Bounded Analytic Functions

474 Pages·1981·7.887 MB·iii-xvi, 1-467\474

by  John B. Garnett (Eds.)

BOUNDED
ANALYTIC FUNCTIONS
John B. Garnett
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CALIFORNIA, LOS ANGELES
LOS ANGELES. CALIFORNIA
1981
ACADEMIC PRESS
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Garnett, John B.
Bounded analytic functions.
Bibliography: p.
Includes index.
1. Analytic functions. 2. Functional analysis.
I. Title.
QA331G32 515.7 80-26298
ISBN 0-1 2-276150-2
PRINTED IN THE UNITED STATES OF AMERICA
81828384 9 8 7 6 5 4 3 2 1
Preface
The primary purpose of this book is to teach technique, and we have
emphasized method rather than generality. Many of the ideas we shall
introduce, from subharmonicity and maximal functions to Littlewood-
Paley integrals, Carleson measures, and stopping time constructions, extend
naturally to Euclidean space and beyond; but for unity and simplicity we
have limited their discussion to one dimension. Some of these ideas are
explored more fully in the books of Stein [ 19701 and Stein and Weiss [1971].
Our secondary purpose is to give a self-contained view of the contem-
porary theory of bounded analytic functions on the unit disc. To do that
we must treat in detail certain notions, such as conformal invariance, the
subharmonicity of log If[, dual extremal problems, and, especially, Blaschke
products, which do not yet generalize well from their classical setting.
Readers interested in higher dimensions or in multiply connected domains
are only advised that the proofs in the text most resisting generalization are
those relying on Blaschke products or dual extremal problems. Freeing
certain single-variable proofs from these notions is tantamount to solving
some of today's most difficult problems on the unit ball of C".
On the other hand, readers patient with one complex variable will be
rewarded with a theory richer in texture. For example, a basic question about
the conjugation operator leads to functions of bounded mean oscillation,
which leads to Carleson measures, and in turn, via Blaschke products or
duality, to interpolating sequences and the corona theorem. Only the last
link of the chain does not generalize. The proof of the H'-BMO duality and
the construction behind the corona theorem, both amenable to higher
dimensions, merge to yield a remarkable characterization, in terms of
Blaschke products, of the closed algebras between H" and L".
This book presents a particular viewpoint, both in method and material;
it is no encyclopedia. Some topics, such as interpolation problems and the
arguments behind the corona theorem, have been pursued at length, while
other topics, such as the applications of Banach algebra theory and the vast
interaction between H" and operator theory, have been minimized. (For the
xi
xii PREFACE
connections with operator theory, we recommend the excellent works of
Douglas [1972] and of Sarason [1979].) Whenever possible we have used
conformal invariance and real variables techniques. Over the past twenty
years, the renewed interest in H“ has been prompted largely by functional
analytic questions, but I believe that solving some of the harder problems
now facing the subject requires returning to the disc or the circle and for-
mulating more constructive arguments.
Prerequisites for this book are basic courses in real and complex analysis;
the first eleven chapters of Rudin’s textbook [1974] should be sufficient. In
Chapter I we present some additional background not usually found in
elementary graduate courses. Chapters 11-V form an introduction to Hardy
space theory, through conjugate functions, dual extremal problems, and some
of the uniform algebra aspects of H”.W e have based the theory on maximal
functions and on subharmonicity. (For different approaches, see the books
of Hoffman [1962a] and Duren [1970].) Chapters VI-X develop the ideas
surrounding the John-Nirenberg theorem, the geometry of interpolating
sequences, and the corona theorem. People already familiar with the field
will notice that these chapters largely grew out of two papers by Carleson
[1958, 1962a). Much of the material in the last five chapters has not
appeared in monograph form before. The book is self-contained, and the
first half is basically a preparation for the second half. However, the early
sections of Chapters VI-VIII contain essential parts of today’s classical HP
theory, while a few specialized items have infiltrated Chapter IV. The notes
by Koosis [ 19803 provide a more elementary and less intense survey of some
of the topics we have considered.
