The Cauchy Transform (Mathematical Surveys and Monographs 125)

(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) to

m(IKp.1 > Y) = o(l/y)· Tsereteli proved the converse, namely, m(lKp.l > y) = o(I/Y)

¢:>

p. «m.

OVERVIEW

6

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

1/-+00

> y) = I1PslI,

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

(Qp)(ei8 ) =

P.V.! (0; t) dp(e cot

and the Hilbert transform

(!Kp)(x) =

p.v.l-

1-

J[l

x-t

it )

dp(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

g(x):=L~' j=1

x - aJ

aj

E

JR,

Cj

> 0,

which is just the Hilbert transform of the positive discrete measure n

p:= LCjda;, j=1

then 1 '!-

LCj,

ml({x E JR: g(x) > y}) = 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. According 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 are of the form (7J H2) l. . The functions

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 U0 mentioned above is the following: let Su be the compression 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 U0 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.), which maps the reproducing kernel

k>.(z)

:= 1 -19(307J(z)

1- AZ

for (iJH2)1. to the function

OVERVIEW

8

in L2 (ua) and extends by linearity and continuity. Clark uses this unitary equivalence, 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») a-D(z)

= [1 -lzl2 dua «(), JT\(-zI 2

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 U a is carried by the set {(. E T: D«() a} and hence U a ..1. m. Furthermore, U a .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)

produces the following formula for ~

: L2(ua } -+ (DH2).!. I

in terms of the 'normalized' Cauchy transform

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 Ua are covered in Chapter 8. Clark measures, as well as Clark's theorem and Poltoratski's weak-type

OVERVIEW

9

theorem

> y). m = j.ts, weak-*,

lim 7rym(IKj.t1

y-oo

are covered in Chapter 9. Poltoratski's theorems on the normalized Cauchy transform 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 dj.t(t) JJR t-z' a close cousin to the Cauchy transform, comes into play.

r

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 transforms 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 capacity 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 perturbation 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 papers [32, 33J are a good place to start. There is also a notion of fractional Cauchy transforms [131J.

CHAPTER 1

Preliminaries 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. • C (complex numbers) • C= C U {oo} (Riemann sphere) • R (real numbers) .1IJ)={zEC:\zi • N = {1,2, .. ·} • No = {O, 1, 2, ... } • Z={· .. -2,-1,O,1,2, .. ·} • When defining functions, sets, operators, etc., we will often use the notation 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 (topologically) linear manifold. 1.2. Lebesgue spaces 1

A non-empty family

:r of subsets of C:= C U {oo} is called an algebra if AuB

E:r

for all

A, B

E

9"

and

C\ A E 9" for all An algebra 9" is called a a-algebra if

A E 9".

00

UAn 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

11/1100 := inf {y ~ 0 : mOIl> y) = O} .

(1.2.1)

As is customary, we equate two measurable n,.nctions that are equal almost every" where. Holder's inequality 1 1 Ilgl dm ~ 1I/1I,,1I9I1q , p ~ 1, -p + -q = 1,

i

Minkowski's inequality

III + gil" ~ as well as the associated inequality

Ilfll" + 11911", p ~ 1,

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 tinuous linear functionall : V - C takes the form

l(f) = for some unique 9 E Lq (1/p+ l/q (1.2.2)

sup

~

1, every con-

if9dm

= 1). Moreover, the identity

{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 references for this are [85, 229]. For I E LO, the function AI: [0, (0)

(1.2.3)

-+

Af(Y):= m(1/1 > V),

[0,1],

is called the distribution function for I and certainly plays an important role in analysis and probability. One can see that Af is a decreasing right-continuous function on [0, (0). There are also the following LP results. PROPOSITION

(1)

1.2.4. For p

11/11: = p 10

00

> 0,

yP-l Af(Y) dy.

(2) Af(Y) ~ y-PII/II~· For

I E LO, the function

(1.2.5)

r :[0,1]

-+

!,,(x) := inf{y

(0,00),

> 0: Af(Y)

is called the decreasing rearrangement of One can check that if

I.

~ x}

If Af is one-to-one, then

f* is A,l.

n

I(() = L)jXAj((), j=l

where the Aj's are pairwise disjoint measurable subsets of 'lr and

b1 > b2 > ... > bn , then n

AJ(X) =

L B jX[b;+l,b )(X) j

j=O

and n

!,,(x) = LbjX[B;_l,B;)(x), j=l

where j

B j :=

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) at least for I that

= ml(f* > x) and ml is Lebesgue measure on R. ~

The second is that, -+ [0,1] so

0, there is a measure preserving transformation h : ']['

(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.

1. PRELIMINARIES

14

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 {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

1/l(A;)1 : {A 1 ,··· , 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 a unique J.L EM such that f = fIJ' where

tJ.l(f) :=

Moreover, II f J.lII

= sup

{II I

E

C(1l') * . Then there is

1

I dJ.L.

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 absolutely 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):= {(e is : -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-tO+

~t().

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

exists lor m-a. e. (

1.3.8 (Lebesgue differentiation theorem). For J.L E 1l' and

(DJ.L)() =

:~ ()

m-a.e.

E

M, (DJ.L)(()

1. PRELIMINARIES

16

THEOREM 1.3.9 (Lebesgue decomposition theorem). Any I-" E M can be decomposed uniquely as

where 1-"0,,1-"8 E M W'ith 1-"0,

«

1-"=1-"0,+1-"8' 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 generalization 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) =

LI

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)

UFn

> O} =

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 11'-11 := sup {ll(x)1 : x E X, IIxll ~ I}.

(1.4.1)

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 sup{ 1I11l : l E :f} <

00.

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).

1. PRELIMINARIES

18

For W C X, define the polar of W to be the set

WO := For Y

c

{i EX* :

sup li(x)1 :l:EW ""\

~ I}.

.

X* define the pre-polar of Y to be the set

0y:=

{x

EX: supll(x)1 lEY

~ I}.

For V C X (or X*) the convex hull of V is the set

{ tc;v;: v;

E

;=1

V,c;;;;= Q,Ec;

;=1

=

I}.

The convex balanced hull of V is the set

{t

CjVj :

v;

;=1

EV, c; EC, t

Ie; I

;=1

~ I} .

Here are some important facts about polars. PROPOSITION 1.4.5.

c Wi; (2) IfY1 eYe X*, then °Y C °Y1 i (3) IfW c X, then O(WO) is the closure of the (1) IfW1 eWe X, then WO

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

Furthermore, there is a

80

w, IIwll ~ I} = distel, w.L). that

1I.e +
= 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 {x k=l n

EX: lik(X-xo)1

< €} c U.

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(x n ) -. 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 topological 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 principle (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. 1.4.'i1. For a Banach space X, the following are equivalent. X is reflexive. X* is reflexive. Every subspace of X is reflexive. Every quotient space of X is reflexive. The closed unit ball {x EX: IIxll ~ 1} is compact in the weak topology.

THEOREM

(1) (2) (3) (4) (5)

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 where Ia

:= {e it :

xa«) := XI.. «), 0 < a < 271", 0 < t < a} and use the previous proposition.

A few results relating separability and reflexivity are the following. 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.

PROPOSITION

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 bounded if and only if its graph

{(x, Ax) : x

E

--+

}.I is

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. 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)

PROPOSITION

If l E }.I* and A : X map A* :}.I* ...... X*, by

--+

}.I is bounded, then loA E x* and this induces a linear

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 conjugate transpose of A. ) If Jelt Je2 are Hilbert spaces, we say a bounded linear operator U : Je 1 is isometric if

--+

Je2

IIUXIl:K2 = IIxlbc1 Vx E Je 1 • We say that U is unitary if UJel = Je2. Notice that a unitary operator U satisfies (Ux, UY}:K 2 = (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 is a unitary U : Je 1 --+ !J{2 such that

--+

Je2 are unitarily equivalent if there

A=U*BU. An operator A : Je vector) such that

--+

Je is cyclic if there is a vector v

E

Je (called the cyclic

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) if and only if the operator

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)

-+

(Mxf)(x) = xf(x).

L2(p,),

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 separable 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 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. X{z}

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-* convergence 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. PROPOSITION 1.6.2. If (fLn)n;:'l eM converges to J-L weak-*, then

< 00

sup IIJ-Lnll n

and IIJ-LII ~ lim IIJ-Ln II· n--+oo

PROOF. By the Principle of Uniform Boundedness, we know that sup IIfLn II n

< 00.

Let n--+oo

and choose a subsequence (J-Lnk)k~l so that 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 IIJ-LII =sup{lf gdJ-L1 :gEball(G(T))}, there is agE ball(G(T)) such that IIJ-LIIBut since J-Lnk

-->

E

<

If

gdJ-Ll·

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 and so for all k

~

9 dfLnk

I ~ IlfLnk II

K,

The result now follows.

o

The Banach-Alaoglu theorem (Theorem 1.4.8) in the setting (M, *) takes the following form.

24

1.

PRELIMINARIES

THEOREM 1.6.3 (Banach-Alaoglu theorem). The closed 'Unit ball := {p; EM: IIp;II ~ I}

ball(M)

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(" h

dp;() ,

nEil,

be the sequence of Fourier coefficients of p;. When dJL = Idm, we write

I()( dm()

f(n):=

for the Fourier coefficients of an L1 function N-partial sum

I.

Also define, for N E No, the

N

L

SN(p;)():=

/l(k)(k

k=-N

and the Cesaro sum (1.6.4) When dp; = Idm, we let uN(f) := uN(fdm). 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, is the unit point mass, that is to say, the measure on T such that

o(

o((A) =

{I,0,

E

~f ( Aj 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

{6, : ( is weak-* dense in the ball

0/

E 1'}

0/ M.

PROOF. If Y = {6, : (E 1'}, one can easily show that oy = ball(C(1'» and so = ball(M). Now use Theorem 1.4.10. 0

(oy)O

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-

For ( E l' and a

(1.7.1)

> 1, let

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)

z-,

= A,

1. PRELIMINARIES

26

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

I(z)

-+

A as z

I is a bounded analytit; function on D. and along some arc lying in)[Jl and terminating at," E T, then

1.7.3 (Lindelof). If -+ (

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 analytic on D and L lim I(z) = 0

I

is

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

h

log IF(O)I =

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!' where b is a Blaschke product

b(z)

=

zm

IT lanl an - z n=l

an 1- anz

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)

where

~ E

= exp ( -

/

~ ~ ; d~(()) ,

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 (a n )n;;:'l. A necessary and sufficient condition that b and all its partial products have nontangential limits of modulus one at ( is that

~ l-Ia n l

~ I( -

ani

<

00.

Ahern and Clark [2, 3] refine Frostman's theorem and extend it to general inner functions. THEOREM

with 1'({(})

1.7.8 (Ahern and Clark). Suppose that {} = bsj.! is inner and ( The following are equivalent.

= O.

E ']f

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

~ 1(-a,,1 +

/

dp(e)

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,

z-,

L lim rfJ(z) - 'f/

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 automatically implies that L lim t/J(z) = 'f/

z-+,

=

and that I'f/I 1. The following result is the key to understanding angular derivatives. A proof can be found in [6, til, 196}. THEOREM 1.7.10 (Ju1i&-Carathoodory). For an analytic function t/J : 0 and (" E T the following statements are equivalent. (1) lim 1 - IrfJ(z)1 = 6 < 00, 1 -lzl

-t

D

z-+,

(2)

z_,

L lim t/J(z) - 'f/ = t/J'«() z- ( exists for some 'f/ E T,

(3) L lim t/J'(z)

*.....,

exists and

z_,

L lim rfJ(z)

= 'f/ E 1'.

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. l-lzl

z ..... ,

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 PROPOSITION

29

1.7.11. If cPl, cP2 are analytic self maps of If} and cP = cP2cP2, then = lcP~(()1

IcP'(()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 and ( E 1l', then b has a finite angular derivative at ( if and only if

(an)n~l

~

1-lanl2 ~ I( - anl2 < 00. 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

JIe , JIe _ (1 ISI'(()I dfJ.(e) (12 <

Moreover,

= 2

00.

dfJ.(e) 2

< 00.

If fJ.({(}) > 0, then the above integral diverges and so Sl' will not have an angular derivative at (. In this case, IslJ.(r() I -+ 0 as r -+ 1- and so slJ. cannot possibly have a finite angular derivative. COROLLARY 1.7.14. An inner function {) at ( E 1[' if and only if

= bsl' has a finite angular derivative

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.

1. PRELIMINARIES

30

THEOREM 1.7.17 (Ahern and Clark [3, 4]). An analytic self map 4J ofD, factored as in eq.(1. 7.16), has a finite angular derivative at (E T if and only if

1-

J

lanl 2

dv(~)

~ 1(-an I2 +2 1{-(12 00

<

00.

Moreover,

1.S. Poisson and conjugate Poisson integrals Define the Poisson kernel

Pz «()

:=

(+z)

~ ( (_ z

l-lzl2 = I( _ z12'

(E T,

zED

2~«(z),

(E T,

zED.

and conjugate Poisson kernel

Qz(() For fixed (

E

:=

~ ( ( + z) (- z

=

I( - zl2

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

(Pp,)(z):=

(1.8.1)

and the conjugate Poisson integral

(Qp,)(z):=

(1.8.2)

J J

Pz «() dp,(()

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,

1 + s~ = 2_1_ _ 1 = 2 ~ snf"n _ 1 l-s~ I-sf" ~ ,

and so

~ (~ ~ :~) = n~oo slnlen where

and 9

(~ ~ :~) = if n < OJ ifn = OJ 1, ifn > O.

-1, 0,

-i n;oo

sgn(n)slnl~n,

1.8. POISSON AND CONJUGATE POISSON INTEGRALS

31

Thus 00

L

(Pp,)(r() =

iL(n)rlnl(n

n=-oo

(1.8.4)

00

(Qp,)(r()

= -i

L

iL(n)sgn(n)rlnl(n,

n=-OO

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-.

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-

1.8.7. (1) From Proposition 1.3.8, Dp, = dp,/dm m-almost everywhere and so the radial 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,

REMARK

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()

r->l-

~

lim (Pp,)(r()

~

(Dp,)«().

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.

1. PRELIMINARIES

32

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. 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+.

THEOREM

From Fatou's theorem (Theorem 1.8.6), we know that PJ.I. has finite nontangential 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.)(e i6 ) 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 L 1 (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 lim (QJ.I.)(re''6 ) = P.v.

r-+l-

127r cot (()-2- t) dJ.l.(e

:= lim E-+O+

0

1,

16-tl~E

t't }

(() - t) .

cot -2- dJ.l.(e't).

for m-a. e. ei6 . 1.9. The classical Hardy spaces For 0 < p < 00, let HP, the Hardy space?, denote the space of functions analytic on D for which the £P integral means

(1.9.1) remain bounded as r

Mp(r; f) :=

i

t£

If(r()IP dm«()

f

riP

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)

is increasing on the interval [0,1), that is,

(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
~p

defines a norm on HP when 1

rtl-

< 00.

When 0

dist(f,g) :=

< p < 1, the quantity

III - gll':ip

defines a translation invariant metric on HP. The pointwise estimate

I/(z)1

(1.9.3)

~ 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 C1-+ I(C) belongs to LP and when o

r-l- iT

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 measure in T. (4) Ifp ~ 1, and f E HP has Taylor series

01 positive

co

f(z):::: Lanzn , n=O

then an

= ( f«()"f dm(C), iT

n E No.

.

(5) For 0 < p < 00, the polynomials are dense in HP. When p :::: polynomials are weak-* dense in Hoo .

00,

the

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

1.

34

PRELIMINARIES

for the HP norm of f, or equivalently the V norm of the boundary function f«,). Throughout this book we will use the following important fact. PROPOSITION 1.9.5 (Smirnov). If 0

valu.es, then

f e yq ..

< p < q and f

E

C1-+

HP has Lq boundary

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

the Fourier

~oefficients

f E V belongs to HP(1') if and only if

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 (,"

Then dJL

dJL«') = 0 whenever n E No.

=
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

(1.9.1O)

'F(z)

(i.e., F has nO,inner factor) can be expressed as

= 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 0h l . 11&2 0 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

r ..... 1-

1Tf log+ I/(r()1 dm«() = J.fr log+ If«()1 dm«().

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 We say

I

(t),

y --+

(t),

y

00.

E L~'oo if

m(1/1

> y)

=

0

--+ 00.

Define the quasi-norms

II/IIL1.OO

:= supym(1/1 y>O

> y).

Let H 1,00 be the analytic functions on 1Dl for which

1I/I1Hl,OO:=

sup

O
IIlrllLl.oo < 00, Ir«() = I(r().

PROPOSITION 1.10.1.

H1,oo

c

n

HP.

O
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 See [111J for more on quasi-norms.

II! + gil " 2(11/11 + IIglI)·

III + gil "

""1 + IIgll but does satisfy

1. PRELIMINARIES

36

(Proposition 1.2.4), tha.t for

100 lA ~ lA

. II/rll: = p

= P

P

I

E

Hl,oo and A :;:: II/rIlLl.oa,

yP-l m(l/rl > y) dy yp-1 m(l/rl yP- 1dy

> y) dy + P £00 yP-2 y m(l/rl > y) dy

+ pA

L

oo 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
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 characteriza.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«(,) = rlim I (r(.) , -lE Hl,oo for which liT E L~'oo. Recall Theorem 1.9.12 which says that if I E N+ (the Smimov class) and liT (the nODtangential boundary values of f) belongs to V, then I E HI'. HElre is the corresponding result for the analytic weak.type spaces.

we can define HJ'oo to be those

I

THEOREM 1.10.4. II I E N+ and IE Hl,oo (respectively I E H~'OO).

liT

E

L1,oo (respectively I E L~'oo), then

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

II ~a 1a-~. az

B(z):=

aE E

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)

s(E):= inf{p(a,b): a,b E E and a"# b} > 0,

where p(a, b) :=

la-bl

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 H oo 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}.

1. PRELIMINARIES

38

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):

(1.11.6)

1 6(E) ~ C(E) ~

(1.11.7)

s(E) ~ c5(E),

where Cl, C2, C3

> 0 are absolute constants.

Cl

'Y(E) 6(E) ~

1

C2 c5(E)

1 s(E) ~ C(E) ,

(

1 ) 1 + log 6(E) ,

6(E) ~ exp

(

'Y(E) )

-C3 s(E)2

'

Interpolation sequences actually exist [101, p. 203]. THEOREM 1.11.8 (Hayman-Newman). A sequence (Zn)n)1 C D such that

sup

IZn+11 : EN} < 1 {I-l-IZnI n

is an interpolating sequence. COROLLARY 1.11.9. If (rn)n)l C (0,1) with rn polating sequence if and only if

sup {

1- rn+1 } 1 _ r.. : n E N

i

1, then (rn )n)l is an inter-

< 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 - re i9 1= (1- 2r cos 8 + r2)1/2

= «1- r)2 + 4rsin2 (8/2»1/2.

Using the estimate

!!.11" ~ sin(9j2) ~ 9

V9 E [0,1I"j,

we get 9: ~ sin2(9/2) ~ 82 V8 E [0,11"]. 11" . Hence we obtain constants Cl, C2 > so that

°

Cl (1- r)2

+(

~ 1- 2rcosO + r2 ~ C2 (1- r)2

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

~

dO ~ 1_~ -;----·;-;;"9:11- ret Ip

c -;-:---:---:(1- r)p-l

for all r E (!, 1). PROOF. Observe that

1_~~ 11 - d(}rei9 p = 1~_~ (1 1

Thus by Lemma 1.12.1,

l_~ 11 1r

d8 re i9 1P

dO 2r cos 0 + r 2)p/2

r

r

= 2 10

( [1-r

dO

~ c 10 «1 - r)2 + 92)p/2

d9

(1- 2r cos 0 + r 2)p/2'

=

C

10

r) .

