Function Theory in the Unit Ball of Cn: Reprint of the 1st Ed Berlin Heidelberg New York 1980

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Classics in Mathematics Walter Rudin

Function Theory in the Unit Ball of

Walter Rudin

Function Theory in the Unit Ball of Reprint of the 1980 Edition

123

Walter Rudin Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 USA

Originally published as Vol. 241 of the series Grundlehren der mathematischen Wissenschaften

ISBN 978-3-540-68272-1

e-ISBN 978-3-540-68276-9

DOI 10.1007/978-3-540-68276-9 Classics in Mathematics ISSN 1431-0821 Library of Congress Control Number: 2008930562 Mathematics Subject Classification (2000): 32-02, 31Bxx, 31Cxx, 32Axx, 32Fxx, 32Hxx © 2008, 1980 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com

Grundlehren der mathematischen Wissenschaften 241 A Series of Comprehensive Studies in Mathematics

Editors

M. Artin S. S. Chern J. L. Doob A. Grothendieck E. Heinz F. Hirzebruch L. Hormander S. Mac Lane W. Magnus C. C. Moore W. Schmidt D. S. Scott J. Tits B. L. van der Waerden S. R. S. Varadhan Managing Editors

B. Eckmann

J. K. Moser

Walter Rudin

Function Theory in the Unit Ball of C"

Springer-Verlag New York Heidelberg Berlin

Walter Rudin University of Wisconsin Department of Mathematics Madison, WI 53706 USA

AMS Subject Classifications: 32-02, 31Bxx, 31Cxx, 32Axx, 32Fxx, 32Hxx

Library of Congress Cataloging in Publication Data Rudin, Walter, 1921Function theory in the unit ball of C" (Grundlehren der mathematischen Wissenschaften; 241) Bibliography: p. Includes index. I. Holomorphic functions. 2. Unit ball. I. Title. II. Series. QA331.R863 515 80-19990 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1980 by Springer-Verlag New York Inc. Printed in the United States of America. 987654321 ISBN 0-387-90514-6 Springer-Verlag New York ISBN 3-540-90514-6 Springer-Verlag BerHn Heidelberg

Preface

Around 1970, an abrupt change occurred in the study of holomorphic functions of several complex variables. Sheaves vanished into the background, and attention was focused on integral formulas and on the "hard analysis" problems that could be attacked with them: boundary behavior, complex-tangential phenomena, solutions of the S-problem with control over growth and smoothness, quantitative theorems about zero-varieties, and so on. The present book describes some of these developments in the simple setting of the unit ball of C". There are several reasons for choosing the ball for our principal stage. The ball is the prototype of two important classes of regions that have been studied in depth, namely the strictly pseudoconvex domains and the bounded symmetric ones. The presence of the second structure (i.e., the existence of a transitive group of automorphisms) makes it possible to develop the basic machinery with a rhinimum of fuss and bother. The principal ideas can be presented quite concretely and explicitly in the ball, and one can quickly arrive at specific theorems of obvious interest. Once one has seen these in this simple context, it should be much easier to learn the more complicated machinery (developed largely by Henkin and his co-workers) that extends them to arbitrary strictly pseudoconvex domains. In some parts of the book (for instance, in Chapters 14-16) it would, however, have been unnatural to confine our attention exclusively to the ball, and no significant simplifications would have resulted from such a restriction. Since the Contents lists the topics that are covered, this may be the place to mention some that might have been included but were not: The fact that the automorphisms of the ball form a Lie group has been totally ignored. There is no discussion of concepts such as curvature or geodesies with respect to the geometry that has these automorphisms as isometrics. The Heisenberg group is only mentioned in passing, although it is an activefieldof investigation in which harmonic analysis interacts with several complex variables. Most of the refined estimates that allow one to control solutions of the 5-problem have been omitted. I have included what was needed to present the

vi

Preface

Henkin-Skoda theorem that characterizes the zeros of functions of the NevanHnna class. Functions of bounded mean oscillation are not mentioned, although they have entered the field of several complex variables and will certainly play an important role there in the future. To some extent, these omissions are due to considerations of space—I wanted to write a book of reasonable size—but primarily they are of course a matter of personal choice. As regards prerequisites, they consist of advanced calculus, the basic facts about holomorphic functions of one complex variable, the Lebesgue theory of measure and integration, and a little functional analysis. The existence of Haar measure on the group of unitary matrices is the most sophisticated fact assumed from harmonic analysis. Everything that refers specifically to several complex variables is proved. I have included a collection of open problems, in the hope that this may be one way to get them solved. Some of these look very simple. The fact that they are still unsolved shows quite clearly that we have barely begun to understand what really goes on in this area of analysis, in spite of the considerable progress that has been made. I have tried to be as accurate as possible with regard to credits and priorities. The literature grows so rapidly, however, that I may have overlooked some important contributions. If this happened, I offer my sincere apologies to their authors. Several friends have helped me to learn the material that is presented here—in conversations, by correspondence, and in writing joint papers. Among these, I especially thank Pat Ahern, Frank Forelli, John Fornaess, Alex Nagel, and Lee Stout. Finally, I take this opportunity to express my appreciation to the National Science Foundation for supporting my work over a period of many years, to the William F. Vilas Trust Estate for one of its Research Professorships, and to the Mathematics Department of the University of Wisconsin for being such a friendly and stimulating place to work in. Madison, Wisconsin March 1980

Walter Rudin

Contents

List of Symbols and Notations

xi

Chapter 1 Preliminaries 1.1 Some Terminology 1.2 The Cauchy Formula in Polydiscs 1.3 Differentiation 1.4 Integrals over Spheres 1.5 Homogeneous Expansions

1 1 3 7 12 19

Chapter 2 The Automorphisms of B 2.1 Cartan's Uniqueness Theorem 2.2 The Automorphisms 2.3 The Cayley Transform 2.4 Fixed Points and Affine Sets

23 23 25 31 32

Chapter 3 Integral Representations 3.1 The Bergman Integral in B 3.2 The Cauchy Integral in B 3.3 The Invariant Poisson Integral in B

36 36 38 50

Chapter 4 The Invariant Laplacian 4.1 The Operator A 4.2 Eigenfunctions of A 4.3 e#-Harmonic Functions 4.4 Pluriharmonic Functions

47 47 49 55 59 vii

Vlll

Contents

Chapter 5 Boundary Behavior of Poisson Integrals

65

5.1 5.2 5.3 5.4 5.5 5.6 5.7

65 67 70 72 79 83 88

A Nonisotropic Metric on 5 The Maximal Function of a Measure on S Differentiation of Measures on S AT-Limits of Poisson Integrals Theorems of Calderon, Privalov, Plessner The Spaces N(B) and HP(B) Appendix: Marcinkiewicz Interpolation

Chapter 6 Boundary Behavior of Cauchy Integrals 6.1 An Inequality 6.2 Cauchy Integrals of Measures 6.3 Cauchy Integrals of L^'-Functions 6.4 Cauchy Integrals of Lipschitz Functions 6.5 ToepUtz Operators 6.6 Gleason's Problem

91 92 94 99 101 110 114

Chapter 7 Some L^-Topics 7.1 Projections of Bergman Type 7.2 Relations between H^ and U n H 7.3 Zero-Varieties 7.4 Pluriharmonic Majorants 7.5 The Isometrics of H\B)

120 120 126 133 145 152

Chapter 8 Consequences of the Schwarz Lemma 8.1 The Schwarz Lemma in 5 8.2 Fixed-Point Sets in B 8.3 An Extension Problem 8.4 The Lindelof-Cirka Theorem 8.5 The Julia-Caratheodory Theorem

161 161 165 166 168 174

Chapter 9 Measures Related to the Ball Algebra 9.1 Introduction 9.2 Valskii's Decomposition 9.3 Henkin's Theorem 9.4 A General Lebesgue Decomposition 9.5 A General F. and M. Riesz Theorem

185 185 187 189 191 195

Contents 9.6 The Cole-Range Theorem 9.7 Pluriharmonic Majorants 9.8 The Dual Space of A(B)

IX 198 198 202

Chapter 10

Interpolation Sets for the Ball Algebra

204

10.1 10.2 10.3 10.4 10.5 10.6 10.7

Some Equivalences A Theorem of Varopoulos A Theorem of Bishop The Davie-0ksendal Theorem Smooth Interpolation Sets Determining Sets Peak Sets for Smooth Functions

204 207 209 211 214 222 229

Chapter 11 Boundary Behavior of / / ^ - F u n c t i o n s 11.1 A Fatou Theorem in One Variable 11.2 Boundary Values on Curves in 5 11.3 Weak*-Convergence 11.4 A Problem on Extreme Values

234 234 237 244 247

Chapter 12 Unitarily Invariant Function Spaces 12.1 Spherical Harmonics 12.2 The Spaces H(p, q) 12.3 ^^-Invariant Spaces on S 12.4 ^//-Invariant Subalgebras of C{S) 12.5 The Case « = 2

253 253 255 259 264 270

Chapter 13

Moebius-Invariant Function Spaces

278

13.1 13.2 13.3 13.4

.#-InvariantSpaceson 5 .^-Invariant Subalgebras of CQ{B) .^-Invariant Subspaces of C{E) Some Applications

278 280 283 285

Chapter 14 Analytic Varieties 14.1 The Weierstrass Preparation Theorem 14.2 Projections of Varieties 14.3 Compact Varieties in C" 14.4 Hausdorff Measures

288 288 291 294 295

X

Contents

Chapter 15

Proper Holomorphic Maps

300

15.1 15.2 15.3 15.4 15.5

300 305 309 314 319

The Structure of Proper Maps Balls vs. Polydiscs Local Theorems Proper Maps from Bio B A Characterization of B

Chapter 16 The g-Problem 16.1 Differential Forms 16.2 Differential Forms in C" 16.3 The a-Problem with Compact Support 16.4 Some Computations 16.5 Koppelman's Cauchy Formula 16.6 The 5-Problem in Convex Regions 16.7 An Explicit Solution in B

330 330 335 338 341 346 350 357

Chapter 17

The Zeros of Nevanlinna Functions

364

17.1 The Henkin-Skoda Theorem 17.2 Plurisubharmonic Functions 17.3 Areas of Zero-Varieties

364 366 381

Chapter 18

Tangential Cauchy-Riemann Operators

387

18.1 Extensions from the Boundary 18.2 Unsolvable Differential Equations 18.3 Boundary Values of Pluriharmonic Functions

387 395 397

Chapter 19

Open Problems

403

19.1 The Inner Function Conjecture 19.2 RP-Measures 19.3 Miscellaneous Problems

403 409 413

Bibliography

419

Index

431

List of Symbols and Notations

The numbers that follow the symbols indicate the paragraphs in which their meanings are explained. For example, 10.4.2 means Chapter 10, Section 4, paragraph 2. Sets C, C" B„,B S = dB„ U,T D(a;r) U\ T" E(a, 8)

e(L

kl

|a|,a! z"

f,

JF,JRF

0(2n) ^ I <Pa ll/lloo U,

M

r

da(z) d(a, b) A, z•w Tc Tf K-lim g* fr

1.1.2 1.1.2 1.1.6 1.1.6 1.2.5 1.3.6 1.4.1 1.4.6 2.1.1 2.2.1 3.2.33.3.4 3.3.6 4.2.1 4.2.2 5.1.1 5.2.2 5.4.2 5.4.2 5.4.6 5.5.8 5.6.1

ll/llp A(C, CO, oc, S)

T co C" (called the derivative of F at z) that satisfies (1)

F(z + /i) = F{z) + F'{z)h + 0(|/ip)

for h near the origin of C". In fact, if the components of F(z + h) are expanded in power series about the point /z = 0, then F'{z)h is just made up from the linear terms in these series. For 1 < /c < n, let /z = Xe^ in (1), and let A -^ 0. It follows that (2)

F\z)e, = (D,F)(z\

a vector in C" with components (A/j) (z)A <j< f^, where F=(fi, . . . , / J . The linear operator F'(z) is thus represented by a matrix A = (ajk), with (3)

aj, = (D,fj)(z)

(l<j<m,l —I, define

'^'^'-LlT

do(0

and (1 - Iwp)'rfv(w)

JB i i - < z , w > r + ! + ' + ' • •*» = J.u' When c < 0, then Ic and Jc,t or^ hounded in B.

1. Preliminaries

When c > 0, then iXz)^(i-\z\T'^JcM Finally, Io(z) ^ log

1-UI

Jo,tizl

The notation a(z) ^ b(z) means that the ratio a(z)/b(z) has a positive finite Hmit as |z| ^ 1. Proof. The binomial series

holds for IXI < 1 when /I is not a negative integer. Choose /I so that 2yl = n + c. Then

r(fc + A) 0. If we apply the shce-integration formula 1.4.7(1) to this integral, we obtain the inequality 1.5.3(1). Hence log | / | is subharmonic.

