Ca > O. then

2 1 ( d - ! ) ( 1 ) 1 a],. - - 10g(1 - ( 2 ) < __4 - log C + C - - -- d

In 4 - A - I 2 a a 4 I - a 2 a

2 5 ( I ) I a< -10 C C - --- . -, . ., 2 g a + a 6 I _ a2 .

Proof We have everything but the second inequality in (b), and that will follow if

we know that the quantity on the right side of the preceding inequality is decreasing

as a function of d. For this we show

Lemma A4.2. Thefunction ¢(a) = 1:~2 A.I_I (Ca + 4log Cal is increasing in

a.

Proof Let y = 1 + ~ ~ ; then ...., l-a2

I 1 I ( I ( I ))¢(a) = - (i + - y + I) + - log - y2 + - y + I • 9y 4 2y 9 4

which is increasing in y since y 2: 1 and the logarithm is negative. • We use the calculator to make the following table.

a 2 ¢(a)

0.36 50.223

0.27 50.152

0.2 50.088

0.13 2:0

0.12 :::0

In particular, ¢(a) ::: 0.223 < 0.25 for all a,.which finishes Lemma A4.l. •

2Case 5. 0.2 ::: a ::: 0.36.

Subcase A. 0.27 ::: a 2 ::: 0.36. By A4.1 (b), A4.2, and ¢(,J0.36) ::: 0.223,

2I 2 ( (5)(0.223) I) a1· - - 10g(1 - a ) < - - -

In 4 - 12 6 I - a 2

< (5)(0.223) _ ~) ~ - 12 6 0.73

I 2 <--a

to


381 380 CHAPTER 15

2When a = 0.27, ~ (..;2 - 1) 12-3/ 4 (4a 2 (1 - ( 2»-1/4 ~ 0.182, and in the usual way we get

1 ( I + ( 2 )lout ~ 8" (I _(2) log 1 _ a 2 + 0.182a2

3 4 2 1 2I < --a + 0 182a - -a- 8 . 10

< O.

Subcase B. 0.2 ~ a 2 ~ 0.27. Exactly the same argument, but using ¢ (JO.27) ~ 0.152 and ~ (v'2 - 1)12-3/4(4(0.20)(0.80»-1/4 ~ 0.192, gives

21lin - 4: Iog(l - ( 2) ~ ( (5)(0.152) I) a

6 0.8

I 2 < --a - 8

83

a 4 + 0 192 . -aI ~ - - a 2 - I 2 8

< O.

Case 6. a 2 ~ 0.2, but ¢ (a) ::: O.

By the table, a 2 ::: 0.12 and ¢ (a) ~ 0.088. We have

21lin - 4: Iog(l - ( 2) ~ ( (5)(0.088) I) a

12 6 0.88

~ -0.147a2

2 lout ~ ~ 8 (l - ( 2) 1og(I --+ a ) + V where by Lemma A2.2,

1 - a 2 '

100 I V ~ 12-3/4(4a2(1 - ( 2»-1/4a 2 [1/t(r)2

21 Jl + r

+ :2((2a2-1+a2C~ + r~))Lrdr

~ 12-3/4 (4a2(l - ( 2»-1/4a 2 (I + r 2)-1/2[1/t(r)2 __ 21°

1

r

6

+ 2r2 + 4J dr + r .

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN JR3

The last line was obtained as follows: The integrand on the previous line was

increasing in a-this uses a 2 ~ 0.2 which implies 2a2 - 1 +a 2(~ + ~) < 0so we could replace a 2 by 0.2. We then changed variables r --+ I/r and used A3.I.

If the (last) integrand is nonzero, then

2 6 2(A.9) 1/t(r) ::: 2" - 2r - 4.

