Ensembles pics pour \(A^{\infty}(D)\) non globalement inclus dans une variété integrale
type: Item
instance of: scholarly article
title: Ensembles pics pour \(A^{\infty}(D)\) non globalement inclus dans une variété integrale
zbMATH Open document ID: 0574.32023
DOI: 10.1007/BF01450680
author: Jacques Chaumat
author: Anne-Marie Chollet
published in: Mathematische Annalen
publication date: 1982-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163593
review text: The authors consider the following situation: Let \(D\subset {\mathbb{C}}^ n\) be a bounded strictly pseudoconvex domain with \(C^{\infty}\)-smooth boundary and let \(A^{\infty}(D)\) be the algebra of functions holomorphic in D and \(C^{\infty}\)-smooth up to bD. A subset \(E\subset bD\) is said to be a peak set for \(A^{\infty}\) if there exists \(f\in A^{\infty}(D)\) such that \(f\equiv 1\) on E and \(| f| <1\) on \(\bar D\setminus E.\) Among other results, in previous papers the authors proved the following theorem: A closed subset \(E\subset bD\) is a peak set for \(A^{\infty}(D)\) if and only if E is locally contained in a submanifold N of bD such that for every \(p\in N\), \(T_ pN\subset T_ p^{{\mathbb{C}}}bD\). (Such an N can be called an integral submanifold for the contact structure on bD.) In this paper the authors construct examples of peak sets for \(A^{\infty}(D)\) which are not globally contained in an integral submanifold for the contact structure of bD.
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32E35
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 3916714
zbMATH Keywords: peak points
zbMATH Keywords: algebras of holomorphic functions
zbMATH Keywords: peak set
zbMATH Keywords: integral submanifold for the contact structure
MaRDI profile type: Publication
cites work: Extending functions from submanifolds of the boundary
cites work: Boundary behavior of holomorphic functions on pseudoconvex domains
cites work: Ensembles pics pour \(A^\infty(D)\)
cites work: Caractérisation et propriétés des ensembles localement pics de \(A^\infty(D)\)
cites work: Ensembles pics dans des domaines strictement pseudo-convexes
cites work: Holomorphic approximation and hyperfunction theory on a C\(^1\) totally real submanifold of a complex manifold
cites work: Zero sets of non-negative strictly plurisubharmonic functions
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description: scientific article; zbMATH DE number 3916714