Pseudoconvex classes of functions. I: Pseudoconcave and pseudoconvex sets
type: Item
instance of: scholarly article
title: Pseudoconvex classes of functions. I: Pseudoconcave and pseudoconvex sets
zbMATH Open document ID: 0693.31006
author: Zbigniew Slodkowski
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: This is a joint review for parts I-III. Natural classes of subharmonic, plurisubharmonic, convex and q- plurisubharmonic functions have a number of common properties which can be reduced to several basic ones. Taking these as axioms, the author defines in Part I the general notion of a pseudoconvex class of functions. The main axioms deal with localization properties and preservation of the class by operations like taking limits of decreasing sequences and maximum of several functions. One of the difficulties is that pseudoconvex classes are usually not closed with respect to addition; however, the set of function whose addition preserves the class is assumed to separate the points of the base space. As an example of a natural result obtained in this context we mention the existence and uniqueness theorem for the Bremermann-Dirichlet problem, with solutions in a pseudoconvex class P. A new and crucial tool used here (and throughout the papers) is the notion of the dual pseudoconvex class \(P^ d\) which consists of functions f such that \(f+g\) satisfies the local maximum principle for all \(g\in P.\) Aside for duality, Part I introduces P-pseudoconvex set and P-maximum sets. The latter are relatively closed sets on which functions of class P satisfy the local maximum principle. Then P-pseudoconvex set (which are analogs of the usual pseudoconvex sets) are defined by a form of Kontinuitätssatz, with P-maximum sets replacing analytic discs. One of the motivations of this study of pseudoconvex classes comes from the applications to complex interpolation theory given by the author [Trans. Am. Math. Soc. 308, No.2, 685-711 (1988; Zbl 0657.46057)]. The connection is based on the fact that the normed spaces \(\{({\mathbb{C}}^ n,\| \cdot \|_ z):\) \(z\in G\}\), \(G\subset {\mathbb{C}}^ k\), form an interpolation family relative to a given pseudoconvex class P if and only if the set \(\{(z,w)\in G\times {\mathbb{C}}^ n:\) \(\| w\|_ z<1\}\) is P-pseudoconvex, and its closure …
Mathematics Subject Classification ID: 31D05
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 46M35
zbMATH DE Number: 4134423
zbMATH Keywords: Bremermann-Dirichlet problem
zbMATH Keywords: maximum principle
zbMATH Keywords: P-pseudoconvex set
zbMATH Keywords: P- maximum sets
zbMATH Keywords: complex interpolation theory
zbMATH Keywords: Lie group
zbMATH Keywords: dual classes
MaRDI profile type: Publication
full work available at URL: pjm.1988.134.343
OpenAlex ID: W2017209191
DOI: 10.2140/PJM.1988.134.343
description: scientific article; zbMATH DE number 4134423