Graduate Texts in Mathematics 137 Editorial Board J.H Ewing EW Gehring P.R Halmas Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTI/ZARINO Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HIT.TON!STAMMBACH A Course in Homological Algebra MAc I ANE Categories for the Working Mathematician HUOHES!PIPER Projective Planes SERkE A Course in Arithmetic TAKEUTI/ZARlNo Axiometic Set Theory HUMPHREYS Introduction to Lie Algebrasand Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed, BEALS Advanced Mathematical Analysis ANoERsON!FULLER Rings and Categories of Modules 2nd ed GOLUBITSKY/GUIT.EMIN Stable Mappings and Their Singularities BERBERIAN Lecturesin Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT RandomProcesses 2nd ed HALMos Measure Theory HALMos A Hilber! 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137) Includes bibliographical referenees (p ) and indexes Harmonie funetions III Tide IV Series QA4OS.A9S 1992 SIS'.S3-de20 I Bourdon, Paul 11 Ramey, Wade 92-16950 Printed on acid-free paper © 1992Springer SeiencetBusiness Media New York Softcover reprint ofthe hardcover 1st edition 1992 Original1y published by Springer-Verlag New York, Ine in 1992 All rights reserved T1tiswork may not be translated or eopied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Ine , 175 Fifth Avenue, New York, NY 10010, USA), exeept for brief exeerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrievaI, electronic adaptation, computer software , or by similar or dissirnilar 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 'Irade Marks and Merehandise Marks Act, may accordingly be used freely by anyone Produetion managed by Henry Krell; manufacturing supervised by Vincent Scelta Photocomposed copy prepared using laTeX 987654321 ISBN 978-1-4899-1186-5 ISBN 978-0-387-21527-3(eBook) DOI 10.1007/978-0-387-21527-3 Preface Harmonie functions-the solutions of Laplace's equation-play a crucial role in many areas of mathematies, physies, and engineering But learning about them is not always easy At times each of the authors has agreed with Lord Kelvin and Peter Tait, who wrote ([12], 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 mathe- maticians, 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 harmonie functions easier The only prerequisite for the book is a solid foundation in real and complex analysis, together with some basic results from functional analysis The first fifteen chapters of Rudin's Real and Complex Analysis, 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 harmonie function is harmonie with some straightforward observations that we believe are more revealing Another example is vi Preface our proof of Böcher's Theorem, whieh is more elementary than the classical proofs We also present material not usually covered in standard treatments of harmonie functions The section on the Schwarz Lemma and the chapter on Bergman spaces are examples For completeness, we include some topies in analysis that frequently slip through the cracks in a beginning graduate student's currieulum, such as real-analytie functions We rarely attempt to trace the history of the ideas presented in this book Thus the absence of a reference does not imply originality on our part In addition to writing the text, the authors have developed a software package to manipulate many of the expressions that arise in harmonie function theory Our software package, whieh uses many results from this book, can perform symbolic calculations that would take a prohibitive amount of time if done without a computer For example, the Poisson integral of any polynomial can be computed exactly Appendix B explains how readers can obtain our software package free of charge This book has its roots in a graduate course at Miehigan State University taught by one of the authors and attended by the other authors along with a number of graduate students The topic of harmonie functions was presented with the intention of moving on to different material after introducing the basie concepts We did not move on to different material Instead, we began