1.5. Homogeneous Expansions

21

Since I/I''= exp(c log | / | ) and since exp(cx) is an increasing convex function of x, the subharmonicity of l o g | / | implies that of | / | ^ 1.5.5. Lemma. Suppose {wj is a sequence of subharmonic functions in Q, and cc, jS are real numbers such that w, (z) < j8 and (1)

lim sup Ui{z) < OL

for every zeQ. IfKciQ

is compact and 8 > 0, then there exists io such that

(2)

Ui(z) < a + £

for allzeK

and all i > IQ.

Proof, The case oc > P is trivial, so assume a < jS. Pick r > 0 so that K + rS c: Q. Put Vi = max(a, w^), put gj = sup{i;i-: i > j}, and define hj on Kby hj(z) =

Qjiz + rw)dv(w).

Each Vi is then subharmonic, and a < i;^ < jS. By (1), Hm Vi{z) = a for all z G Q. Hence ^/z) decreases monotonically to a as; -> oo. The same is true of hj(zX by the dominated convergence theorem. Since OL < QJ < P, each hj is continuous. But if a sequence of continuous functions converges monotonically to a continuous limit, on a compact set, then the convergence is uniform. Thus /i/z) -> a uniformly on K. Now the subharmonicity of Vj, combined with the obvious inequality Vj < Qj, shows that oc < Vj < hj. Hence Vj -^ a uniformly on K, and since Uj < Vj, the lemma is proved. 1.5.6. Theorem. Suppose (a) Q is an open set in C", (b) for s = 0,1,2,... ,Fsis a homogeneous polynomial of degree s, and (c) sups I Fs(z) I < 00 for every zeQ. The series Y,o ^si^) converges then uniformly on every compact subset ofQ. Corollary. The sum of the series is holomorphic. Proof. Pick a compact K cz Q, and pick t > 1, so close to 1 that t^K a Q. By (c) and Baire's theorem, there exist a e Q , r > 0 , M < o o , such that (1)

|F,(a + z ) | < M

( | z | < r , 5 = 0,1,2,...).

22

1. Preliminaries

Lemma 1.5.2, applied to the polynomial (2)

P(X) = M-'F,(a

+

Hl

where C^S, shows that (3)

|F,(a + z ) | < M r - 1 z r

(|z| > r).

Put Us = |i^sl^^*- By Proposition 1.5.4, each Ug is subharmonic. By (1) and (3), {wj is uniformly bounded on every compact subset of C". By assumption (c), (4)

lim sup Ms(z) < 1 s-ôo

(ze Q).

Lemma 1.5.5 implies therefore that there is an SQ such that Us < t on t^K for all s > SQ. Thus (5)

\Fsiz)\ 2, make the induction hypothesis (which is vacuously true when m = 2) that Fj = 0 for 2 < s < m. Then F^ has the homogeneous expansion F\z) = z + kFîz) + • • • 23

24

2. The Automorphisms of B

in r^B, as is easily proved by induction on k. The homogeneity of the maps F^ impHes that kFJ,z) = i ^ J '

F\e''z)eie-^"V0

(|z| < r i ) .

(This is a C"-valued integral.) Since F^ maps Q into Q, we have | F^(e'^z) | < r2 for all z G riB and for all 0. Thus k\FJ^z)\ 1. There is a well-known open question related to this (Magnus [1]): Suppose F is a holomorphic map of C" into C" whose components are polynomials and whose Jacobian is 1 at every point. Does it follow that the range of F is all of CI If the question is asked with entire functions in place of polynomials, the answer is negative (for every n > 1). This was shown by Fatou and Bieberbach; see Bochner-Martin [1], pp. 45-48.

2.2. The Automorphisms 2.2.1. Recall that to every a in the unit disc of C corresponds an automorphism (p^ of the disc that interchanges a and 0, namely (pM = (a — X)/{1 — a A). The same can be done in the unit ball B of C". Fix aeB. Let P^ be the orthogonal projection of C" onto the subspace [a] generated by a, and let Qa = I — Pa he the projection onto the orthogonal complement of [a]. To be quite exphcit, PQ = 0 and (1)

PaZ = p ^ a \a, a}

ifaÔ.

P u t s , = (1 - |ap)^/^ and define (2)

(paiz) =

] 7 r 1 - 1, the integrand converges uniformly to f(il/(rj)X by 3.3.4(a). Hence (3)

n / ] ( ^ ( 0 ) ) = P[/oiA](0),

which establishes (1) at the origin. For any zeB, two applications of (3) give

ThusP[/](./.(z)) = P [ / o ^ ] ( 2 ) . Remark. Since P(0, 0 = 1, (3) is tiie same as (4)

(p(HO),OnC)d 0, where Uxit) ^ 0) and if w = u^v, one finds that Lw = kw and that w(0 is unbounded as t -^ 0. Since the solution space of (1) is 2-dimensional, the solutions of (1) that stay bounded as t -> 0 form a space of dimension 1. This proves the uniqueness asserted by the theorem. The corollary follows, since 4.2.2(1) is unchanged if a is replaced by 1 — a. 4.2.4. Theorem. Every f eX^ satisfies (1)

{fmrQ)da{Q Js

=

gj,rn)mm

for every xj/ e Aut(J5), 0 < r < l,rjeS, Equivalently (2)

f f((p, Uw)dU = gMf(z)

(z, w G B).

Conversely, iffe C(B) and f satisfies (1), thenfeC'^(B)

andfeXx-

Proof. I f / G X;t, so is its radialization

(3)

/ * = Uf^WdU.

By Theorem 4.2.3,/'*' = /(O)^^. This is (l)with/in place of/o i/^. The general case of (1) follows from the ^-invariance of XÎf we take il/ = (pîn (1) and put r = | w|. Proposition 1.4.7(3) gives (2).

52

4. The Invariant Laplacian

For the converse, pick a radial function h e C°°(5), with compact support in B, such that f^ hg^^ dx = 1, where T is the ^-invariant measure of Theorem 2.2.6. Multiply (2) by h(w)dT(w% integrate, invert the order of integration, and use the ^-invariance of T to obtain

(4)

f(z) = f dU \f{ w)/z(i/-Vz(w)yT(w).

Since h is radial, the inner integral is independent of U, and (4) simplifies to (5)

/ ( z ) = \ fiw)h(cpM)dT{w). JB

This makes it evident that / e CîB). We can now differentiate (2) with respect to w, and find that (6)

^ficpXO)) = A/(z),

i.e., that A/ = Xf.

ThusfeX;,.

Corollary 1. Every X^ is a closed subspace ofC(B), in the topology of uniform convergence on compact sets. Proof. (1) is preserved if fj -^ / uniformly on compact sets. CotôUary 2. Every ^-harmonic function in B has the invariant mean value property, and every feC(B) with the invariant mean value property is Ji-harmonic. Proof. This is the case A = 0 of the theorem. Note that go = Qi = ^' Corollary 3. For every A 6 C, X;i c C°°(B). Actually, more is true: 4.2.5. Theorem. Every fe Xj^ is real-analytic in B. The term real-analytic refers of course to analyticity relative to R^" rather than C". A real-analytic function is thus one that is locally the sum of a convergent power series in the 2n real variables Xi, y i , . . . , x„, y„. Proof. We will see that S meets the requirements of the following theorem about partial differential equations (Hormander [1], Theorem 7.5.1): If L is an elliptic differential operator with real-analytic coefficients, then every solution M of Lw = 0 is real-analytic.

4.2. Eigenfunctions of A

53

Theorem 4.1.3(ii) allows us to represent S in the form (AM)(Z) = p{z, D)u

where (replacing d/dxj by Uj, d/dy^ by v^, Wj = Uj + iv^ P(Z, W) = (1 -

| z p ) < X WfWf -

^ Z,.Z;tW^.Wfc>

= (l-UP)(|wp-|p}. This is a polynomial in the variables Wj, vvj^, whose coefficients are evidently real-analytic functions of the z-variables. For ZEB, p(z, w) = 0 only when w = 0. By the definition of "elliptic" (Hormander [1], p. 177) this proves the ellipticity of S. 4.2.6. Definition. Co(B) is the space of all fe C(B) such that f(z) ^ 0 as

1^1-1. 4.2.7. Theorem, (i) Z^ n Co(B) ^ {0} if and only if (1)

4n^Re:i + (lmXy

< 0.

(ii) X;^ contains a bounded function (not identically 0) if and only if (2)

4n^ Re A + (Im A)^ < 0.

Proof. Let X — — 4wâ(l — a), as before. Then (1) and (2) are equivalent to 0 < Rea < 1 and 0 < Rea < 1, respectively. Suppose /eX;^. If / e CQ{B\ or / is bounded, the same is true oi f* = f(0)g^ (Theorem 4.2.3). The following three assertions about g^ therefore prove the theorem. (Recall that6fi-a = â-)

(a) If 0 < Re a < i then g^ e Co(B). (b) If Re a = 0 then I â I < 1 but ^« ^ Co{B). (c) If Re a < 0 then g^ is unbounded in B. To prove (a), write formula 4.2.2(3) more explicitly:

Proposition 1.4.10 shows that the integral in (3) is bounded if Re a < ^ and that it is 0( —log(l — |z|)) if Re a = i. In either case, (a) follows from (3), as does part of (b), namely that \go,(z)\ < 1 if a is pure imaginary.

54

4. The Invariant Laplacian

Put j8 = —n(x. Then (c) and the second part of (b) will be proved once we show that the integral

(4)

^ l l - — w/2, by analytic continuation. 4.2.8. A Digression. Every function / with domain B "lifts" to a function / * on the group Aut(5), by setting

(1)

f%^) = mm

(^6Aut(B)).

Clearly, f\\j/U) = /*(iA), for every UeÂffis invariance property (2)

/*(l/i 0 with the following property: If C e 5, w e S, | C — w | < d, then w = UQC for some UQ e N. The in variance of the Haar measure dU leads from (4) to (9)

GiUoz)=

\

F(Uz)h(UUo')dU,

If we combine (4) and (9), and use (8), we obtain the desired equicontinuity: (10)

\GXO - Gr(w)| <M,8

(0 < r < 1),

whenever |C — >v| < 5; note that Mi < Mp. Since {GJ is equicontinuous and uniformly bounded (see (6)), there is a sequence ri -^ 1 such that {G^..} converges uniformly to a function g e C(S). Let (11)

8, = s u p | G ( r , 0 - P M ( r , 0 | .

Since \G(riO - g(0\ and \g(0 - P[gUriO\ tend to 0 uniformly, as / -^ oo, we see that Si -^ 0. We now use the fact that AF = 0. By (4), Ao = 0. Thus G - P[^] is ^-harmonic, and the maximum principle (Theorem 4.4.2), combined with (11), shows, for every i, that (12)

|G(z)-PM(z)| (d) follows from Theorem 4.1.3(iii), and Theorem 4.4.4 shows that (d) -> (a). It is trivial that (a) -> (e). If (e) holds, then (Au)(a) = A(uocpJ(0) = 0 for all aeB, since uocp^ is harmonic. Thus (e) -^ (c), and the proof is complete. The implications (e) -»(a) and (c) -> (a) occur in Nagel-Rudin [1], p. 865, and in Rudin [10], respectively. As is clear from the above proof, ForelH's theorem is now the only nontrivial step in these equivalences. If B is replaced by any region Q ci C", the implication (b) -> (a) holds in every ball in Q, hence it holds globally whenever Q is simply connected. We conclude this section by clarifying a remark made prior to the proof of Theorem 4.3.3, to the effect that dilates of ^-harmonic functions need not be ^-harmonic: 4.4.10. Theorem. Iff is Ji-harmonic in B and if there is one r, 0 < r < 1, such thatf is also M-harmonic in B, then f is pluriharmonic in B. Proof. Put g{z) = /(rz), for zeB. Since A/ = 0 and A^ = 0, both / and g are solutions of the equation (Aw)(z) = (AwJ(l), by 4.1.3(iii). Since gX^) = frzW, it follows that 0 = iAg)(z) - (Ag,)(l) = r^(A/)(rz) - ( A / J ( l ) = - ( 1 - r^)(A/)(rz). Thus A/ = 0 in rB. Being ^-harmonic, / is real-analytic in B (Theorem 4.2.5), hence so is A/, and it follows that A/ = 0 in all ofB. Thus/has property (c) of Theorem 4.4.9.

