Topics in Complex Analysis

167 Pages·1996·2.82 MB·English

by  Mats Andersson

TOPICS IN
COMPLEX
ANALYSIS
Universitext: Tracts in Mathematics
Editorial Board
(North America):
S. Axler
F.W. Gehring
P.R. Halmos
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BoossBleecker: Topology and Analysis
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(continued after index)
Mats Andersson
Topics in
Complex Analysis
Springer
Mats Andersson
Department of Mathematics
Chalmers University of Technology
Goteborg University
S-412 96 Goteborg
Sweden
Editorial Board (North America):
S. Axler F.W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Michigan State University of Michigan Santa Clara University
University Ann Arbor, MI 48109 Santa Clara, CA 95053
East Lansing, MI 48824 USA USA
USA
Mathematics Subject Classification (1991): 30-01, 30C20, 30C15, 32A35
With 4 figures.
Library of Congress Cataloging-in-Publication Data
Andersson, Mats.
Topics in complex analysis/Mats Andersson.
p. cm. - (Universitext)
Includes bibliographical references and index.
ISBN 0-387-94754-X (soft: alk. paper)
1. Functions of complex variables. 2. Mathematical analysis.
1. Title.
QA331.7.A52 1996
515'.9-dc20 96-11793
Printed on acid-free paper.
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Preface
This book is an outgrowth of lectures given on several occasions at Chalmers
University of Technology and Goteborg University during the last ten years.
As opposed to most introductory books on complex analysis, this one as-
sumes that the reader has previous knowledge of basic real analysis. This
makes it possible to follow a rather quick route through the most fundamen-
tal material on the subject in order to move ahead to reach some classical
highlights (such as Fatou theorems and some Nevanlinna theory), as well
as some more recent topics (for example, the corona theorem and the Hi-
BMO duality) within the time frame of a one-semester course. Sections 3
and 4 in Chapter 2, Sections 5 and 6 in Chapter 3, Section 3 in Chapter 5,
and Section 4 in Chapter 7 were not contained in my original lecture notes
and therefore might be considered special topics. In addition, they are
completely independent and can be omitted with no loss of continuity.
The order of the topics in the exposition coincides to a large degree
with historical developments. The first five chapters essentially deal with
theory developed in the nineteenth century, whereas the remaining chapters
contain material from the early twentieth century up to the 1980s.
Choosing methods of presentation and proofs is a delicate task. My aim
has been to point out connections with real analysis and harmonic anal-
ysis, while at the same time treating classical complex function theory. I
also have tried to present some general tools that can be of use in other
areas of analysis. Whereas these various aims sometimes can be incom-
patible, at times the scope of the book imposes some natural restrictions.
For example, Runge's theorem is proved by the "Hahn-Banach method,"
partly because it is probably the simplest way to do so, but also because it
is a technique that every student in analysis should become familiar with.
However, a constructive proof is outlined as an exercise. Complex anal-
ysis is one of the origins of harmonic analysis, and several results in the
latter subject have forerunners in complex analysis. Fatou's theorem in
Chapter 6 is proved using standard harmonic analysis, in particular using
the weak-type estimate for the Hardy-Littlewood maximal function. How-
vi Preface
ever, most standard tools from harmonic analysis are beyond the scope of
this book, and therefore, the LP-boundedness of the Hilbert transform and
the Hr-space theory, for instance, are treated with complex analytic meth-
ods. Carleson's inequality is proved by an elementary argument due to B.
Berndtsson, rather than using the LP-estimate for the maximal function,
and Carleson's interpolation theorem is proved using the beautiful and ex-
plicit construction of the interpolating function due to P. Jones from the
1980s. However, a proof based on the 86-equation is indicated in an exer-
cise.
Necessary prerequisites for the reader are basic courses in integration
theory and functional analysis. In the text, I sometimes refer to distribution
theory, but this is merely for illustration and can be skipped over with no
serious loss of understanding. The reader whose memory of an elementary
(undergraduate) course in complex analysis is not so strong is advised to
consult an appropriate text for supplementary reading.
As usual, the exercises can be divided into two categories: those that
merely test the reader's understanding of or shed light on definitions and
theorems (these are sometimes interposed in the text) and those that ask
the reader to apply the theory or to develop it further. I think that for
optimal results a good deal of the time reserved for the study of this subject
should be devoted to grappling with the exercises. The exercises follow the
approximate order of topics in the corresponding chapters, and thus, the
degree of difficulty can vary greatly. For some of the exercises, I have
supplied hints and answers.
At the end of each chapter, I have included references to the main results,
usually to some more encyclopedic treatment of the subject in question, but
sometimes to original papers. If references do not always appear, this is
solely for the sake of expediency and does not imply any claim of origi-
nality on my part. My contribution consists mainly in the disposition and
