Analysis-of-Several-Complex-Variables-Takeo-Ohsawa.pdf

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MATHEMATICALMONOGRAPHS

Volume 211

Analysis of SeveralComplex VariablesTakeo Ohsawa

American Mathematical Society

Titles in This Series

211 Takeo Ohsawa, Analysis of several complex variables, 2002210 Toshitake Kohno, Conformal field theory and topology. 2002209 Yasumasa Nishlura, Far-from-equilibrium dynamics, 2002208 Yukio Matsumoto, An introduction to Morse theory. 2002207 Ken'ichi Ohshika, Discrete groups. 2002206 Yuji Shimizu and Kenji Ueno, Advances in moduli theory, 2002205 Seiki Nishikawa, Variational problems in geometry, 2001204 A. M. Vinogradov, Cohomological analysis of partial differential

equations and Secondary Calculus, 2001203 Te Sun Han and Kingo Kobayashi, Mathematics of information and

coding, 2002

202 V. P. Maslov and G. A. Omel'yanov, Geometric asymptotics fornonlinear PDE. I. 2001

201 Shigeyuki Morita, Geometry of differential forms, 2001200 V. V. Prasolov and V. M. Tikhomirov, Geometry, 2001199 Shigeyuki Morita, Geometry of characteristic classes, 2001198 V. A. Smirnov, Simplicial and operad methods in algebraic topology,

2001

197 Kenji Ueno, Algebraic geometry 2: Sheaves and cohomology, 2001196 Yu. N. Lin'kov, Asymptotic statistical methods for stochastic processes,

2001

195 Minoru Wakimoto, Infinite-dimensional Lie algebras, 2001194 Valery B. Nevzorov, Records: Mathematical theory. 2001193 Toshio Nishino, Function theory in several complex variables, 2001192 Yu. P. Solovyov and E. V. Troitsky, C'-algebras and elliptic

operators in differential topology. 2001191 Shun-Ichi Amari and Hiroshi Nagaoka, Methods of information

geometry, 2000190 Alexander N. Starkov, Dynamical systems on homogeneous spaces,

2000189 Mitsuru Ikawa, Hyperbolic partial differential equations and wave

phenomena, 2000

188 V. V. Buldygin and Yu. V. Kozachenko, Metric characterization ofrandom variables and random processes. 2000

187 A. V. Fursikov, Optimal control of distributed systems. Theory andapplications, 2000

186 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Numbertheory 1: Fermat's dream, 2000

185 Kenji Ueno, Algebraic Geometry 1: From algebraic varieties to schemes,1999

184 A. V. Mel'nikov, Financial markets. 1999

Analysis of SeveralComplex Variables

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Editorial BoardShoshichi Kobayashi (Chair)

Masamichi Takesaki

TAHENSU FUKUSO KAISEKI(A MODERN INTRODUCTION TOSEVERAL COMPLEX VARIABLES)

by Takeo OhsawaCopyright © 1998 by Takeo Ohsawa

Originally published in Japaneseby Iwanami Shoten, Publishers, Tokyo, 1998

Translated from the Japanese by Shu Gilbert Nakamura

2000 Mathematics Subject Classification. Primary 32Axx.

ABSTRACT. This is an expository account of the basic results in several complexvariables that are obtained by L2 methods.

Library of Congress Cataloging-in-Publication DataOhsawa, T. (Takeo)

[Tahensu fukuso kaiseki. English]Analysis of several complex variables / Takeo Ohsawa ; translated by Shu

Gilbert Nakamura.p. cm. - (Translations of mathematical monographs, ISSN 0065-9282

v. 211)(Iwanami series in modern mathematics)Includes bibliographical references and index.ISBN 0-8218-2098-2 (soft cover : acid-free paper)1. Functions of several complex variables. 2. Mathematical analysis. I. Title.

II. Series. III. Series: Iwanami series in modern mathematics.

QA331.7.03713 2002515'.94-dc2l 2002019351

© 2002 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rights

except those granted to the United States Government.Printed in the United States of America.

® The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Information on copying and reprinting can be found in the back of this volume.Visit the AMS home page at URL: http://vwv.ams.org/

10987654321 070605040302

Contents

Preface

Preface to the English Edition

Summary and Prospects of the Theory

ix

xi

Chapter 1. Holomorphic Functions 1

1.1. Definitions and Elementary Properties 1

1.2. Cauchy-Riemann Equations 81.3. Reinhardt Domains 18

Chapter 2. Rings of Holomorphic Functions and 8 Cohomology 232.1. Spectra and the 8 Equation _ 232.2. Extension Problems and the 8 Equation 252.3. c? Cohomology and Serre's Condition 27

Chapter 3. Pseudoconvexity andPlurisubharmonic Functions 35

3.1. Pseudoconvexity of Domains of Holomorphy 353.2. Regularization of Plurisubharmonic Functions 413.3. Levi Pseudoconvexity 47

Chapter 4. L2 Estimates and Existence Theorems 554.1. L2 Estimates and Vanishing of 8 Cohomology 554.2. Three Fundamental Theorems 75

Chapter 5. Solutions of the Extension andDivision Problems 83

5.1. Solutions of the Extension Problems 835.2. Solutions of Division Problems 875.3. Extension Theorem with Growth Rate Condition 935.4. Applications of the L2 Extension Theorem 100

VII

viii CONTENTS

Chapter 6. Bergman Kernels 1056.1. Defi nitions and Examples 1056.2. Tran

to Hsformation Law and an Applicationolomorphic Mappings 107

6.3. Bou ndary Behavior of Bergman Kernels 110

Bibliography 115

Index 119

Preface

This book does not intend to explain the whole theory of complexanalysis in several variables as it stands today. The goal of the bookis to introduce methods of real analysis and see these methods producea variety of global existence theorems in the theory of functions basedon the characterization of holomorphic functions as weak solutions ofthe Cauchy-Riemann equations.

Chapter 1 starts with the definition and elementary propertiesof holomorphic functions, and in Chapter 2 the problem of extensionof functions and the division problem are converted to the problemof solving the Cauchy-Riemann equations of inhomogeneous form.These are called a equations. The theme to observe up to Chapter 3is that the solvability of the a equation on an open set 11 in C" imposeson 11 a geometric restriction called pseudoconvexity. Chapter 4 shows,to the contrary. the solvability of the a equation on a pseudoconvexopen set; and, as an application, we generalize to several variables theMittag-Leffler theorem, Weierstrass' product theorem, and the Rungeapproximation theorem, which are included in many textbooks forcomplex analysis in one variable. This approach is called the methodof L2 estimates. By virtue of this method, in Chapter 5, we solve theextension and division problems. The point of this argument is thatthe solutions are evaluated by the estimates, and thus the applicationimmediately becomes wider. The content stated so far is like the viewdown from a high place, while Chapter 6 invites the reader to climbthe untrodden mountains, which is, so to speak, the view of the greatmountains gazed upon from the base camp. The reader will see theauthor break down from exhaustion there. It is left to the reader todecide whether he has fallen down forward or backward.

Several people offered help with the present publication. In par-ticular, Professor Kazuhiko Aomoto, a great pioneer in analysis, rec-ommended that I write the book and provided useful advice. Theeditorial board of Iwanami Shoten, Publisher, paid careful attention

ix

x PREFACE

to the appearance of the book. Mr. Tetsuo Ueda and Mr. HaruoYokoyama pointed out many mistakes. I am grateful to these people.I regret that just before the approval of the book, my teacher, Pro-fessor Shigeo Nakano, who had saved me from dropping out and in-troduced me to this field, passed away. I would like to offer this bookon the altar with my respect. Further, criticism on the book fromreaders will be considered as my teacher's reprimands from heaven,which I look forward to hearing.

Takeo OhsawaMay 1998

Preface to the English Edition

Voluminous textbooks have already appeared in several complex vari-ables. In this concise booklet the author assumes a basic knowledgeof analysis at the undergraduate level, and gives an account for the L2theory of the 6 equation. Emphasis is put on recent results which havebrought us a deeper understanding of pseudoconvexity and plurisub-harmonic functions, and opened a major new way of developing com-plex analysis.

Xi

Summary and Prospects of the Theory

The concept of analytic function was introduced by L. Euler, J. L.Lagrange and others during the 18th century, and it was A. L. Cauchy,C. F. Gauss, G. F. B. Riemann, K. T. W. Weierstrass and others ofthe 19th century who made the theory of complex functions of onevariable as complete and elegant as it is today.

Entering the 20th century, breakthroughs to the world of complexfunctions of several variables were made by F. Hartogs, E. E. Levi,P. Cousin and others. The problems which they proposed in the fieldwere extremely difficult at the time, but it did not take even a halfcentury to settle all these problems and establish the foundations forthe theory of analytic functions of several variables.

As a matter of common knowledge, all the core problems in thisarea were solved by one mathematician. His name is Kiyoshi Oka(1901-1978).

He grasped all the central problems as a system in the course of so-lution and gave the last polish to this system by solving affirmativelythe so-called Levi problem, which asserts the crucial proposition thatif a domain satisfies a geometric condition called pseudoconvexity,one can construct a holomorphic function such that every boundarypoint of the domain is an essential singularity of this function.

The methods created by Oka were of striking originality. (One ofthe methods was neatly named "the hovering principle.")

On the other hand, some of the methods contained expressionsthat were difficult to understand and became obstacles which werehindrances to the succeeding development.

However, it is fortunate that Oka's theorems have been widelyaccepted today as a lucid fundamental theory, due to H. Cartan'sformulation by virtue of cohomology with coefficients in sheaves, andH. Grauert and L. Hormander's application of methods of functionalanalysis.

X iii

xiv SUMMARY AND PROSPECTS OF THE THEORY

analysis

Theory of analytic functionsof several variables

Based on Oka's work, the theory of analytic functions of severalvariables has developed in a variety of directions. Oka himself sug-gested how groups of problems should be located by using the schemaabove.

Borrowing this schema of his, the subjects raised in this bookmostly belong to the intersection of the theory of analytic functionsof several variables with mathematical analysis. The book hardlytouches the other four parts, as noted in the Preface.

As for mathematical analysis, what comes to mind first as anobject is those functions dependent on space and time variables thatsatisfy some sort of differential equations. In order to describe prop-erties of these functions and determine their exact formulae, we needto analyze the distribution of values taken on by these functions andtheir behavior at points at infinity or singular points, as is often ex-perienced in solving even elementary problems. This methodologyforms a complete theory by restricting the range of functions underconsideration to analytic ones. This is exactly the theory that wasfounded by Weierstrass and others within the theory of complex func-tions of one variable, and the basis for this methodology consists of theMittag-Lefer theorem, the Weierstrass product theorem, the Runge

SUMMARY AND PROSPECTS OF THE THEORY xv

approximation theorem, and so forth. The Oka theory has had thisbasis transplanted into the soil of the theory of analytic functions ofseveral variables. On this account, the purpose of this book is to tellthe detailed story about how the basis of one variable took root inthe soil of several variables, while deferring the sight of what treesgrew and blossomed on it.

In what follows, the precise contents of the book are explained.Chapter 1 gives the definition and fundamental properties of holo-

morphic functions. P. Montel's theorem and Weierstrass' double se-ries theorem guarantee that the space of holomorphic functions isclosed with respect to the topology induced by uniform convergenceon compact sets. This is similar to the completeness of the real num-bers, and is quite fundamental in deriving existence theorems for holo-morphic functions.

From thorough investigation on the structure of the space of holo-morphic functions by methods of real analysis, one can derive the fun-damental existence theorem. This is grounded on the characterizationof holomorphic functions as weak solutions of the Cauchy-Riemannequations in the sense of distributions. Namely, this allows the spaceof holomorphic functions to be identified with a closed subspace in thespace of locally square integrable functions, where functional analyticmethods are applicable.

As to individual problems, I. M. Gel'fand, A. Grothendieck andothers pointed out that geometric problems of specific spaces can beinterpreted as algebraic problems of the function rings on these spaces,and the latter formulation offers an unbroken vista to approach theproblems. This relies on the general tendency that the ring structurecan be studied well through its ring extensions. From this point ofview, the fundamental problem of the ring of holomorphic functionsconcerns the relation between the set of all the maximal closed idealsof the ring and the domain of definition for the functions. More pre-cisely, the question remains whether 1 belongs to a closed ideal gen-erated by a system of holomorphic functions which have no commonzero point, and also whether one can construct a holomorphic functionwhich takes values determined beforehand over a given discrete set.In Chapter 2, these problems are converted into the Cauchy-Riemannequations of inhomogeneous form, or the so-called a equation. It istoo early to treat the a equation to the fullest, but we introduce theconcept of '5 cohomology group and make an elementary observationon conditions to solve the above problems. This observation results

xvi SUMMARY AND PROSPECTS OF THE THEORY

in a necessary condition that the function ring must contain a func-tion which cannot holomorphically extend beyond a given boundarypoint of the domain of definition. An open set that has this propertyis called a domain of holomorphy, and Chapter 3 connects this toa certain geometric concept called pseudoconvexity. The definitionand fundamental properties of plurisubharmonic functions, a gener-alization of subharmonic functions to several variables, are described.This is done by Oka in a 1942 paper based on Hartogs' results. Inaddition, we prove basic results on the regularization of plurisubhar-monic functions, as differentiable plurisubharmonic functions becomeimportant later. The approximation theorem due to J.-P. Demaillybelongs to the same family of results and hence is introduced here.However, this is rather a deep result, and the proof of the theorem isnot elementary and thus is postponed until Chapter 5. We also in-troduce the concept of Hartogs function, due to Bochner and Martin.This enables us to feel as if we were observing the development of theLevi problem around 1948.

In case an open set in Cn has a smooth boundary, the pseudocon-vexity implies some property of the boundary as a real hypersurface.This is what is called the condition of Levi pseudoconvexity. Opensets of this sort enable one to argue minutely about the boundarybehavior of holomorphic functions and mappings. This is a subjectof Chapter 6, but Chapter 3 also prepares for this subject._ Chapter 4 explains a new methodology for L' estimates of thea operator. We begin with basics of closed operators on a Hilbertspace, establish estimates involving the a operator and its adjointoperator on the completion of the space of differential forms withrespect to the L2 norm with some weight function, and from theseestimates derive the existence theorem on solutions of the a equationin Theorem 4.11. Further, we apply the theorem and generalize theM Iittag-Leffier theorem, the Weierstrass product theorem, and theRunge approximation theorem to several variables. The thread ofour argument itself is not different at all from that of Hormander'sbook [27), the standard textbook in the theory of analytic functionsof several variables. But it is worth emphasizing that what is differentbetween our approach and his is the vehicle in which we are traveling,in spite of the same path.

That is to say', the calculations that provide the foundation ofthe arguments in this book are a re-formation of S. Nakano's for-mulae used for the proof of the vanishing theorem of cohomology in

SUMMARY AND PROSPECTS OF THE THEORY xvii

the theory of complex manifolds, and these calculations contain newestimates. Differing from Hormander's, in the background of our esti-mates, there is W. V. D. Hodge's book [24] that discusses the theoryof harmonic integrals on projective algebraic manifolds, grounded onthe symmetry of complex Laplacians with respect to the complex con-jugate. It would be of no use if the new vehicle were cheap, but inChapter 5 by virtue of this new method we show in Theorem 5.10 thatholomorphic functions defined on a closed subspace can be extendedunder an estimate given by the L2 norm with some weight function.This estimate is what Hormander's method could not reach, and isthe main theme of this book. The calculations of Chapter 4 were,in fact, designed by the author together with Kensho Takegoshi inorder to prove this extension theorem. (Refer to [37].) The proof ofDemailly's approximation theorem is an application of the extensiontheorem.

Although we cannot treat this topic in this book, the extensiontheorem has recently shown to have some applicability to some subtleproblems in algebraic geometry and the theory of complex manifolds.(Refer to [44], [46], and [47].)

Chapter 5 touches on H. Skoda's division theorem. With this, allthe problems posed in Chapter 2 have been solved. We do not givefull details of the proof of Skoda's theorem, but only introduce theessential part of the argument. The author would put an emphasisalso on this point as a special feature of this book which is not foundin any other books.

We throw in Chapter 6 as an extra, "just for fun". It containsa lighthearted approach to the difficult open problem of determiningthe singularity of the Bergman kernel, which is a reproducing kernelof the space of L2 holomorphic functions. After describing C. Fef-ferman and Bell-Ligocka's example of applying the Bergman kernelto holomorphic mappings, we show our own recent results about theBergman kernel on a general Levi pseudoconvex domain. This mightseem too scanty and miserable, but may rouse those who want to dosome research in this field from this point forward. That is why theauthor did not shrink from "cutting a sorry figure".

CHAPTER 1

Holomorphic Functions

To begin with, we define holomorphic functions as convergent powerseries, describe elementary properties of them, and achieve the maingoal of characterizing them as weak solutions of Cauchy-Riemannequations. Our proof restricts itself to locally square integrable func-tions, while the concept of holomorphy of weak solutions is knownto be extendible as far as hyperfunctions. This choice is made so asto keep our argument as simple as possible. At the end, we mentionthe Reinhardt domain, finding it important to study some propertiespossessed by domains of convergence for power series.

1.1. Definitions and Elementary Properties

Let C be the complex plane and consider the n-dimensional com-n

plex number space Cn := C x x C. Let z = (zl, , zn) be thecoordinate system of Cn. Essentially z is a vector valued functionon Cn, but we also denote a point of Cn by the same symbol z aslong as there is no fear of confusion. Write x2j _ 1 and x2j for thereal part Re zj and the imaginary part Im zj of zj, respectively. LetR be the real line, and identify Cn with R2n by the correspondence

(z1,... ,zn) ~' (x1,x2,... ax2n)

For z E Cn, set

IzImax = max Izj1 and IzI = (1z112 + + Jzn12)'i

These are norms on C' that are topologically equivalent to each other.Let Z4. be the set of all nonnegative integers, and for an element

a= (al,- an) in Z n' setn n

a! f aj! , (a) > aj,j=1 j=1

z° z1°1...zn°^.

2 1. HOLOMORPHIC FUNCTIONS

Furthermore, we use the following notation:

a 1( a Vaaz2 2 (9X- 2i _ 1 ax2i

a 1 ( a+ a

azj 2 8x2j-1 ax2j

Also, f o r $ = (Q1. , )32n) E Z+1, let

From now on, let (1 denote a nonempty open set in

;31 (aan)$2"

C.

DEFINITION 1.1. A complex valued function f on (1 is holomor-phic if for each point a of P. there exists a power series

P. (z) := E c. (z - a)° with c" E CaEZ+

that converges to f on some neighborhood of a, where the convergenceis regarded with respect to some linear order of Z+, or a bijectionZ+ E) k a(k) E Z+. f is called antiholomorphic if the complexconjugate of f, z'-- f(z), is holomorphic.

Let A(R) denote the set of all holomorphic functions on R.First, let us describe briefly the most elementary properties of

holomorphic functions:1. A(Q) is a C algebra with the usual four rules of arithmetic.

(This is obvious from the definition.)2. Holomorphic functions are of class C'°.

In fact, (2) is included in the following proposition:

PROPOSITION 1.2. If a power series

E b"(z - a)" with 6" E C"EZ+

around a point a = (a1, , a point a' = (a...... aft)with a' # aj (1 < Vj <= 9a), then for any $ E Z+ , a power series

6"(z - a)°Y°EZ+

1.1. DEFINITIONS AND ELEMENTARY PROPERTIES 3

converges absolutely and uniformly on compact subsets of{zEC' Ijzj_ail<la- aiIforI j n}.

In particular, for f E A(fl) and a E fl,

f(z) _ E f ()(z - a)°

0,EZ+

on some neighborhood of a, uhere f (:=8z()°f.

PROOF. This follows from the comparison with geometric series.The reader should be familiar with the technique from the theory ofcomplex analysis of a single variable.

COROLLARY 1.3 (Theorem of Identity). If f E A(Q) \ {0} and flis connected, then f -1(0) has no interior point.

PROOF. This follows from the fact that the set

{a E f2 l f(°)(a) = O for b'a E Z}}

is both open and closed in P. D

When f E A(fl) \ {0}, a E fl and f (a) = 0, we call

inf{(a) I f(°)(a) 54 0}

the order of the zero of f at a.

Before proceeding further, we explain holomorphic mappings,polydiscs, and complex open balls.

A mapping F = (fl? ... , fm) from P to an open set fl' in C"'is said to be holomorphic if every component fj of F is a holomor-phic function on P. From Definition 1.1 and Proposition 1.2 (inparticular, absolute convergence of power series), it follows that thecomposite of holomorphic mappings is holomorphic. In particular.1

fE A (fl \ f (0)) if f E A(Q) \ {0}. A holomorphic mapping

F : P fl' is said to be biholomorphic if it has the holomorphic in-verse mapping F-1 : fl' P. fl and fl' are, by definition, holomor-phically equivalent to each other if there is a biholomorphic mappingfrom fl to fl'. A bijective holomorphic mapping between domainsis biholomorphic (see § 5.4 (a)). Biholomorphic mappings from fl toitself are called holomorphic automorphisms. They form a group withthe composition of mappings being the group product. This is calledthe holomorphic automorphism group of fl and denoted by Ant R.

4 1. HOLOMIORPHIC FUNCTIONS

An n-dimensional polydisc around a = (a,,... , an) E Cn (or simplyan n-polydisc) is, by definition, a nonempty set

{zECnI1zj-ajI <rj for'<j<n},

and is denoted by A(a, r), where r := (rI, , rn). For the sake ofbrevity, we write A for 0(0, 1). It is clear that 0(a, r) is holomorphi-

n

cally equivalent to On := A x x A.A subgroup of Aut On whose elements fix the origin is isomorphic

to a semidirect product of U(1)n and the n-dimensional symmetricgroup 6n. In fact, if F E Aut An with F(0, , 0) = (0, . , 0), then

N

the application of (1.2) below to the N times composite F o o Fof F shows that each component of F must be a linear form withrespect to z.

Next, for R > 0 we call an open set {z E Cn I Iz - al < R} acomplex n-dimensional open ball around a of radius R (or simply an n-open ball) and denote it by B(a, R). n-open balls are holomorphicallyequivalent to each other. We express 13((0, . , 0), 1) by 3n for short.

As in the case of Aut On, by knowing that elements of Aut 3nthat fix the origin are linear, we obtain

{o E Aut l3n I Q(0) = 0} = U(n).

If n > 2, clearly U(n) is not equal to the semidirect product of U(1)1and Can. From this, it is seen that in general On and 3n are notholomorphically equivalent to each other'.

Aut 13n acts on l3n transitively. To see this, it is sufficient to notethat l3n is holomorphically equivalent to the open set

n

Dn c=( 1,...,(n)ECn Im(I>EICi12j=2

under the Cayley transformation:

(I - -,/--1 2(2 2(nz1

_(1+V-122

1In general, the group of holomorphic automorphisms on a bounded domainis known to be a Lie group with respect to the compact open topology (see 1331).

I.I. DEFINITIONS AND ELEMENTARY PROPERTIES 5

In fact, the transitivity of Aut B' is obvious from the facts thatU(n) C Aut B" and Aut Dn contains the transformations:

fort>0,((1 +t,(2i ,(n) fort ER.

One of the holomorphic automorphisms on Bn that map a pointw (0, , 0) to the origin is given by the formula

1 --I w (z - (iwl2) w) - w + (Iwl2) w

1 - (z, w) '

nwhere (z, w) E zjwj.

j=1It is an interesting task to calculate the groups of holomorphic

automorphisms on various bounded domains, but we do not studymore than these examples in this book.

Let us return to the exposition of elementary properties of holo-morphic functions. We wish to describe topological properties of thering A(.R) of holomorphic functions. For this purpose, it is necessaryto clarify the range in which the Taylor expansion of a holomorphicfunction on !l is convergent.

An immediate consequence from Proposition 1.2 is that the sumof a power series that converges on 0(a, r) is a holomorphic functionon A (a, r). The converse statement to this does not hold for realanalytic functions in general, and this difference between holomorphicand real analytic functions is fundamental.

PROPOSITION 1.4. A holomorphic function on A(a, r) is equalto the sum of a power series whose convergence holds on the samedomain 0(a, r).

PROOF. It is enough to show the proposition for 0(a, r) = An.Let

f (z) = E caza with c,, = f (0)n a!

be the power series expansion of an element f in A(A") at the origin.Since there exists a positive number a such that the power seriesconverges for IzImax < c, cQ is expressed by

Zw 2a

(1.1) cQ = I f (re`B' , ... , re'8^) r-(a) e-i(a,e) d9,


where r is an arbitrary real number with 0 < r < e, and we set0:= (01, - ,B") and

However, since f is holomorphic on 0", it follows that the righthand side of (1.1) takes on finite determinate values for all r with0 < r < 1, and is of class C" with respect to r. Hence, (1.1) holdsfor all r with 0 < r < 1.