Chapter 5

Boundary Behavior of Poisson Integrals

The principal result of the present chapter is Koranyi's Theorem 5.4.5—the fact that certain maximal functions associated to invariant Poisson integrals are of weak type (1, 1). The existence of what we call "K-Hmits" at almost all points of S follows easily from this. The chapter begins with the definition of a certain '^-invariant metric and the associated maximal function Mjn of a measure /a on S. The corresponding maximal theorem 5.2.4 is true for many other choices of metrics. As the proof will show, the finiteness of the constant A^ in Lemma 5.2.3 is basically all- that is needed. The reason for the particular choice made is that Mjx dominates Plfi] in a way that Theorem 5.4.5 makes precise. The chapter contains some appUcations to holomorphic functions that do not depend on the more dehcate theorems concerning the boundary behavior of Cauchy integrals. The latter will be studied in Chapter 6.

5.1. A Nonisotropic Metric on S 5.1.1. Definitions. For

aeB,beB,

(1)

d(a,b)=\l-(a,b}\'^\ For C e 5, 5 > 0,

(2)

Q(i:,d) =

{rieS:d(i:,rj) ^/l. Throughout this chapter, the letter A will denote a positive finite constant that depends only on the dimension n. Any dependence on other parameters will be explicitly indicated. 5.1.2. Proposition, (i) The triangle inequality d(a, c) < d(a, b) + d(b, c) holds for all a,b,ce B. (ii) On S, d is a metric; the sets Q{C, ^) cire the corresponding balls. Proof. Since d is ^-invariant, one may take b = rei(0 < r < l\ and one then has to prove that (1)

|i_| 1, apply formula 1.4.5(2) to the characteristic function of 2(î, 3\ to get

^(&) = ^ ^ ^ f (l-|Ap)"-2^m2(A) where E{3) = {A:|A| < 1 and |1 — /l| < 3^}. The change of variables 1 — A = (5^/z turns this into

^

= !Lzi r 0

n

(2x-3^y-^\z\-^^dxdy

JE'{3)

where E'(3) = {z = x -{- iy:2x > 3^ and |z| > 1}. As ^ decreases, both the integrand and the domain of integration increase. The monotone convergence theorem shows therefore that the hmit O{(T(Q^)/3^", as ^ ^ 0, exists and equals 1)2""^

Ao = -

n

/'OO

Ji

J

^

/.71/2

J-K/2

This proves the proposition. Computation of these integrals gives the more explicit result jrjn + 1) ^ o = f;27

n(n/2 + 1)

5.2. The Maximal Function of a Measure on S 5.2.1. Terminology and Notation. If/< is a complex measure on 5 ("measure " will always mean "Borel measure") then, as usual, (i) (ii)

I ju I is the total variation measure of fi, M = \ti\iS),

68

5. Boundary Behavior of Poisson Integrals

(iii) jii < (T means that fi is absolutely continuous with respect to cr, i.e., that fi(E) = 0 for every Borel set £ c S with (T(E) = 0, and (iv) ju J_ (T means that jii is singular with respect to a, i.e., that there is a Borel set £ cz 5 with (j(E) = 0, |/x|(E) = ||/x||. We recall the theorem of Lebesgue and Radon-Nikodym: dfi = f da + d^s^ uniquely, with / e L^(cr), /x^ 1 cr. 5.2.2. Definition. The maximal function of a complex measure // on 5 is the function M/i: S ^ [0, oo] defined by (Miu)(0 = sup

-——.

Note that M/i = M|/i|. For each fixed 5 > 0, the above quotient is easily seen to be a lower semicontinuous function of C- Hence M/x is lower semicontinuous. The proof of Theorem 5.2.4 will use the following simple covering lemma, that involves the constant

.

^(36)

^3 = s u p - — - . Proposition 5.1.4 implies that ^3 < oc. 5.2.3. Lemma. IfE is the union of a finite collection O of halls Q f^, stop. If not, let ifc+i be the first index such that Q^^^^ is disjoint from g^^. Since O is finite, this process stops, say at /c = m. Put F = {Q^^,..., Q,-^}. To every Qi-eO corresponds some /c such that i^ fc^, since there is no im+i)- Since Si < Sf^ and Q,- intersects g;^, we have gj c: 3Qi^. This proves (1), and (2) follows from (1) and the definition of^3.

5.2. The Maximal Function of a Measure on S

5.2.4. Theorem. Iffiisa (1)

69

complex measure on 5, then (T{M//>r}< ^3^"'11/^11

for every t > 0, The left side of (1) replaces the more cumbersome (2)

t}).

We shall often simplify notation in this way. Proof. Fix fi and t. Let K he 2i compact subset of the open set {M/i > t}. Each C e X is the center of a ball Q such that | /x(Q) | > to-(Q). Some finite collection O of these g's covers K. If F is as in Lemma 5.2.3, then t}. 5.2.5. Weak L\ Iffe L\a) and t > 0, then the inequality (1)

t} 0. There is an open set Q ZD K, having (T(Q) < e. There is a finite collection O of balls Q cz Q, with centers in K, whose union covers K. Choose F as in Lemma 5.2.3. If g e F, then 3Q is a ball with center in X cz £^, so that | A* 1(3(2) < tcr(3g). Lemma 5.2.3 implies therefore that \f^\(K) < X \fi\(3Q) < t X (J(3Q) < ^3? Z ^(Q) < A,tG(n) < A,ts r r r by the disjointness of F. Thus \fi\(K) = 0.

5.3. Diflferentiation of Measures on S Theorem 5.2.4 gives easy access to differentiation theorems. We shall deal separately with the absolutely continuous case and with the singular case. 5.3.1. Theorem. IffeL\o) (1)

then

li^^T^f

l/-/(OM^ = 0

for almost every C^S. Hence (2)

/(0=lim-i-f

fda

(a.e.)

Note: The points C for which (1) holds are called the Lebesgue points of / . If £ c: S is measurable and / is the characteristic function of E, then every point of E that is a Lebesgue point of / is called a point of density of E. The theorem implies that almost all points of E are points of density of E.

5.3. Differentiation of Measures on S

71

Proof. Define T / Q to be the left side of (1), but with Um sup in place of lim. Choose t > 0and e > 0. Chooseg e C(S) with \\f - g\\i < £• Put h = f - g. Then (i) (ii) (iii)

7} < T, + r „ Tg = 0 since g is continuous, and T, < |/i| -f M/i.

Thus Tf t} is a subset of

By Theorem 5.2.4, and 5.2.5(1), (T(E,) 0. Letting t -^ 0, we see that ^JJ. = 0 a.e. [cr].

72


5.3.3. The two preceding theorems can be combined: If fi is any complex Borel measure on S, then its derivative (^/i)(0 = lim^((2(C,(5)) exists a.e. [ a ] ; ifdfi = fdo + d^^ is the Lebesgue decomposition offi, then

i^fiXO = m

(a.e. M).

5.4. X-Limits of Poisson Integrals 5.4.1. Approach Regions. For a > 1 and C e S, we let DJiJi) be the set of all zeV such that

(1)

|i-aC>l = 0. The preceding discussion shows that the vector iC lies in T^ but that iC is perpendicular to Tf. To return to our regions D^(C), they are tangential to S in the direction of Tf. In formula 5.4.1(1), the left side is the euclidean distance from z to the affine set C + Tf; the easiest way to see this is to refer to 5.4.1(3). The right side is essentially proportional to 1 — | z |, the euclidean distance from z to S. This interpretation of 5.4.1(1) suggests generalizations which led E. M. Stein [2] to extend Koranyi's theorem to a much wider class of regions. When n = 1, this discussion is vacuous, since {0} is then the only complex subspace of codimension 1. 5.4.3. Lemma. Ifrj G S, C ^ S, z e DJ£), |z| = r, then (1)

|l-| 1 and coeV^, Lemma 5.4.3 gives

76


by (4). Hence (5) implies that (9)

f

1 corresponds an r < 1 such that (1)

{|z|>r}nD,(Oc:Q.

(Q = Bis the simplest and most important example.) We say that a function F : Q ^ C has X-limit X at C, and write (2)

(K-\imF)(0

=^

if the following is true: For every a > 1 and for every sequence {Zi} in Da(C) ^ ^ that converges to C, F{zj) -^ /I as i ^ oo. The case X= cois not excluded. But usually we will of course be interested in finite iC-limits. Koranyi [2] used the terms "admissible limit" and "admissible convergence" in this context. The next three theorems are consequences of the maximal theorems 5.4.5 and 5.2.4. 5.4.7. Theorem. If ii is a positive measure on S and if(^fi)(0 C eS, then (1)

= Ofor some

(X-limPM)(O = 0.

Proof. Pick £ > 0. Choose Qo = 6(C, ô) so that (2)

MG(C, d)) < 8(7(e(C, d))

(0<S
0, 0 < ffc < 1, for /c = 1, 2, 3 , . . . , so that ^ m^t < oo but mj(l — tkY -^ 00. Choose 0^ so that 11 — e'^^\ = 1 — r^. Let ja be the sum of the point masses nik at

(e'\OX

-m^

at

fe,^!

- ti).

Then fi{Q{ei, 3)) = 0 for every 5 > 0, hence (^fj)(ei) = 0. Next,

For 0 < r < 1, each term in this sum is positive. When r = t^, thefethterm is (2tt + tl)m^

(1 - tlf which tends to oo as /c -> oo. The proof of Theorem 5.4.5 shows that the constant ^(a) is at most AoiP. Our third example shows that this is the correct order of magnitude^ as far as its dependence on OL is concerned. Let /z be the unit point mass at e^. Then P M ( z ) = P{z, Ci). Pick (,eS, with 0 3 and r = (a — t)/((x + t) then re^ e DJÔ (a computation shows this), hence M « P M ( 0 > P(re,, e,) = H

> A-â"(M/i)(0.

5.5. Theorems of Calderon, Privalov, Plessner This section extends two classical theorems of Privalov [1] and Plessner [1], concerning the nontangential behavior of harmonic and holomorphic functions' in the unit disc, to analogous theorems about ^-harmonic and holomorphic functions in the approach regions DJ^Q. (Theorems 5.5.7 and 5.5.8.) Their first extension to several variables was made by Calderon [1]—to harmonic and holomorphic functions in half spaces and their behavior on cones with vertices on the boundary. The material of the present section is directly based on Calderon's ideas, although many details are different.

80


5.5.1. Definition. If E cz S and a > 1, then Q(£,a) = U ^ a ( 0 . For any set E, XE denotes its characteristic function: XEM = 1 if X G £, 0 if xÊ. 5.5.2. Lemma. If E a S, a{E) = m < 1, and a > 1, there is a constant c = c(a, m) > 0 such that P{IS\E'\{Z)>C

for every z outside Q(E, a). Proof. Put V = S\E, f=Xv' Pick ZEB,Z^ Q(E, a). Without loss of generality, z = rci, 0 < r < 1. If r < 1/a, a trivial lower bound for the Poisson kernel shows that

(1)

pum > (|^,)"'^(^) ^ (^)"(i - '")•

So assume 1/a < r < 1. Let (^^ e Aut(B) be as in §2.2. L Then / o (^^ is the characteristic function of (pjy), and the ^-invariance of Poisson integrals (Theorem 3.3.8) implies that (2)

P [ / ] (z) = P [ / o (pJ(0) = f ( / o (^JrftT = G{cp,{V)). Js Put G„ = {^ 6 5:1 + 1/a < 11 - fî I < 2}. Pick rjeGo,, put C = ^PzC'?)- Since r > 1/a, |1 -rf],\>

2 |1 - ^ / i l - ( 1 - r ) > - . a

The definition of (p^C^) implies therefore that

Thus z = rî G D^(C). Since z ^ Q(£, a), C e K We have shown that (pXGJ ^ V. Hence G« c (^^(F), (7((p^(F)) > (T(GJ > 0, so that (2) yields

(3)

n/](z)>a(Gj

('-^)-

5.5. Theorems of Calderon, Privalov, Plessner

81

The lemma follows from (1) and (3). 5.5.3. Lemma. Suppose E a S is measurable, C is a point of density of E, a > 1, jS > 1. Then there exists r = r(E, C, a, jS) < 1 such that {\z\>r}nDf,(O^Q(E,oc). (The interesting case is jS > a.) Proof. Without loss of generality, a(E) < 1. By Theorem 5.4.8, the Poisson integral in Lemma 5.5.2 has X-limit 0 at C- Since this integral is > c outside Q(£, a), it follows that every sequence {zj that converges to C within Dp(C) must eventually be in Q(£, a). 5.5.4. Lemma. Fix r, 0 < r < 1. Every g e C{rS) extends then to a function G e C(rB) that satisfies AG = 0 in rB. Every uniformly bounded collection of these G's is equicontinuous on every compact subset ofrB. (The first part of the lemma asserts the solvability of a certain Dirichlet problem.) Proof Let ju be a complex measure on S that satisfies