adaptation of some material and proofs, previously found only in papers
or encyclopedic texts addressed to experts, into a form that hopefully will
be accessible to students.
Finally, I would like to take this opportunity to express my apprecia-
tion to all of the students and colleagues who have pointed out errors and
obscurities in various earlier versions of the manuscript and made valu-
able suggestions for improvements. For their help with the final version,
I would like to thank in particular Lars Alexandersson, Bo Berndtsson,
Hasse Carlsson, Niklas Lindholm, and Jeffrey Steif.
Goteborg, Sweden Mats Andersson
Contents
Preface v
Preliminaries 1
§1. Notation ......................................................................... 1
§2. Some Facts .................................................................... 2
1. Some Basic Properties of Analytic Functions 5
§1. Definition and Integral Representation ....................... 5
§2. Power Series Expansions and Residues ....................... 12
§3. Global Cauchy Theorems .............................................. 18
2. Properties of Analytic Mappings 28
§1. Conformal Mappings ..................................................... 28
§2. The Riemann Sphere and Projective Space .................. 33
§3. Univalent Functions ...................................................... 35
§4. Picard's Theorems .......................................................... 38
3. Analytic Approximation and Continuation 46
§1. Approximation with Rationals ..................................... 46
§2. Mittag-Leffler's Theorem and the Inhomogeneous
Cauchy-Riemann Equation .......................................... 48
§3. Analytic Continuation ................................................... 51
§4. Simply Connected Domains .......................................... 53
§5. Analytic Functionals and the Fourier-Laplace
Transform ...................................................................... 55
§6. Mergelyan's Theorem .................................................... 58
4. Harmonic and Subharmonic Functions 67
§1. Harmonic Functions ...................................................... 67
§2. Subharmonic Functions ................................................ 71
viii Contents
5. Zeros, Growth, and Value Distribution 82
§1. Weierstrass' Theorem .................................................... 82
§2. Zeros and Growth .......................................................... 85
§3. Value Distribution of Entire Functions ....................... 88
6. Harmonic Functions and Fourier Series 97
§1. Boundary Values of Harmonic Functions .................... 97
§2. Fourier Series ................................................................. 104
7. HP Spaces 112
§1. Factorization in IF Spaces ............................................ 112
§2. Invariant Subspaces of H2 ............................................ 116
§3. Interpolation of H°° ....................................................... 118
§4. Carleson Measures ......................................................... 121
8. Ideals and the Corona Theorem 130
§1. Ideals in A(12) .................................................................. 130
§2. The Corona Theorem ..................................................... 131
9. H' and BMO 141
§1. Bounded Mean Oscillation ............................................ 141
§2. The Duality of H' and BMO ......................................... 146
Bibliography 151
List of Symbols 153
Index 155
Preliminaries
§ 1. Notation
Throughout this book the letters 12 and K always will denote open and
compact sets, respectively, in 1R2, and w will denote a bounded open set
with (when necessary) piecewise C' boundary 8w, which is always supposed
to be positively oriented; i.e., one has w on the left-hand side when passing
along Ow. The notation w CC St means that the closure of w is a compact
subset of St, and d(K, E) denotes the distance between the sets K and
E. Moreover, D(a, r) is the open disk with center at a and radius r,
and U denotes the unit disk, i.e., U = D(0, 1), and T is its boundary
aU = {z; Izj = 1}. The closure of a set E E 1R2 is denoted by E and its
interior is denoted by int E.
The space of k times (real) differentiable (complex valued) functions in
St is denoted by Ck(Sl) (however, we write C(1) rather than CO(Q)) and
C°°(S2) = nCk(Sl). Moreover, &(D) is the subspace of functions in Ck(S2)
whose derivatives up to the kth order have continuous extensions to St, and
Co (St) is the subspace of functions in Ck(c) that have compact support
in Q. Lebesgue measure in 1R2 is denoted by dA, whereas do, denotes arc
length along curves. We use the standard abbreviation a.e. for "almost
every(where)." We also use u.c. for "uniformly on compact sets." If f, q5
are functions, then "f = O(0) when x -+ a" means that f/0 is bounded
in a neighborhood of a and "f = o(O) when x -p a" means that f /0 0
when x a. Sometimes we also use the notation f N g, which means that
f is less than or equal to some constant times g. Moreover, f - g stands
for f N g and f ? g.
We will use standard facts from basic courses in integration theory and
functional analysis. Sometimes we also refer to distribution theory (mainly
in remarks), but these comments are meant merely for illustration and
always can be passed over with no loss of continuity. In the next section
we have assembled some facts that will be used frequently in the text. In the
first chapters we refer to them explicitly but later on often only implicitly.