1

r

Since 1/t is increasing and fz - 2r2 - 4 decreasing, the set where (A.9) happens is an interval [R, I). By A3.9 and calculator, R ::: 0.87. So

1 1 V ~ IT3/ 4 (4a2 (l - ( 2»-1/4a 2 (1 + r 2)-1/2 -1/t(r)2dr

0.87 r

11 1

~ IT3/4(4a2(1 _ ( 2»-1/4a 2(1 + (0.87)2)-1/2 -1/t(r)2dr. 0.87 r

Now by (A3.5), (A3.I), (A3.2), (A3.6)

11 0 87

1 I 1 .1 1-1/t(r)2dr ~ -1/t(r)2dr - rdr 0.87 r oro

4 (0.87)2 < - 3 2

If we use a 2 ::: 0.12 to estimate (4a2(1 - ( 2»-1/4, the calculator now gives

V ~ 0.I386a2 ~ 0.I4a2 and, therefore,

3I < -0.I47a2 - _a4 + 0.14a 2

- 8

< O.

Case 7. ¢(a) < O.

Then a 2 < 0.13. By Lemma AA.I,

21 1 a j. < - Iog( I - ( 2) - - --.

10 - 4 6 1 - a 2

We estimate lout using the first alternative in A2.2 and dropping the 0+:22)r2

term, i.e.,

I ( 2 ( I + ( 2 )lout ~ - (l - ) log --2 + V 8 I - a

V ~ ~ a 2 {00[1/t(r)2 + ~ (2a 2 - 1 + a 2( ~ + ~ ))J rdr2 2 44 11 a r r +


383 382

CHAPTER 15

1 [1 2 2a2 I 12= 4: a2 Jo [t/r Cr)2+ a2( r2- +a (2+ r2»))L-;dr.

Nowa 2 < 0.13 implies that

2 (2a2 - 2 21 2 )

2" 2 + a C2 + r ) < 2" C5a2 - 1) < -5.38 a r a

and

1 drV::s -a21 1

[t/rCr)2-5.38]+-.4 0 r

Let s satisfy t/rCs)2 = 5.38. By A3.9, s > 0.96 so

1 [1V ::s 4:a2CO.96)-2 s Ct/rCr)2 - 5.38)rdr

1 (3 ) 1 ::s 4: a 2CO.96)-2 2" C5.38) 1/2 + 1 2" (l - s2)

by A3.II and the estimates J ::s i, K ::s 1 that we have been using. So

21 (3 ) 1 V ::s 4: a CO.96)-2 2" C5.38)1/2 + 1 2" (l - CO.96)2)

::s 0.05a2.

We conclude that

21 a 3J < - 6" 1 _ a2 Sa4 + 0.05a2

< 0,

and no more cases remain.

University ofCalifornia, Berkeley

REFERENCES

[I] A. B. Aleksandrov. "Inner functions: results, methods, problems." ICM Proceedings. Berkeley, 1986.

[2] A. B. Aleksandrov, P. Kargaev. Letter to author, 1988. To appear. [3] 1. Bourgain. "On the Hausdorffdimension ofharmonic measure in higherdimensions."

Inv. Math. 87 (1987),477-483. [4] J. Bourgain, T. Wolff. "Note on gradients of harmonic functions in dimension ~ 3."

Colloq. Math. 40/41, I (1990), 253-260. [5] A. P. Calderon, A. Zygmund. "On higher gradients of harmonic functions." Studia

Math. 24 (1%4), 211-226.


[6] L. Carleson. "On the support of harmonic measures for sets of Cantor type." Ann. Acad. Sci. Fenn. 10 (1985),113-123.

[7] B. Dahlberg. "Estimates of harmonic measure." Arch. Rat. Mech. Anal. 65 (1977), 275-288.

[8] S. Gleason, T. Wolff. "Lewy's harmonic gradient maps in higher dimension." Comm. Part. Diff. Eq. 16 (1991),1925-1968.

[9] D. Jerison, C. Kenig. "Boundary behavior of harmonic functions in nontangentially accessible domains." Adv. Math. 46 (1982), 80-147. .

[10] . "Boundary value problems on Lipschitz domains." In Studies in Partial• Differential Equations, edited by W. Littman. MAA Studies in Mathematics, Vol. 23, 1982.