(1)

[PUWOdKO = 0

for every / G C(S). By Fubini's theorem, (2)

jpifi](rcD)f(co)dc7(co) = 0

for every / . Thus P[//] = 0 on rS, hence in rB (by the maximum principle for ^-harmonic functions), hence in B by real-analyticity. Thus ju = 0, by Theorem 3.3.4(d). It follows now from the Hahn-Banach theorem that there is a sequence {y;.} c C(S) whose Poisson integrals converge to g, uniformly on rS. By the maximum principle, {P^fi]} converges uniformly on rB to GeC(rE). If z e rB and t > 0 is so small that (pJitB) cz rB, then G inherits the mean value property (3)

Giz)=

{Giq>M))d(î.Q Js

from Plfi]. To finish the proof, confine z to a compact set K c: rB, let T be the invariant measure of Theorem 2.2.6, and pick h e C°^(B) whose support

82


is a very small neighborhood of 0, such that j^hdr (4)

= 1. Then (3) implies

G(z) = {(Go (p^)hdx = f (/io (p^)G dx. JB

JB

The last integral shows that GeC'^(K); by Theorem 4.1.3(i), (3) implies now that (AG)(z) = 0. The equicontinuity assertion follows from the representation of G by the last integral in (4). 5.5.5. Theorem. Suppose E cz S is measurable, a > 1, and u is a bounded function in Q(£, a) such that Au = 0. Then u has K-limits at almost every point ofE. [Although it is not obvious that Q(£, a) satisfies the geometric condition that is needed before it makes sense to talk about X-limits at points of E, Lemma 5.5.3 shows that the condition does in fact hold at every point of density of £.] Proof. Suppose 0 < w < 1, without loss of generahty. If we replace a by a sUghtly smaller value, and write Q for Q(£, a), then u is continuous onQn B, hence u extends to a continuous function on B, with 0 < w < 1. Let rj/" 1, rj < r^+i- By Lemma 5.5.4 there are functions QJG C{r^B\ such that Qj = uon rjS, A^j = 0 in rjB, 0 < gj < L By equicontinuity, some subsequence, again denoted by {QJ}, converges uniformly on compact subsets of B, to an ^-harmonic function g in B. Let V be the Poisson integral in Lemma 5.5.2. (We may of course assume, without loss of generahty, that (T(E) < L ) Then v>c>0 on Bn 5Q, and |w — ^jl < 1 in rjB. Also, u — gj = 0 on rjS. Thus (1)

\u~gj\ P2Suppose g is real-valued function in B. If there is an ^-harmonic u'mB such that g < u, "WQ call u an J^-harmonic majorant of g. If, furthermore, every other ^-harmonic majorant M^ of ^ satisfies the inequality u(z) < Ui(z) at every zeB, then w is said to be the least ^-harmonic majorant ofg in B. 5.6.2. Theorem. Let fe H(B), let cp be as above, put v = (p(\og | / | ) , and define (1)

/ , = sup 0 l o g | / * | allows us to apply Fatou's lemma and to conclude that f(w* - \og\ f*\)da Js

< liminf f (w, - log\f,\)d(7. r-*l

Js

Since both u^ and log|/J tend to fi in the weak*-topology, the integral of the nonnegative function w* — l o g | / * | is 0. This proves (c). Since u* = ^fi a.e. on S (Theorem 5.4.9), (d) follows from (c) and §5.3.3. 5.6.5. Theorem. To every a > 1 corresponds a constant A((x) < oo such that

b

\MJ\^d(7 2, and Theorem 5.6.5 shows that F e L\(J). The inequalities

^ ^y>

-\^P

+ yP

1),

(0 0, show that (3)

* ....[2^^^ \r-fr\'< [2F^

(P^l) (0 3^2/4. Discard all members of r2 that intersect ^2 > let T^ be the remaining collection, etc. (The process stops if some F^ is empty, otherwise it continues through the natural numbers.) We thus get disjoint balls qi. Put Qi = 4^^. It is then easy to find disjoint Borel sets P^ such that q^ cz Vi cz Q. for each i, and [jiVi= [jiQi, If some qeT^ was discarded at the ith stage, then q intersects qi, and r(q) < 4r(î)/3, where r{q) is the radius of q. Since 1 + f + f < 4, we have q c 4qi = Qi. Thus E^ cz | J . Q^., and (i) is proved. Next, (3)

(y(Qi) < AMqd

< A^r

'Kqtl

by (1). This proves (ii). If we add the inequahties (3), the disjointness of {qi} proves (iii). Finally, fi(Qi) < tcr(gf), by (2), so that (iv) follows from KVd < KQi) < t(T{Qd < A^taiqd. 6.2.2. Theorem. To every a > 1 corresponds a constant A((x) < 00 such that (1)

t} t}, and A is the constant A^ in Lemma 6.2.1. Choose Qi.Vi'm accordance with that lemma, and put E^ = {Mfi > t}. Pick some g, = g(C, S), If co e Q,-, it follows that Qi cz Q(a), 23). Hence ( M M ) M > ^ (6(0), 2^)) > ^ ^ ^ > - 2 , by6.2.1(ii). Hence (Jf ^ ^ {^/^ > t/A"^}. By 6.2.1(iv),

KEt) oo. Hence //(£,) -• 0 as t -^ 00, by (1). Since fi^ is concentrated on nE^ (Theorem 5.2.7), we conclude that fi^ = 0, hence dfi = fda for some / e L^(cr), and Mf = Mfi. At every Lebesgue point of/, / ( O < (Mf)(Q. Thus / < t a.e. outside £,, or {/ > r} t

fdc7;

the dot-product is as in §5.4.2. Comparison of (1) and (3) leads to the following characterization: A C^-curve y in S is complex-tangential if and only if y is perpendicular to every circle F^ that it intersects. For any n > 1 there is a large supply of such curves. To illustrate this, consider the case n = 2. Let r = r(0, ^ =- 0(0 be of class C^ on /, so that y^ = re'^ defines a C^curve 7o iî the complex plane. Choose real functions w, t; on / such that (4)

t;' = j - ^ T ^ ,

u=

v-e,

and define

(5)

re" y='

Then 7 is a curve on S. Consider the map of S onto the Riemann sphere (more appropriately, in the present context, onto the complex projective space of dimension 1) that sends (2, w) to w/z. Its fibers are the circles F^. It carries y onto our prescribed curve yo, since t; — w = 6, by (4).

6.4. Cauchy Integrals of Lipschitz Functions

103

Differentiation of (5) leads to (6)

=

i(u' + r^v') 1 + r2

which is 0, by (4). Hence y is complex-tangential This construction can be used to connect any point (a, b)eS to any (z, w) G 5 by a complex-tangential curve: Perform a unitary change of variables, if needed, so that none of a,fc,z, w are 0. Let / = [0,1]. Find a positive function r e C\I) with r(0) = \b/a\, r(l) = |w/z|, then find 0 e C\I) so that e(0) = arg('^j,

0(1)

= a,gQ

and

j\l+r'r'Q'dt

= SiTg(^\.

Put t!(0) = arg b, define u and v by (4), and y by (5). Then y(0) = (a, b), y(l) = (z,w). 6.4.3. A theorem of Hardy and Littlewood (see, for example, Duren [1], p. 74) characterizes the holomorphic function in U that belong to Lip a (0 < a < 1) by the growth condition

i/'(z)i = o((i - izir-i). We shall reprove this in the n-variable context, but mention it here because it furnishes much of the motivation for the work in this section. 6.4.4. The Radial Derivative ^ / . L e t / e H{B) have the homogeneous expans i o n / = ZJF^. We define (1)

(^/)(z) = f/cF,(z)

(zeB).

k= 0

^f is related to the derivative of the slice functions /^ by

(2)

m)W) = ¥lW

(CeSAeUy

The advantage of ^f over f[ is that the former is a holomorphic function in 5.

104

6. Boundary Behavior of Cauchy Integrals

If / is a Cauchy integral, say / = C[JK] for some measure fj, on S, one can use (2) to derive a Cauchy formula for ^ / , namely

(3)

^""^^^'^-U-tor^''^^^-

6.4.5. Lemma. Iffe

H{B) and P is a multi-index, then

(1)

D^^f-diDPf=\p\D''f

and, for 0 < r < l,CeS, P ^ 0,

(2)

r^^Wf)(,rO= fV^/)(tOf"""'rft.

Proof. By linearity, it suffices to prove (1) and (2) for / = F^, a homogeneous polynomial of degree k. Then D^F^ has degree k — \p\ (negative degree indicates the zero-function), so that - ^D^JF, = [feD" - (/c - \P\)D':\F, =

\p\D'F„

which gives (1). Next,

so that the integral in (2) is r\D^F,)(0

=

r\^WF,)(rO.

6.4.6. Lemma. Ifg e H(B), c > 0, and (1)

|^(z)| 1 - |zp. Let F be the circle on which |Ap = ^(1 - r^). The Cauchy formula (5)

;,W(0) = ^ 2ni

(x-'-'h(X)dX JY

gives the estimate (6)

\h\0)\ 2)\\gi\\^ < 2'-\ m^ < 2'-\ Then g = YA 9i is in C, /i = ^T h^ is in H'^, and hence / = ^ + ft is in if °° + C.

114

6. Boundary Behavior of Cauchy Integrals

6.5.6. Remarks. The case n= 1 of Theorem 6.5.5 was proved by Sarason [1]. When n = 1, then //°° + C is the smallest closed subalgebra of L°° that contains //°^ and is larger than i/°°. (See Sarason [2].) This is false when n > 1. For example, let E be an arbitrary circular subset of 5 with 0 < (7(E) < 1; then £ is a union of circles F^ introduced in §6.4.2. Let XE be the subalgebra of H°°(S) + C(S) consisting of all / whose restriction to almost every F^ c= £ belongs to i/°°(T). It is clear that XE is closed, that H'îS) ^ XE ^ H'îS) + C(5), and that both inclusions are strict. Aytuna and Chollet [1] have extended Theorem 6.5.5 to strictly pseudoconvex domains. If B and S are, however, replaced by a polydisc W (n > 1) and its distinguished boundary T", the situation is different: H'^{T'') + CiT"") is still a closed subspace of L°°(T"), but it fails to be an algebra (Rudin [7]).

6.6. Gleason's Problem 6.6.1. Let X be some class of holomorphic functions in a region Q cz C". Gleason's problem for X is the following: IfaeQ and fe X, do there exist functions g^,.. .,g„eX such that (1)

/(^) - m

= E (z, - a,)g,(z) k=l

forallzeQl Gleason [1] originally asked this for Q = B, a = 0, and X = A(BX the ball algebra. (In other words, do the coordinates functions generate the maximal ideal consisting of 3\\fEA(B) with /(O) = 0?) The difficulty of the problem depends on Q. For instance, if B is replaced by a polydisc, Gleason's question becomes trivial: take n = 2, for simpHcity, and decompose a n y / e H(U^) that is continuous on U^ into a sum (2)

/(z, w) = /(O, 0) + zg,(z, w) + wg2(z, w)

where ,,, (3)

, , /(z,0)-/(Q,Q) î(z, w) = , z

^ ^ /(z,w)-/(z,0) g2(z, w) = . w

The point is that z # 0 and w 7^ 0 on the torus T^, so that the functions (3) belong to the prescribed class. Put in different terms, i f / G //°°(L/^) and ^2 is as in (3), then ^2 e H'îU^). This is not true in B:w^/(1 — z) is bounded in B2, but w/(l — z) is not; to see the latter, take (z, w) = (cos t, sin t).

6.6. Gleason's Problem

115

Gleason's original question was solved by Leibenson (see Henkin [4]), who observed that (4)

/(z) - /(O) = C ^f(tz)dt JQ

at

= t z, f^=i

C(DJ)(tz)dt JQ

and then proved that the functions (5)

g,(z) = f (DJ)(tz)dt Jo

(l 0).

122

7. Some L^-Topics

Since 21 sin(7rs/2)| > | e^^'^ — e~'^^'^ |, G is a bounded holomorphic function in the half-plane c > 0. Since H vanishes at every positive integer, G vanishes at every odd positive integer. This zero-set of G violates the Blaschke condition for bounded holomorphic functions in a half-plane. Hence G is identically 0, and so is H. Thus (1) holds, and T^f = f. To evaluate 7^/, it is enough to consider the case /(O) = 0. Insert / into 7.1.1(6), use the explicit formula for K^z, w), and integrate in polar coordinates (§1.4.3). It is then easy to see that the integrals over S are 0, by the mean-value property of conjugate-holomorphic functions. 7.1.3. The Case s = n -\- 1. This case is particularly simple, because

(1)

\K„^,{z, w)\ = {j^^

l_|w|2

T T ^ ^

>+i

=

(JR - . P

7.1. Projections of Bergman Type

(b)

123

/ / (1) holds then T, projects L^(v) onto {U n H){B): in fact

(2)

TJ = f and Tj = W)

for every

fe{UnH){B).

