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Analysis of several complex variables. Transl. from the Japanese by Shu Gilbert Nakamura. (English) Zbl 1005.32001

Iwanami Series in Modern Mathematics. Translations of Mathematical Monographs. 211. Providence, RI: American Mathematical Society (AMS). xvii, 121 p. (2002).
In this concise booklet, the author gives an account for the \(L^2\) theory of the Cauchy-Riemann equations called the \(\overline\partial\) equations. Emphasis is put on recent results, including an \(L^2\) extension theorem for holomorphic functions, that have brought a deeper understanding of pseudoconvexity and plurisubharmonic functions. Based on Oka’s theorems and his schema for the grouping of problems, topics covered in the book are at the joint of the theory of analytic functions of several variables and mathematical analysis. The author gives a lucid presentation of how the methods originating in real analysis produce a variety of global existence theorems in the theory of functions based on the characterization of holomorphic functions as weak solutions of the \(\overline\partial\) equations.
Ch. 1, Holomorphic functions, starts with the definition and elementary properties of holomorphic functions, and in Ch. 2, Rings of holomorphic functions and \(\overline\partial\) cohomology, the problem of extension of functions and the division problem are converted to the problem of solving the \(\overline\partial\) equations of inhomogeneous form. The theme to observe up to Ch. 3, Pseudoconvexity and plurisubharmonic functions, is that the solvability of the \(\overline\partial\) equation on an open set \(\Omega \in\mathbb C^n\) imposes on \(\Omega\) a geometric restriction called pseudoconvexity. Ch. 4, \(L^2\) estimates and existence theorems, shows, to the contrary, the solvability of the \(\overline\partial\) equation on a pseudoconvex open set; and, as an application, the author generalizes to several variables the Mittag-Leffler theorem, Weierstrass’ product theorem, and the Runge approximation theorem, which are included in many textbooks for complex analysis in one variable. This approach is called the method of \(L^2\) estimates. By virtue of this method, in Ch. 5, Solutions of the extension and division problems, are given. The point of this argument is that the solutions are evaluated by the estimates, and thus the application immediately becomes wider. In Ch. 6, Bergman kernels, the reproducing kernel of the space \(A^2(\Omega)\) is explained. The definitions and basic facts are given, and the boundary holomorphy theorem on biholomorphic mappings between strongly pseudoconvex domains with boundaries of class \(C^\infty\) is proved. This theorem was obtained by Fefferman, but the proof introduced here is the one due to Bell and Ligocka’s idea, a skillful use of the transformation law of Bergman kernels. Next, a few results on the boundary behavior of Bergman kernels are explained. Most of the results included here are restricted to elementary cases.
The book would make a fine supplementary text for a graduate-level course on complex analysis.

MSC:

32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
32A10 Holomorphic functions of several complex variables
32B05 Analytic algebras and generalizations, preparation theorems
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32U05 Plurisubharmonic functions and generalizations
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