Harmonic function theory.

*(English)*Zbl 0765.31001
Graduate Texts in Mathematics. 137. New York: Springer-Verlag. xii, 231 p. (1992).

In the preface, the authors state that the main purpose of the text is to make learning about harmonic functions easier. This goal has certainly been accomplished. The book is a well written text, suitable for a one semester beginning graduate level course on harmonic function theory in \(\mathbb{R}^ n\). Although the authors assume some knowledge of measure theory and elementary results in functional analysis, this requirement is quite minimal and is only significant in the later chapters. A solid foundation in complex analysis will help the reader appreciate some of the subtleties of harmonic function theory for \(n\geq 3\).

Chapters 1-3 contain the basic properties of harmonic functions in \(\mathbb{R}^ n\), including the maximum principle, the Poisson integral formula and the Dirichlet problem for the unit ball \(B\). These chapters also contain results concerning bounded and positive harmonic functions in various settings. Chapter 3 contains an elementary proof of BĂ´chers theorem characterizing the positive harmonic functions on \(B\setminus\{0\}\). The Kelvin transform is the topic of chapter 4. This is used to solve the Dirichlet problem for the exterior of the ball. Chapter 5 contains a detailed discussion of harmonic polynomials and the decomposition of \(L^ 2(S)\) as the direct sum of \({\mathcal H}_ m(S)\), the spherical harmonics of degree \(m\) on \(S\).

Chapter 6 is devoted to the harmonic Hardy spaces \(h^ p(B)\), \(1\leq p\leq\infty\), which is defined as the space of harmonic functions \(f\) on \(B\) for which \[ \sup_{0<r<1} \int_ S f|(rt)|^ p d\sigma(t)<\infty, \qquad 1\leq p<\infty. \] This chapter also contains a proof of Fatou’s theorem on the existence of nontangential limits of functions in \(h^ p(B)\), \(p\geq 1\). The analogous questions on a half space \(H\), including the local Fatou theorem, are considered in chapter 7.

Chapter 8 contains a discussion of the harmonic Bergman spaces \(b^ p(\Omega)\), \(p\geq 1\), on a domain \(\Omega\) in \(\mathbb{R}^ n\), including a computation of the reproducing kernel of \(b^ 2(B)\) and \(b^ 2(H)\). The decomposition theorem for harmonic functions is proved in chapter 9, and chapter 10 deals with harmonic functions on annular regions. The final chapter is devoted to a brief discussion of the Perron method for solving the Dirichlet problem on general domains.

The subject matter of the text is greatly enhanced by the numerous exercises (at various levels of difficulty) at the end of each chapter. To supplement the text, the authors have developed a software package using “Mathematica” to manipulate many of the symbolic expressions that arise in the study of harmonic functions. The software for example can be used to produce a basis for \({\mathcal H}_ m(S)\), and examples of its application are given in the text.

Chapters 1-3 contain the basic properties of harmonic functions in \(\mathbb{R}^ n\), including the maximum principle, the Poisson integral formula and the Dirichlet problem for the unit ball \(B\). These chapters also contain results concerning bounded and positive harmonic functions in various settings. Chapter 3 contains an elementary proof of BĂ´chers theorem characterizing the positive harmonic functions on \(B\setminus\{0\}\). The Kelvin transform is the topic of chapter 4. This is used to solve the Dirichlet problem for the exterior of the ball. Chapter 5 contains a detailed discussion of harmonic polynomials and the decomposition of \(L^ 2(S)\) as the direct sum of \({\mathcal H}_ m(S)\), the spherical harmonics of degree \(m\) on \(S\).

Chapter 6 is devoted to the harmonic Hardy spaces \(h^ p(B)\), \(1\leq p\leq\infty\), which is defined as the space of harmonic functions \(f\) on \(B\) for which \[ \sup_{0<r<1} \int_ S f|(rt)|^ p d\sigma(t)<\infty, \qquad 1\leq p<\infty. \] This chapter also contains a proof of Fatou’s theorem on the existence of nontangential limits of functions in \(h^ p(B)\), \(p\geq 1\). The analogous questions on a half space \(H\), including the local Fatou theorem, are considered in chapter 7.

Chapter 8 contains a discussion of the harmonic Bergman spaces \(b^ p(\Omega)\), \(p\geq 1\), on a domain \(\Omega\) in \(\mathbb{R}^ n\), including a computation of the reproducing kernel of \(b^ 2(B)\) and \(b^ 2(H)\). The decomposition theorem for harmonic functions is proved in chapter 9, and chapter 10 deals with harmonic functions on annular regions. The final chapter is devoted to a brief discussion of the Perron method for solving the Dirichlet problem on general domains.

The subject matter of the text is greatly enhanced by the numerous exercises (at various levels of difficulty) at the end of each chapter. To supplement the text, the authors have developed a software package using “Mathematica” to manipulate many of the symbolic expressions that arise in the study of harmonic functions. The software for example can be used to produce a basis for \({\mathcal H}_ m(S)\), and examples of its application are given in the text.

Reviewer: M.Stoll (Columbia)

##### MSC:

31-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to potential theory |

31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |

31-04 | Software, source code, etc. for problems pertaining to potential theory |