Several Complex Variables and Integral Formulas

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.

Sample Chapter(s)
Chapter 1: Pseudoconvexity and Plurisubharmonicity (465 KB)

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• Pseudoconvexity and Plurisubharmonicity
• The Bar {Partial} Problem in Pseudoconvex Domains
• Integral Formulas for Strictly Pseudoconvex Domains
• Integral Formulas with Weight Factors
• The Classical Theory in Several Complex Variables
• Compact Operators