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.
Author(s): Kenzo Adachi
Publisher: World Scientific Publishing Company
Year: 2007
Language: English
Pages: 378
Contents......Page 11
Preface......Page 9
1.1 The Hartogs Theorem......Page 13
1.2 Characterizations of Pseudoconvexity.......Page 31
2.1 The Weighted L2 Space......Page 59
2.2 L2 Estimates in PseudoconvexDomains......Page 65
2.3 The Ohsawa-Takegoshi Extension Theorem......Page 112
3.1 The Homotopy Formula......Page 129
3.2 Holder Estimates for the P Problem......Page 155
3.3 Bounded and Continuous Extensions......Page 164
3.4 Hp and Ck Extensions......Page 195
3.5 The BergmanKernel......Page 208
3.6 Fefferman’sMapping Theorem......Page 222
4.1 The Berndtsson-Andersson Formula......Page 257
4.2 Lp Estimates for the Problem......Page 267
4.3 The Berndtsson Formula......Page 276
4.4 Counterexamples for Lp (p > 2) Extensions......Page 282
4.5 Bounded Extensions by Means of the Berndtsson Formula......Page 293
5.1 The Poincare Theorem......Page 303
5.2 The Weierstrass Preparation Theorem......Page 310
5.3 Oka’s Fundamental Theorem......Page 319
5.4 The Cousin Problem......Page 338
Appendix A Compact Operators......Page 343
Appendix B Solutions to the Exercises......Page 355
Bibliography......Page 371
Index......Page 377
