American Mathematical Society
American Mathematical Society
4.4. Schauder bases
Chapter 5. Which functions are Cauchy hitegrals? 5.1. General remarks 5.2. A theorem of Havin 5.3. A theorem of 'I\llnarkin 5.4. Aleksandrov's characterization 5.5. Other representation theorems 5.6. Some geometric conditions
Chapter 6. Multipliers and divisors 6.1. Multipliers and Toeplitz operators 6.2. Some necessary conditions 6.3. A theorem of Goluzin~ 6.4. Some sufficient conditions 6.5. The !T-property 6.6. Multipliers and inner functions
Chapter 7. The distribution function for Cauchy transforms 7.1. The Hilbert transform of a measure 7.2. Boole's theorem and its generalizations 7.3. A refinement of Boole's theorem 7.4. Measures on the circle 7.5. A theorem of Stein and Weiss
Chapter 8. The backward shift on H2 8.1. Beurling's theorem 8.2. A theorem of Douglas, Shapiro, and Shields 8.3. Spectral properties 8.4. Kernel functions 8.5. A density theorem 8.6. A theorem of Ahern and Clark 8.7. A basis for backward shift invariant subspaces 8.8. The compression of the shift 8.9. Rank-one unitary perturbations
Chapter 9. Clark measures 9.1. Some basic facts about Clark measures 9.2. Angular derivatives and point masses 9.3. Aleksandrov's disintegration theorem 9.4. Extensions of the disintegration theorem 9.5. Clark's theorem on perturbations 9.6. Some remarks on pure point spectra 9.7. Poltoratski's distribution theorem
Chapter 10. The normalized Cauchy transform 10.1. Basic definition 10.2. Mapping properties of the normalized Cauchy transform 10.3. Function properties of the normalized Cauchy transform 10.4. A few remarks about the Borel transform
95
163 163
179 179 180 184 185 186 192 192 194 196
201 201 208 211 212 218 221 222
227 227 227 230 241
CONTENTS
10.5. A closer look at the ~-property
Chapter 11. Other operators on the Cauchy transforms 11.1. Some classical operators 11.2. The forward shift 11.3. The backward shift 11.4. Toeplitz operators 11.5. Composition operators 11.6. The Cesaro operator
List of Symbols
255
257
267
Preface
This book is a survey of Cauchy transforms of measures on the unit circle. The study of such functions is quite old and quite vast: quite old in that it dates back to the mid 1800s with the classical Cauchy integral formula; quite vast in that even though we restrict our study to Cauchy transforms of measures supported on the circle and not in the plane, the subject still makes deep connections to complex analysis, functional analysis, distribution theory, perturbation theory, and mathematical physics. We present an overview of these connections in the next chapter.
Though we hope that experienced researchers will appreciate our presentation of the subject, this book is written for a knowledgable graduate student and as such, the main results are presented with complete proofs. This level of detail might seem a bit pedantic for the more experienced researcher. However, our aim in writing this book is to make this material on Cauchy transforms not only available but accessible. To this end, we include a chapter reminding the reader of some basic facts from measure theory, functional analysis, operator theory, Fourier analysis, and Hardy space theory. Certainly a graduate student with a solid course in measure theory, perhaps out of [182], and a course in functional analysis, perhaps out of [49] or [183], should be adequately prepared. We will develop everything else.
Unfortunately, this book is not self-contained. We present a review of the basic background material but leave the proofs to the references. The material on Cauchy transforms is self-contained and the results are presented with complete proofs.
Although we certainly worked hard to write an error-free book, our experience tells us that some errors might have slipped through. Corrections and updates will be posted at the web address found on the copyright page.
We welcome your comments.
J. A. Cima - Chapel Hill A. L. Matheson - Beaumont W. T. Ross - Richmond cima~email.unc.edu matheson~ath.lamar.edu wross~richmond.edu
Overview
Let X denote the collection of analytic functions on the open unit disk j[]) = {z E C : Izl < I} that take the form
(KI-')(z):= r ~ dl-'«(), iT 1- (z
where I-' belongs to M, the space of finite, complex, Borel measures on the unit circle 'Jl' = aD. In the classical setting, as studied by Cauchy, Sokhotski, Plemelj, Morera, and Privalov, the Cauchy transform took the form of a Cauchy-Stieltjes integral
r 1 1_'8 dF(O), i[O,21f] - e • z
where F is a function of bounded variation on [0,211"].
In this monograph, we plan to study many aspects of the Cauchy transform: its function-theoretic properties (growth estimates, boundary behavior); the properties of the map I-' 1-+ KI-'; the functional analysis on the Banach space X (norm, dual, predual, basis); the representation of analytic functions as Cauchy transforms; the multipliers (functions cP such that cPX c X); the classical operators on X (shift operators, composition operators); and the distribution function y 1-+ m(l KI-' I > y) (where m is Lebesgue measure on 1['). We will also examine more modern work, beginning with a seminal paper of D. Clark and later taken up by A. B. Aleksandrov and A. Poltoratski, that uncovers the important role Cauchy transforms play in perturbations of certain linear operators. To set the stage for what follows, we begin with an overview.
We start off in Chapter 1 with a quick review of measure theory, integration, functional analysis, harmonic analysis, and the classical Hardy spaces. This review will provide a solid foundation and clarify the notation.
The heart of the subject begins in Chapter 2 with the basic function properties of Cauchy transforms with special emphasis on how these properties are encoded in the representing measure 1-'. For example, a Cauchy transform f = K I-' satisfies the growth estimate
I f(z)j~ ~ D ....., l-Izl' z E ,
(111-'11 is the total variation norm of J.I.) as well as the identity
lim (1- r)f(r() = I-'({(}), (E 'Jr. r ..... l-
This last identity says that Cauchy transforms behave poorly at places on the unit circle where the representing measure I-' has a point mass. Despite this seemingly
1
2 OVERVIEW
poor boundary behavior, Smirnov's theorem says that Cauchy transforms do have some regularity near the circle in that they belong to certain Hardy spaces HP. More precisely, whenever f = Kp. and 0 < p < 1,
sup f If(r()IPdm«() < 00, o<r<liT
where dm = d() /2tr is normalized Lebesgue measure on the unit circle. Let HP be the space of analytic functions f for which the- above inequality holds and let
IIfllHP := (sup f If(r()IP dm«») l/p . O<r<l iT
By standard Hardy space theory, Cauchy transforms haVe radial boundary values
f«):= lim f(r() r-+l-
for m-almost every ( E T. In fact, the formulas of Fatou and Plemelj say that the analytic function f on C \ l' (where C = C u {qo}) defined by
satisfies
r~T- (J(r() - f«/r» = :~ «)
lim (J(r() + f(C/r» = 2P.V.! dp.(~) r .... l-· '1-{(
In this chapter we also discuss when f = K Il can be recovered from its boundary function (1-+ f(C) via the Cauchy integral formula
fez) = f f«l 'dm(C), zED. iT 1- (z
For a general f = KIl, the boundary function (1-+ fCC), although belonging to lJ' for 0 < p < 1, need not be integrable and so the above Cauchy integral representa­ tion may not make Sense. A result of Riesz says that the Cauchy integral formula holds if and only if f belongs to the Hardy space HI, that is,
sup (If(rC)1 dm(C) < 00. o<r<liT
Interestingly enough, there is a substitute Cauchy 'A-integral formula' due to Ul'yanov which says that if p.« m and f = KIL, then
fez) = lim f f(r.}, dm«(), ZED. L .... oo il/lft.L 1 - (z
This Cauchy A-integral formula has been recently used by Sarason and Garcia to further study the structure of certain HP functions.
In Chapter 3 we treat the Cauchy transform not merely as an analytic function, but as a linear mapping p. 1-+ K P. from the space of measures on the circle to the space of analytic functions on the disk. From Smirnov's theorem, we know that
K(M)~ n HP. O<p<l
OVERVIEW 3
In fact,
IIKJlIIHP = 0 (1 ~ p)' p -d-.
We first cover the well-studied problem: if 1 belongs to a certain subclass of Ll, what type of analytic function is
1+ := K(fdm)?
Probably the earliest theorems here were those of Privalov (if 1 is a Lipschitz function on the circle, then f+ is Lipschitz on JI)-), and of Riesz (if 1 < p < 00
and 1 E LP, then 1+ E HP). Then there are the more recent theorems of Spanne and Stein which say that if 1 E Loo, then 1+ E BMOA (the analytic functions of bounded mean oscillation) while if 1 is continuous, then 1+ E VMOA (the analytic functions of vanishing mean oscillation). When 1 E L2 has Fourier series
n=-oo
then 00
n=O
belongs to the Hardy space H2 and the mapping 1 ~ f+ is the orthogonal projec­ tion, the 'Riesz projection', of L2 onto H2.
Riesz's theorem says that the Riesz projection operator 1 ~ 1+ and the asso­ ciated conjugation operator 1 ~ 1:= -2il+ + if(O) + il are continuous on £P for 1 < p < 00, that is to say,
1I111LP ~ ApII/IILP, 1I/+IIHP ~ BpII/IILP, 1 E LP,
for some constants Ap and Bp that are independent of I. An old theorem of Pichorides identifies the best constant Ap as tan(7r/2p) if 1 < p ~ 2 and cot(7r/2p) if p > 2, while a relatively recent theorem of Hollenbeck and Verbitsky identifies the best constant Bp as 1/ sin(7rp).
This chapter also covers the important weak-type theorem of Kolmogorov
m(IKJlI > Y) = O(I/y), y -+ 00,
that gives an estimate of the distribution function for KJl. It will turn out, quite amazingly, that one can recover information about the measure from this distribu­ tion function. For example, Tsereteli's theorem says
Jl« m # m(IKJlI > y) = o(l/y), y -+ 00.
Other work of Hruscev and Vinogradov, covered in Chapter 7, as well as some relatively recent work of A. Poltoratski, covered in Chapter 9, shows even more is true.
In Chapter 4 we treat the Cauchy transforms X = {K Jl : Jl E M} as a Banach space. Since
KJll = KJl2 # Jll - Jl2 E HJ, where HJ are the measures fJ dm : 1 E Hl, 1(0) = O}, X can be identified in a natural way with the quotient space Mj HJ, by means of the mapping KJl ~ [JlJ. Here [Jll is the coset in M / HJ represented by Jl. One defines the norm of K Jl to be
4 OVERVIEW
the norm of the coset lJ.t] in the quotient space topology of M j HJ. Equivalently, the norm of an 1 E X is
11/11 = inf {11J.t1l : 1 = KJ.t}.
Equipped with this norm, X becomes a Banach space and furthermore, the previous growth estimate can be improved to
I/(z) I ~ JltlL 1n\ "" 1 -izi ' z E JUl.
Thus X becomes a Banach space of analytic functions in that the natural injection i : X -+ Hol(D) (the analytic functions on D with the topology of uniform convergence on compact sets) is continuous. From here, one can ask some natural questions. Is X separable? Is it reflexive? What is its dual (predual)? How do the weak and weak-* topologies act on X? Is X weakly complete? Is X weakly sequentially complete? Does X have a basis? What type? These questions are thoroughly addressed in this chapter.
