Several complex variables and integral formulas

Kenzo Adachi9789812705747, 9812705740

This volume is an introductory text in several complex variables, using methods of integral representations and Hilbert space theory. It investigates mainly the studies of the estimate of solutions of the Cauchy Riemann equations in pseudoconvex domains and the extension of holomorphic functions in submanifolds of pseudoconvex domains which were developed in the last 50 years. We discuss the two main studies mentioned above by two different methods: the integral formulas and the Hilbert space techniques. The theorems concerning general pseudoconvex domains are analyzed using Hilbert space theory, and the proofs for theorems concerning strictly pseudoconvex domains are solved using integral representations. This volume is written in a self-contained style, so that the proofs are easily accessible to beginners. There are exercises featured at the end of each chapter to aid readers to better understand the materials of this volume. Fairly detailed hints are articulated to solve these exercises.

---

Table of contents :
Contents……Page 10
Preface……Page 8
1.1 The Hartogs Theorem……Page 12
1.2 Characterizations of Pseudoconvexity…….Page 30
2.1 The Weighted L2 Space……Page 58
2.2 L2 Estimates in PseudoconvexDomains……Page 64
2.3 The Ohsawa-Takegoshi Extension Theorem……Page 111
3.1 The Homotopy Formula……Page 128
3.2 Holder Estimates for the P Problem……Page 154
3.3 Bounded and Continuous Extensions……Page 163
3.4 Hp and Ck Extensions……Page 194
3.5 The BergmanKernel……Page 207
3.6 Fefferman’sMapping Theorem……Page 221
4.1 The Berndtsson-Andersson Formula……Page 256
4.2 Lp Estimates for the Problem……Page 266
4.3 The Berndtsson Formula……Page 275
4.4 Counterexamples for Lp (p > 2) Extensions……Page 281
4.5 Bounded Extensions by Means of the Berndtsson Formula……Page 292
5.1 The Poincare Theorem……Page 302
5.2 The Weierstrass Preparation Theorem……Page 309
5.3 Oka’s Fundamental Theorem……Page 318
5.4 The Cousin Problem……Page 337
Appendix A Compact Operators……Page 342
Appendix B Solutions to the Exercises……Page 354
Bibliography……Page 370
Index……Page 376