Proof. By Proposition 7.1.2, part (b) is an immediate consequence of (a), since H°°(B) is dense in {U n H)(B). The estimate (3)

e-^''l/2|X,(z, w)| < |K,(z, w)| < ^^'^'/^|X,(z, w)|

(where s = a + it, as before) will be used in the proof of (a); it holds because 1 — has positive real part, so that its argument Hes between —n/2 and n/2. We first take p = 1, in which case (1) becomes a > 0. If (7 < 0 and zeB, then X/z, w) is not a bounded function in B, and therefore 7^/fails to exist for some feL^(B). So assume a > 0. The duahty between L^ and L** shows that 7^ is bounded on L^ if and only if (4)

sup

|X,(z, w)\dv(z) < 00.

By (3), we can replace s by a in (4), and now Proposition 1.4.10 implies that (4) fails when a = 0 but that (4) holds when a > 0. This proves (a) in the case p = I. If 1 1 < oo such that IIT,^ lip < AII ^11 p for every g e L^(v). Apply this tog =f and g = f^

7.1. Projections of Bergman Type

125

(where /^z) = f(rz), 0 < r < 1, as usual). By Theorem 7.1.4, X = TXfr) = Ufr + /.) = ITXUrX SO that { IfA-dv = 2" \ \ Uu.Wdv

JB

JB

< {lAf

f luj" dv

JB

or (2)

f \f\Pdv < (2Ay f \u\P dv JrB JrB

(0 < r < 1).

Now (1) follows as r / * 1 in (2). 7.1.6. The real part of the function /(z) = f l o g ( l - z O is bounded in B, but its imaginary part is not. Theorem 7.1.5 can therefore not be extended to the case /? = oo. As in the proof of 7.1.5 shows, Theorem 7.1.4 also fails for p = oo, no matter what s we choose. 7.1.7. The Koranyi-Vagi theorem shows that Theorem 7.1.5 has an analogue in which integration over B is replaced by integration over S: There are constants Mp^„ < co, for l < p < o o , n = l , 2, 3 , . . . such that

(1)

{\f\''da<M,^J\u\''da

Js

Js

for every f = u -\- iv in H^{S) with v(0) = 0. In contrast to Theorem 7.1.5 this fails when p = 1 and n = 1; the functions (2)

/Xz) = y ^ T ^

(0 < r < 1)

show this. However, it is not known whether there exist constants M^ „ < oo for n = 2, 3, 4 , . . . , such that (1) holds. In other words, if 5 is (for example) the unit sphere in C^, it is not known whether (3)

sup|4q^ M(a + lb)

is subharmonic in V. Less precisely, the restrictions of u to all complex Hues should be subharmonic. The most obvious examples of plurisubharmonic functions are l o g | / | and l / r ( c > 0), for any / e //(Q). The slice-integration formula shows that every plurisubharmonic function M in Q is subharmonic in Q (see the proof of Proposition 1.5.4) and that therefore AM > 0, provided, of course, that u e C^. When Q. = B, the class of plurisubharmonic functions is Moebius invariant. It follows that every plurisubharmonic u e C^(B) satisfies (2)

AM

> 0 and

AM

> 0.

Contrary to what one might expect by analogy with Theorem 4.4.9, (the equivalence of (b) and (c)), this last statement does not have a converse (at least not when n > 2): The function (3)

M(Z) = ZjZi + Z2Z2 -

Z3Z3

has AM = 4 and Au = 4(1 - |z|^)(l - M) > 0 in B, but M(0, 0, W) = - 1 wl^ is not subharmonic. Plurisubharmonic functions will play an important role in Chapter 17. They are also closely related to the characterization of domains of holomorphy (see, for instance, §2.6 in Hormander [2]). At present, we only need the following inequality. Its statement uses the familiar notation z = (z', z„) for zeB„, where z' = (z^,..., Z „ _ I ) E B „ _ I . 7.2.2. Proposition. Ifu is plurisubharmonic in JB„ (n > 1) and M > 0, then (1)

f ^Bn-i

M(ZÔMV„_I(Z')
0 can be removed. It is made because its presence assures the existence of all integrals that occur in the proof and it allows us to apply Fubini's theorem without any argument. (ii) The integrals on the right of (1) are nondecreasing functions of r, since u is subharmonic. The supremum is therefore equal to the limit, as Proof. To each z'sB^-i 0 < r < 1, (2)

corresponds a

Z „ G C SO

that {z\zêS.

For

u{rz\ 0) < ^ J " u{rz\ rzê'^Q,

since u is plurisubharmonic. By formula 1.4.7(2), it follows that (3)

f

u(rz\ 0)dv„_ i(zO < f u(rOdcT(0.

JSn-i

JS

Setting rz' = w\ the left side of (3) is seen to be I

JrBn- 1

u(w\, l , 0 < p < o o . (a) (b)

The extension E is a linear isometry of(L^ n H)(B„_ j) into H^(B„). The restriction p is a linear norm-decreasing map of H^(B„) onto (UnH)(B,.,).

Proof Apply Proposition 7.2.2 to the plurisubharmonic function u = | / | ^ where / 6 //(B). This gives (a) and most of (b). The fact that p(H%B„)) is all of (U n H)(B„-1) follows from (a) and the identity pEg = g.

128

7. Some L^-Topics

Note: An analogue of this result holds in polydiscs, for restrictions to the diagonal, but it is not quite so easy. See Rudin [1], pp. 53, 69, DurenShields [1], Horowitz-Oberlin [1], J. H. Shapiro [1], Moulin-Rosay [1], and Detraz [1]. 7.2.5. Theorem. Suppose n > 1, 0 < p < oo. (a)

lffeH\B,)then

(1)

1/(^)1 0, Fubini's theorem implies now that (6)

[gii. {^J)d^(0 = 0

Js

for almost all {[/J G^°^. For any such choice of {(7J, ^(C, {t/J) = 0 for almost all êS, and therefore (2) holds. 7.2.8. Proposition. For z = ( z j , . . . , z„) G C", define F,(z) = n''"(zi-.-2„)^''

(/c = l , 2 , 3 , . . . ) .

130

7. Some L'-Topics

Then (i) \F^(z)\ there is also the integrated counting function f

dt n/0Jo '

Nf(r)=

(1)

(0 | , we have 00

CO

(2) \P{z)\ < n of{1I's+ in4'"|zr2'} - 1expansion + ^4'"^|zr where c{t) is the number the binary of t. Thus c{t) < log t + 1/log 2, so that

(3)

^

c

ip(z)i> 1 + I (r + i)''"izr pm2' >

4 > 1 — r^

2 -. 1— r

The left side of (7) is thus oo, and (5) is proved. Note. Each of the m2^ discs in Ej, has radius (1 -riy/2

^2-^'-'^^^

so that the area of E^ is about 2nm, for every k. Thus (b) shows that E u D is a zero-variety for A(B\ of infinite area. No such example seems to be known when n > 2 (where "area" must of course be replaced by (2n — 2)-dimensional volume), even with H'^{B) in place of A{B). The following lemma will be used in the proof of Theorem 7.3.8. 7.3.7. Lemma. Suppose (a) n is a finite positive measure on a set Q; (b) V is a real measurable function on Q, with 0 < y < 1 a.e., whose essential supremum is 1; (c) 0 is a continuous nondecreasing real function on [0, oo), 0(0) = 0, 0(x) -^ CO as X -^ ao\ (d) 0 < ^ < 00, 0 < t < 00. Then there exist constants c^ e (0, oo\for k = 1, 2, 3 , . . . , such that (1)

{ ^(c,v')dfi = S.

//I a I < 1 and i/1^ = 1^(0 is the set of all coeQat which Ckv\aj) > t, then (2) lim Ck(x^ = 0 k->ôo and (3)

hm f ^(CkV^)dfi = S.

140

7. Some L^-Topics

Proof. The monotone convergence theorem shows that (4)

c-^

\^(cv^)dfi

is, for each k, a continuous mapping of [0, oo) onto [0, oo) which carries 0 to 0. Hence (1) holds for some c^ e (0, oo). Assume |a| < j5 < 1, and let E be the set of all co e Q at which v(co) > fi. Since the essential supremum of r is 1, ix{E) > 0. Also, (5)

{cJ^)ii{E) < [{c,v')dii log | r-y(O) | > 0.

Next we note that

(25)

n(i-2--)>i 1=1

If C e Y; and I A| = r, it follows from (9) and (11) that 4|/(A0I > \FjiXO\ - 1 = r*'|f/OI - 1 > \FjiO\ - 2, by (22). Since IF/OI > tj on Yj, (21) implies now that (26)

8|/(A0 + b(AOI > 2 | F / 0 I - 4 - 8||Z>|L > 1 + 1^/01

By (20), (24), and (26), (27)

^ J " log I ch(re'''01 ^6 > log{e(l + | F/QI )}•

Since ij/ is convex and nondecreasing, Jensen's inequality leads from (27) to (28)

^ I " ./.(log I cHre'Ô I )rfe > W( \ F / 0 1 )•

This holds for every C^Yj. If we integrate (28) over Yj and recall the rotation-invariance of a, (12) shows that (29)

f il/(\og\cHrO\)d(T(0

>I

JYJ

This was done for fixed r = r(j). But (29) impHes obviously that (30)

sup

U(\og\cHrO\)d(T(0

>j

for every j e J. The left side of (30) is therefore infinite, and we have proved that chis not in H^. 7.3.9. Remarks, (a) The appearance of a "constant multiple" may be a bit puzzHng in the conclusion of Theorem 7.3.8. The phrase may be omitted

7.4. Pluriharmonic Majorants

145

when \l/ satisfies the growth condition Mt + 1) (1)

Hm s u p —77-T— < 00

since H^(B) is then closed under scalar multiplication. In that case, the conclusion is simply that h ^ H^(B), The case (p(t) = exp(eO, il/(t) = (p(t + 1), shows the need for the "constant multiple": For every / G i / ^ we have e'^feH^. Thus H^ and H^ have the same zero-varieties, although \j/{t)/q){t) ^ 00 as t -> 00. (b) Theorem 7.3.8 states that no "bounded perturbation" f -\- b oi f has the same zeros as any member of H^(B), Letting b range over the constants, this gives some information about the level sets of/, or, in other words, about the distribution of values of/. (c) Fixp,0 p. Since ij/ satisfies (1), Theorem 73.S furnishes anfe H^(B) whose zero-variety is a determining set for every H\B) with q > p. (d) With a gap series in place of a product. Theorem 7.3.8 appears in Rudin [9]. An earlier result, for polydiscs, is due to Miles [1]; it strengthened Theorem 4.1.1 of Rudin [1]. Similar theorems have also been proved for Bergman spaces, by Horowitz [1] and J. H. Shapiro [2].

7.4. Pluriharmonic Majorants 7.4.1. Introduction. If Q is a region in C" and 0 < p < 00, Lumefs Hardy space (LHYiQ) is defined to consist of all feH(a) such that | / | ^ < w for some pluriharmonic u. (Lumer [1].) Whenn = 1, pluriharmonic is the same as harmonic, so that this definition coincides with the classical one that involves harmonic majorants. But when n > 1, then (Lif)^(Q) is a proper subclass of what is usually called H^(Q); see Stein [2], Stout [5], for instance. The use of pluriharmonic majorants leads to some appeahng function-theoretic properties of (LH)^(Q); from the standpoint of functional analysis, however, (LHy(B) has some unexpectedly pathological properties, as we shall see presently; for example, {LHy{B) is not even a Hilbert space. One appealing property is what may be called holomorphic invariance: IfQ>: Qi ^ Q2 is holomorphic and f e (LHy(Q2) then / o O 6 (LHf (Qj). This is trivial, for if u is pluriharmonic, so is w o O, simply because u is locally the real part of a holomorphic function (Theorem 4.4.9).