From this we obtain the estimate of cQ:

(1.2) Ic0I < sup If(z)Ir-(a) for 0 < r < 1.IZ-Imax=*

Therefore, >2c,, z° converges on 0". l7Q

The estimate (1.2) implies a fundamental fact of the topology ofA(11):

THEOREM 1.5 (Alontel's Theorem). A sequence of holomorphicfunctions on i2 that is uniformly bounded on compact subsets of .Rcontains a subsequence that converges uniformly on compact subsetsof 12.

PROOF. Let {fk}A I C A(Q) be a sequence in the theorem. Foran arbitrary 0(a, r) C (1 (relatively compact), the uniform bounded-ness on 0(a, r) implies that

Al := sup{ Jfk(z) I I z E 0(a, r) f o r k = 1, 2. . } < c.

From (1.2), it follows that for any j = 1. , 11,

supk

afk(z)azj

<2TI forzEA(a,2)

Therefore, the Ascoli-Arzela theorem concludes that { fk}k 1 has asubsequence converging uniformly on compact subsets of Q. 0

THEOREM 1.6 (Weierstrass' Double Series Theorem). If a se-quence of holomorphic functions on !2 converges uniformly on com-pact sets in .f1, then the limit function is holomorphic on P as well.

PROOF. If fk - f uniformly on compact sets in .R and fk EA(Q), then, as in the proof of Montel's theorem, for any a E 7L+,there exists f(') such that fka) - f(') uniformly on compact setsin (2. Also, if we take 0(a, r) C !2, then the application of (1.2) toA(a, r/2) and the uniform convergence of fk to f on 0(a, r/2) show

1.1. DEFINITIONS AND ELEMENTARY PROPERTIES 7

that for any e > 0, there exists a positive integer N such that, forany z E 0(a, r/2),

E 1Aa)i a) (z - a)a(et) >N a!

and

Hence,

<2

for k =

Ifk(z)-f(z)I<2 for k>N.

(a)

z f (a) z - aa

(a) a< I : f,a;)(z

(a)<N

+ fk(a) a)a

(a)<N

<2+2+ E f(a)(a)(z - a)a

(a)<N a!

Accordingly, as k -+ oo, it follows that

f (a)(a) a )c'

(a)<N a!< 6.

- a) a

(a) aTherefore, the power series E f () (z-a)' converges on 0(a. r/2),

a.and its sum is equal to f (z). 0

From now on, the origin (0, . 0) is denoted simply by 0 if thereis no fear of confusion.

THEOREM 1.7. If 17 is connected, nonconstant holomorphic func-tions on .R are open mappings.

PROOF. For f E A(il) \ C and a E C, take a holomorphic map-ping h : i. -+ Si satisfying h(0) = a and f o h E A(s) \ C. Then onsome neighborhood U of 0, there exists a decomposition

f(h(()) = (mg(() + f(a) for ( E U,

where m E N, the set of positive integers, and g E A(U) with g(0) # 0.Since the mapping --' (m is locally a differentiable homeomor-phism except for ( 0. there is a holomorphic function u satisfy-ing f (h(()) - f (a) _ ((u(())"' on some neighborhood of 0. Since


u(O) 54 0, the inverse mapping theorem implies that Cu(() has a dif-ferentiable inverse on some neighborhood of 0. Therefore, Cu(() is anopen mapping, and so is f (h(()) (= ((u(())m + f (a)). Hence, f mapsneighborhoods of a to neighborhoods of f (a).

COROLLARY 1.8 (Maximum Principle). If .R is connected, theabsolute value of a nonconstant holomorphic function on 12 has nomaximum in .fl.

PROOF. The reasoning is that any open disc of C contains a pointwhose distance to 0 is larger than the absolute value of the center ofthe open disc.

From this, an important proposition in the theory of holomorphicmappings follows:

THEOREM 1.9 (Schwarz's lemma). If f E A(An), f (0) = 0, andsup If (z) I = M, then

zEA

(1.3) If W1 < MIZImax

on An.

PROOF. For a = (a1i , an) E 8A n, consider a holomorphicmapping

Ira : A - On

W W

C ~- (a1C, ... , an(),and apply the maximum principle to f o 7ra(()/C. Then we obtain

If o 7ra(C)I < MICI,

where the equality holds if and only if f o 7r0(()/t; E C. Hence, (1.3)follows from BSI = I7ra(()Imax. 0

1.2. Cauchy-Riemann Equations

Differential equations

of =0forJ =1,.. ,n

on f2 are called the Cauchy-Riemann equations. Holomorphic func-tions are all solutions of the Cauchy-Riemann equations since 8z'/O,= 0 for any a E Z+, and convergent power series are differentiated

1.2. CAUCHY-RIEMANN EQUATIONS 9

term by term. The aim of this section is to characterize holomor-phic functions as locally square integrable functions that satisfy theCauchy-Riemann equations.

To begin with, we introduce some notation and terminology.Let L'(17) denote the Hilbert space of all complex valued mea-

surable functions on .fl that are square integrable with respect to theLebesgue measure, where two functions whose values coincide every-where except on measure zero sets are considered to be the same.

We identify the Lebesgue measure on Cn with the volume elementwith respect to the Euclidean metric

dX1 A dx2 A ... A dX2n I = ( 2 )n

dz1 A dz1 A ... A dzn A dzn

and denote it by dV, dVn, or dVz for simplicity.The inner product

JfgdV

nof L2(11) is denoted by (f, g) (= (f, g)n), and the norm of it byIifII (= IIfIIQ).

Set A2(i1) := A(Q) n L2(Q), and call the elements of this setL2 holomorphic functions. A2(!2) will be shown to be a closed sub-space of L2(f2) in the present section.

For k E Z+ U fool U {w}, let Ck(.R) be the set of all complexvalued functions of class Ck on 11, and Co (.R) the set of those whosesupports are compact. It is clear that Co (.R) C L2(Q). Let us recallthat L2 (.fl) is a separable space and that Co (.R) is dense in L'(9).

Let LiC(!l) be the set of all locally square integrable functions on

.R, that is, complex valued measurable functions on Q that are squareintegrable on compact subsets of .R.

Evidently, A(f2) C COO(f2) C L (.R).In general, let V' be the set of all complex linear functionals on

a vector space V over C.If we define an element t(f) in C01(9)' for f E L °(.R) by

t(f)(g) := (f, g) for g E Co (.R),

then t is an injection from Li C(!1) to C000(0)', since Co (.R) is densein L C(Q) with respect to the topology induced by the L2 convergenceon compact sets. Hence, L(Q) is identified with a subset of C01'0 (.R)'under t.


R

For an element u in Co (.fl )', define an element C a/

u in

C0 (f2)' by

(-)u(g) :_ (-1)ca>u l a9J

The definition of I 7 I u is a natural generalization of the con-

cept of the usual derivative, since t I h = (ax) 3t(h) for anax19 )

element h in C°°(Il), by integration by parts.The following theorem provides a characterization of holomorphic

functions.

THEOREM 1.10 (Theorem of L2 Holomorphy).

aA(.R) = f E L1 °(fl)

a j= 0 for j = 1, n .

us review, below, complex differential forms in order to pre-pare some calculations that are used in the proof of this theorem. Adifferential form of degree r (or r-form) on !l (C C) can be writtenas

E uIJ dzj Adz J, where ujj = sgn I1 I sgn ( j1 ) uj,j,IJ

for multi-indices I = (i1, ... , ik) and J = (ii,... , j!) with k + I = rwhose components are taken from the natural numbers from 1 to n,and

dzl = dz;, A ... A dz;k ,

dzJ=dzj,A...Adzj,.A differential form is often written as

u = F,'uIJdzj AdzJ,I,J

in which the multi-indices I and J are only allowed to have strictlyincreasing components. In this case, u is said to be a differential formof type (p, q), or simply a (p, q) form, if the lengths k and 1 of I andJ are equal to the given constants p and q, respectively.

Let C(")(fl) be the set of all r-forms of class C°° on fl, and letCp,9(fl) be the set of all (p, q)-forms of class C°° on 12. In general,


let C°(K) be the set of all complex valued functions of class C'° ona subset K of C". When K = !l, we define CM(72) C CM(Q) andC(r) (.R) C C(') (,f1) in a similar fashion.

Let us give an example of a calculation in which differential formsare effectively used.

Given a holomorphic mapping F = (f1, , fn) :.fl --> C", wehave

F' (dz1 A ... A dzn A dz 1 A ... A dzn )2

det(af') ZIn...AdznAdz1n...ndzn\azkjk

From this, the Jacobian of F with respect to the real coordinates2

(x1. x2, ... . X2n) is equal to det af3\ a`k 7,k

The complex exterior derivative operators

a : Cp,q(Q) _ Cp+1,q(l?)

of type (1, 0) anda : CP,q(Q) .4 Cp.q+I(Q)

of type (0,(El1) are defined respectively by

aulJdzl AdzJl = F_': aulJdzk Adzj Ad4J,Id

\/Id k azk

(>'u1dzJ A dJ I >' ° ! iiA dzl n dj.Id / I,J k k

From this definition, it is obvious that the ordinary exterior derivatived is equal to a + D.

Let L ,q(17) denote the set of all differential forms of type (p, q)whose coefficients ujj are elements in L2 C(0), and let LP?9(j?) bethe set of those whose coefficients are in L2(fl). L1oC(!2) and L(r)(Q)are defined similarly. Also, Cop, q(12) expresses the subset of CP.q(.R)whose elements have compact supports, and C0(') (f2) the subset ofC(r) (,f1) whose elements have compact supports.

Using this notation, through the involution

L(r)(f2) XC012n-r)(n) -+ C

W W

(u, v) uAv,Jo


L joy (fl) is identified with a subspace of Co(2n-T)(fl )', and similarlyL o'C(fl) with a subspace of Co -p'n-Q(fl)'. Accordingly, the domainsof definition for the exterior derivative operators d, 0, and a extendto &)(fl) or L oC(fl).

The proof of Theorem 1.10 needs the mean-value property fordifferentiable solutions of the Cauchy-Riemann equations.

PROPOSITION 1.11. Assume that f E C1(fl) and 8f = 0.

1. For an arbitrary n-open ball B(a, R) C= 12, it follows that

(1.4) 1 f dS = f (a),Vol(OB(a, R)) JB(Q, R)

where dS denotes the volume element of 8B(a, R) induced bythe Euclidean metric, and we set

LB0.Vol(818(a, R))

R)dS = (n - 1)!

2. f E A(fl).

PROOF OF (1). Set g(z) := f (Rz+a) -f (a); then (1.4) is equiv-alent to

r(1.5) J gdS=0.

OB"

n-1Now that the restriction of a (2n -1)-form Slog I z I A ( A 08 log IzI

to 8En is not equal to 0 but unitarily invariant, (1.5) is equivalent to

Jean g(z) 8log Izl A (A109

log IzI I = 0.

In general, for a function u of class C1 onStokes' formulaimplies

-r n-1(1.6) /eBn u(z) 8log Izl A A aalog Izl

J d{u(z)alogIzIA ("A'oThogIzI)}^ \B^ (0, e )

n-1+ f u(z) 8log IzI A A 070 log IzI ,

as^(o,e>


for 0 < e < 1. In the case u = g, since the first term of the right handside is 0 by the condition ag = 0, it follows that

Jag(z)n 1

s alogIzl n A aalogIzIn

/n 1Caf g(z)alogIzI A \( A logIzl .

8B" (0, e)

Therefore, we obtain (1.5) by letting e \ 0, since g(0) = 0.

REMARK. When n = 1, in the above argument, take g(z) = 1,and replace z by an element f in C1 (a) fl Ker a that has no zero pointon aA. A similar calculation provides

Argument Principle:

JUlogIf(z)l=owhere n f denotes the sum of the orders of zeros of f in A.

Before proceeding to the proof of (2), we need to prepare thefollowing proposition:

PROPOSITION 1.12. Under the same assumption as in Proposi-tion 1.11,

1 f dV = f(a),(1.7)Vol(BB(a, R)) JB(a, R)

where 7nR2nVol(B(a, R)) _

n!

PROOF. This is due to (1.4) and Fubini's theorem.

Apply the Cauchy-Schwarz inequality to the left hand side of(1.7); then we obtain

(1.8) If(a)12 <Vol(3(a,R)) JB(a,R)

1112 dv.

This inequality is called Cauchy's estimate. In (1.8), the conditionfor B(a, R) may be relaxed to 3(a, R) C R. (The right hand side isallowed to be oo.)

Combine Cauchy's estimate with Theorem 1.6; then it turns outthat A(Q) is a closed subspace of L C(0) with respect to the topol-ogy induced by the L2 convergence on compact sets. From this, theseparability of A(.R) and A2 (.R) follows.


PROOF OF (2). If f were of class C' on fl, Of = 0, and f ¢A(fl), then there would exist lE$(a, R) C= fl such that the orthogo-nal projection P : L2(B(a, R)) -' A2(13(a, R)) does not map u :_f 113(a, R) to itself. Hence, g := Pu - u satisfies g # 0, 8g = 0, andg 1 A2(1B(a, R)).

However, if we fix an arbitrary element o in Aut IB(a, R), then

J2

0 = f g(()h(() dVV = g(cr(z))h(a(z)) det (8zk) dVz(a, R) (a, R)

for any h E A2(3(a, R)). Hence,t

det (p!) E A(1B(a, R)) fl C°`(B(a, R))

implies

g(a(z)) det (7k) 1 A

Since clearly

( H (a, R)).

8(g('(z)) det ()) = 0,it follows from the mean-value property that

g(a(a)) det ( 8zk) (a)

fB(R)det ()dVVol((a,

R)) 84=0 (. 1 E A2(3(a, R))).

Now that Aut 3(a, R) is transitive, it follows that g = 0. Thiscontradicts that Pu 0 u. _

Therefore, from the assumption that f E C'(fl) and Of = 0, itmust follow that f E A(fl).

Let us review some fundamental facts on the regularization ofelements in L oc(fl) before getting into the proof of Theorem 1.10.

Take a monotone decreasing function (in the broad sense) aR -i [0,1] of class C° with supple C (-oo,1) and

00µ(t)t2n-' dt = 1.f

0


and let

(1.9) µE(z) .- E2nVol(a

The main properties that uE possesses are µE E C' (C'), 0,

supp µE C r (0, e),

J pE dV = 1,nand that µE depends only on lzt as a function. The monotonicity ofp will be convenient for later use.

Also, set

.f2E:= jzE.f2l inf Iz - wl > F}

for a positive number e.If for an element f in L'10C(D), we put

fE(z) L. f (z + ()pE(() dVV .

hen fE E CO-(.RE) and fE converges to f with respect to the L2 normton compact sets. That is to say, for any relatively compact opensubset fl' of fl,

(1.10) Einolife-I11W=0.

(For the proof, see (28] for instance.)fE is called the E-regularization of f. Later, this terminology will

be used for differential forms, with the same meaning.

PROOF OF THEOREM 1.10. If f E L2I.C(D) and 8f = 0, then

a_ z) =fn f(()-.pE((-z)dV( _- f f(()19

pE((-z)dVV=0.c , (;

Hence, what has been previously shown implies fE E A(12E). There-fore, the mean-value property becomes applicable to fE and resultsin

1

(fE)b(z) = f fE(z +()' a(() dVC = fE(z) for z E È+a'n


On the other hand, the right hand side of these equations is equalto

.f(z+(+C)pE(C)dVCP6(()dVCJ 1 LnCn

=f (I f (z + + dV(

f f6(z + C)µf(C) dVt = (f6)f(z)C^

Therefore, ff = f6 on Q. Additionally, if (1.10) is taken intoaccount, then f, = f on fl f. This proves f E A(Q), since we haveshown that ff E A($?f).

As an application of Theorem 1.10, we obtain a continuation the-orem that describes a sufficient condition, in terms of the Lebesguemeasure, for a closed subset E of f? to satisfy A2(f2 \ E) = A2(Q).Below, let m(B) denote the Lebesgue measure of B.

THEOREM 1.13. Assume that for an arbitrary point zo in a closedsubset E of .fl, there exists a neighborhood U of zo in .R such that

(1.11) liminf E-2m({z E U I inf Iz - wI < E}) < oo.C-0 wEE

Then A2(fl \ E) = A2(f2).

PROOF. Set dE(Z) := ti E Iz - wI. Also, take a C°° function

p : R --+ [0,1] such that PI (- oo, 2) = 1 and pI(1, oo) = 0, and define

a function Xf on fl by

Xe(Z) :=(dE(z))

where E > 0.E

dE(z) is almost everywhere differentiable, since it is Lipschitz contin-uous. Accordingly, so is XF, and

si9x

(z)< E sup

I P (t)I

almost everywhere on Q.Suppose that f E A2(f2 \ E). Since the given condition implies

L2(fl \ E) = L2(1?), it suffices to show that for any element u inCo (fl),

ff.dV=0fori=1,... ,n .8 zj


For this purpose, divide the left hand side of the above equationinto

fn f (XEu)dV +Jnf i((1-XE)u)dV.

First, since D f = 0 on Si \ E, integration by parts implies

(1.12) fa ((1 - XE)u) dV = 0.fa-ZiOn the other hand,

r rJf(XeU)dVLH J

IFin azi azi

The first term on the right hand side satisfies2

fL'-6udVI < E-2 supIU12 suPIP I2 -m(Eu,,)J IfI2dV ,

n "J u

Eu,E := {z I dE(z) < e} fl supp u.

From the assumption, the inferior limit of the right hand side equals0 as c -' 0. Moreover, the second term satisfies

117xe dV

J

2

<suplau 12

La.. If12dV -+0.

Hence, by combining these results we obtain

(1.13) lim ionf I f f a (XEu) dVn J

=0.

Now (1.12) and (1.13) imply the desired conclusion.

For a holomorphic function f on Si, the zero set f (0) of fis denoted by V (f) or V (f (z)) . An important example of applyingTheorem 1.13 is given in the case E = V (f) as follows (though thisresult will not be used in this book):

PROPOSITION 1.14. If V (f) does not contain an interior point,then A2(.f2 \ V(f)) = A2(Q).

SKETCH OF THE PROOF. If for a point a in V (f ),

f (z) = E ca(z - a)a, where ca 0 for some (a) = m,(a)>m


on some neighborhood of a in f2, then by applying an appropriatecoordinate transformation

z - a = Bw for ut E C" and a complex it x n regular matrix B,

we obtain

f (Bw + a) = w'g(w) + c1(w')wn-1 + ... + c,,,(w'),

where w':=(w1,...,w,,-1),c1(0)= =c,,,(0)=0,andg(0)00.Therefore, by restricting the projection w -4 w' to a neigh-

borhood of 0, the intersection of the preimage of each point withV (f (Bu; + a)) consists of at most m points (by the argument prin-ciple). Hence, from Fubini's theorem it turns out that for an e-neighborhood V(f)E of V (f) and a relatively compact subset U off2, the Lebesgue measure of V (f )E fl U is evaluated to be the infini-tesimal of order 2 with respect to E. This means that V(f) satisfiesthe condition of Theorem 1.13.

REMARK. We describe two facts that are related to Theorem 1.13.

1. In the case n = 1, a necessary and sufficient condition forA2(f2 \ E) = A2(12) is known. (We refer the reader to Theo-rem 5.13in§5.4.)

2. Shiffman [41] has shown that A(.f2 \ E) = A(fl) in the specialcase when the left hand side of (1.11) is equal to 0.

1.3. Reinhardt Domains

It is fundamental that the convergence range of the Taylor series at theorigin for a holomorphic function defined on A" is a set containingA". In general, however, the convergence range of a power seriesin several variables can take various forms other than A". In thissection, we will mention general properties that such sets possess.

Let fl be a domain in C", namely, a connected open set.

a ifDEFINITION 1.15. fl is said to be a Reinhardt domain with center

(1.14) (al + (1 - (zl al), ... , an .}. (" - (zn - an)) E 12

for any z E fl and any C E (8:,)". Also, fl is called a completeReinhardt domain with center a if (1.14) holds for any z E fl and any(EA".

1.3. REINHARDT DOMAINS 19

Polydiscs and complex open balls are examples of complete Rein-hardt domains. Clearly, a complete Reinhardt domain contains thecenter in it.

Let D be a complete Reinhardt domain. Assume that the centerof D is 0 for simplicity. The next proposition follows immediatelyfrom Proposition 1.4.

PROPOSITION 1.16. For a holoinorphic function f on a completeReinhardt domain D with center 0, the power series

P(0, f) = E f (0) z°Q a!

converges on D.

Define the logarithmic image log D of D by

log D:= {x E (]RU{-oo})" I e= ._ (e", ,eX ) E D}.Let (log D)^ be the convex hull of log D. and set

D:= {z E C" I (log Iz1I, , log Iznl) E (logD)^}.

THEOREM 1.17. For f E A(D). P(0, f) converges on D.

PROOF. Take any ( E D, and set r := ICI := (1(1 1, , I(n I).

Then, from the definition of D. r can be written asr = r' tr111-t ritrn1-t

1 1 n n

for some two points r' and r" in D n [0, oc)' and some 0 S t <_ 1. Letc° be the coefficient of z° in P(0. f). Since

E Ic°Ir'° < c and IcaIr"° < oc.a °

there exists a constant Al such that

j c° I r'° < Al and I c° I r"° <A1

for any a. Hence, it follows that

Ic° Ir = Ic° I (r'tr111-t )°

= (Ic°Irb°)t(Icol.r»°)1-t < Aft Afl-t

This implies the convergence of P(0, f) on 0(0, r). Moreover, notingthat

D= U A(0, I(I),'ED


we conclude that the convergence of P(0, f) holds on D as well.

A Reinhardt domain whose logarithmic image is convex is said tobe logarithmically convex. Let us show one property of logarithmicallyconvex complete Reinhardt domains.

PROPOSITION 1.18. For a logarithmically convex complete Rein-hardt domain D with the origin centered and an exterior point a ofD, there exists a monomial ma(z) such that

sup ima(z)I < ma(a) = 1.-ED

PROOF. Since log D is convex, there are y E Z' and 6 E R suchthat

f sup J (X, y) + 8 I x E log D} < 0,j ebla7j=1.

Hence, it suffices to put ma(z) = e6la''la_y z''.

COROLLARY 1.19. Let D be the same as above. For an arbitrarypoint a of 8D, there exists an element F in A(D) that satisfies

lim IF(z)I = oo.z-aPROOF. Take an increasing sequence Dj of relatively compact

subdomains of D that are logarithmically convex complete Reinhardtdomains. Take a sequence a j of points in D that converges to a andsatisfies aj E Dj+1 \Dj. Let mad (z) be such a monomial as above thatis determined for each domain Dj and point aj. Choose sequencescj E R and vj E N such that

(1.15)1

sup Ickmak (Z) "k I <2k

.

zEDk

Then the series00

E cjma, (z)-'j=1

converges uniformly on compact sets in D. Denoting the sum of theseries by F(z), Theorem 1.6 implies F E A(D). On the other hand,from (1.15), we clearly have IF(ak)I > k - 1 for each k. This resultsin lim IF(z)j = oo.z-a

1.3. REINHARDT DOMAINS 21

The above proof shows that the conclusion of Corollary 1.19 maybe strengthened as follows:

"For any sequence {aj } of points in D that has no ac-cumulating point in D, there exists an element F inA(D) such that

I F(aj) I = co ."3 00

General open sets that possess this property will be discussedlater. It seems interesting, at this point, to study the convergence

mrange of a series > fk whose terms are homogeneous polynomials fk

k=0of degree k as a slight generalization of a power series, but we do notdo so in this book. H. Cartan wrote an article [8] about this problem,and we refer the reader to it.

CHAPTER 2

Rings of Holomorphic Functions and8 Cohomology

From Weierstrass's theorem, it follows that A(Q) is a complete topo-logical ring with respect to the topology induced by uniform conver-gence on compact sets. Let us consider the structure of the spectrumof A(Q), i.e., the space of all maximal closed ideals of A(fl) equippedwith the weak topology. J? can be regarded as a subset of the spec-trum of A(Q), since each point of fl corresponds to a maximal closedideal consisting of functions whose value is zero at that point. Inwhat case does fl coincide with the spectrum? If this is the case,from the Banach-Steinhaus theorem, it follows that for a discrete se-quence Vk}k 1 of points in fl, there always exists an element f inA(R) such that lim If (G) I = oc: does there exist any f that satisfies

k- wf(tk) = k(k = for instance?

By replacing these problems with those of solving the Cauchy-Riemann equations of inhomogeneous form. let us connect the spec-trum, a concept of topological algebra, with 8 cohomology, an analyticconcept.

