Partial Differential Equations in Several Complex Variables: 19 (AMS/IP Studies in Pure Mathematics) - Softcover
Inhaltsangabe
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.
Die Inhaltsangabe kann sich auf eine andere Ausgabe dieses Titels beziehen.
Reseña del editor
This book is intended as both an introductory text and 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 study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables. After the background material in complex analysis is developed in Chapters 1 to 3, 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 Hórmander's L2 existence progress on the global regularity and irregularity of the \bar\partial-Neumann operators. The second part of the book gives a comprehensive study of the tangential Cauchy-Riemann equations, another important class of equations in several complex variables first studied by Lewy. An up-to-date account of the L2 theory for \bar\partial b operator is given. Explicit integral solution representations are constructed both on the Heisenberg groups and on strictly convex boundaries with estimates in Hölder and L2 spaces. Embeddability of abstract CR structures is discussed in detail here for the first time. Titles in this series are co-published with International Press, Cambridge, MA.
„Über diesen Titel“ kann sich auf eine andere Ausgabe dieses Titels beziehen.