+ 11-r

Estimating these two integrals, we get

[l-r

10 and

r

dO «1 - r)2 + (}2)p/2

r

[l-r

~ 10

d() «1 - r)2)p/2

1

= (1 -

r)p-l

d9 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

1. PRELIMINARIES

40

o 1.12.3. There are constants C1,C2 > 0 independent ofr E (~, 1) such

LEMMA

that 1 cllog 1- r ~

From Lemma 1.12.1,

PROOF.

71"

Cl

171" 1 1 -71" 11- reiBl dB ~ C2log 1- r'

1o

d()

v'(1 - r)2 + ()2 By integrating, we get

1 71"

~

171" 1 1'11" . '81 dB ~ Cl 0 -71" I1 - ret

d()

v'(1 - r)2 + ()2

•

d()

--;:;:=~=:;<: = log (71' + V7r 2 + (1- r)2) -log(1- r),

o V(1- r)2 +()2

and the estimate follows.

o

CHAPTER 2

The Cauchy transform as a function 2.1. General properties of Cauchy integrals For IL EM, the analytic function

(KIL)(Z) :=

(2.1.1)

J~dlL(() 1- (z

on ID> is called the Cauchy transform of IL and the set of functions

X:={KIL:ILEM} is called the space of Cauchy transforms. Note that

and so K IL has the power series expansion co

(KIL)(Z) =

(2.1.2)

L ji(n)zn n=O

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

n~oo

la n l1/ n

=1

2 and "lanl ~

= 00,

n=O

then for cJ.most every choice of signs (€n)n~O, the analytic function on ID> defined by 00

f(z) ::;:

L €nan Zn n=O

41

2. THE CAUCHY TRANSFORM AS A FUNCTION

42

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 everywhere. 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:

be the set of measures that represent

1= Kp.}

f.

Observe that Rf is always an infinite set. To see this, notice that if t/> E

{f E H1 : 1(0) = O}, then

¢(n)

=

!"(

t/>«() dm«()

=0

HJ =

'v'n E No,

and so 00

K (dp. + 4)dm) (z) =

....

00

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 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. REMARK

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 ::;;

This follows from the inequalities

I(Kp.)(z)l::;;

!

1

Il_(zldIJLI«()::;;

For any ( E T, observe that

(1- r)(Kp.)(r() =

11~:~1'

! 1_lzl 1

dIJL1 «()::;;

1Ip.1I 1-lzl'

! /_-~~(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 =

r-l-

00

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

dJL

=L

en 15,,, ,

n=l

where 15,,, is the unit point mass at (n. 3 Despite the fact that for certain measures IL, lim I(KIL)(r()1 =

r_l-

00

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
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~ = J.f

r

If«()lPdm«)

= r_llim f If(r()lPdm«(). iT

THEOREM 2.1.10 (Smimov). If JL E M, then

KJLE

n

HP

O
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 noncontinuability 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 = (/-L1 - /-L2) + i(/-L3 - /-L4),

/-L

E M as

~j EM;,

and noting that

lR(K/-Lj)(z) = /

1- ~('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 and 0
> O. Then for all 0 < r < 1

Moreover,

Ap=OC~p)'

p-tl-.

PROOF. Since 1RF> 0, then F = IFleit/J, where -Tr/2 < t/> < Tr/2. Since F has no zeros in the disk, the function FP (the branch which has positive real part at the origin) is also analytic on lD and ,

~

FP =

IFIP (cos~l + i sin(ptJ») .

For 0

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
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

GENERAL PROPERTIES OF CAUCHY INTEGRALS

2.1.

belongs to HP for all 0 condition

< p < 1.

However,

f

does not satisfy the necessary growth

0,

If(z)1 ~ 1 -Izl'

zED,

in eq.(2.1.7) to be a Cauchy transform. One can also see that transform by using Proposition 2.1.3 and the observation that

I(z) =

45

f

is not a Cauchy

(n 1)

~ zn (; k ' 00

and hence has unbounded Taylor coefficients. ~f

PROPOSITION 2.1.13. If f is analytic on JI)) and

> 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 (Theorem 1.8.9) to see that

I(z)

=

J~~:

dJ1.«()

for some J1. E M+. A little algebra shows that

(+z 1 - - =2--_--1 (- z 1- (z and so f

= K(2J1. -

m).

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

IIKJ1.lIp =

0(1 ~ p)' p

--+

r.

For certain measures, we can do a bit better. PROPOSITION 2.1.15. II J1.

«

m, then

IIKJ1.lIp=O(I~P)' PROOF. Let dl'

= 9 dm for some 9 E Ll. -1

h(() = L n=-N

p--+r.

For

E

> 0 given,

let

N

h(n)C

+ Lh(n)(n,

(E 'lI',

n=O

be a trigonometric polynomial with IIg-hll l < E (an appropriate Cesaro polynomial of 9 will work - see Theorem 1.6.5). Observe that N

K(hdm)(z) = Lh(n)zn. n=O

2. THE CAUCHY TRANSFORM AS A FUNCTION

46

Thus, N

P

IIK(hdm)ll~ = ~h(n)zn n=O

P P

I

N

(

~lh(n)12

~ ~h(n)Zn N

=

•

2 ) P/ 2

~ IIhll~ ~ IIhll~

and so certainly

1 = 0(-1-), -p Using Smirnov's theorem (Theorem 2.1.10), IIK(hdm)lI~

p

-+

1-.

1 IIK(hdm) - K(gdm)lI: ~ 01--lIhdm - gdmll P

-p 1

=0 1- pllh - gill ~

Finally, Letting

€ -+

1 0-1-€P, -p

(1 - p)IIK(gdm)lI~ ~ 0(1) + O€p. 0 yields the result.

o

2.2. Cauchy integrals and HI, The classical Cauchy integral formula4 says that if I is analytic in a neighborhood of 0-, then

I(z) = ~ 1 1«(,) d(,. 21rt .11'1=1 (, - z Making the observation that d(, 211'i(, = dm«(,) ,

we will write the Cauchy integral formula

I(z)

=f

88

1('2 dm«(,).

JT 1- (,z,

. The question now is: what is the 'largest', of analytic functions on III that can be written via the Cauchy integral formula? For I E HI, Theorem 1.9.4 says that

class

1(') == r-+llim I(r(,) exists for almost every ( E or and defines an integrable 'function. Thus for Hl functions, the integral on the right. hand' side 'of the Cauchy integrSJ. formula makes sense. It turns out that the left-hand side is equal to the right-hand side. 4See [201] for a historical overview of Ca""chy.

2.2.

CAUCHY INTEGRALS AND

HI

47

PROPOSITION 2.2.1 (Cauchy integral formula). For I E HI,

I(z) that is to say,

J.f 1- (z

dm«(),

1= K{fdm).

PROOF. For 0 slightly larger disk

< r < 1, let Ir(z)

= I(rz) and note that Ir is analytic on the By the classical Cauchy integral formula,

{Iz\ < l/r}.

J

Ir(z) = Note that Ir

1«(1

= (

--+

Ir('.) dm«().

1- (z

I in L1 (Theorem 1.9.4) and so for each z E ID>,

J

Ir('.) dm«)

J 1«(1

--+

1- (z

1- (z

dm«) as r -d-.

Clearly Ir --+ I pointwise in ID>. Combine these two limits to obtain the Cauchy integral formula. 0 By Holder's inequality, HP C HI for all p ~ 1, and so we can combine Theorem 2.1.10 and Proposition 2.2.1 to see that (2.2.2) O
p~1

We have already seen why the second containment is strict. The first containment is strict since I = (1- Z)-I = K8 I but does not belong to HI. Indeed, the boundary function l(e iO ) is 1/{1- eiO ) and 1

'0

\f(e' )\

rv

TOT'

(J

rv

0,

and hence is not integrable. From Proposition 2.2.1, every I E HI can be written as the Cauchy integral of its boundary function. This next proposition says, in a sense, that H1 functions are the only ones which can be written in this way. PROPOSITION 2.2.3. Let I be analytic on ID>. Then the lollowing two conditions hold il and only il I E HI: (1) The junction I«():= lim I(r() r--+l-

exists lor m-almost every ( E 11' and is integrable. (2) For all Z E ID>,

I(z)

=

J 1«(1

1- (z

dm«).

PROOF. If I E HI, then the two conditions hold by Theorem 1.9.4 and Proposition 2.2.1. Conversely, suppose the two conditions hold. Then I = K{fdm) E HP for all 0 < p < 1 (Theorem 2.1.10). But since the boundary function belongs to Ll, then I E HI (Proposition 1.9.5). 0

2. THE CAUCHY TRANSFORM AS A FUNCTION

48

2.3. Cauchy A-integrals In this section, we prove a generalization of the Cauchy integral formula

I(z) =

j 1(9.

1-(z

dm«)

involving the theory of A-integrals as studied by Denjoy, Titchmarsh [215], Kolmogorov, Ul'yanov [225J, and Aleksandrov [9J. A Lebesgue measurable function 9 on 11' is A-integrable if the following two conditions hold. The first is that 9 E L~'oo, that is to say,

m(lgl > y) =

(2.3.1)

o(l/y),

y

-+ 00.

The second is that

(A)jg«)dm«):= lim f g«()dm«() y.... oo J1g1
(A) j u + g)dm

= (A) 1

Idm+ (A) 1 gdm

(A) j(a:J)dm = a: (A) 1 Idm). PROOF.

For y

> 0, let I(y,() :=

{/«), 0,

if I/«)J < y; otherwIse.

and observe that

(A) 1 I«) dm«) The functions

(2.3.3)

= yl~ 1

I(y, () dm«).

U + g)(y, () and I(y, () + g(y, () equal 1(') + gee) off'the set {II + gl > y} u {III> y} u {Igl > y}.

But since

{II + gl > y} c {If I > y/2} u {Igl > y/2}, and m(I!1 > y) and m(lgl > y) are both o(l/yK the measure of the set in eq.(2.3.3) is o{l/y). Also observe that for all ( E 11',

IU + g)(y, ()I

~

y, I/(y, () + g(y, ()I ~ 2y.

Therefore,

1(1 I(y,()dm«) + 1 g(y,()dm«») - 1(J+g)(y,()dm«()1

~

f

3ydm«()

J{If+gl>Y}U{I/I>y}u{lgl>lI}

(

3y o(l/y)

-+

0

Swe thank A. Poltoratllki. for showing us this proof.

49

2.3. CAUCHY A-INTEGRALS

as Y - 00. This proves the first identity of the proposition. The second identity is obvious. 0 The growth condition in eq.{2.3.1) is essential for linearity (see Remark 2.3.18 below). For J.I. E M with J..L « m, the Cauchy transform I = K J.I. has non-tangential boundary values m-almost everywhere and, as a consequence of Kolmogorov's theorem (see Theorem 3.4.1 and Proposition 3.4.11 below), I satisfies the growth condition in eq.{2.3.1). However, this Cauchy transform may not belong to Hl (see eq.(3.2.6) below) and so cannot be recovered from its boundary function via the Cauchy integral formula. A theorem of Ul'yanov [225] is the substitute 'Cauchy A-integral formula'. THEOREM 2.3.4 (UI'yanov). For J.I. E M with J.I.

«

m, the function

I

= KJ.I. is

A-integrable and

I(z) = (A)

(2.3.5)

1(0.

(

iT 1- (z

dm{() , zED.

The version of Ul'yanov's theorem we wish to prove is a generalization due to Aleksandrov [9] concerning the weak-type class H6'00. See Chapter 1, especially Theorem 1.10.4, for a reminder of the definition of H6'00. THEOREM 2.3.6 (Aleksandrov). II IE HJ'oo then

I(~) =

1{0.

(A) (

iT 1- (z

dm{(),

zED.

PROOF. We begin our proof by showing that

I{O)

(2.3.7)

= (A)

J

Before computing

(A) Jldm

= Llim ..... oo

I dm.

1

Ifl~L

Idm,

we make some preliminary observations. Recall from eq.(1.2.3) the distribution function A: [0,00) - [0,1], A(Y) = m{1/1 > y) for I and, from eq.(1.2.5), the decreasing re-arrangement

¢: [0,1}- [0,00),

¢(x):= inf{y > 0: A{Y)

~

x}

of f. Notice that m is normalized Lebesgue measure on 'the circle and so m(T) = 1. We will also make several uses of the inequality (2.3.8)

¢(x)

=0(;) ,

X-+O+.

Indeed, a geometric argument shows that when y

x¢{x)

~

= ¢(x),

ym(¢ > y) = YA{y),

where the last equality follows from eq.(1.2.6) which states that the distribution functions for ¢ and III are the same. The estimate in eq.(2.3.8) now follows from the assumption that YA(Y) == 0(1) as Y -+ 00.

2. THE CAUCHY TRANSFORM AS A FUNCTION

50

From eq.(1.2.7), there exists a measure preserving'map h: 'll' -. [0,1] such that

4> 0 h = III ..

(2.3.9)

Since h is measure preserving, it must therefore satisfy

!

(2.3.10)

10

G(h«(» dm«() =

1

G(x) dx 1

for every G E Ll[O, 1]. For large L > 0 and our measure preserving map h, let

, {O

log ~ig

gL«):= Notice that

iT( gLee) dm«) =

if h«() ~ '\(L), if h«)

1h'~(L)

log

~ '\(L).

~«2) dm«)

which by eq.(2.3.10) equals (>'(L)

10

(2.3.11)

x

log >.(L) dx

= ->.(L).

Thus 9L satisfies -00

< gL

~ 0 and 9L ELI.

Let 9L = PgL , 9L =:= QgL be the Poisson6 and conjugate Poisson integrals of gL and note that 9L(0) = O. The function

FL

= exp(gL + i9L)

is analytic on lD, IFLI ~ 1 (since 9L ~ 0), and is outer7 • Furthermore, I FL E HI. To see this last fact, first observe that I FL E HP for all 0 < p < 1 (since I E H~'oo c HP for all 0 < p < 1 and FL is bounded) and so, by Proposition 1.9.5, it suffices to show that I FL has integrable boundary values. Using the identities eq.(2.3.9)"and eq.{2.3.1O), ,

!

IIFLI dm =

1h'~{L) I/«)I ~((2)

dm«)

+(

lh>>'(L)

If«)1 dm«()

= lh'~(L) ( 4>(h«)) ~«2) dm«) + ( 4>(h«»dm«() lh>>'(L) (>'(L)

= 10

(I , 4>(x) >.(L) dx + l~(L) 4>(x) dx. x

.

Now use eq.(2.3.8) to show that the first integral converges. The fact that 4> is decreasing shows that the second integral also ,converges. Th~s f FL E HI. 6We abuse notation a little here and identify gL on the circle with its harmonic extension to the disk via the Poisson integral. Notic.e that both functions are the same almost everywhere on ~~

70bserve

that!

log IFLI dm =

!

,

9L dm

=(P9L)(O) =log !FL(O)I (see eq.(1.9.9)).

51

2.3. CAUCHY A-INTEGRALS

Since fFL E Hi,

J

f FL dm = f(O)FdO)

(2.3.12)

= f(0)e 9dO ) = f(O)e->-"(L).

This last equality follows from the identity

=

9L(0)

J

9L dm

= -"\(L)

and eq.(2.3.11). We are now ready to compute lim

L~oo

1

fdm.

III~L

We have

1

III~L

fdm=

rfFLdm+lI/I~L f(l-FL)dm-lI/I>L fFLdm

JT

= (I)

+ (II) -

(III).

By eq.(2.3.12), the first quantity (I) is equal to

f(O)FL(O)

= I(O)e->-"(L) -

since '\(L) = o(l/L) by our assumption that For the second quantity (II); (2.3.13)

r

JI/I~L

Ifill - FLI dm

~ (1

I/r~L

f(O) as L -

f

E

00,

HJ·oo.

1/12 dm) 1/2

(1I/r~L

11 _ FLI2 dm) 1/2

Let us estimate each of the factors in eq.(2.3.13). To this end,

r

JI/I~L

Ifl2 dm

={

¢(h)2 dm (from eq.(2.3.9»

=r

¢(X)2 dx (from eq.(2.3.1O»

J4>(h)~L J¢(Z)~L

~

1

r ¢(x)2dx. J>-"(L)

This last inequality follows from the containment {x: ¢(x) ~ L} C ("\(L), 1]

(2.3.14)

which follows from the definition of ¢. Apply the distributional equality (with k ~ 0)

10

1

k(X)2dx

=

1

00

tml(k > t)dt

to the function X¢, where X = X[>-"(L).lj, to get 1

¢(x?dx= rOOtm1(X¢>t)dt. r J>-"(L) Jo

2. THE CAUCHY TRANSFORM AS A FUNCTION

From the definition of 4>, it follows that 4>(.~(L)) ~ L. But since 4> is decreasing, it also follows that x4> ~ L and so

1

00

tml(X4>

> t)dt =

lL tml(X4> > t)dt

~ fo\m 1(4) > t)dt =

foL t .x(t) dt

(byeq.(1.2.6»

~ Lsup{t.x(t): t E ~

[O,oo)} cL (since .x(t) = o(l/t), t

-+

(0).

Combine all of this to get (2.3.15)

, We will come back to this inequality in a moment. Now let us estimate the second factor in eq.(2.3.13}. indeed,

1

11-FLI2dm=1

IIIE;L

Il-e9Lei9LI2dm

IflE;L

=

1

IflE;L

'

11- ei9L 12 dm

(9L =

0 on {If I ~ L}, eq.(2.3.14))

~ ill-ei9Ll2dm

~i ~.

IgLf2 dm

hIg~12

dm

(since

11- eizl ~ Ixl)

(by Parseval and eq.(1.8.4»

=lE;L (lOg ~~2r dm«() =

fo>'(L) (lOg ~tL») 2 dx

(byeq.(2.3.1O»

= 2.x(L). Putting this together with eq.(2.3.13) Bll:d eq.(2.3.15), ~ get

1

IJI~L

which goes to zero as L

-+ 00.

Ifill -

FLI dm

~ c';L>.l,L)

I

,

.'

2.3. CAUCHY A-INTEGRALS

For quantity (III),

1f1fl >L Ifl IFL Idm =

1

If Ilog A(~) dm

Ifl>L

4>(h«()) ~«f» dm«()

=f

1q,(h»L

=

53

(byeq.(2.3.9»

4>(x) A(~) dx

f

1';(a:»L

x

. = l0 4>(x) >.(L) dx. Using the fact that 4>(x) = o(l/x), one shows that this quantity approaches zero as ~(L)

L-

00.

Thus

(2.3.16)

=

(A)jf dm

lim

1

L-oo Ifl~L

fdm

= f(O).

To prove the Cauchy A-integral formula .

fez) = (A)

f f('l

1T 1- (z

dm«(), ZED,

fix zED and assume for a moment that fez)

= O.

Apply eq.(2.3.16) to the function

= wf(w)

F(w)

w-z

to get

0= fez)

(2.3.17)

= F(O) = (A) f

1T

F{() dm{()

= (A) f f«(2 dm«(). 1T 1- (z

When fez) is not necessarily equal to zero, apply eq.(2.3.17) to the function

G(w) = few) - fez) to get

G(z) = (A) ( G(C] dm«()

1T 1- (z

= (A) f fee) = (A)

1T

fez) dm{() 1- (z

f

f«(l dm«() - j{z).

J.r 1- (z

o

The result now follows.