146

7. Some L^-Topics

As regards zeros, (LHy(Q) behaves much better than H%B) does: / / Q is simply connected, then every fe(LHy(Q) has the same zeros as some heH'îO). This is a corollary of the following result: 7.4.2. Proposition. Suppose Q is a simply connected region in C" and f e H(Q). Then f has the same zeros as some h G H * ( Q ) if and only if there is age H(Q) such that (1)

log|/| 1. In addition to the difference in zero-varieties that we just saw, there are at least two other ways of seeing that {LHy{B) is a rather small subclass of H\B). The first of these involves rates of growth. If w is a positive pluriharmonic function in J5, then each slice function u^ is a positive harmonic function in U, hence u^{X) < 2w(0)/(l - \X\). lffe(LHy(B), it follows that (1)

\f(z)\

(15)

(l-xMx,0)>X4 m=l

/2m\ ~ ^ \ ^ /

(1+x)2m+l

As X / * 1, the sum of this series tends to infinity, by (3) and (7). But M(Z, 0) is a positive harmonic function in (7, so that (1 — x)u(x, 0) must be bounded. This contradiction shows that | / | ^ has no pluriharmonic majorant in B. Hence / is not in {LH)\B). 7.4.5. Notation. As usual, /°° is the Banach space of all bounded complex sequences, with the sup-norm, and CQ is the subspace of/°° consisting of those sequences that converge to 0. The following theorem describes some pathological features of (LHy(B): 7.4.6. Theorem (Rudin [11]). Fix n > 1,1 0. (a)

There exists a linear map of /°° into (LHy(B) which assigns to each yel"^ a function fy that satisfies

(1)

lly|loo i h m supjcj,

which proves (c). We conclude this section with an observation which is just as easily proved for arbitrary regions: 7.4.7. Proposition. Suppose Q is a region in C", feH(Q), 0 < t < n/2. Then there is an heH(Q) such that

| I m / | < 1, and

Qxp{t\f\} and (1 + zi)/(l - zi) is not in

1 + Zi

H\U),

7.5. The Isometrics of H\B) 7.5.1. Introduction. The isometrics in question are the linear maps T of H\B) onto H\B) that satisfy

(1)

l|T/||,= 11/11,

for every feH^(B).

These will be completely described, for all M > 1 and

for all p 7^ 2, 0 < p < 00.

It should be clear why the case p = lis special: H^(B) is a Hilbert space, isometrically isomorphic to every other separable Hilbert space (of infinite dimension), and its isometrics are therefore the unitary operators; the fact that H^(B) is also a space of holomorphic functions plays no role here. For any p, the following construction yields isometrics: Choose xj/ e Aut(B), c e C, | c | = 1, put i^ ~ ^(0) = a, and define

(2)

(T/)(z) = c^^^_^^j^^^,„,^/Wz)). It is easily seen that these satisfy (1), since f IT/1" dcr = r P{a, Q \ m{Q) Js Js

I" do{0

= i'CI/r-i/'DCa) = PUfnma))

=

[ifl'da.

7.5. The Isometrics of HP{B)

153

This computation made use of the ^-invariance of Poisson integrals (Theorem 3.3.8) and the fact that \l/{a) = 0; we have identified / and Tf with their boundary values / * and {Tf)* and will continue to do so, unless there seems to be some danger of confusion. It is also easily seen that the range of any T given by (2) is all of H\B\ for if ^G/f ^(5) and /(^) = ^

(1 - \a\yip

^(^ ^^)^'

then/GH^(B)and7y = gf. The principal result of this section, Theorem 7.5.6, is that every linear isometry ofH^{B) onto H\E) has the form (2), when p ^ 2. The first case of this (n = 1, p = 1) was proved by deLeeuw-RudinWermer [1], using the fact that the extreme points of the closed unit ball of H^(U) were completely known. Forelli [2] devised a different method that handled H^(U) for all p, and which was then appHed by Schneider [1] to HP(U"X and by ForeUi [3] to HP(B) for p > 2, The proof of Theorem 7.5.6 (Rudin [8]) is based on these ideas, in combination with some very general facts about L^-isometries on arbitrary finite measure spaces, namely Theorems 7.5.2 and 7.5.3. As shown by Koranyi-Vagi [2], this proof extends without difficulty to arbitrary bounded symmetric domains in place of balls. Stephenson [1] has studied the isometrics of the Nevanlinna class. In the following two theorems, fi^ and JLI2 will be finite positive measures, on some sets that will play no role at all and that will not even be named. 7.5.2. Theorem. Assume 0 2,

J l l + A/iTd^. = J l l + A/^rd/i^

for every XeC, then

(2)

jihof,)dfi, = jihof^)dti2

for every bounded Borel function h:C^C, function h on C.

and also for every positive Borel

Proof. It is enough to prove that (2) holds for every heCo = Co(C), the space of all continuous functions on C that vanish at infinity. Let X be the set of all heCo such that (1) implies (2). If/j, /2 satisfy (1), then so do c -h / i , c -h /2 and c/j, c/2, for any ceC. It follows that J!f is a uniformly closed subspace of CQ which is invariant under translations, dilations, and rotations. We have to prove that X = CQ.

154

7. Some L^-Topics

Define (3)

11 + ê'^ r ^9

u(X) = -^ f

(^ e C).

By Fubini's theorem,

(4)

jiu o y;.)d^, = _L J " rfe J11 + e'«/;. i" rf/i,

for 7 = 1, 2, so that (1) impHes (5)

J (u o f^)dfii "" J (" ° -^2)^/^2 •

Equation (5) remains true if u is replaced by w,, where w^^) = w(tA), t > 0, XeC, hence also if w is replaced by any finite linear combination (6)

v{X)=

Y.î B^ with 0(0) = 0, the following are equivalent: (i) O has property (*). (ii) O is a linear isometry. (iii) There is a multiplicative linear operator

such that (Ef) o^=.ffor

every fe WîB^).

Proof. Assume (i). Pick CeC", |CI = 1, and put f(z) = (z,0' Then feH^(B„l li/ll^ = 1. Hence there is a geH'îBJ, with H^filoo = 1, such that g( be a conformal map of U onto S, with (p(0) = 0, such that, setting T = cpoy^ we have Im r(t) -> + oo as t -^ 1, Replace/by F =fo cp'K Then F e H'^^L), \F\ < 1, F(r(r)) ^ 0 as r ^ 1. Given d e (0, 1), we have to prove that F(x + iy) ^ 0 SiS y ^ -f oo, uniformly in |x| < 1 - d. Fix e, 0 < e < 1. Choose any >; > Im r(0), so large that |F(r(0)| < e whenever Im r(t) > y. We claim that then (2)

|F(x + i » | S. I f — l + ( 5 < x < XQ, replace 1 + z by 1 — z in (3); this leads to the same conclusion. Thus (2) holds, and the proof is complete. 8.4.2. Remark. We stated in Lindelof's theorem in the disc U but proved it in the strip Z. Other conformal maps will of course transfer the theorem to other regions in C. For example, let U^ = {z = re'^: r > 0, | e | < a}. If / G H ° " ( n j , / - > L along some curve yo ^^ n^ that approaches 0, and p < CL, then / tends to L along every curve y that approaches 0 within 11^. 8.4.3. Approach Curves in B. A curve in B that approaches a point C e 5 will be called a C-cwn;^. More precisely, a C-curve is a continuous map F: [0, 1) -> 5 such that F(0 -> C as t ^ 1. Usually, however, it will not be necessary to refer to any parametrization. With each C-curve F we associate its orthogonal projection (1)

y =

P-oc

l-|yp

P

\r-y\

which tends to oo as t ^ 1, since F is special. Now define g(X) = / ( ( I - A)7 + XT) for \X\ < R, use the fact that / is bounded in Dp, and estimate g(l) — g(0) by the Schwarz lemma, as in the proof of Theorem 8.4.4. This leads to (1).

8.5. The Julia-Caratheodory Theorem 8.5.1. In the present section, the following one-variable facts will be generalized to holomorphic maps from one ball into another: Suppose f:U ^ U is holomorphic. If there is some sequence {zj in U, with Zi -> 1 and /(z,) -^ 1, along which

1 - i/fe)i 1 - k.-i is bounded, then / maps each circular disc in U that has 1 in its boundary into a disc of the same sort. This (in a more quantitative form) is Julia's

8.5. The Julia-Caratheodory Theorem

175

theorem. Caratheodory added that /'(z) then has a nontangential positive finite Hmit at z = 1. Full details of this may be found in vol. 2 of Caratheodory's book [3]. The generalizations to several variables will be proved directly, without any reference to the theorems just mentioned. In fact, if one takes n = 1 and m = 1, the proofs that follow are the classical ones. 8.5.2. The Setting. Throughout this section, m and n will be fixed, F will be a holomorphic map of B„ into B^, C will be a fixed boundary point of B„, and we define (1)

L = limi„f-^-|^(^>l^ 1 - |z| Z-; The basic assumption we make is that L < oo. There is then a sequence {aj in J5„ that converges to C, such that

and such that F{ai) converges to some boundary point of B^. By unitary transformations we may choose coordinates so that C = î and F(a;) converges to ^1. (The symbol e^ is here used with two meanings; it designates the first element in the standard basis of C" as well as C". It is unlikely that this will cause any confusion.) Let / i , . . . , /^ be the components of F. The Schwarz lemma (Theorem 8.1.4) states that ^^^

|l-P^l-|F(a,)P

l-|f(z)|^

-

l-|a..p

|l-p

l-|zp

for all zGfi„. As i -> 00, 2,

(21) ^^^m^^ = —p(~^•Yn]_;—i^^^-^t^—'

^®-

The integrands are bounded, by the bounds of (i) and (iii) in D^{e^. In view of (19), the left sides of (20) and (21) are therefore bounded in DJê^. To finish, we have to prove that the left side of (20) has restricted iC-Hmit 0 at e^. By Theorem 8.4.8 it is enough to prove this for the radial limit. Moreover, it involves now no loss of generality to assume n = 2, m = 1, in which case / i = F. Writing (z, w) in place of (z^, Z2), we can expand F in the form (22)

F(z, w) = /(z) + 2w(l - zy^^Q{2) + £ QJ^z)v^K

Then (D2F)(z, 0)/(l — zY'^ = 2g(z). It is therefore enough to show that (23)

^(x)^0

as

x/l.

We know that (1 — /(z))/(l - z) -^ Las z ^ 1 nontangentially, and that g is nontangentially bounded at 1, and we make one further reduction: //I YJO ^k^^\ < 1 i^ ^ certain disc with center at 0, then also \CQ + ^C^W | < 1 in this same disc. This is so because CQ + ^C^W is the arithmetic mean of the first two partial sums of the power series. If we apply this to (22), we see that (23) is a consequence of the following proposition (in which there is some redundancy in the hypotheses): 8.5.7. Proposition. Suppose h: B2 -^ U has the form (1)

h(z, w) = f(z) + w(l -

zy^'g(z)

where / , g e H(U\ (1 —/(z))/(l — z) has finite nontangential limit L at z = 1, and g is nontangentially bounded at 1. Then (2)

^(x)-^ 0 as X y 1.

Proof. Choose e > 0, put c = L^/s^, let z tend to 1 along the Hne z = x 4ic(l — x). Then 1 — z = (1 — ic)(l — x\ hence (3)

|l-z|>c(l-x),

182

8. Consequences of the Schwarz Lemma

and also 1 — \z\^ > 1 — x if c^/(l + c^) < x < 1, an assumption that will be made in the rest of this proof. Note that (4)

/(z) = 1 - (L + o(l))(l - ic)(l - X)

SO that (5)

R e / ( z ) = l - ( L + o(l))(l-x).

Associate with every z under consideration a w e C with |wp = l — | z p > 1 — X, whose argument is so chosen that (6)

w(l - zY^'giz) = |w(l - zy/'g{z)\ > c'^\l

- x)\g(z)U

by (3). Hence, by (5) and (6), (7)

1 > Re h(z, w) > 1 + {c'^^\g(z)\ - L - o(l)}(l - x). Consequently,

(8)

lim sup \g(x + ic(l - x))\ < Lc~^'^ = G.

The same estimate holds on the Hne z = x — ic{\ — x). Since g{z) is bounded as z -^ 1 between these two lines, it follows that (9)

limsup|6f(x)| < e ,

which proves (2), since £ was arbitrary. 8.5.8. Examples. We shall now show that the conclusions of Theorem 8.5.6 are optimal. The numbers (i) through (vi) will refer to Theorem 8.5.6. The first two examples will use the function (1)

g{z) = exp^ - ^ - / log(l -z)\

Note that \g\ < \mU, (2)

{ze U).

and that g'{z) =

1- z

As z -• 1, g(z) spirals around the origin without approaching it.

8.5. The Julia-Caratheodory Theorem

183

First Example. Take n = m = 2, define F : B2 ^ ^2 by (3)

F(z, w) = (z, wg(z)).

The hypotheses of Theorem 8.5.6 hold, with L = 1. Since D^/i = 1 and D2/1 = 0, we have (4)

(JFKz, w) = (D,f,)(z,

w) = g(z).

Therefore the radial limit 0/1)2/2 cind ofJF does not exist at e^. This dealt with (vi). As regards (iv),

(5)

IW

(1 - zyi\D, f,)iz, w) = ^^^r^2 • 9(z).