[11] P. Jones. "A geometric localization theorem." Adv. Math. 46 (1982), 71-79. [12] P. Jones, T. Wolff. "Hausdorff dimension of harmonic measures in the plane." Acta

Math. 161 (1988), 131-144. [13] R. Kaufman, J.-M. G. Wu. "On the snowflake domain." Arkiv for Math. 23 (1985),

177-183. [14] J. L. Lewis. "Note on a theorem of Wolff." In Holomorphic Functions and Moduli,

edited by D. Drasin, et al. MSRI Publication, Vol. 10-11. Springer Verlag, 1988. [15] N. G. Makarov. "Distortion of boundary sets under conformal mappings." Proc.

London Math. Soc. 51 (1985), 369-384. [16] N. Martin, J. England. "Mathematical Theory of Entropy." Encyclopedia of Mathe

matics and its Applications, Vol. 12. Addison-Wesley, 1981. [17] V. G. Maz'ja, V. P. Havin. "The Cauchy problem for Laplace's equation." Vestnik

Leningrad Univ. 23, 7 (1968),146-147. [18] S. N. Mergelyan. "Harmonic approximation and approximate solution of the Cauchy

problem for Laplace's equation." (In Russian.) Uspekhi Mat. Nauk (N. S.) 11 (1956), no. 5 (71), 3-26.

[19] B. Oksendal. "Null sets for measures orthogonal to R(X)." Amer. J. Math. 94 (1972), 331-332.

[20] . "Brownian motion and sets of harmonic measure zero." Pacific J. Math. 95 (1981),179-192.

[21] N. V. Rao. "Uniqueness theorems forharmonicfunctions." Math. Notes USSR 3 (1968), 159-162.

[22] Ya. G. Sinai. Introduction to Ergodic Theory. Translated by V. Scheffer. Princeton University Press, 1976.

[23] M. Spivak. "A Comprehensive Introduction to Differential Geometry." Vol. 4, Publish or Perish, Berkeley, 1979.

[24] E. M. Stein. Singular Integrals andDifferentiability Properties ofFunctions. Princeton University Press, 1970.

[25] E. M. Stein, G. Weiss. "On the theory of harmonic functions of several variables, I: the theory of HP spaces." Acta Math. 103 (1960), 25~2.

[26] A. Uchiyama. "A constructive proof of the Fefferman-Stein decomposition of BMO(Rn)." Acta Math. 148 (1982), 215-241.

[27] . "The singular integral characterization of HP on simple martingales." Proc. Amer. Math. Soc. 88 (1983), 617~21.

[28] . "The Fefferman-Stein decomposition of smooth functions and its applications to HP(Rn)." Pacific J. Math. 115 (1984), 217-255.


~":~~~;'''.'~~",-",==~~'''_'''''C'''''''-'''-,,"~'';;'''~~-=-~-_ :=-__-_-:-::=~~~~~_=_".,:_~._=::::::::=__=:::.:::_ ..,.~.-.:='O;~.0,,~;;:;;..;.,,- '_·"'''C';;;·'="=-':-'i'''''c'''''~0'',~~~~~..,'-H,:i'''''""",~"",~·"·,_.. -·,,,~~-,...,~~:~._~,,"·:v,..'''',","'_:''_.''', C"'-~"'-~-":;;-'-~"'""'-"'"'"_'''''_-''=' ;'-'""'.c-,-''''' • _ _ "'''''"~:""_._'"'':'''''~'".rc.,,,-__·" - .. ""c~.__ -"',-- ._-_._'-.,.~-~""---

384 CHAPTER 15

[29] G. Verchota. "Layer potentials and regularity for the Dirichlet problem for Laplace's equation on Lipschitz domains." J. F'unct. Anal. 59 (1984), 572--611.

[30] W. S. Wang. To appear. [31] T. Wolff. "Gap series constructions for the p-Laplacian." Typescript, 1984. [32] J.-M. G. Wu. "On singularity of hannonic measure in space." Pacific J. Math. 121

(1986),485-496.

, .

Documents

· Princeton Mathematical Series EDITORS: LUIS A. CAFFARELLI, JOHN N. MATHER, and ELIAS M. STEIN I. The Classical Groups by Hermann Weyl 3. An Introduction to Differential Geometry