2.3.18. (1) The little-oh condition in eq.{2.3.1) is important in the statement of Aleksandrov's theorem (Theorem 2.3.6) and the theorem fails without it. For example, consider the function

REMARK

fez)

=i 1+ z

1-z

and observe (see Theorem 7.4.1) that

m(lfl

~ L) =

; arctan

(~)

54

2. THE CAUCHY TRANSFORM AS A FUNCTION

which is not 0(1/L) as L

-+ 00.

Furthermore, for all. (E 'Jr \ {I},

-2

I( -112 ~«() E R..

I«() = Thus the quantities

1

I«)dm«()

IJI~L

are real-valued for every L. However, 1(0) = i and hence the formula in eq.(2.3.5) is not valid for this I. (2) Theorem 2.3.4, and various other results about conjugate functions, were explored by Ul'yanov [224, 225]. For example, if IE L1, then the nontangential boundary function j of the conjugate function (Qldm) , although defined almost everywhere, need not be integrable, making the quantity (Qjdm) undefined as a Lebesgue integral. However,

(Qjdm)(z)

= (A)

I

Qz«)f«)dm«), zED.

A good treatment of this is found in {26, Vol II]. (3) See [20, 184] for other A-integral theorems. (4) The paper [771 contains an applica.tion of A-integrals to the study of real outer functions. 2.4. Fatou's jump theorem So far I we have been considering the, Cauchy transform K J1. as a function on the unit disk. However I the function

JfT ~dJ1.«) 1- (z

(CJ1.)(z):=

is analytic on C \ 'Jr with an analytic continuation across the complement of the support of J.t. Let us use the notation

(C/l)i := CJ.tID,

(CJ.t)e:= C/lIDe ,

where De := {z E C : 1 < Izi ~ oo} is the extended exterior disk. Of course one notices that (C/l)i = KJ.t. One can check, using an analog of Smirnov's theorem Theorem 2.1.10, that

(C/l)e E

n HP(lIJ)e) ,

0
where F E HP(De) means the analytic function z ~ F(l/z) on D belongs to HP. The norm of an F E HP(D e) is

JlFIIHP(D.) := IIF(l/z)IIHP' One can also prove, as before, that

II(CJ.t)eIlHP(De)=O(l~P)'

p-+1-.

This means that (CJ1.)e has (exterior) non-tangential limits almost everywhere on 1'. It is also true that

55

2.4. FATOU'S JUMP THEOREM

and that (CJ-L)e(oo) = o. One can even compute, in a similar way as in eq.(2.1.2), the Taylor series of (C J-L)e about the point at infinity as

(CJ-L)e(z)

= _

~ j1( -n).

L.J

zTl.

n=l

Define, for almost every ( E T,

(CJ-L)i«):= lim (CJ-LMr() , r~l-

(Cp.)e«):= lim (Cp.)e«jr). r~l-

Two important quantities to consider are (2.4.1) For the first, a computation shows that

(Cp.)(r() - (CJ-L)«jr) =

i Prd~)

dJ-L(e),

the Poisson integral of J-L (see eq.(1.8.1)). By Fatou's theorem (Theorem 1.8.6) we obtain the following 'jump theorem'. COROLLARY 2.4.2 (Fatou's jump theorem). For J-L E M, dJ-L (CJ-LM() - (CJ-L)e«() = dm «) m-a.e. ( E T. As we shall see later (see Theorem 5.3.1 and Theorem 5.4.5), this 'jump' theorem will help us characterize the analytic functions f on C\ 'lr which can be written in the form f = CJ-L for some J-L E M. For now though, we summarize the above ideas. PROPOSITION 2.4.3. For J-L E M, the following are equivalent. (1)

r~~- (Kp.) (r() = :~ «) m-almost everywhere. (2) (CJ-L)e == O. (3) The Poisson integral

is analytic on 11). PROOF. (1):::} (2): By Fatou's jump theorem (Corollary 2.4.2)

(CJ-LM() - (CJ-L)e«)

=

:~ «)

almost everywhere. So statement (1) implies that (CJ.L)e«) = 0 almost everywhere. By Theorem 1.9.4, we get (CJ.L)e == O. (2) :::} (3): For Z E 11), one can verify the identify

(2.4.4)

u(z) =

J

Pz«)dJ-L«) = (CJ-L)(z) - (CJ-L)(1jz).

Thus if (CJ-L)e == 0, then u = (CJ-L)i and thus is analytic. (3) :::} (1): Suppose u is analytic on D. By using the identity in eq.(2.4.4), we know that the function (CJ-L)e(1jz) is analytic on De. But since this function is also

56

2.

THE CAUCHY TRANSFORM AS A FUNCTION

conjugate analytic, it must be constant (Cauchy-Riemann equations). However, since (CtL)e(OO) = 0, this constant must be zero. Thus u = (CtL)i and so, by Fatou's jump theorem,

«) ddtL m

=

lim u(r()

1' ..... 1-

= .......lim1- (CtLMr() o

m-almost everywhere. 2.5. PlemeJj's formula

We now return to the second quantity in eq.(2.4.1), namely

This quantity is closely related to the conjugation operator. Before getting to this though, we bring in an auxiliary operator dating back to as early as 1873 with the work of Sokhotski. For tL EM, the Cauchy integral

f ~dtL(e) 1- {z is defined for all z E]I)). At least formally, replace z E ]I)) with a boundary point ( E '][' and consider the integral

f ~dtL({). 1-{( Technically this integral is not always defined since the function { 1-+ (1 _ {()-l may not belong to Ll(tL). Perhaps a more precise definition of the integral should be the principal value (or singular) integral

P.V.

f~ 1 - {(

dtL(e)

:=

lim (

E-+O+

~ dtL({)

Jl{-(I~E 1 - {(

whenever this above limit exists. This is indeed the correct one. THEOREM 2.5.1 (Privalov8). For tL E M, (2.5.2) exists for m-almost every ( E '][' and is equal to 1

2 «CtLM() + (CtL)e«»' We will give a proof of the general Plemelj formula in a moment. But for now, we give a very elementary proof in a special case. Suppose that f is analytic on 0, continuous on 0-, and satisfies the Lipschitz condition

8Theorem 2.5.1 is known as the Plemelj or the Sokhotski-Piemelj formula. Sokhotski [202J in 1813 was the first to prove this formula when dp, ,., fd(, where f is a. Lipschitz function. Plemelj [160] refined this result in 1908. The 'version here is due to Privalov as part of this thesis [168]. See also [169] where one can find a more general version of Theorem 2.5.1 for Ca.uchy integrals on general curves.

2.5. PLEMELJ'S FORMULA

for some a

57

> 0 and AI independent of the points z and w in l!)-.9 For Izl '# 1 let (Cf)(z) = ~

r f«() d(

27r~ JT ( - Z

and for

Iwl =

1 and to

> 0 let

r

~

(Cd)(w) =

de.

f«()

2n J{I'-wl~E}nT ( - w

We will show that lim (Cf)(rw)

r-+l-

+ r-+llim (Cf)(w/r)

=

2 lim (Cd)(w) E-+O+

= P.v.~ 7r~

f

f«() d(

(- w

which is a special case of the Plemelj formula. To see this, observe from the classical Cauchy integral formula that for 0 < r < 1 and Iwl = 1, we have the following two identities (Cf)(rw) = ~ f«() - f(rw) d( + f(rw) 27rt

(Cf)(w/r)

=

r

JT

(- rw

rf«() - w/r f(rw) d(.

~

JT

27rt

(-

Using the Lipschitz condition,

If«() - f(rw)1

~

AII( -

rwlQ

and the dominated convergence theorem, one can prove the identities lim (Cf)(rw) =

~ 27rt

r-+l-

lim (Cf)(w/r) =

rf«() - wf(w) d( + f(w)

JT

~

(-

r f«() -

f(w) d(.

JT (- w Notice, by the Lipschitz condition, that these two integrals are absolutely convergent. Now observe that 27rt

r-+l-

(Cd)(w)

=~ 27rt

r

J{I'-wl~E}nT

r

f«() - f(w) d( + f(w.) ~. ( - w 27rt J{IC-wl=E}nD ( - w

Again using the Lipschitz condition and the dominated convergence theorem, we see that lim (Cd)(w) = ~ f«() - f(w) d( + f(w). E-+O+ 27rt T 2 ( - w Combine this with the two identities above to get Plemelj's formula.

r

J

To prove the Plemelj formula for the general case, use the power series expansions 00

(CJLMr() =

L jL(n)rn(n, n=O

00

(CJL)e«(/r)

=- L

n

jL( -n) ~n'

n=l

and eq.(1.8.4) to get

(CJLMr()

+ (Cf.L)e«(/r) = i(Qf.L)(r() + JL(T).

9A theorem of Hardy and Littlewood [82] says that the above condition is equivalent to being continuous on][)- and 1/«1) - /«2)1 ~ A/I(l - (21" for all (1,(2 E T.

1

2. THE CAUCHY TRANSFORM AS A FUNCTION

58

Thus for m-almost every (, (2.5.3) where, by Theorem 1.8.10, (Q",)«) is given by the principal value integral

P.V.

J~ (~ ~ ~)

But since

d",(e)·

(e+()

2 . --= 1 +~~ c-;: , l-e( ... -."

the principal value integral

P.V.J~d",(e)

l-e<

exists m-almost ~verywhere and is equal to ",Cll') +i(Q",)«). From eq.(2.5.3) we get lim «C",Mr() + (C"')e«/r» = P.V.

r-l-

J~e( 1-

d",(e)

which is Plemeij's formula (Theorem 2.5.1). 2.6. Tangential boundary behavior From our previous work, we know that X c HP for all 0 < p < 1 and such functions have finite non-tangential limits at almost every point of the circle. Generally, this is about the best one can do since by Theorem 1.7.4, there are bounded analytic functions, which are certainly Cauchy transforms by Proposition 2.2.1, which do not have tangential limits at any point of the circle. If one is willing to make some extra assumptions, there is something more to be said. First, let us be clear by what we mean by tangential limits. Consider the following types of contact regions:

A-y,c«)

:=

{z ED:

I( - zl < c(1 _Izl)lh } ,

(E T,

E'¥,c«):={ZED:I(-ZI
c, 'Y > 0

(ET,

c,'Y>O.

The first types of regions .A.y,c«) are the finite order of contact regions with contact point ( E T, while the second type E.y,c{() are the exponential order of contact regions. For example, the regions A1,c are the non-tangential approach regions and are triangular-shaped regions with vertex at (. When 'Y = 2, these regions become (essentially) circles tangent" to T at ( and are called oric1lclic approach regions. We say, for an analytic function J on D, that J has an A-y-limit L at if J{z) --+ Las z --+ ( within: .A.y,c«) for every c> O. The definition of an E..,-limit is similar. We state here without proof some results from [148, 223] concerning the tangential limits of Cauchy integrals of the form

<

1 hF(Z) := -2 'It'

12'1r 1 F(t) 0

-

e-\'t z dt,

where F is an integrable function on [0,2'1t']. 2.6.1. Suppose F is integrable on [O,2'1t'] and 80 E [0, 2'1t']. (1) If F'(Oo) exists, then hF has an El-limit at ei90 •

THEOREM

2.7. CAUCHY-STIELTJES INTEGRALS

59

(2) If

F«()o + t) - F«()o) = o(ltla), t ~ 0, for some a E (0,1), then hF has an Ea-limit at ei90 . If F is of bounded variation on [0, 211'j, then F must be differentiable almost everywhere and we have the following corollary. COROLLARY 2.6.2. If F is of bounded variation on [0,211'], then hF has an El -limit almost everywhere on 'TI.".

For other global theorems, we need a notion of capacity. We follow [223] and refer the reader to [1] for a modern treatment of capacity. Define the functions Ha : lR \ {O} ~ [0,00) by

Ha (X ) .._- {

Isin l~xla ' 1 log I . 1 I' SlO2'X

0< a < 1;

a

= O.

For a E [0,1), a Borel set E C [0,211'J is said to have a-capacity zero if there is a positive measure of total mass one and carried by E, i.e.,

r

JE for which sup

dl.l =

r

dl.l = 1,

J[O,21r]

r 'Ha(x - t)dl.l(t) < 00.

xER J[O,21r]

It is a well-known fact that any set of a-capacity zero has Lebesgue measure zero. However, there are sets of Lebesgue measure zero but with positive a-capacity. The O-capacity is usually called the logarithmic capacity. Here is a global theorem from [223] on tangential boundary limits of Cauchy transforms.

2.6.3. Suppose F is of bounded variation on [0,211']. (1) For each a E (0,1), there is a set Wa C [0,211'] of zero a-capacity such that hF has an Ea-limit at all points ei9 for which () E [0,211'] \ Wa' (2) There is a set W ofO-capacity (or logarithmic capacity) zero such that hF has A-y-limits for every 'Y > 1 at all points ei9 for which () E [0,211'] \ W.

THEOREM

Finally, we mention that these results are best possible in that the types of tangential approach regions can not be increased. See [223, Thm. 3] for an exact statement of this.

2.7. Cauchy-Stieltjes integrals For a measure I.l EM, we call the function K I.l the Cauchy transform of I.l. In the classical setting, for a function F of bounded variation on [0,211'], the function (CF)(z):=21 11'

r

J[O,21rj

1

-

I_it dF(t) e z

was called the Cauchy-Stieltjes integral of F. Equating a function F of bounded variation with a measure I.lF (see [99, p. 331]), these two integrals are the same. Though we pose the main function theoretic properties of these integrals in terms of measures, we mention that in this original setting, these integrals were studied

2o THE CAUCHY TRANSFORM AS A FUNCTION

60

by Cauchy, Morera [145], Sokhotski [202], Plemelj [160], and Privalov [168, 169]. Cauchy examined these integrals in proving the 'Cauchy integral formula' when

dF(t) = f(eit)dt and f was the boundary function of an analytic function on a neighborhood of the closed. unit disk. Sokhotskii and Plemelj examined these integrals when dF(t) = 4>(e't)dt and 4> was a Lipschitz function on the circle. Privalov, examined the general case. Collectively, their theorems say (in the context of Cauchy-Stieltjes integrals) that the limits (CF)i(e i6 ):= lim (CF)(re i6 ) r-l-

(CF)e(ei6):= lim (CF)(e,6Ir) r-l-

exist for almost every () E [0, 27r] and (CFli{ei6 ) - (CF)e(e i6 ) = F'(6)

(CFMe t°6 ) for almost every 6.

11

+ (CF)e(e''6 ) = P.v.-7r

S

°t

e' i6 dF (t) e' - e 't

CHAPTER 3

The Cauchy transform as an operator In the previous chapter, we explored the function theoretic properties (growth rates, boundary values, etc.) of

(KJ.I-)(z) =

f ~dJ.l-«(), 1- (z

the Cauchy integral of a measure J.I- EM. In this chapter, we examine the properties of the linear mapping J.l-1-+ KJ.Ifrom the space of Borel measures M on the unit circle to the space of analytic functions on the disk. In particular, we examine the question: For a given class of measures X C M, what can be said about the functions in K(X) = {KJ.I-: I" EX}? We already know from Smimov's theorem (Theorem 2.1.10) and the discussion following that

K(M)

n

c;

HP.

O
In this chapter, we take a closer look at K J.I- when J.I- « m, that is to say, dJ.lfor some f E L1. For notational convenience we write

= f dm

f+ :=K(fdm)

whenever f E £1. Most of the results we plan to cover here are stated in the literature in terms of the conjugate function (see Theorem 1.8.10)

Reie ) = lim E..... O+

For

f

f

J18-tl ~E

cot (0 -2

t) f(e

dt it )2 7l"

.

E L1, recall from eq.(1.8.4) the Poisson and conjugate Poisson integrals (Pf)(r():=

i

Pr((e)f(e)dm(e)

=

l(n)r 1nl (n,

n=-oo

T

(Qf)(r() :=

f f

1

Qrc(e)f(e)dm(e) = -i

']['

/(n)sgn(n)rlnl(n.

n=-oo

Since

~ = -21 (1 + Pz «() + iQz«()), 1- (z

(E 'f,

and lim (Pf)(r() = f«(),

lim (Qf)(r()

r-+l-

r-+l-

61

ZED,

= 1«)

3.

62

THE CAUCHY TRANSFORM AS AN OPERATOR

for almost every ( E 'lL', then

I+«) = ~ {1(0) + f(O + in()} a.e. questions about the operator I 1-+ 1+ (on

Thus, continuity spaces of functions defined on 'lL') are equivalent to continuity questions about the conjugation operator 11-+

1=

-2il+ + ij(O)

+ if.

For example, by Smirnov's theorem (Theorem 2.1.10), whenever I E L1. The following result now follows. PROPOSITION 3.0.1 (Smirnov). II IE Ll, then

f+

E

HP for all 0 < p < 1

1E LV for all 0 < p < 1 and

IIllip :::; Cpllfili' Moreover,

3.1. An early theorem of Privalov Perhaps the earliest theorem concerning the mapping properties of the Cauchy transform is due to Privalov and deals with the Lipschitz classes Aa := {f: 'lL' -+ c: I/«) - f(e)1 :::; AII( - el a V(,e E t}, 0 < 0: < 1. The Lipschitz classes are clearly linear spaces of continuous functions and when given the norm

. .. ( IIfllAa .= IIflloo + sup { If(~) I( _- ell(e)1 a

=j:.

e} ,

Aa becomes a Banach space. There is also the associated space

A! = {f E Aa : 1( -n) = 0 Vn E N} which can be considered as a space of analytic functions on JI)). We will see in a moment that A! is the space of functions f that are continuous on JI))-, analytic on JI)), and such that 11'lL' E Aa. THEOREM 3.1.1 (Privalovl ). II IE Aa , then then f+ EAt· PROOF. To prove this theorem, we let

IU+),(z)1 = 0

f

E

1 E Aa.

Consequently, if f E Aa.

Aa and first show that

CI -1~Dl-a

) .

Indeed, a computation shows that

U+)'(z) =

f«)( dm«). 1Tf (1(z)2

lThe original proof that the conjugation operator maps Aa to Aa is due to Privalov (1916) (167). The proof here is adapted from [82, p. 411], which ultimately comes from a theorem of Hardy and Littlewood [88]: An analytic function f on ][) belongs to At if and only if II'(z}1 = 0«1 -Izl)a-l} (see also [66]). Another proof of Privalov's theorem, as well as generalizations to other spaces of smooth functions, is found in Chapter 7 of [234]. There is also a several variable analog of this theorem in [181] (see also [232]).

3.1. AN EARLY THEOREM OF PRlVALOV

63

Moreover, if c is any constant, then

h ~~Z)2 (1

dm«() = 0,

since the above integral represents the derivative of the constant function Z t-+ c. Letting z = re it and ( = eifJ , we take advantage of this observation to obtain the formula , it _ [21f U(e i8 ) - l(e it ))e- i8 dO U+) (re ) (l-re i (t-8»)2 211"'

10

Now use the Lipschitz property

If(ei8 ) - l(eit)1

~ OJle i8 - eitlCl: = Of 11- ei (t-8)1Cl:

to estimate IU+)'(reit)1 as follows: ,.t

IU+) (re~ )1 ~ Of

[21f

10

11- ei(t-8)10 dO 11- re i (t-8) 12 211"'

The estimate 11- ei (t-8) I ~ 211 - rei(t-fJ) 1 yields .

(3.1.2)

I(f+),(re d ) I ~ Of

dO 11- rei(t-8)12-0 = OJ

[21f

10

[21f

10

du 11- rei'U12-0'

Lemma 1.12.2 gives us the estimate [21f

10

du 11 - re i 'U1 2-0

0

~ (1 - r)l-o'

Combining this with eq.(3.1.2), we obtain (3.1.3)

IU+)'(r()1

~

(1

_0~1_0'

(E 11',

0
< 1.