2.1. Spectra and the a EquationThe spectrum of A(fl) is denoted by Sped. A(Q).

PROPOSITION 2.1. If for an arbitrary sequence { fk}k 1 C A(fl)of functions that has no common zero point, there exists a sequence{gk}k 1 of functions in A(fl) such that

(2.1)

xE fkgk = 1,k=1

then fl = Spec,, A(fl).

PROOF. This is clear from the paracompactness of fl and thedefinition of Spec, A(0). O

23

24 2. FUNCTION RINGS AND 8 COHOMOLOGY

The converse of Proposition 2.1 may seem self-evident, but infact it is not.' The proof of the converse requires a good amount ofpreparation, and is deferred until Chapter 5.

Let us characterize a sequence {gk}' 1 of functions that satisfies(2.1) as a solution of the 8 equation with some restraints.

We can assume that the seriesa

h rIfkl2k=1

converges uniformly on compact sets by replacing the given sequenceof functions fk with Ekfk (Ek # 0) if necessary. The estimate (1.2)and the Ascoli-Arzela theorem imply h E CI (R), and h has no zeropoint by assumption. Hence,

hk := hk E C°°(,f1),

and

Therefore,

0CF, fkhk = 1 uniformly on compact sets.k=1

00

fkfhk = 0.k=1

The uniform convergence of the left hand side on compact sets is dueto the same reason that h E COO (12).

If there exists a sequence {uk}k1 of functions in C°°((1) suchthat

o°

(2 2)E fkuk = 0 uniformly on compact sets, and

. k=18uk=8hkfor

then by setting 9k = hk - Uk, we obtain a sequence of functionsthat meets the condition (2.1). Conversely, given 9k, the functionsUk := hk-gk provide a solution of (2.2). Therefore, the equation (2.1)for the system of unknown functions gk is equivalent to the equationsin (2.2) for the system of unknown functions Uk.

As the second problem, we consider a condition for the restrictionmap A(Q) -+ Cr to be surjective for a given set r of points thathas no accumulating point inside .R. _Let us convert this simplestinterpolation problem into one for the 8 equation.

001Because E fkgj,k 1(j - oo) does not necessarily imply the convergencek=1

of {gj.k}-1.

2.2. EXTENSION PROBLEMS AND THE 6 EQUATION 25

Fix a system {U{}W of mutually disjoint open sets with t; EUU C 12, and take a function p of class C°° on f2 such that supp p CU Ut and p = 1 on some neighborhood of r. For an arbitrary b E Cr,

CEr

define b E C' (0) by

b(z) _

b(Z;)p(z) for z E U{,

0 for zE IUU£\l

I

Then b is a C°° extension of b to (1 with the property that D b = 0on some neighborhood of r. Therefore, a necessary and sufficientcondition for an element f in A(S7) that satisfies f I r = b to exist isthat the 8 equation

(23) Du=ab,{ u!r=0

have a C°° solution u.

2.2. Extension Problems and the D Equation

The proper way to describe the extension problem of holomorphicfunctions should be more general, as seen below.

DEFINITION 2.2. A closed subset X of (1 is said to be an analyticsubset if for any point x0 of X, there exist a neighborhood U of x0 inCn and a system of functions {fa}aEA C A(U) (A may be an infiniteset) such that

XnU={zEUI fa(z)=0 for aEA}.We call { fa b EA a system of local defining functions of X on U orsimply around x0.

A discrete subset and the intersection of Si with a complex m-dimensional hyperplane are examples of analytic subsets.

For a function on an analytic subset X of 12, the concept ofholomorphic function is generalized as follows:

DEFINITION 2.3. A function f on X is holomorphic if for everypoint x0 of X, there exist a neighborhood U of xo in Cn and anelement F in A(U) such that F j U fl X= f I U fl X.

By this definition, the interpolation problem can be grasped asthe extension problem, to extend holomorphic functions defined on a

26 2. FUNCTION RINGS AND a COHOMOLOGY

'lower-dimensional' subset inside 12 to functions on Q. This under-standing is geometrically richer and more interesting.

For the sake of brevity, we call a complex 1-dimensional hyper-plane a complex line and a hyperplane of complex codimension 1simply a hyperplane.

It is self-evident that when X is the intersection of a complex m-dimensional hyperplane with fl, the above definition of holomorphicfunction coincides with the usual definition, in which X is regardedas an open set in C1.

Putting L := {z E C' I z1 = = zn_m = 0}, let us convert thesurjectivity problem of the restriction map

A(fl) -A(flnL)

into the D equation.Let f = f (zn -.n+ 1, , zn) be a holomorphic function on f1 n L.

Then a function f (z) f (0, , 0, zn-m}1+ , zn) is holomorphicon an open set

Therefore, by taking a C°° function pw : W -+ [0, 1] that satisfies

f supp (pw - 1) n L = 0,1 supppwn8.R=0,

and by letting f be the trivial extension of pw f to fl, we see thatf E C°° (fl) and f I f2 n L= f.

Hence, the existence problem of a holomorphic extension of f tofl can be replaced by the solvability problem of

au=af,{ ulflnL=O

in the same way as (2.3).For a general analytic subset X, it is difficult to construct directly

and precisely a local extension of f that corresponds to f in the aboveargument. For this reason, careful consideration on a system of localdefining functions is inevitable in order to discuss, from a generalpoint of view, the extension problem of holomorphic functions withoutrestricting ourselves to the case of discrete subsets (see § 5.1).

2.3. 0 COHOMIOLOGY AND SERRE'S CONDITION 27

2.3. a Cohomology and Serre's ConditionLet L be as defined in § 2.2. Let us develop the argument to extendholomorphic functions on .fl n L to Q. The result will relate to thestructure problem of spectra.

DEFINITION 2.4. The 8 cohomology group of type (p, q) on ,fl is,by definition,

HP.q(Q) := Ker a n Cp,4(Q)/Im a n Cp,9(Q).

where the domain for the operator 8 is restricted to the space of C'°differential forms on Q, and in general the kernel and image of a linearmap T are denoted by Ker T and Im T, respectively.2

From the above definition and the result in Chapter 1, it followsthat H°.0(0) = A(fl).

When there is an inclusion relation fli D ,f12 between open sets f11and f22i the restriction map Cp,q(fli) -i is well-defined and,being commutative with a, induces a homomorphism from HP'Q(fli)to Hp,q(fl2). This is called the restriction homomorphism.

On the one hand, the restriction map Cp.q(fl) - C""q(fl n L)is well-defined as the restriction of differential forms on !1 to thesubmanifold fl n L and, again by the commutativity with 8, inducesa homomorphism from HP,q(fl) to HP.q(fl n L), which is also calledthe restriction homomorphism.

Set Lj := {z E Cn I z1 01 for 0 j 5 n. (Hence,L = Lm.)

THEOREM 2.5. The restriction map

A(fl) -p A(.f1 n L)

is a surjection if H°.q(fl) = {0} for all q with 1 <_ q <- n - m.

This is called Serre's criterion.For the proof, we need one lemma.

LEMMA 2.6. The restriction homomorphism

a : HP,q(fl) -, HP,q(fl n L1)is a surjection if Hp,q+l(fl) = {0}. In particular. n Ln_1) _{0} if HP,q+k(fl) = {0} for all k = 0,1.

ZIm overlaps the notation for the imaginary part of a complex number, butthere should not be any confusion.

28 2. FUNCTION RINGS AND 8 COHOMOLOGY

PROOF. Take v E CP' (.fl n L,,_1) n Ker B. As in the case of aholomorphic function, there is a C°° extension v of v to .fl such thatav = 0 on some neighborhood of ,R n L,,_1. Since

E Cp,4+1(,R) n Ker 8,z1

the assumption implies that there is an element u in CM(i7) suchthat

8u= av.z1

Hence, it follows that v - z1 u E Ker a and u - z1 u I fl n Li_ 1 = v,which shows that a is surjective.

PROOF OF THEOREM 2.5. By applying Lemma 2.6, the assump-tion implies

H°"1(i?nL;)={0}for m+1<j<-n.Hence, from the same lemma again, the restriction map

A(Sl n L3+,) - A(.R n L3)

is a surjection for j >_ m. Therefore, the restriction map A(12)A(.fl n L) is also a surjection.

For the closure ,fl of (1, let HP,q(.R) := 1 HP (U), where theUDn

inductive system {HP'q(U)} is regarded with respect to the restric-tion homomorphisms, and U runs in the fundamental system of openneighborhoods of fl. For the sake of consistency in notation, we setA(!2) := Ho,°(?).

From Serre's criterion, the next theorem follows immediately.

THEOREM 2.7. The restriction map

A(12) - A(12 n L)

is a surjection if H°'Q(D) = {0} for the range of 1 < q:5 n - m.

If (l is a polydisc, one can easily show that H0,9(32) = 10) forq > 1 by an elementary method. Although the proof of this statementis included in many books, we will describe it in detail, consideringthe important role that this result will play when we generalize thetheorem of L2 holomorphy to the one for a cohomology.

We begin with the following lemma.

2.3. 3 COHOMOLOGY AND SERRE'S CONDITION 29

LEMMA 2.8. For a bounded closed set K of C,

(2.5) lil H°'1(U) _ {0}.UDK

PROOF. Let U be a neighborhood of K, and v E C°'1(U) (=C°'1(U) fl Ker a). It suffices to show that the a equation au = v hasa solution under the assumption that v E Coo" (U), by multiplying v,if necessary, by a function whose value is 1 on a neighborhood of Kand whose support is contained in U.

Let v denote also the coefficient function in the given (0, 1)-formv. Set

d(2.6) u(z) = 27r y( z)d,A

Then it follows that u E COO (C), and

(2 7)au 1 [v(4 + z) d A daz 27r

Stokes' formula implies

(2.8) v(z)21r -l JC

Noting that (t; + z) = vZ( + z), from (2.7) and (2.8) we derive

au= v.

az

0When v contains a parameter w, if v is of class C°O or holomorphic

with respect to w, so is the above solution u, as we clearly see from(2.6). This fact also will be used in the following argument.

THEOREM 2.9.

Ho. (On) = {0} for all q > 1.

PROOF. Take a neighborhood U of On that provides a repre-sentative v E C°. (U) fl Ker a of an arbitrary element in H°()(q > 1). Set

v = L,' Vrdzn A dzI + >' vjdz jIT, Jon

Ill=9-1 IJI=9

with VI', v'j E CO°(U),

30 2. FUNCTION RINGS AND 8 COHOMMOLOGY

where III denotes the length of the multi-index I, and I n meansthat I does not contain n.

From Lemma 2.8 and the succeeding remark, there is an elementu11i in C°° (Cn) satisfying

NO)(2.9) r =

a -z"vi

on some neighborhood of On. By setting

UM u1lidz1,In

(II=q-1

from (2.9) v is transformed into the following form that does notcontain dzn any more:

v - Du(1) w'1dzn-1 A dz1 + >' ZUJdwJ,1 on,n-1 Jj1n,n-111I=q-1 IJI=q

where w'1 and w; on the right hand side are holomorphic with respectto Zn.

Therefore, we can take, this time, a C°° function u (2) that satisfies

tv'1 and is holomorphic with respect to zn. Set u(2)-Zn -1athen v-auili -aui2i contains neither dzn nor dzn_1.

f3Jn.n-1I1I=q-1In this way, if we keep producing new forms starting with v, theneventually the form reaches the (0, q)-form that does not contain any

of dzn, , dzl, or the form 0 at which v = d E(w).j=1

COROLLARY 2.10. f E C°°(0) if both f E Li C(fl) and of EC0.11Q).

PROOF.

Since the equation au = of for an unknown function uhas locally a C°° solution, f is holomorphic modulo C° functions.Hence, in particular, f E C'(17).

Let us show one consequence of Serre's criterion.

PROPOSITION 2.11. If HO,q(Q) = {0} for 1 < q n - 1 and ifan is a real hypersurface of class C', then for every boundary point


zo of 12, there exists an element f in A(f2) that satisfies

lim If(z)I =oo.zU

PROOF. Let L be a complex line that intersects transversely with8.2 at zo. Take a holomorphic function on L \ {zo} that has a poleat zo, and extend it to .f2. Then this holomorphic extension f willsatisfy the above condition. 0

This proposition naturally poses the problem of relating the van-ishing of a cohomology groups on .fl with such a geometric conditionas the logarithmic convexity of complete Reinhardt domains.3 Forbrevity, we call the condition

H°'9 (Q)=0 for all 1 <q<n-1Serre's condition.

The a cohomology groups are related to the spectrum problemas follows:

THEOREM 2.12. If .f2 satisfies Serre s condition, then for everypoint a of B.2, there exist elements gi, g in A(!2) such that

n

E (zj - aj)gj(z) = 1.j=1

COROLLARY 2.13. An open set .R that satisfies Serre's conditionis closed in Spec,,, A(f2) with respect to the weak topology.

PROOF OF THEOREM 2.12. Let V be a module C°°(.R)®n overCOD (!2), and define a C°° (!2) homomorphism a : V -> C°° (.R) by

ita(vi ... v,,) _ E zjvj.

j=1

Let. {e1, , en} be the standard basis of V. In terms of a. definek k-i

contractions ak : AV -i AV pointwise by

k 1.7

ak(eil A ... A eik) = L (-1)3a(ei, )ei1 A ... Aeikj=1

and by linearity. Then the following exact sequence is obtained:

0-,nV an, n1Vck- ... AV V

3This will be described in detail starting in the next chapter.

32 2. FUNCTION RINGS AND $ COHOMOLOGY

If we extend the range of definition for the operator 8 to vector-valueddifferential forms, noting that ak commutes with the operation of 8,then 8 cohomology groups HM(Ker ak) with coefficients in Kerakare defined by

HP,q(Kerak) 4:= (CM(0) ® Ker ak) fl Ker B/8 (Cp,q-1(12) (9 Ker ak) .

Then the same argument as for (2.2) induces that if H°,k+I (Ker ak+1)

_ {0}, then the homomorphism derived from ak+I,

HO,k(H)®(k+1) _, H°A(Kerak) ,

becomes surjective.Therefore, it follows that

Im(aI A(.fl)®")=A(,fl)H°"1(Ker a) = {0}HO-1 (0) = {0} and {0}

H°, I (f2) = H°,2 (f2) = = HO,n-1(.12) = {0}and H°""(Keran) = {0}.

Since Ker a" = {0}, the proof is complete. 0

PROOF OF COROLLARY 2.13. If a sequence of points {z(µ)}°Oµ=1C 11 is a convergent sequence with respect to the weak topology,then the z(µ) are clearly a bounded sequence and must converge tosome point inside .fl. Otherwise, as there is a subsequence z(µk) thatconverges to some point of 8,11, by choosing this point as a and takinggi , , g" produced in Theorem 2.12, it would follow that

lim F, Ig;(z(µk))I = 00,k-oc 1=1

which contradicts the fact that the z(µ) form a weakly convergentsequence.

The above argument tells us that the smoothness assumptionon 817 in Proposition 2.11 is redundant. Let us emphasize this factbecause of its importance:

4The tensor products on the right hand side are regarded as COO (g)-modules.


THEOREM 2.14. If Serre's condition holds for fl. then for everyzo E OR and every sequence {pµ} of points in fl that converges to zo,there exists f E A(f2) such that

limoI f(p,,)I = 00.

In general, an open set fl of C' is said to be a domain of holomor-phy5 if there is no connected open set U that possesses the followingproperty:

(2.10) U !2, and there is a nonempty open subset Vof f2fU such that for any element f in A(Q), thereis an element g in A(U) that satisfies f I V = g I V.

Let us leave to the reader the verification that logarithmicallyconvex Reinhardt domains, convex domains, and open sets in thecomplex plane are all domains of holomorphy.

By virtue of this terminology, it follows from Theorem 2.14 thatevery open set satisfying Serre's condition is a domain of holomorphy.The converse statement is also true, but the proof of it needs furtherpreparation, and we postpone the details until the next chapter. Notethat the following fact is derived from this converse statement, whichwe accept for the present.

PROPOSITION 2.15. A necessary and sufficient condition for f2 tobe a domain of holomorphy is that for any zo E 8f2 and any sequence{pµ} of points in f2 that converges to zo, there exists f E A(f2) suchthat

slim If(p,.)I = oo.-CC

What we wanted to make clear in this chapter is that a region ofdefinition f2 of holomorphic functions must be a domain of holomor-phy in order to solve affirmatively the extension and division prob-lems. Later on, we will indeed investigate whether these problemscan be solved on domains of holomorphy. In deference to KiyoshiOka's methods, we will first characterize domains of holomorphy bythe concept of pseudoconvexity, and then treat solutions of all theproblems on domains of holomorphy as consequences of pseudocon-vexity. Our lines are in imitation of what has been done repeatedly inmathematics, such as the reconstruction of Euclid's theory by meansof Descartes' methods.

5Connectedness is not imposed on the definition of "domain" of holomorphy.

CHAPTER 3

Pseudoconvexity andPlurisubharmonic Functions

As seen in the preceding chapter, f2 must be a domain of holomorphyin order that the propositions corresponding to the Euclidean algo-rithm on the ring of integers and to Lagrange's interpolation on thering of polynomials of one variable hold on the ring A(f2) of holo-morphic functions. On the other hand, these propositions are closelyrelated to Serre's condition, the vanishing of a cohomology on f2, andthis condition provides a unified grasp of various phenomena on adomain of holomorphy. a cohomology is essentially under control ofthe pseudoconvexity of an open set. Pseudoconvexity is a conceptsimilar to geometric convexity, but is much weaker as a condition.

This chapter begins with the definition of Hartogs pseudoconvex-ity and verifies that a domain of holomorphy is Hartogs pseudoconvex.Secondly, we show that some canonical function on a Hartogs pseudo-convex open set expressed by a distance function is plurisubharmonic.In consequence of this, it is derived that a domain of holomorphy ispseudoconvex. Hartogs and Oka's discovery of this relevancy gives aunique vitality to the theory of analytic functions of several variables.

In Chapter 4, in order to show that a pseudoconvex open set is adomain of holomorphy, some kind of differentiable plurisubharmonicfunctions will be needed. For this purpose. we detail the regulariza-tion of plurisubharmonic functions in the present chapter.

Finally, we mention the Levi pseudoconvexity and introduce basicfacts and important examples of pseudoconvex open sets that havesmooth boundaries. This also serves as an introduction to Chapter 6.

3.1. Pseudoconvexity of Domains of Holomorphy

A handhold is a domain

T={ (z l, Z2) E A 2I I z l I< E or 1- c.< I Z21 < 1}.

35

36 3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS

which is called a Hartogs figure.

DEFINITION 3.1. fl is said to be pseudoconvex in the sense ofHartogs, or, for short, Hartogs pseudoconvex, if any holomorphic map-ping from a Hartogs figure TE with a arbitrary to !l always extendsto a holomorphic mapping from 02 to P.

THEOREM 3.2. C is Hartogs pseudoconvex.

PROOF. It suffices to show that the restriction mapping A(02) -A(TE) is surjective. Let f E A(TE). From Proposition 1.4, f(z)expands into the power series

Ck,lzk1 z

t2

k.1

which is convergent on 02(0, (e,1)). After changing the order of thesummation, let us observe the range on which the following equationholds:

.f (z) _ > (Eckzz) Z

By setting Ck(Z2) := E ck,tz2 for Iz21 < 1, we regard the right hand

side as a series with terms of holomorphic functions ck(z2)zi. Sincethe left hand side is holomorphic on A x (Z2 I 1 - e < I Z21 < 11, fromthe same argument as in (1.2) it follows that when 1 - e < Iz2I < 1,

(3.1) Ick(z2)I < sup If (zI, z2)Ir-k for 0 < r < 1.Izi I=r

Hence, from the maximum principle, when Iz21 < r < 1,

(3.2) ICk(z2)I :-5 SUP If (Z1, Z2)Ir-k121 I=Iz21=r

aTherefore, the series E ck (z2) zi converges uniformly on compact sets

k=1in 02, and Theorem 1.6 says that this series is a holomorphic exten-sion off to 02 0

COROLLARY 3.3. C" is Hartogs pseudoconvex.

THEOREM 3.4. A domain of holomorphy is Hartogs pseudocon-vex.

PROOF. Let f be a holomorphic mapping from TE to P. FromCorollary 3.3, f has a holomorphic extension f : 02 _ C. If it were

3.1. PSEUDOCONVEXITY OF DOMAINS OF HOLOMORPHY 37

true that f (A2) fl, then, letting T. be the maximum among con-nected open sets U that contains T. and satisfies f (U) C fl, it wouldfollow that 8TE n A2 54 0. The maximum property of TE impliesf (p) E 89 for a given point p E 8T£ fl A2. _Hence, from Proposi-tion 2.15, choosing a sequence of points p in TE fl A2 that convergesto p, there exists an element g in A(fl) such that

(3.3) lim 1g(f(pµ))J = oo.

This contradicts the fact that g o f has a holomorphic extension to2 O

COROLLARY 3.5. The following are all Hartogs pseudoconvexopen sets:

1. Logarithmically convex Reinhardt domains.2. Convex domains.3. Open sets in the complex plane.

Next, observe arfunction 6'(z) on 1fl defined by12

6n(z):=inf{iS z+II 1 ¢fl}, where vEC'1\{0},

in order to relate Hartogs pseudoconvexity to a metric character.From the definition, 6n(z) is lower semicontinuous as a function fromf2 to (0, oo]. In addition to this, the following remarkable property of6a1' (z) emerges if 12 is Hartogs pseudoconvex.

THEOREM 3.6 (F. Hartogs, K. Oka). If fl is Hartogs pseudocon-vex, then for z E fl, v E Cn \ {0}, w E C", and r > 0 that satisfy

(3.4) {z+tesew10<t<r and 0<_B<27r}Cfl,

we havezir

(3.5) log 69' (Z)> 27r J log 6$2- (z + re=Bw)d9.

0

PROOF. Since log bn (z + re1Bw) is lower semicontinuous with re-spect to 0, this is the limit of an increasing sequence {FR(B)}R 1 ofcontinuous functions. Set

iB' 1 2a r2 _ t2uR(te )

27C JoV)R(8)r2 - 2rtcos(9 - 01) +

t2dO,

38 3. PSEUDOCONVEXITY AND PLURISUBHARMMONIC FUNCTIONS

then Fatou's theorem in the theory of Lebesgue integrals yields that

fo

2a(3.6) Rim UR(0) =

1log S (z + reiew)d8.

-oc

Now take hR E A(0(0, r)) with UR = Re hR, and consider the map-ping

aR on

wZ + ZIChR(-s)v + 22w.

From the continuity of WR(B), we see that

lim uR(te'o) _'OR(e)t-r

Therefore, given a positive number e. we can choose an appropriatepositive number 8 such that

{QR(z) I Iz1I < 6or 1 - 5 < Iz2I < 1 - e} C f1.

At this point, apply the Hartogs pseudoconvexity of f1, then it followsthat aR(A2) C Q.

Hence, in particular, we obtain e"R(°) < 5v (z), or

(3.7) UR(0) < log 6' (z).

From the combination of (3.6) and (3.7), the desired inequality(3.5) follows.

For the reader's convenience, let us review the basics of subhar-monic functions without proof. Let .f1 be an open set in the complexplane for a while.

The following two conditions on an upper semicontinuous func-tion i from fl to [-oc, oc) are equivalent:

(3.8) If A(z, r) C f1, then

2-,r ?P(z+ re'o)d9.to < 1

In0

(3.9) If A(z, r) C fl, h E A(-A (z, r)), and

Re hI5. (z, r) >- r'Ia0(z, r) ,

then

3.1. PSEUDOCONVEXITY OF DOMAINS OF HOLOMORPHY 39

Re h(z') >_ for z' E A(z, r).

When these conditions are met. tr} is called a subharmonic function onV. When 0(z, r) C .R, given a subharmonic function i on !1, definea function AI(v, t) by

1 2n2

AI(t t) := f t (z + te`B)dO.

Then AI (i', t) is monotone increasing on t. This is obtained from (3.9)and the mean-value property of harmonic functions. We can take thiscondition on AI( , t) as the definition of subharmonic function.

The next four properties follow immediately from the definitionof subharmonic function:

(3.10) If Y and 2' are subharmonic, so are w + and aY (a >- 0).

(3.11) For a family {V.,,\}A of subharmonic functions that are boundedfrom above uniformly on compact sets in Q. set th := sup v'a

A

and r/,*(z) := lieu sup Then tb' is subharmonic one-.0 z'ESC(z._)

Si.

