Graduate Texts in Mathematics S Axler 137 Editorial Board F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 TAKEUTUZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAc LANE Categories for the Working Mathematician 2nd ed HUGHESIPIPER Projective Planes SERRE A Course in Arithmetic TAKEUTUZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSONIFULLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUlLLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKIISAMUEL Commutative Algebra Vol.I ZARISKIISAMUEL Commutative Algebra VoU! JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra Ill Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed 35 ALEXANDERiWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONIC Mathematical Logic 38 GRAUERTIFRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C·-Algebras 40 KEMENy/SNELLIKNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nded 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEvE Probability Theory I 4th ed 46 LoEvE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SAOISlWu General Relativity for Mathematicians 49 GRUENBERGIWEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANlN A Course in Mathematical Logic 54 GRAVERiW ATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWNIPEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELLIFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOvIMERLZJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed (continued ajier index) Sheldon Axler Paul Bourdon Wade Ramey Harmonic Function Theory Second Edition With 21 Illustrations t Springer Sheldon Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu Paul Bourdon Mathematics Department Washington and Lee University Lexington, VA 24450 USA pbourdon@wlu.edu Wade Ramey Bret Harte Way Berkeley, CA 94708 USA wramey@home.com Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 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 Harmonic functions I Bourdon, Paul II Ramey, Wade QA405 A95 2001 515'.53-dc21 © 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 III Title IV Series 00-053771 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by llQyone This reprint has been authorized by Springer-Verlag (BerlinlHeidelberg/New York) for sale in the People's Republic of China only and not for export therefrom 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 Basic Properties of Harmonic Functions Definitions and Examples lnvariance Properties The Mean-Value Property The Maximum Principle The Poisson Kernelfor the Ball The Dirichlet Problem for the Ball Converse of the Mean-Value Property Real Analyticity and Homogeneous Expansions Origin of the Term "Harmonic" Exercises 1 12 17 19 25 26 CHAPTER Bounded Harmonic Functions liouville's Theorem Isolated Singularities Cauchy's Estimates Normal Families Maximum Principles Limits Along Rays Bounded Harmonic Functions on the Ball Exercises v 31 31 32 33 35 36 38 40 42 Contents vi CHAPTER Positive Harmonic Functions Liouville's Theorem Harnack's Inequality and Harnack's Principle Isolated Singularities Positive Harmonic Functions on the Ball Exercises 45 45 47 50 55 56 59 59 61 62 63 66 67 71 Harmonic Polynomials Polynomial Decompositions Spherical Harmonic Decomposition of L (5) Inner Product of Spherical Harmonics Spherical Harmonics Via Differentiation Explicit Bases of 1fm (R n ) and 1fm (5) Zonal Harmonics " The Poisson Kernel Revisited A Geometric Characterization of Zonal Harmonics An Explicit Formula for Zonal Harmonics Exercises 73 74 78 82 85 92 94 97 100 104 106 CHAPTER The Kelvin Transform Inversion in the Unit Sphere Motivation and Definition The Kelvin Transform Preserves Harmonic Functions Harmonicity at Infinity The Exterior Dirichlet Problem Symmetry and the Schwarz Reflection Principle Exercises CHAPTER CHAPTER Harmonic Hardy Spaces Poisson Integrals of Measures Weak* Convergence The Spaces h P (B) The Hilbert Space h (B) The Schwarz Lemma The Fatou Theorem Exercises 111 III 115 117 121 123 128 138 Contents CHAPTER vii Harmonic Functions on Half-Spaces The Poisson Kernel for the Upper Half-Space The Dirichlet Problem for the Upper Half-Space The Harmonic Hardy Spaces h P (H) From the Ball to the Upper Half-Space, and Back Positive Harmonic Functions on the Upper Half-Space Nontangential limits The Local Fatou Theorem Exercises CHAPTER 143 144 146 151 153 156 160 161 167 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 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 Contents viii The Perron Construction Barrier Functions and Geometric Criteria for Solvability Nonextendability Results Exercises 226 227 233 236 APPENDIX A Volume, Surface Area, and Integration on Spheres Volume of the Ball and Surface Area of the Sphere Slice Integration on Spheres Exercises 239 239 241 244 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 harmonic functions easier We start at the beginning of the subject, assuming 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 sufficient preparation In several cases we simplify standard proofs For example, we replace the usual tedious calculations showing that the Kelvin transform of a harmonic function is harmonic with some straightforward observations that we believe are more revealing Another example is our proof of Bacher's Theorem, which is more elementary than the classical 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 ApPENDIX B J-{armonic Junction TFzeory ami :M.atliematica Using the computational environment provided