We will use this inequality to show that the boundary function ( satisfies the Lipschitz condition 1/+«(d - 1+«(2)1 ~ OJI(1 - (21 0 '1(1, (2 E 11'.

t-+

f+ «()

First we show that this boundary function actually exists for every ( E 11' (By the fact that 1+ E HP for all 0 < p < 1, we already know that I+«() exists for almost every ( E 11'). Indeed, for any ( E 11', note that (3.1.4)

f+(r() = f+(0)

+

lor (U+)'(s() ds.

Also observe from eq.(3.1.3) and the fact that 0

10

1

< a < 1, that the integral

1(f+)'(s()1 ds

converges for every (. Thus,

exists for every ( E 11'. We conclude, from eq.{3.1.3) and eq.(3.1.4), that 1+ is bounded on JI)l and hence is the Poisson integral of its boundary function ( 1--+ 1+ {(). We will show in a moment that the boundary function for 1+ is continuous - and even Lipschitz. With this assumption, it follows, from solving the Dirichlet problem (Proposition 1.8.5), that 1+ is continuous on JI)l-.

3. THE CAUCHY TRANSFORM AS AN OPERATOR

To establish the Lipschitz inequality, it suffices to show that

1/+«1) - 1+«2)1 ~ G,I(1 - (21 a

V(1,(2

E

1(1 - (21 < l.

T,

To prove this, we observe that when h = 1 -1(1 - (21, we obtain the formula

l

1+«(1) - 1+«2) = (I+)'(w) dw, where 'Y is the piecewise smooth curve consisting of the straight line segment from (1 to h(l followed by the arc of the circle of radius h subtended by h(l and h(2 followed by the straight line segment from h(2 to (2. 2 This integral can be estimated in three parts as

1/+«t) - 1+«2)1

~

r \U+)'(r(l)1 dr + l lh 1

arg

C;2 argC;l

r

hi (f+)'(heit)1 dt +

1 dr h ~ 2G'lh (1 _ r)1-a + (1 _ h)1-a G,

l

arg C;2 argC;l

r IU+)'(r(2)1 dr lh 1

dt (byeq.(3.l.3))

~ G,I(1 - (2I a .

o Zygmund went on to show that the conjugation operator

A:

:=

1 t-+ f

maps

{f : l' -+ C : fen) E Aa}

1(

to itself and hence 1 t-+ 1+ maps A~ onto (A~)+ := {f E A~ : -k) = 0 V kEN}. The conjugation operator also maps the Zygmund space A! to itself, where A'Z is the set of functions l(e i9 ) so that I(n) is continuous and the symmetric differences satisfy I/(n)(ei (8+t» _ 2/
Hence I t-+ 1+ maps A: onto (A'Z)+. All of the above spaces can be endowed with appropriate norms that make them Banach spaces. With these norms, the maps 1 t-+ and 1 t-+ f+ are continuous. See [234] for details.

J

3.2. Riesz's theorem If 1 E L2 has Fourier series

n=-oo

Parseval's theorem says that 00

L

IIfll~ =

11(n)12 •

n=-oo

Furthermore, byeq.(2.l.2), 00

I+(z) =

L j(n)z"'. ",=0

2Notice, from the estimate in eq.(3.1.3), that this integral converges.

3.2. RIESZ'S THEOREM

65

Again, by Parseval's theorem, 00

1I/+1I~ =

L li(n)1 2 n=O

and so

111+112 ~ 11/112. This makes the map I f-+ 1+ a projection of L2 onto H2. In fact, an orthogonality argument shows that I f-+ 1+ is the orthogonal projection of L2 onto H2, often called the Riesz projection. In a similar way, 00

L

I'" -i

i(n)sgn(n)(n

n=-oo

and so

1I111~

=

L li
making the conjugation operator I f-+ generalizes this to £P for 1 < p < 00.

1continuous on L2.

The following theorem

1

THEOREM 3.2.1 (M. Riesz). III < p < 00, the conjugation operator I f-+ is a continuous map from £P to £P. Consequently, I 1-+ 1+ is a continuous onto map from £P to HP.

PROOF.

3

Let us first set some notation. For 1 < p < 00 and F(z)

I

E

LP, let

= (~ + z I«(}dm«() = (Hldm)(z)

iT" -z

be the Herglotz integral of the measure Idm (see eq.(1.8.3». Let u(z}

= (Pf)(z),

v(z}

= (Qf)(z)

be the Poisson and conjugate Poisson integrals of I. Note that v(O) = 0 and F = u + iv. Recall that u(r() -+ I«() almost everywhere and in the norm of £P (Proposition 1.8.5) and v(r(} -+ almost everywhere as r -+ 1- (Theorem 1.8.1O). Let us first assume that 1 < P ~ 2 and IE £P is non-negative. Notice that u is a positive harmonic function and IFI > O. A somewhat tedious computation using the fact that the Laplacian operator

i«)

d

= o:J::J: + Oyy

can be written as shows that and

3The proof here is found in [65, p. 54) which was ultimately adapted from a proof due to P. Stein [209). Other proofs of Riesz's theorem are found in [36), [229, p. 256), or [234, p. 253). See [174) for the original proof. 48z := ~(8., - i811 ), 8z:= ~(8., + i8y )

3. THE CAUCHY TRANSFORM AS AN OPERATOR

66

But since u/IFI ~ 1 and p - 2 ~ 0, we see that up- 2 ~

IFlp-2 and so

AIFIP ~ 2-Aup • p-1 Apply Green's theorem in polar coordinates (3.2.2)

r

1

211" ~ 8¢

o

ur

= !~

d(}

A¢dxdy

j.z\,r

to eq.(3.2.2) to get ddr

dd rlu(r()I iTrIF(r()IP dm«() ~ P2-1 r iT

Pdm«().

Now integrate the previous inequality from 0 to r and use the facts that F = u + iv and v(O) = 0 to see that lIF(r()I Pdm«() - u(O)P

~ P ~ 1 {llu(r()I Pdm«() -

U(O)p}

and consequently,

iTrIF(r()I

(3.2.3)

Pdm«()

~

iTrlu(r()IP dm«().

2p-1

Thus

rIn()IPdm«()

iT

= ( lim Iv(r()lPdm«() (by Theorem 1.8.10) iTr ..... l-

~ r~~-lr Iv(r()lPd~«() ~ r~ ~

lr

IF(r()lPdm«() (since Ivl

lim 2-1 (lu(r()lPdm«() (byeq.(3.2.3» r ..... l-

p-

, ~ -P-lIfll~ p-1 For a general real-valued

f

~ IFI)

E

iT '

(by Proposition 1.8.5).

V we write

f

=

h - 12, h, 12 ~ 0, to see that

1I111~ ~ 2p(lIhll~ + IIhll~) ~ Ap(llhll~

+ IIhll~)

= ApllfII~.

The proof for complex-valued f now follows. Thus, we have shown that for 1 < P ~ 2, (3.2.4) We will now show that if eq.(3.2.4) works for some 1 < p < 00, then eq.(3.2.4) holds for the associated conjugate index q (l/p + l/q = 1). Since we have already shown eq.(3.2.4) for all 1 < p ~ 2, we will have completed the proof. To this end, let 9 be any trigonometric polynomial N

g«() =

2: n=-N

an(n

3.2. RIESZ'S THEOREM

and

67

N

h = Q(gdm) = -i

L

sgn(n)an(n

n=-N

be its conjugate function. Notice that 9 + ih is an analytic polynomial. Apply the mean value property for harmonic functions to the imaginary part of the analytic function (u(rz) + iv(rz» (g(z) + ih(z)) to get

i

(u(r()h«()

+ v(r()g«(»

dm«()

= u(O)h(O) + v(O)g(O) = O.

Rearranging the terms and putting in absolute values, we conclude that

Ii

I Ii

v(r()g«() dm«() =

u(r()h«()dm(OI

~ IIhllpllu(r ·)IIq

~ Apllgllpllu(r ·)IIq·

In the last inequality, we are assuming that 1 < p ~ 2 and so, using eq.(3.2.4), IIhilp ~ ApIIglip. Now take a supremum over all trigonometric polynomials 9 with IIglip ~ 1 and use the fact that trigonometric polynomials are dense in V' (via Cesaro means - Theorem 1.6.5) to get, via eq.(1.2.2),

IIv(r·)l\q ~ Apllu(r ·)lIq which, after taking limits as r -+ 1- and using Fatou's lemma and Proposition 1.8.5, shows that IIl11q ~ Apllfll q · The proof is now complete.

o

3.2.5. (1) Riesz's theorem says the Cauchy transform is a projection operator, the Riesz projection, f 1-+ f + from LP onto HP. Furthermore, when 1 < p < 00 and f E V', then f = !I + 12, where !I E HP('lI.') and 12 E Hg('lI.'). Also note that by the F. and M. Riesz theorem (Theorem 1.9.7)

REMARK

HP('lI.') n Hg('lI.') = {O}. In other words, HP('lI.') is complemented in V'. When 0 have the decomposition

< p < 1, we still

V = HP('lI.') + Hg(T) although

HP('lI.')

n Hg('lI.') =I- {O}.

In fact HP(T) n H&(T) is the closed linear span in V of {K5, : ( E 'lI.'} [8, 10] [44, p. 116]. (2) There are several proofs of Theorem 3.2.1. The one presented here has the advantage of keeping track of the constant in the norm inequality. We were not too careful here - and often went from one line to the next using the same symbol Ap - since there is a result of Hollenbeck and Verbitsky (see Theorem 3.7.3 below) which computes the best constant in 1\f+l\p ~ Apllfllp as Ap = 1/ sin(1r/p). If one is not particular about the constant Ap in Riesz's theorem, there is an alternate proof in [36].

68

3.

THE CAUCHY TRANSFORM AS AN OPERATOR

The endpoint cases p = 1 and p = 00 are different. For example [65, pp. 6364], the 1 E L1 whose Fourier series is cosn(J

00

~

logn

00

=~

ein9

+ e-in9

2logn

has Cauchy transform equal to 1

(3.2.6)

f+(z) =

00

zn

2~ logn

which does not belong to H1. The reason for this is 'Hardy's inequality' (see [65, p. 48]) which says that

1=

f

anz n E H1 =?

f

~~Il ~ '11'11/111,

n=O

n=O

Notice that the Taylor coefficients of 1+ in eq.(3.2.6) do not satisfy Hardy's inequality. With a little bit more work, one can create examples of 1 E L1 for which (equivalently the boundary function for f+) is not integrable on any interval [234, p. 257]. It is worth remarking here that not only does the Riesz projection 1 1-+ f+ fail to be bounded from L1 onto Hl, but there is no other bounded projection of L1 onto H1 [152]. Equivalently, H1 is not complemented in L1. H the function 1 is slightly better than Ll, there is the following theorem of Zygmund [233] (see also [65, p. 58] and [234, p. 254]) which we· state without proof.

1

THEOREM 3.2.7 (Zygmund). If 1 E LlogL, that is to say,

h1/1(1 + then

log+ I/l)dm <

00,

1+ E HI.

The bounded function (3.2.8) has Fourier series

l(e i9 )

:=

f

{-'11'/2 - 9/2 ~ 9 < 0, '11'/2 - (J /2 0 < (J ~ '11', -7('

sin n9 =

n=1

n

f n=1

ein9

-

e- in9

n2i

and thus has Cauchy transform

I+(z)

1

00

2~

n=1

= -; L

-znn = --;2'£1 log(l- z)

which is unbounded. Not only does the Riesz projection 1 1-+ f+ fail to be a bounded projection of Loo onto Hoo, there is no other bounded projection of LOO onto Hoo [101, p. 155J. Equivalently, Hoo is not complemented in Loo.

3.3. BOUNDED AND VANISHING MEAN OSCILLATION

69

3.3. Bounded and vanishing mean oscillation

DEFINITION 3.3.1. A function 9 E £1 is of bounded mean oscillation, or BMO, if

11911. := sup { m~I) 119 - 911 dm : I

is a subarc of

T} <

00,

where

is the 'average' of 9 on I.

With the norm

If gdm\ + IIgli.

one can check that BMO is a Banach space of functions on T. The functions in BMOA:= BMOnH1 (T) = {I E BMO: f(-n) = OVn EN}, the analytic junctions

01 bounded mean oscillation, form a closed subspace of BMO.

Recall from our previous discussion (see eq.(3.2.8» that (L oo )+ = {/+ : I E L oo }

is not contained in HOCl. However, the following theorem, discovered independently by Spanne [203] and Stein [206], is true. THEOREM 3.3.2 (Spanne, Stein). (1) The operator f (2) The operator f

t---7 t---7

J is continuous from Loo to BMO.

1+ is

continuous from Loo onto BMOA.

There are some standard proofs of this theorem (see [79, 118]). The one we wish to present here involves the Garsia norm. For f E L2, a routine computation shows that (3.3.3) (PI/1 2)(z) - (Plfl)(z)2 =

~

r r I/(e

2J-~J_~

is ) _

f(eitW PAeiS)Pz(eit) ds 2dt .

2w w

One also checks that (3.3.4)

(PI/12)(Z) - (PI/I)(Z)2

= POI -

(Pf)(zW)(z) ~

o.

Define the Garsia norm of I E L2 to be

9(J)

:=

sup {(PI/12)(z) - (PI/I)(z)2} 1/2 zED

and observe the simple inequality, (3.3.5)

9(J)

~

Cll/lloo,

IE Lr;>o.

Technically, the Garsia norm is not really a norm since it does not distinguish the constants. However, it is equivalent to II . I •. PROPOSITION 3.3.6. The Garsia norm is equivalent to the BMO norm. Equivalently, there are constants Ct, C2 > 0, independent 01 I E L2, such that

c111/11. ~ 9(f) ~ ~lIfll*·

3. THE CAUCHY TRANSFORM AS AN OPERATOR

70

We will only prove one of the inequalities. It ,is the easier of the two inequalities and it is the only one we really need to prove Theorem 3.3.2. The other inequality depends on the John-Nirenberg inequality

~ Om(I) exp C~~),

m(19 - 911 > y)

y

> 0, 9 E BMO.

The interested reader can consult [79, 118]. The one direction we present below is from [118, p. 222]. Indeed, let J = (-a,a), where 0 < a ~ 71'. When r

=1-

sin ~

and t E [-a, a],

we have

p. ( it) r

e

1 - r2

= l+r2 -2rcost (l+r)(l-r)

= (1-r)2+4rsin2(t/2) 1

~ 5sin(a/ 2) 2

~ 5a' Thus

(PlfI 2)(r) - (Plfl)(r)2

= 8:2 ~ =

i: i:

If(e is ) - f(eit)12pr(eiS)Pr(eit)dsdt

12 2ja, ja l'f(eiS ) 5071' a - a - a

(5~Y 21(~)2

ii

_

f(e it )12 ds dt

If(e iS ) - f(e it )1 2 dsdt,

where I(J) is the length of the interval J. We observe that when

fJ

= l(~)

i

f(e it ) dt,

the average of f on J, we have the inequality

1f

it

l(J) J)f(e ) -

fJl dt ~

{I

f

it

l(J) JJ If(e ) -

~ {2l(~)21 i

fJl

If(e it )

2dt }1/2 -

f(e iS )12 dtdS} 1/2

Combine this with our previous estimate to get

(3.3.7) .

1 f it I 5'11' ( ) l(J) JJ If(e ) - fJ dt ~ 2 9 f .

Now take the supremum over all intervals J to see that

571'

IIfli. ~ 2 Our next step, to prove that

f -

9(f).

1is bounded from Loo to BMO, is to show

that

9(1) = 9(f).

3.3. BOUNDED AND VANISHING MEAN OSCILLATION

71

For a E 11)), let

a-z tPa(Z) := --az 1 and notice that tPa is a conformal self map of II)) with (3.3.8) LEMMA

3.3.9. For 9

E

L1,

i(gotPa)Pzdm =

!

gP"'a(z)dm.

PROOF. If 9 E G(T), then both sides of the above equation are harmonic functions of z which tend continuously to g(tPa «» as z -+ (. By the maximum principle, these two harmonic functions must be equal. Hence the result is true when 9 E G(1'). Now use the density of C('JI') in L1 to obtain the full result. 0 LEMMA

3.3.10. If f

E

L2, then

Q(f 0 tPa) = (Qf) 0 tPa - (Qf)(tPa(O». PROOF.

From Lemma 3.3.9,

and so the harmonic function

(Qf)(tPa(Z» =

!

fQ"'a(z) dm

is a conjugate function for f 0 tPa. It may not vanish at zero to be the conjugate function Q(f 0 'I/la)(z). We will fix that in a moment. Since conjugate functions must differ only by a constant,

(Qf)(tPa(Z»

= Q(f

0

tPa)(z) + c.

o

But Q(f 0 'I/la)(O) = 0 and so c = (Qf)('I/la(O». PROPOSITION 3.3.11. For

f

E L2,

9(1) PROOF.

(3.3.12)

= 9(f).

From Lemma 3.3.9 and he fact that 'I/la{a)

= 0 we have the identity

i (h 0 tPa)Padm = i hdm

for all h E L1. For 9 E L2, apply this identity to h = fact that 'I/la(O) = a to get

Ig - (Pg)(a)1 2

and use the

3. THE CAUCHY TRANSFORM AS AN OPERATOR

72

This last identity applied to 9 =

! 11-

I shows that

! 11 = ill f 1~12 JII = ill f II -

o'I/Ja - (Pl)('l/Ja(0»1 2dm

(Pl)(a)1 2Pa dm =

0

'l/Ja - (Qf)('l/Ja(0»1 2dm (since QI = pl)

= =

0

dm (Lemma. 3.3.10)

'l/Ja - P(f 0 'l/Ja)(O) 12 dm (Pa.rseval) P(f 0 'l/Ja) (0) 12Padm (by (eq.(3.3.12»

(PI)(a)1 2Pa dm (by eq.(3.3.12) and 'I/J;;1 = 'l/Ja).

=

Now take a supremum (in a E D) over both sides of the above string of equalities and use eq.(3.3.4) to get 9(1) = 9(f).

o PROOF OF THEOREM 3.3.2. If 1 E Loo, then

11111. ~ 09(1)

(Proposition 3.3.6) = 09(f) (Proposition 3.3.11) ~

011/1100

(eq.(3.3.5».

o A natural subspace of BMO is the space VMO, the functions of vanishing mean oscillation. These are the functions 1 E BMO satisfying lim {sup

a-+O

m(I)";:a

m

if II - /II dm} = O.

1(1) {

A routine argument shows that VMO is a closed subspace of BMO. As we did for BMO, we consider the analytic functions of vanishing mean oscilla.tion VMOA:= VMO n H1(1l').

1

If f E 0(1l'), then may not be continuous [79, p. 127]. The following result of Sarason [187] is the analog of Stein's (Spanne's) result for BMOA.

THEOREM 3.3.13 (Sarason). (1) The operator 1 t-+ (2) The operator I t-+

1 is continuous from 0("Jl.") to VMO. f+ is continuous from 0("Jl.")

onto VMOA.

Though we will not present it here, a proof of this theorem depends on the following alternate characterization of VMO using the Garsia norm: 1 E L2 belongs to VMO if and only if lim {(PI/12)(z) - (PI/I)(z)2} = O.

1%1-+1

3.4.