So far, we have discussed the basic properties of a Cauchy transform 1 = K J.t. An interesting and still open question is to determine whether or not a given analytic function 1 on the disk takes the form 1 = K J.t. From what was said above, certain necessary conditions hold. For example, a Cauchy transform 1 must have bounded Taylor coefficients; must satisfy the growth condition I/(z)1 = 0«1- Izl}-l); the boundary values ofthe function 1 must satisfy the £P condition II/IILI' = 0«1 - p)-l) for ° < p < 1; the boundary values for 1 must also satisfy the weak­ type inequality m(l/l > Y) = O(ljy). Unfortunately, none of these conditions is sufficient.
A more tractable question is: if 1 is not merely analytic on ID> but instead is analytic on the larger set C \ T with I(oo} = 0, when is I equal to
f ~ dJ.t«() , z E C \ T, 1-(z
for some measure J.t on the circle? Tumarkin answered this question with the following theorem: if I is analytic on C \ T with 1(00) = 0, then 1 is the Cauchy integral of a measure on the circle if and only if
sup [ II(r() - 1«(jr)1 dm«() < 00. O<r<lJT
Aleksandrov refined this theorem and identified the type of measure (whether abso­ lutely continuous or singular with respect to Lebesgue measure) needed to represent I. These representation theorems are covered in Chapter 5.
At the end of this chapter we examine the question: which Riemann maps 1/J : D -+ n are Cauchy transforms? For example, it is relatively easy to see that if 1/J(D) is contained in a half-plane, then 1/J is a Cauchy transform. What is more difficult to see is that 1/J is a Cauchy transform whenever 1/J(D) omits two oppositely pointing rays. What happens when 1/J(D) is a domain that spirals out towards infinity?
An important class of functions associated with a function space X are the 'multipliers'. Here we mean the set of functions <P for which <pX C X. The multipli­ ers constitute the complete set of multiplication operators 1 1-+ <PIon X and there is quite a large literature on the subject. One can show that when X is a space
OVERVIEW 5
of analytic functions, a multiplier of X must be a bounded analytic function. For the Hardy spaces HP, the multipliers are precisely the bounded analytic functions. However, for other function spaces, such as the classical Dirichlet space or the ana­ lytic functions of bounded mean oscillation, not every bounded analytic function is a multiplier. Furthermore, even when a complete characterization of the multipliers is known, it is often difficult to apply to any particular circumstance.
Chapter 6 deals with the multipliers of X. Despite some interesting results, these multipliers are still not thoroughly understood. For example, a multiplier of X must be bounded, must have radial limits everywhere (not just almost everywhere), and the partial sums of its Taylor series must be uniformly bounded. However, these conditions do not characterize the multipliers.
In this chapter we also cover the !f-property for X. A space of functions X contained in the union of the HP classes, as the Cauchy transforms are, satisfies the J'-property if whenever f E X and f) is inner with f /iJ E HP for some p > 0, then f If) E X. By the classical Nevanlinna factorization theorem, the Hardy spaces have the :f-property. It turns out that X, as well as the multipliers of X, enjoy the J'-property.
For the Hardy space, every inner function is a multiplier. On the other hand, there is the deep result of Hruscev and Vinogradov which says that an inner function is a multiplier of X if and only if it is a Blaschke product
m 11°O lanl an - z z ---- an 1- anz
n=l
~ I-Ianl sup L II" I < 00. (E11' n=l '> - an
The proof of this is quite complicated but still worthwhile to present since it involves many earlier results about Cauchy transforms as well as the well-known Carleson interpolation theorem.
There is also an interesting connection between multipliers and co-analytic Toeplitz operators, namely, a bounded analytic function <p on JI)) is a multiplier of X if and only if the co-analytic Toeplitz operator
(T"4>f)(z):= ("¢(()f(() dm(() = ("¢f)+(z) 111' 1- (z
is a bounded operator from the space of bounded analytic functions to itself.
Kolmogorov's weak-type estimate m(lKp.1 > y) = O(l/y) has been re-examined recently yielding some fascinating results on how this distribution function y 1-+
m(lKp.1 > y) can be used to recover the singular part of the measure p.. Chapter 7 is devoted to these ideas. For example, it is relatively easy to show that when p. « m, the Kolmogorov estimate can be improved from
m(IKp.1 > Y) = O(l/y)
6 OVERVIEW
The relationship between the distribution function and the singular part of the measure goes well beyond the improved Kolmogorov estimate. The first of two important theorems here is one of HrusCev and Vinogradov which says that
lim 1rym(IKpl > y) = I1PslI, 1/-+00
where Ps is the singular part of p with respect to Lebesgue measure. Notice that when p « m, or equivalently Ps = 0, we obtain Tsereteli's theorem. The other more striking, and more recent, theorem of Poltoratski says that
thus recovering the actual singular part of the measure and not merely its total variation norm.
These distributional results are closely related to the distribution functions
y f-+ m(IQpl > y) and y f-+ ml(\!Kp\ > y)
of the conjugate function
and the Hilbert transform
(!Kp)(x) = p.v.l-1- dp(t), J[l x-t
where p is a finite measure on JR. Some. of these distribution theorems are quite classical. For instance, an 1857 theorem of Boole says that if
n
aj E JR, Cj > 0,
which is just the Hilbert transform of the positive discrete measure
then
n
Y j=1
where ml is Lebesgue measure on lR.
Though the material in the first !Several chapters is certainly both elegant· and important, our real inspiration for writing this monogra.ph is the relatively recent work beginning with a seminal paper of Clark which relates the Ca.uchy transform to perturbation theory. Due to recent advances of Aleksandrov and Poltoratski, this remains an active area of research rife with many interesting problems. Chapters 8, 9, and 10 cover this connection between Cauchy transforms and perturbation theory.
Let us take a few moments to describe the basics of Clark's results. Accord­ ing to Beurling's theorem, the subspaces 1JH2, where 1J is an inner function, are
OVERVIEW 7
all of the (non-trivial) invariant subspaces of the shift operator Sf = zf on H2. Consequently, the invariant subspaces of the backward shift operator
S*f = 1-1(0) z
k>.(z) := 1 -19(307J(z) , A, Z E Jl)), 1 - AZ
are the reproducing kernels for (7JH2)1. in the sense that k>. E (7JH2)1. and
f(A) = (I, k>.) V f E (7JH2)1..
Here we are using the usual 'Cauchy' inner product
(I,g) := h f«()g«)dm«)
on H2. Clark's work was inspired by the question as to whether or not a given sequence of kernel functions (k>'n)n~l has dense linear span in (7JH2)1.. Clark showed that for certain ( E T, the kernels kC; belong to (iJH2)1. and are eigenvectors for an associated unitary operator Uo. on (7JH2) 1. . Using the spectral properties of Uo., Clark determined when these eigenvectors kC; form a spanning set for (iJH2)1. and then used a Paley-Wiener type theorem to say when the k>'n's were 'close enough' to the kC;'s to form a spanning set.
The unitary operator U 0 mentioned above is the following: let Su be the com­ pression of the shift S to (iJH2)1.; that is,
Su := PuSI( iJH2)1.
where Pu is the orthogonal projection of H2 onto (iJH2)l.. All possible rank-one unitary perturbations of Su, under the simplifying assumption that iJ(O) = 0, are given by
Uof:= SuI + (I,~) 0:, 0: E T.
It turns out that U 0 is also cyclic and hence the spectral theorem for unitary operators says that Uo is unitarily equivalent to the operator 'multiplication by z', (Zg)() f--t (g«), on the space L2(170.), where 170. is a certain positive singular measure 011 T. It is quite remarkable, as we shall discuss in a moment, that 170. can be computed from the inner function 7J.
The unitary equivalence of Z on L2(170.) and Uo on (iJH2)1. is realized by the unitary operator
:Ja : (iJH2)1. ~ L2(170.),
k>.(z) := 1 -19(307J(z) 1- AZ
8 OVERVIEW
in L2 (u a) and extends by linearity and continuity. Clark uses this unitary equiv­ alence, as well as the structure of the associated space L2( U a), to further examine whether or not the kernels (k.\ .. )nfll fortn a spanning set for (t1H2).!..
This spectral measure U a for Ua arises as follows: for each fixed a E T the function
Z 1-+ R (ex + D(Z») ex - D(z)
is a positive harmonic function on D, which, by Herglotz's theorem, takes the form
!R (a + D(Z») = [1 -lzl2 dua «(), a-D(z) JT\(-zI2
where the right-hand side of the' above equation is the Poisson integral (Pua)(z) of a positive measure U a on T. Without too much difficulty, one can show that the measure Ua is carried by the set {(. E T: D«() = a} and hence Ua ..1. m. Further­ more, Ua .l. U(3 for a I- {3. Though many mathematicians, and some physicists, have used the measures described above, we think it is appropriate to call such measures 'Clark measures' since they are frequently referred to as such in the literature.
This idea extends beyond inner functions D to any cP E ball{HOO) to create a family of positive measures {J.'a : ex E T} associated with cP. It is becoming a tradition to call this family of measures the 'Aleksandrov measures' associated with cP. A beautiful theorem of Aleksandrov shows how this family of measures provides a disintegration of normalized Lebesgue measure m on the circle. Indeed,
lrJ.'adm{a) = m,
where the integral is interpreted in the weak-* sensej that is,
Ir (i f«() dJ.'a«(.») dm(a) = 1r f«(.) dm«()
for all continuous functions / on T.
The identity 1
(Kua)(z) = 1- aD(z)
~ : L2(ua} -+ (DH2).!. I
T./ = K(fdua) a Kuo ·
Poltoratski showed that several interesting things happen here. The first is that for uo-almost every (. E T, the non-tangential limit of the above normalized Cauchy transform exists and is equal to f«(.). On the other hand, for 9 E (DH2).!., the non-tangential limits certainly exist almost everywhere with respect to Lebesgue measure on the circle (since (DH2).!. C H2). But in fact, for ua-almost every (, the non-tangentialliInit of 9 exists and is equal to (~ag)«().
The compression S-{J and its rank-one unitary perturbation U a are covered in Chapter 8. Clark measures, as well as Clark's theorem and Poltoratski's weak-type
OVERVIEW 9
y-oo
are covered in Chapter 9. Poltoratski's theorems on the normalized Cauchy trans­ form
are covered in Chapter 10. At the end of Chapter 10, we briefly mention an independent and parallel
'Clark-type' theory, starting with some early papers of Aronszajn and Donoghue and continued in more recent papers of Simon and Wolff, involving the spectral measures for the rank-one perturbations
AA := A + AV ® v, A E JR.,
of a self-adjoint operator A with cyclic vector v. Here, the Borel transform
r dj.t(t) JJR t-z'
a close cousin to the Cauchy transform, comes into play.
In Chapter 11 we survey some results about the classical operators on X. These operators, which have been studied quite extensively on the Hardy spaces HP, include the shift, backward shift, composition, Toeplitz, and Cesaro operators. We also discuss versions of the Hardy space theorems, Beurling's theorem for example, in the setting of Cauchy transforms.
Conspicuously missing from this book is a discussion of the Cauchy transform
J w ~ z dj.t(w)
of a measure j.t compactly supported in the plane. Certainly these Cauchy trans­ forms are important. However, broadening this book to include these opens up a vast array of topics from so many other fields of analysis such as potential theory, partial differential equations, polynomial and rational approximation [212, 213, 214], the Painleve problem, Tolsa's solution to the semi-additivity of analytic ca­ pacity 1216, 217], as well as many others, that our original motivation for writing this monograph would be lost. Focusing on Cauchy transforms of measures on the circle links the classical function theory with the more modern applications to per­ turbation theory. If one is interested in exploring Cauchy transforms of measures on the plane, the books [27, 50, 73, 78, 146, 154, 169J as well as the survey pa­ pers [32, 33J are a good place to start. There is also a notion of fractional Cauchy transforms [131J.
