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Softcover ISBN:  9780821829615 
Product Code:  AMSIP/19.S 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $54.40 
eBook ISBN:  9781470438104 
Product Code:  AMSIP/19.E 
List Price:  $63.00 
MAA Member Price:  $56.70 
AMS Member Price:  $50.40 
Softcover ISBN:  9780821829615 
eBook ISBN:  9781470438104 
Product Code:  AMSIP/19.S.B 
List Price:  $131.00 $99.50 
MAA Member Price:  $117.90 $89.55 
AMS Member Price:  $104.80 $79.60 

Book DetailsAMS/IP Studies in Advanced MathematicsVolume: 19; 2001; 380 ppMSC: Primary 32; Secondary 35
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 CauchyRiemann and tangential CauchyRiemann operators. This book gives an uptodate 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 CauchyRiemann equations using Hilbert space techniques. The authors provide a systematic study of the CauchyRiemann 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 CauchyRiemann equations. Chapter 7 introduces the tangential CauchyRiemann 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 Hölder and \(L^p\) spaces. Embeddability of abstract \(CR\) structures is discussed in detail in the last chapter.
This selfcontained book provides a muchneeded introductory text to several complex variables and partial differential equations. It is also a rich source of information to experts.
Titles in this series are copublished with International Press of Boston, Inc., Cambridge, MA.
ReadershipGraduate students and research mathematicians interested in several complex variables and PDEs.

Table of Contents

Chapters

Real and complex manifolds

The Cauchy integral formula and its applications

Holomorphic extension and pseudoconvexity

$L^2$ theory for $\overline \partial $ on pseudoconvex domains

The $\overline \partial $Neumann problem on strongly pseudoconvex manifolds

Boundary regularity for $\overline \partial $ on pseudoconvex domains

CauchyRiemann manifolds and the tangential CauchyRiemann complex

Subelliptic estimates for second order differential equations and $\square _b$

The tangential CauchyRiemann complex on pseudoconvex $CR$ manifolds

Fundamental solutions for $\square _b$ on the Heisenberg group

Integral representations for $\overline \partial $ and $\overline \partial _b$

Embeddability of abstract $CR$ structures


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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 CauchyRiemann and tangential CauchyRiemann operators. This book gives an uptodate 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 CauchyRiemann equations using Hilbert space techniques. The authors provide a systematic study of the CauchyRiemann 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 CauchyRiemann equations. Chapter 7 introduces the tangential CauchyRiemann 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 Hölder and \(L^p\) spaces. Embeddability of abstract \(CR\) structures is discussed in detail in the last chapter.
This selfcontained book provides a muchneeded introductory text to several complex variables and partial differential equations. It is also a rich source of information to experts.
Titles in this series are copublished with International Press of Boston, Inc., Cambridge, MA.
Graduate students and research mathematicians interested in several complex variables and PDEs.

Chapters

Real and complex manifolds

The Cauchy integral formula and its applications

Holomorphic extension and pseudoconvexity

$L^2$ theory for $\overline \partial $ on pseudoconvex domains

The $\overline \partial $Neumann problem on strongly pseudoconvex manifolds

Boundary regularity for $\overline \partial $ on pseudoconvex domains

CauchyRiemann manifolds and the tangential CauchyRiemann complex

Subelliptic estimates for second order differential equations and $\square _b$

The tangential CauchyRiemann complex on pseudoconvex $CR$ manifolds

Fundamental solutions for $\square _b$ on the Heisenberg group

Integral representations for $\overline \partial $ and $\overline \partial _b$

Embeddability of abstract $CR$ structures

Anyone planning to do research in this area will want to have a copy of the book.
Mathematical Reviews