KOLMOGOROV'S THEOREM

73

In fact, one can see one direction from the proof of eq.(3.3.7). rem 3.3.13, note that if IE C('ll'), then

To see Theo-

(PliI 2)(z) - (Plil)(z)2 = (PI/12)(z) - (PI/I)(Z)2 and so from the proof of Proposition 3.3.11, lim {(PI11 2)(z) - (Pll1)(z)2} Izl ..... 1

= 1zlim {(PI/12)(z) I"'" 1

(PI/I)(z)2}

=0 since I is continuous. The continuity of the operators from Theorem 3.3.2.

I

~

i

and

I

~

1+ follows

3.4. Kolmogorov's theorem Despite the fact that (L1)+ is not contained in L1, there is a well-known and often revisited theorem about the Cauchy transforms of L1 functions due to Kolmogorov. In fact, we will prove a stronger result. THEOREM 3.4.1 (Kolmogorov). For"" EM,

~ AJJcl, y

m(IK",,1 > y)

y > O.

A similar result holds lor the conjugate function

/1(0 =

lim (Q",,)(r().

r ..... 1-

REMARK 3.4.2. (1) We write, as is traditional in many probability books, m(IK",,1 > y) in place of the more proper m({( E 1': I(K",,)«()I > y}). (2) We will treat the distribution function y ~ m(IK",,1

> y)

in greater detail in Chapter 7. PROOF OF THEOREM 3.4.1. (3.4.3)

5

We will first prove that if "" E M+ and

F(z) :=

l

(+z

-r-

'll',,-z

d",,«(),

then (3.4.4)

m(lFI

> y) ~

211",,11 11",,11 +y'

y> O.

Indeed, for y > 0, the map

w-y g(w):= 1+-w+y maps {!Rw > O} to {Iw -11 < I} and, since RF > 0, the function <j)(z) := 1 + F(z) - Y F(z) + y 5For the original proof, in terms of the distribution function for the conjugate function, see [116]. The proof here is modified from [118, p. 92].

74

3. THE CAUCHY TRANSFORM AS AN OPERATOR

is bounded and analytic on D. By the mean-value property for harmonic functions,

(3.4.5)

h

iRq,(O) =

'iRq, dm.

From the definition of F in eq.(3.4.3), we know that F(O) q,(O) = 1 +

IIJLII -

y =

+y

/IJL/I

= IIJLII and so

211JLII •

IIJLII + Y

Combining this with eq.(3.4.5) we obtain

f

ir Thus, since SRq,

~

lRq,dm =

211JLII . IIILII + Y

0 we get m(SRq, ~ 1) ~

(3.4.6)

iTf lRq,dm =

211JLII IIJLII + Y'

Using the identity

we see that lRq,«() ~ 1 {:}

IF«()I ~ Y

and so by eq.(3.4.6)

m(IFI ~ y) ~

211ILII /IJLII + y'

This proves eq.(3.4.4). Still for JL E M+, we notice, from the observation

(+z =2~-1, (- z

1- (z

that (3.4.7)

(KJL)(z) =

1

2 (IIILIl + F(z»

and, by eq.(3.4.4),

~ y) ~ cy IIJLII. Here we are using the fact that if I = h + h, then m(IKILI

{III

~ y}

c {Ihl ~ y/2} U {Ihl ~ V/2}.

Hence (3.4.8)

m(1/1 ~ y) ~ m(lh I ~ y/2) + mOhl

~

V/2).

Kolmogorov's theorem for general JL E M follows from the Jordan decomposition of JL as JL =

(JLl -

and the trick in eq.(3.4;8).

JL2)

+ i(JLa -

JL4),

ILj

E

M+, 0

3.4. KOLMOGOROV'S THEOREM

75

REMARK 3.4.9. If we are willing to be a bit more careful, we can, in some sense, keep track of the constant A in Kolmogorov's inequality

m(IKI1I

~

> y)

:!.111111. y

Start with the observation from eq.(3.4.7) that

1

IKI1I ~ '2"1111

1

+ '2IFI

and apply it, for 11 E M+, to the inequality

m(WI > y) from eq.(3.4.4) to get, for y

~

2111111 111111 +y

> 111111/2,

m(IKI1I > y)

~ m (~IFI + ~III1I1 > y) = m(IFI

> 2y - 111111)

211JLII ~ 111111 + 2y - 111111

&

=M y Hence

m(IKJLI > y)

~~, y

11 E M+,

y>

1I1L1I/2.

Recall the analytic weak-type space H1,OQ from Chapter 1. From Theorem 1.10.4, Theorem 3.4.1, and the fact that KI1 E N+, we get the following corollary.

H1,OQ.

COROLLARY 3.4.10. Xc

Thus we can refine the string of containments in eq.(2.2.2) a bit to

UHP ~ X ~ H 1,OQ ~ p~l

PROPOSITION 3.4.11.

that is to say, K(L1)

HP.

For 11« m, m(IKI1I > y)

(3.4.12)

n

O
= 0 (~) ,

C H~'OQ.

PROOF. Suppose dJL = I dm, I ELl. Let e > 0 be given and choose a trigonometric polynomial p so that III - pill < e (Theorem 1.6.5). Notice that

ym(If+1 ~ y) ~ ym(lp+1 ~ y/2)

+ ym(l/+ -

p+1 ~ y/2).

For y sufficiently large, m(lp+1 ~ y/2) = 0 (since p+ is a analytic polynomial and hence bounded). By Kolmogorov's theorem (Theorem 3.4.1)

m(lf+ - p+1

~ y/2) ~

C III _ pill y

~

C e. y

Thus and the result follows.

o

3. THE CAUCHY TRANSFORM AS AN OPERATOR

76

More surprising however, is that the converse of this is true, namely

~I-'«m.

m(IKI-'I >y)=o(t)

See [218] for details. We will not prove this here since we will be proving this, as well as a stronger result, in Chapter 7. One can work out the distribution function for K61 = (1- Z)-l. Indeed, IK61(e i9 )1 = 11_1

ei9 1= 12ie!9/2 sin(~/2) 1=

Thus

m(IK61 1 > y) =

~ 1sin(~/2) I·

~ Sin- 1 (21 ). Y

'Tr

Notice that this quantity is O(l/Y) but not o(l/y) and moreover, lim ym(IK61 1 > y)

y-+oo

=.!.'Tr = .!.'Tr 1161 II·

We will see later in Chapter 7 that lim y'Trm(IKI-'I

y .....oo

> y) = III-'Bli.

In Chapter 9 we will prove the stronger result

Y'Trm(IKI-'I > y) . m weak-* as y

-+

I-'B

-+ 00.

3.5. Weighted spaces What about Cauchy transforms of functions from weighted £P spaces? The question here is the following: given 1 < p < 00, what are the conditions on a measure I-' E M+ such that there is a constant C> 0 with the inequality

!

Ig+IPdl-'

~C

!

IglPdl-'

holding for all trigonometric polynomials g? A theorem of Helson and Szego [97] says that such a measure I-' must satisfy I-' « m and so the problem can be rephrased as: given 1 < p < 00, what are the necessary and sufficient conditions on a nonnegative weight function w on 'lr such that

!

(3.5.1)

Ig+IPwdm

~C

!

IglPwdm

for all trigonometric polynomials g? In the special case where the weight function is w( ei9 ) = 11- ei9lQ, a classical result of Hardy and Littlewood [89) says that when p = 2, eq.(3.5.1) holds if and only if -1 < a < 1. For general w, Helson and Szego [97) proved that eq.(3.5.1) holds in the case p = 2 if and only if logw = u + V, where u, v E Loo, IIvll oo < 'Tr /2, and is the conjugate function for v. The definitive theorem here is one of Hunt, Muckenhoupt, and Wheeden [106] (see also [79, p. 253]) which says that when 1 < p < 00, eq.(3.5.1) holds if and only if the weight w satisfies the condition