(3.12) A real-valued function 0 of class C2 on Si is subharmonic if andonly if the following differential inequality holds everywhere:

a2 i.

a:ati '-°

(3.13) If a sequence {j} I of subharmonic functions on I? is mono-tone decreasing, then lim bj is subharmonic.

y-.00

A fact that is immediate from (3.12) and often applied is that if Vis subharmonic and of class C2. then is subharmonic for anyincreasing convex function A of class C2 on R. Combining this with

(3.13), for instance, it is easily seen that log E I f j 12 and Ul I fj I2j=1

(a ? 0) are subharmonic on ft for f1.... f,,, E A(Q).Let us return to the topics on C". The formula (3.5) indicates

the subharmonicity of - log bv(z + cup) with respect to (. Namely,- logJv is subharmonic on L fl Si for any complex line L.

DEFINITION 3.7. An upper semicontinuous function

!':.R - (-oc, oc)


is said to be plurisubharmonic if O(z + (w) is subharmonic as a func-tion of ( for a given (z, w) E f1 x C".

Denote by PSH(fl) the set of all plurisubharmonic functions onfl. For simplicity, f1 may be omitted from this notation.

The basics of plurisubharmonicity are totally the same as in thecase of subharmonic functions, but we list them for convenience:(3.14) If pand0areinPSH(Q),soarecp+0 andacp(a>0).(3.15) If 0 E PSH(fl), then, given z E fl,

111(z), t) ==Vol 8I8(z, t) J8B(z,t)

dS

is monotone increasing on t with B(z, t) C 17.(3.16) If {z'A}AEA C PSH(.fl) and if t?', are bounded above uniformly

on compact sets in fl, then

V)`(z) = lim sup sup 7/iA(z') E PSH(Q).e-0 z'EB(z,e) A

(3.17) Given a real-valued function , of class C2 on fl, 0 E PSH(12)if and only if the n x n matrix

8zj8zkCa20)is semipositive definite everywhere on Q.

(3.18) If a sequence of functions {Oj}'?O I C PSH(fl) is monotonedecreasing, then lim 1' E PSH(Q).

From Theorem 3.6 and (3.16), it is seen that

fl is Hartogs pseudoconvex - log bn E PSH(Q),

where bn(z) := inf Iz - wI. Unlike bn, bn is a continuous functionwE8l1

that has some finite determinate values on .fl as long as fl 0 C".

DEFINITION 3.8. Q is said to be pseudoconvex if there exists acontinuous plurisubharmonic function Eli: fl - R such that the set

fly,,c :_ {z E fl 1 '(z) < c}

is relatively compact inside 9 for each c E R.

It is clear from the definition and either (3.14) or (3.16) that if flland f22 are pseudoconvex, then so is S11 fl f12. Pseudoconvexity andHartogs pseudoconvexity are equivalent, but before giving the proof,

3.2. REGULARIZATION OF PLURISUBHARMONIC FUNCTIONS 41

we will describe the regularization of plurisubharmonic functions inthe next section.

REMARK. From the proof of Theorem 3.6, it turns out that ifIl is not pseudoconvex, then there exists a biholomorphic mapping tfrom 0" onto an open set U C C" such that

t(TE x 0"-2) C fl and U 0 .R.

This means, by Theorem 3.2, that all elements in A(V) extend toU as holomorphic functions. But the value of the analytic continua-tion of a holomorphic function does not necessarily coincide with thevalue of the original function at a point in 12nU except c(TE x An-2).Therefore, in order to discuss, in general, the global theory of holo-morphic functions, it would be insufficient to restrict the domains offunctions to open sets in C. However, as long as we stay only withinthe theory explained in this book, the replacement of .fl with a moregeneral space such as a complex manifold will not affect the frame-work of the theory (although this replacement would put things inmuch wider perspective). Hence, in order to give priority to brevity,we will be restricting ourselves to open subsets of C" from now on,too.

3.2. Regularization of Plurisubharmonic Functions

The existence of differentiable plurisubharmonic functions is impor-tant, as we will need to differentiate formulae that contain plurisub-harmonic functions later on to solve the 8 equation on pseudoconvexopen sets. As preparation for this, we describe below the regulariza-tion of plurisubharmonic functions.

Let .RE and µE be as defined after (1.9) in § 1.2.Let be a plurisubharmonic function that is locally integrable

on fl, and set

E(z) := f 'Y(z + ()µe(()dV( for z E f? .

^

Clearly, & E C°°(QE). Furthermore, 'NE has the following property:

PROPOSITION 3.9. WE E PSH(!lE), and pE \ 0 as e \ 0.

PROOF. The monotonicity of the family of functions ,0E followsfrom that of Ill (i , t). The plurisubharmonicity of Y'E is shown as


follows:1 2.

1 J te(z + re'Bw)dO27r 0

J;ii (()dV0(z + re0w - e()dO

rf27T

=

e()dV(

where r is chosen to be sufficiently small.

COROLLARY 3.10. Let V) E PSH(f1). If A is an increasing convexfunction (in the broad sense) defined on an interval that contains therange of V;, then A(0) E PSH(fl).

PROOF. There is a family of increasing convex functions Au ofclass C''° with A. \ A. For this family. (3.17) implies a,1(O5) EPSH(fQE) by direct differentiation, while since \,,(V,) \ A(v), (3.18)yields A(t,1') E PSH(fl).

COROLLARY 3.11. Let f11 C Cm and f12 C C" be open sets, andlet F: f11 - f12 be a holomorphic mapping. Then, for any t' EPSH(f22). one has that v o F E PSH(f11).

PROOF. If ,IL E C2 (02) n PSH, then tai o F E PSH(f11) followsfrom taking its derivatives. For the general case, use an approximatefamily as in Corollary 3.10.

THEOREM 3.12. Hartogs pseudoconvexity and pseudoconvexityare equivalent to each other.

PROOF. Hartogs pseudoconvexity Pseudoconvexity: Astake I z 12 for the case f1 = C', and 1z12 - log 5n otherwise.

Psuedoconvexity Hartogs pseudoconvexity: From the prop-erty (3.15) of plurisubharmonic functions and Corollary 3.11, it suf-fices to repeat an argument similar to the proof of Theorem 3.4.

COROLLARY 3.13. For an increasing sequence {flk}k I of pseu-x

doconvex open sets, U flk is pseudoconvex.k=1

If plurisubharmonic functions are continuous, a stronger approx-imation theorem holds. To describe this, let us introduce the conceptof strictly plurisubharmonic function.

3.2. REGULARIZATION OF PLURISUBHARMMONIC FUNCTIONS 43

In general, given a locally integrable function cp on 12 and z° E Q.let L[yp](zo) (E [-oo. oo]) be the supremum of real numbers E such thatV(z) -EIzl2 is plurisubharmonic on some neighborhood of z°. Clearly,

L[V + 01 > L[4o] + L[O]

and

L[max(ip,11b)] >- L[V;])

It is also obvious that L[V] is lower semicontinuous.If L[ i J(zo) > 0, 0 is said to be strictly plurisubharmonic at zo,

and if L[y,] > 0 on fl, 4' is called a strictly plurisubharmonic functionon f2.'

Denote by PSH*(f2) the set of all strictly plurisubharmonic func-tions on fl.

THEOREM 3.14 (Richberg's theorem). If 4 E PSH*(f1) n C1 (fl)?then for any positive-valued continuous function E on fl, there existsa function V E PSH*(fl) n C' (0) that satisfies the inequalities

0<y0<4+E.PROOF. Take countably many n-dimensional open balls 3(pj, RR)

(j = 1, 2, ) that satisfy all the following three conditions:(3.19) 3(p;, R,) C .f2.

°(3.20) UB (pj,

Rj=fl.

i=1

/2

(3.21) For an arbitrary compact set K in f1, there are only finitelymany j's such that 3(p., R;) n K # 0.

Also, fix an increasing convex function A (in the broad sense) of class

C°° on R such that supp A C 13 , oc) and A(1) = 1.

For 4' and e, let us construct inductively an element 4(k) ink R.

PSH*(fl) that is of class C°° on a neighborhood of 6 B pi,j=1

and satisfies 4' < 4,(k) < 0 + e on f2.For the case k = 1, since 4' E PSH*(fl), there is a positive number

,h such that

(3.22) L [4, - 7,A ( R2 )] >2

3L['U]

1

'In the literature, the condition "of class C2" is also frequently included.


on IR (pi, R1). Hence, given a positive number 5, set

1P a := 06 - 771,\1z - R2112

I

then, for a sufficiently small 6,

(3.23) L[ib] >3

L[ei] ands < -0 + E

on B(pj, R1). In addition, we have

(3.24) b<*on some neighborhood of OB(pi, RI ). Fix such a 6, and set

(3.25) 0(1) (z) := max{Ifi(z), ib(z)} for z E B(pl, RI),

10(z) for z E fl \ B(pi, RI ).

Then V) < (1) < t/i + E and 0(1) IB (p1, 21) E C°°(12). From the

2construction, it is also obvious that L[ip(1)] > 2L[7p] on B(p1,RI).

The method of producing V,(k+1) from 0(k) is described as follows:First, by a method similar to the above, transform 1'(k) to ilk)

on B(pk+I, Rk+1). But in this process, replace the condition (3.22)by

(3.26) L {b(k) - 7]k+1X CIz RPk+112

/ J> -

k+1 7=1

so that z<i(k) possesses the properties corresponding to those of -06 on

B(Pk+1, Rk+1)Let Xk be a nonnegative real-valued function of class C°° on .fl

kwith a compact support such that Xk U B (pj, 2' 1 and I/i(k) is

i=11of class C°C on supp Xk. Set

(3.27)

1maX{7G(k)(z), (1- Xk(z))t 'k) (Z) + Xk(z) (k)(z)}0(k+1)(z) for z E B(pk+1, Rk+1),

0(k) (Z) for z E 12 \ B(pk+1, Rk+1)

Clearly, 0(k+1) > 0(k) >_ 0. By taking a sufficiently small 6, we get

3.2. REGULARIZATION OF PLURISUBHARMONIC FUNCTIONS 45

(3.28)k+1

L[(1 - Xk) (k) + Xk'I/l(k)] > 1 - and V)(k+1) < V)+ sj=1

on B(pk+1,Rk+1), and furthermore

(3.29)a(k) < P(k)

on some neighborhood of 8B(pk+1, Rk+1). In this case, (k+1) coin-cides with 'tl k) on some neighborhood of .fl \ 18(pk+1, Rk+1). Also,

Rk+1 1since >on B t

2J , it follows that z/i(k+1) coincides

with (1-Xk)O(k)+Xklk(k) and is of class C°° on this open ball. Hence,all the requirements are met.

r lk

lFrom the construction, we have 0(k) on U lE$ pj,2j=1

m 1 (k) exists, satisfying i& < cp < 0 + c andTherefore, cp :=00k

cp E PSH* n COO (17). 0In general, a real-valued continuous function cp on a topological

space X is said to be an exhaustion function on X if the subset {x I'p(x) < c} of X is relatively compact for every real number c that isless than the supremum of the values of 'p.

By virtue of R.ichberg's theorem, the definition of pseudoconvex-ity can be strengthened as follows:

THEOREM 3.15. A pseudoconvex open set has an unboundedstrictly plurisubharmonic exhaustion function of class C°°.

In recent years a remarkable result on the approximation of plurisub-harmonic functions has been obtained. For the present, let us intro-duce the statement of this result while putting off its proof.

Let V) be a plurisubharmonic function on Q. The Lelong numberv(0, x0) of i at a point x0 of .R is defined by

sup V)

v(7/,, xo) := liminf = limz-.x0 log Iz - xol r\0 logr

Recall that log If I E PSH(Q) for f E A(Q). It is easy to see that

v(log If I, xo) = sup{k E Z+ I (a) < k implies f (a) (xo) = 0}.


For a locally integrable plurisubharmonic function define aHilbert space

(( lA2my (n) { f E A(n) f IfI2e-2mL4dV < oo }

ll n JJJ

Take an orthonormal basis {Q1 }i ° I of A2,.,,t (.R), and set

X00log L IQ1I2.2m. 1=1

THEOREM 3.16 (J.-P. Demailly, 1992). There exist constants CIand C2, independent of m, which satisfy the following conditions:

a. ?'(z) - CI v,,,(z) Sup L'(() + 1 log Cn , where z E Qfin K-zI<r m r

and r < bn(z).b. v(,, z) -

m< V(z m, z) S v(,, z), where z E ,fl.

As an application of Demailly's theorem (whose proof will begiven in § 5.4 (b)), we immediately obtain a deep result on the Lelongnumber.

COROLLARY 3.17 (Siu's theorem). Let !1 and V be as definedabove. Then, given a positive number c, the set

Ec(') :_ {z E 12 I v(Vi, z) >_ c}

is an analytic subset of Q.

PROOF. From Theorem 3.16 (b),

Ec(') = n Ec-n/m(Wr)m> 1

However, since

Ec-n/m(1m)_{zI (a) <me-nimplies a")(z)=0forl=1, 2, },

it follows that, the Ec_n/m(t&m) are analytic subsets, and so is theintersection EE(') of these sets.

REMARK. Ec(ip) is clearly monotone decreasing with respect toc, but not much more than this is known. From a property of analyticsubsets, it has been known that Ec( r/)) is left continuous on c and hasonly countably many discontinuous points. At a discontinuous pointc', there occurs the phenomenon that a family of analytic subsets of

3.3. LEVI PSEUDOCONVEXITY 47

EE(?P) as c \ c' is absorbed by a higher-dimensional analytic subsetin EE' (0).

REMARK. The first conjecture on the approximation of plurisub-harmonic functions was made by Bochner and Martin [4, p.145] inrelation to the Levi problem in the following form:

DEFINITION 3.18. A Hartogs function on fl is by definition anelement of the smallest among the families _F(fl) of functions on P_with values in IR U {-oo} that satisfy the following conditions:

1. f E A(fl) implies log If E F(fl).2. fl, f2 E .F(.R) and c E [0, oo) imply fI + f2, cf1 E .F(fl).3. If { fA}.EA C F(fl), and if the fA's are uniformly bounded

from above on compact subsets in fl, then sup fa E .F(fl).

4. Both {fj}j- I C.F(fl) and fj fj+I imply slim f? E F(fl).-00

5. f E F (S?) implies lim f (z') E .F(fl).Z' z

6. Letting 1-1(fl) be the smallest of the .T(fl)'s that satisfy (1) -(5)1 if f satisfies f I fl* E f(fl*) for any relatively compactopen set fl* of fl, then f E .F(12).

Bochner-Martin conjecture:Plurisubharmonic functions should be Hartogs functions.

If fl is pseudoconvex, we see that the B-M conjecture is correctby using the solution of the Levi problem (Bremermann [6]). Onthe other hand, clearly the B-M conjecture would affirmatively solvethe Levi problem, but there is a domain that gives a counterexampleagainst the B-M conjecture ([ibid.]).

Roughly speaking, Bremermann's result corresponds to the poly-hedral approximation of figures, while Demailly's theorem corre-sponds to the approximation by surfaces.

3.3. Levi Pseudoconvexity

Open sets that have boundaries of class C2 are mainly described.A boundary point p of fl is said to be of class Ck if there exist aneighborhood U of p in C' and a real-valued function ru of class Ckon U that satisfy the following two conditions:

(3.30) U n P = {z I rj(z) < 0}.(3.31) {zEUIdru(z)=0}=0.


In this case, ru is called a defining function of .R on U or simplyaround p. When every point of 8.R is of class Ck, .R is said to haveboundary of class Ck, and we write al? E Ck. By means of a partitionof unity, we can construct a real-valued function r of class Ck definedon a neighborhood of Si that satisfies

(3.32) .R = {z I r(z) < 0},

(3.33) {zE0(11dr(z)=0}=0.In general, we call such a function r a defining function of .R.

Let p be a C2 boundary point of .R. Since the tangent space to8f2 at p is of real dimension 2n - 1, this space contains a complexhyperplane. This is called a complex tangent space of 8(l at p, andis denoted by Tp. Given a defining function r of (1 around p, thequadratic form

z _(3.34)

j,k 8 8jazk (P)CjZk

on the vector space {l; E Cn I l; + p E Tp} (= KerOr(p)) is said to bethe Levi form of r at p.

DEFINITION 3.19. A C2 boundary point p of .R is said to bestrongly pseudoconvex if either n = 1 or the Levi form of r at p ispositive definite.

.R is called a strongly pseudoconvex open set if f1 is bounded andits boundary points are all strongly pseudoconvex.

The positive definiteness of the Levi form does not depend onthe choice of a defining function r, since another defining function iswritten as a multiple ur by a positive-valued function u of class C',and, given p E 012, we have

(3.35) 8(ur)(p) = (u8r)(p),

and

(3.36) 08(ur)(p) = (u8ar + 8u A Or + Or A 5u) (p).

In particular, take u = eBr; then the right hand side of (3.36) becomes(88r + 2B8r A 8r) (p). Hence, if the Levi form of r is positive definiteat p, then ur is strictly plurisubharmonic near p for a sufficiently largeB.

From this the next proposition follows.


PROPOSITION 3.20. A strongly pseudoconvex open set is pseudo-convex.

PROOF. For a strongly pseudoconvex open set 12, from the aboveargument, there is a defining function r of (2 that is strictly plurisub-harmonic on a neighborhood of 8.(1. Since 12 is bounded, there is asufficiently large positive number C such that - log(-r) + Cjzl2 isexhaustive and strictly plurisubharmonic on .R.

From (3.36) it follows that the signature of the Levi form doesnot depend on the choice of a defining function.

(2 is said to be Levi pseudoconvex at p if the Levi form of a definingfunction is semipositive definite at p. An open set whose boundarypoints are all Levi pseudoconvex is called a Levi pseudoconvex openset.

THEOREM 3.21. A Levi pseudoconvex open set is pseudoconvex.

PROOF. For p E an, set

12p,e := {z I r(z)+EIz-p12 <0}, where e> 0.

From the assumption, there are a neighborhood U of p and a positivenumber co that satisfy the following two conditions:

1. Given z E U and e < co, there is a point w in anp,E fl U such

that 5Q, (z) = I z - w1.2. Given e < e'o and q E a12p,e fl U, the Levi form of r + elz - p12

is positive definite at q.

In this case, since every point of a12p,e fl U is a strongly pseudocon-vex boundary point of f2p,e i it follows that - log 8n, is plurisubhar-monic on ,(2p,e fl U. Moreover, since - log 5np.r \ - log 5n as e \ 0,(3.18) implies that - log bn is plurisubharmonic on .(? fl U. As p waschosen arbitrarily, there is some neighborhood W of a12 such that- log bn E PSH(f2 fl W). Finally, compose - log 5n with an appro-priate increasing convex function; then we obtain a plurisubharmonicexhaustion function on 12. (See Corollary 3.11.)

That the Levi pseudoconvexity is a fundamental concept is alsounderstood from the following:

PROPOSITION 3.22. A pseudoconvex open set that has a C2boundary is Levi pseudoconvex.

50 3. PSEUDOCONVEXITY AND PLURISUBHARAIONIC FUNCTIONS

PROOF. Let .0 be a plurisubharmonic exhaustion function on fl.If there were a boundary point p at which f1 is not Levi pseudoconvex,then there would be some v E Tp such that

(3.37)82r

(p) (v - p)iv - P)k < 0.,.k uZj0Tk

Let vp be the unit inward normal vector WOO at p, and considera family of holomorphic mappings:

7i : C -p C'

W W

--1 p + tvp + ( (v - p) fort>0.Then from (3.37) there is a positive number E such that

(3.38)J70(0(0, E) \ {0}) C fl,

7rt(A(0,E))Cf2 for0<t<<-

Note that o 7t is subharmonic on its domain, since is plurisub-harmonic. From this, it follows that

sup v = sup{y o 7t(() E p(0, E) for 0 < t <_ E}

< sup{v/ 07rt ) for0<-t<_E}< sup V,

J7

which is a contradiction. Therefore, f1 must be Levi pseudoconvex.0

It is evident that Levi pseudoconvexity does not necessarily implystrong pseudoconvexity, but we call attention to the following twofacts:

PROPOSITION 3.23. If 8f1 E C2 and !2 is bounded. then .R has astrongly pseudoconvex boundary point.

PROOF. It suffices to set R := sup jzj and choose a point p of 0 1zEf2

such that IpI = R.

PROPOSITION 3.24. If a pseudoconvex domain f1 is bounded, and8.2 E C' , then the set of all the strongly pseudoconvex boundarypoints of !1 is a dense open subset of W.


PROOF. If n = 1, from the definition 12 is strongly pseudoconvex.In the case n >_ 2, aS? must be connected. Otherwise, there wouldbe a most inside component of OR, and from Proposition 3.23 itsinside would have a strongly pseudoconvex boundary point. Thismeans that the same point cannot be a Levi pseudoconvex boundarypoint of .R, hence a contradiction. Now that 8.f1 is connected, hasa strongly pseudoconvex boundary point, and is real analytic, thedesired consequence follows from the theorem of identity. 0

The two examples stated below are well-known, and explain muchabout general Levi pseudoconvex domains.

EXAMPLE 3.25 (Kohn-Nirenberg). Set

:= {z r := Re z2+IZ118+715 Iz1I2Re ri <0}

Since r = Re z2 + zi zi + 14 (z1 z1 + z1 z1), it follows that

rz2z2 = rz122 = rz211 = 0,

and

rz1z1 = 16jz116 + 15Re z1

> 161z116-151x116 = Iz116 >0.

Hence, r E PSH(C2), and .f2KN is pseudoconvex. Moreover, B.RKr4I isof class CW on some neighborhood of 0, since 0 E 812Kr.1 and i9r(0) =dz2(0). Therefore, from Proposition 3.22, 12KM is Levi pseudoconvex.Notice that .f2Kr.I shows, at first glance, a singular property at 0 asfollows:

(3.39) If f E A(13(0, e)) with e > 0 and f (0) = 0, then

V(f)nQKN 6 0.For the proof of this fact, we refer the reader to either [20] or [31].

The Kohn-Nirenberg example presents a striking contrast to thefollowing self-evident fact:

PROPOSITION 3.26. If p is a strongly pseudoconvex boundarypoint of .(l, then there exist a neighborhood U of p and a biholomorphicmapping 7r: U -+ On such that 7r(p) = 0 and

7r(Unf2)e {zRe z+> Izj12<0j=1


In particular, putting f := z o ir, we obtain f E A(U), f (p) = 0, andV(f)nQ=0.

This means that `strong pseudoconvexity - convexity in the nar-row sense' modulo biholomorphic equivalence, and .f0KN provides anexample showing that `strong' cannot be replaced by 'Levi,' and 'nar-row' cannot be replaced by `broad.'

EXAMPLE 3.27 (Diederich-Fornaess). Let A: R -+ R be a C°°function that satisfies the following four conditions:

1. a>0.2. A(t)=0when t50,and)(t)> 1 when t - 1.3..1"(t) ? 100A'(t) > 0 when t > 0.

4. A'(t) > 100 for t with A(t) > 2

The following domain is called a worm domain, due to Diederich andFornaess:

I'2

.fQDF,r {z E C2I z2 1

+A(Iz112

- 1) + A(Iz1I2 - r2) < OT

where r > 1.i?DF,r is a bounded pseudoconvex domain with boundary of class

C°°. If r >_ e', then

1DF,r D {(zl,0)I15IzlI<e"}U{(zl,z2)I Izll=1ore',Iz2+1151}.Hence, by an argument similar to the proof of Proposition 3.22, anypseudoconvex domain containing S2DF,r must contain {(z1, z2) 11 <Izi I < e" and 1z2 + 11 < 1}. In order to construct .fQDF,r, first takea disk on the z2 plane whose boundary contains the origin, and thenlet this disk revolve around the origin, changing the center and radiusappropriately at the same time the disk travels along the zI plane.The point of deriving the pseudoconvexity of l1DF,r is that the angleof rotation is a harmonic function of zI. (For the details, refer to theoriginal article or [25].)

REMARK. Recently, a remarkable result on the a equation ona worm domain has been obtained. Namely, due to M. Christ [9],it is known that P(C0 (.fQDF,r)) 0 C'(QDF,r) for the orthogonalprojection

P: L2(.RDF,r) A2(I1DF,r)


This property of worm domains contrasts finely with the nextresults.

THEOREM 3.28. The closure of a pseudoconvex domain whoseboundary is of class C" possesses a system of pseudoconvex neigh-borhoods.

For the proof, refer to [14].

THEOREM 3.29 (Kohn's theorem). If (l is a strongly pseudocon-vex domain with boundary of class C°°, then P(C°D (f1)) C C'(77),where P: L2(37) - A2(fl) denotes the orthogonal projection.

For the proof, refer to [18].As it will be seen in Chapter 6, Theorem 3.29 is useful in studying

the boundary behavior of holomorphic mappings.