CHAPTER 1
1.1. Basic notation
There is a complete list of symbols towards the end of the book. Here are some basic symbols and some remarks to help get the reader started.
and
• C (complex numbers) • C = C U {oo} (Riemann sphere) • R (real numbers) .1IJ)={zEC:\zi<l}
• 1l' = BID> • N = {1,2, .. ·} • No = {O, 1, 2, ... } • Z={· .. -2,-1,O,1,2, .. ·} • When defining functions, sets, operators, etc., we will often use the nota­
tion A := xxx. By this we mean A 'is defined to be' xxx. • As is traditional in analysis, the constants e, e', e", ... el, C2, ..• can change
from one line to the next without being relabled. • Numbering is done by chapter and section, and all equations, theorems,
propositions, and such are numbered consecutively. • If J is a set in some topological vector space,
- V J is the closed linear span of the elements of J. - J- is the closure of J.
• If A c C, then ]I = {a : a E C}, the complex conjugate of the elements of A. From the previous item, note that A-is ,the closure of A.
• A linear manifold in some topological vector space is a set which is closed under the basic vector space operations. A subspace is a closed (topolog­ ically) linear manifold.
1.2. Lebesgue spaces
1 A non-empty family :r of subsets of C := C U {oo} is called an algebra if
A u B E:r for all A, B E 9"
C \ A E 9" for all A E 9". An algebra 9" is called a a-algebra if
00
U An E 9" whenever {An: n E N} c 9". n=l
1 A complete treatment of this standard real analysis material can be found in many texts. Several that come to mind are [68, 117, 149, 182, 229].
11
12 1. PRELIMINARIES
Given any collection !T of subsets of C, there is a smallest a-algebra containing !T. The Borel algebra, or the Borel sets, is the smallest a-algebra containing the open subsets of C.
A function I : 'lI' - C is a Borel function if 1-1(G) is a Borel set whenever Gee is open. If follows that l-l(B) is a Borel subset of 'II' set whenever B is a Borel subset of C. If (fn)n~1 is a sequence of real-valued Borel functions, then the functions
are Borel functions.
Let m denote standard Lebesgue measure on 'lI', normalized so that m(1') = 1. This normalization will help us avoid an extra 27T in our formulas. Let LO denote the Lebesgue measurable functions I : T - C and, for 0 < p < 00, let V denote the space of IE LO for which
11/11" := {i III" dm} 1/" < 00.
When p = 00, V~O will denote the (essentially) bounded measurable functions with the (essential) sup norm
(1.2.1) 11/1100 := inf {y ~ 0 : mOIl> y) = O} .
As is customary, we equate two measurable n,.nctions that are equal almost every- where. "
Holder's inequality
Minkowski's inequality
III + gil" ~ Ilfll" + 11911", p ~ 1,
as well as the associated inequality
111+ 911: ~ 11/11: + 11911:, 0 < p < 1,
imply that for 1 ~ p ~ 00, the quantity II/lIp defines a norm on V which makes it a Banach space (complete normed linear space) while for 0 < p < 1, the quantity d(f,g) := III - gll~ defines a metric on L" that makes it a complete, translation invariant, metric space.
A classical representation theorem of F. Riesz says that for p ~ 1, every con­ tinuous linear functionall : V - C takes the form
l(f) = if9dm for some unique 9 E Lq (1/p+ l/q = 1). Moreover, the identity
(1.2.2) sup {Ii f9dml : I E £P, II/lIp ~ 1} = 11911q
implies that the norm of this functional is IIgliq. Thus when p ~ 1, one equates (V) "', the set of continuous linear functionals on L", with Lq. When 0 < p < 1, we have (U)'" = (0) [56].
1.2. LEBESGUE SPACES 13
We will now review distribution functions and rearrangements. Two nice ref­ erences for this are [85, 229]. For I E LO, the function
(1.2.3) AI: [0, (0) -+ [0,1], Af(Y):= m(1/1 > V),
is called the distribution function for I and certainly plays an important role in analysis and probability. One can see that A f is a decreasing right-continuous function on [0, (0). There are also the following LP results.
PROPOSITION 1.2.4. For p > 0,
(1) 11/11: = p 1000 yP-l Af(Y) dy.
(2) Af(Y) ~ y-PII/II~·
(1.2.5) !,,(x) := inf{y > 0: Af(Y) ~ x}
is called the decreasing rearrangement of I. If Af is one-to-one, then f* is A,l. One can check that if
n
I(() = L)jXAj((), j=l
where the Aj's are pairwise disjoint measurable subsets of 'lr and
then
and
where
n
j
Bj := L m(Ai)' i=1
Note that bo := 00, bn+1 := 0, Bo := O. The first important fact about r is that
(1.2.6)
where Ar (x) = ml(f* > x) and ml is Lebesgue measure on R. The second is that, at least for I ~ 0, there is a measure preserving transformation h : '][' -+ [0,1] so that
(1.2.7) I=!"oh. See [185] for details2 •
2In our presentation here, we are really considering the decreasing re-arrangement of III. If one is willing to expand the definition of decreasing re-arrangement, one can prove eq.(1.2.1) for general real-valued J.
14 1. PRELIMINARIES
1.3. Borel measures
A (finite) Borel measure IL on 'll' is a function which assigns to each Borel set A c 'll' a complex number IL(A) such that IL(0) = 0 and
IL (Q1 An) = ~IL(An)' whenever (An)n~l c 'll' is a sequence of pairwise disjoint Borel sets. Unless we say otherwise, our measures will be complex-valued. We will denote the linear space of Borel measures on 'll' by M. A measure IL E M is positive (denoted IL ~ 0) if IL(A) ~ 0 for all Borel sets A c 'll'. We set M+ := {/l EM: /l ~ o}.
THEOREM 1.3.1 (Jordan decomposition theorem). Any /l E M can be written uniquely as
(1.3.2)
For /l EM, define the total variation of IL to be the number
(1.3.3) II/lil := sup {t 1/l(Aj) I : {AI," . , An} is a Borel partition of 'll'} . 3=1
For a measure IL, define the total variation measure l/ll by
(1.3.4) 1/lI(A) := sup {t 1/l(A;)1 : {A1,··· , An} is a Borel partition of A} . 3=1
Note that l/ll('ll') = II/lil
and that if /l is real with /l = /l1 - /l2, /lj E M+, then
l/ll = /l1 + /l2· For a general /l E M with
/l = (/l1 - /l2) + i(/l3 - /l4), /lj E M+,
it follows from the inequality
. a+b la ± 2bl ~ ..j2' a, b > 0,
that for all Borel sets A C 'll',
~ {t,~;(A)} " I~I(A)" t,M;(A). PROPOSITION 1.3.5. The space M, endowed with the total variation norm 11·11,
is a Banach space.
Let G('ll') denote the Banach space of complex-valued continuous functions on 'll' endowed with the supremum norm
1111100 = sup {II(()I : ( E 'll'}.
The identification of G('ll')* (the dual space of G(11')) with the Borel measures M is a classical theorem of F. Riesz.
1.3. BOREL MEASURES 15
THEOREM 1.3.6 (Riesz representation theorem). Let f E C(1l') * . Then there is a unique J.L EM such that f = fIJ' where
tJ.l(f) := 1 I dJ.L.
Moreover,
II f J.lII = sup {II I dJ.L1 : I E C(1l'), 11/1100 ~ I} = IIJ.LII·
The Riesz representation theorem implies that the map J.L 1-+ fJ.l is an isometric isomorphism between M and C(1l')* and one often identifies C(1l')* with M.
A measure J.L E M is absolutely continuous (with respect to Lebesgue measure m), written J.L « m, if JL(A) = 0 whenever A is a Borel set with m(A) = O. A measure JL is singular (with respect to m), written J.L .1 m, if there are disjoint Borel sets A and B such that AU B = 1l' and J.L(A) = m(B) = O.
THEOREM 1.3.7 (Radon-Nikodym theorem). A Borel measure J.L E M is abso­ lutely continuous with respect to Lebesgue measure m il and only il dJ.L = I dm lor some I E Ll, that is to say,
J.L(E) = L/dm,
lor all Borel sets E c 1l'.
The function I in the above theorem is called the Radon-Nikodym derivative of J.L and is often denoted by
dJ.L ._ -.-f. dm
It is a standard fact that the Radon-Nikodym derivative of J.L can be computed as a symmetric derivative. We spend a little time with this idea since it will become important in Chapter 9. We follow [68, 182]. For each ( E 1l' and t > 0 (sufficiently small), let
I((,t):= {(eis : -t < s < t} be the arc of the unit circle subtended by the points (e it and (e- it . If J.L E Mis real, define, for each ( E 1l',
~t(() = J.L(I((,t)) m(I((, t))
and note that ( 1-+ ~t(() is a Borel function on 1l'. Define
(l2Jt)(():= lim ~t(() t-O+
(DJ.L)():= lim ~t(). t-tO+
When (l2Jt)(() = (DJ.L)(() < 00 we say that J.L is differentiable at ( and we write (DJ.L)() := (l2Jt)() = (DJ.L)((). For a complex measure J.L = J.Ll +iJ.L2, where J.Ll,J.L2 are real measures, we say that (DJ.L)() exists if both (DJ.Ll)(() and (DJ.L2)(() exist. Here is a collection of important properties of DJ.L.
PROPOSITION 1.3.8 (Lebesgue differentiation theorem). For J.L E M, (DJ.L)(() exists lor m-a. e. ( E 1l' and
(DJ.L)() = :~ () m-a.e.
16 1. PRELIMINARIES
THEOREM 1.3.9 (Lebesgue decomposition theorem). Any I-" E M can be de­ composed uniquely as
1-"=1-"0,+1-"8' where 1-"0,,1-"8 E M W'ith 1-"0, « m and 1-"8 .1. m.
As a consequence of the Lebesgue decomposition theorem and the definition of the total variation norm, one has the following.
COROLLARY 1.3.10. II I-" = 1-"0, + 1-"8 is the Lebesgue decomposition 01 J.L, then
1IJ..t1l = 111-"0,11 + 111-",11·
Define Ma:={I-"EM:I-"«m} Ms:={I-"EM:I-".1.m}.
Note from Proposition 1.3.5 that M, when endowed with the total variation norm, is a Banach space and by the Lebesgue decomposition theorem,
M=Ma$Ms.
In particular, 111-"11 = 111-"0,11 + 111-"811, J..ta E Ma, 1-"8 EM,
and so Ma and M, are closed subspaces of M.
For J.L EM, consider the union 'U of all the open subsets U c T for which I-"(U) = O. The complement T \ 'U is called the support 01 1-". A Borel set H c T for which I-"(H n A) = I-'(A) for all Borel subsets AcT is called a carrier 01 J..t. Certainly the support of I-' is a carrier but a carrier need not be the support and need not even be closed. For example, if I is continuous and dl-' = I dm, then a carrier of J..t is T \ I-I({O}) (which is open) ~hile the support of I-" is the closure of this set. The following facts are found in [68, 182].