v

~~~ (m~I)

h

w(()dm((»)

(m~I) h(W((»-1/(p-1)dm((»)P-1 < 00,

77

3.6. THE CAUCHY TRANSFORM AND DUALITY

where the supremum is over arcs I of the circle. We refer the reader to the references in [106] for other sufficient conditions on w. See [150] for a shorter proof of this result (at least when p = 2).

3.6. The Cauchy transform and duality In this section we will use Riesz's theorem (Theorem 3.2.1) to equate the norm dual of HP with Hq. Here 1 < p < 00 and q is the HOlder conjugate index to p. From the Riesz representation theorem for J;P, we know that every bounded linear functional on J;P takes the form

I

f-4

f

[gdm

for some unique 9 E Lq. Moreover, the norm of the above linear functional is 1!91!q, i.e., (3.6.1) An application of the F. and M. Riesz theorem (Theorem 1.9.7) says that

(HP)J.. := {9 E Lq :

i

Igdm = 0 VI E HP('lr)}

is equal to Hg and so we can apply Theorem 1.4.6 to conclude the following. THEOREM

3.6.2. For 1 ~ p <

00,

(HP)* is isometrically isomorphic to Lq / Hg.

We can use Riesz's theorem (Theorem 3.2.1) to identify, in an isomorphic (but unfortunately not isometric) way, the dual of HP with Hq when 1 < p < 00. One can see this as follows. By HOlder's inequality, the linear functional

I

f-4

i[gdm

is continuous on HP for fixed 9 E Hq.

On the other hand, if i E (HP)*, the

Hahn-Banach extension theorem says that

i(f) =

h

f9i dm

for some 91 E Lq. Using the continuity of the Riesz projection operator h from Lq onto Hq and the identity

h

191 dm

=

i

t-+

h+

1(91)+ dm

(which follows from Proposition 1.8.5 and Theorem 1.9.6), one can replace the above 91 E Lq with a unique function 9 := (91)+ E Hq. Thus every i E (HP)* takes the form

ig(f) :=

h

[gdm

for some unique 9 E Hq. Hence we can identify (HP)* with Hq when 1 < p < 00. We can apply Riesz's theorem again to say something about norms. Clearly, by Holder's inequality, the norm of the functional i g , that is, sup

{Ih

[g dml : I E ball(HP) } ,

78

3. THE CAUCHY TRANSFORM AS AN OPERATOR

is no bigger than 11911q. For the other direction, let A" be the norm of the Riesz projection, i.e., the smallest constant A" so that 11/+11" ~ A"IIfIl" for all 1 E Y, and observe from eq.(3.6.1) that

IIgllq = sup

= sup

{ll.f9 1 Hi 1 {Ii ~:Ydml 1 {Ii dm \ :

E ball(U)}

I+Ydml :

:

= Apsup

~ A" sup =

E ball(U) } E ball(LP)}

Fgdml : F E ball (HP) }

Aplligll·

Thus

IIgllq ~ Aplligll ~ IIgllq· This makes the sesquilinear map 9 -ig continuous and invertible from H9 to (HP)* and hence one identifies (HP)* with Hq with comparable norms. If one changes the dual pairing slightly to Lg(/) :=

i

I«()g(() dm«()

the mapping 9 l-+ L9 becomes an isomorphi8m. We summarize this with the following corollary. COROLLARY

and only

3.6.3. Let 1 < p < 00. A linear functionall belongs to (HP)* agE Hq such that

if there is

i(/) =

Jf9

il

dm.

Moreover, this 9 is unique and satisfies

When p = 1, we can still say that (Hl)* is isometrically isomorphic to Loo / H~. However, (Hl)* is not identified with HOC) via the above dual pairing. Instead, (Hl)* is identified with BMOA, the analytic functions of bounded mean oscillation. Certainly, by the Hahn-Banach extension theorem, if i E (Hl)*,

t(f) =

Jf9dm

for some 9 E Loo. However, when 9 is replaced by g+, the above integral ma.y not converge since g+ may not belong to Hoo. By the Spanne/Stein theorem (Theorem 3.3.2), 9+ E BMOA and moreover

i(/) where Ir{z)

= I(rz).

= r lim ..... l-

J

Irg+ dm,

Conversely if 9 E BMOA then

19(/) =

r~T-

JIrY

dm

3.7.

BEST CONSTANTS

79

defines a continuous linear functional on HI and c11IgIIBMO::;;; Illgll ::;;; c21IgIIBMO. This is the Fefferman-Stein duality theorem [71, 70J (see also [79] and [118]). Finally, we would like to mention an alternate and useful representation of the dual pairing between HI and BMOA involving truncations [211] (see also [44, p. 32]).

3.7. Best constants In the previous sections of this chapter, we examined the mapping properties of the operators JL 1----+ KJL and JL 1----+ QJL on various subspaces contained in M. In this section, we discuss, without proof, the norms of these operators. A nice survey paper on this material is [158]. We begin with a relatively easy one. From Smirnov's theorem (Theorem 2.1.10) we have

IIKJLI/p ::;;; CpIlJLI/, where Cp :=:: (1 - p) -1. This first result computes, at least for positive measures, the smallest such constant cpo We thank Al Baernstein for showing us this proof. PROPOSITION 3.7.1. For fixed 0

< p < 1,

sup {I/KJLI/p : JL E M+, I/JLII = I} =

111 ~ z

L·

PROOF. The function ¢(z) := (1 - Z)-l maps ID> onto {lRz > 1/2} and is univalent. Furthermore, for JL E M+, IIJLII = 1, we have the containment

(KJL)(ID»

c {lRz > 1/2}.

Thus KJL is subordinate to ¢ and so by Littlewood's subordination theorem [65, p.

10]6,

The equality in the statement of the proposition is achieved when JL = 81,

0

REMARK 3.7.2. For any complex measure JL = (JL1 - IL2) +i(IL3 - JL3), ILj E M+, one can use a slight variation of the above argument four times to prove Smirnov's theorem: IIKILllp::;;; Apl/ILI/. However, the best constant Ap is not known for general complex measures, only for positive ones. The Riesz theorem (Theorem 3.2.1) says that for fixed 1 < p < 00, the operator f+ = K(fdm) (the Riesz projection) is a continuous operator from V onto HP. This next result computes the norm of this projection.

f

1----+

THEOREM 3.7.3 (Hollenbeck and Verbitsky). For fixed 1 < p

< 00,

1 sup{IIf+llp:llfl/ p ::;;;l)}= . ( /)' sm 7r p 6Littlewood subordination theorem: If I, 9 are analytic on lD> and 1= go w, with w analytic on lD> and Iw(z)1 ~ Izl, then M(r; f) ~ M(r; g} (see eq.(1.9.1}) for all 0 < r < 1.

3. THE CAUCHY TRANSFORM AS AN OPERATOR

80

A proof of this theorem can be found in [102] where they also discuss the norms (and even essential norms) of some other classical operators. For

1 E L1, the conjugate function

f

1<ei8 ):= lim e-+O+

JI9-tl~f

cot(9 -2 t)J(eit )2dt 7r

is defined almost everywhere and by Riesz's theorem, the operator J t-+ 1 is continuous on LV for 1 < p < 00. Moreover, for 0 < p < 1, 1 t-+ is continuous from L1 to £P (Proposition 3.0.1). The norms of these operators are computable [159].

1

3.7.4 (Pichorides). (1) For fixed 1 < p < 00,

THEOREM

(2) For fixed 0 < p < I, (cos{p7r/2»-1/V ~ sup {1IJ1lv We know, for

: 1I/11t ~ I} ~ 2 1/V- 1(cos(p:n"/2»-1/V.

I-" EM, that the conjugate function (QI-")(z)

=

J;}(~ ~

:)dl-"«()

is harmonic on JI) and has radial limits

l1{ei8 ):= lim (QI-")(re i8 ) r-+l-

almost everywhere. Moreover, a variant of Kolmogorov's theorem (Theorem 3.4.1) says that for fixed 0 < p < 1, 11 belongs to LV and

111111, ~ CpIlI-"I1· B.

Davis7

Cp, at least for real measures MR. (Davis). For fixed 0 < p < 1, sup {lIl1l1p : I-" E MR, 111-"11 ~ I} = IIi/lip,

computes the best constant

THEOREM

3.7.5

where

1 1 v:= 261 - 26-1.

Kolmogorov's theorem says that

m(lK1-"1

~ y) ~ eM. y

e

Though the smallest constant is unknown here, it is known when K I-" is replaced by the conjugate function 11 [53, 54]. 7The original proof is found in [55) and involves a Brownian motion argument. A more analytic proof can be found in a paper of Baernstein [24J. See also [14).

3.8.

THE HILBERT TRANSFORM

THEOREM 3.7.6 (Davis).

(1) For

m(ll1

81

IE L1,

~ y) ~ ~ 1I~11,

y

> 0,

where 1 - 3- 2 + 5- 2 - ••. . 1 + 3- 2 + 5- 2 + ... Moreover, 9- 1 is the smallest possible constant. (2) For 1-£ E M+,

8=

(3.7.7)

m(liLl

~ y) ~ 1· 111-£11 , y

y > O.

Moreover, 1 is the smallest possible constant.

81

Notice that

9=2" 1f

0

00

tet dt 8 -1 2t =2"C,

+e

1f

where C := 1 - 3- 2 + 5- 2 - •.• is the Catalan constant. It is unknown whether or not the Catalan constant is rational. We will say more about the distribution function for K 1-£ in Chapters 7 and 9.

3.8. The Hilbert transform For

I

E L1(lR), the functions

('J'f)(z):=

~ f ~ (_I_)'/(S)d8, 1f

JR

8- Z

(Of)(z):=

1(8)d8 JR !R (_1_) Z - 8

~ f 1f

are harmonic on the upper-half plane C+ := {z E C : ~z

('J'f)(z)

> O} and

+ i(Of)(z) = ~ f _1_/(s) d8 1fZ JR. 8 - z

is analytic on C+ and is often called the Borel transform of I. As it turns out, the function 'J'I plays the role of the Poisson integral PI (f E L1 (T)) in the disk setting (it is in fact called the Poisson integral) in that lim ('J'f)(x + iy) = I(x)

y--+o+

almost everywhere and when 1 < p < 00, ('J' f)(. + iy) - I in the norm of LP(lR) as y - O. The function 01 is the harmonic conjugate of 'J'I and plays the role of the conjugate Poisson integral QI (f E L1(T)) in the disk setting. One can show that lim (Of)(x + iy) = P.V.

y--+o+

f 1(8) d8 JR. X - 8

almost everywhere. The above singular integral is called the Hilbert translorm of I and is denoted by (~f)(x). Notice how the boundary function for the conjugate function 01 yields a singular integral similar to that of Q I. Many of the theorems, for example Privalov's theorem (Theorem 3.1.1), Riesz's heorem (Theorem 3.2.1), Kolmogorov's theorem (Theorem 3.4.1), Spanne's (Stein's) heorem (Theorem 3.3.2), etc., have direct analogs for the Hilbert transform. In ummary,

I E Aa(lR) =} ~I E Aa(lR) f

E

LP(lR), 1 < p

< 00 =} ~I E V(lR)

82

3. THE CAUCHY TRANSFORM AS AN OPERATOR

Ll(JR) =? ml({I~'1 > y}) ~ Cllflhy-l , E LOO(JR) n Ll(JR) =? ~, E BMO(JR) , E C(JR) n Ll(JR) => 'J(, E VMO(JR). In the above, ml is Lebesgue measure on JR. , E

Just as in the conjugate function case, the best constants are known. For example [159], if 1 < p < 00,

sup{II'J('"p: "'"p

~ I} =

tan( ~ ), {

1 < p ~ 2;

: cot(2p)'

2 < p < 00.

The smallest constant C in the weak-type inequality

ml({I'J('1 > y}) ~ C"'"ly-l is a-I, where e is the constant defined in eq.{3.7.7). The best constant C above for non-negative functions is one [53, 54]. As was the case on the circle, the mapping properties of the Hilbert transform playa crucial role in determining the mapping properties of the Cauchy transform. In Chapter 7 will be looking at the function y

1-+

ml(I~1-'1

> y),

the 'distribution function' of the Hilbert transform

('J(I-')(x) = P. v.

f

_1_ dl-'{s)

x-s

of a measure I-' on the real line. We refer the reader to [207] for a thorough treatment of the Hilbert transform as well as other singular integral operators.

CHAPTER 4

Topologies on the space of Cauchy transforms In this chapter, we discuss several natural topologies one can place on the space of Cauchy transforms X. For notational purposes, the reader might want to review the basic functional analysis facts covered in Chapter 1.

4.1. The norm topology For

I

E X recall from Definition 2.1.4 the set Rj := {v EM:

of 'representing measures' for

I.

I

= Kv}

From Proposition 2.1.5 we know that

I-L, v E Rj => dl-L - dv = 4>dm, where ¢ E HJ ('lI') and HJ is the subspace of H1 consisting of functions which vanish at the origin. As is customary, we will abuse notation slightly and let HJ denote the following subspace of M,

HJ := {"¢dm : 4> E HJ} . Since II4>dmll = 114>111 and HJ is a closed subspace of £1, subspace of M. Thus the quotient space M / HJ of cosets [ILl:= I-L+ HJ

then

HJ

is a closed

is a Banach space with norm II[ILlll := dist(l-L. HJ) = inf {lldl-L + "¢dmll : 4> E

HJ}.

By identifying a Cauchy transform with its set of representing measures and then using the previous discussion, it makes sense to associate the Cauchy transform K I-L with the coset [ILl in M / HJ. The mapping

KI-L

t-t

[ILl

is a vector space isomorphism from X onto M / HJ. If we equip X with the quotient space norm (4.1.1)

IIKI-LII:=II[l-LllI,

the map K IL t-t [ILl becomes an isometric isomorphism, making X a Banach space. Let us say a few words about this norm topology on X. PROPOSITION

4.1.2. For

I

E

X,

11/11 = inf{llvll : v E Rj}. 83

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

84

PROOF. Proposition 2.1.5 says that -

1

Rf = {dJ-to + ¢>dm: ¢> E Ha},

J.Lo E Rf·

o

Now use the definition of IIfll from eq.(4.1.1).

Normally, computing the norm of a Cauchy transform is quite difficult. However, we can compute some easy ones. COROLLARY 4.1.3. If J.L E M+ and PROOF. For any

II

E Rio

I!ILII =

!

I

= KIL, then 11111

= I\J.LII.

"'"

=

dJ.L = f(O)

!

dv = v('II')

~ IIvll·

o

Now use Proposition 4.1.2. Corollary 4.1.3 says that for certain Cauchy transforms such that IIfil = IIILII· This turns out to be true in general. PROPOSITION 4.1.4. For each IIILI! = IIfll·

I

E

f,

there is aILE R,

X, there is a unique J.L E Rf such that

This proposition is really just a Cauchy transform version of the following result from the theory of dual extremal problems (see Theorem 1.4.6 and Theorem 1.4.7). Originally shown by Doob [63], we present the proof from [79] (see also [65)). The papers [94, 95, 176J also relate Cauchy transforms to dual extremal problems. PROPOSITION 4.1.5. For a given

I

E Ll, there is a unique 9 E HI with

PROOF. Our first step is to show that a best approximant 9 E HI exists. Let lif - gnlll --+ dist(f,HI) as n --+ 00. This means that the HI norms of gn are uniformly bounded and so from eq.(1.9.3), (gn)n~l is a normal family on D. Passing to a subsequence if necessary, we can assume that 9n converges to an analytic function 9 pointwise on D. Moreover, for any 0 < r < 1, (gn)n~l C HI with

!

Ig{r()1 dm«)

~ n~! 19n(r()1 dm«) ~ n~! 19n«)1 dm{()

(integral means increase in r)

~ sup IIgnlit n

=c for some constant c independent of 0 < r < 1. Now take a supremum in r to conclude that 9 E HI. If P(f - 9)(r() is the Poisson integral of f - 9 evaluated at

4.1. THE NORM TOPOLOGY

r(, note that P(gn)(r()

85

= gn(r() --+ g(r() as n --+ 00 and so

J IP(f - g)(r()1 dm«()

~ !~~ J

IP(f - gn)(r()1 dm«()

= !~ J

IJ

Pr,(~)(f(~) - gn(~)) dm(~) Idm«()

~ n~~J J Prd~)I!(~) - gn(~)1 dm(~) dm«()

n~~ J

=

=

(J

Prd~) dm«(») I!({) -

gn({)1 dm({)

lim IIf - gnlll

= dist(f, Hl).

Observe from Theorem 1.8.6 (Fatou's theorem) that IIf - gill

~ r~- J

IP(f - g) (r() I dm«()

and so, using the obvious inequality dist(f, Hl) ~ II! - gll1, we have II! - gll1 = dist(f,H l ).

(4.1.6)

Thus a best approximant 9 exists. We now argue that it is unique. By the F. and M. Riesz theorem (Theorem 1.9.7), the annihilator of Hl in Ll via the dual pairing

Jl9dm,

fELl,

gEL OO

is H(f. Thus, from Theorem 1.4.7, dist(f,Hl) = sup {IJ !Fdml : F E ball(HOO)}. Let Fn E ball(HIf) with

J fFn dm

--+

dist(j,H l ).

By the Banach-Alaoglu theorem, the sequence (Fn)n~l has a weak-* limit point F E ball(H(f). With 9 being a best approximant in eq.(4.1.6), note, since F E HOO, that

J gFdm=O and so dist(f,H l ) = J(f - g)Fdm

~ II! -

gllllWiloo

~ II! -

This string of inequalities says that

JU-9)Fdm= Jlf-9l dm and so (4.1.7)

U - g)F = I! - 91 a.e.

gilt = dist(j,H 1 ).

86

4.

TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

H f e HI, clearly the best approximant 9 must be equal to f and it is unique. Otherwise, the above equation says that F is the unique function in ball{H8") satisfying

!

fFdm = dist(f,H 1 ).

If 9 is any best approximant in eq.{4.1.6), we can use eq.(4.1.7) to say that

$;.)(gF)

= 9(fF)

and so $;.)(gF) is unique. But since gF e H8", this determines gF uniquely. Finally, IFI > 0 almost everywhere and so 9 is uniquely determined. 0 COROLLARY 4.1.8. For a given fELl, there is a unique 9

inf{IIf+hlll:

e HJ

so that

h E HJ} = IIf+gIl1.

REMARK 4.1.9. Before proceeding to the proof of Proposition 4.1.4, we would like to mention a particular extremal problem we will make use of later. If

p(z) = Co + C1Z + C2Z2 + ... + cnz n is an analytic polynomial, one can consider the extremal problem

L=

inf {lip - gill:

9 e HJ}.

We know from the previous corollary that this extremal problem has a unique solution go e What is interesting here is that one can actually compute go and L. When Cj = 1 for all j, this problem was explored by Landau as far back as 1913 [121, 122] where he was able to compute L Ln as

HJ.

=

n

Ln = ~)AjI2, j=O

where

(2j)!

Aj

= tV 0!)2 .

A estimate using Stirling's formula yields

(4.1.10)

Cl

log(n+ 2) ~ Ln ~ c210g(n+2),

n E N,

for some universal constants Cl, C2 > O. Putting this in another way, the Cauchy transform norm of 1 + z + Z2 + ... + zn can be estimated as

111 + z + z2 + ... + znll =

Ln

x log n.

See [79, p. 175] for a nice exposition of this. PROOF OF PROPOSITION 4.1.4. First note that for any ve Rf, (4.1.11)

IIfil = inf{IIdll + gdmll : 9 e HJ}

~

IIvll·

By the definition of the norm on X, there is a sequence (IIn )-n;;;':l of measures from. RI such that lllinil ~ Ilfil + lin. Since this sequence is uniformly bounded in total variation norm, it has a weak-* cluster point v e M (Banach-Alaoglu theorem). Hence, we can pass to a subsequence (Vn)n~l such that lim !9dVn = !gdV 't/g E C(1l').

n-+oo

4.1. THE NORM TOPOLOGY

Thus, for each

Z

E lIJ>,

f

= ~ dvn «() ~ 1- (z

fez)

f~ 1- (z

87

dv«(),

n

~ 00,

and so v E Rf. We now need to prove the equality IIvll = Ilfli. One direction (Ilfll : ; ; IIvll) comes from eq.(4.1.1l). For the other direction, observe from Proposition 1.6.2 that

For uniqueness, first notice that when v E Rf with

IIvll = IIfll,

I :~" + IIvsll = Ilvll = inf{lIv + hdmll : h E HH

= IIvs II + inf {II :~ + hill: hE HJ}

(by Corollary 1.3.10).

Thus

and by Corollary 4.1.8, this infimum is achieved precisely when h == O. Hence (4.1.12) If IL, v E Rf with

IIILil = IIvll = IIfll, then by Proposition 2.1.5,

(4.1.13)

ILs =

Vs

and consequently' (4.1.14) Since IL, v E Rf, Proposition 2.1.5 says that

dIL = dv +h, hE -HJ. dm dm By eq.(4.1.14) and eq.(4.1.15) along with eq.(4.1.12) we know that h == 0 and so (4.1.15)

dIL

dv

dm = dm' By eq.(4.1.13), ILs measure is unique.

= Vs

and hence IL = v. Thus the norm attaining representing

o

REMARK 4.1.16. One should be careful as to not over interpret this theorem. It does not say, for a particular measure IL, that IIKILII = UILII, although this is the case when IL E M+ (Corollary 4.1.3). Instead, it says that there is some (perhaps other) measure v with KIL = Kv such that IIKILII = IIvll.

This next result relates the growth near the boundary with the norm.

88

TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

4.

PROPOSITION 4.1.17. If f E X, then

If(z)]

(4.1.18)

~ II~l~I'

z E ]D).

PROOF. H f = Kp., then for all z E ]D),

IfCz)1 =

If

1

I

~ (z dl-'C') ~ f 11 ~ (zl dll-'I(() ~ II~l~I'

The desired estimate follows by taking I-' to be the unique I-' E RJ with (Proposition 4.1.4).

11111 = 1Ip.1I 0

The previous proposition implies that if (fn)n~l C X converges in norm to f, then this sequence also converges to f uniformly on compact subsets of D (as is the case with all the well-known Banach spaces of analytic functions on ]D). Thus ball (X) forms a normal family of functions on D. By the Lebesgue decomposition theorem (Theorem 1.3.9),

M=Ma+M., where

Ma:={I-'EM:I-'«m}.

The

Ms:={I-'EM:p.1.m}.

+sign here means that M

= {VI + ZI:z : VI E Ma. ZI:z EMs}

and Ma

n Ms = {OJ.

Moreover (Corollary 1.3.10),

where Thus

M=MaEBMs. We can use the above decomposition to rewrite X as

X=Xa+X s • where

Xa:= {Kp.: p. E Ma}, Xs:= {KI-': I-' EMs}. We will now see that in fact X = Xa ED X B • PROPOSITION 4.1.19. If V EMs, then

IIKvlI = IIvlI.

PROOF.

9E HJ} = inf {1I1111 + I/g/ll : 9 E HJ}

IIKp.11 = inf {1I 11 + gdml\

:

(since

II

1. m)

= \111\1.

o We leave it to the reader to generalize the above proof to obtain the following proposition.

4.1.

THE NORM TOPOLOGY

89

PROPOSITION 4.1.20. Suppose J.I. = J.l.a + J.l.s. Then

IIKJ.l.II = IIKJ.l.a;1I + IIKJ.l.sl1 = IIKJ.l.a;1I + IIJ.l.slI· In particular, Xa is isometrically isomorphic to L1 / HJ while Xs is isometrically isomorphic to Ms. . The above says that Xa and Xs are indeed closed subspaces of X and that

X=Xa$Xs' PROPOSITION 4.1.21. (1) Xa is separable. In lact, the analytic polynomials are dense in Xa. (2) Xs is not separable. PROOF. For any I E LI, note that the Cesaro sums uN(f) approximate the L1 norm (Theorem 1.6.5) and so, by the definition of the norm on X,

lIuN(f)+ -

