Modern Birkhï½user Classics

By: Lars HÃ¶rmander

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Paperback | 27 December 2006

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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, leading up to Trepreau's theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category.

At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodifferential calculus is required, but these sections can be bypassed with no loss of continuity. The major part of the book should therefore be 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.

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Preface	p. iii
Contents	p. v
Convex functions of one variable	p. 1
Definitions and basic facts	p. 1
Some basic inequalities	p. 9
Conjugate convex functions (Legendre transforms)	p. 16
The [Gamma] function and a difference equation	p. 20
Integral representation of convex functions	p. 23
Semi-convex and quasi-convex functions	p. 26
Convexity of the minimum of a one parameter family of functions	p. 28
Convexity in a finite-dimensional vector space	p. 36
Definitions and basic facts	p. 36
The Legendre transformation	p. 66
Geometric inequalities	p. 75
Smoothness of convex sets	p. 94
Projective convexity	p. 98
Convexity in Fourier analysis	p. 111
Subharmonic functions	p. 116
Harmonic functions	p. 116
Basic facts on subharmonic functions	p. 141
Harmonic majorants and the Riesz representation formula	p. 171
Exceptional sets	p. 203
Plurisubharmonic functions	p. 225
Basic facts	p. 225
Existence theorems in L[superscript 2] spaces with weights	p. 248
Lelong numbers of plurisubharmonic functions	p. 265
Closed positive currents	p. 271
Exceptional sets	p. 285
Other convexity conditions	p. 290
Analytic functionals	p. 300
Convexity with respect to a linear group	p. 315
Smooth functions in the whole space	p. 315
General G subharmonic functions	p. 324
Convexity with respect to differential operators	p. 328
P-convexity	p. 328
An existence theorem in pseudoconvex domains	p. 332
Analytic differential equations	p. 344
Convexity and condition ([psi])	p. 353
Local analytic solvability for [Partial]/[Partial]z[subscript 1]	p. 353
Generalities on projections and distance functions, and a theorem of Trepreau	p. 372
The symplectic point of view	p. 375
The microlocal transformation theory	p. 382
Appendix	p. 391
Polynomials and multlinear forms	p. 391
Commutator identities	p. 396
Notes	p. 403
References	p. 407
Index of notation	p. 411
Index	p. 413
Table of Contents provided by Ingram. All Rights Reserved.

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