to ask natural questions about harmonie functions Lively and illuminating discussions ensued A freewheeling approach to the course developed; answers to questions someone had raised in dass or in the hallway were worked out and then presented in dass (or in the hallway) Discovering mathematies in this way was a thoroughly enjoyable experience We will consider this book a success if some of that enjoyment shines through in these pages Acknowledgments Dur book has been improved by our students We take this opportunity to thank them for catching errors and making suggestions while attending courses at Michigan State University based on material in this book Among the many mathematicians who have influenced our outlook on harmonie function theory, we give special thanks to Dan Luecking for helping us to better understand Bergman spaces, and to Elias Stein and Guido Weiss for their book [10], which contributed greatly to our knowledge of spherical harmonics Lastly we thank the typists, who labored endlessly on this project Although they produced some of the worst typing we have seen, the number of errors from one draft to the next did, on occasion, actually decrease The typists were: Sheldon Axler, Paul Bourdon, and Wade Ramey Contents Prefaee Acknowledgments v vii CHAPTER Basie Propereies of Harmonie Functions Definitions and Examples Invariance Properties The Mean-Value Property The Maximum Principle The Poisson Kernel for the Ball The Diriehlet Problem for the Ball Converse of the Mean-Value Property Real Analyticity and Homogeneous Expansions Origin of the Term "Harmonie" Exercises 1 12 16 18 24 26 x Contems CHAPTER2 Bounded Harmonie Functions Liouville's Theorem Isolated Singularities Cauchy's Estimates Normal Families Maximum Principles Limits Along Rays Bounded Harmonie Functions on the Ball Exercises 31 31 32 33 34 36 38 40 41 CHAPTER3 Positive Harmonie Functions Liouville's Theorem Harnack's Inequality and Harnack's Principle Isolated Singularities Positive Harmonie Functions on the Ball Exercises 45 45 47 50 54 56 CHAPTER4 The Kelvin Transform Inversion in the Unit Sphere Motivation and Definition The Kelvin Transform Preserves Harmonie Functions Harmoni city at Infinity The Exterior Diriehlet Problem Symmetry and the Schwarz Reflection Principle Exercises 59 59 61 62 63 65 66 70 CHAPTER5 Spherieal Harmonies L 2(S) = E9~=o 'limeS) Zonal Harmonies The Poisson Kernel Revisited An Explicit Formula for Zonal Harmonies A Geometrie Characterization of Zonal Harmonics Spherical Harmonics Via Differentiation Explicit Bases for 'lim(Rn) and 'limeS) Exercises 13 74 78 82 85 87 89 92 94 216 Appendix A Volume, Surface Area, and Integration on Spheres spherical slices This formula is an immediate consequence of A.2 and A.4 (We state A.5 in terms of normalized surface-area measure because that is what we have used most often.) A.5 Theorem: Let f be a Borel measurable, integrable function on Sn If ::; k < n, then JSn f dun equals Slice Integration: Special Cases Some cases of Theorem A.5 deserve special mention We begin by choosing k = n -1, which is the largest permissible value of k This corresponds to decomposing Sn into spheres of one less dimension by intersecting Sn with the family of hyperplanes orthogonal to the first coordinate axis The ball Bn-k is just the unit interval (-1, 1), and so for xE Bn-k, we can write x instead of Ix/ Thus we obtain the following corollary of A.5 A.6 Corollary: Let f be a Borel measurable, integrable function on Sn Then JSn f dun equals n n V(B V(B - ) ) n n 1 -1 (1 - X 2) n231 Sn-l f (V x, - x 2) ( dun-l (( ) dx At the other extreme we can choose k = This corresponds to decomposing Sn into pairs of points by intersecting Sn with the family of lines parallel to the n th coordinate axis The sphere SI is the two-point set {-1, 1}, and dul is counting measure on this set, normalized so that each point has measure 1/2 Thus we obtain the following corollary of A.5 A.7 Corollary: Let f be a Borel measurable, integrable function on Sn Then JSn f dunequals nV(Bn) r JBn-l f(x, Jf=lXF) + f(x, -Jf=lXF) dlt () Jf=lXF n-l X Slice Integration: Special Cases 217 Let us now try k = (assuming n > 2) Thus in A.5 the term (1 _lxI ) (k- 2)/ disappears The variable ( in the formula given by A.5 now ranges over the unit circle in R2, so we can replace ( by (cosO,sinO), which makes du2(() equal to dO/(21r) Thus we obtain the following corollary of A.5 A.8 Corollary (n> 2): Let! be a Borel measurable, integrable function on Sn Then I sn ! dun equals n V~B) n r JBn - 11\" !