1+11 ~ lIuN(f) -

1111

-+

0 as N

I

in

-+ 00

and so the analytic polynomials are dense in Xa. Since Xs is isometrically isomorphic to Ms and, by means of eq.(1.6.6), Ms is not separable, we conclude that Xs is not separable. 0 One can use the F. and M. Riesz theorem along with Theorem 1.4.7 to prove the following. THEOREM 4.1.22. The dual ql L1 / HJ is isometrically isomorphic to HOO. A pairing between these two spaces is given by

([/l,g):= As a consequence, the dual

J

fgdm,

01 Xa

[/1 E L1/HJ, g E H oo •

can be identified with HOO by the pairing

(I+,g):=

J

fgdm.

Let us compute, as it will be used later, the X-norms of the functions 1 zl-+-1 _,

-az

aElIr.

When a E 1', then 1 -1 - = (KcSa)(z)

-az

and so, since

When

cSa

.1. m, we can use Proposition 4.1.20 to get

lal < 1,

1!az = (1 ~a() + (z) and so, using Proposition 4.1.20 again and the definition of the norm in L1/ HJ, we obtain

4.

90

TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

Using the dual pairing in Theorem 4.1.22 along with the dual extremal setting of Theorem 1.4.7, we conclude that

inf

{Ill ~

a( -

9111 : 9 E HJ}

{If

= sup I«() 1 ~ a( dm«()1 : 1 E OO11(H';")} = sup {If(a)l : 1 E ball(HOO)} =1. Thus for a E D-, (4.1.23)

111

~azll = 1.

REMARK 4.1.24. One can avoid the above argument by noticing that for each

aED,

and so

1 -1 - =K(Padm)(z).

-az

But since Pa dm E M+. we can apply Corollary 4.1.3 to see that

111 ~ az II = IlPa dmll = IlPa III == 1. We end this section with a remark about reflexivity. PROPOSITION 4.1.25. X is not reflexive. PROOF. Since X is isometrically isomorphic to M / HJ and reflexivity is preserved under isometric isomorphisms, we just need to show that M / HJ is not reflexive. By Theorem 1.4.11 (subspaces of reflexive spaces must be reflexive), we reduce the problem to showing that L1 / HJ is not reflexive. By Theorem 4.1.22 we can use Theorem 1.4.11 once again (the dual of a reflexive space is also reflexive) to reduce the problem to showing Hoo is not reflexive. Since L1 / HJ is separable (L1 is separable), we can use Proposition 1.4.13 to reduce the problem to showing that Hoo is not separable. For this last detail, let

be a family of atomic inner functions. A computation shows that

o

and so by'Proposition 1.4.12, Hoo is not separable.

=

1Actually, if one is willing to work a hit harder, one can show that II" - !{lIoo 2. To see this, note that ft.. is analytic at ( with I!{«)\ = 1 while" has JI) as its cluster set at (.

4.2. THE WEAK-* TOPOLOGY

91

4.2. The weak-* topology Let us apply the basic functional analysis layed out in Theorem 1.4.6, i.e.,

X* IS1. ~ S*, to the case where X = G(T), the continuous functions on T. By the Riesz representation theorem (Theorem 1.3.6), the mapping from M to G(T)* defined by J.1.~ Lf,.t, where Lf,.t(f) :=

J

fdJ.1.,

is a conjugate linear isometric isomorphism. If A, the disk algebra, denotes the functions f E G(T) which have continuous extensions to 0- which are also analytic on D, one notices that

A1. =

{J.£ EM:

J

fdJ.£

=

0 for all

I E

A} .

Letting I«) = (n for n E No in the previous equation, we see, by the F. and M. Riesz theorem (Theorem 1.9.7), that A 1. -- HIo' Thus from Theorem 1.4.6,

A* ~ MIHJ 2and the dual pairing between A and M I HJ is

(I, [Jt]) =

(4.2.1)

J

fdJ.1..

By the discussion in the previous section, the map KJ-L ~ [JLl is an isometric,isomorphism between X and M 1HJ. Put this all together to obtain the following. THEOREM

4.2.2. A * is conjugate linearly isomorphic to X with dual pairing

(f, KJL) =

J

fdJL.

A computation with power series shows that this pairing can be written in the more familiar Cauchy pairing (4.2.3)

Indeed,

r~xr- ~ f(n)jL(n)r n ~ r~- ~ f(n) (J (ndJ.£(») rn =

r~- J(~1
=

r~xr-

=

J

J

f(r()dJL«)

fdJL.

2The duality A* e! M/ HJ has an analog in higher dimensions [181, p. 202).

92

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

We can use the above pairing to prove an interesting result about the norms of Cauchy transforms. For f E HP, recall from eq.(1.9.2) that when fr(z) := f(rz), the integral means, II/rilp increase as r --+ 1- and Ilfrilp ~ 11/111" The same is true for the norms of Cauchy transforms. PROPOSITION

4.2.4. If 1= KJ.L, and 0

We will first show that IIfrll ~

PR.OOF.

II/rlll ~ 1111"211· 11/11 for each 0 < r < 1. If 1= KJ.L and

< rl < r2

~

I, then

9 E A, then 00

Ir{z) ::::

L rnji(n)zn, n=O 00

gr(z) =

E rng(n)zn n=O

and so by eq.{4.2.3), Furthermore,

Ilfril = sup {1(/r,9)1 : 9 E ball (A)} = sup {I(I,gr)1 : 9 E ball (A)} ~ sup {l1/l1l1grlloo : 9 E ball(A)} ~ IIfll· Now suppose r1 < T2 < 1. Then with 8 = rl/r2' the above estimate shows that

o 4.2.5. A sequence (9n)n;;n C X converges to 9 E X weak-* if and 9 pointwise on II) as n - 00 and the sequence (1I9nll)n~1 is bounded.

PR.OPOSITION

only if gn

-+

Let 9n = KJ.Ln and 9 = KJ.L. Suppose gn -+ 9 weak-*. This means (I, g) for each lEA. In particular, for each f E A, the sequence «(f, gn) )n;;~l is bounded. By the principle of uniform boundedness (Theorem 1.4.2), (1I9nll)n~1 is a bounded sequence. To show pointwise convergence, notice thB:t for each fixed z E 11), the function ( t-+ (1 - (Z)-1 belongs to A and PROOF.

that (f, gn)

(4.2.6)

--+

/1

\ 1- (Z,gn - 9

)Jl =

--

1- (Z d(J.Ln - J.L) = 9n(Z) - g(z)

and so (since gn --+ 9 weak-*), 9n -+ 9 pointwise in]()). Conversely, suppose that 119nll ~ c for all n and that gn(z) - g(z) for each z E 11). Let B be the set of all finite linear combinations of the functions

1

(t-+ 1- (Z'

z

E 11),

and notice from eq.(4.2.6) that (h,gn) --+ (h,g) for each h E B. Assume for a moment that B is a dense subset of A (we will prove this shortly), and observe that

4.2.

if I E A and choice of h,

f

THE WEAK-* TOPOLOGY

93

> 0 are given, we can find an h E B with III - hll oo < f. With this

l(f - h,gn - g} + (h,gn - g)1 ~ III - hlloollgn - gil + i(h,gn ~ f(C+ IIgll) + I(h,gn - g)l·

l(f,gn - g}1 =

g}1

Since I(h,gn - g)1 -+ 0, the proof (except for the proof that B is dense in A) is complete. To show B is dense in A, note that a measure v belongs to B1. if and only if 0=

f

1 ~ 1- (Zdv = ~znv(n)

Vz ED.

n=O

By the F. and M. Riesz theorem, this takes place precisely when v E HJ Thus B1. = A1. and, by the Hahn-Banach theorem, the proof is complete.

= A1. . 0

REMARK 4.2.7. Another proof of Proposition 4.2.5, which can be applied to other spaces of analytic functions, can be found in [34, Prop. 2]. In the previous section we saw that X, endowed with its norm topology, is not separable (Xa is separable but Xs is not). PROPOSITION 4.2.8. X, endowed with the weak-* topology, is separable. In fact, Xa and Xs are each weak-* dense in X. PRoliF. When M and M / HJ are endowed with their respective weak-* topologies, eq.(4.2.1) shows that the natural map 1r :

M

-+

M / HJ ~ X

is weak-* continuous. The fact that Ma and Ms are weak-* dense in M (Proposi0 tion 1.6.7) finishes the proof. REMARK 4.2.9. (1) If 9 = KJL and O"n(g) is the n-th Cesaro mean of g, that is O"n(g) = O"n(JL), then O"n(g) -+ 9 weak.-* in X as n -+ 00. One can see this directly by the identities

(f,O"n(g)} = where

f

100n(J.L) dm =

f

O"n(J) dJ.L,

I E A [101, p. 20]. This last integral converges to

f

IdJL = (f,g).

(2) From the proof of Proposition 4.1.21, we saw that if IE L1, then O"n(f)+, the n-th Cesaro mean of i+, converges to 1+ in the norm of Xa. However, if JL .1 m and JL t= 0, then O"n(KJL), although it converges weak-* to K J.L, does not converge in norm to anything. The reason for this is that O"n(KJL) E Xa and Xa is a norm closed subspace of X (Proposition 4.1.20). Thus if O"n(KJ.L) has a norm limit F, it must belong to Xa. However, it is easy to check that O"n(KJL) converges pointwise to KJL and so KJL = F E Xa which is a contradiction to the fact that KJL E Xs.

94

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

4.3. The weak topology, A topological vector space ~ is complete if every Cauchy net (see [113, p. 190] for a definition) in 11 converges to an element of~. If ~ is a Banach space, it is complete by definition. On the other hand, if ~ is either (X, wk) or (X*, Wk4), that is, a Banach space X endowed with its weak/weak-* topology, then ~ is complete if and only if '0 is finite dimensional [142, p. 215, p. 226]. Thus in the weak and weak-* topologies, questions about completeness are irrelevant. There is, however, a more interesting notion of weak/weak-* sequential completeness. DEFINITION

4.3.1.

(1) A sequence (Xn)n;?;l in a Banach space X is weak Cauchy if the numerical sequence (i(Xn))n;?;l is a Cauchy sequence for every i E X*. A sequence (in )n;?;1 c X* is weak4 Cauchy if the numerical sequence (in (X))n;?;1 is a Cauchy sequence for each x E X. (2) X is weakly sequentially complete if every weak Cauchy sequence in X converges weakly to some element of X. X* is weak-* sequentially complete if every weak-* sequence converges to some element in X*. As a consequence of the Principle of Uniform Boundedness and the BanachAlaoglu theorem, X* is always weak-* sequentially complete whenever X is a separable Banach space. Thus the more interesting topic to explore is weak sequential completeness. The classical Lebesgue spaces LP, 1 < p < 00, are weakly sequentially complete since they are reflexive and so the ~eak and weak-* topologies coincide. For the same' reason, the Hardy spaces HP, 1 < p <.00, are weakly sequentially complete. The non-reflexive space L1 is weakly sequentially complete but for a different reason [231, p. 140][60, p. 91]. From here one can argue that Hl is weakly sequentially complete. In fact, an arbitrary L1(Q, E, /-t) space is weakly sequentially complete. This will be important in a moment. The space e[O, I], however, is not weakly sequentially complete. One can see this by observing that the functions fn(t) = (1 - t)n satisfy lim jfnd/-t = /-t({I})

n--oo

for each measure /-t on [0,1] and so (fn)n;?;1 is a weak Cauchy sequence. However there is no continuous f such that

!

fd/-t = /-t({l})

for every measure /-t. When 0 < p < 1, the Hardy spaces are not weakly sequentially complete [66]3. Although the space of Cauchy transforms X does not have a readily identifiable dual space (as a Banach space of analytic functions on JI}), it is weakly sequentially complete. We would like to very briefly discuss this result, which is often called Mooney's theorem. The key to this is the following. 3Technically, these spaces are not Banach spaces. However, the dual of HP can be identified with a non-trivial algebra of analytic functions on ]() and from here, the weak topology can be defined. The reader can find all of this done quite precisely in [66].

4.4. SCHAUDER BASES

95

THEOREM 4.3.2 (Mooney). Let (I"/Jn)n;;;'l be a sequence in Ll such that the limit lim [ I"/JnJ dm = L(f) n-+oojT exists for every

I

E Hoc. Then there is a function 4> E Ll such that

L(f)

=

h

I"/JJdm Vf E H oo .

Mooney's theorem was discovered independently by Mooney [144] and Havin [93]. Proofs can also be found in [79, pp. 206-209] or [118]. How does this prove that X is weakly sequentially complete? The above theorem, along with Theorem 4.1.22, is just the statement that Ll / HJ is weakly sequentially complete. Now notice from Proposition 4.1.20 that X ~ Ll / HJ EEll Ms

and so (Theorem 4.1.22)

X*

~

HOC

EEll

M:.

sum4 •

We use EEll to denote an exterior direct Suppose (4)n)n;;;'1 is a weak Cauchy sequence in X. Each 4>n has a unique decomposition as 4>n = tPn + Vn , where tPn E Xa and Vn E Xs. Because of the direct sum decomposition of X*, it is evident that each of the sequences (tPn)n-;;;'l and (Vn )n-;;;'1 is a weak Cauchy sequence. By Mooney's theorem, the first sequence converges weakly to some tP E Xa. On the other hand, a theorem of Kakutani says that Ms is isometrically isomorphic to Ll(n, lJ, 1-') for some abstract ,measure space (n, lJ, 1-') [110]. It follows that Ms is weakly sequentially complete since every such space Ll (n, E, 1-') is weakly sequentially complete [60]. The second theorem we wish to present on the weak topology in X is a deep result due independently to Delbaen [59J and Kisljakov [115]. For each f E X, recall from Proposition 4.1.4 that 1-'1 is the unique measure such that I = KI-'I and

11/11

=

111-'111·

THEOREM 4.3.3. Let W be a weakly compact set in X, and let

W = {I-'I:

fEW}.

Then W is relatively weakly compact in M, that is to say, the weak closure of W is weakly compact.

A thorough discussion of this theorem can be found in [157, Ch. 7] or [231]. This theorem derives its significance from the Dunford-Pettis characterization of the weakly compact subsets in M: W c M is weakly compact if and only if it is norm bounded and unifonnly absolutely continuous (cf. [60]). 4.4. Schauder bases A sequence (X n )n;;;'l in a Banach space X is called a Schauder basis for X if every x E X can be written uniquely as

4A EBl B = ({a, b) : a E A, bE B} with norm II (a, b) II = lIaliA

+ IIbIiB.

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

96

where c,.,. are complex numbers and the = sign means convergence in the norm of X, i.e., N

lim

N-oo

x - L..J '" c,.,.Xn = O. n=1

The sequence space lP, 1 ~ P < 00, has a Schauder basis (en)n~b where en(j) = 6n,j' The space of continuous functions e[o, 1] has the classical Schauder basis discovered by J. Schauder [189J (see also [142, p. 352]). Schauder [190J (see also [142, p. 361]) also proved that the LP spaces for 1 ~ P < 00 have a basis, the Haar basis. As a consequence of this, and the fact that HP is isomorphic to V, one can show that HP, 1 < p < 00, has a basis. Maurey [138] proved that HI has an unconditional basis. Carleson [39] explicitly constructed such a basis. Wojtaszczyk [231J provided further improvements. Boekarev [29] discovered a basis (bn)n~1 for the disk algebra A with the additional property that

J

bnbk dm = 6n,k'

(4.4.1)

We will make use of this in a. moment. The existence of a Scha.uder basis is not automatic since there are separable Banach spaces (even reflexive ones) without a basis [67J. We refer the reader to [60, 142, 157, 231] for more on bases. A sequence (.en)n~l C X* is called a weak-* Schauder basis if every I. EX· can be written uniquely as

n=1

where dn are complex numbers and = in the above equation means weak-* convergence, i.e.,

for each x E X. For a Schauder basis (Xn)n~I' there is a natural sequence linear functionals defined by

(X~)n~l

of continuous

00

x~(x)

= c,.,.,

where x

=L

CnXn

n=1

(remember that the expansion of x is unique and so x~ is well-defined). The fact that the functionals x~ are continuous is a deep theorem of Banach [60, p. 32]. PROPOSITION 4.4.2. If (Xn)n~1 is a Schauder basis for X, then weak-* Schauder basis for X· . PROOF.

For lEX·, let dn:= i(xn ) and for each N N

iN:=

LdnX~. n=1

E

N let

(X~)n~l

is a

4.4. SCHAUDER BASES

97

Then for any 00

X= LCnxn EX, n=1 N N iN(X) = ~dnx~(X) = ~dnCn Thus iN

-+

l weak-* as N

-+ 00

N (N) = ~i(Xn)Cn =£ ~XnCn

.

and so every l E X* can be written as 00

l= LdnX:. n=1 The uniqueness follows from the identities X:(Xk) = On,k' Hence (X:)n;)1 is a weak-* Schauder basis for X*.

o

What is more difficult to prove (and we refer the reader to [60, p. 36] for the details) is the following. PROPOSITION 4.4.3. If {Xn)n~1 is a Schauder basis for X, then {X:)n~1 is a Schauder basis for its closed linear span in X* .

We will now apply the previous two propositions to X = A and X* ~ X to identify a weak-* Schauder basis for X and a Schauder basis for Xa' We will use the Boekarev basis (bn)n~l for A. To clarify notation, we let

Bn := (bn )+. Certainly Bn = bn as analytic functions on IDl. We use this notation to avoid confusing, bn , the element of the disk algebra A, with B n , the element of X (in fact Bn E Xa). The result here is the following. PROPOSITION

4.4.4.

(Bn)n~1

is a weak-* Schauder basis for X and a Schauder

basis for Xa. PROOF.

From eq.(4.2.3), the dual of A can be identified with X using the

pairing

(j, KJ-L)

= / f dJ-L.

So if f E A is written in terms of its Schauder basis (the Boekarev basis)

n=1

then for each k E fIl, we can use the orthogonality in eq.{4.4.1) to see that

(j,Bk)

=/

(~Cnbn) bkdm = ~Cn /

bnbkdm = Ck·

Note that passing the sum through the integral is justified since the sum converges in the norm of A (i.e., uniformly). Thus, via our linear pairing, Bk can be identified with the linear functional bt: (f) = Ck. Applying Proposition 4.4.2 we have shown that (Bnk~l is a weak-* Schauder basis for X. In order to prove that (Bn)n~1 is a Schauder basis for X a, we notice that Bn = (bn )+ E Xa and so, using Proposition 4.4.3, it suffices to show that the closed

98

4. TOPOLOGIES ON THE SPACE OF CAUCHY TRANSFORMS

linear span of (Bn)n~l is all of Xo. From Theorem 4.1.22, with Hoo by means of the dual pairing

(/+,g) =

f

Jgdm,

f

E

L1,

X: can be identified

9 E H oo .

Thus if 9 E Hoo and annihilates every B n , then 0= (Bmg)

= «bn )+, g) =

f

bnydm 'tin E N.

But since the closed linear span of the bn's is the disk algebra A, we have

f

(ny dm = 0 'tin E No

and so g(n) = 0 for all n E No. This me~ that 9 == O. An application of the Hahn-Banach separation theorem completes the proof.

o

CHAPTER 5

Which functions are Cauchy integrals? 5.1. General remarks Which analytic functions on ]])) belong to the space of Cauchy transforms X? Gathering up our observations from the previous three chapters, here are some necessary conditions a Cauchy transform must satisfy. PROPOSITION

5.1.1. Suppose f = KiJ. for some iJ. E M. Then

(1) f satisfies the growth condition

If(z}1

~ 1 ~iz"

z E ]])).

(2) f has finite non-tangential limits m-almost everywhere on T and

m(lf! > y} ~ C" y

y > O.

(3) f E HP for all 0 < p < 1 and

IIfllp = (4) If f

= En;ll:oanz n ,

0(1 ~p)' p-1-.

then (an)n;ll:O is a bounded sequence of complex num-

bers. None of the above conditions is sufficient. The above necessary conditions can only be used to determine which analytic functions on ]])) are definitely not Cauchy transforms. Known necessary and sufficient conditions are difficult to apply and in a way, the very question is unfair. For example, suppose that f is analytic on ]])) with power series

fez} = ao + atZ + a2 z2 + ... and we want to determine whether or not

f=

K iJ. for some IL EM. Since

(KiJ.)(z) = j1(0) + jL(l)z + j1(2}Z2 ... , we would be trying to determine, by equating an with j1( n) for n E No, the measure IL from only 'half' its Fourier coefficients, the non-negative ones.

5.2. A theorem of Havin If one is willing to settle for a functional analysis condition, there is an old characterization of X [91], albeit difficult to apply. 99

5. WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

100

THEOREM

5.2.1 (Havin). Suppose 00

1= Lak zk k=O

is analytic on D. Then the lollowing statements are equivalent. (1) There is some constant C > 0, depending only on I, such that

I~Akak ~cmax{ ~Ak(k

(5.2.2)

:(ET}

lor any complex numbers Ao,· .. , An. (2) 1= Kp lor some p E M. PROOF.

Recall that if I = K p, then

I(z)

=

f ---dJ.L«() = 1

1- (z

00

Lzk

J

00

tdp«() = LI1(k)Zk

k=O

k=O

and so the Taylor coefficients of I are equal to the (non-negative) Fourier coefficients of p. Also recall from eq.(4.2.3) that the dual of A (the disk algebra) can be identified with X via the Cauchy pairing 00

gEA.

More specifically, 00

r~~- Ly(n)p(n)rn ~ Cpllgll oo •

(5.2.3)

k=O

To prove (2) ~ (1), let I = Kp and Ao, At.··· , An be given complex numbers. With g(z) = Ao + AIZ + ... + Anzn, the inequality in eq.(5.2.2) follows from the inequality in eq.{5.2.3). To prove (I) ~ (2), the hypothesis imply that the linear functional i, defined first on polynomials p{z) = Ao + AIZ + A2z2 + ... + Anzn by

10;=0

extends to a bounded linear functional on the disk algebra A. Hence i(P) = (p, f) for some I = K p. Thus

o

and the result follows.

5.3. A theorem of Tumarkin Instead of asking whether or not an analytic function defined only on D is a Cauchy transform, suppose we were to ask whether or not an analytic function I on T is a Cauchy transform