CHAPTER 4

L2 Estimates and Existence Theorems

We have already observed that the division and extension problemson A(.(1) are reduced to the problem of solving the a equation underappropriate constraints, and have shown that in order for the a equa-tion to be solvable, Q must be pseudoconvex. In this chapter, let usgo ahead and solve the a equation on a pseudoconvex open set oncewithout any constraint. The argument stated here follows basicallythe approaches by J. J. Kohn, L. Hormander, and others, but thecore method of deriving the L2 estimates is due to the article [371.This modification has practically no effect on the existing results in-cluded in this chapter, but will make an essential difference in thenext chapter. _

In § 4. 1, we derive the vanishing of the a cohomologyon a pseudo-convex open set by means of the L2 estimates. Here the 0 equation issolved under estimates with respect to the L2 norm with weight func-tion. Fundamental formulae and inequalities are shown in this sec-tion. From the vanishing of the a cohomology, it follows that Serre'scondition and pseudoconvexity are equivalent. In § 4.2, we apply theexistence theorem proved in § 4. 1 to generalize the classical results offunction theory in one variable to several variables. These results wereall established by Kiyoshi Oka, but we follow Hormander's methodshere.

4.1. L2 Estimates and Vanishing ofd Cohomology

The goal of this section is to show the solvability of the a equationon a pseudoconvex open set by means of the so-called L2 estimate.This method is one of the fundamental ideas in functional analysiswhich originates with the Fredholm alternative, and uses an infinite-dimensional system that keeps the equivalence between the existenceof solutions for linear equations and the uniqueness of solutions oftheir adjoint equations.

55

56 4. L2 ESTIMATES AND EXISTENCE THEOREMS

Let us review closed operators briefly before starting the details.Let HI and H2 be Hilbert spaces. A closed operator from HI

to H2 is by definition a linear mapping T that is defined on a denselinear subset D in H1 and has values in H2 such that the graph

GT:={(u,Tu)IuEV}is a closed set of the direct sum H1 ® H2. V is called the domain ofT and denoted by Dom T. Also, T (V) is the image of T and denotedby Im T. The denseness of Dom T determines a closed operator ffrom H2 to H, whose graph is the orthogonal complement of GT inH1 ® H2. -T is called the adjoint operator of T and denoted by T.The definition of T* may be rephrased in terms of the inner product(, ); of H; as follows:1

(4.1) T'v = u:' (Tw, v)2 = (w, u)I for any element w in Dom T.

From (GT) l = UT = GT, we obtain (T')* = T.In what follows, let II Ili denote the norm of Hi.

THEOREM 4.1. The following two statements on an element v inH2 and a positive number C are equivalent:

1. There is an element u in Dom T such thatTu=v,

1 Ilulll -< -C-

2. For any element w in DomT',

I(V, W)21:5 C 1 .

PROOF. 1 : 2: Let w E Dom T'. From (4.1) it follows that

I(V,w)21= I(Tu,w)21= I(u,T'w)IIIIull1 IIT*wII I C IIT`wIII

2 1: Define an antilinear mapping over Im T' by

1 : Im T' Cw w

y = T'w (v, w)2 .

From the assumption, Il(y)I < C IlyIII for any y E Im T*. Therefore,from the Hahn-Banach theorem, we obtain an extension T of I to H1

I A :tea B reads 'A is defined by B.' This is a common usage nowadays.

4.1. L2 ESTIMATES AND VANISHING OF & COHOMOLOGY 57

such that

(4.2) Il(x)I<_Clixil1 for anyxeH1.

From the Riesz representation theorem, there exists u E H1 such that

(4.3) l (x) = (u, x) 1 for any element x in H1.

In particular, this implies that

(4.4) (v,w)2 = (u,T*w)1 for any element w in DomT*.

By applying (4.1) to this statement, we derive (T*)*u = v, and thusTu = v since (T*)* = T. Moreover, it follows from (4.2) and (4.3)that IIulI1 < C. 0

We would like to apply Theorem 4.1 to the 8 equation:

6u=vforvELoop(fl)f1Ker8 (q>0)in order to show that there exists a solution u in L °' -1(Il) under anappropriate condition.

For this purpose, we consider the Hilbert space

L° (n) {f EL i feIfI2 dV < oo }

for a real-valued function p of class C2 over fl. Also, let Lo , q(Q) bethe set of all (0, q)-forms with coefficients in L' (.fl), which inheritsthe structure of Hilbert space as the direct sum of L2,(Q)'s. Letu:= E' uI dd1 and v E' v, dz1 be elements in L°, q(,fl). We definethe weighted L2 norm f fl

on L°W q(Q) with weight function W by

(4.5) IIutI , := E' IIuiV ,, where 11U111, J e-(Iuj12 dV.

(This definition of norms of differential forms is not, in general, com-patible with the patching of coordinate neighborhoods in a manifold,but the advantage of fixing a local coordinate system is an enormoussimplification of arguments.) Similarly, the weighted inner product(, ),y on L°,q(Q) with the weight function cp is defined by

(4.6) (u, v),1, :=v) dV, where (u, v) :_ E' ujUj .I

Also, for a multi-index I:= (i1, ... , ip), we set

(4.7) I :_ {il,... , ip}


and

(4.8) Ik" :_ (i1, ... , k, ... , ip), where iµ is replaced by k.

µ

For u E Co'q-1(Q) and v E C°,q(0),

if-`,' (au, v) dV

E, E' sgn (J) auJvi dV

n I kI azkiu{k} = J

eE-' E' sgn (:)Iavj - aW vr/1 dV_ -J

n I ur Ozk 994iu{k} = J

_- fur sgn

/.Il(av, - acdV.vi)

In kIJ azk &kiu{k} = J

Therefore, if we define a differential operator `pt9 of the first order by

(,) (avJ -acp vj) dzr.(4.9) "19v :_ sgn U &k azk

iu{k} =J

it follows that

Je f(a u, v)= (au, v) dV = J e(u, W19v) dV = (u, t9v).

nSimilarly to the case of a, we extend the domain of `pT9 to L

°C(Q).

Define an operator a from Lw, Q-1(.f2) to L°p, q (.R) by

Dom`65:= {u E Lo,q-1(.fl) I DU E LOP, q(.R)}

w

U -- èau := au E L°q(f2).

Then it turns out that a is a closed operator2. In fact, Domais dense in L Q-1(Q), since clearly Co' q-1(Q) C Dom "a. In orderto show that the graph of 'a is closed, it is sufficient to show that(u, v) E G,,3 provided that (u, v) E i.e. there is a sequence(u,,, 4'c9-u,) E Gj9 such that

u u and v (v -+ co).

2Closed operators of this kind are, in general, called 5 operators.

4.1. L2 ESTIMATES AND VANISHING OF t'3 COHOMOLOGY 59

From(u, `°19w),, = lim (u,,, "'19w)w

V 00

= lim (au,,. W)"' = (v. W)" .V 0C

it follows that au = v. Therefore, (u, v) = (u, 8u) = (u, `58u), whichtells us that Gj = C.

The adjoint operator of `08 is written by In a similar fashionto `18, by restricting '10 to the following subset of L°,-4(f2):

{v E L° 4(12) I "19v E L0, Q-1(12)}

we obtain another closed operator, which we denote by the samenotation I19 for convenience.

In this section, we show that some differential inequality holdsby applying integration by parts to an element in Co'1(1l), and thatthe range for this inequality to hold, in fact, extends to Dom va * flDom "o n L°,9(0) by means of a sort of approximation theorem.

First of all, let us prepare this approximation theorem.Let p : 11 (0, 1] be a C°° function such that. 5n > p

min {bn

,1.

and sup I dp(z) I < oo.l 2 ZeR

THEOREM 4.2. For any element v in Dom `'8f1Dom ''19fL°,q(.R).there exists a sequence {v,}µ 1 in C0'Q(.fl) such that

(4.10)

lim (IIP' (av11 - 8v) II, + IIp. (`'19vp - `P19v) II', + iIvµ - V11 ,P) = 0.{10

PROOF. Take a Cx function X : l1 - l such that X I (-oo, -2)=1andXI(-1,oc)=0,andset

XR(Z) := X(-RP(z)) . X ( IR - 3)

for R > 0. Then, as R - oo,

PD(XRV) -' PZ)V,(4.11) P'"l)(XR v) - p "i9v,

XR Z' i TJ

in the sense of convergence with respect to the normIn fact, by calculation, we obtain

(9(XR v) = aXR A V ± XR COV.


For the second term of the right-hand side, we have

IIP (v - XROV) III; Ii5v-XR3VIIy, - 0

as R oo. As for the first term OXR A v, we have

8XR = - Rx' (-RP(z)) X (IRI - 3) aP

+aRIX(-RP(z))x'(IR 3).

Since p < R on supp x'(-Rp(z)), from the boundedness of I apl, it

follows that IPOXR A vI is bounded when R -' oo. On the otherhand, because of the choice of p, for any compact set K of fl, thereis a sufficiently large R such that aXR I K = 0. Therefore, sinceIIP'XR A vli, -' 0 as R --+ oo, the combination of these results yieldsP0(XR V) - P&-

The other two cases in (4.11) follow similarly.In order to deduce the desired conclusion from (4.11), it suffices

to use the following three kinds of convergence with respect to II Ilp

a(xR v)e 0 a(XR v),(4.12) `°'9(XR V)E I '19(XR v),

(XR V)E -' XR v

as e --+ 0, where (XR v)E is the e-regularization of XR v. This conver-gence is nothing but a general property of regularizations. (Refer to[281.)

For the time being, we exhibit several formulae in order to arrangeefficiently those terms produced by integration by parts.

For a continuous (a, b)-form won fl, an operator e(w) : L oq (fl) -LI a,4+b(!l) is defined by e(w)(u) := w A u. Let t(w) denote theadjoint operator of e(w), that is, t(w) is an operator from L oq(fl) toLp-a,Q-6(p) such that

(w A u, v) = (u, t(w)(v)) for u E Llp-"9-b(fl) and v E L oq(fl).

c(w)(v) is simply written as w _j v.

4.1. L2 ESTIMATES AND VANISHING OF 5 COHOMOLOGY 61

PROPOSITION 4.3. For B >2Badza E C°"1(17) and uC1

uldzl E C°'q(n),1

(4.13) BJu=E'EE'sgn( J)BauldTi.

PROOF. For v = E'vjdzj E C°'q-1(9),

(B Av, u)

()_ E' E E' sgn I BavJut

EvJ (F- "Baut sgn (aj_ (v,E'EE'Bautsgn( j)dz.1) .

\ J a I

Set V :_ °z9, and define t9 by :u := (T9u). Then from (4.9) it iseasy to see that

(4.14)ai9 +,9a = at9 + We,8t9+t9a=o,a,9+t9a=0.

PROPOSITION 4.4. For a C2 junction?? on (l and an element uin Cn,q(fl),

(4.15) an n Wu + au + a u)

_ Z' E 8277ujdz1 A ... A dzn A dz1k

I j 6 j k OZjazk

= E' E(9277

urkdzl A ... A dzn A d'21,I j,k OZjazk j

where we set ulf = 0 for j and k when I is not defined.

PROOF. From (4.9),

(4.16) N A :aua aut

dz1 A ... A dzn A dz j.1 j azj azj

62 4. L` ESTIMATES AND EXISTENCE THEOREMS

From (4.13),

(4.17) art, au = F_' E azaul

dz1 A A dzn A dzlI jgi j j

a I dzI A ... A dz A dztk .j¢IkElazk azj

Similarly,

(4.18) a(art _j u)/

ulsgn` I )dzl.I j I azj jJ

a277 uldzl A ... A dzn A dzljI j.k aZja zk k

art au j+E'E. _I jElak azj azk

E' E 19277 uldzl A ... A dzn A d7-IAAI j.k azja.Zk

'Iq+E'

auldzl n ... A dz,, A dz jaz a z Ik

I jEIkOI 7 k

1: ±1azJ

aut dzI A ... A dzn A d'51.+EI AEI J

By adding these formulae, the desired (4.15) is obtained.

From (4.14) and (4.15) it is easy to derive the following:

PROPOSITION 4.5. If 77 is a positive-valued C2 function on 11,then

(4.19) II 1auII2 + IIv' 9uII22

2Re (t9u,art_j u) - aaAUjkfjdVI 1j. k

aj

for u E Co "(Q).

In what follows, (4.19) will be proven again in a more generalform for the L2 norm 1111,, with weight function o.

For this purpose, define an operator Pa by

(4.20) ''au := au - A u.


Then, for u E C"'(Q) and v E COP}1*9(Q).

(4.21) (Wau, v)s, _ (e-1' 'au, v)ov)o = (e-yu,19v)o

(u,

Namely. the adjoint operator of a with respect to II II, is equal to 0on Cop" (0).

TO-:=On the other hand, for the complex conjugate a - e(0 ) of'P,Y, we have

(VC9u, v)0 = (e Y(a - e(acp))u. v)-Y

_ (a(e 'u), Z')_ , = -'Ov)_

(u. -w19v)o

Therefore, the adjoint of 'a with respect to II IIo is equal to -Pi9 onCop,q(Q).

Direct calculation shows that

(4.22) z9 pa + "'a19 =19a + a19 - z9e(8(p) - e(acp)19 .

Now take the complex conjugate of both sides of (4.22), consider theiradjoint operators, and change the sign of cp so as to obtain

(4.23) V19a + v x'19 =19a + a19 + &(a0.

For simplicity, given a C2 function 5= on n and a differential formu = E'uldzl A A dzn A dz1 or u = E'u jdz1, we define

1 1

2

(4.24) Ly,u := E' E as a4k ulkdzl A ... A dzn A dz1I 3,k '

a2

or F_' F, ulkdzl,I ;,k az;azk

respectively.

PROPOSITION 4.6. Let V and p be C2 real-valued functions on !1.Then, for any element u in Co,4(.R),

(4.25) IIPauI12 +IIP19uII = 11p:jU112 + 4Re (apJ u.

+(p2L,u, u),, - (Lp2u, u),..


PROOF.

lipauII1 + IIP"tullw - IIPtuil,= (p28u, 8u)W + (`°t9u, p2 `°t9u),, - (t9u, p2t9u),,

= (p2u, Pt98u),, - (8p2 A U, 7u),, + (u, p28'Pt9u),,

+(u, 8p2 A °jqu)W - (u, P2'°8t9u),P - (u, 8p2 A t9u)

= (P2 U, au + e(eW _j u) + A 9u),,

+ 2 Re (u, aP2 A `'ft),, - (Lp2u, u)W= 4 Re (8p J u, p `Pt9u),, + (p2Lwu, u)ti, - (Lp2 u, u),p,

where we have used both (4.22) and (4.23) for the third equality and(4.16) for the last equality.

COROLLARY 4.7. Under the same condition as above, for everypositive number C,

(4.26) IlPeull, + (1 + C)IIPpt9uiI ,

> (p2L ,u, u),, - (Lp2u, u), - 2 U 1 1 2

The calculation passed through (n, q)-forms, but by looking into(4.26) we see that the same result also holds for (0, q)-forms. Since wepreviously set up the 8 equation for (0, q)-forms, though overlappinga little, Corollary 4.7 can be restated in this form:

PROPOSITION 4.8 (Fundamental inequality). If cp and p are C2real-valued functions on (1, then, for any element u in C0'q(.R),

(4.27) Ilpeu1l,2p + (1 + C)IlppVuII ,

>_ (P2 - llwu, u)v - (Lp2u, u),p 112)P u ,

where C is an arbitrary positive number.

In practice, we have to estimate the right-hand side of (4.27) frombelow in order to derive the existence theorem from this inequality.

A (1, 1)-form w = Ew,kdzj A dzk on fl is said to be nonnegativej'k

(or positive) at a point xo E Q if the matrix (wjk(xo)) is a semipositive(or positive) definite Hermitian matrix, respectively. We write w > 0(or w > 0) when w is nonnegative (or positive), respectively.3 In

3The condition wl - w2 > 0 is written as wi > w2. (wi > w2 is understoodsimilarly.)

4.1. L2 ESTIMATES AND VANISHING OF a COHOMOLOGY 65

terms of this expression, from the definition of L,p it is clear that(L,Du, u) >_ 0 for any u E CO.9(12) if and only if 809W ? 0 on Q.

Also, set

dzJ(4.28) wVu:=wjkuIk

I j,k

for u = >'ujdzl. Then from this definition and (4.13) it is easy tosee that

(4.29) IN-j ul12 = ((ap A p) V u, u),

Therefore, the fundamental inequality is written as follows:

(4.30) IIp au11 , + (1 + C) Ilp p9ull2

> ((p2o_oP2......aPAP)VU,U).

From now on, let (1 be pseudoconvex, and fix an unboundedstrictly plurisubharmonic exhaustion function 0 of class C°° on 11.

LEMMA 4.9. Let p be a strictly plurisubharmonic function of classC2 on fl with L[W] >_ 1 everywhere. Then, given any c E R and anycontinuous function T : R -+ R, there exists a function A : R -+ R ofclass C2 that satisfies the following four conditions:

1. A(t) = 0 when t < c.2. A(t) > r(t) when t > c + 1.3. A'?0 andA"?0.4. Letting W,\ := cp + A(' b), if u E Dom `Pa a n Dom wa t9 n Lwq (.fl),

then

IIauliv, + Il'' 9uIIW>

PROOF. For a natural number v, we can take a C°° function

p :.fl -+ (0, 1]that satisfies both

(4.31)

and

(4.32)

min { 26,?,1} min{v6,o,1}

ap9zj

<v (j=1,...,n),

because of uniform approximation of the continuous function

min{vbn(z),1}


on f1 by a C' function.Given c E R and choosing v so that p = I on we can take

A that satisfies not only (1)-(3) but also the inequality

(4.33) aapV - 2vap A app, > 0.

since v E PSH'.On the other hand, from the condition 4o] > 1, we conclude

that aa,;, aaI-I2. Therefore, for any element u in Co'9(f1), (4.30)yields

(4.34) II Pv UII A + 1+ 1) Ilp.v U112, > qjI P"uII2

Since, by Theorem 4.2, (4.34) holds for any element in Dom''aa nDom P'A V n L°4 (f1), we obtain (4) by letting v - oc.

Let us write down what follows immediately from Lemnia 4.9 andTheorem 4.1.

PROPOSITION 4.10. Let fl and 4;a be as above. If u' E Kera nL°Q(f1) (q > 0). then there exists an element v in n

such that

(435)2

at' = ur,

gIlz'll a i--1 IIWIL

Likewise. if w E KerY'd n L°,4(R) (q > 0), then there exists anelement v in Dom s'' 0 n L°' (fl) such that

val3v=w,(1.36)

gIIL'I12 Ilu'll

PROOF. We prove only the first statement, since the second canbe (lone by the same argument. It suffices to show that

(4.37) I(u',u')wa12 < g1lu'II21I'aa*u1il

a

for any element u' in Dom n Decompose u' as

u' = u1 + u2, where ul E Ker a and u21Ker a.

Since E Ker a, it follows that

(4.38) (uw, u') a = (w. u1)ti,A.


On the other hand, since u21Ker a,

(4.39) v' E 0 = (c?v', u2),,A = (v','°''a*u2)a,, .

Therefore, U2 E Ker èa 8 * , which in particular implies that u l EDom a * and u' = w) 8 * ul. Hence, Lemma 4.9 can be appliedto ul to yield

(4.40) II`Pxa*u'IIA

Combining this with (4.38), the Cauchy-Schwarz inequality implies(4.37), as we wish.

The following existence theorem is a fundamental theorem witha wide variety of applications.

THEOREM 4.11 (Hormander's theorem). Assume that f2 is pseu-doconvex, and a C2 function ep :.R , ]R satisfies L[,p] > 1.

1. For any w E Kerafl L°,4(0) (q > 0), there exists an elementV in L°-9-1(Q) such that av = w and gllvll;

2. If w E Ker B fl L a9 (.R) (q < n) and the support of w iscompact, then there exists an element v in L o9-1(Tl) suchthat the support of v is compact, 8v = w, and (n - q)IIvII?< IIwI1?,,

PROOF. (1) For c E R and T - 0, let ac denote, expressing thechoice of c, a function \ that satisfies the condition of Lemma 4.9.Since Ac is nonnegative, IIwII.A, IIwIIw and w E L°Q (!l).

Therefore, Proposition 4.10 implies that there is some v, ELO,? 1(Q) such that

4.41Javc = w,

( ) gIIvcll;aC < IIwI12 (- IIwII2)

Since l l vc l l v,,, is bounded with respect to c, {v} has a subsequencethat converges weakly on compact sets. It is sufficient to choose thelimit of this subsequence as v.

(2) For u = E'uldzl, set* , 12 nu := sgn ( I J ujdzj.

Then the defining equation (4.9) of 'PO is written as

`°19u = -e`°(ae-`Pu*)* .


Therefore, if w E Ker 8, then &A- w* E Ker VA-d. Also, since thesupport of w is compact,

IleW.%C W*

Ilsv.,C = II eS'w II, (= IIwII-w)

for a sufficiently large c.Now that the latter half of Proposition 4.10 is applicable to

&ac w*, there is vc such that

W.%,Ovc = &acw*,

1(n - IIwII',

for a sufficiently large c.Apply the above result to co,a, (s > 0) instead of spa,, and take a

sequence of numbers sµ -> oo and a sequence Iv,,}-, that convergeswith the LZ norm on compact sets such that

f `R' 6v,, = e`° '>w*,

(n - IIwII' ,

where we set cp,µa, If we choose, in advance, ac(t) > 0 whent > c, then, for v,,,, := lim vµ,µo0

I supp voo C 11 ,,c ,

(4.42) '*Vv,,. = ePw*,

(n - 9)IIv00II IIwII'

Hence, it is enough to put v := a-`°v;. 0Supplement. It is readily seen from the above proof that if

supp w C 11 y,c ,

then we can take v with supp v C .fl,,,c as solutions of the 8 equation.As an application of Theorem 4.11, we can derive fundamental

results on the representation of 8 cohomology groups.Let us begin by setting up our notation. Define

(4.43) W (.fl) u E L o'" (.fl) I E L o'q+I (!2)

Then from the complex

woe(') ...the cohomology groups Hpo, (.fl) are determined by

(4.44) H('c(Q) := Kerafl Woc (.fl)/{8u I u E Woc-'(fl)} .


If Si is pseudoconvex, then, given u E L ,q([2), there is some C2function o with L[p] >_ 1 such that IIuJI,p < oo. Therefore, fromTheorem 4.11, we obtain, in particular, a vanishing theorem of coho-mology.

THEOREM 4.12. If Si is pseudoconvex, then Ho,9(12) = {0} forq>0.

From the theorem of L2 holomorphy, we get

(4.45) Hoo(Q) = Hp,o(fl) .

In effect, this correspondence holds in general.

THEOREM 4.13. For any open set .fl C C", the homomorphisms

a : Hp,q(0) - Hoc (Q)induced by the injections Cp,q(f1) L oq(!1) are bijections.

PROOF. 4 We can assume that q > 0 from the above observation.Also, clearly it is sufficient to prove only the case p = 0.

Proof of surjectivity of a. Let v E Ker 8 n L°oq (.R). It is enoughto show that there is an element u in L °C-1(.R) such that v - au ECO,q(,R). For this purpose, fix a locally finite covering {Ui}901 of flwith U, C Si, where the Ui are open balls; then construct inductivelyvia...;, E Loq-t-1(U. n n U,) (0 < 1 < q) that satisfy the next twoconditions:

(4.46) vUi=&i,(4.47) Uio n ... n ui, = 5vio...it

V=0

where Uio n n U, # 0, and i means the exclusion of the index i,,.This process is possible since

(4.48) ((_1YV10......11)0=o

I

V<µ(-1) vio...

V_p

0,

4The proof is self-evident if the knowledge of the theory of cohomology withcoefficients in sheaves is assumed, but the argument is hands-on, so we give it indetail.


and H° " (Uio n n Ui,) = {0} for q > 0 from the pseudoconvexity

Set

1

(4.49) uio...ii L I Uio n ... n Ui, .

V=0

For l = q, (4.48) implies that

(4.50) uio...iq E Ker a n LI c(Uio n ... n Uiq) = A(Uio n ... n Uiq) .

Also, (4.49) yields

q+1(4.51) E

(-1)Vuio...;V

...i +i = 0.v=0

q

Use the partition of unity {pi} subordinate to {Ui} to set

uio...iv-i Piuiio...iq-i

then, first, by (4.50) we have uio...iq_i E COO(Uion nUiq). Secondly,

q(4.52)

V=0

q

(-1)VPiuiio...i,,...iqV=0 i

piuio...iq by (4.51)

rPiZ(-1)Vuiio...!2,...iq

= uio...iq .

