Table Of Content137
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Springer Science+Business Media, LLC
Graduate Texts in Mathematics
T AKEUTUZARING. Introduction to 35 ALEXANDERiWERMER. Several Complex
Axiomatic Set Theory. 2nd ed. Variables and Banach Algebras. 3rd ed.
2 OXTOBY. Measure and Category. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear
3 SCHAEFER. Topological Vector Spaces. Topological Spaces.
2nd ed. 37 MONIC Mathematical Logic.
4 HILTON/STAMMBACH. A Course in 38 GRAUERTIFRITZSCHE. Several Complex
Homological Algebra. 2nd ed. Variables.
5 MAc LANE. Categories for the Working 39 ARVESON. An Invitation to C·-Algebras.
Mathematician. 2nd ed. 40 KEMENy/SNELLIKNAPP. Denumerable
6 HUGHESIPIPER. Projective Planes. Markov Chains. 2nd ed.
7 SERRE. A Course in Arithmetic. 41 ApOSTOL. Modular Functions and Dirichlet
8 TAKEUTUZARING. Axiomatic Set Theory. Series in Number Theory.
9 HUMPHREYS. Introduction to Lie Algebras 2nded.
and Representation Theory. 42 SERRE. Linear Representations of Finite
10 COHEN. A Course in Simple Homotopy Groups.
Theory. 43 GILLMAN/JERISON. Rings of Continuous
II CONWAY. Functions of One Complex Functions.
Variable I. 2nd ed. 44 KENDIG. Elementary Algebraic Geometry.
12 BEALS. Advanced Mathematical Analysis. 45 LoEvE. Probability Theory I. 4th ed.
13 ANDERSONIFULLER. Rings and Categories 46 LoEvE. Probability Theory II. 4th ed.
of Modules. 2nd ed. 47 MOISE. Geometric Topology in
14 GOLUBITSKy/GUlLLEMIN. Stable Mappings Dimensions 2 and 3.
and Their Singularities. 48 SAOISlWu. General Relativity for
15 BERBERIAN. Lectures in Functional Mathematicians.
Analysis and Operator Theory. 49 GRUENBERGIWEIR. Linear Geometry.
16 WINTER. The Structure of Fields. 2nd ed.
17 ROSENBLATT. Random Processes. 2nd ed. 50 EDWARDS. Fermat's Last Theorem.
18 HALMOS. Measure Theory. 51 KLINGENBERG. A Course in Differential
19 HALMOS. A Hilbert Space Problem Book. Geometry.
2nd ed. 52 HARTSHORNE. Algebraic Geometry.
20 HUSEMOLLER. Fibre Bundles. 3rd ed. 53 MANlN. A Course in Mathematical Logic.
21 HUMPHREYS. Linear Algebraic Groups. 54 GRA VERiW ATKINS. Combinatorics with
22 BARNES/MACK. An Algebraic Introduction Emphasis on the Theory of Graphs.
to Mathematical Logic. 55 BROWNIPEARCY. Introduction to Operator
23 GREUB. Linear Algebra. 4th ed. Theory I: Elements of Functional
24 HOLMES. Geometric Functional Analysis Analysis.
and Its Applications. 56 MASSEY. Algebraic Topology: An
25 HEWITT/STROMBERG. Real and Abstract Introduction.
Analysis. 57 CROWELLIFox. Introduction to Knot
26 MANES. Algebraic Theories. Theory.
27 KELLEY. General Topology. 58 KOBLITZ. p-adic Numbers, p-adic Analysis,
28 ZARISKIISAMUEL. Commutative Algebra. and Zeta-Functions. 2nd ed.
Vol.I. 59 LANG. Cyclotomic Fields.
29 ZARISKIISAMUEL. Commutative Algebra. 60 ARNOLD. Mathematical Methods in
VoU!. Classical Mechanics. 2nd ed.
30 JACOBSON. Lectures in Abstract Algebra I. 61 WHITEHEAD. Elements of Homotopy
Basic Concepts. Theory.
31 JACOBSON. Lectures in Abstract Algebra II. 62 KARGAPOLOvIMERLZJAKOV. Fundamentals
Linear Algebra. of the Theory of Groups.
32 JACOBSON. Lectures in Abstract Algebra 63 BOLLOBAS. Graph Theory.
Ill. Theory of Fields and Galois Theory. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed.