<\

(CJ.£){z) =

J~ l-(z

dp«(),

z E C\ T.

A THEOREM OF TUMARKIN

53

101

This is a more tractable question since we would be comparing 00

00

(fID)(z) = Lan zn with (CplD)(z) = L n=O

and 00

(fIDe)(z) = L n=l

jL(n)zn

n=O

a- n zn

with (CpIDe)(z) = -

f

jL(-n)

n=l

ZR

which would involve knowing all of the Fourier coefficients of p and not just the non-negative ones as before. An early result which answers this question is one of Tumarkin [220J (see [133J for a generalization). f

THEOREM 5.3.1 (Tumarkin). Let f be analytzc on C\ T unth f(oo) = O. Then for some p E M zf and only If

= Cp

(5.3.2)

sup

f

If(r() - f«fr)! dm«()

< 00.

O
PROOF. Writing 00

fez) = LllnZn,

ZED,

n=O 00 b few) = ~~,

~wn n=l

w EDe ,

a power series computation shows that for any 0 < r < 1,

f

(5.3.3)

If

f(r() - f«(/r) dm«() = {f(rz), 1 - (z f(zlr) ,

z E ]I), z E De.

Assuming the hypothesis in eq.(5.3.2), the measures dJLr = (f(r() - f«(/r»dm«()

are uniformly bounded in the norm of M and so, by the Banach-Alaoglu theorem, some subsequence dPr.. (where rn /' 1) converges weak-* to a measure dp. Passing to the limit in eq.(5.3.3) says f = Cp. Conversely, if f = Cp, a computation shows that for any (E '][' and 0 < r < 1, f(r() - f«(fr) =

and so

1r

If(r() - f«(fr)! dm«() :E;;

J ih

Prc(w) dp(w)

Prc(w) d!pl(w) dm«().

Now use Fubini's theorem along with the identity

h

Prdw) dm«() = 1

to obtain the result.

o

5 WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

lO2

5.4. Aleksandrov's characterization We now present a refinement of Tumarkin's theorem due to Aleksandrov [10] which identifies the type of measure used to represent a Cauchy transform. Before doing this, we need a definition and a few reminders. For 0 < p < 00, let HP(C\ 'J!') denote the analytic functions on the disconnected domain C\ 'll' for which

IIfIlHP(C\T)

:=

{sup

f

rio11T

If(r()IP dm(()}l

/P

< 00.

Recall from Smirnov's theorem (Theorem 2.1.10) that if f := CJ-L for some J-L E M, then f E HP(C \ 'll') for all 0 < p < 1. Also recall the jump function (Jf)(():= lim (f(r() - f((/r)) r-+l-

which exists for almost every (E 'll' and, via Fatou'sjump theorem (Corollary 2.4.2), is equal to the Radon-Nikodym derivative dJ-Lldm. In summary, a Cauchy transform f = C J-L on C\'ll' satisfies the four conditions (5.4.1)

f(oo) = 0,

(5.4.2)

fEn HP(C\ 'll'), O
(5.4.3)

C~ p)'

p

r,

~ IIfIlHP(C\T) = 0 (1 ~ p)'

p

IlfIlHP(C\T) = 0

-+

(5.4.4) Also recall from Proposition 2.1.15 that

J-L« m

-+

1-,

and, from Fatou's jump theorem, that

J-L -1 m

~

J f = 0 m-a.e.

THEOREM 5.4.5 (Aleksandrov). If f is an analytic function on C\ 'll' satisfying the conditions in eq.(5.4.1} through eq.(5.4.4} above, then f = CJ-L for some J-L E M. Moreover, if the conditions in eq.(5.4.1} and eq.(5.4.2} are satisfied, then (1) f = CJ-L for some J-L« m if and only if lim II f II HP(t\T)(1 - p) = 0

p-.1-

and Jf ELI. (2) f = CJ-L for some J-L -1 m if and only if lim IIfIlHP(c\T)(I- p) p-+l-

and J f = 0 m-almost everywhere.

< 00

5 4 ALEKSANDROV'S CHARACTERIZATION

103

The proof of this theorem requires a few preliminaries. We first need some basic facts about subharmonic functions. Two good references for this are [79, 96]. A function u : nee -+ [-00, 00) is subharmomc if u is upper semicontinuous on n, that is, u(zo) ~ lim u(z), Zo E n, z---+%o

and satisfies the following sub-mean value property: given Zo E ~(Zo,s)- en, the following inequality holds

u(zo) ~

(2w

10

n and s > 0 with

dt

U(Zo + sed) 21r·

The mean-value property for harmonic functions says that u is harmonic if and only if equality holds for every s. Here are two important examples of subharmonic functions. If I is analytic on n and p > 0, then I/IP is subharmonic on n. If u is subharmonic on C and I is analytic on n, then u 0 f is subharmonic on n. For a subharmonic function u on n, we say that a function U on n is a harmomc ma)orant for u if U is harmonic and u ~ U. A harmonic function U on n is called the least harmomc ma)orant of u if U is a harmonic majorant and U ~ V for any other harmonic majorant V of u. The Perron construction [7, p. 248] says that if u has a harmonic majorant, it has a least harmonic majorant If we focus our attention to the unit disk ]D), determining whether or not a harmonic majorant exists and computing its least harmonic majorant, when it does, is not too difficult. Indeed, if u is subharmonic on]D), then the integral means M(r; u):=

increase as r /

h

u(r() dm«)

1-. This next result is found in [79, p. 38]

LEMMA 5.4.6. A subharmomc functwn u on]D) has a harmomc ma)orant ~f and only ~f the zncreaszng zntegral means M (r; u) are bounded as r -+ 1-. The least harmomc ma)orant ~s then

U(z):= hm (pz«()u(r() dm«(). r-+l-

1T

With these preliminaries about subharmonic functions in place, we begin the proof of Aleksandrov's theorem by proving a few technical lemmas. Define, for o < p < 1, the following function Gp on C.

(5.4.7)

Gp(z)

:= {lzlP cos (p9(z))

o

if z =f:. 0, , if z = O.

where (5.4.8)

arctan(Y/lxl) { 9(z).= 1r/2 -1r /2

Note that 19(z)1 ~ 1r/2 and 0 < p

(5.4.9)

x =f:. 0,

x=O,Y>O, x = 0, Y

< o.

< 1 and so Gp

~

O.

The following technical lemma of Pichorides [159] was used to find the best constants in the LP estimates for the conjugation operator (see Theorem 3.7.4 presented earlier).

5 WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

104

LEMMA

5.4.10. The function Gp is subharmonic on C.

PROOF. Clearly Gp is continuous on C and so it suffices to show that given any z E C, there is an r(z) > 0 such that

Gp(Z) ~ - 1 111" Gp(Z + re it ) dt, for all r E (0, r(z)). 211" -11"

(54.11) Since

G (z) =

Gp is harmonic on {iRz {iRz f. OJ. If Z = 0, then

{~(ZP)

for )arg(z») < 11"/2, for! arg( -z)! < 11"/2,

~(-z)P

p

f. O} and thus the inequality in eq.(5.4.11) holds on the set

. dt = 2 ( - 1 111"/2 r Pcos(pt) dt ) = -2rP sin( p7r - 1 111" Gp(re't) -2 ) ~ 0 211" -11" 211" -11" /2 p7r since 0 < p < 1. But Gp(O) = 0, by definition, and so the inequality in eq.(5.4.11) is satisfied. Letting Hp(z) = iRzP on {z : ! arg z! < 11", z f. OJ, we first notice that Gp -Hp ~ o on this set. To see this, observe that

Hp(z) = iR(zP) = Gp(z),

Z

!¢! ~

= rei"',

11"/2, r > O.

11"/2 < ¢ < 11", then Gp(z) - Hp(z) = r P cos (p (¢ - 11"» - r P cos(p¢)

If z = re i , where r > 0 and

= 2rP sin (p (¢ -

i) )Sin(p~

)

~o.

If z = rei , where r > 0 and -11" < ¢ <

-11"/2,

then

Gp(z) - Hp(z) = r Pcos (p(¢ + 11"» - r Pcos(P¢) = - 2rP sin

(P (¢ + ~) ) sin(p~ )

~O.

But Hp is harmonic on its domain, and so, for y

f. 0 and r

E

(O,ly!),

Gp(iy) = Hp(iy)

which proves the inequality in eq.(5.4.11).

o

The next lemma is a standard real analysis exercise. LEMMA

n E N. If hn as n ~ 00.

5.4.12. Suppose q > 1 and (hn)n~l c Lq with IIhnll q ~ C for all ~ h almost everywhere as n ~ 00, then h E Lq and hn ~ h weakly

5 4 ALEKSANDROV'S CHARACTERIZATION

105

By Fatou's lemma,

PROOF.

f

Ihl qdm

iT

= f lim Ihnl q dm ~ lim f Ihnl qdm ~ C q < 00

Jr n->oo

n->oo iT

andsohELq. Let f > 0 and 9 E LP be given (lip + 11q = 1) and note, by basic properties of the integral, there exists a 0 > 0 such that for any measurable set A c 1l' with meA) < 0, we have

(L IglPdm) By Egorov's theorem, there is a set E uniformly on E. With this in place,

f


c 1l' with m(1l' \

The last integral converges to zero smce hn

and, since

IIp

~

E) < 0 such that hn

~

h

h uniformly on E. Thus

o

was arbitrary, the result follows.

The following technical lemma is the key to proving Aleksandrov's theorem. LEMMA 5.4.13. Suppose f E HP lor all 0

< p < 1, f(O)

lim (1 - p)ll/lI~ <

(5.4.14)

E JR, 'fRf E Ll, and

00.

p->l-

Then

fez)

=

f ~ + z dJl(()

Jr",-z

for some real measure Jl E M sat'tSfyzng

ilJlII

~

lI'fRfll£1

+ ~ p~~_ (1 - p)II/II~·

Moreover, tf lim (1 - p)lIfll~ = 0, p-+l-

then dJl =

~fdm.

PROOF. Notice that Gp 0 I ~ 0 on JI)) (since Gp ~ 0 - see eq.(5.4.9». Moreover, since Gp is subharmonic on C (Lemma 5.4.10) and I is analytic on JI)), Gp 0 f is subharmonic on JI)) •

106

5 WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

Assume for a moment that !RI = 0 almost everywhere on T. For q E (I,l/p) and r E (0,1), the Lq integral means of Gp 0 I satisfy

h

(Gp 0 f)(r()qdm«()

h : :; h

I/(r()lpq (cospO(f(r()))q dm«()

=

I/(r()lpqdm

=

Ilfll:g

which is finite since 0 < pq < 1. Thus the function Gp 0 I is subharmonic on D with uniformly bounded Lq integral means. By Holder's inequality, these integral means are uniformly bounded in L1. By Lemma 5.4.6, the function

a~

a r~~- h

r~~-l P

is the least harmonic majorant for G p

(Gp 0 f)(a) :::;

«() (Gp 0

0

f) (r() dm«()

f. Thus for all a E D, Pa«()(Gp 0 f)(r() dm«().

-+ f«() almost everywhere as r -+ 1- and Gp is continuous, (G p 0 (Gp 0 f)«() almost everywhere as r -+ 1-. This, together with the uniform boundedness of the Lq means of Gp 0 I, say, via Lemma 5.4.12, that (G p 0 f)(r·) -+ Gp 0 I weakly in Lq. Thus

Since I(r()

f)(r()

-+

f =f

(Gp 0 f)(a) :::;

Pa«()(Gp 0 f)«() dm«() Pa«()!f«()IP cos (P27r ) dm«(),

a E D.

In the last integral above, notice that we are assuming R/«() = 0 and so argf«() = ±7r/2. Since cos(p7r/2) ;:::: 1 - p, we can use our assumption in eq.(5.4.14) to observe that

p~~_

h

If«()I P cos

This means that for some sequence Pn /

(~) dm«() < 00. 1, the measures

dVPn := Ifl Pn cos

(P;7r) dm

satisfy lim IIvpn II

(5.4.15)

n---+oo

= lim (1/«()iP cos (P7r) dm«(). p---+l- iT

2

Since these measures are uniformly bounded in total variation norm, they have a weal<-* cluster point dv. However, by the definition of Gpn from eq.(5.4.7), 0:::; lim (G pn n---+oo

0

f)(a) = I/(a)1 cos(argf(a» = R/(a).

From the estimate 0:::; (G pn

0

f)(a) :::;

J

Pa«() II «()iPn cos

(P; 7r) dm«(),

5 4 ALEKSANDROV'S CHARACTERIZATION

we get, from weak-* convergence,

o ~ rRf(a) ~

(5.4.16)

l

107

Pa«) dv«().

Moreover,

= nI:.~

~

hIfI

P"

cos

(P;1f) dm

[lflPcos (~) dm

lim

p--+l- iT

= lim cos(p7r/2) (1 - p)lIfll~ 1- P

p--+11f

•

= 2" hm (1 -

p)lIfll~·

p--+l-

But, by eq.(5 4.16), rRf is a positive harmonic function that is majorized by the Poisson integral of a positive measure. Thus the sequence of positive measures

{(rRf)(Tn ·)dm . Tn / I}

(5.4.17) satisfies

lI{lRf)(rn ·)dmll =

i

~

(lRf){Tn()dm«)

hh

Pr"dw)dv(w)dm«() = IIvll

and hence is uniformly bounded in r. Letting dJt be a weak-* cluster point of the sequence in eq.(5.4.17), we conclude that

(lRf)(a) = lim (rRf)(ra) r--+l-

=

r~T-

=

J

i

(lRf)(r()Pa«) dm«)

Pa«) dJt«).

Moreover, IIJtIl ~ lim

II (rRf)(rn ·)dmll

(by Proposition 1.6.2)

~ IIvll

~ ~ lim (1 - p)lIfll~. 2

p--+l-

The condition f(O) E R says that the harmomc conjugate of lRf is

(~f)(a) = and so

f(a)

= rRf(a) + iCJf(a) =

J

Qa«) dJt«)

J

(Pa«) + iQa«») dJt«)

=

J~ ~:

dJt«)

Notice how this last integral is the Herglotz integral H Jt of the measure Jt. For the general case, when rRf 0, let

t

h(z):= [ ; + z rRf«) dm«). JT .. -z

5

108

WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

f - h

We claim that the function with the extra property that Fatou's theorem,

~(f(()

satisfies the hypothesis of the theorem along - h(() = 0 m-almost everywhere. Indeed, by

i

~h(() == r~~- Pr,(~)~f(~) dm(~) = ~f«() for m-almost every ( E 'f. Furthermore, by using the fact, from Proposition 2.1.15 that lim (1 - p)llhllPp = 0, p .... l-

we see that lim(l- p)lIf - hll~ :s:; lim (1 - p)llfll~

p .... l

f-

p .... l

< 00.

From our above analysis applied to the function f - h, we see that the function h = H u for the measure u which is a weak-* cluster point of the measures ~(f

- h}(r .)dm

and, as such, satisfies lIull :s:;

~2 lim (1 - p)llf - hll~ :s:; ~ lim(l - p)lIfll~· 2 p--+l

p .... l

~fdm

Putting this together, we have f = HJ.L, where dJ.L:= IIJ.LII :s:;

+ du.

Also,

lI~fllLl + ~2 lim(l- p)lIfll~. p--+l

Finally, if limp--+l (1 - p)lIfll~ = 0, then u = 0 and f = H(~fdm).

0

PROOF OF THEOREM 5.4.5. Assume that f E HP(C \ 'f) for all 0 < p < 1, J f E L1, and f( (0) = O. Without loss of generality, assume further, by subtracting off a constant multiple of Cm, that 1(0) = O. Define the following analytic functions h,h onDby

h(z)

:= -i

(f(z)

+ f(l/Z)) , h(z):= I(z) - l(l/z)

and note that both h and h satisfy the hypothesis of the previous lemma. Hence there are real measures J.Ll and J.L2 such that

h(z) := Set J.L = iJ.Ll

fez) since

+ J.L2

J

(+z ( _ z dJ.Lj«().

and observe that

= -21 J ~+z dJ.L«() = J~dJ.L«() .. - z 1 - (z

J

dJ.L

1 -2

JdJ.L

= J~dJ.L«() 1 - (z

= iiI (0) + h(O) = iO + 0 = O.

This says that I = C J.L. If, in addition to conditions eq.(5.4.1) through eq.(5.4.4), we assume that lim (1 - p)II/Il~,,(c\ll') = 0,

p .... l-

then lim (1 - p)II/J II~ = 0 p--+l-

55

OTHER REPRESENTATION THEOREMS

109

and by Lemma 5.4.13, we can take the measures p,} above to be absolutely continuous. Assume f satisfies conditions eq.(5.4.1) through eq.(5.4.4) and also that J f = 0 almost everywhere. From the first part of the proof, we know that f = C p, for some p, E M. Now, by Fatou's jump theorem (Corollary 2.4.2), dp,

dm = J f = 0 m-a.e and so JL -L m. This completes the proof of Aleksandrov's theorem.

o

There is a related characterization of Cauchy integrals, also due to Aleksandrov

[9], that involves the space Ll,oo (see Chapter 1 for a definition). THEOREM 5.4.18 (Aleksandrov) Suppose f 1,S analyt?'c on Then f = C p, for some p, E M 'tf and only 1,f sup IIf(r() 11£1.= <

iC\1f wzth f(oo) = O.

00.

r,eI

5.5. Other representation theorems Let us refine the problem further. Suppose that E is a closed subset of'][' and f is analytic on C\ E with f(oo) = O. When is f = Cp, for some p, E M that is supported in E? This next theorem of Havin [91J (see also [78, p. 52]) provides an answer. THEOREM 5.5.1 (Havin). Let E be a closed subset of'][' and let f be analytzc on C\ E wzth f( (0) = O. Then there 1,8 a p, E M supported m E such that f = C p, 1,f and only 'tf there 1,8 a constant C such that

whenever all' .. ,an

f/. E

and AI,'" ,An E C.

See [78J for equivalent versions as well as refinements of Havin's theorem. Though we stated the Havin result for Cauchy transforms of measures on the circle, the theorem solves the more general problem: when is a functIon f analytic on C\ E, where E is a compact subset of the plane, equal to a Cauchy transform

J

_l_ d p,(w)

w-z for some finite measure p, supported in E?

There is even a further refinement. Let X be a class of analytIC functions on and E be a closed subset of T. Let !T(X, E) denote the functions f E X such that f = Kp" where p, EM and has support in E. Under what conditions on E is !T(X, E) =F (O)? For the Hardy spaces HP, the result is known. For 0 < p < 1, notice that !T(HP, E) =F (0) for every non-empty closed set E (Theorem 2.1.10). For 1 ~ P ~ 00, there is the following result of Havin [92]. Jl}

THEOREM 5.5.2 (Havin). For 1 m(E) > O.

~ P ~ 00,

!T(HP,E) =F (0) 1,f and only 'tf

5

110

WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

PROOF. Suppose m(E) = 0 and there is a measure J.L bupported in E with KJ.L E HP. Note, since J.L is supported E and m(E) = 0, that J.L .L m. Since fdm, where I is a boundary function for K J.L, is also a representing measure for K J.L (Cauchy integral formula - Proposition 2.2.1), and all representing measures have the same singular part (Proposition 2.1.5), then J.L must be the zero measure. Conversely, if m{E) > 0, then, by a theorem of Ahlfors [78, p. 6], there is a non-trivial bounded analytic function 9 on C\ E and, by subtracting off a constant, we may assume g( 00) = o. Define f m-almost everywhere on '][' by f(():= lim (g(r() - g«(Jr)) r-+l-

and note that the measure Idm is supported in E. A power series computation shows that K(fdm) = glD is non-trivial. 0 Though the analysis is more complicated, HruSCev [103J provides an answer to this question for other spaces of analytic functions on D. We state these results without proof. THEOREM 5.5.3 (HruSCev).

(1) If A is the disk algebra, then 9='(A, E) -=I- (0) il and only if m(E) > O. (2) II Aoo is the space 01 functions f which are analytic on D such that f(n) E A lor all n = 0,1,2,·· , then 9='{Aoo, E) -=I- (O) if and only il E contains a closed subset F 01 positive Lebesgue measure satisfying the Carleson condition 00

L m{In) logm(In) > -00, where

{In)n~l

n=l is the sequence of complimentary arcs of F.

5.6. Some geometric conditions So far, we have investigated the question as to whether or not a particular analytic function on D (or on C \ 1!') is a Cauchy transform in terms of growth conditions near the boundary. We could also attempt to answer this question in terms of some geometric conditions For example, if f{D) is contained in a halfplane, then f E X (Proposition 2.1.13). In this section, we provide some other geometric conditions. LEMMA 5.6.1. II I E X and ¢ : D - D is analytic, then f

0

¢ E X.

PROOF. If J.L E M+, then !R{KJ.L 0 ¢) > 0 on II} and so by Proposition 2.1.13, K J.L 0 ¢ E X. Write any J.L E M as J.L = (J.Ll - J.L2) + i(J.L3 - J.L4), J.Lj E M+, and apply the above argunIent four times to conclude that K J.L 0 ¢ E X for any J.L EM. 0 This simple lemma yields the following corollary. COROLLARY 5.6.2. Suppose that n is a proper simply connected subset of C and tf; : D - n is a Riemann map. If tf; E X (this may not always happen!), then any analytic map f : D _ n 1 belongs to X PROOF. If tf; = KJ.L, apply Lemma 5.6.1 to ¢ := tf;-l

f = K J.L 0 ¢ E X. INote that f{D} need not be equal to

n

0

f : D - D to see that 0

SOME GEOMETRIC CONDITIONS

56

111

The following [31] is a nice geometric cond1tion for membership in X. THEOREM 5.6.3 (Bourdon and Cima). Suppose f 1.S analyt1.c and f(lO) 'tS contamed m a regwn that om1.ts two opposztely pomted half-lmes (see Fzgure 1), then

fEX. PROOF. Assume feD) is contained in a region that omits two oppositely pointed half-lines and let n be the complement of those two oppositely oriented lines. In a moment, we will construct an invertible analytic 1/J : D -+ n and show that the harmonic function h := ~1/J satisfies sup

(5.6.4)

O
( Ih(r()1 dm«()

J.Jr

< 00.

By standard harmonic analysis [65, p. 2] (also see the proof of Theorem 9.1.1), h = PIL, the Poisson integral of a real IL E M. This will imply that ~1/J = QIL

+ e,

where QIL is the conjugate Poisson integral of IL, and so

1/J = PIL + ZQIL + te =

HIL +

ze,

where H IL is the Herglotz integral of IL. It follows now that v EM. Now apply Corollary 5.6.2. We now construct '1/;. Let cjJ(z) := 1 + z

1/J

= K v for some

l-z

be the usual analytic map from D onto {!Rz > O} and notice that

cjJ(elt ) For b > 0 and c E

= iy, y

> 0, whenever 0 < t < 71",

cjJ(ed ) = zy, y < 0, JR., fixed, let

1/J(z)

=

cjJ(z) -

whenever

cjJ~Z)

71"

< t < 271".

- 2cilog (zcjJ(z».

A computation shows that

1/J(e't)

= {

zk(y) + 2C7r, 0 < t < 71"; 71" < t < 271",

ik( -y),

where k(y) = y

b

+ -y -

2clog Iyl.

Notice that

key)

+00 as y -+ 0+ or y -+ 00, -00 as y -+ 0- or y -+ -00.

-+

key) -+

Let

Ym = min{k(y) : 0 < y < oo}, and observe that

YM = max{k(y) : -00 < y < O}

1/J maps the umt circle to the two half lines

L1 := {(2C7r, y) : y ;;::: Ym},

L2 = {(O, y) : y ~ YM}.

Since 'I/; covers each of these half lines twice, it follows from the argument principle that 'I/; is univalent. The parameters band c adjust the geometry of these two lines

112

5

WHICH FUNCTIONS ARE CAUCHY INTEGRALS?

(distance they are apart, where they start). Any map from D onto a region omitting two oppositely oriented half-lines takes the form

w{z) = ei9 'I/J{z)

+ d,

where () is a rotation and d is a translation. Finally, we leave it to the reader to show that

h(z) = ~'I/J{z) =

1-lz12 11- zl2

1-lz12

(1+Z)

- bl1 + zl2 - 2carg 1- z

The first two terms are constant multiplies of Poisson kernels and thus have uniformly bounded integral means as in eq.{5 6.4). The last term is bounded since