Similarly, for 1 with 0 1 q - 1, we can construct u;o,,,it_1 En . . . n Ui,) such that

1+1 _a

(4.53) 1 (-1)Vu1 = Vui0...i1V_0

From (4.49) and (4.52), it follows that

q(4.54) (-1)V(vio...i,,...iq - uio...i....iq) = 0.

V=0

When q = 1, this means that

.(4.55) vi-u;=V3-u1

4.1. L2 ESTIMATES AND VANISHING OF 5 COHO`MOLOGY 71

When q > 1. from (4.54) in the same way that we produced u;,..;,_,we can take vio...;q_2 E W10C (U;0 n - - . n U{ q_2) such that

(4.56)

q-1'Uio ...t u. =

(_)Vv,. ,4-1

V=010...1"...44-1

By applying a to both sides of this equation, we obtain

_= q-1 -(4.57) avi... _ au'... _E(-1)"av:0 t9 1 ip 2q_2 i,,...i,...t4_1

V=°

Rewrite this in terms of (4.47); then

q-1

51?(4.58) E u: - : 0.V_° q-1 4p...i"...i4_1 io...

Repeat the process of producing (4.58) from (4.54) until we even-tually reach the formula

(4.59) vi - ui - t ' i = Z'; - u - av;

Therefore, finally we define an element u in L °C-1((1) by

(4.60) u:= vi-ui ifq=1,Vi - u= - av; ifq > 1;

then this u satisfies

(4.61) v-au(=ai)EC°"q(fl).Hence, the surjectivity of a is proven.

Proof of the injectivity of a. From Corollary 2.10, this is truewhen q = 1. Let us assume that the assertion holds for all k with1 < k < q - 1. Choose any w E Kera n C°,q(Q) with w = agfor some g E Recalling that H°-I(0") = {0} for l > 0,select a similar covering {Ui} as above so that for each i there issi E C°'q-1(Ui) satisfying asi = w I U2. Hence, using Theorem 4.12and the induction hypothesis at the same time. we can inductivelyconstruct elements s10...i, in n ... n Ui,) such that

t

(4.62) (-1)"si I U10 n ... n U1, = 5si....i1V=°

f o r 0: 51 :5q-1 .


Note that w = 8g yields 8(si-g) = 0. Hence, from Theorem 4.12,there are tio...i, E L oc-i-1(Uio fl . fl Ui,) (1 < 1:5 q - 1) such that

(4.63)

ti = Si - 9,{

- (-1)Ytio...v=0

8tio...if = sio...it

In addition, for the case l = q - 1,

(4.64) 8 (io...iq_q-1

1 - ` (%v...iv-1} = 0'

and so the L2 holomorphy theorem implies thatq-1(

-1)Ytio...

is of class C°°.

Therefore, applying the gluing in terms of the partition of unityused for the proof of surjectivity, first take

tio...iv-2 E C°°(Uio fl ... n Uiv-2)

such thatq--1

Y

q-1

vE (-1) -1 - -1)vY=0 Y=0

and consecutively choose tio...ii E C°'q-1-2 (Uio fl . . fl Ui,) inductivelyin the descending order of 0 5 1 < q - 2 such that

sia ...i1 1 tio...t,,...ttY=0

Then eventually we obtain si - R, = sj - &I'. This represents anelement of C°,q-'(fl), say, v, which results in 8v = 8(si - 8ti) =w. 0

From Theorems 4.12 and 4.13, the following vanishing theoremfor 8 cohomology groups is obtained:

THEOREM 4.14. If 17 is pseudoconvex, then

HP,q(.fl) = {0} for q > 0.

COROLLARY 4.15. Pseudoconvex open sets satisfy Serre's condi-tion.

4.1. L2 ESTIMATES AND VANISHING OF $ COHOMOLOGY 73

With this, we have the following implications:

Serre's condition = Hartogs pseudoconvexity (Theorem 3.4),Hartogs pseudoconvexity e=*- Pseudoconvexity (Theorem 3.12),Pseudoconvexity Serre's condition (Theorems 4.12, 4.13).

Therefore, the condition in Theorem 2.14, for instance, may be re-placed by pseudoconvexity. That is to say,

pseudoconvex open sets are domains of holomorphy.

This assertion, named the Levi problem (or the inverse problem ofHartogs), had been a central conjecture of long standing in the theoryof analytic functions of several variables, but was solved by KiyoshiOka for the case n = 2 in 1942, and by K. Oka [39], H. J. Bremermann[6], and F. Norguet [35] independently for general n.

The first half of Theorem 4.11 has played an active part, but alsofrom the second half we can derive an explicit consequence on theanalytic continuation of holomorphic functions.

THEOREM 4.16. Let K be a bounded closed subset of C", and flan open set that includes K. If n >- 2 and if fl \ K is connected, thenthe restriction mapping A(Q) --i A(Q \ K) is a surjection.

PROOF. Fix a neighborhood U of K that is relatively compactin Sl, and let X be a real-valued function of class C° on fl withX I K= 1 and supp X C U. Then, given a holomorphic function f onfl \ K, the trivial extension of 8((1- X) f) to C' belongs to C0011(C" )Take an open ball 3(0, R) for which supp 8((1 - X) f) C 3(0, R) and83(0, R) fl 17 0. From the latter half of Theorem 4.11, there is asolution u of the equation 8u = 8((1 - X) f) such that

(4.65) u E L2(3(0, R)) and supp u C= 3(0, R).

In this case, the theorem of L2 holomorphy implies (1 - X) f - u EA(Q), while, by the condition 83(0, R) fl 17 0 0 and the theorem ofidentity, (1 - X) f - u coincides with f on fl \ K.

REMARK. Intuitively, K may seem to be extinguished thoroughlyby repeating the process of embedding the biholomorphic image of aHartogs figure into 17 \ K and extending this image to the biholo-morphic image of 02. However, it is uncertain whether the functionextended by this method is single-valued. In fact, there is an examplein which this process cannot be continued without allowing the imageof A2 to stick out from fl in the course. (Refer to [19].)


Essentially, the same content as Theorem 4.16 can be describedas the extension theorem for functions on a real hypersurface. (Theassumption that n > 2 is kept valid successively.)

DEFINITION 4.17. Let n be a bounded domain whose boundaryis of class C'. A complex-valued function f of class CI on an is saidto satisfy the tangential Cauchy-Riemann equation if there exists anelement F in C'(77) such that

F I 8f2= f andaFABrI aft=0,where r is a defining function of n.

THEOREM 4.18 (Bochner-Harvey). Let an E C'. If an elementf in C' (an) satisfies the tangential Cauchy-Riemannequation, thenthere exists an element f in CI(fl) fl A(fl) such that f 180 = f.

PROOF. We prove this only in the case that an E COO and f ECOC(an). For the general case, refer to [22] and [27].

Let F E COO (fl) with F I an = f. From the assumption,

aF = afar+13,r for some a, E C°°(n) and Q1 E

Hence, setting FI := F - alr, we see that

F, I an = f and aF, = 0'r, where Ql := 01 -Dal

.

From a(,3ir) = 8(aF,) = 0,

01AOr Ian=o,

and so, Ql = agar +,32r for some a2 E C'(17) and 32 E Co" 1(f2)Setting

2F2:=F-a1r- 2 r2 ,

we see that

F2 I an = and OF2 =,32r2, where , 1f 2 = Q2 -2

aa2

As the same operation can be repeated, there is a sequence of func-tions {ak}k , C C°°(n) such that, given a natural number N,

a F - k k, rk = QNrN for some ON' E C°" (n)k=1

4.2. THREE FUNDAMENTAL THEOREMS 75

Therefore, there is an element F in C" (.R) such that k 1 00 = fand the derivatives of OF of all orders are equal to 0 on X2. If we set

w:= 8F on D.10 onCf2

then w E CO, I (CI) fl Ker a and supp w C fl. Hence, the rest of theproof is similar to that of Theorem 4.16. 0

REMARK. As to Theorem 4.13, it appears that the proof con-nects the world of C°° functions with that of locally square integrablefunctions in terms of holomorphic functions. The generality that hasdeveloped from arguments of this kind is the so-called theory of co-homology with coefficients in sheaves; and, further, the unobstructedview that has grown by applying this theory to analyzing the singu-larities of solutions for linear partial differential equations is nothingbut the microlocal analysis of M. Sato, T. Kawai. and M. Kashiwara[40].

4.2. Three Fundamental Theorems

In the classical general theory there are results that display the perfec-tion of the world of complex functions; here we find the Mittag-Lefertheorem, the Weierstrass theorem and the Runge theorem, and theirgeneralizations to several variables are derived from the establishedexistence theorems, Theorems 4.11-4.14.

4.2.1. Distribution of Poles and Zeros. When a function fis defined on an open set (l in C' except a null set E, f is saidto be a meromorphic function on .fl if each point xo E fl has someneighborhood U (= U(f , x0)) such that on U \ E. f can be expressedas the quotient of two holomorphic functions defined on U. Let M(Q)denote the set of all meromorphic functions on 11. Given f E M(.R),we call

f.,c:={pE.Rl limIf (--)I=x}

the pole of f. From the definition, it is obvious that f, is a closedset that is included in E. From Theorem 1.13, f extends over !l \ f«,as a holomorphic function. We identify this extension with f.

DEFINITION 4.19. For an open set U in .fl, the subset

(f I U\ f.)+A(U)


of M(U) is called the principal part of f on U and is denoted byP(f,U)

In the case of one variable, the sum of terms of negative powerin the Laurent expansion of a meromorphic function at a pole wascalled the principal part of the function. The above definition is ageneralization of this.'

The Mittag-Leffier theorem can be generalized to the case ofseveral variables as follows:

THEOREM 4.20. Let .R be a pseudoconvex open set, U an opensubset of .R, and g E M(U). If go. is a closed subset of 12, then thereexists an element f in M(.R) such that g E P(f, U).

PROOF. From the condition, there is a C°° function p on .R whosevalue is 1 on some neighborhood of go. and such that supp p C U.Set

Igap on U,(4.66) V:=

0 onQ\U.

Then v E Ker 8 f1 (.R). From the pseudoconvexity of .R, there isa solution u E C°° (.R) of the equation au = v.

Hence, it suffices to define f := pg - u, where pg I .R \ U = 0. O

The pole of a meromorphic function is an analytic subset, al-though we do not prove this fact in the present book.

DEFINITION 4.21. For a holomorphic function f on .R and anopen set U C .0, f A(U) is called the divisor class of f on U and isdenoted by D(f,U).

A generalization of the Weierstrass product theorem to severalvariables is made possible on a pseudoconvex open set whose secondBetti number is 0.

THEOREM 4.22. Let .R be a pseudoconvex open set, and H2(12, Z)_ {0}. If V(g) is a closed set in .R for an open set U in .R and g EA(U), then there exists an element f in A(.R) such that g E D(f, U).

51n the above situation, unlike the case of one variable, there does not existanything that corresponds to the Laurent series. Hence, we are obliged to definethis concept as an equivalence class, as in Definition 4.19.


PROOF. Take a locally finite family {B} of open balls BjB(pj, Rj) in fl such that

(4.67)00

fl= UBjj=1

and

(4.68) Bj n V (g) # 0 implies Bj C U.

Define gj E A(Bj) by

0 ,(4.69) gj g I Bj if Bj n V(9) 341 ifBjnV(g)=0,

and define gjk E A(Bj n Bk) by 9jk := 9jlgk, where Bj n Bk # 0.Note that Bj nBk is simply connected since it is convex, and that

from the definition, gjk does not have any zero point. Hence, we canhave some branch ujk of log gjk be in one-to-one correspondence to(j, k). Then uijk := uij + ujk + uki E on Bi n Bj n Bk.Adjusting the Ujk'S in advance so that ujk = -ukj, we can assumeUijk + 'ujk, + Ukli + ulij = 0. Therefore, from the assumption on fl,there is a set { m consisting of elements of Z such that

(4.70) -1(mij + mjk + mki) = uij + ujk + Uki

on each Bi n Bj n Bk. If we set uij :=uij - 21rmij, then

(4.71) uij+ujk+uki=0.Let {pj} be a partition of unity subordinate to the open covering{Bj }, and define

(4.72) ui pjuij , where pjuij I Bi \ Bj 0.

Then it follows that

(4.73) ui - uj = E Pkuik - F, Pkujkk k

_ >Pk(uik - ujk) _ Pkii:jk k

Hence, since uij E A(Bi n Bj), we have c3ui = 0u, on Bi n Bj, andso this determines an element in Ker c? n C°-'(fl). Therefore, fromTheorem 4.14, there is an element u in C°°(fl) such that &u = aui.


If we set h= := e"--", then h; has no zero point and h= E A(131), whilefrom

(4.74) (u; - u) - (uj - u)

log(gi /gj) mod 27r Z

it follows that he/h, = g;/gj.Consequently, if we define f := g;/h=, then f E A(R) and g E

f A(U), which completes the proof. O

4.2.2. Approximation Theorem. According to the Runge ap-proximation theorem in the theory of functions of one variable, anecessary and sufficient condition for the polynomial ring C[z] to bedense in A(Q) for a given open set .fl C C is that C \ .f1 be connected.This topological condition is related to the theory of functions by thefollowing proposition:

LEMMA 4.23. A necessary and sufficient condition for C \ .R tobe connected is that for every compact set K of .R. there exists acontinuous subharmonic exhaustion function cp : C -+ R such that

KC{zE.Rlcp(z)<0}CR.PROOF. Necessity: By the connectedness of C \ .fl, there is an

open set .fl' with a C°° boundary such that K C .(2' C= f2 and C \ f2'is connected. For this !Y, fix a homeomorphism 4? : C C of classC°° such that

k4)(ffl')= {zEC I r/'(z):= L logIz - vj < -R(k)I

ll v=Ifor some R(k) >> 1, where k denotes the number of connected com-ponents of !Y. Since w is subharmonic on C, it does not have anymaximal value, and this property is transmitted to 0 o 4b. Hence, wecan choose an increasing convex function A : R R (in the broadsense) of class CO° that grows so rapidly that A(i' o (D) satisfies all therequirements.

Sufficiency: If C \ f2 were not connected, let E be one of itsbounded components. Take a sufficiently large compact set K of .flsuch that some bounded component of C \ K contains E, and let besuch a function as stated in the proposition. Then the set {z I 'p(z) >-0} would have a bounded component k (D E). However, from thisit follows that fp is subharmonic on the open set {z I cp(z) < 0} U Ewhile o has its maximum (greater than or equal to 0) at an interiorpoint. Hence, a contradiction. 0


The above interpretation of connectedness is essential to a gen-eralization of the Runge theorem to several variables:

THEOREM 4.24. For a pseudoconvex open subset fl of C. a nec-essary and sufficient condition for C[z] to be dense in A(fl) is that forevery compact set K C !l, there exists a continuous plurisubharmonicexhaustion function 4; defined on C" such that K C {z I cp(z) < 0} CQ.

The following generalization of this theorem makes the proofclean-cut.

THEOREM 4.25. For pseudoconvex open subsets fll and fl2 of C"with fll C f12, a necessary and sufficient condition for A(f22) to bedense in A(f11) is that for every compact set K of Q. there existsa continuous plurisubharmonic exhaustion function p defined on f22such that KC{zI ip(z)<0}Cf11.

PROOF OF SUFFICIENCY. Richberg's theorem enables us to as-sume that % is of class C. Let f EA(Q1). Take a C" function

R ---# R such that x I I -x, 1 sup 0 1 and Y I (0, oc) = 0, then2 K

consider a solution u E C'(Q2) of the C equation du = O(x(r) f) suchthat(4.75)

122e-iIul2 dV < f e-IZ12-a( (Z)) 12 dV ,

2where A is an increasing convex function (in the broad sense) of class

COC. Since 4% > 1 sup on supp d(x((i;) f ), for a given 6 > 0 we can2 K

take A, with the condition that A I -0C, 1 sup = 0. such that.2 K

(4.76)1Q2

<e.

Note that u is ltolomorpllic on { z I cp(z) < 2 sup ca so, froml K JJJ

Cauchy's estimate, there is a constant C that depends only on K andsuch that

r(4.77) sup Jul < C J

e-1--12 1U12 dV.K in,


Therefore, defining f := X(cp) f - u, we see that f E A (f22) and

(4.78) sup If- f I = sup Iul < CE.K K

0

Next, we prepare a lemma to show the necessity.

LEMMA 4.26. If fl is a strongly pseudoconvex open set, then forany boundary point x0 of !l, there exists an element f in A(fl) suchthat

(4.79) lim If (z)I = oc .Z-XO

PROOF. From the strong pseudoconvexity, Proposition 3.26 im-plies that there is a neighborhood U 3 xo and g E A(U) such thatV(g) fl fl = {xo}. If we choose a C°° function p : Cn -' [0, 11 suchthat supp p C U and p = 1 on a neighborhood of xo, then since 8(p/g)is of class C°° on some strongly pseudoconvex neighborhood (2' of Si,form Theorem 4.14 there is it E C°°(Q') such that 8u = C3(p/g) onSi'. In this case, f := p/g - it is a holomorphic function on Si andsatisfies (4.79).

PROOF OF NECESSITY. Let 1' be a strictly plurisubharmonic ex-haustion function of class C°° on .R1, and for a given compact set Kin fll, take a real number c such that K C f11,c := {z I &(z) < c} andf11,° is strongly pseudoconvex. Then from Lemma 4.26, for each pointx0 of 8f11,c there is an element f in A(i71,c) such that lim If (z) I =

Z 4X0oo. Hence, the density of A(122) in A(fl1), that of A(fl1) in A(f11,°),and the compactness of 9171,E all together enable us to choose ele-

mments fI, , f,, in A(Q2) and construct ;(z) E I fk(z)I2 - 1 so

k=1that some connected component W of f22,o {z I ;5(z) < 0} satisfiesKCWCfl1.

Therefore, the desired function cp is obtained by setting

y0-2sujpcp on W,'P ( 1 1

max t - - sup c', cp` - - sup cp on !l2 \ W.2 K 2 K0


REMARK. A pseudoconvex domain .R such that C[z] is dense inA(f2) is said to be polynomially convex. As the condition for polyno-mial convexity in one variable was topological, this became a problemin several variables as well, but K. Oka, J. Wermer, and others foundcounterexamples.

Wermer's counterexample:1. K := {(z,w) E C2 I w = IRe zI <_ 1, IImzI <- 1} K has

a fundamental system of neighborhoods consisting of domainsUJ that are biholomorphically equivalent to double discs.

2. c(z, w) := (z, (1+Vl--l)w-Vl--l-zw2-z2w3) the Jacobianof 4(-1+x)34 0.

3. 1 is one-to-one on a neighborhood of K.4. j >> 1 (D is biholomorphic on UJ.5. KJy:={(e'B,e-ie)EC2I0<0 27r}

d(y) ={(e:e,0) I0<_0<-27r}

{(z,0) E C2 I IzI 1}

C {(zw) If(z, w)I < supf1, df E C[zw] } .

6. z. (z,'z) = (z2, IzI2{(1-IzI4)+VI-j-(1-IzI2)}) the secondcomponent of z) is not equal to 0 in the range of values0<IzI<1, IzI>1.

7. From (6), in particular, (o) (K) (o)(j>> 1).

8. From (5) and (7), ,t(UJ) is not polynomially convex.However, the following question remains unsettled.Bremermann's Problem. Is 11 polynomially convex, provided that

for any complex line 1 C C'°, l \ .fl is connected?The problems of generalizing the Mittag-Leffler theorem and the

Weierstrass product theorem to several variables were formulated byCousin in a more abstract form, and thus they are sometimes calledthe first and second Cousin problems, respectively. Oka solved theseproblems first on domains of holomorphy, and eventually establishedthem as the theory on pseudoconvex open sets by settling the Leviproblem.

CHAPTER 5

Solutions of the Extension andDivision Problems

In the present chapter, the extension and division problems are solvedon pseudoconvex open sets; namely, the 8 equation is solved undercertain constraints. In § 5. 1. we show how to omit the differentiabilityfrom the conditions imposed on weight functions in Theorem 4.11. Bythis omission, the constraints stemming from the extension problemscan be replaced by some conditions on the integrability of relevantweight functions. which produces a general extension theorem. In§ 5. 2, the division problem is solved by restricting the domain of the8 operator to an appropriate subspace and inducing an L2 estimate onthis new operator. This approach is due to H. Skoda. § 5.3 presentsan extension theorem which is equipped with a growth rate condition.The content of this theorem asserts that L2 holomorphic functions de-fined on the intersection of a hyperplane and f1 extend to f1 under anestimate on the norm which is allowed to involve a plurisubharmonicweight function. § 5.4 introduces two applications of this L2 exten-sion theorem; one is a characterization of the removable singularitiesof L2 holomorphic functions (Theorem 5.18), and the other the proofof Demailly's approximation theorem (Theorem 3.16).

5.1. Solutions of the Extension Problems

For a general plurisubharmonic function V on f1, the Hilbert spacesL,2p(fl) and LP-9(fl) can be defined as before, because e-`°dV is a(Lebesgue) measure on f1. Noting that L2 (f1) C L2 c(f1), if the8 equation on such a general L49(fl) is solvable with the norm esti-mate, then we expect to approach even a deeper structure of A(f2)by applying the estimated solutions.

Let us generalize the first half of Theorem 4.11 to this form.

83

84 5. EXTENSION AND DIVISION PROBLEMS

THEOREM 5.1. Let 11 be a pseudoconvex open set, and W : 1.1 -'I-oe, oo) a plurisubharmonic function with L[cp] >_ 1. Then for anyv E KerB fl L°p9(Q) (q > 0), there exists an element u in L°pq-'(f1)such that 0u = v and IIuII,, < IIvII,,

PROOF. Let cp£ be the e-regularization of cp, and .flf£1 a pseudo-convex open set with .R(,) C Q. Since 00WE 00(IzI2)£ = 001x12,Theorem 4.11 is applicable and implies that there is an element of inL°,q-1(.f1(,)) such that 0u, = v and 11u,11,,, < IIvIIp,, where the normis regarded on 11(e).

Since o£ >- cp, IIvI1,,, -< 11vII,,. Therefore, from the above estimatefor u, there is a subfamily of uE that is weakly convergent on anycompact set in 0. If we denote the limit of this subfamily by u, thenDu = v and IIuII, < IIvll,. 0

Theorem 5.1 is due to Hormander, but the literature often quotesit in the following form:

COROLLARY 5.2. Let .f1 be a bounded pseudoconvex domain inC ", and cp a plurisubharmonic function on .fl. Then for any v EKerB fl L0,9 (0) (q > 0), there exists an element u in L°p4-1(0) suchthat 0u = v and IIuII,, 5 CIIvll,,, where C is a constant that dependsonly on the diameter of .fl (:= sup Iz - z'I).

z.z'E1

The interpolation problem raised in Chapter 2 can be solved per-fectly as an application of Theorem 5.1.

THEOREM 5.3.1 The following are equivalent:

1. !1 is pseudoconvex.2. For any discrete set t c (1, the restriction mapping A(Q) -

Cr is a surjection.

PROOF. (1) (2): Take h E C°°(!1) such that ah = 0 onsome neighborhood of F. Then it suffices to show that there is anelement g in L C(Q) such that 0g = 0h and g I I' = 0.

Let p and UU (l; E r) be as defined in § 2. 1, and decompose p as

(5.1) p = E pt with supp p C UU.FEr

1 Onecan see also from this theorem that pseudoconvex open sets are domainsof holomorphy.

5.1. SOLUTIONS OF THE EXTENSION PROBLEMS 85

Then for a function c defined by

(5.2) 4 (z) := 2n> Pa(z) log Iz - I,CEr

there is a continuous function -r : 11- R such that

(5.3) O (z) > T(z)aa1z12 for z E .R\r.

Hence, if we choose an appropriate exhaustion function on . ? with'4' E C°°(Q) fl PSH`, then

(5.4) L[(D + ] ? 1

and Ilahll,+ ,< 1. _ _

From Theorem 5.1, there is g E Lb+,y(fl) such that ag = 8hand II9Il4,+' < II Also, from Theorem 2.7, g is of class C°°.Noting that a-'-'0 is not integrable around r, it follows that g I r =0, which is what we wished to show.

(2) (1): This follows from Theorems 3.4 and 3.12.