33 HIRSCH. Differential Topology. 65 WELLS. Differential Analysis on Complex
34 SPITZER. Principles of Random Walk. Manifolds. 2nd ed.
2nd ed.
(continued ajier index)
Sheldon Axler
Paul Bourdon
Wade Ramey
Harmonic Function
Theory
Second Edition
With 21 Illustrations
t
Springer
Sheldon Axler Paul Bourdon
Mathematics Department Mathematics Department
San Francisco State University Washington and Lee University
San Francisco, CA 94132 Lexington, VA 24450
USA USA
[email protected] [email protected]
Wade Ramey
8 Bret Harte Way
Berkeley, CA 94708
USA
[email protected]
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East Hall University of California
University University of Michigan at Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840
USA USA USA
Mathematics Subject Classification (2000): 31-01, 31B05, 31C05
Library of Congress Cataloging-in-Publication Data
Axler, Sheldon Jay.
Harmonic function theory/Sheldon Axler, Paul Bourdon, Wade Ramey.-2nd ed.
p. cm. - (Graduate texts in mathematics; 137)
Includes bibliographical references and indexes.
ISBN 978-1-4419-2911-2 ISBN 978-1-4757-8137-3 (eBook)
DOI 10.1007/978-1-4757-8137-3
1. Harmonic functions. I. Bourdon, Paul. II. Ramey, Wade. III. Title. IV. Series.
QA405 .A95 2001
515'.53-dc21 00-053771
© 200 I, 1992 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2001
Softcover reprint of the hardcover 2nd edition 200 I
All rights reserved. This work may not be translated or copied in whole or in part without the
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This reprint has been authorized by Springer-Verlag (BerlinlHeidelberg/New York) for sale in
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Reprinted in China by Beijing World Publishing Corporation, 2004
98765 432 I
ISBN 978-1-4419-2911-2 SPIN 10791946
Cantents
Preface ix
Acknowledgments xi
CHAPTER 1
Basic Properties of Harmonic Functions 1
Definitions and Examples ....................... 1
lnvariance Properties . . . . . . . . . . . . . . . . . . . . . . . . .. 2
The Mean-Value Property. . . . . . . . . . . . . . . . . . . . . . .. 4
The Maximum Principle. . . . . . . . . . . . . . . . . . . . . . . .. 7
The Poisson Kernelfor the Ball . . . . . . . . . . . . . . . . . . .. 9
The Dirichlet Problem for the Ball . . . . . . . . . . . . . . . . .. 12
Converse of the Mean-Value Property . . . . . . . . . . . . . . .. 17
Real Analyticity and Homogeneous Expansions . . . . . . . . .. 19
Origin of the Term "Harmonic" . . . . . . . . . . . . . . . . . . .. 25
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
CHAPTER 2
Bounded Harmonic Functions 31
liouville's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . .. 31
Isolated Singularities .. . . . . . . . . . . . . . . . . . . . . . . .. 32
Cauchy's Estimates ........................... 33
Normal Families . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
Maximum Principles. . . . . . . . . . . . . . . . . . . . . . . . . .. 36
Limits Along Rays . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38
Bounded Harmonic Functions on the Ball. . . . . . . . . . . . .. 40
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
v
vi Contents
CHAPTER 3
Positive Harmonic Functions 45
Liouville's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
Harnack's Inequality and Harnack's Principle . . . . . . . . . .. 47
Isolated Singularities ....... . . . . . . . . . . . . . . . . . .. 50
Positive Harmonic Functions on the Ball . . . . . . . . . . . . .. 55
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. 56
CHAPTER 4
The Kelvin Transform 59
Inversion in the Unit Sphere. . . . . . . . . . . . . . . . . . . . .. 59
Motivation and Definition . . . . . . . . . . . . . . . . . . . . . .. 61
The Kelvin Transform Preserves Harmonic Functions . . . . .. 62
Harmonicity at Infinity . . . . . . . . . . . . . . . . . . . . . . . .. 63
The Exterior Dirichlet Problem . . . . . . . . . . . . . . . . . . .. 66
Symmetry and the Schwarz Reflection Principle. . . . . . . . .. 67
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
CHAPTER 5
Harmonic Polynomials 73
Polynomial Decompositions . . . . . . . . . . . . . . . . . . . . .. 74
Spherical Harmonic Decomposition of L 2 (5) ........... 78
Inner Product of Spherical Harmonics. . . . . . . . . . . . . . .. 82
Spherical Harmonics Via Differentiation . . . . . . . . . . . . .. 85
Explicit Bases of .1fm(Rn) and .1fm(5) ............... 92
Zonal Harmonics . . . . . ". . . . . . . . . . . . . . . . . . . . . . .. 94
The Poisson Kernel Revisited . . . . . . . . . . . . . . . . . . . .. 97