PROPOSITION 1.3.11. II I-' E M+ and J.L = l-"a+J.L, is the Lebesgue decomposition of 1-', then
(1) DJ..t, = 0 and DJ..t = DI-"a I-"a-a.e. (2) 1-"0, is carried by {O < !J.p < co}. (3) 1-'8 is carried by {l2J.t = co}.
REMARK 1.3.12. From time to time, we will be using the following generaliza­ tion of the Lebesgue decomposition theorem (see [99] for example): for v,J..t E M, we say that v is absolutely continuous with respect to 1-", written v « J.L, if
II-"I(E) = 0 =? veE) = o. If v = (VI -v2)+i(V3 -V4), Vj E M+, is the Jordan decomposition of v, the following are equivalent: (i) V « 1', (li) Vj « 1-", j = 1,2,3,4, (iii) Ivi « 1-', (iv) Ivi « 11'1. The Ra.don-Nikodym theorem becomes: if v « 1-", then there is an fELl (II-'I) such that
YeA) = L I dp.
for all Borel subsets AcT. We say 1-", v E M are mutually singular, written I' .1. v, if there are disjoint
Borel sets A and B with Au B = T and II-'I(A) = Ivl(B) = O. The following are equivalent: (i) I' .1. v, (ii) 11-'1.1. 1"1, (iii) J.Lj .1. "k for j, k = 1,2,3,4.
1.4. SOME ELEMENTARY FUNCTIONAL ANALYSIS 17
The Lebesgue decomposition theorem says that for J.L, /I E M,
/I = /I!: + /If, where /I!: « J.L and /If .1 J.L. Furthermore, this decomposition is unique.
For J.L E M+ and n E N, let Fn:= {( E 'll': J.L({O) > lin} and observe, since J.L is a finite measure, that Fn is a finite set. Also observe that
00
{( E 'll': JL({O) > O} = U Fn n=1
and so the set of atoms of a measure (Le., those ( E 'll' for which JL( {O) > 0) must be at most a countable set. A measure J.L E M is a discrete measure if it has a carrier that is at most countable. A measure J.L E M is continuous if J.L( {O) = 0 for all ( E 'll'. There is the following refinement of the Lebesgue decomposition theorem [99, p. 337].
THEOREM 1.3.13. If JL E M, then
J.L = J.La + J.Lc + JLd,
where J.La « m, J.Lc, J.Ld .1 m, J.Lc is continuous, and J.Ld is discrete. Furthermore, J.La, J.Lc, J.Ld are pairwise mutually singular.
1.4. Some elementary functional analysis
We expect the reader to know the basics of functional analysis and so this brief section is merely to set the notation. For a reader needing a review, we recommend the books [49, 142, 183, 231].
For a complex Banach space X, with norm 11·11, let X" denote the dual space of continuous linear functionals '- : X t-+ C. Note that X" is a Banach space when endowed with the norm
(1.4.1) 11'-11 := sup {ll(x)1 : x E X, IIxll ~ I}.
We will make several uses of the uniform boundedness principle.
THEOREM 1.4.2 (Principle of uniform boundedness). Let!J be a family in X". If for each x E X,
sup{ll(x)1 : l E!J} < 00,
then
We will also make several uses of the Hahn-Banach theorems.
THEOREM 1.4.3 (Hahn-Banach extension theorem). Suppose W is a closed subspace of X and'- E W*. Then there is an L E X* such that LIW = '- and
IILII = 1I11l· THEOREM 1.4.4 (Hahn-Banach separation theorem). Suppose W is a closed
subspace of X and x E X \ W. Then there is an l E X* such that '-(W) = {O}, 11'-11 = 1, and '-(x) = dist(x, W).
18 1. PRELIMINARIES
For W C X, define the polar of W to be the set
WO := {i E X* : sup li(x)1 ~ I}. :l:EW
""\ . For Y c X* define the pre-polar of Y to be the set
0y:= {x EX: supll(x)1 ~ I}. lEY
For V C X (or X*) the convex hull of V is the set
{ tc;v;: v; E V,c;;;;= Q,Ec; = I}. ;=1 ;=1
The convex balanced hull of V is the set
{ t CjVj : v; E V, c; E C, t Ie; I ~ I} . ;=1 ;=1
Here are some important facts about polars.
PROPOSITION 1.4.5.
(1) IfW1 eWe X, then WO c Wi; (2) IfY1 eYe X*, then °Y C °Y1i (3) IfW c X, then O(WO) is the closure of the conv~ balanced hull olW.
For a closed subspace W of a Banach space ~, let ~/W be the space of cosets [yJ := y + W. When given the usual (pointwise) vector space operations
[YIJ + [112] := [Yl + 112], c[y]:= fcy},
where Yl, 'Y2 E ~ and c E' C, and the norm
Il(yjll := dist(y, W) = inf{lIy + wll : w E W}.
the quotient space ~ /W becomes a Banach space. Let W.L, the annihilator of W, be the subspace
W.L := {l E ~* : leW) = Q}. '
Note that w.l is a closed subspace of ~*. The following two results follow from the Hahn-Banach theorems.
THEOREM 1.4.6. For a closed subspace W 01 a Banach space X, the quotient space X* /W.l is isometrically isomorphic to W*. In fact, for each l E X* ,
sup{II(w)1 : w E w, IIwll ~ I} = distel, w.L).
Furthermore, there is a <P E W.L 80 that
1I.e + <PI! = dist(i, W.l).
THEOREM 1.4.7. Por a closed subspace W of a Banach spaCe X, the Banach space (X/W)* is isometrically isomorphic to W.L. Moreover, lor fl$ed x E X,
sup{ll(x)I : I E Wi, IIlli ~'1} = dist(x, W).
Furthermore, this supremum is achieved.
1.4. SOME ELEMENTARY FUNCTIONAL ANALYSIS 19
We now consider other topologies on X and X*. We say U C X is weakly open if given any Xo E U, there are i1. ... , in E X* and an € > 0 such that
n n {x EX: lik(X-xo)1 < €} c U. k=l
We mention a few important facts about the weak topology on X. First, X, endowed with its weak topology, is a locally convex topological vector space. Second, a weakly closed subset of X is normed closed but the converse is generally not true. However, as a consequence of Mazur's theorem, a convex subset of X is weakly closed if and only if is it norm closed. Third, the weak and norm topologies on X are the same if and only if X is finite dimensional. A sequence (Xn)n~l C X converges to x E X weakly if i(xn) -. i(x) for each i E 1:*.
The dual space 1:* is endowed with the norm given by eq.(1.4.1) which makes it a Banach space. There is another important topology on X*. A set U c X* is weak-* open if for any io E U, there are Xl> ... ,Xn E X and an € > 0 such that
n n {i E X* : I(i - io)(xk)1 < €} c U. k=l
The space (X*, *), X* endowed with this weak-* topology, is a locally convex topo­ logical vector space. A sequence (in)n~l c X* converges to i weak-* if and only if in(x) -. i(x) for each x E X. An application of the uniform boundedness princi­ ple (Theorem 1.4.2) says that a weak-* convergent sequence (in)n~l is uniformly bounded, that it to say, sup{IIinil : n ~ I} < 00. There is also the important Banach-Alaoglu theorem.
THEOREM 1.4.8 (Banach-Alaoglu). For a Banach space X, the closed unit ball
ball(X*) := {i E X* : IIill ~ I}
is compact in (X*, * ). REMARK 1.4.9. If X is also separable (Le., contains a countable dense set),
then ball(X*) (with the weak-* topology) is metrizable. Thus compactness, in the weak-* topology, of ball(X*) is equivalent to the fact that if (in)n~l is a sequence in ball(X*), then there is an i E ball(X*) and a subsequence in" -. i weak-*. We will be applying this result to the unit ball in the space of measures many times. This also says, using an elementary property of the metric topology, that if E c ball(1:*) and i belongs to the weak-* closure of E then there is a sequence (in)n~l C E converging weak-* to i. In several applications, we will have a subset E of ball(X*) for which we can identify the weak-* closure using the Hahn-Banach separation theorem. Using only this Hahn-Banach argument, we can say that given an i in the weak-* closure of E, there is a net in E converging to i weak-*. The above argument using the Banach-Alaoglu theorem says there is a sequence in E converging to i weak-*.
THEOREM 1.4.10. If Y c 1:*, then (oY)O is the weak-* closure of the convex, balanced hull of Y.
If X is a Banach space, then so is X* and hence one can consider its second dual X** := (1:*)*. For x E X, let Q(x) be the element of X"* defined by
(Q(x))(i) = i(x)
20 1. PRELIMINARIES
and observe from the Hahn-Banach theorem that the map x ~ Q(X) is an isometric linear map from X into X**, often called the cannonical embedding of X into X**. The space X is said to be reflexive if this map x ~ Q(x) is onto. One can show that V', for 1 < p < 00, is reflexive while L1 is not. We point out some basic facts about reflexive spaces.
THEOREM 1.4.'i1. For a Banach space X, the following are equivalent.
(1) X is reflexive. (2) X* is reflexive. (3) Every subspace of X is reflexive. (4) Every quotient space of X is reflexive. (5) The closed unit ball {x EX: IIxll ~ 1} is compact in the weak topology.
The last of the above equivalent conditions is a consequence of Goldstine's theorem [81].
A Banach space X is separable if it contains a countable dense set. For example, the V', 1 ~ p < 00, spaces are all separable (the trigonometric polynomials are dense) while Loo is not. A topological vector space lJ (for example X* endowed with the weak-* topology), is separable if it contains a countable dense set. The following proposition is useful in proving a Banach space is not separable.
PROPOSItIoN 1.4.12. If X is a Banach space and {xa : a E A} is an uncountable subset of X satisfying
IIXa - xbll ~ 1, a,b E A, a =F b,
then X is not separable.
PROOF. The hypothesis says that the open balls
~(a, 1/2) := {x EX: IIx - all < 1/2}
are disjoint. H J were a countable dense subset of X then each ball ~(a, 1/2) would contain at least one element of J, making J uncountable. 0
For example, to see that Loo is not separable set
xa«) := XI .. «), 0 < a < 271",
where Ia := {eit : 0 < t < a} and use the previous proposition.
A few results relating separability and reflexivity are the following.
PROPOSITION 1.4.13.
(1) Let X be a Banach space. If X* is separable, then X is also separable. (2) If X is a reflexive Banach space, then X is separable if and only if X* is
separable.
1.5. Some operator theory
Here are a few reminders from operator theory. The sources [49, 173, 183] will have the details. For Banach spaces X, lJ, a linear operator A: X -+}I is bounded if
(1.5.1) 8up{IIAXII~ : IIxlix ~ I} < 00.
The quantity in the previous line is called the opemtor nonn of A and is denoted by IIAII. Note that A is continuous if and only if it is bounded.
1.5. SOME OPERATOR THEORY 21
THEOREM 1.5.2 (Closed graph theorem). A linear operator A : X --+ }.I is bounded if and only if its graph
{(x, Ax) : x E X}
is a closed subset of X x}.l. Equivalently, the graph of A is closed if and only if given a sequence Xn --+ x such that AXn --+ y, then Ax = y.
If A: X --+ X is a bounded linear operator, we define u(A), the spectrum of A, to be the set of complex numbers ~ such that (AI - A) is not invertible.