This has no K-limit at e^, although its restricted X-Umit is 0. We see also that the boundedness assertion made about (iv) becomes false if the exponent ^ is replaced by any smaller one. Second Example. Take n == 2,m = 1, put (6)

F(z, w) = z +

Wg(z)'

Example 8.1.5 shows that F maps B2 into U. The hypotheses of Theorem 8.5.6 hold again with L = 1. Since F = f^ ,we now have

1 -z (8) (9)

(DiA)iz,^ w) = (D2A)(Z , w) (1_

2)1/2

'

^V-h

2 ( 1 - z)

iw^ 1 1 •QiA ^ ' 2(1 - z) w -(l_z)l/2

.

f

,(r\

Hence (i), (ii), and (v) n^^rf /tare no K-limit at e^, and the boundedness assertion made about (v) becomes false if\ is replaced by any larger exponent. Third Example. This will show that the exponent | is best possible in (iii). Take n = 1, m = 2. Pick a > 0, put

(10)

''(^)=^rj-î«i"^^«'

and note that h hes in the disc algebra, that h{\) = 0, and that Re h(z) > 0 for all other zeU.

184

8. Consequences of the Schwarz Lemma

Put c = lIlTi^^' and define F = (/i,/2) by (11)

f,{z) = ze-^'^'^K

Mz) = c'^\l

-

zr^-^^'z.

Let u = Re[ch]. Since |1 -^^"^l < | e | if |e| < TT, we have I/2P < w on the unit circle, hence I/21^ < w in t/, because I/2 P is subharmonic. Therefore (12)

l/iP + l / 2 p < " + e - ^ " < l

in U; the last inequahty holds because 0 0. By (14) and (11), Um f\(x) is also finite. (Ahern and Clark [1] have proved much more general theorems about derivatives of functions of the form (10).) By (11), fzM/il — xY^^^^'^^ is unbounded as x/" 1. The boundedness assertion concerning (iii) becomes therefore false if^ is replaced by any larger exponent. Finally, we note that the map F defined by (3) furnishes an example in which the function (iii) has no X-Hmit at e^.

Chapter 9

Measures Related to the Ball Algebra

9.1. Introduction This chapter deals with two types of topics. The material of Sections 9.2 and 9.3 is function-theoretic. The measures that are discussed there are intimately related to the holomorphic functions in B. On the other hand, Sections 9.4 and 9.5 describe some measure-theoretic aspects of the theory of function algebras in general. These would not become any simpler by specializing to the ball algebra. Both aspects are used in the proof of the Cole-Range theorem (Section 9.6), which is one of several modern generalizations of the classical theorem of F. and M. Riesz. The results of this chapter will be used in Chapters 10 and 11. Here are some of the relevant definitions. 9.1.1. Function Algebras. Let X be a compact Hausdorff space. A function algebra on X is a. subalgebra A of C(X) which is closed in the sup-norm topology, which contains the constants, and which separates points on X. In the case of greatest interest to us, X will be S, and A will be A(SX the restriction of the ball algebra A(B) to the sphere S. Let us note that A(S) and A{B) are isometrically isomorphic Banach algebras. This is an immediate consequence of the maximum modulus theorem. 9.1.2. Representing Measures. If X is a compact Hausdorff space, the set of all regular complex Borel measures on X will be denoted by M{X). With respect to the total variation norm, M{X) is a Banach space which can be identified with C(X)*, the dual space of C(X). Hence M(X) also has the corresponding weak*-topology. Suppose ^ is a function algebra on X, and /i is a multipHcative linear functional on A,h ^ 0. (I.e., /i is a homomorphism of A onto C.) Since h(l) = 1 and \\h\\ = 1, there is at least one probability measure p e M(X) which represents h, in the sense that (1)

Kf)={

fdp

(fBA).

Jx 185

186

9. Measures Related to the Ball Algebra

The set M^ of all such p is clearly a convex subset of M(X) which is also weak*-compact. In the special case A = A{SX we associate to every z e B the set M^ of all probability measures p e M(S) that "represent z" in the sense that

(2)

f(z)=

[fdp Jss

for every fe A(B) = A(S). For example, if PÔ = P(z^ 0 (the invariant Poisson kernel), then Pâ e M^.ln particular, a e MQ. There is an interesting difference here between the cases n = 1 and n > l.Whenn = 1, every point of the open unit disc (7 has a wm^we representing measure on the circle T. But when n > 1, every z e B has many representing measures on S. For example, for every C e S we have

(3)

/(O) = ^ J ' fie^'OdQ

(fe A(B)).

Thus there is a p e MQ that is concentrated on the circle {e'^C- —7i cP^ on S, and p^ e M^. Define

Then p^ e M^ and p^ ^ = 0 for some y' e K. The convexity of F^' shows therefore that

o 0, without loss of generality. Let G be the set of all continuous functions taking X into [0, 1]. Define F on G x K by (1)

F(K p)=

[hdv+ Jx

f (1 - h)dp. Jx

By the definition of the weak*-topology in M(X), F is a continuous function of p, for every fixed h. Since F is affine in each variable, F satisfies the hypotheses of the minimax theorem. Now fix pe K and recall that v _L p. This implies that there are disjoint compact sets in X, one of which carries most of the mass of p, whereas the other carries most of the mass of v. Urysohn's lemma provides therefore an heOfor which both integrals in (1) are very small. In other words, (2)

inf F(h, p) = 0 heG

for every p e K,so that (3)

inf sup FQi, p) = 0 heG

peG

194


by the minimax theorem. By (3), there are functions hie G (i = 1, 2, 3,...) such that (4)

{ hidv-\- f (1

-hi)dp ti, and there are Borel sets A^ (^ X such that pp. = /x|^.. Put A = \Ji Ai, and define Pa = IÎASetting PQ = Yj ^'Pt^ the convexity and compactness of K imply that Po G K. Suppose E cz X satisfies Po{E) = 0. Then pf(E) = 0 for all i, hence also \pp.\(E) — 0, or \p\(E n Ai) = 0. Taking the union over all i, it follows that \p\(E n A) = 0, or |jU„|(£) = 0. This proves that p^ \\Pp.\\ > t^ for all I Consequently, Finally, put p^ = p — p^ = p |^y^. Pick p EK. Then (2)

Pa + {lisX < i(Po + P) e X,

so that \\pa -\- (ps)p\\ < t. Since p^ and (ps)p are concentrated on A and X\A, respectively, the norm of their sum is the sum of their norms, and since

9.5. A General F. and M. Riesz Theorem

195

ll/^flll = t, we conclude that ||(iuj^|| = 0. Thus fi^ J- p, for every p e K, and an appeal to Rainwater's lemma shows that fi^ is concentrated on a set of type F„ that is X-null.

9.5. A General F. and M. Riesz Theorem 9.5.1. In this Section, X is a compact Hausdorff space, as before, A is a function algebra on X, O is a multipHcative linear functional on A, and M^ is the set of all probability ipeasures p e M(X) that represent O. As was pointed out in §9.1.2, M^ is nonempty, convex, and weak*-compact. Hence M^ can play the role of K in Theorem 9.4.4. The main result (Theorem 9.5.6) states that if v e A^ and if v = v^ + v^ is its GKS decomposition relative to M^, then both v^ and v^ He in A-^. (In the original version of the F. and M. Riesz theorem, A was the disc algebra on the unit circle, and the conclusion was that v^ = 0, i.e. that V f. Then (1)

Q>u = \ udp Jx

for every p e M^, simply because every such p is a real measure. The set of all real parts of members of ^ will be denoted by Re A. Clearly, Re A is a subspace of

CR(X).

Observe that if p G M(X) is a probability measure that satisfies (1) for every w G Re ^4, then p e M^. For iffeA, then (1) holds for the real parts of / and of // 9.5.3. If ^ G Cj^(X), ueRe A, u > g, and p e M^, then obviously (1)

Jx

dp < Ow.

Hence the supremum of the left side, over all p, is at most equal to the infimum of the right side, over all ueRe A such that u > g. The following lemma asserts that equahty actually holds. 9.5.4. Lemma. If g e CR(X) then there exists po G M^ such that (1)

g dpo = infjOw: w > ^, w G Re A}. Jx

196


Proof. For h e (2)

CR(XI

define p(h) = inf{OM: M > fc, w e Re A}.

Then pQi^ 4- /12) < p(hi) -\- K/iiX and p(th) = tpQi) for scalars t >0. In particular, 0 = p(0) < p(h) + pi — h), hence — K^) < p{ — h\ so that fp(/i) < p(th) for all real scalars t and all /i e C^{X), If we define A(fgf) = tp{g\ it follows that A is a linear functional on the one-dimensional space spanned by g, and that A < p there. One of the standard versions of the HahnBanach theorem (see, for example, Theorem 3.2 in Rudin [2]) asserts now that A extends to a linear functional on CR{X) that satisfies (3)

-p(-h) A/i= ihdpo Jx

(heCîX)).

Since Ag = p(g), the proof is complete. 9.5.5. Lemma (ForelH [1]). Suppose E a X is a set of type F^, and p(E) = 0 for every p e M^. Then there is a sequence {/„} in A, with \\f^\\ < I for all m, such that (i) Um„_oo /mW = ^fo^ every xe E, but (ii) lim^-.oo /mW = 1 a.e. Ip^for every peM^. Proof By assumption, £ = U £^, where each £^ is compact, and £^ ci £;^+j. Fix m, for the moment. By the minimax theorem, (1)

inf sup I hdp = sup inf h

p

Jx

p

h Jx

h dp,

where p ranges over M^ and h ranges over all h G CR(X) that satisfy h > m on E^,h>0 on X. Since p(Ef„) = 0 for every p, the right side of (1) is 0. Hence (1) shows that for some ft = /z^ in our class, (2)

\h„dpmonE^,u^>()on X, and Ow^ < 1/m^. Now choose gê A so that u^ = Re g^ and O^^ is real, and put f^ = exp(-0. Then \fj = e x p ( - M j < 1 on X. On £^, l/^l < e"'", which proves (i). If p G M^ then (3)

\ fmdp = ^fm = e x p ( - % J = e x p ( - 0 i i j , Jx

and since Ow^ < 1/m^, (4)

£(l-/J.p^l-exp(-i,)g,uE

RP(B)}

if ^ is a real function with domain B, and (2)

Pig) = inf(M(0): M > ^, M G Re A(B)}

if g is bounded above on B, In terms of this notation, the norm |||/|||p of a function/G(LH)^(J5), as defined in §7.4.3, satisfies (3)

lll/lll? = a(|/|'').

Here 0 0. Since | / 1 ' ' is uniformly continuous on 5, there exists r < 1 such that ii/r-i/.ns

+

\fA''>\ff

on B, and u(0) = e + w(0) < |||/|||^ + 2e. Hence i ? ( i / n < Ill/Ill?. Since a < j5 is trivial, the lemma follows from 9.7.1(3). 9.7.3. Lemma. Iffe

(LH)^(B) then

(1)

lim IIIXIII, = Ill/It = sup |||/,|||,.

w>\ff,

200


Note that the first of these equaUties holds although |||/ —Xlllp need not tend to 0 as r -^ 1 (Theorem 7.4.6). The second equahty in (1) impHes that III X III p is a nondecreasing function of r. Proof. If w > l/l^ then u, > |/,|^ hence

(2)

\\\fr\K ^ uxo) = u(oy

Taking the infimum over all such u e RP(B) gives (3)

|||/,|||p < lll/lllp

(0 < r < 1).

Next, associate to each r < 1 a function i/^''^ e RP(B) such that (4)

u^''>\fr\'

and

u^'KO)OdQ.

W i t h / 6 A(B) in place of f, the fact that the slice functions/^ are in A(U) shows therefore that j fdp — /(O). Thus p e MQ, as asserted. If fe H(B) and v is replaced by | / J ^ in (2), consideration of the slice functions shows that (3)

Js

flfrl'dp

is a nondecreasing function of r. But if p G MQ is not circular, then (3) may fail to be a monotonic function of r, for certain/, in spite of the monotonicity of |||/|||p (Lemma 9.7.3) and Theorem 9.7.4. Theorem 9.7.6 will show this. To construct some noncircular p e MQ , take n = 2, for simplicity, and let T be any probability measure on [7 c= C that satisfies (4)

f9dT = g(0)

for every g e A(Uy For example, T might be concentrated on a simple closed curve F in U that surrounds the origin, in such a way that T solves the Dirichlet problem at 0 relative to the domain bounded by F. The measure p that satisfies

(5)

jvdp = jdTiz). ^ I" viz, e ^ ' ^ y r ^)dd

202


for every v e C(S) belongs then to MQ . To see this, simply note that the inner integral on the right side of (5), with v replaced b y / e A(B\ equals/(z, 0). The support of this p is the set of all (z, w)e S for which z lies in the support of T. 9.7.6. Theorem. Put f(z, w) = (1 — z)w in C^. Then there exists p e MQ such that Js | / . T ^P i^ ôt a monotonic function ofr in [0, l],/or any p e (0, oo). Proof. Let / be the interval [|, ^] on the real axis in C, let Q = U\I, and let T be the probability measure on ^Q = / u T that satisfies (1)

hdT = h(0) Jdi Jen

for every h e C(Q) that is harmonic in Q. Use this T to define p as in 9.7.5(5). If we replace y by | X1^ in that formula, the inner integral is (2)

|r-r^z|^(l -

\z\y^\

This vanishes on T, where \z\ = 1. Hence (3)

J \fr\' dp = j(r - r'xni

- xy^' dz(x).

