Harmonic Function Theory

1 Harmonic FunctionTheorySecond EditionSheldon AxlerPaul BourdonWade Ramey26 December 2000 This copyrighted pdf file is available without charge only toindividuals who have purchased a copy ofHarmonic Function Theory ,second edition. Please do not distribute this file or its password toanyone and do not post it on the web. 2001 Springer-Verlag New York, Issues In your Adobe Acrobat software, go to the File menu, select Preferences , then General , then change the setting of SmoothText and Images to determine whether this document looks bet-ter with this setting checked or unchecked. Some users reportthat the text looks considerably better on the screen with SmoothText and Images unchecked, while other users have the oppositeexperience.

2 Text in red is linked to the appropriate page number, chapter,theorem, equation, exercise, reference, etc. Clicking on red textwill cause a jump to the page containing the corresponding item. The bookmarks at the left can also be used for navigation. Clickon a chapter title or section title to jump to that chapter or section(section titles can be viewed by clicking on the expand icon to theleft of the chapter title). Instead of using the index at the end of the book, use Acrobat sfind feature to locate words throughout the Properties of Harmonic Functions1 Definitions and Mean-Value Maximum Poisson Kernel for the Dirichlet Problem for the of the Mean-Value Analyticity and Homogeneous of the Term Harmonic .

3 Harmonic Functions31 Liouville s s Along Harmonic Functions on the Harmonic Functions45 Liouville s s Inequality and Harnack s Harmonic Functions on the Kelvin Transform59 Inversion in the Unit and Kelvin Transform Preserves Harmonic at Exterior Dirichlet and the Schwarz Reflection Polynomials73 Polynomial Harmonic Decomposition ofL2(S)..78 Inner Product of Spherical Harmonics Via Bases ofHm(Rn)andHm(S)..92 Zonal Poisson Kernel Geometric Characterization of Zonal Explicit Formula for Zonal Hardy Spaces111 Poisson Integrals of * Spaceshp(B)..117 The Hilbert Spaceh2(B)..121 The Schwarz Fatou Functions on Half-Spaces143 The Poisson Kernel for the Upper Dirichlet Problem for the Upper Harmonic Hardy Spaceshp(H).

4 151 From the Ball to the Upper Half-Space, and Harmonic Functions on the Upper Local Fatou Bergman Spaces171 Reproducing Reproducing Kernel of the inbp(B)..181 The Reproducing Kernel of the Upper Decomposition Theorem191 The Fundamental Solution of the of Harmonic cher s Theorem Sets for Bounded Harmonic Logarithmic Conjugation Regions209 Laurent Residue Poisson Kernel for Annular Dirichlet Problem and Boundary Behavior223 The Dirichlet Perron Functions and Geometric Criteria for , Surface Area, and Integration on Spheres239 Volume of the Ball and Surface Area of the Integration on Function Theory andMathematica247 References249 Symbol Index251 Index255 PrefaceHarmonic functions the solutions of Laplace s equation play acrucial role in many areas of mathematics, physics.

5 And learning about them is not always easy. At times the authors haveagreed with Lord Kelvin and Peter Tait, who wrote ([18], Preface)There can be but one opinion as to the beauty and utility of thisanalysis of Laplace; but the manner in which it has been hithertopresented has seemed repulsive to the ablest mathematicians, anddifficult to ordinary mathematical quotation has been included mostly for the sake of amusement,but it does convey a sense of the difficulties the uninitiated 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 complexanalysis along with a knowledge of some basic results from functionalanalysis.

6 The first fifteen chapters of [15], for example, provide suffi-cient several cases we simplify standard proofs. For example, we re-place the usual tedious calculations showing that the Kelvin transformof a Harmonic Function is Harmonic with some straightforward obser-vations that we believe are more revealing. Another example is ourproof of B cher s Theorem, which is more elementary than the classi-cal also present material not usually covered in standard treatmentsof Harmonic functions (such as [9], [11], and [19]). The section on theSchwarz Lemma and the chapter on Bergman spaces are examples. ForixxPrefacecompleteness, we include some topics in analysis that frequently slipthrough the cracks in a beginning graduate student s curriculum, suchas real-analytic rarely attempt to trace the history of the ideas presented in thisbook.

7 Thus the absence of a reference does not imply originality onour this second edition we have made several major changes. Thekey improvement is a new and considerably simplified treatment ofspherical harmonics (Chapter5). The book now includes a formula forthe Laplacian of the Kelvin transform ( ). Another ad-dition is the proof that the Dirichlet problem for the half-space withcontinuous boundary data is solvable ( ), with no growthconditions required for the boundary Function . Yet another signifi-cant change is the inclusion of generalized versions of Liouville s andB cher s Theorems ( ), which are shown to beequivalent. We have also added many exercises and made numeroussmall addition to writing the text, the authors have developed a soft-ware package to manipulate many of the expressions that arise in har-monic Function Theory .

8 Our software package, which uses many resultsfrom this book, can perform symbolic calculations that would take aprohibitive amount of time if done without a computer. For example,the Poisson integral of any polynomial can be computed exactly. Ap-pendixBexplains how readers can obtain our software package free roots of this book lie in a graduate course at Michigan StateUniversity taught by one of the authors and attended by the other au-thors along with a number of graduate students. The topic of harmonicfunctions was presented with the intention of moving on to differentmaterial after introducing the basic concepts. We did not move on todifferent material. Instead, we began to ask natural questions aboutharmonic functions.

9 Lively and illuminating discussions ensued. Afreewheeling approach to the course developed; answers to questionssomeone had raised in class or in the hallway were worked out and thenpresented in class (or in the hallway). Discovering mathematics in thisway was a thoroughly enjoyable experience. We will consider this booka success if some of that enjoyment shines through in these book has been improved by our students and by readers of thefirst edition. We take this opportunity to thank them for catching errorsand making useful the many mathematicians who have influenced our outlookon Harmonic Function Theory , we give special thanks to Dan Lueckingfor helping us to better understand Bergman spaces, to Patrick Ahernwho suggested the idea for the proof of , and to EliasStein and Guido Weiss for their book [16], which contributed greatly toour knowledge of spherical are grateful to Carrie Heeter for using her expertise to make oldphotographs look our publisher Springer we thank the mathematics editors Thomasvon Foerster (first edition) and Ina Lindemann (second edition)

10 For theirsupport and encouragement, as well as Fred Bartlett for his valuableassistance with electronic Properties ofHarmonic FunctionsDefinitions and ExamplesHarmonic functions, for us, live on open subsets of real Euclideanspaces. Throughout this book,nwill denote a fixed positive integergreater than 1 and will denote an open, nonempty subset continuously differentiable, complex-valued functionudefinedon isharmonicon if u 0,where =D12+ +Dn2andDj2denotes the second partial derivativewith respect to thejthcoordinate variable. The operator is called theLaplacian, and the equation u 0 is calledLaplace s that a functionudefined on a (not necessarily open) setE Rnisharmonic onEifucan be extended to a Function Harmonic on an openset letx=(x1.)