One of the approaches to the study of functions of several complex variables is to use methods originating in real analysis. In this concise book, the author gives a lucid presentation of how these methods produce a variety of global existence theorems in the theory of functions (based on the characterization of holomorphic functions as weak solutions of the Cauchy-Riemann equations). Emphasis is 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, the book covers topics at the intersection of the theory of analytic functions of several variables and mathematical analysis. It is assumed that the reader has a basic knowledge of complex analysis at the undergraduate level. The book would make a fine supplementary text for a graduate-level course on complex analysis.
Author(s): Takeo Ohsawa
Publisher: American Mathematical Society
Year: 2002
Language: English
Pages: 144
Front Cover......Page 1
Title Page......Page 6
Copyright......Page 7
Contents ......Page 8
Preface ......Page 10
Preface to the English Edition ......Page 12
Summary and Prospects of the Theory ......Page 14
1.1. Definitions and Elementary Properties ......Page 20
1.2. Cauchy-Riemann Equations ......Page 27
1.3. Reinhardt Domains ......Page 37
2.1. Spectra and the \bar{partial} Equation ......Page 42
2.2. Extension Problems and the \bar{partial} Equation ......Page 44
2.3. \bar{partial} Cohomology and Serre's Condition ......Page 46
3.1. Pseudoconvexity of Domains of Holomorphy ......Page 54
3.2. Regularization of Plurisubharmonic Functions ......Page 60
3.3. Levi Pseudoconvexity ......Page 66
4.1. L^2 Estimates and Vanishing of \bar{partial} Cohomology ......Page 74
4.2. Three Fundamental Theorems ......Page 94
5.1. Solutions of the Extension Problems ......Page 102
5.2. Solutions of Division Problems ......Page 106
5.3. Extension Theorem with Growth Rate Condition ......Page 112
5.4. Applications of the L^2 Extension Theorem ......Page 119
6.1. Definitions and Examples ......Page 124
6.2. Transformation Law and an Application holomorphic Mappings ......Page 126
6.3. Boundary Behavior of Bergman Kernels ......Page 129
Bibliography ......Page 134
Index ......Page 138
Back Cover......Page 144
