Synopses & Reviews
The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.
Synopsis
The term convexity used to describe these lectures given at the Univer sity of Lund in 1991-92 should be understood in a wide sense. Only Chap ters I and II are devoted to convex sets and functions in the traditional sense of convexity. The following chapters study other kinds of convexity which occur in analysis. Most prominent is the pseudo-convexity (plurisubh- monicity) in the theory of functions of several complex variables discussed in Chapter IV. It relies on the theory of subharmonic functions in R DEGREES, so Chapter III is devoted to subharmonic functions in R" DEGREES for any n. Existence theorems for constant coefficient partial differential operators in R' DEGREES are re lated to various kinds of convexity conditions, depending on the operator. Chapter VI gives a survey of the rather incomplete results which are known on their geometrical meaning. There are also natural classes of "convex" functions related to subgroups of the linear group, which specialize to sev eral of the notions already mentioned. They are discussed in Chapter V. The last chapter. Chapter VII, is devoted to the conditions for solvability of microdifferential equations, which can also be considered as a branch of convexity theory. The whole chapter is an exposition of a part of the thesis of J.-
Synopsis
The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.
Synopsis
This book first details convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed, leading up to Trépreau's theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category. The major part of the book is accessible to graduate students so that it can serve as an introduction to complex analysis in one and several variables. The last sections, however, are written mainly for readers familiar with microlocal analysis.
Table of Contents
Preface.- Convex Functions of one Variable.- Convexity in A Finite-Dimensional Vector Space.- Subharmonic Functions.- Plurisubharmonic Functions.- Convexity with Respect to A Linear Group.-Convexity with Respect to Differential Operators.- Convexity and Condition.- Appendix.- Notes.- References.- Index of Notation.- Index.