by Mathematica,* the authors have written software to manipulate many of the expressions that arise in the study of harmonic functions This software allows the user to make symbolic calculations that would take a prohibitive amount of time if done without a computer For example, Poisson integrals of polynomials can be computed exactly Our software for symbolic manipulation of harmonic functions is available over the internet without charge It is distributed as a Mathematica package that will work on any computer that runs Mathematica This Mathematica package and the instructions for using it are available at http://math.sfsu.edu/axler/HFT_Math.html and in the standard electronic mathematical archives (search for the files HFT m and Harmoni cFuncti onTheory nb) Comments, suggestions, and bug reports should be sent to axler@sfsu.edu Here are some of the capabilities of our Alathematica package: symbolic calculus in R n, induding integration on balls and spheres; solution of the Dirichlet problem for balls, annular regions, and exteriors of balls in Rn (exact solutions with polynomial data); solution of the Neumann problem for balls and exteriors of balls in Rn (exact solutions with polynomial data); "Mathematica is a registered trademark of Wolfram Research, Inc 247 248 ApPENDIX B Harmonic Function Theory and Mathematica computation of bases for spaces of spherical harmonics in Rn; computation of the Bergman projection for balls in Rn; manipulations with the Kelvin transform K and the modified Kelvin transform X; computation of the extremal function given by the Harmonic Schwarz Lemma (6.24) for balls in Rn; computation of harmonic conjugates in R2 New features are frequently added to this software References Harmonic function theory has a rich history and a continuing high level of research activity MathSciNet lists over five thousand papers published since 1940 for which the review contains the phrase "harmonic function", ""ith over fifteen hundred of these papers appearing since 1990 Although we have drawn freely from this heritage, we list here only those works cited in the text [1] V Anandam and M Damlakhi, Bacher's Theorem in R2 and Caratheodory's Inequality, Real Analysis Exchange 19 (1993/94),537-539 [2] Sheldon Axler, Harmonic functions from a complex analysis viewpoint, American Mathematical Monthly 93 (1986), 246-258 [3] Sheldon A.'Cler, Paul Bourdon, and Wade Ramey, Mcher's Theorem, American Mathematical Monthly 99 (1992), 51-55 [4] Sheldon A.xler and Wade Ramey, Harmonic polynomials and Dirichlet-type problems, Proceedings of the American Mathematical Society 123 (1995), 3765-3773 [3] Stefan Bergman, The Kernel Function and Conformal Mapping, American Mathematical Society, 1950 [6] Ronald R Coifman and Guido Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogenes, Lecture Notes in Mathematics, Springer, 1971 [7] John B Conway, Functions of One Complex Variable, second edition, Graduate Texts in Mathematics, Springer, 1978 249 250 References [8] P Fatou, Series trigonometriques et series de Taylor, Acta Mathematica 30 (1906), 335-400 [9] L L Helms, Introduction to Potential Theory, Wiley-Interscience, 1969 [10] Yujiro Ishikawa, Mitsuru Nakai, and Toshimasa Tada, A form of classical Picard Principle, Proceedings of the japan Academy, Series A, Mathematical Sciences 72 (1996), 6-7 [11] Oliver Dimon Kellogg, Foundations of Potential Theory, Springer, 1929 [12] Steven G Krantz, Function Theory of Several Complex Variables, John Wiley, 1982 [13] Edward Nelson, A proof of Liouville's Theorem, Proceedings of the American Mathematical Society 12 (1961), 995 [14] Walter Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill, 1976 [15] Walter Rudin, Real and Complex Analysis, third edition, McGraw-Hill, 1987 [16] Elias M Stein and Guido Weiss, Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971 [17]· William Thomson (Lord Kelvin), Extraits de deux lettres adressees aM Liouville, journal de Mathematiques Pures et Appliques 12 (l847), 256-264 [18] William Thomson (Lord Kelvin) and Peter Guthrie Tait, Treatise on Natural Philosophy, Cambridge University Press, 1879 [19] John Wermer, Potential Theory, Lecture Notes in MathematiCS, Springer, 1974 [20] Hermann Weyl, On the volume of tubes, American journal of Mathematics 61 (1939),461-472 Symho[ Ind£x Symbols are sorted by ignoring everything except Latin and Greek letters For sorting purposes, Greek letters are assumed to be spelled out in full with Latin letters For example, QE is translated to "Omega£", which is then sorted with other entries beginning with "0" and before o (n), which translates to "On" The symbols that contain no Latin or Greek letters appear first, sorted by page number \\ IIbP, 171 \ \, 'V,4 EB,76,81 [ ],77 ( , ), 79 1., 136 3, 206 b P (Q),I71 C~, 192 Cc (Rn-l),148 C(E),2 XE, 18 Ck(O),2 c m ,87,217 A,215 for n = 2, 107 Cn , 144 C"(O),2 Co(Rn-l),146 Cm lX, 15 lX!, 19 \lX\, 19 A[u),51 d(a,E),34 13,5 DC(, 15 ~, B(a, r), D m ,3 E(a,r),5 b m ,215 Dn,4 'D y ,31,45 Bn , dS,4 B,5 251 252 da, dan, 240 ds n , 240 dV,4 dVn , dX,144 dy,192 E*,60 f, 245 fa, 38 f~, 161 H,32 h m,95 J{m(Rn),75 J{m(S),80 H n ,32 h (D),200 1111hP, 117, 151 OO h P (B),117 h P (H),151 K,61 X,155 3K,133 K«(, 6),131 LP(Rn-l),146 LP(S),112 L2(5),79 :M,131 M(Rn-l),146 M(S), III IIpll, III Pf,112 n,l n,4 N,103 Na[u], 129 Symbol Index D,1 Da «(l,128 D E ,67 O(n),95 II lip, 112, 146 PA [J],217 PA (x,(),216 p(D),85 Pe[f],66 Pe(x'(l,66 P[J], 12, 112 P[f], 226 PH[J],146