(x,V1-\xI2cosO,Vl-lxI2sinO)dOdVn_2(x) -1\" An important special case of the last result occurs when n = In this case B n - is just the interval (-1,1), and we get the following corollary A.9 Corollary: Let! be a Borel measurable, integrable function on S3' Then Is3 ! dU3 equals -I 41r 111\" -1 -1\" !(x, VI - x cos 0, VI - x 2sin 0) dO dx 218 Appendix A Volume, Surface Area, and Integration on Spheres Exercises Prove that if and only if p kn (lxi ~ l)P dV(x) < 00 > n (a) Consider the region on the unit sphere in R 3lying between two parallel planes that intersect the sphere Show that the area of this region depends only on the distance between the two planes (This result was discovered by the ancient Greeks.) (b) Show that the result in part (a) does not hold in Rn if n =1= and "planes" are replaced by "hyperplanes" Let f be a Borel measurable, integrable function on the unit sphere 84 in R Define a function q; mapping the reetangular box [-1,1] x [-1,1] x [-11",11"] to 84 by setting q;(x,y,O) equal to (x, J1- x y, J1- x 2J1- y2 cosO, J1- x 2J1- y2 sinO) Prove that f } 84 f du 4=2\jl J1_x jlj11" f(q;(x,y,O))dOdydx 11" -1 -1 - 11" Without writing down anything or using a computer, evaluate Let m be a positive integer Use A.6 to give an explicit formula for 219 Exercises For readers familiar with the gamma function T: Prove that the volume of the unit ball in Rn equals 7fn/2 r(j + 1)' APPENDIXB Mathematiea and Harmonie Funetion Theory Using Mathematica,· a symbolic processing program, the authors have written routines to manipulate many of the expressions that arise in the study of harmonie functions These routines allow 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 routines for symbolic manipulation of harmonie functions are distributed free of charge by electronic mail They are designed to work on any computer that runs Mathematica Requests for our software package should be sent to axler@math.msu.edu Comments, suggestions, and bug reports should also be sent to the same electronic address • Mathematica is a registered trademark of Wolfram Research References [1] Sheldon Axler, Harmonie funetions from a eomplex analysis viewpoint, American Mathematical Monthly 93 (1986), 246-258 [2] John B Conway, F'unctions oi one complex variable, Springer-Verlag, New York, 1973 [3] P Fatou, Series trigonometriques et series de Taylor, Acta Mathematica 30 (1906), 335-400 [4] L L Helms, Introduction to Potential Theory, Wiley-Interscienee, New York, 1969 [5] Oliver Dimon Kellogg, Foundations oEPotential Theory, SpringerVerlag, Berlin, 1929 [6] Steven G Krantz, F'unction Theory oE Several Complex Variables, John Wiley, New York, 1982 [7] Edward Nelson, A proof of Liouville's Theorem, Proceedings oE the American Mathematical Society 12 (1961), 995 [8] Walter Rudin, Principles oEMathematical Analysis, third edition, MeGraw-Hill, New York, 1976 [9] Walter Rudin, Real and Complex Analysis, third edition, MeGrawHill, New York, 1987 [10] Elias M Stein and Guido Weiss, Fourier Analysis on Euclidean Spaces, Prineeton University Press, Prineeton, 1971 224 [l1J References William Thomson (Lord Kelvin), Extraits de deux lettres adressees Appliques 12 (1847), 256-264 a M Liouville, Journal de Mathematiques Pures et [12J William Thomson (Lord Kelvin) and Peter Guthrie Tait, Treatise on Natural Philosophy, Cambridge University Press, Cambridge, 1879 [13J John Wermer, Potential Tbeoty, Lecture Notes in Mathematics 408, Springer-Verlag, New York, 1974 Symbol Index A[u],50 A,189 B(a, r), B(a,r),5 B,5 B,5 B n,5 bP(O), 151 bm,189 C k(0),2 COO(0),2 C(E),2 Cn,126 Co(Rn- 1), 128 Cc(Rn-l),131 