PROPOSITION 1.5.3. If A: X --+ X is a bounded linear operator, then
(1) u(A) is a non-empty compact subset ofC. (2) u(A) c {z : Izl ~ IIAII}. (3)
If l E }.I* and A : X --+ }.I is bounded, then loA E x* and this induces a linear map A* :}.I* ...... X*, by
A*(l) := to A.
The map A * is called the adjoint of A.
PROPOSITION 1.5.4. If A: X J---+}.I is bounded, then so is A* and IIAII = IIA*II. FUrthermore, if the dual pairing between X and X* is written as l( x) = (x, t}x, then
(x,A*t}x = (Ax,t}lI, x E X, t E }.I*.
Notice that when X,}.I are Hilbert spaces, then A* is the usual Hilbert space adjoin in that A * : }.I --+ X and
(Ax,Y)lI = (x,A*y}x, x E X, Y E}.I.
In particular, if A is represented by a n;tatrix, then A * is represented by the conju- gate transpose of A. )
If Jelt Je2 are Hilbert spaces, we say a bounded linear operator U : Je1 --+ Je2 is isometric if
IIUXIl:K2 = IIxlbc1 Vx E Je1•
We say that U is unitary if UJel = Je2. Notice that a unitary operator U satisfies
(Ux, UY}:K2 = (x, Y}!Kl Vx, Y E Je1
and U* = U-1 . Moreover, if U : Je --+ Je is unitary, then
u(U) c 'Jr.
Two operators A : Je1 --+ Je1 and B : Je2 --+ Je2 are unitarily equivalent if there is a unitary U : Je1 --+ !J{2 such that
A=U*BU.
An operator A : Je --+ Je is cyclic if there is a vector v E Je (called the cyclic vector) such that
22 1. PRELIMINARIES
Here V denotes the closed linear span. If a E M, a theorem of Szego [101, p. 49] says that
(1.5.5)
Me. : L2(a) -+ L2(a), (Me.f)«():= (f(()
has the constant function X = 1 as its cyclic vector. Since Me = M" this operator is unitary. As it turns out, this operator is the 'model' for all cyclic unitary operators.
THEOREM 1.5.6 (Spectral theorem for unitary operators). If:J{ is a separable Hilbert space and U : 1C -+ 1C is unitary and cyclic with cyclic vector v, then there is a measure a E M satisfying eq.{1.5.5} and a unitary T : 1C -+ L2(a) such that Tv = 1 and
T*Mc.T= U.
If A : 1C -+ 1C is self-adjoint, that is, A* = A, then it is well-known that a(A) c JR. If fL is compactly supported measure on JR one can consider the operator
M", : L2(fL) -+ L2(p,), (Mxf)(x) = xf(x).
Since M; = M x , M", is self adjoint. Moreover, by the Stone Weierstrass theorem, the vector ifJ == 1 is cyclic for Mx. It turns out that Mx is the 'model' for all cyclic self adjoint operators.
THEOREM 1.5.7 (Spectral theorem for self-adjoint operators). If 1C is a sep­ arable Hilbert space and A : 1C -+ 1C is a cyclic self-adjoint operator with cyclic vector v, then there is a finite compactly s'upported measure fL on JR and a unitary T : 1C -+ L2(fL) such that Tv = 1 and
T*MxT=A.
DEFINITION 1.5.8. If A : 1C -+ 1C is either self-adjoint or unitary, we will say that A has pure point spectrum if the corresponding spectral measure (from the spectral theorem) is discrete, that is p, = fLd (see Theorem 1.3.13).
Notice that p, has a point mass at z if and only if the characteristic function X{z} is an eigenvector for Mz on L2(fL). Thus fL is discrete if and only if the characteristic functions on the point masses of fL span L2(fL). Since the eigenvectors for M", correspond to the eigenvectors for A (or U) via the intertwining operator, the operator A (or U) has pure point spectrum if and only if its eigenvectors form a spanning set. This observation will become important in Chapter 8 and Chapter 9.
1.6. Functional analysis on the space of measures
Recall from Section 1.3 that M denotes the space of finite, complex, Borel measures on T and G(T) denotes the complex-valued continuous functions on T. By the Riesz representation theorem (Theorem 1.3.6) the mapping fL H f/-, is an isometric isomorphic mapping from M to G('JI')· which, from our remarks in the previous section, gives rise to the weak4 topology. As before, we write (M, *) to denote M, endowed with the weak-* topology. A net (fL>.h .. EA converges to fL weak-* if and only if
f fdp.>. -+ f fdfL
1.6. FUNCTIONAL ANALYSIS ON THE SPACE OF MEASURES 23
for every f E G(T). An equivalent and useful characterization of weak-* conver­ gence in M comes with the following [156, 210].
PROPOSITION 1.6.1. A net (J-L>..)>"EA C ball(M) converges weak-* to J-L if and only if
J-L.\(A) --> J-L(A)
for each Borel set A C 1f' with J-L(8A) = O.
This next lemma is a general fact about weak-* limits and works in a variety of settings. We state and prove it in the special setting of measures.
and
Let
PROPOSITION 1.6.2. If (fLn)n;:'l eM converges to J-L weak-*, then
sup IIJ-Lnll < 00 n
PROOF. By the Principle of Uniform Boundedness, we know that
sup IIfLn II < 00. n
n--+oo
lim IIJ-Lnk II = L. k--+oo
Given E > 0, there is a KEN so that
IIJ-Lnk II ~ L + E 't/ k ~ K.
Since
there is agE ball( G(T)) such that
IIJ-LII- E < If gdJ-Ll·
But since J-Lnk --> J-L weak-*, we can assume the above K was chosen so that
IIfLlI- E < If gdJ-Lnkl 't/k ~ K.
However, since 9 E ball(G(1l')),
If 9 dfLnk I ~ IlfLnk II
and so for all k ~ K,
The result now follows. o The Banach-Alaoglu theorem (Theorem 1.4.8) in the setting (M, *) takes the
following form.
ball(M) := {p; EM: IIp;II ~ I}
is compact in (M, *). In particu.lar, il (JLn)n~l is a sequence from ball(M), there is a subsequence (P;nk)k~l and a JL E ball(M) s'Uch that lor each I E Cpr),
J IdJLn k ---+ J Idp;.
We also make a few remarks about separability and density. For p; E M we let
/len) := h (" dp;() , nEil,
be the sequence of Fourier coefficients of p;. When dJL = Idm, we write
f(n):= h I()( dm()
for the Fourier coefficients of an L1 function I. Also define, for N E No, the N-partial sum
N
and the Cesaro sum
THEOREM 1.6.5.
(1) (Fejer) II I E C(T), then IIuN(f)IIoo ~ 11/1100 and uN(f) ---+ I unilormly on T as N ---+ 00.
(2) (Lebesgue) II IE LP, 1 ~ p < 00, then UN (f) ---+ I almost everywhere and in LP-norm as N ---+ 00.
(3) II I E Loo, then IIuN(f)lloo ~ 11/1100 and uN(f) ---+ I weak-* as N ---+ 00.
(4) For general p; E M, UN(p;) dm ---+ dJL weak-* as N ---+ 00.
A computation with the total variation norm shows that the uncountable set {oeit : 0 ~ t < 211'} satisfies
(1.6.6)
and so by Proposition 1.4.12, M is not separable in the norm topology. Here, for ( E T, o( is the unit point mass, that is to say, the measure on T such that
o((A) = {I, ~f ( E Aj 0, If ( fj!' A.
However, since every element of MOo (the absolutely continuous measures) is of the form Idm, IE L1, and
III dml\ = II/III, we can apply statement (2) of Theorem 1.6.5 to say that
UN (f) dm ---+ I dm, N ---+ 00
in the norm of M and so, since the trigonometric polynomials with complex rational coefficients are a countable dense subset of L1, MOo is a separable subspace of M.
1.7. NON-TANGENTIAL LIMITS AND ANGULAR DERIVATIVES 25
On the other hand, (M, *), the space of measures endowed with the weak-* topology, is separable. One can see this in several ways. First, by part (4) of the above theorem, UN (f) dm -+ dl' weak-*. We can also see this with the following.
PROPOSITION 1.6.7. Both Ms and Ma are dense in (M, *).
PROOF. For f E C(1') and ( E'1', we have
! f d6, = f«()·
It follows that the only / E C(1') that annihilates the linear span of the point masses is the zero function. Thus, by the Hahn-Banach separation theorem, the linear span of Ms is dense in (M, *).
To see the density of Ma in (M, *), define
1 dllh := 2h Xl" dm, h > 0,
where h is the arc of the circle subtended by e-ih and eih and observe that "h -+ 61
weak-* (Lebesgue differentiation theorem). Now use the density of Ms in (M, *) as argued in the first part of the proof. 0
We will also make use of the following.
PROPOSITION 1.6.8. The convex balanced hull 0/ {6, : ( E 1'}
is weak-* dense in the ball 0/ M.
PROOF. If Y = {6, : (E 1'}, one can easily show that oy = ball(C(1'» and so (oy)O = ball(M). Now use Theorem 1.4.10. 0
REMARK 1.6.9. We can combine Proposition 1.6.8 with Remark 1.4.9 to prove the following: given I' E M, there is a sequence (J.I.n)n;;>.1 c M such that each I'n is a finite union of point masses, IIl'nll ~ 111'11 for all n, and J.I.n -+ I' weak-*.
There is also the following refinement (see [40, p. 221])
PROPOSITION 1.6.10. Suppose I' E M+ with support on a closed set F c 1'. Then there is a sequence J.I.n -+ I' weak-* such that for each n, J1.n E M +, is supported in F, is a finite linear combination 0/ point masses, and IIl'nll = 111'11.
1.7. Non-tangential limits and angular derivatives
For an analytic function / on D and ( E 1', we say that / has a mdiallimit L at (, if
lim f(r() = L. r-l-
(1.7.1) r a«() := {z ED: Iz - (I < a(l -Izl)}
be a non-tangential approach region (often called a Stoltz region). Note that r a«() is a triangular shaped region with its vertex at ( (see Fig. 1). We say that / has a non-tangential limit value A at (, written
L lim f(z) = A, z-,
26 1. PRELIMINARIES
FIGURE 1. Non-tangential approach region with vertex at (E 'lI'
if I(z) -+ A 88 Z -+ ( within any non-tangential approach region r a«)' Let us mention a few well-known results about non-ta.zi.gentia11imits. We refer the reader to [48] for the proofs.
THEOREM 1.7.2 (Fatou). II I is a bounded analytic function on)[Jl, then the non-tangential limit of I exists and is finite lor almost every '" E T.
For bounded analytic functions, the existence of radial and non-tangential limits are the same.
THEOREM 1.7.3 (Lindelof). If I is a bounded analytit; function on D. and I(z) -+ A as z -+ ( along some arc lying in)[Jl and terminating at," E T, then
L lim I(z) = A. z-C
Unfortunately, for bounded analytic functions, non-tangential limits is about the best we can do. -
THEOREM 1.7.4. Let C be a simple closed Jordan curve internally tangent to T at the point ( = 1 and having no other points in common with T. For 0 < f) < 21[", let CfJ be the rotation 01 C through an angle (J about the origin. Then there is a bounded analytic function I on D which does not approach a limit as z approaches any point eifJ from the right or the left along CfJ.