Next, coming to the extension of holomorphic functions on ananalytic subset X of dl, two new problems arise if we apply the sameargument as in the case of discrete sets:

1. Can holomorphic functions on X be extended to holomorphicfunctions on some neighborhood of X?

2. Does there exist a function that corresponds to (D + 0 in theproof of Theorem 5.3?

As to (1), in general there does not exist any holomorphic map-ping from a neighborhood of X to X that coincides with the identitymapping when restricted to X. This problem has already been as dif-ficult as the extension of functions to the whole .fl. Now let us thinkin a more adaptable way: Given a holomorphic function f on X,construct an extension f of class C°° by patching local holomorphicextensions of f in terms of the partition of unity, and apply Theo-rem 5.1 to 5f. Then in order to ensure the finiteness of the norm ofa f by adjusting 0, a-1 15f I2 must be locally integrable in the firstplace. Since a-D is also required not to be locally integrable along X,it turns out, in turn, that (2) is quite a subtle problem. Precise ar-gument for this point calls for two fundamental theorems on analyticsubsets.

THEOREM 5.4 (for the proof, see [34] or [25]). Let .fl be an openset in C", and X an analytic subset of (l. Then there exists a family


{XQ}QEi of analytic subsets of .R that satisfies the following condi-tions:(5.5) X = UXQ, each XQ is non-empty, and

Q

#{XQIXQnK#0}<xfor any compact set K of Q.

(5.6) Every XQ contains a connected differentiable manifold Xn as adense open set in it, and XQ fl X3 = 0 for a : 3.

An Xa that appears in Theorem 5.4 is called an irreducible com-ponent of X. Also, the maximum open differentiable manifold con-tained in XQ is called the regular part of XQ and is denoted by Reg Xa.Reg XQ is a locally closed complex submanifold of Q. Namely, for anypoint x E Reg X. there are a neighborhood U of x in I? and a bi-holomorphic mapping F to On such that

F(U fl RegXa) = {z E An I Z'.+1 = = Zn = 01,

where rn.Q is an integer that is independent of the choice of x andis called the dimension of X. The dimension of XQ is denoted bydim Xa. In addition, U Reg XQ is called the regular part of X and

denoted by Reg X.

THEOREM 5.5 (for the proof, see [34]). Let Si and X be as de-fined above. Then for any x E X. there exist a neighborhood U xin Si and a system {wQ}a=1 (1 # oc) of local defining functions of Xon U that possess the following property:(5.7) For any y E U fl X and any system {h,3}- 1 (1 m <_ oo) of

local defining functions of X around y, there exist a neighbor-hood V of y in U and a system {gai3}a"'1 p=1 of holomorphicfunctions on V such that, for any 0,

I

h,3 = E waga,3a=1

on V.

From now on, such a system { wQ } is called a reduced system oflocal defining functions of X.

THEOREM 5.6. Holomorphic functions defined on an analyticsubset X of a pseudoconvex open set Si are the restrictions of holo-morphic functions on Si.

5.2. SOLUTIONS OF DIVISION PROBLEMS 87

OUTLINE OF THE PROOF. For the reason described above, it suf-fices to show the existence of a function 4) : P -i [-oo, oo) thatsatisfies the following conditions:

(5.8) There is some continuous function T : f2 -p IR such that

aa4) >_ Taa1z12 .

(5.9) Given a reduced system {wa} of local defining functions of X.the function e-4' >2 Jwa12 is locally square integrable on the do-main of wa.

(5.10) a-"' is not integrable around any point of Reg X.

In fact, given a holomorphic function f : X -i C, construct fby patching local extensions of f. If a 4) that satisfies (5.8)-(5.10)is obtained, then there are elements zi in PSH' n C°` (!2) and u inL+v(f1) such that au = af. Since u I X = 0 from (5.10) andX = Reg X, we see that f - u is the desired extension of f .

Construction of 4). As it is sufficient to construct 4) for eachirreducible component Xa of X, let us assume from the beginningthat X is irreducible. Take a locally finite open cover {Uj } of fl sothat there is a reduced system {wQ} of local defining functions of Xon each Uj, and define

(5.11) 4):=(n-m)log >IpjwQ12 (in:=dimX)j,a

by means of a partition of unity { pj } associated with {U}. Then(5.9) and (5.10) clearly hold. We leave it to the reader to verify (5.8).(Recall the Gauss-Codazzi formula.)

5.2. Solutions of Division ProblemsGiven a vector f = (f1, , f,,,) E A(f1)3)"' of functions that haveno common zero point on .fl, as described in § 2.1, a necessary andsufficient condition for there to exist holomophic functions gj thatsatisfy the equation

m(5.12) >2 fjgj = 1

j=1

is that the vector-valued a equation

(5.13) au = u :_\a CIf121 1

a 0-0)

88 S. EXTENSION AND DIVISION PROBLEMS

has a solution u E L'10C(Q)e- with fjuj = 0.j=1

Let us proceed with the calculation under the assumption that cpis a C°° function with 8(f j/I f I2) E L°W,1(Q). Define

(5.14) S,°'q := a E L°q(Q)®"` I fjaj = 0j=1

Then, by the holomorphy of Ii, the D operator a '- (Da I, , Da"' )becomes a closed operator from S,°p,q to SO q+', which is denoted by

DS for distinction.In order to solve the division problem, it is enough to choose an

appropriate w so that the ratio of I (w, v),, I to 1IDs w1j,' is bounded onKer Ds (l Dom Ds.

We will try to express Ds in terms of '"D * (which operates com-ponentwise) and f. If the problem is solvable, this calculation shouldnaturally produce an L2 estimate.

Let us first introduce the following notation: For elements w1 andw2 in Lwq(f2)®m, set

(5.15)

(5.16)

(5.17)

m(WI, W2) E (w1j,w2j),

j=1

(WI, w2),p Jn e-4'(wl, w2) dV,

IIwl IILP ( wl )lp

The orthogonal complement of the set SO,'0 in LO,°(f2)®m is denotedby (4'0)1. Set

(5.18) (S°p'0)0 -L := {w I w = (cf1, cfm) for c E Co (.R)}.

Then (S °)o is dense in (S°p'0)1. In fact, since f1, , fm do not haveany common zero point, any element in L°Y,O(Q)®m that is orthogonalto S0.'0 must be a function-multiple of (f 1, , f1).

Go back to the definition of adjoint operator; then for an elementw in Dom Ds and an element h in Dom Ds,

(5.19) (as w, h),p = (w, ah),P.


From this, we see that as satisfies Dom'B * fl SO.,I C Dom Bs, andthat

(5.20) BSw=P`°Xw

by using the orthogonal projection P : Loo(n)®m -. So'°This formula can be expressed as

(5.21) BSw=vB*w--(`°Xw,ej),pejj=I

in terms of an orthonormal basis {ej} I of (S°'0)1. We may as-sume ej E in advance. Using the equation ej) , _(w, Bej )r the self-duality of Hilbert space allows us to identify Bejwith an element in (LO,I(.f1)Om)`, which results in the expression

(5.22) BSw =' ''B - f ej ®BejI w.j=1

Since ej = (c171/If, ... , cjfm/Ifl), cj E Co (.fl), and IIcjII'v =1, it follows that

(5.23)

L(-)cj l cj l

D ej = (f15Ifl

, Ifl+ (I,...

,Ifl afm)

If W E Dom'B * fl S-1, the inner product of w with the first term ofthe right hand side of the above formula is equal to 0. Hence,

(5.24) BS w 0a w- [.. L. (wk'I f I a f k ej

j=1k=1_ oo m Bfk

j=1k If I /Jwk,Cjl,p

ej=1 (

lafk-Jwk f f .'°BOw-(k'=

Set for simplicity

_ f maf (w) if 12157k -J Wk-

k=

Then from the above formula, for p E C°°(Q) and X E Co (.R),

IIpa;(XW)II ,

= IIp`,a*(Xw)II , - 2R.e (pS°a (Xw), pf3i(Xw)),, + IIPQf(Xw)II , .


Therefore, for any positive number r,

IlPas(xw)II ? (1 -

r := 11

E(E > 0), we get

IIPas(Xw)III >= -EIIPaf(xw)II,

Combine this with the fundamental inequality (4.27); then, whenp > 0, we obtain

IIPas(Xw)II; + IIPa(X011,

i + E IIP °a*(xw)II , + IIPa(xw)II2 - EIIPQf(Xw)III

6 2

E 1 + C Ilp ((P2 L, - LP2) XW, XW ),P C I I aP j X W 112> 1 +

-6IIPQf(Xw)II(C > 0).Hence, from this point on, as in deriving Theorem 4.11, we obtain

the following existence theorem by running p and X by means of theauxiliary weight function cpa.

THEOREM 5.7. Let fl be a pseudoconvex open set. Assume thatelements fl, , f,,, in A(fl) (m < oo) have no common zero pointand that a Cx plurisubharmonic function o on .f1 satisfies both

Flfne-°Ie( f12)I'dV <oo

andL[p] >(l+E){Ifl_2>k=1 JJJ

18fk12+1}forsomeE>0.

Then the re exist elements gk in A(fl) (k = 1, , m) such thatm

fkgk = 1, andk=1

fe_lg2dV (I+ftE) mEl

la (di;)

2

dV.

COROLLARY 5.8. If .(1 is pseudoconvex, then for any system{ fk}"0

1of holomorphic functions on (1 that has no common zero

point, there exists a system {gk}k 1 of holomorphic functions on f1

such that E fkgk = 1. In particular, Spec,,, A(f1) = Q.ock=1


ao

PROOF. We may assume that 0 < E IM2 < oo. Then fromk=1a

Cauchy's estimate it follows that E Iefki2 E C°(f1). Hence, therek=1

is a plurisubharmonic exhaustion function cp of class CO° on fl suchthat

"0 r \fI2)l dV<oo,L[yo] 2

(If-2 E Iefkl2 + 1) .

k=1

For any c < sup gyp, we can take a sufficiently large integer m = meso that fl, , f, have no common zero point in f1,,c. From Theo-rem 5.7, there are holomorphic functions 9c,k on f1,,, (k = 1, , m)such that

mEfk9c,k=1,k=1

feEIgc,k2dV < Al.1 k=1

where Al is a constant that does not depend on c or nl.Therefore, from Cauchy's estimate and Montel's theorem, there

is a sequence of numbers c. with cµ / sup cp such that the sequence[9,,_k}0'1 of functions is uniformly convergent on compact subsetsof f1 for each k. Define

9k := lira

then from Weierstrass' double series theorem it follows that 9k EA(fl) and

CIO

E fk9k = 1.k=1

0If Spec,,, A(fl) = fl, then in particular, since for any point a of

811 there are elements gj in A(ft) (j = 1, , n,) such thatn

E (zj - aj)gj(z) = 1,j=1

it follows that fl is a domain of holomorphy, and thus pseudoconvexdue to Oka's theorem. Hence, the combination of this with Corol-lary 5.8 shows the converse of Proposition 2.1.


The condition of Theorem 5.7 involves the derivatives of fk. Theabove argument does not especially have anything artificial, and thisis good enough. However, the result will be neatly stated if we findthe calculation described below.

Let v be as defined in (5.13). For w E SS'1 fl Co'1(Q)em, it isdesirable to evaluate I(w, v),PI by a constant multiple of II wll,p +II c7w I I, from above; but since

(w, v)" = (w, c'3u),P

(''O *w, u),y

= (as w, u)', + (of (w), u)'a,

it suffices to evaluate Ilpf(w)IIw. What is readily seen from the formof Of (w) is that if we take cp+p log If I2 as a weight function instead ofcp, then 11Qf(w)II,+P loglfl2 is absorbed in (Ly,+P loglfl2W,W)w+P logJfJ2when p >> 1. Afterwards, accurate evaluation of the quadratic formimplies the following theorem:

THEOREM (Skoda's Theorem [45]). Let Q be a pseudoconvexopen set, and cp a plurisubharmonic function on Q. Suppose that weare given p holomorphic functions gl, , gP (or a sequence {gj }?_-1of holomorphic functions) on (1. Let a > 1, and q := inf{n, p - 1}(or q := n). If a holomorphic function f on !l satisfies

fn If121gl-2«q-2e-wdV < 00,

then there exist p holomorphic functions hj (or there exists a sequence{hj}?_1 of holomorphic functions) on 11 such that

P

00f = > gjhj or f = > gjhj andj=1 j=1

Ih121gl-gage-1,dV <

aJn

a f I fI2Ig'I-2aq-2e-`°dV

,-1 n

00respectively, where gjhj is the sum in the sense of the uniformj=1

convergence on compact subsets in Q.

5.3. EXTENSION THEOREM WITH GROWTH RATE CONDITION 93

5.3. Extension Theorem with Growth Rate Condition

5.3.1. L2 Extension Theorem. In what follows fl is assumedto be a pseudoconvex open set.

As in § 4. 1, for a general plurisubharmonic function cp on .fl, con-sider a Hilbert space L2 (fl), and set A,2o(f1) A(f1)nL ,(.fl). A,2o(fl)is a closed subspace of L ,(f1) due to Cauchy's estimate.

Put H {z E C' I z,, = 0} and f1' := fl n H. Then consider

A12P(f2') _ {f E A(fY) I f fI2dV < oo I

as a subspace of A(fl'). From Theorem 5.5 (or Theorem 2.5 + The-orem 4.14), there is a mapping

I : A,2D (f') ---+ A(fl )

such that I(f) I fl' = f for every f E A2 (fl').The problem arising here is about the existence of such an I

that is also a bounded linear mapping from A ,(fl') to some subspaceA2 (fl) of A(fl). When this condition is satisfied, I is said to be an in-terpolation operator from A , (fl') to A2y (fl). Of course, it depends onthe relation between V and 0 whether or not there is an interpolationoperator. _

Before interpreting this into the problem of 8 equation, let usreduce the situation to a specific case.

Take an increasing sequence { flk } k I of strongly pseudoconvexopen sets of f1 such that flk C= flk+1 and Uk I f1k = fl. Then setf4k := flk n H. Also, given two plurisubharmonic functions VI and02 on Q, let Vi,, be the e-regularization of Vi.

The following will be self-evident:

PROPOSITION 5.9. Let {flk }k I be as above. If for some sequenceof positive numbers Ek that converge to 0 there exist interpolationoperators

2 2Ik : A,s.ck (11k+I) AV,." (Qk)

whose norms form a bounded sequence on k, then there exists aninterpolation operator I : 4 (f1') - A,2, (R) whose norm does notexceed lim, IIIkII, where IIIkII denotes the norm of Ik.

Therefore, in order to finish this argument, it suffices to constructa linear mapping I : A,2p2 (f22) -' A ,1(f11) that satisfies 1(f) I fY =


f I fl' with a certain estimate on the norm for two given stronglypseudoconvex domains f 2i Q2 and cpl, W2 E PSH n c, v22).

Let Qi and gyp, (i = 1, 2) be as above, and take f E A W2(02). Thea equations that are necessary in the procedure for construction of Iare obtained as follows:

By means of the projection

p: Cn -- Hw wz ,-, z' := (zI,... ,zn-I),

we extend f = f (z') to a function p* f (z) := f (p(z)) on p-I (Il2).Choose a positive number 6 such that

p-I(Il2) n {z I Iznl < 6} D fll,b := Ill n {z I Iznl < 6}.

Take a C°° function x : IR - [0, 1] that satisfies1

(5.25) x(t) = f1 fort < 2 ,

0 fort > 1,

d f I nl torX6(z):=X ( ),sean

V6forzElll,5,

1 0 forzEIll\J7,6.

Then it follows that v6 E Ker a n C"(121) and

suppv6C{z2<Iznl <61-

At this point, if there is a u6 E LW1(c11) such that

aub = v6,

(5.26)IIu611.. < CIIv61I,2for some constant C independent of f and 6, andU6

E I'2loc(2I

n

and if the correspondence f i - u6 can be made linear, then for asufficiently small 6, the linear mapping

16 : A2 (372) A (Ill)w w

f '-' p*f-X6-u6


will obviously satisfy

IIo(f)If2i=fIQ.I f12dt;,IIIo(f)IIY, (C + 1) f

z

This argument has clarified what kind of 8 equation should betreated, but here we will solve the equation only for the case 4%t = y;2for simplicity, and will refer the reader to the literature for the generalcase.

The next theorem is contained in the author's joint paper [37]with K. Takegoshi. The proof in the original article was written in theframework of differential geometry as a development of the Kodaira-Nakano vanishing theorem, but as described below, the proof canbe done without displaying the concepts of metric and curvature forsome sequence of positive numbers Ek that converge to 0. (A similarapproach can be seen in [3] and [44].)

THEOREM 5.10 (L2 extension theorem). Given a plurisubhar-monic function p on a bounded pseudoconvex open set f1, there existsan interpolation operator from A (Q') to A' (Q) whose norm doesnot exceed a constant that depends only on the diameter of 5l.

PROOF. Recall the result of Proposition 4.6, that for any elementu in Con" (f2)1

+ 4Re (5p _j u. p 'I'd u).:,(5.27) Ilpaull2 +11P411)

u11= IIPd1111241

+ (p2L ,u. u)4 - (Lpsu,

where p and 4P are arbitrary real-valued functions of class C2 on I?.We assume p > 0 below. _

As the èrror term' Ilap _j u112 in the fundamental inequality isnot easy to evaluate in this case. we will use, instead, the followinginequality:

(5.28) IIP P3+l11Ilull41+llpaull41> ((P2L,,p - Lp2)u.u)4, -411P-28P ull241

=: Q,,,,P(u) ,

which is obtained by modifying (5.27) in terms of

11Re (api u.pfidu)i,1 IIP3 dull, + 411P-2ap_ t1Ij241,

It is obvious that. (5.28) can apply to an element in C°''(.fl).


Let us consider the 8 equations (5.26) under the assumption thatf2lc=f22c12andcp1=V2EPSHnCOO(f22).

Since p is contained as a factor in both norms on the left hand sideof (5.28), as in deriving Theorem 4.11, the approximation principlebased on Theorem 4.2 implies the following: If there exists a constantCn such that for any w E Co'1(.f21) we have

(5.29)K

2

va,wi I <CCQP',(w) f e-`°'I.fl2dVzn q, s?z

then there exists a unique element u'6 E L ,(f21) that satisfies

(5.30) a(P p3+1u'a) = zn,

(5.31) 112 <Cn e-°' Ifl2dvn-1,f

nZ

(5.32) u'61Ker(8op p3+1).

If the diameter of f2 is denoted by dn, then

i(z) := -logIzn12+21ogdn > 0 (z E .fl).

Hence, putting 71.. (z) := - log(Izn12 + e2) + 2logdn + 3, if thepositive number a is sufficiently small, then il£ > 2 on R.

In order for (5.29) to hold, we first take 8 so that > 2 inadvance, and then set

p:= 076 +109,76,

In this case, since

(5.33) Qp,4.(w) (aa(log(Izn12 + 52) - log 77) V w, w)",1

- 411ii la(ia + log 776) i wllv1

> (881og(Izn12 + 52) V w, w)w,

b2> e-w' (IznI2 + 52)2IwnI2 dV

J{ <Iz.,l<I}nnl

nwhere w wjdzj

j=1


the Cauchy-Schwarz inequality implies

(5.34)L6-, Wzn

'P1

2e-v, ( I -+- 52)2 I

dV52 IznI2

e-`°'62

Iwn12 dV(Izn12+62)2I

25< f e-v dV QP.a,(w) -4n,

Therefore, if we take C$7 _ sup then (5.29) holds when 5 is

sufficiently small.In this case, for u' that satisfies (5.30)-(5.32), if we set

ub := znub ,

then it follows that Dub = vb, zn Nub E L10 2 C(01), and

(5.35) Ilubli,,, = Ilzn P P3 + 1 ub j ,

Sup IznIP P3+lIIi4IL:1-En1

Vt-(-log e+log (-log s)+1)512IIu<sup 6

O<t<dn

C'nllu'alj .1

where C" is a constant that depends only on dn.Combination of (5.31) with (5.35) shows that ub is determined in

the way of being linearly dependent on f and satisfies (5.26) forC :_CICn . from which Proposition 5.9 yields the desired conclusion.

0

5.3.2. Generalizations of the L2 Extension Theorem. InTheorem 5.10, it is clear that the boundedness of 1? can be replacedby sup I zn I < oc. When P is a general unbounded pseudoconvex

nopen set, there may or may not exist an interpolation operator de-pending on the condition about the weight function p. We will give aclass of weight functions that guarantee the existence of interpolationoperators.


First we consider

go := {G : fl - [-oo, 0) IC is continuous, and

G - log E C2(.R)}

as an auxiliary family of functions, and

11,;(,R) :_{yo E PSH(17) I ' + G, p + nG E PSH(Q) for some G E gal

as a practical class of weight functions. Set lG (G - log Izn12) 112,for an element G of !go, and

IK := sup {mf lG (So + G, p + nG E PSH(fl) }

for an element p of HK(fl).

LEMMA 5.11. Assume that V E IIK0(fl) and lK 34 -oc for someno > 1. Then for arbitrary strongly pseudoconvex open set !2' C f2and e > 0, there exists a positive number 60 such that for 5 with0 < 6 < 60 and an element v of L,+,,,,2(fl') that satisfy

(5.36).36)ot,-0 (9'=1 ,n-1)l8-2j

there exists an element u of (fl ') such that

8u=vdz(5.37) IIu II,o+El:l2 < Cl IwIIs+e z 2 ,

ZnE LIOC(n') ,

where Cl is a constant that depends only on no and h1*.

PROOF. Let V + G, s + KoG E PSH(fl), and G E go. V may betaken to be locally integrable. If we define µa by putting n = 1 andc = 5 in (1.9), then since

8a-(log IznI2)a = 2/25(zn)dzn A din,

from the assumption, for any e > 0 we can take a sufficiently small 5such that for any n, with 1 < n < no,


(5.38) aa( s + hG6 + EIzl2)

> 1p5(z,,)dzn (z E f2').

Set T1A :_ -G6+A (A > 0), and define a quadratic form Qp,.D(w)to be the one for

p : = 77A + log r7A .

P := tpr + GT + E1z12 (T < 5).

For such p and 4b, we have

(5.39) p2ad' - aap2 - 4p -4,9P A dp

A(aa r+aaGr+EIzI2)

(77A + log ??A)-3a(7]A + log 7)A) A a((7IA + log r)A)

= A(aa,jor + aaGr + EI ZI2) + C1 + 1 I aaG67JA/

+ a77A A d77A - r1 + 1 ) 2 aTJA A a77A

r]A \. 17A J (rlA + 109 77A)3

On the one hand. from the condition +KG E PSH(Q) (1 < rc <- tco),it follows that 4; + K(G - log Iz,tI2) E PSH(fl). In fact, since

'p+r,(G-log Iz,, I E PSH(Q)

for any a > 0, it is enough to let a \ 0.Therefore, if 8 is sufficiently small, for any r with 0 < r <- 8,

'p + K(G - log IznI2)5 + CIZI2 E PSH(Q*).

Hence, if we take A so large that

(1 + A) < no.

then

A(da'r + a8Gr + EIzI2) + I 1 +17A

1)07G6

11p5(z,,)dz,, Ad-,1.

Also, if A > 4. it is obvious that the difference between the 3rdand 4th terms in the right hand side of (5.39) is >_ 0.


Therefore, for w E Co11(Q

(5.40) Qp"t(w)>

1 f e-'vt-Gt-c'.j'µd(zn)Iwnj2dV2 .

(w:=>widi)

From this, in order to assert the existence of u that satisfies (5.37)for v that satisfies (5.36), it is sufficient to solve the a equation inL 9+GT+£I=12 (f2 *) and let T --+ 0. (Note that C1 does depend on Ifas well, because of the appearance of Gr in the right hand side of(5.40).)

We use Lemma 5.11 and execute limit operations similar to thosein the proof of Theorem 5.10 in order to deduce the following exten-sion theorem:

THEOREM 5.12. For a pseudoconvex open set fl and cp E II,..(fl)(ho > 1), there exists an interpolation operator I.: A ,(f?) --+ A ,(fl)if lK # -oo. The norm of I, depends only on ho and lK .

If sup Iznl < oc, then PSH(fl) C II,,(fl) for any r, > 0, andn

pp ? -tc sup log lzn 12 . Therefore, Theorem 5.12 is a generalization ofn

Theorem 5.10.A similar method can prove the following extension theorem. (For

the proof, see [36].)

THEOREM 5.13. Let Q be a pseudoconvex open set, and let both cpand ip be plurisubharmonic functions on fl. If there exists an elementG of Cn such that +G is plurisubharmonic and bounded on fl, thenthere exists an interpolation operator from A2 (fl') to A2 (fl).+v v

This allows us to evaluate the Bergman kernel (Chapter 6) interms of a geometric invariance of 8f1.