C~, 170 Cm,191 Dm , Dn , dV,4 dVn , ds,4 du, Da, 15 o; 31, 45 d(a, E), 34 dm,81 dx,126 dY,170 ds n, 214 dUn, 214 E*,60 H,32 H n,32 1im(R n), 73 1im(S ), 73 h m,79 hP(B),103 hP(H),132 h OO(0),176 K[u],61 K[u],135 Symbol Index 226 L 2(S) ,75 V(S) ,98 V(Rn - ) , 128 M(S) ,97 M[JL], 112 M(Rn - ) , 128 n, n ,4 N,88 No [u], 11 O(n ),74 P( x , () , 12 P [f ], 12,98 PE(x ,(),65 Pdf], 65 P m(Rn),76 P[JL],97 PH(Z, t) , 127 PH[JL], 128 PH[f] ,129 PB,137 P(x, y), 157 PH(Z, w), 163 PA(x ,( ), 190 PA[f], 191 P[f] ,200 Q ,l54 R(y) ,19 R n U{oo} ,59 R [u], 111 Rn , 152 Res(u , a), 187 S, S ,5 S+ ,106 S-,106 S,134 supp , 170 SI' 200 Sn, 214 Sn, 214 U,106 Un , 106 U y , 125 Ur, V, Vn ,4 ib, 163 x O , 19 x*, 60 XE, 66, 67 X *,101 ZT/ ' 78 Z((, Tl ), 78 Zm(( , Tl) , 78 Zm(x , y) , 154 Ixl,l la l, 19 IIJLII,97 Ilfl lp , 98 IlullhP , 103, 132 Ilfllp , 128 Ilullp , 151 ,1 A, 'V,4 0',5 a ,15 XE, 18 a l,19 Symbol Index 227 r a(a), 38 ~((, 6), OE, 66 114 J ,118 ~, 134 r~(a) , 142 ä,181 Un , 214 r,219 ffi, 74 ( , ), 75 ( , )m, 76 [m/2],76 J.Lf,98 Oa((),110 3~, 112 Index annular region, 183 approximate identity, 13, 126 Arzela-Ascoli Theorem, 35 Baire's Theorem, 42 balls internally tangent to B, 167 barrier, 201 barrier funetion, 201 barrier problem, 203 basis for 'Hm (Rn), n: (S), 92 Bergman space, 151 Bergman, Stefan, 151 Bloch space, 42, 167 boundary data, 15 bounded harmonie function, 31 bounded harmonie function on B, 40, 105 Böcher's Theorem, 50, 57, 175 dilate,2 direct sum, 75 Dirichlet problem, 12, 197 Dirichlet problem for annular regions, 189 Dirichlet problem for annular regions (n = 2), 195 Dirichlet problem for convex regions, 204 Dirichlet problem for H, 128 Dirichlet problem for smooth regions, 204 divergenee theorem, dual space, 101 Cauchy 's Estimates, 33 computer package, 74, 87, 93, 96, 107, 158, 192, 195, 221 cone, 206 conformal map, 60 conjugate index, 98 convex region, 204 eovering lemma, 114 equieontinuity, 102 essential singularity, 185 essential singularity (n = 2), 193 essential singularity at 00, 194 exterior Dirichlet problem, 65 exterior Poisson integral, 65 exterior Poisson kemel, 65 extemal ball condition, 203 extemal cone eondition, 206 extemal segment condition, 211 extremal function, 106 extreme point, 122 decomposition theorem, 172, 173 decomposition theorem for holomorphie funetions, 181 degree, 22, 73 Fatou Theorem, 110, 140 finitely connected, 178 Fourier series, 73-75 , 81 fundamental pole, 185 230 Index " fundamental pole (n = 2), 193 fundamental pole at 00, 194 fundamental solution of the Laplacian, 171 gamma function, 219 generalized annular Dirichlet problem, 195 generalized Dirichlet problem, 96 Green's identity, Hardy, G H., 103 Hardy-Littlewood maximal funetion, 112 harmonie,! harmonie at 00, 63 harmonie Bergman space, 151 harmonie Bloch space, 42, 167 harmonie conjugate, 178 harmonie functions, limits of, 15, 49 harmonie Hardy space, 103, 132 harmonie measure, 211 harmonie motion, 24 harmonics, 24 Harnack's Inequality, 48 Harnack's Inequality for B, 47, 56 Harnack's Principle, 49 holomorphie at 00, 70 homogeneous expansion, 23, 84 homogeneous harmonie polynomial, 24,73 homogeneous polynomial, 22, 76 Hopf Lemma, 28 inversion, 60 inversion map, 134 isolated singularity, 32, 185 isolated singularity at 00 , 61 isolated singularity of positive harmonie funetion, 50 isolated zero, Kelvin transform, 59, 61, 135 Laplace's equation, Laplacian, Laurent series, 171, 183 law of eosines, 111 Lebesgue decomposition, 118 Lebesgue Differentiation Theorem, 146 Lebesgue point , 123 limits along rays, 39 Liouville's Theorem, 31 Liouville's Theorem for positive harmonie functions, 45, 56 local defining function, 204 Local Fatou Theorem, 142 loca1ly connected, 209 loca1ly integrable, 17 logarithmie conjugation theorem, 179 Mathematica, 221 maximum principle, 6, 36 maximum principle for subharmonie funetions , 198 maximum principle, local, 22 mean-value property, mean-value property, eonverse of, 16 mean-value property, volume version, minimum principle, Morera's Theorem, 187 multi-index, 15 non-extendability of