Results are numbered lexicographically within each chapter, so that
“Theorem 1.3” is the third item of Section 1 of the same chapter, whereas
“Theorem 1.3 of Chapter I” or “Theorem I. 1.3” is in Section 1 of Chapter 1.
Independently, the same convention is used to number formulas, such as
“( 1.10) of Chapter 111.”
There are 31 figures in the text. Understand the figures and you understand
the book.
Each chapter ends with some bibliographical notes and a section called
Exercises and Further Results.” Some exercises are intended for beginners,
“
while others, the “further results,” are theorems not in the text. They usually
include references, which serve also to suggest that they may not be elemen-
tary. Sometimes extensive hints have been given, and occasionally an exer-
*.
cise with thorough hints is referred to later in the text. Especially satisfying
exercises have been marked with one, two, or three stars
Acknowledgments
Without the help and encouragement of many friends this book would
never have been finished. I am fortunate to have had two students, Donald
Marshall and Peter Jones, each of whom has turned this subject in his own
direction. I thank them for their numerous improvements on the text and for
the mathematics they have taught me. Over the years I have found the advice
and encouragement from Ted Gamelin, Paul Koosis, and Nicholas Varo-
poulos most heartening. Irving Glicksberg, still my teacher, bombarded me
with corrections, mathematical and stylistic. Were it not for him, this book
would have been a great deal harder to read. Others who have provided
valuable assistance with the subject or the text include Anthony Carbery,
Lennart Carleson, David Drasin, John Fagarason, Michael Frazier,
Gregory Gibbons, Leslie Kay, Steven Krantz, Robert Latter, Robyn Owens,
Donald Sarason, and Allen Shields. The manuscript, including myriad
scribbled revisions, was typed by Debra Remetch, who was aided by Sarah
Remetch. I thank the staff of Academic Press for their patience during
production.
I am very grateful to the University of California, Los Angeles, with its
pleasant atmosphere, efficient staff, congenial colleagues, and stimulating
students, for giving me considerable professional liberty. Much of this book
was written while I was professeur ussocik at Universitt de Paris-Sud, which
provided me with an excellent library, exciting seminars, and even greater
professional liberty. I thank its kquipe dhnulyse hurmonique for a most
pleasant year.
...
XIII
List of Symbols
J
Symbol Page Page
A,, 125 62
A" 127 16
A" 127 333
A, 90 280
A-' 182 57
A 186 110
A1 203 410
% 59 3 80
B 282 105
283 110
BO
BLO 281 128
BMO 223 128
BMOA 270 50
BMO(T) 225 51
BMO, 274 50
BUC 2 50 133
CB. HI 278 275
9 = (J analytic, I f(z)l 5 1) 1 244
@, a set of inner functions 373 66
373 137
%cA
= C(T) 133 373
COP 442 3 14
CA 376 5
CKf, 4 80 228
D 1 414
D 1 328
dist(J, Hp) 134 2
dist, 249 411
9 400 243
aiaz 321 222
aiaz 318 243
a
364 66
E* 332 401
1 E I = Lebesgue measure of E 24 35
d 30 1 287
f (4 186 403
f(m) = Gelfand transform 186 I86
xv
xvi LIST OF SYMBOLS
Page Symbol Page
20 uc 250
133 QA 374
22 0,(2) 37
29 VMO 250
2 50 VMOA 283
47 VMO, 385
189 X 187
x,
184 214
190
BN 186
403
337
69
A(c, R) 3
31
Au 11
71
ra(0 22
348 r,(e'') 24
50,51 rw
389
223
1, 240
258
A* 442
244
A, 106
11
A* 282
12
402 P'.p 240
"I 380
103
102 P(Z7 w) 2
Ph,, md 401
377
P(S, T) 314
376
CPI 222
12
46) 105
80
w,@) I05
380
vg 235
343
IVgI2 235
159
* 12
102
290 I = complex conjugate of z
121 = complex conjugate off
28 8 = {f:f~%}w,h en 8 is a set of functions
102 E = closure of E, when E is a point set
Preliminaries
As a preparation, we discuss three topics from elementary real or complex
analysis which will be used throughout this book.
