Lectures On Counterexamples In Several Complex Variables
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Lectures on Counterexamples in Several Complex Variables
Counterexamples are remarkably effective for understanding the meaning, and the limitations, of mathematical results. Fornaess and Stensones look at some of the major ideas of several complex variables by considering counterexamples to what might seem like reasonable variations or generalizations. The first part of the book reviews some of the basics of the theory, in a self-contained introduction to several complex variables. The counterexamples cover a variety of important topics: the Levi problem, plurisubharmonic functions, Monge-Ampere equations, CR geometry, function theory, and the $\bar\partial$ equation. The book would be an excellent supplement to a graduate course on several complex variables.
Analysis of Several Complex Variables
This monograph describes real analysis approaches to the study of functions of several complex variables, and describes how these methods produce global existence theorems in the theory of functions. The book brings particular attention to recent results with implications for the understanding of pseudoconvexity and plurisubharmonic functions. Based on Oka's theorems and his schema for the grouping of problems, the discussion integrates the theory of analytic functions of several variables and mathematical analysis. Annotation copyrighted by Book News, Inc., Portland, OR.
Partial Differential Equations in Several Complex Variables
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, thenext 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 globalregularity 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 L2spaces. 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.