The integral on the right is 0 when r = 0. It is positive for every r e (0, 1]. It decreases as r increases from f to 1, since the integrand is then a decreasing function of r, for every xe I.

9.8. The Dual Space of ^(5) 9.8.1. The dual X* of a given Banach space X can often be described in several ways. For instance, A(B) is isometrically isomorphic to A(S\ a closed subspace of C(S). Since C(5)* = M(S), the space of complex Borel measures on 5, standard duality theory gives one description of A{B)'^ as a quotient space, namely (1)

A(B)* = M(S)/A\

(In this context, equality is understood to mean isometric isomorphism.) Here is another description: Let HM be the space of all Henkin measures on S, and let TS be the space of all totally singular ones. If we combine the theorems of Henkin and ColeRange, we see that HM consists of precisely those measures that are abso-

9.8. The Dual Space of A(B)

203

lutely continuous with respect to some representing measure p e MQ . The GKS decomposition theorem (with respect to MQ) says therefore that there is a direct sum decomposition (2)

M(S) = HM@ TS.

Observe next that A^ c= HM. Thus (1) can be replaced by (3)

A(B)* = (HM/A^) @ TS.

By Valskii's theorem, HM = L^(a) + A^. (Note that this sum is not direct, since L^(o-) n A-^ is far from being {0}.) The first summand on the right of (3) is thus the image of L^((T) under the quotient map with null space A-^. In particular, HM/A^ is separable. If/i G r S and A 2 and A(U^) is the polydisc algebra in k variables, then A(U^)'^ is not isomorphic to any closed subspace ofA(BJ*. Here "isomorphic" means: linearly homeomorphic. The proof uses more Banach space theory than can be presented here. We therefore omit it. But let us state the dual formulation of the result: 9.8.3. Theorem. If n> \ and /c > 2, then A{U^) is not isomorphic to any quotient space of A(B^. In other words, there is no continuous linear mapping of any A{B^ onioA{U^)'\{k> 2. Henkin [1] proved earher that A{Bj) and A{U^) are not isomorphic as Banach spaces. In fact, it was apparently for this purpose that he introduced his class of measures.

Chapter 10

Interpolation Sets for the Ball Algebra

10.1. Some Equivalences 10.1.1. Definitions. We shall be concerned with compact sets K c= S. For convenience, we introduce six labels, (Z), (P), (/), PI), (AT), and (TN), to denote certain properties that K may or may not have in relation to the ball algebra AiB). X is a (Z)-set (zero set) if there is a n / e A(B) such that/(C) = 0 for every C e X and/(z) ^ 0 for every z e B\K. X is a (P)-set (peak set) if there is a n / e A(B) such that/(C) = 1 for every Ce K and | f(z) | < 1 for every z e B\K. (Such a n / i s said to peak on K,) K is an (/)-set (interpolation set) if every complex continuous function on K extends to a member of A(B). X is a (P/)-set (peaknnterpolation set) if the following is true: to every g E C(K) (g ^ 0) corresponds a n / e A(B) such that / ( O = g(Q for every CG iCand|/(z)| < ||^|||j^ for every z e S\iC. (Here ||gf|||5: denotes the maximum

of 1^(01 on X.)

K is an (iV)-set if X is a null set for every v e A^. More exphcitly, the requirement is that |v|(X) = 0 whenever v e M(S) satisfies j fdv = 0 for every/G A(B). X is a (TN)-SQt if K is totally null, i.e., if p(K) = 0 for every representing measure p G MQ. (See §9.1.2.) These six properties turn out to be equivalent: 10.1.2. Theorem. / / a compact set K (^ S has one of the six properties (Z), (P), (/), (P/), (N), (TN), then it has the other five. This will be proved in accordance with the following diagram, in which single arrows indicate easy implications, whereas double arrows indicate substantial theorems.

204

10.1. Some Equivalences

205

It is trivial that (PI) implies (/) and (Z). Suppose/e A(B) has K as its zero set in B, and | / | < 1. Since B\K is simply connected, / = exp g for some g that is continuous on B\K and holomorphic in B. Clearly, Re ^ < 0 and Re g(z) -> — oo as z approaches any point of K. Hence h = g/(g — 1) is in A(B) and peaks on K. This shows that (Z) -^ (P). If/6yl(B) peaks on X, and p e M o , then/'"(O) = j / ' " ^p for m = 1, 2, 3, The integral converges to p(K) as m ^ oo, by the dominated convergence theorem, and/'^CO) ^ 0, since | / ( 0 ) | < 1. Thus p(K) = 0. This shows that (P) -> (TAT). That (TAT) =^ (N) follows from the Cole-Range theorem: Suppose K is totally null, and v e A^. Then v is a Henkin measure, hence v (PI) are special cases of the theorems of Varopoulos and Bishop which will be proved in Sections 10.2 and 10.3. 10.1.3. Some Background. The case n = 1 of Theorem 10.1.2 (dealing with the disc algebra and with compact sets K on the unit circle) represents theorems that were proved over a rather long span of time. [When n — 1, (TN) is simply the property of having Lebesgue measure 0. We shall nevertheless continue to write (TN\ for the sake of uniformity and brevity.] In his thesis, Fatou [1] showed that (TN) imphes (Z). The brothers F. and M. Riesz [1], in their only joint paper, proved that (TN) implies (N). That (TN) also imphes (P) is implicit in their proof. About 40 years later, Carleson [2] and Rudin [4] showed, independently, that (TN) implies (/). Since it is easy to see that no set of positive Lebesgue measure has any of the properties (Z), (N), (I), these three were thus known to be equivalent in the disc algebra context, but only because each of them was known to be equivalent to (TN). Bishop [1] threw an entirely new light on this subject by proving that (N) implies (PI) in very general situations that have nothing to do with holomorphic functions or with (TN). This theorem occupies Section 10.3. In the polydisc setting (where K a T") it is known that (Z), (P), (/), (PI), and (N) are equivalent. Proofs may be found in Chap. 6 of Rudin (1) and in §21 of Stout [1]. Valskii [1] proved part of Theorem 10.1.2, namely the equivalence of (PI), (Z), (P), and (N), for smoothly bounded, star-shaped, strictly pseudoconvex domains in C . Chollet [1] showed that Theorem 10.1.2 holds in arbitrary strictly pseudoconvex domains (not necessarily simply connected) by giving a different proof of (Z) -^ (P). 10.1.4. Among the implications marked in the diagram that follows Theorem 10.1.2 there is only one, namely (TN) => (N), whose proof involves holomorphic functions in a significant way. The others (except for (Z) -> (P), where holomorphic functions do play a small role) are function algebra

206

10. Interpolation Sets for the Ball Algebra

theorems, and it is reasonable to ask questions such as the following: Does (P) imply (N) in every function algebra? Do (P) and (/) always imply each other? Here is a simple example that answers these negatively. LetX = KÛ 1^2 be the following compact subset of C x i^:Ki consists of all points (z, 0) with \z\ < 1, K2 consists of all points (0, t) with 0 < t < 1. Thus X looks hke a thumbtack, and K^ can be identified with the closed unit disc in C. Let A consist of all/G C(X) whose restriction to K^ is holomorphic in the interior ofK^. Then A is a. function algebra on X (and, in fact, X is the maximal ideal space of A). Clearly, K^ is a (P)-set of A which is neither (/) nor (A^). (Note that X^ is a (P)-set which contains compact sets that are not (P)-sets!) On the other hand, K2 is an (/)-set for A which is neither (P) nor (AT). In view of Varopoulos' theorem (Section 10.2), it should be mentioned that only one point of K2, namely (0, 0), fails to be a peak point for A. That K2 is not (P) is due to the maximum modulus theorem. To see that K2 is not (AT), let V be the sum of normalized Lebesgue measure on the boundary of K^ and the point measure of mass — 1 at (0, 0) G K^ n X2 • Both Ki and K2 are (Z)-sets for A. 10.1.5. Consequences of Theorem 10.1.2. Properties (AT) and (TAT) are obviously hereditary (if K has one of them, so does every compact K^ cz K\ and they are preserved under the formation of countable unions. Since this is not so obvious for the other properties under consideration. Theorem 10.1.2 leads to some nontrivial conclusions, such as the following. (a) If a compact K cz S is the union of countahly many (Piysets, then K is a (Piyset. (b) If K c: Sis a (Z)-set, then every compact K^ a K is a (Z)-set. (c) For explicitly described sets K, properties (P) or (Z) are often the ones that are most easily verified. For example, let K be the set of all z E S = dB„ all of whose coordinates are real. Thus K is an (n — l)-sphere. Define

g(z) = zl + --' + zl Then |(1 -j- g) peaks on K. Conclusion: K is totally null, X is a (P/)-set, etc. (d) Here is an elaboration of (c): Let £ be a compact subset of the unit circle, of Lebesgue measure 0, let g be as in (c), and put K =

Sng-\E).

The case E = {1} occurred in (c). Since E has measure 0, £ is a peak-set for A(U). Let he A(U) peak on E. Then ho g e A{B) and hog peaks on K. If E is a Cantor set of measure 0, we conclude that there exist "Cantor sets of (n — l)-spheres" in S that are (P/)-sets.

10.2. A Theorem of Varopoulos

207

(e) Let E be as in (d), but this time define g(z) =

n"^^z,'"Z„.

The set of all z G 5 where \g(z)\ = 1 consists of the n-torus defined by

For any a 6 £, g~^((x) n S is thus an (n — l)-torus in S. Setting K = S n g~^(E\ it follows, as in (d), that there exist "Cantor sets of (n - l)-tori" in S that are (P/)-sets.

10.2. A Theorem of Varopoulos 10.2.1. The original proof of this theorem, as given by Varopoulos [1], relied on the fact that every compact Hausdorff space can be homeomorphically embedded in some (large) compact abelian group, and used some harmonic analysis on such groups. Glicksberg [2; p. 9] simplified the proof a great deal: the only group that appears is a finite cychc one, and no harmonic analysis is needed beyond the ability to sum a finite geometric progression. 10.2.2. Theorem. Suppose (i) (ii) (Hi) (iv)

A is a function algebra on a compact Hausdorff space X, K cz X is a compact interpolation set for A, every point ofK is a peak-point for A, veM{X)andveA'-.

Then |v|(K) = 0. Proof Fix e > 0. There is an open set Q c= X such that K cz Q and IVI {Q\K) < s. By (iii) there corresponds to every point x e K a, function fx^ A such that f^(x) = 1 but | fîy) | < 1 for every y ^ x,y e X, We can replaceXc by/^, for some large integer m, so that | /^ | < s outside Q. The sets where \1 —fj < s form an open cover of K. Hence (writing f in place of fx) there are finitely many functions/i,... ,/v 6 A, with WfW = 1, such that I /• I < e outside Q, and such that the sets (1) cover K.

E, = { | 1 - / , ! < £ }

(i=l,...,iV)

208


Next, there are pairwise disjoint compact sets Ki (^ E^n K such that

("\y K}J |v|(Q\yxJ 0 on S. Hence |^,| < 1. Fix z e K. Then z e H^ for some t e E. Hence there are infinitely many / with t E Li, so that \t — ti\ < djl. For any such f, |1 - h,i^z)\ = \hiz) - Ki^z)\ < c\t -U\
converge, as 3 \ 0, to

(2)

W(x)(v - u)\'

for all xeQ and all u,ve R"". Note that the denominator in (1) is 3^, not 3. Proof. Fix X, u, v. If 3 is small enough then (since Q is open) all points (3)

y(t) = X + (1 - t)3u -^ t3v

(0 < t < 1)

he in Q. Fix such a 3, for the moment, and define 7: [0, 1] ^ 5 by y(t) = ^(y(t)). Then (4)

y'(0 = 30mmv

- u).

216


Since O is complex tangential, it follows that (5)

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Function Theory in the Unit Ball of Cn: Reprint of the 1st Ed Berlin Heidelberg New York 1980