Littlewood [1241 proved the 'almost everywhere' version of this theorem while Lohwater and Piranian [126] proved the stronger 'everywhere' result above.
THEOREM 1.7.5 (Privalov's uniqueness theorem [48, 118, 169]). Suppose I is analytic on D and
L lim I(z) = 0 z--+C
lor ( in some subset ofT of positive Lebesgue measure. Then f == O.
Non-tangential limits are important in the statement of Privalov's theorem since there are non-trivial analytic functions on D which have r.adiallimits equal to zero almost everywhere on T [25]. There are no non-trivial analytic functions on D which have radial limits equal to zero everywhere on 'lI' [44, p. 12].
1.7. NON-TANGENTIAL LIMITS AND ANGULAR DERIVATIVES 27
We know that bounded analytic functions have non-tangential limits almost everywhere. To focus on the question as to whether or not a bounded analytic function has a non-tangential limit at a specific point ( E ']f, we need the following factorization theorem [65].
THEOREM 1.7.6. If f is a bounded analytic junction on]D), then
f = 1JF,
where 1J is a bounded analytic junction that has boundary values of unit modulus almost everywhere and F is a bounded analytic junction that satisfies
log IF(O)I = h log IF(()I dm(().
The function 1J is called the inner factor of f and the function F is called the outer factor of f. We can factor {} further as
{} = bSI!'
b(z) = zm IT lanl an - z an 1- anz
n=l
whose zeros at z = 0 as well as {an} C ]D)\{O} (repeated according to multiplicity) satisfy the Blaschke condition
n=l
(which guarantees the convergence of the product) and sl! is the (zero free) singular inner factor
SI!(z) = exp ( - / ~ ~ ; d~(()) , where ~ E M+ and is singular. Furthermore, the outer factor F can be written as
F(z) = ei'r exp (i ~ ~; log IF(()ldm(()) .
Note that log IFI E £1 (see Theorem 1.9.4 below) and so the above integral makes sense.
The following theorem of Frostman [48, p. 33] [72], discusses non-tangential limits of Blaschke products.
THEOREM 1.7.7 (Frostman). Let b be a Blaschke product with zeros (an )n;;:'l.
A necessary and sufficient condition that b and all its partial products have non­ tangential limits of modulus one at ( is that
~ l-Ian l
~ I( - ani < 00.
Ahern and Clark [2, 3] refine Frostman's theorem and extend it to general inner functions.
THEOREM 1.7.8 (Ahern and Clark). Suppose that {} = bsj.! is inner and ( E ']f
with 1'( {(}) = O. The following are equivalent.
28 1. PRELIMINARIES
(1) Every divisor3 of fJ has a non~tangentiallimit of modulus equal to one at (.
(2) Every divis()r of fJ has a finite non-tangential limit at (. (3)
~ 1 -Ianl / dp(e) ~ 1(-a,,1 + le-(I < 00.
DEFINITION 1.7.9. For an an~ytic function fP: D -+ 0 4 and a point <" E T, we say that t/J has an angular derivative at , E T if for some 'f/ E T,
L lim rfJ(z) - 'f/ z-, z - ( exists and is finite. We d~ote the above limit, whenever it exists, by rfJ'(,).
The first thing to notice is that the existence of an angular derivative automat­ ically implies that
L lim t/J(z) = 'f/ z-+, and that I'f/I = 1. The following result is the key to understanding angular deriva­ tives. A proof can be found in [6, til, 196}.
THEOREM 1.7.10 (Ju1i&-Carathoodory). For an analytic function t/J : 0 -t D and (" E T the following statements are equivalent.
(1)
(2)
z-+, 1 -lzl
L lim t/J(z) - 'f/ = t/J'«() z_, z - ( exists for some 'f/ E T,
(3) L lim t/J'(z) * ..... ,
exists and L lim rfJ(z) = 'f/ E 1'. z_,
FUrthermore, (a) 6> 0 in (I). (b) The points." in (S) and (9) is the same. (c) t/J'«() = ('f/6 and
L lim t/J'(z) = t/J'«(). z_, (d) If any of the above conditions hold, then
6 = L lim 1 -1t/J(z)l. z ..... , l-lzl
We now focus on specific results on the existence -of iLllgnlar derivatives. We begin with a simplifying proposition which is a corollary of Theorem 1.7.10.
Swe say an inner function'I/J is a ditMOf" of tJ if tJ{'IfJ is also inner. "such t/J are often called analytic sell-maps 0/ D. .
1.7. NON-TANGENTIAL LIMITS AND ANGULAR DERIVATIVES 29
PROPOSITION 1.7.11. If cPl, cP2 are analytic self maps of If} and cP = cP2cP2, then
IcP'(()1 = lcP~(()1 + IcP~(()1 for every ( E 1['.
If we focus our attention on inner functions {) = bsl" where b is a Blaschke product with zeros (an)n~l and slJ. is the singular inner factor with singular measure fJ., the above proposition says we can consider the Blaschke factor and singular inner factor separately. Here are two classical theorem that do this.
THEOREM 1.7.12 (Frostman [72]). Ifb is a Blaschke product with zeros (an)n~l and ( E 1l', then b has a finite angular derivative at ( if and only if
~ 1-lanl2
Moreover,
THEOREM 1.7.13 (M. Riesz [175]). The s'ingular inner function sl' has a finite angular derivative at ( E 1l' if and only if
J dfJ.(e) Ie - (12 < 00.
Moreover, , J dfJ.(e) ISI'(()I = 2 Ie _ (1 2 < 00.
If fJ.({(}) > 0, then the above integral diverges and so Sl' will not have an angular derivative at (. In this case, I slJ.(r() I -+ 0 as r -+ 1- and so slJ. cannot possibly have a finite angular derivative.
COROLLARY 1.7.14. An inner function {) = bsl' has a finite angular derivative at ( E 1[' if and only if
Moreover,
For conditions on the existence of angular derivatives for general self maps cP, we need the following factorization theorem.
PROPOSITION 1.7.15. If cP: II) -+ II) is analytic, then
(1.7.16)
where b is a Blaschke product with zeros (an)n~l and v E M+.
If v 1- m, then the second factor is a singular inner function.
30 1. PRELIMINARIES
THEOREM 1.7.17 (Ahern and Clark [3, 4]). An analytic self map 4J ofD, fac­ tored as in eq.(1. 7.16), has a finite angular derivative at (E T if and only if
00 1 - lanl2 J dv(~) ~ 1(-an I2 +2 1{-(12 < 00.
Moreover,
Define the Poisson kernel
and conjugate Poisson kernel
Qz(() := ~ (( + z) = 2~«(z), (E T, zED. (- z I( - zl2
For fixed ( E T, the functions
z t-+ Pz«() and Z t-+ Qz«()
are harmonic on the open unit disk D and so, for /-L E M j the Poisson integral
(1.8.1) (Pp,)(z):= J Pz «() dp,(()
and the conjugate Poisson integral
(1.8.2) (Qp,)(z):= J Qz«() d/-L«()
are harmonic on D. An obvious closely related kernel is the Herglotz kernel
Hz«():= ~ +z .,,-z
which is an analytic function of z with IRHz «() = Pz «() > 0 and so the Herglotz integral
(1.8.3)
is analytic on D and has positive real part whenever /-L E M+.
Observe that for 0 < s < 1 and ~ E T,
and so
~ (~ ~ :~) = n~oo slnlen and 9 (~ ~ :~) = -i n;oo sgn(n)slnl~n, where
-1, 0, 1,
1.8. POISSON AND CONJUGATE POISSON INTEGRALS
Thus 00
where, as before,
iL(n):= h (dp,«()
are the Fourier coefficients of p,.
Here are some standard facts about Poisson integrals [101, p. 32 - 33].
PROPOSITION 1.8.5. For an fELl and 0 < r < 1, let
fr«() := (P fdm)(r(), (E 1r.
(1) If f is continuous, then fr --+ f uniformly on 1r as r --+ 1-. (2) If f E V, 1 ~ p < 00, then fr --+ f in V as r --+ 1-. (3) If f E Loo, then fr --+ f weak-* as r --+ 1-, that is to say
hfrgdm --+ hf9dm, r --+ 1-,
for every 9 ELI. (4) For a general p, E M, (PIL) (r·) dm --+ dIL weak-* as r --+ 1-.
31
Here are two important results that will be used many times throughout this book. The first is Fatou's theorem5.
THEOREM 1.8.6 (Fatou). If p, E M, and (Dp,)«() exists, then
lim (Pp,)(r() = (Dp,)«() . . r->l-
REMARK 1.8.7. (1) From Proposition 1.3.8, Dp, = dp,/dm m-almost everywhere and so the ra­
dial limit of the Poisson integral is equal to the Radon-Nikodym derivative m-a.e.
(2) If IL 1. m, or equivalently Dp, = dp,/dm = 0 m-a.e., then the above limit is zero m-a.e.
(3) The radial limit in Fatou's theorem can be replaced by a non-tangential limit, that is to say,
L.lim(Pp,)(z) = (DIL)«() z-+(
whenever (Dp,)«() exists. (4) If ( E 1r and p, is a real measure, then [182]
(1.8.8) (QJ.L)(() ~ lim (Pp,)(r() ~ lim (Pp,)(r() ~ (Dp,)«(). r->l- r->l-
5Fatou's original proof in terms of Poisson-Stieltjes integrals is in [69). The references [65, p. 39) or [101, p. 34) have modern proofs.
32 1. PRELIMINARIES
If J.I. E M+, then certainly PJ.I. ~ 0 on D. Also note that HJ.I. is analytic on D with ~HJ.I. = PJ.I. ~ O. This following theorem of Herglotz 6 is the converse.
THEOREM 1.8.9 (Herglotz).
(1) If u ~ 0 on D and harmonic, then u = PJ.I. for some J.I. E M+. (2) II I is analytic on D, ~I ~ 0, and 1(0) > 0, then I = HJ.I. for some
J.l.EM+.
From Fatou's theorem (Theorem 1.8.6), we know that PJ.I. has finite non­ tangential boundary values m-almost everywhere and we will see in the next chapter (Lemma 2.1.11) that HJ.I. does as well. Since HJ.I. = PJ.I.+iQp., then QJ.I. has boundary values and the m-almost everywhere defined boundary function
(QJ.I.)«():= lim (Qp.)(r() r-+l-
is called the conjugate junction. At least formally (replacing z with ei6 and ( with eit in the eq.(1.8.2», this boundary function (QJ.I.)(ei6 ) is equal to
(QJ.I.)(ei6) = 121r ~ (::: ~ :::) dJ.l.(eit) = 12ff cot (9; t) dJ.l.(eit).
Unfortunately, for fixed (), the function cot«({ .... t) may not belong to L1 (J.I.), making the integral possibly undefined. In terms of principal value integrals, we do have the following standard fact.
THEOREM 1.8.10. II J.I. E M, then
i6 for m-a. e. e .
'6 127r (() - t) 't lim (QJ.I.)(re' ) = P.v. cot -2- dJ.l.(et } r-+l- 0
1, (() -t) . := lim cot -2- dJ.l.(e't).