The first topic is the invariant form of Schwarz’s lemma. It gives rise to the
pseudohyperbolic metric, which is an appropriate metric for the study of
bounded analytic functions. To illustrate the power of the Schwarz lemma,
we prove Pick’s theorem on the finite interpolation problem
f(zj)= wj, j = 1, 2, . . . , n,
with If(z)l I 1.
The second topic is from real analysis. It is the circle of ideas relating
Poisson integrals to maximal functions.
The chapter ends with a brief introduction to subharmonic functions and
harmonic majorants, our third topic.
1. Schwarz’s Lemma
Let D be the unit disc {z : (zI c l} in the complex plane and let 9J denote
the set of analytic functions from D into B. Thus If(z)l I 1 iffEW. The
simple but surprisingly powerful Schwarz lemma is this :
Lemma 1.1. Zff(z)E 9J, and iff(0) = 0, then
Equality holds in (1.1) ut some point z ifand only iff (z) = eiVz,c p a real constant.
The proof consists in observing that the analytic function g(z) =f(z)/z
satisfies 19 I I 1 by virtue of the maximum principle
We shall use the invariant form of Schwarz’s lemma due to Pick. A Miibius
transformation is a conformal self-map of the unit disc. Every Mobius
2 PRELIMINARIES Chap. 1
transformation can be written as
with cp real and lzol < 1. With this notation we have displayed zo = z-’(O).
Lemma 1.2. Iff(z) E 9, then
and
Equality holds at some point z if and only iff (2) is a Mobius transformation.
The proof is the same as the proof of Schwarz’s lemma if we regard z(z)
as the independent variable and
f(4- -f(zo )
1 -f(zo)f(z)
as the analytic function. Letting z tend to zo in (1.2) gives (1.3) at z = zo, an
arbitrary point of D.
The pseudohyperbolic distance on D is defined by
Lemma 1.2 says that analytic mappings from D to D are Lipschitz continuous
in the pseudohyperbolic distance :
P(f(ZXf(W)) 5 P(Z, w).
The lemma also says that the distance p(z, w) is invariant under Mobius
transformations :
P(Z7 4 = P ( mt( W)>.
We write K(zo,r ) for the noneuclidean disc
K(zo,r)= {z:p(z,zo) c r}, 0 < r < 1.
Since the family W is invariant under the Mobius transformations, the
study of the restrictions to K(zo,r ) of functions in W is the same as the study
of their restrictions to K(0, r) = {lwl < r}. In such a study, however, we
must give K(zo, r) the coordinate function w = z(z) = (z - zo)/(l - 5,~).
Sect. 1 SCHWARZ’S LEMMA 3
For example, the set of derivatives of functions in Bdo not form a conformally
invariant family, but the expression
( 1.4) If’(z)1(1 - 1--12>
is conformally invariant. The proof of this fact uses the important identity
which is (1.3) with equality for f(z)= z(z). Hence if f(z)= g(s(z)) = g(w),
then
If’(z)1(1 - lz?) = Ig’(w)I lz’(z)l(l - M2) = Ig’(w)l(l - lWl2)
and this is what is meant by the invariance of (1.4).
The noneuclidean disc K(zo, r), 0 < r < 1, is the inverse image of the disc
IwI < r under
w = T(Z) = --z. - 20
1 -zoz
Consequently K(z,, r) is also a euclidean disc A(c, R) = {z : )z- CJ < R},
and as such it has center
and radius
1 - lz0lZ
(1.7) R = r
I - r21zo12’
These can be found by direct calculation, but we shall derive them geometri-
cally. The straight line through 0 and zo is invariant under z, so that dK(zo, r)
= T - ‘(lwl = r) is a circle orthogonal to this line. A diameter of K(zo, r) is
therefore the inverse image of the segment [ - rzo/l zo 1, rzo/l zo I]. Since
+ +
z = (w zo)/( 1 Z0 w), this diameter is the segment
The endpoints of (1.8) are the points of K(z,, r) of largest and smallest
+
modulus. Thus c = (a /3)/2 and R = (ID1 - 1a1)/2 and (1.6) and (1.7)
-,
hold. Note that if r is fixed and if Jzo( 1, then the euclidean radius of
K(zo, r) is asymptotic to 1 - (zoI.