Integral representations and boundary value problems in multidimensional complex analysis. (Integral’nye predstavleniya i kraevye zadachi v mnogomernom kompleksnom analize.) (Russian) Zbl 1088.32001

Dzebisov, Kh. P.

Moskva: Nauka (ISBN 5-02-033711-0/hbk). 255 p. (2005).

The book is devoted to the description of different integral representations for holomorphic functions of several complex variables. Besides, statements of certain boundary value problems for holomorphic functions in \({\mathbb C}^2\) are given and their solutions are obtained in term of integral representations. The book consists of an Introduction, a main part divided into 7 Chapters and a list of references (115 items mainly by Russian authors).
In the short Introduction a brief history of integral representations in several complex variables is presented as well as the organization of the book. Chapter 1 “Integral representations of special type for analytic functions” describes mainly known representations valid in domains of special type, namely, Cauchy, Bergmann-Weyl, Martinelli-Bochner, Lerey (or Cauchy-Fantapier), and Temlyakov integral representations. Solution of special type of boundary value problems are presented in Chapter 2 “Methods of solution of certain two-dimensional boundary value problems”. The main attention is paid to the generalization of the classical \({\mathbb C}\)-linear conjugation (or Riemann) problems and Hilbert (or Riemann-Hilbert) problems. An attempt to state certain analogs to the problems of mathematical physics in \({\mathbb C}^2\) is made in Chapter 3. These are oblique derivative problems, Dirichlet and Neumann problems.
The question of analytic continuation is discussed in Chapter 4 “Analytic continuation of functions represented by integrals”. The remaining short Chapters contain the results on the following topics: “Behavior of certain classes of functions with unbounded definitive domains in \({\mathbb C}^2\)”, “Differential properties of certain classes of functions in \({\mathbb C}^2\)”, “Two-dimensional boundary value problem of Riemann-type for the bi-disc”.
Part of the material of the book is already known. Its machinery is based on certain generalizations of Temlyakov-type integral representations.
Remark. The book contains a lot of misprints (and mistakes) and cannot be recommended as a good reading on this subject.

Reviewer: Sergei V. Rogosin (Minsk)

MSC:

32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32A26	Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32A40	Boundary behavior of holomorphic functions of several complex variables

Keywords:

functions of several complex variables; integral representations; boundary value problems; analytic continuation