E-+O+ 16-tl~E
1.9. The classical Hardy spaces
For 0 < p < 00, let HP, the Hardy space?, denote the space of functions f analytic on D for which the £P integral means
(1.9.1) Mp(r; f) := t£ If(r()IP dm«() riP remain bounded as r i 1-. This definition can be extended to p = 00 by
Moo(rj f) := sup{lf(r()1 : ( E 'f}
and so Hoo is the set of bounded analytic functions on D. The function
r~ Mp{r;f)
(1.9.2) M(rl; f) ~ M(r2; f), 0 ~ rl ~ r2 < 1,
6The reference [98] contains the original proof while [101, p. 34] or [65, p. 2] have more modern proofs.
7We refer the reader to several classic texts [65, 79, 101, 118, 234] for the proofs of everything in this section.
1.9. THE CLASSICAL HARDY SPACES 33
and the quantity
II/IIHP:= sup Mp(r; f) = lim Mp(r; f) O<r<l rtl-
defines a norm on HP when 1 ~ p < 00. When 0 < p < 1, the quantity
dist(f,g) := III - gll':ip defines a translation invariant metric on HP. The pointwise estimate
(1.9.3) I/(z)1 ~ 21/PII/IIHP (1 _ ~I)l/P' zED,
can be used to show that HP (1 ~ P < (0) is a Banach space while HP (0 < p < I) is an F -space (a complete translation invariant metric space). In particular, if In -+ I in HP, then In -+ I uniformly on compact subsets of D.
The following standard facts about functions in HP spaces will be used many times throughout this book.
THEOREM 1.9.4. For 0 < p ~ 00 and I E HP,
(1) I(C) := L lim I(z),
z-(
the non-tangential limit of f at (, exists for almost every ( E T. (2) This m-a.e. defined boundary function C 1-+ I(C) belongs to LP and when
o <p < 00,
Hence II/IIHP = 11/1Ip· (3) II f E HP \ {O}, then
i iog I/«()I dm«() > -00
and hence the function ( 1-+ f«() can not vanish on any set 01 positive measure in T.
(4) Ifp ~ 1, and f E HP has Taylor series co
f(z):::: Lanzn, n=O
an = ( f«()"f dm(C), n E No. iT .
(5) For 0 < p < 00, the polynomials are dense in HP. When p :::: 00, the polynomials are weak-* dense in Hoo .
Every I E HP has an associated boundary function which belongs to V' and has the same norm. We denote this set of boundary functions by
HP(1') ::::: {I E V' : f(C) = lim f(r() a.e. for some f E HP} . r-.l-
Frequently we will not make a distinction between HP and HP (1'). As such, we will also use the notation
34 1. PRELIMINARIES
for the HP norm of f, or equivalently the V norm of the boundary function C 1-+
f«,). Throughout this book we will use the following important fact.
PROPOSITION 1.9.5 (Smirnov). If 0 < p < q and f E HP has Lq boundary valu.es, then f e yq ..
We know that HP(T) is a closed subspace of V. Turning this problem around, one can ask: when does a given f E V belong to HP(T}? At least forp ~ 1, there is an answer given by a theorem of F. and M. Riesz.
THEOREM 1.9.6. For p ~ I, a function f E V belongs to HP(1') if and only if the Fourier ~oefficients h f«()'r dm«,)
vanishJor all n < O.
Actually, the following is the most useful version of this theorem.
THEOREM 1.9.7 (F. and M. Riesz theorem). Su.ppose JL E M satisfies
f (," dJL«') = 0 whenever n E No.
Then dJL = <pdm, where <p E HJ = {f E H1 : f(O} = O}.
Every f E HP can be factored as
(1.9.8) . f =. O,If·
The function 0" the outer factor, is characterized by th~ property that 0, belongs to HP and
(1.9.9)
Every H.P outer function F (i.e., F has nO,inner factor) can be expressed as
(1.9.1O) 'F(z) = ei'Yexp (1 ~ =: 10g,p(C) dm(c}) , where 'Y is a real number, ,p ~ 0, log,p E L1, and ,p E V. Note that F has no zeros in the open unit disk and IF(C)I = ,p(C) almost everywhere. Moreover, every such F as in eq.(1.9.1O} belongs to HP and is outer. The inner factor, If, is characterized by the property that If is a bounded analytic function on D whose boundary values satisfy II,(C)I = 1 for almost every (,. Fu,rthermore, as seen Section 1.7, the inner factor I, can be factored further as the product of two inner functions
(1.9.11) I, == ba",.
where b is a Blaschke product and sp. is a singular inner function. A meromorphic ~ction f on.D is said to be of bounded type if f = hl/~' where
hll h2 are bounded analytic functi~ on D. From Theorem ,1.9.4 and .eq.(1.9.8), a function of bounded type must have finite non-tangential limits almost everywhere on 'll' and can be fact0red as '
f = I1&1 0hl.
11&20 11.2
The set N, the Nevanlinna class, will be the functions f of bounded type which are analytic on J) (equivalently Ih2 is a singular inner function). The set N+, the
1.10. WEAK-TYPE SPACES 35
Smimov class, will be the set of lEN for which h2 is a constant. It is a standard fact that
lEN {::} r~rp- i log+ I/(r()1 dm«() < 00
and that for lEN, the boundary function satisfies
h log+ 1/«()1 dm«() < 00.
For lEN, we have
IE N+ {::} lim f log+ I/(r()1 dm«() = f log+ If«()1 dm«(). r ..... 1- 1T J.r
Note also that
UHPCN+. 11>0
We also have the following generalization of Proposition 1.9.5.
THEOREM 1.9.12 (Smirnov). II IE N+ with lJ' boundary junction, then IE HP.
1-10. Weak-type spaces
We say a function I E LO (the Lebesgue measurable functions on'll') belongs to L 1,00, or weak-L1, if
mOil> y) = 0 (t), y --+ 00.
We say I E L~'oo if
m(1/1 > y) = 0 (t), y --+ 00.
Define the quasi-norms
II/IIL1.OO := supym(1/1 > y). y>O
Let H 1,00 be the analytic functions on 1Dl for which
1I/I1Hl,OO:= sup IIlrllLl.oo < 00, Ir«() = I(r(). O<r<l
PROPOSITION 1.10.1. H1,oo c n HP.
O<p<l
PROOF. It follows from the distributional identity
IIgl\~ = p f yP:lm(lgl > y) dy, 9 E LO, 1[0,(0)
&rhls quasi-norm does not satisfy the triangle inequality III + gil " ""1 + IIgll but does satisfy II! + gil " 2(11/11 + IIglI)· See [111J for more on quasi-norms.
36 1. PRELIMINARIES
(Proposition 1.2.4), tha.t for I E Hl,oo and A :;:: II/rIlLl.oa,
. II/rll: = p 100 yP-l m(l/rl > y) dy
= P lA yp-1 m(l/rl > y) dy + P £00 yP-2y m(l/rl > y) dy
~ P lA yP- 1dy + pA Loo yP-2dy
=AP+~AP I-p
AI' = I-p'
o The following deep result is an equivalent characterization of Hl,oo [9].
THEOREM 1.10.2. For an analytic function Ion D, the/ollowing are equivalent. (1) IE H1,oo. (2) The radial maximal function
{M!)«():= sup I/(r(.) I O<r<1
belongs to L1,oo. (3) The non~tangential maximal function
(No !)«():= sup I/(z)1 zer .. (c}
belongs to Ll,oo .
REMARK 1.10.3. Compare this theorem to the following equivalent character­ iza.tion of HP by Hardy and Littlewood [87] (1 ~ p ~ 00) and Burkholder, Gundy, and Silverstein [35] (0 < p < 1) (see also [79, 118)): if 0 < p ~ 00 and I is analytic on D, then the conditions (i) I E HI', (ii) MI E V, (iii) Nol E V, are equivalent.
Since every I E Hl.oo has boundary ~ues, defined almost everywhere by
1«(,) = lim I (r(.) , r-l-
we can define HJ'oo to be those I E Hl,oo for which liT E L~'oo. Recall The­ orem 1.9.12 which says that if I E N+ (the Smimov class) and liT (the nOD­ tangential boundary values of f) belongs to V, then I E HI'. HElre is the corre­ sponding result for the analytic weak.type spaces.
THEOREM 1.10.4. II I E N+ and liT E L1,oo (respectively I E L~'oo), then IE Hl,oo (respectively I E H~'OO).
1.11. Interpolation and Carleson's theorem
It will be important for the work in Chapter 6 to gather up some well-known results about interpolating sequences. We quickly review these ideas and refer the reader to sources like [21, 65, 79, 191, 1D3} for the formal proofs. We will write E
1.11. INTERPOLATION AND CARLESON'S THEOREM 37
to indicate a sequence in lD>. For simplicity we will always assume 0 ¢. E. Associated to E is the discrete measure J1.E on lD> given by
J1.E(A):= L (1- laD, A c lD>. aEEnA
The Blaschke condition on Ej that is,
E(1-lal) < 00,
aEE
simply asserts that J1.E is a finite measure. This condition is equivalent to the convergence of the Blaschke product
B(z):= II ~ a-~. E a 1- az
aE
uniformly on compact subsets of lD>. We write ba for the individual Blaschke factor
ba(z) = ~ a -.: , a 1-az
and let B(z)
Ba(z) = ba{z)
be the Blaschke product with one of its factors divided out. We say a sequence E is sepamted if
(1.11.1)
where
s(E):= inf{p(a,b): a,b E E and a"# b} > 0,
la-bl p(a, b) := 11- abl
is the pseudo-hyperbolic distance between a and b, and unilormly sepamted if
(1.11.2) 6(E) := inf IBa(a)1 > O. aEE
Let I be an arc on the unit circle, and define the Carleson square on I to be the set
(1.11.3) Q = { z E lD>: j:l E I and 1 - Izi < m(I) }
(see Figure 2). A positive measure J1. on lD> is a Carleson measure if there is a constant cp. depending only on J1. such that
J1.(Q) ~ cp.m(I)
for each Carleson square Q. We define "Yp. to be the infimum of all such constants Cp.o We say that E is a Carleson sequence if JLE is a Carleson measure and we set "Y(E) := "YP.E'
The sequence E is an interpolating sequence if, whenever g E too (E), the bounded functions on the sequence E, there is a function I E Hoo such that liE = g. By the open mapping theorem, there is a constant C such that for each 9 E lOO(E), a function I E Hoo can be chosen so that
(1.11.4) 11/1100 ~ Csup{lg(z)1 : z E E}.
38 1. PRELIMINARIES
FIGURE 2. A Carleson square Q over the arc I c T
We define C(E) to be the infimum of such constants C above. It is easy to see that E must be the zero set of a BlaSchke sequence, and not too difficult to see that E is separated. The main theorem here is one of Carleson.
THEOREM 1.11.5 (Carleson). Let E be a countable subset of D. Then the following are equivalent.
(1) E is an interpolating sequence; (2) E is uniformly sepamtedj (3) E is sepamted and f..LE is a Carleson measure.
In case any of these conditions hold, we have the following relationships between the constants s(E),6(E),'Y(E), and C(E):
1 'Y(E) 1 ( 1 ) (1.11.6) 6(E) ~ C(E) ~ Cl 6(E) ~ C2 c5(E) 1 + log 6(E) ,
(1.11.7) 1 ( 'Y(E) ) s(E) ~ c5(E), s(E) ~ C(E) , 6(E) ~ exp -C3 s(E)2 '
where Cl, C2, C3 > 0 are absolute constants.
Interpolation sequences actually exist [101, p. 203].