A Geometric Characterization of Zonal Harmonics . . . . . . . . 100
An Explicit Formula for Zonal Harmonics ............. 104
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
CHAPTER 6
Harmonic Hardy Spaces 111
Poisson Integrals of Measures . . . . . . . . . . . . . . . . . . . . . III
Weak* Convergence ........................... 115
The Spaces hP (B) ............................ 117
The Hilbert Space h 2 (B) ........................ 121
The Schwarz Lemma .......................... 123
The Fatou Theorem ........................... 128
Exercises .................................. 138
Contents vii
CHAPTER 7
Harmonic Functions on Half-Spaces 143
The Poisson Kernel for the Upper Half-Space ........... 144
The Dirichlet Problem for the Upper Half-Space .......... 146
The Harmonic Hardy Spaces hP (H) ... . . . . . . . . . . . ... 151
From the Ball to the Upper Half-Space, and Back ......... 153
Positive Harmonic Functions on the Upper Half-Space ...... 156
Nontangential limits .......................... 160
The Local Fatou Theorem ....................... 161
Exercises .................................. 167
CHAPTER 8
Harmonic Bergman Spaces 171
Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . 172
The Reproducing Kernel of the Ball . . . . . . . . . . . . . . . . . 176
Examples in bP(B) ............................ 181
The Reproducing Kernel of the Upper Half-Space ......... 185
Exercises .................................. 188
CHAPTER 9
The Decomposition Theorem 191
The Fundamental Solution of the Laplacian ............ 191
Decomposition of Harmonic Functions ............... 193
Bacher's Theorem Revisited ...................... 197
Removable Sets for Bounded Harmonic Functions .'. ...... 200
The Logarithmic Conjugation Theorem . ; ............. 203
Exercises . . . . . . . . . . . . . . . . . . . . .....' . . . . . . . . . 206
CHAPTER 10
Annular Regions 209
Laurent Series ............................... 209
Isolated Singularities .......................... 210
The Residue Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 213
The Poisson Kernel for Annular Regions . . . . . . . . . . . . . . 215
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
CHAPTER 11
The Dirichlet Problem and Boundary Behavior 223
The Dirichlet Problem .......................... 223
Subharmonic Functions ......................... 224
viii Contents
The Perron Construction ........................ 226
Barrier Functions and Geometric Criteria for Solvability .... 227
Nonextendability Results ........................ 233
Exercises .................................. 236
APPENDIX A
Volume, Surface Area, and Integration on Spheres 239
Volume of the Ball and Surface Area of the Sphere ........ 239
Slice Integration on Spheres ...................... 241
Exercises .................................. 244
APPENDIX B
Harmonic Function Theory and Mathematica 247
References 249
Symbol Index 251
Index 255
Preface
Harmonic functions-the solutions of Laplace's equation-playa
crucial role in many areas of mathematics, physics, and engineering.
But learning about them is not always easy. At times the authors have
agreed with Lord Kelvin and Peter Tait, who wrote ([18], Preface)
There can be but one opinion as to the beauty and utility of this
analysis of Laplace; but the manner in which it has been hitherto
presented has seemed repulsive to the ablest mathematicians, and
difficult to ordinary mathematical students.
The quotation has been included mostly for the sake of amusement,
but it does convey a sense of the difficulties the uninitiated sometimes
encounter.
The main purpose of our text, then, is to make learning about har
monic functions easier. We start at the beginning of the subject, assum
ing only that our readers have a good foundation in real and complex
analysis along with a knowledge of some basic results from functional
analysis. The first fifteen chapters of [15], for example, provide suffi
cient preparation.
In several cases we simplify standard proofs. For example, we re
place the usual tedious calculations showing that the Kelvin transform
of a harmonic function is harmonic with some straightforward obser
vations that we believe are more revealing. Another example is our
proof of Bacher's Theorem, which is more elementary than the classi
cal proofs.
We also present material not usually covered in standard treatments
of harmonic functions (such as [9], [11], and [19]). The section on the
Schwarz Lemma and the chapter on Bergman spaces are examples. For
ix
