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Volume 19

# Partial Differential Equations in Several Complex Variables

Original Publication: 2001

Publisher: American Mathematical Society / International Press

## Description

This book is intended both as an introductory text and as a reference book for those interested in studying several complex variables in the context of partial differential equations. In the last few decades, significant progress has been made in the fields of Cauchy-Riemann and tangential Cauchy-Riemann operators. This book gives an up-to-date account of the theories for these equations and their applications. The background material in several complex variables is developed in the first three chapters, leading to the Levi problem. The next three chapters are devoted to the solvability and regularity of the Cauchy-Riemann equations using Hilbert space techniques.The authors provide a systematic study of the Cauchy-Riemann equations and the $\bar\partial$-Neumann problem, including $L^2$ existence theorems on pseudoconvex domains, $\frac 12$-subelliptic estimates for the $\bar\partial$-Neumann problems on strongly pseudoconvex domains, global regularity of $\bar\partial$ on more general pseudoconvex domains, boundary regularity of biholomorphic mappings, irregularity of the Bergman projection on worm domains. The second part of the book gives a comprehensive study of the tangential Cauchy-Riemann equations. Chapter 7 introduces the tangential Cauchy-Riemann complex and the Lewy equation. An extensive account of the $L^2$ theory for $\square_b$ and $\bar\partial_b$ is given in Chapters 8 and 9. Explicit integral solution representations are constructed both on the Heisenberg groups and on strictly convex boundaries with estimates in Ho$lder and$L^p$spaces.Embeddability of abstract$CR\$ structures is discussed in detail in the last chapter. This self-contained book provides a much-needed introductory text to several complex variables and partial differential equations. It is also a rich source of information to experts.

## Publications

 Pub. Date ISBN-13 ISBN-10 Medium Binding Size, Etc. Status List Price 2015 Jan 9780821888230 0821888234 Print Out of Print 2015 Jan 9780821829615 0821829610 Print paperback Out of Print 2001 9780821810620 0821810626 Print Out of Print