harmonie functions, 206 nontangential approach region, 38, 110 nontangentiallimit, 38, 111, 140 nontangential maximal funetion, 111 nontangentially bounded, 142 normal family, 34 north pole, 88 one-point compactification, 59 one-radius theorem, 28 open mapping, 27 open mapping theorem, 159 operator norm, 101 order of a pole, 185 order of a pole (n = 2), 193 orthogonal projection, 154, 157 orthogonal transformation, 3, 74 orthonormal basis for b2(B2), 167 parallel orthogonal to 1], 80 Perron family, 200 Perron function, 200 Picard's Theorem, 187 point evaluation, 151 point of density, 145 Poisson integral, 12 Poisson integral for annular region, 191 231 Index Poisson integral for H, 128 Poisson integral of a measure, 97 Poisson integral of a polynomial, 83,87 Poisson kernel, 9, 12, 84, 138, 157 Poisson kernel, expansion into zonal harmonies, 84 Poisson kernel for annular region, 189 Poisson kernel for H, 126, 138, 162 Poisson modifieation, 200 Poisson's equation, 171 polar coordinates, integration in, polar coordinates, Laplacian in, 26 pole, 185 pole (n = 2), 193 positive harmonie funet ion, 45 positive harmonie function on B, 54,105 positive harmonie function on H, 136 positive harmonie function on R \ {O}, 46 positive harmonie function on Rn \ {O}, 54 power series, 18 principal part, 185 principal part (n = 2), 193 produet rule, 13 punetured ball, 192 radial derivative, 123 radial funet ion, 27, 51, 57 radial harmonie funetion, 51, 57 radial limit , 162 radial maximal funetion , 111 real analytie, 19 reflection about a hyperplane, 66 refleetion about a sphere, 67 removable sets, 176 removable singularity, 32, 166, 185 removable singularity at 00 , 61, 194 reproducing kernel, 152 reproducing kernel for B, 154 reproducing kernel for H, 162 residue, 187 residue (n = 2), 195 residue theorem, 187 residue theorem (n = 2), 195 Riemann-Lebesgue Lemma, 160 Riesz Representation Theorem, 97 rotation, Schwarz Lemma, 105 Schwarz Lemma for V'u, 108 Schwarz Lemma for h2 , 123 Schwarz Reflection Principle, 66 separable normed linear space, 102 simply eonnnected, 178 singular measure, 118 singularity at 00, 194 slice integration, 215 smooth boundary, 204 software for harmonie funetions, 74, 87, 93, 96, 107, 158, 192, 195,221 spherieal average , 50 spherieal cap, 112 spherieal coordinates, Laplacian in, 26 spherieal harmonie, 25, 73 spherieal harmonies via differentiation, 89 Stone-Weierstrass Theorem, 77, 94, 190, 191 subharmonie, 198 submean-value property, 198 support, 170 surface area of S , 215 symmetry about a hyperplane, 66 symmetry about a sphere, 67 symmetry lemma, 10 total variation norm, 97 translate, uniform boundedness principle, 120 uniformly eontinuous funetion, 130, 147 uniformly integrable, 121 upper half-space , 125 volume of B , 213 weak* eonvergenee, 101, 131 zonal harmonie, 153 zonal harmonie expansion of Poisson kernel, 84 zonal harmonie, formula for, 85 zonal harmonie, geometrie eharacterization, 87 zonal harmonie with pole 1], 78 Graduate Texts in Mathematics continu.dfrom page ü 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 SAOlS/WU General Relativity for Mathematicians GRUENBERO/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERO A Course in Differential Geometry HARTSHORNE AlgebraicGeometry MANIN A Course in Mathematical Logic GRAVER!WATKINS Combinatorics with Emphasis on the Theory of Graphs BROWNIPEARCY Introduction to OperatorTheory I: Elements of Functional Analysis MASSEY AlgebraicTopology: An Introduction CROWEU/FOX Introduction to Knot Theory KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed, LANG Cyclotomic Fields ARNOLD Mathematical Methods in Classical Mechanics 2nd ed WHITEHEAD Elements of Homotopy Theory 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Mathematics 13 7 Editorial Board J.H Ewing EW Gehring P.R Halmas Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43... Subjeet Classifieations (19 91) : 31B05, 31- 01, 3IJ05 Library of Congress Cataloging-in-Publication Data Axler, Sheldon Harmonie funetion theory Sheldon Axler, Paul Bourdon, Wade Ramey p em - (Graduate