THEOREM 1.11.8 (Hayman-Newman). A sequence (Zn)n)1 C D such that
sup {I- IZn+11 : n E N} < 1 l-IZnI
is an interpolating sequence.
COROLLARY 1.11.9. If (rn)n)l C (0,1) with rn i 1, then (rn)n)l is an inter­ polating sequence if and only if
{ 1- rn+1 } sup 1 _ r.. : n E N < 1.
Just in case the reader might think that interpolating sequences must approach the unit circle exponentially, there is this curious result of Naftalevic in [147].
1.12. SOME INTEGRAL ESTIMATES 39
THEOREM 1.11.10 (Naftalevic). If (rn)n~1 C (0,1) satisfies
n=l then there is a sequence of angles (9n)n~1 C [0,211") such that (rnei9n )n~1 is an interpolating sequence.
1.12. Some integral estimates
.We end this chapter with some trivial but very useful integral estimates that will be used often throughout the book. The first estimate, through rather easy, drives everything.
LEMMA 1.12.1. There are universal constants Clo Cl > ° such that
cl«1 - r)2 + (2)1/2 ~ 11 - rei9 1 ~ c2«I- r)2 + (2)1/2
for all r E (!, 1) and all 8 E [0,11"].
PROOF. Note that
11 - rei9 1 = (1- 2r cos 8 + r2)1/2 = «1- r)2 + 4rsin2(8/2»1/2.
Using the estimate
we get
. Hence we obtain constants Cl, C2 > ° so that
Cl (1- r)2 + (2) ~ 1- 2rcosO + r2 ~ C2 (1- r)2 + (2)
for all r E (~, 1) and () E [0,11"].
LEMMA 1.12.2. Given p > 1, there is a positive constant c > ° so that
1~ 1 c -;----·;-;;"9:- dO ~ -;-:---:---:­
_~ 11- ret Ip (1- r)p-l
for all r E (!, 1).
PROOF. Observe that
1~ d(} 1~ dO r d9 _~ 11 - rei9 1p = _~ (1 - 2r cos 0 + r2)p/2 = 2 10 (1- 2r cos 0 + r2)p/2'
Thus by Lemma 1.12.1,
l 1r d8 r dO ( [1-r r) _~ 11 - rei9 1P ~ c 10 «1 - r)2 + 92)p/2 = C 10 + 11-r .
Estimating these two integrals, we get
[l-r dO [l-r d() 1
10 «1 - r)2 + (}2)p/2 ~ 10 «1 - r)2)p/2 = (1 - r)p-l
and r d9 r dO 1 C
11- r «1 - r)2 + (}2)P/2 ~ 11-r (02)p/2 = C + (1 - r)p-l ~ (1 - r)p-l .
o
40 1. PRELIMINARIES
o LEMMA 1.12.3. There are constants C1,C2 > 0 independent ofr E (~, 1) such
that 1 171" 1 1
cllog 1- r ~ -71" 11- reiBl dB ~ C2log 1- r'
PROOF. From Lemma 1.12.1,
1 71" d() 171" 1 1'11" d() Cl ~ I . '81 dB ~ Cl •
o v'(1 - r)2 + ()2 -71" 1 - ret 0 v'(1 - r)2 + ()2
By integrating, we get
o V(1- r)2 +()2
CHAPTER 2
The Cauchy transform as a function
2.1. General properties of Cauchy integrals
For IL EM, the analytic function
(2.1.1) (KIL)(Z) := J ~dlL(() 1- (z
on ID> is called the Cauchy transform of IL and the set of functions
X:={KIL:ILEM}
co
where
ji(n):= J (dlL(() , nEZ,
are the Fourier coefficients of the measure IL. From the elementary inequality
lji(n) I ~ IIILII. we can say the following.
PROPOSITION 2.1.3. The Taylor coefficients of a Cauchy transform are bounded.
Having bounded Taylor coefficients does not automatically gain one entrance into the space of Cauchy transforms. One need only consider the following theorem of Littlewood [65, p. 228]: If (an)n~l is a sequence of complex numbers such that
co
lim lan l1/ n = 1 and "lanl2 = 00, n~oo ~ n=O
then for cJ.most every choice of signs (€n)n~O, the analytic function on ID> defined by
00
41
42 2. THE CAUCHY TRANSFORM AS A FUNCTION
does not have radial limits on a set of full measure on T.1 We will see momentarily (Theorem 2.1.10) that a Cauchy transform must have radial limits almost every­ where. From here, one can create an analytic function on II} with bounded Taylor coefficients that is not a Cauchy transform.
DEFINITION 2.1.4. For a fixed I E X, let
Rf := {p. EM: 1= Kp.}
be the set of measures that represent f. Observe that Rf is always an infinite set. To see this, notice that if t/> E HJ =
{f E H1 : 1(0) = O}, then
¢(n) = !"( t/>«() dm«() = 0 'v'n E No,
and so 00 .... 00
K (dp. + 4)dm) (z) = L (J1(n) + 4)(n)) zn = L J1(n)zn = (Kp.)(z). n=O n=O
Thus P. E Rf :::::} dp. + 4)dm E Rf 'v',p E HJ,
making Rf an infinite set. We leave it to the reader to use the F. and M. Riesz theorem (Theorem 1.9.7)
to prove the following proposition.
PROPOSITION 2.1.5. Let lEX. (1) Kp. == 0 il and only i/dp. = 4)dm lor some,p E HJ. (2) For p., v E Rj, dp. - dv = 4)dm for some t/> E HJ. (3) If p., v E Rj, then P.s = vs •2
REMARK 2.1.6. Using (2) above we have an equivalence relation on the space of measures M and each element of X corresponds to a coset in M. We will discuss this further in Chapter 4.
Let us say a few words about the boundary behavior of a Cauchy transform. A simple estimate shows that K p. satisfies the growth condition
(2.1.7) I(Kp.)(z)1 ::;; 11~:~1' This follows from the inequalities
! 1 ! 1 1Ip.1I I(Kp.)(z)l::;; Il_(zldIJLI«()::;; 1_lzl dIJL1 «()::;; 1-lzl'
For any ( E T, observe that
(1- r)(Kp.)(r() = ! /_-~~(dP.(~)' A routine exercise using the dominated convergence theorem will show that
(2.1.8) lim (1- r)(Kp.)(r() = JL({(}). r-+l-
IThere is a rich history of such types of functions. See [234, p. 380] and [109]. 2Recall that p.. is the singular part, with respect to m, of p. (see Theorem 1.3.9).
2.1. GENERAL PROPERTIES OF CAUCHY INTEGRALS 43
Thus lim I (KIL) (r() I = 00
r-l-
whenever JL( { (}) f 0, which can indeed be a dense subset of 1'. In fact, Poincare noticed the poor behavior of Cauchy transforms of certain discrete measures back in 1883 when he observed that the analytic function defined by the series
00
(2.1.9) fez) = ~ 1-~nZ' where (en)n~ 1 is an absolutely summable sequence of non-zero numbers and «(n)n~l is a sequence of distinct points that are dense in T, does not have an analytic continuation across any portion of the unit circle. Observe that the above example of Poincare is the Cauchy transform of the discrete measure
00
where 15,,, is the unit point mass at (n.3
Despite the fact that for certain measures IL,
lim I(KIL)(r()1 = 00 r_l-
for ( in some dense subset of T, this pathological set must be of Lebesgue measure zero. Indeed, there is some regularity in the boundary behavior of the Cauchy transform. Recall from Chapter 1 the definition and basic properties of the classical Hardy space HP (0 < p < (0) of analytic functions f on the unit disk for which
IIfllp:= {sup [If(r()lPdm(()}l/P < 00. o<r<liT
For example, by TheOrem 1.9.4, functions f E HP have radial boundary values
f«):= lim f(r() r ..... l-
for almost every ( E l' and
IIfll~ = f If«()lPdm«) = lim f If(r()lPdm«(). J.r r_l- iT THEOREM 2.1.10 (Smimov). If JL E M, then
KJLE n HP O<p<l
and moreover,
where
3Poincare's example In eq.(2.1.9) is more general than what we stated here. He proved, using a different method, since the Lebesgue theory was not available to him, the same non­ continuability result with the circle replaced by a curve bounding a convex set in the plane [1611. In fact, there is quite a large literature on creating analytic functions on ~ which have all sorts of pathological properties near the boundary. Several representative examples are [25, 126, 127].
44 2. THE CAUCHY TRANSFORM AS A FUNCTION
PROOF. Using the Jordan decomposition to write /-L E M as
/-L = (/-L1 - /-L2) + i(/-L3 - /-L4), ~j EM;,
and noting that
lR(K/-Lj)(z) = 11- (z12 d/-Lj«() > 0, zED,
the result follows from four applications of the following standard fact [79, p. 114]. o
LEMMA 2.1.11. Let F be analytic on D with lRF > O. Then for all 0 < r < 1 and 0 <p< 1,
Moreover,
Ap=OC~p)' p-tl-.
, ~
For 0 <p < 1, lR(FP) = IFIP cos(pq,) ~ IFIJI cos(pTr /2).
We conclude that
fo21r IF(rei9)IJld9 ~ Ap fo21r lR(FP(rei9»d8 = ApR(FP(O».
The last equality follows from the mean-value property of harmonic functions. The desired inequality follows from the observation that R(FP(O» ~ IF(O)IP, Finally notice that
o COROLLARY 2.1.12. If f E X, then the non-tangential limit of f exists and is
finite for almost eveT1J ( E 'f.
PROOF. Since X C HP for all 0 < p < 1 (Theorem 2.1.10), the result follows from the existeDce of non-tangential limits of HP functions (Theorem 1.9.4). 0
Observe that the containment
X~ n lfP O<p<l
is strict since one can cheek, by using the estimate in Lemma 1.12.1, that the function
fez) = log (_1_) _1_ l-z 1-z
2.1. GENERAL PROPERTIES OF CAUCHY INTEGRALS 45
belongs to HP for all 0 < p < 1. However, f does not satisfy the necessary growth condition
0, If(z)1 ~ 1 -Izl' zED,
in eq.(2.1.7) to be a Cauchy transform. One can also see that f is not a Cauchy transform by using Proposition 2.1.3 and the observation that
00 (n 1) I(z) = ~ zn (; k '
and hence has unbounded Taylor coefficients.
PROPOSITION 2.1.13. If f is analytic on JI)) and ~f > 0, then lEX.
PROOF. Without loss of generality, assume that 1(0) > O. If this is not the case, replace I by 9 = I -i~f(O). If we can show 9 = KJ1., then I = K(i~/(O)dm+dJ1.).
With the assumption that f(O) > 0, we can apply Herglotz's theorem (Theo­ rem 1.8.9) to see that
I(z) = J ~~: dJ1.«()
for some J1. E M+. A little algebra shows that
and so f = K(2J1. - m).
(+z 1 -- =2--_--1 (- z 1- (z
D
We will see in Theorem 5.6.3 that if I is analytic on D and C \ I(D) contains two oppositely oriented half-lines, then lEX.
REMARK 2.1.14. Theorem 2.1.10 is due to Smirnov [200] (see also [65, p. 39]). In Proposition 3.7.1, we will begin to look at the 'best' constant Cp in the inequality
IIKJ1.llp ~ CpIlJ1.II·
Smirnov's theorem yields the estimate