5.4. Applications of the L2 Extension Theorem

5.4.1. Locally Pluripolar Sets. A set of boundary points of apseudoconvex open set may happen to be removable for functions withlow growth rate, such as L2 holomorphic functions. The problem ofcharacterizing such a function-theoretically small set has been deeplystudied in the case of one variable, and in particular, the character-ization of removable singularities of bounded holomorphic functions

5.4. APPLICATIONS OF THE L2 EXTENSION THEOREM 101

in terms of analytic capacity and that of L2 holomorphic functions interms of logarithmic capacity are well-known.

In what follows the latter will be generalized to several variables.

DEFINITION 5.14. We say that a subset E of the complex plane islocally polar, or the logarithmic capacity of E is 0, if for each point xof E, there exist a connected neighborhood U of x and a subharmonicfunction cp $ -oo on U such that E n U C {y E U I cp(y) = -oo}.

THEOREM 5.15. For an open set f2 in the complex plane and aclosed subset E of f1, we have A2(f2) = A2(f1\E) if and only if E islocally polar.

For the proof, see (7] and [42].The concept of local polarity naturally extends to several vari-

ables.

DEFINITION 5.16. A subset E of Cn is said to be locally pluripo-lar if each point x of E has a connected neighborhood U and 0 EPSH(U)\{-oo} such that E n U c {x E U 1,0(x) = -oo}.

An analytic subset X of a domain f2 is locally pluripolar. In fact,for a system {f} of local defining functions of X, it will do to set

:= log E If. I2 .a

The next statement can easily be proved by applying Theorem5.15 to a function with parameter (the details are omitted).

THEOREM 5.17 (J. Siciak). If a closed subset E of f2 is locallypluripolar, then A2(f2\E) = A2(fl).

As an application of this theorem, we show that a bijective holo-morphic mapping F : f21 - f12 between domains is biholomorphic.In fact, if f denotes the Jacobian of F, then F-1 is holomorphic oni72\F(V (f )). However, setting

floglf(F'-1(z))I for z ¢ F(V(f)).(z) 1-oo for z E F(V(f)),

since Sard's theorem implies -0 # -oo, F(V(f)) turns out to be locallypluripolar, and from Theorem 5.17, it follows that F- I is holomorphicon $22.

A generalization of Theorem 5.15 in a rigorous sense is as follows:

THEOREM 5.18. For two bounded pseudoconvex open sets f11 Df12 # 0, we have A2(f11) = A2(f12) if and only if for any point zo of


fl2 and any complex line l through zo, l n (nl\n2) is locally polar in1.

PROOF. Sufficiency is a direct consequence of Theorems 5.15 and5.10, and necessity is obvious from the same theorems with p = 0.

REMARK. Due to Josefson's theorem, given a pluripolar set E.there exists an element i of PSH(C)\{-oo} such that E C {z E Cn Iz':(z) = -oo}. From this, in particular. it follows that a countableunion of locally pluripolar sets is locally pluripolar.

5.4.2. Proof of Demailly's Theorem. Siu's theorem hasshown a similarity between locally polar sets and analytic subsets.Demailly's theorem, used for the proof of that theorem, is a beautifulapplication of the L2 extension theorem as stated below.

PROOF OF DEMAILLY'S THEOREM. Let the notation be that of§3.2. First, regarding the sum of the series E Ia,(z)12, since {al}

is the orthonormal basis, E Iai(z)12 coincides with the square of thenorm of the following linear function on A2mtP(fl):

am: A2W W

f f(z).From this by Cauchy's estimate it follows that E 1at12 converges uni-formly on compact sets in P and is of class C° on fl, and that thefollowing equation holdds:

(5.41) zitm(z) = sup j log If(z)I i f E A2mv(n), 1If II2mC' = 1 } .

For 0 < E < an(z) and f E A2.1,..,(17), since If 12 E PSH(fl),

nl If(()12 dl'.nE2n J It-=I<f

111 2 -2mI f I e dV.< nE2n exp 2m sup s'(() )19

IK-zI<E

Hence, by taking the supremum of the left hand side within the rangeIIf I12mt, = 1, we obtain

1 I

(5.42) '0m(z) <_ sup b(() + 2m log . e2nItzl<e

5.4. APPLICATIONS OF THE L2 EXTENSION THEOREM 103

Therefore, the second inequality of (a) is shown.Next, from n applications of Theorem 5.10 as the dimension in-

creases, it follows that for an arbitrary constant a E C, there existsan element f of A2mti, (.R) that satisfies

(5.43)17

If I2e-2m ' dV < CIaI2e-2mtP(=)

where C is a constant that depends only on n and the diameter of Q.If we choose a such that the right hand side of (5.43) = 1, then

since from (5.41) we must have

m(z) log Ial,

the first inequality of (a)

log C

is obtained. From this and the definition of the Lelong number, weget v(i m, z) S v(z/), z).

It still remains to prove the first inequality of (b). In order todeduce this, since from (5.42), letting C' be the appropriate constant,it follows, in particular, that

sup ?Gm (°) < sup v(() + log - ,

lx-zl<e !C-2j<2E

mif we divide both sides of this inequality by loge and let Er - 0, thenthe definition of the Lelong number at x implies

nv(om, X) v(v, x) -

m.

0

CHAPTER 6

Bergman Kernels

We have already stated several fundamental propositions about thefunction space A2 (Q). In this chapter, we will explain the Bergmankernel, which is the reproducing kernel of the space A2(0). First, wegive the definitions and basic facts, and then we prove the boundaryholomorphy theorem on biholomorphic mappings between stronglypseudoconvex domains with boundaries of class CO°. This theoremwas obtained by Fefferman, but the proof introduced here is the onedue to Bell and Ligocka's idea [2), a skillful use of the transformationlaw of Bergman kernels. Next, a few results on the boundary behaviorof Bergman kernels are explained. This is the central problem in thetheory of reproducing kernels, but many results on this are beyondthe scope of the present book. Hence, we can say that most of theresults included here are restricted to elementary cases, and, even so,parts of some proofs are omitted.

6.1. Definitions and Examples

Again, let f2 be a general open set in C". For an orthonormal basis{aµ},° 1 of A2(f2), Cauchy's estimate implies that the series

xEa. (Z) 0". (W)µ=1

converges uniformly on compact subsets of f2 x (l, and is holomor-phic and antiholomorphic on z and w, respectively. This is called aBergman kernel function or Bergman kernel of .R, and is denoted byKo(z, w).

A2(Q) is a function space whose reproducing kernel is Kim. Thatis, for any f E A2(17), the value off at a point z E 17 is expressed by

105

106 6. BERGMAN KERNELS

the inner product of Kn with one variable z fixed and f , precisely as

f(z) = JKri(zw)f(w)dV

= (Kn(z,.),jTY).

On the analogy of the significance of the Cauchy kernel C(z, c) _1 1

27rv"'--l((-z)- in the case of one variable, we know that in general,

analysis of reproducing kernels will bring many good results.In particular, the Bergman kernel has properties useful in the

study of holomorphic mappings, as we will see later, and it is an im-portant research object in the theory of functions of several variables.Since Kn is the sum of an infinite series, it is also an interesting objectin numerical analysis.

We will give examples of domains in which the exact formulae forKg are given.

A2(EXAMPLE 6.1. In the case 17 = V, as an orthonormal basis of

1H ), we can take

/(n + (a))!za

a! 7rnQEZ+

(The L2 norms of the zQ may be obtained by induction on the di-mension.)

Hence, the Bergman

(6.1) KBn(z,w)

kernel of Bn is given by

F-q(n + (a))!

z°wQQEZ+ a! 7rn

(n+Y)!-7.- E zaT

n

e

Y7rv=0 lal=V a!

n F (n +'Y)! (Z' w)v7, v=0

rY.

n do 1 1 /- X lx=(Z,W)

n! (1 _ (z'w))-n-1

7rn

EXAMPLE 6.2. In the case 12 = On, from the above calcula-tion and the general formula (which is obvious from the definition

6.2. TRANSFORMATION LAW AND AN APPLICATION 107

of Bergman kernel)

,KaiXn2 ((z,z'),(w,w')) = K71(z,w)Kn2(z',W')

it follows that

(6.2) KAn (z, w) = f KA(zj, wj) = fj (1 -.7=1 7r 9=1

Besides these, Bergman kernels of various domains have beencalculated (see [29]).

6.2. Transformation Law and an Applicationto Holomorphic Mappings

In a case such as f2 = fl' x C, we have K0(z, w) = 0, and this isnot of interest. Hence, we assume below for simplicity that f2 is abounded domain.

If there exists a biholomorphic mapping F : fl - !2' to anotherdomain f1 , then from the integration formula on variable transforma-tion, we obtain the transformation formula for the Bergman kernel:

(6.3) Kn(z, w) = det (aa (z) Kn (F(z), F(w)) det I81, (w)\ A atVk

Since in particular, for a holomorphic automorphisnl a of fl,

(6.4) K0(z, z) = K0(a(z), a (z)) det Iav

\OZ. )1

2

and Kn(z, z) > 0, it follows that. 881ogKo(z, z) is a (1, 1)-form thatis invariant under the action of Aut S2. In other words, as it is clearthat 88 log Kn(z, z) > 0, the action of Aut .f2 is isometric with respectto the Hermitian metric:

n

E82 log Kn(z, z)

dxj o d---k .

1,k=1 azjazk

This metric is called the Bergman metric of V.

THEOREM (Bremermann's Theorem). If fl is complete as a met-ric space with respect to the Bergman metric, then fl is obviously aconnected domain of holomorphy. Hence, a bounded homogeneousdomain in en turns out to be pseudoconvex.


We will describe an application of the Bergman kernel to holo-morphic mappings.

Caratheodory's theorem in the theory of conformal mappingsstates that if there exists a biholomorphic mapping F between do-mains 111 and 112 of the complex plane, and if each of 0111 and 0172consists of a finite number of simple closed curves, then F can beextended to a homeomorphism from 111 to 112.1

A generalization of this to several variables is the following theo-rem:

THEOREM 6.3 (Fefferman's theorem). Assume that there exists abiholomorphic mapping F between strongly pseudoconvex domains 1l1and 02 in Cn , and 0111 E C°°. Then F can be extended to a diffeo-morphism of class C°° from j71- to 112.

PROOF. Let Pi : L2(112) -p A2(11i) be the orthogonal projection.If we set

u := det0Fj84

then from the transformation formula (6.3) we see that

dgEL2(f12).

By the method of indeterminate coefficients, for an arbitrary h Ethere exists an element v of CO.1(?12)f1Dom0* such that any

derivative of h - e* v is equal to 0 on 0112, where we apply the methodof indeterminate coefficients to the coefficients of the formal powerseries of the defining functions of 112. (See the proof of Theorem 4.18.)

Therefore, for any h E C°°(172), there exists an element ho ofCo (C) such that suppho C 722 and P2(ho 1112) = P2 h.

Regarding an element g of C°°(111), we write, for simplicity, g EC'(771-) when g can be extended to .71 as a C°° function.

Since the transformation formula implies

in order to show that Fj E C°D (111) (1 n), it suffices to provethat

(6.5) u (ho o F) E C°°(f11) .

'See Einar Hille, Analytic Flunction Theory, vol. II, Theorem 17.5.3, Ginnand Co., Boston, Mass., 1962.

6.2. TRANSFORMATION LAW AND AN APPLICATION 109

In fact, in this case, since Kohn's theorem (see Theorem 3.29)implies P, (u (ho o F)) E C' (71), we have

(6.6) u (h o F) E C°° (511) .

Hence, by putting h = 1, we obtain u E C°°(521).The same argument is applicable to F-1, and in particular, it

turns out that u does not have any zero point on 511. _Therefore, from (6.6), it follows that h o F E C°°(711).If we use this for h = zj (1 < j < n), then F3 E C°O(Q1), and F

can be extended to .f11 as a C°° function.As a similar argument applies to F'1, it follows that F can be

extended to a diffeomorphism of class C°° from n1 to 512.

PROOF THAT u (ho o F) E C°° (5l2). Since F3 is a bounded holo-morphic function, by Cauchy's estimate we have

IFIfa)(z)1 < cabnI (z)-(a), z E 111,

where ca is a constant that does not depend on z. _Since from ho E Co (C') we see that supp ho C .R2, in order to

say that every derivative of u (ho o F) is bounded, it is sufficient toshow that there exists a constant C such that

(6.7) JR2 (F(z)) < C5n, (z), z E 121.

However, since for a strictly plurisubharmonic defining function rof l1,, r o F-1 is both negative-valued and subharmonic on 512, fromHopf's lemma we get

(6.8) r o F-1(w) < -C1bn2(w), WE 172,

for some constant C1 > 0. (For Hopf's lemma and its proof, seeProposition 6.4 below.)

From (6.8) and the self-evident inequality

-r(z) < C26n, (z) for some constant C2,

if we set C = CIC2, then (6.7) holds.

Hopf's lemma used in the above proof is the following proposition:

PROPOSITION 6.4 (E. Hopf). Let f2 be a bounded domain in 1R''whose boundary is of class C2, let a be a boundary point of 12, andlet v(a) be the inward unit normal line to 80 at a. Then for an ar-bitrary negative-valued subharmonic function u(x) on .R, there exists


a positive number c such that

(6.9) lim u(x) < -c,Ix - al -

where the superior limit on the left hand side is taken when x E v(a)and x - a.

PROOF. If we take a point x0 on v(a) sufficiently close to a, thereexists an open ball B(xo, R) with center x0 of radius R within .R whoseboundary is tangent to OR at a. For a fixed R' with 0 < R' < R. wecan take a sufficiently large positive number A such that a function

v(x) := e-ajX_b0 2 - e-AR2

is subharmonic2 on B(xo. R) \ B(xo, R'), and satisfies v 18B(xo, R) _0. If we choose a positive number e such that

sup{u + ev I Ix - xol = R'} < 0,

the maximum principle for subharmonic functions implies u < -Evon B(xo, R) \ B(xo, R').

From this, (6.9) is evident.

6.3. Boundary Behavior of Bergman Kernels

The proof of Fefferman's theorem contained in the original article wasbased on a rigorous analysis of the boundary behavior of Bergmankernels. This would be the main road in the sense of tackling the sin-gular points of reproducing kernels directly, but this much analysis ofBergman kernels requires some treatment of the so-called degenerateelliptic boundary value problems that involve slightly more precisetools than merely the L2 estimates. In fact, Kohn's theorem used inthe above proof is one of those precise tools, and we have no space forthem in the present book. However, as the method of L2 estimatesis able to deduce interesting general properties on the singularity ofBergman kernels, we will describe these below.

The next characterization of the value of Kn(z, z) is often used:

(6.10) Kn(z. z) = suP{if(z)I2 I f E A2(Q) 11f 11 = 1}.

2This means Av ? 0 only for this place.

6.3. BOUNDARY BEHAVIOR OF BERGMAN KERNELS 111

In fact, Kn(z, z) is nothing but the square of the norm of the followinglinear mapping:

A2 (.fl) Cw w

f '' f (z)Also, since Kn is a reproducing kernel, it turns out that the

function that realizes the right hand side of (6.10) is the followingone:

eie Kn(.. z) (0 E R).V K0(z, z)

It is clear from (6.10) that for an open subset (1* of (1,

(6.11) (zE.fl`).Concerning the boundary behavior of Bergman kernels, the fol-

lowing fact is the most fundamental:

THEOREM 6.5. If rt E C2(C') satisfies lim

I z1 -2 n(z) = 0, and ifan open set

fl1 = zECnImzn+1zj1 2+n(z)<01

j=1

is pseudoconvez, then

(6.12) limo Kn,, (z, z)(Im zn)n+1 = n! .

Re Zn=0

OUTLINE OF THE PROOF. Since from the Cayley transforma-tion, 120 is biholomorphic to W', the transformation formula (6.3)implies

-n-1

Kno(z,z) = 4?"-Imzn - E 1z312

j=1

Hence, (6.12) is equivalent to

limKn (z, z)

..-.0 Kg.(z,z)Re z,=0

Cauchy's estimate implies

KR,, (z, z)lira < 1.

kn0(z, z)_


Now we deduce the reverse inequality. For ( with Re 0 andCEf1q,ifweset

n

-n-1

7r"Kno (z,1Y zn (z/,() 2_fe(z)

/))Kno(C, 2 (-Im(" - I('I2)-n-1

(z _ (z1, ... , zn-1)) ,

since OR,, and 800 contact properly of degree 2 or more at 0, as( 0, there exists a constant neighborhood U of 0 such that ff isholomorphic on f2,7 fl U, II ft I I no = 1, and

0.

Therefore, by solving, with L2 estimate, the 8 equation

fDu = D(XfO ,

1UM = 0

for a C°° function X whose support is contained in Uand whose valuein a neighborhood of 0 is 1, there exists an element ff of A2(f1,7) suchthat ff(() = ff(() and IIftIIn,, -' 1 as - 0, from which it mustfollow that

lim (z, z) > 1.Re Kno(z, z) _

0COROLLARY 6.6 (L. Hormander, K. Diederich). Let fl be a pseu-

doconvex domain in Cn, and zo a strongly pseudoconvex boundarypoint of f1. Let r be a defining function of fl around zo, and k(zo)the Jacobian of the Levi form of r at zo. If Igradr(zo)I = 1, then

(6.13) lim Kn(z, z)bn(z)"+1 =47r" k(zo).

For general domains, no clear relation, as seen above, betweenthe Levi forms of defining functions and the boundary behavior ofKn is known. But the next statement is fundamental in a differentsense from the above.

THEOREM 6.7. If fl is a bounded pseudoconvex domain withboundary of class C2, then

(6.14) lim Kg (z, z)bn(z)2 > 0.z-8n

6.3. BOUNDARY BEHAVIOR OF BERGMAN KERNELS 113

PROOF. In the case n = 1, from the condition we conclude thatfor each point zo of Oft there exists a circle of constant radius thatis contained in C \ 12 and is tangent to 01? at zo. Since the Bergmankernel for the outside of the closed disk satisfies (6.14), from (6.11),(6.14) holds even for .f1. In the case n ? 2, it suffices to make use ofthe L2 extension theorem, Theorem 5.10 (we leave the details to thereader). 0

Concerning the Bergman metric, the following is basic:

THEOREM 6.8 (K. Diederich). If f1 is a pseudoconvex domain,and zo is a strongly pseudoconvex boundary point of f2. then thereexist a neighborhood U of zo and a positive number C such that forany z E f1 fl U.

00 log Ks? (z, z) >85 5 as - e6

± COOT z I 2

(the double symbols read in the same order), where b := Sn.

For the proof, see [12].A natural question arises on the estimate of the distance function

d(z, w) with respect to the Bergman metric:

CONJECTURE. If 1? is a bounded pseudoconvex domain withboundary of class C', then for any zo E f1,, there exists a positivenumber C such that

d(zo. z) > 1 1 logbn(z)I - C.

At present the following is known in this direction:

THEOREM 6.9. If f1 is a bounded pseudoconvex domain withboundary of class C2, then for any zo E f1. there exists a positivenumber C such that

d(zo, z) >C

log (I log bn(z)I) - C.

For the proof, see [15].For strongly pseudoconvex domains with boundary of class C",

the following decisive result is obtained, and there are several studiesmodeled on this:

THEOREM 6.10 (C. Fefferman [17]). If ft is strongly pseudocon-vex, and if OS? E CO°, then there exist functions cy, W E Cx(f1) anda number c > 0 such that

KQ(z. z) = y7(z)8n"-1(z) + i'(z) logbn(z). z E fl\f1E .

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Index

dim Xa. 86Dom T, 56log D, 19(log D) ", 10PSH(n), 40PSH-(9)? 43Reg X, 86R.eg Xa, 86A2(f2)1 9A2 (fl), 46, 93Ck(fl), 9Co(n), 9C(r) (fl), 10C(r) (n), 11Corl(17), 11Cp,9(n),10Cp.4(n), uCo.e(n), 11D(f, U), 76D, 19e(w), 60f(a). 3

l K , 98

P(f, v), 76V(f), 17za, 1Gn, 98M(fl), 75N, 7Z+. 1(a). 1a!, 10,40(a, r), 40", 4t(w), 60i(f). 9ms, nn,(17), 98fly, 15, 16, 41fY, 93w-j v,60a a

azl 'j 2

'fo., 75' , 2. 10('

58I !' Xt ,

I. 58/

2L[co](zo), 43 (&i.LO. Q(9), 57 afl E Ck, 48

L2(fl), 9 C=, 6

Li (f1). 9 (, )v, 57L2 (n), 57 g) 0' 2L(rl (fl), 11Lloc(f1), 11LP,a(fl), 11L oC(n), 11

),5,57Ifln, 9IzI, 1

Izlmax, 1

88

lG, 98 II Ilv, 57

119

120

adjoint operator, xvi, 56analytic subset, 25antiholomorphic, 2argument principle, 13

Bell, xviiBergman

kernel, xvii, 19Q M5kernel function, 105metric, 14Z

biholomorphic mapping, 3Bochner-Martin conjecture, 47boundary of class Ck, 48Bremermann, 47 73

problem, 81theorem, 107

Caratheodory's theorem, 108Cartan, xiiiCauchy, xiii

estimate, 13.Cauchy-Riemann

equation, xv, fitangential, Z4

Cayley transformation, 4Christ, 52closed operator, 56complex line, 26complex open ball, 4complex tangent space, 48Cousin, xiii

first, second problems, 81

a cohomology group, xv8 equation, xv8 operator, xvi

cohomology group, 27. 28 31. 32a equation, 29a operator, 58defining function, 48

local, 25Demailly, xvi

theorem, 46Diederich-Fornaess example, 52differential form, 14

of type (p, q), 10distribution, xvdivisor class, Z6domain of holomorphy, xvi, 33

INDEX

Euclidean metric, SEuler, xiiiexhaustion function, 45exterior derivative operator, U

Fefferman, xvii, 195theorem, 108, 113

fundamental inequality, 64

Gauss, xiiiGel'fand, xvGrauert, xiiiGrothendieck, xv

Hartogs, xiii, xvifigure, 36function, xvi, 47inverse problem of, 73pseudoconvex, 36

Hodge, xviiholomorphic, 25

automorphism, 3automorphism group, 3function, 2mapping, 3

holomorphically equivalent, 3Hopf's lemma, 144Hormander, xiii, xvi, xvii, 55

theorem, 67hyperplane, 26

idealmaximal closed, xv

interpolationoperator, 23problem, 24

irreducible component, 86dimension of, 86

Jacobian, 11Josefson theorem, 102

Kohn, 55theorem, 53, 110

Kohn-Nirenberg example, 51

L2 convergence on compact sets, 15L2 estimate, xviL2 extension theorem, 95L2 holomorphic function, fi

L2 holomorphytheorem of, 14

Lagrange, xiiiLelong

number, 45Levi, xiii

form, 48problem, xiii, xvi, 73pseudoconvex,49pseudoconvex domain, xviipseudoconvexity, xvi

Ligocka, xviilocally closed complex submanifold,

86locally pluripolar, 111locally polar, 111locally square integrable function, 9logarithmic

capacity, 111image, 19

logarithmically convex, 24

maximum principle, 8meromorphic function, 75Mittag-Leffler theorem, xiv, xvi, 76Montel's theorem, xv, 6

Nakano, xvi

Oka, xiii-xviorder of zero, 3orthogonal projection, 14

plurisubharmonic, 44function, xvistrictly, 43

pole, 75polydisc, 4polynomially convex, 81principal part. 76pseudoconvex,44

Hartogs, 35Levi, 49strongly, 48

pseudoconvexity, xiii, xvi

reduced system, 85regular part, 8firegularization, 14 31 Al

E-, 15

INDEX 121

Reinhardtcomplete - domain, 18domain, 18

reproducing kernel, 1.45restriction homomorphism, 27Richberg's theorem, 43Riemann, xiiiRunge

approximation theorem, xv, xvitheorem, 72

Schwarz's lemma, 8Serre

condition, 31criterion, 27

Siu's theorem, 46Skoda

division theorem, xviitheorem, 92

spectrum, 23subharrnonic, 1151

function, 39

Weierstrass, xiii, xivdouble series theorem, xv, 6product theorem, xiv, xvi, 76

weight function, xvi, xvii, 51weighted inner product, 57weighted L2 norm, 57Wermer's counterexample, 81worm domain, 52

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One of the approaches to the study of functions of several complexvariables is to use methods originating in real analysis. In thisconcise book, the author gives a lucid presentation of how thesemethods produce a variety of global existence theorems in thetheory of functions (based on the characterization of holomorphicfunctions as weak solutions of the Cauchy-Riemann equations).

Emphasis is on recent results, including an L' extension theoremfor holomorphic functions, that have brought a deeper under-standing of pseudoconvexity and plurisubharmonic functions.Based on Oka's theorems and his schema for the grouping ofproblems, topics covered in the book are at the intersection of thetheory of analytic functions of several variables and mathematicalanalysis.

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