Potential theory in the complex plane. (English) Zbl 0828.31001

Ransford, Thomas

London Mathematical Society Student Texts. 28. Cambridge: Univ. Press. x, 232 p. £ 32.50; $ 49.95/hbk, £ 13.95; $ 19.95/pbk (1995).

This book is an introduction to the subject suitable for beginning graduate students, concentrating on the important case of potential theory in two dimensions. The author claims that the book is the one that he would have liked to have read when he was learning the subject many years before: in writing it, he has succeeded in saving his readers a great deal of time-consuming effort in locating intelligible explanations of the material.
Chapter 1 outlines the connection between harmonic and analytic functions, the Dirichlet problem and various useful material concerning positive harmonic functions and Harnack’s theorem.
Chapter 2 introduces upper semi-continuous functions and subharmonic functions and discusses the Maximum Principle, integrability, convexity and smoothing, and criteria for subharmonicity. In Chapter 3 this leads on to potentials, polar sets, equilibrium measures, removable singularities, the generalised Laplacian and thin sets.
Chapter 4 deals with the Dirichlet problem, its solution, regularity, harmonic measure and Green’s functions. Chapter 5 introduces the notion of capacity, discusses capacity as a set function, looks at ways of computing or estimating capacities, and relates capacity to thinness and to transfinite diameter.
Finally Chapter 5 gives a number of applications of the theory in the previous chapters to interpolation in \(L^p\)-spaces, homogeneous polynomials, uniform approximation, Banach algebras and complex dynamics.
The book includes a helpful appendix on Borel measures, useful literature references at the ends of chapters, and a range of exercises from the routine to thinly-disguised theorems in the research literature.
The interplay between potential theory and complex analysis is a key feature of this interesting textbook.

Reviewer: D.A.Brannan (Milton Keynes)

MSC:

31-02	Research exposition (monographs, survey articles) pertaining to potential theory
31A05	Harmonic, subharmonic, superharmonic functions in two dimensions
31B15	Potentials and capacities, extremal length and related notions in higher dimensions
30C85	Capacity and harmonic measure in the complex plane

Keywords:

maximum principle; interpolation in \(L^ p\)-spaces; capacity; analytic functions; Dirichlet problem; harmonic functions; subharmonic functions; polar sets; equilibrium measures; removable singularities; generalised Laplacian; thin sets; transfinite diameter; homogeneous polynomials; uniform approximation; Banach algebras; complex dynamics