The Cauchy Transform, Potential Theory and Conformal Mapping

221 Pages·2015·3.893 MB·\221

by  Steven R. Bell (Author)

Mathematics
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K25868
www.crcpress.com
K25868_cover.indd 1 10/9/15 11:27 AM

The Cauchy Transform,
Potential Theory and
Conformal Mapping
2nd Edition

The Cauchy Transform,
Potential Theory and
Conformal Mapping
2nd Edition
Steven R. Bell
Purdue University
West Lafayette, Indiana, USA
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
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No claim to original U.S. Government works
Version Date: 20150930
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Contents
Preface vii
Table of symbols xi
1 Introduction 1
2 The improved Cauchy integral formula 3
3 The Cauchy transform 9
4 The Hardy space, Szego˝ projection, and Kerzman-Stein
formula 13
5 The Kerzman-Stein operator and kernel 17
6 The classical definition of the Hardy space 21
7 The Szego˝ kernel function 27
8 The Riemann mapping function 33
9 A density lemma and consequences 35
10 Solution of the Dirichlet problem in simply connected
domains 43
11 The case of real analytic boundary 47
12 The transformation law for the Szego˝ kernel 51
13 The Ahlfors map of a multiply connected domain 59
14 The Dirichlet problem in multiply connected domains 65
15 The Bergman space 69
v
vi Contents
16 Proper holomorphic mappings and the Bergman
projection 77
17 The Solid Cauchy transform 87
18 The classical Neumann problem 93
19 Harmonic measure and the Szeg˝o kernel 97
20 The Neumann problem in multiply connected domains 107
21 The Dirichlet problem again 111
22 Area quadrature domains 113
23 Arc length quadrature domains 123
24 The Hilbert transform 131
25 The Bergman kernel and the Szeg˝o kernel 135
26 Pseudo-local property of the Cauchy transform 141
27 Zeroes of the Szeg˝o kernel 149
28 The Kerzman-Stein integral equation 153
29 Local boundary behavior of holomorphic mappings 159
30 The dual space of A∞(Ω) 165
31 The Green’s function and the Bergman kernel 177
32 Zeroes of the Bergman kernel 183
33 Complexity in complex analysis 187
34 Area quadrature domains and the double 191
A The Cauchy-Kovalevski theorem for the
Cauchy-Riemann operator 197
Bibliographic Notes 199
Bibliography 203
Index 207
Preface
The Cauchy integral formula is an old and beautiful result. Anybody
with an undergraduate mathematics education knows something about
this formula. Since the Cauchy integral has been studied for so long by
so many people, it is tempting to believe that everything there is to
know about the integral is known. Such beliefs are always wrong. In
1978, Norberto Kerzman and Elias M. Stein [K-S] discovered a very
basic, yet previously unknown, property of the Cauchy integral. They
discoveredthattheCauchyintegralisnearly self adjoint. Thisdiscovery
constituted a shift in the bedrock of complex analysis. The vibrations
were felt immediately in the areasof boundary behavior ofholomorphic
functions, Hardy spaces, and mapping problems. I hope this book will
help dissipate the shock wave through some areas that were thought to
be peacefully settled forever.
Students encountering complex analysis for the first time are de-
lighted to see how many of the basic theorems follow directly from the
Cauchy integral formula. In the first course on complex variables, the
Cauchy integral is applied only to holomorphic functions. This book is
intended to be part of a second course on complex variables. In it, the
Cauchy integral will be applied to nonholomorphic functions. Since the
Cauchy integral of a nonholomorphic function does not reproduce that
function, this process gives rise to a transform. The Cauchy transform
maps functions defined on the boundary of a domain to holomorphic
functions on the domain.
The Kerzman-Stein breakthrough allows many of the basic objects
of a second course in complex analysis to be described in terms of the
Cauchy transform. More theorems than ever before can be seen to be
directcorollariesofsimplefactsabouttheCauchyintegral.Inthisbook,
wewilldeducetheRiemannmappingtheorem,wewillsolvetheDirichlet
and Neumann problems for the Laplace operator, we will construct the
Poisson kernel, and we will solve the inhomogeneous Cauchy-Riemann
equations. We will do all this in a very constructive way using formulas
stemming from the Kerzman-Stein theorem.
Kerzman and Stein made their discovery while studying the Szego˝
kernel in several complex variables. Discussion of the Szego˝ projection
andkernelhavealwaysbeenconsideredtobetooadvancedtoincludein
vii
viii Preface
a second course in complex analysis.However,because of the Kerzman-
Stein result, we will be able to discuss these objects in very simple and
concreteterms.WewillalsobeabletostudytheBergmankernel.These
classicalobjects are veryuseful inthe study ofconformalmappings and
we explore many of their applications.
Another motivation for writing this book has been that the new for-
mulas expressingthe classicalobjects ofpotential theory and conformal
mapping in terms of Cauchy transforms and Szego˝ projections have led
to new numerical methods for computing these objects. In the past, it
has been difficult for nonspecialists to find accessible descriptions of the
background material needed to understand the new formulas. A reader
who is interested only in understanding the Kerzman-Stein numerical
methodforcomputingconformalmappingsmayreadChapters1–8,then
skip to Chapter 28.
Many classicalproblems in conformal mapping and potential theory
on finitely connected domains in the plane can be reduced by means of
the Riemann mapping theorem to problems on domains whose bound-
ariesconsistoffinitely manyrealanalyticcurves.Inthisbook,westudy
manyproblemsforwhichthisstandardreductionisapplicable,however,
the fullforceofthe realanalyticityofthe boundarycurvesisonlyrarely
needed. In most situations we will encounter, it will suffice to study
problems on a domain whose boundary curves are merely C∞ smooth.
ThisiswhywefixateonproblemsofC∞ behaviorofobjectsondomains
withC∞smoothboundaries.Althoughmanycomplexanalystsmaylook
uponsuchafixationasbeingunwholesome,Ibelieveitisjustifiedinthe
context of this book. I should also mention that many results that we
prove in the C∞ category can be quite routinely generalized to apply
undermildersmoothnesshypotheses.Ididnotfeelcompelledtoexplain
thesemoregeneralresultsbecausetheyareirrelevanttothemaintheme
of the book.
I have written this book thinking that my readers have completed
a one-semester course on complex analysis and that they know some
rudimentary facts about realanalysisand Hilbert space. The facts from
Hilbert space and measure theory are so few and so basic that they
could easily be learned along the way. I also assume that my reader
knowsaboutC∞ partitionsofunityandalittleaboutdifferentialforms,
enoughtounderstandGreen’sTheoreminplanardomains.Anyvoidsin
the reader’s backgroundcan be filled by consulting Ahlfors [Ah], Rudin
[Ru1, Ru2], and Spivak [Sp].
I wrote the first edition of this book thinking of it as a tour of a
beautifulregionofclassicalanalysisviewedthroughthenewlensesofthe
Kerzman-Stein theorem.Writing the first edition propelled me to study
quadraturedomains,complexityincomplexanalysis,andimprovements