Une classe de domaines pseudoconvexes. (A class of pseudoconvex domains)
type: Item
instance of: scholarly article
title: Une classe de domaines pseudoconvexes. (A class of pseudoconvex domains)
zbMATH Open document ID: 0622.32016
DOI: 10.1215/S0012-7094-87-05516-5
author: Nessim Sibony
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: In this interesting paper, the author calls a compact set K in \({\mathbb{C}}^ n\) B-regular if every continuous real function on K is the uniform limit on K of continuous plurisubharmonic functions in neighbourhoods of K. The case of interest is \(K=\partial D\), D a bounded pseudoconvex domain in \({\mathbb{C}}^ n\) with differentiable boundary. It is shown that if \(\partial D\) is B-regular, then D behaves in many respects like a strictly pseudoconvex domain, although the set of points of weak pseudo-convexity can have positive measure in \(\partial D\), so that D need not be of finite type. For example, it is shown that, if \(\partial D\) is \(C^{\infty}\), then \(\partial D\) is B-regular if and only if it satisfies the condition (P) of \textit{D. Catlin} [Proc. Symp. Pure Math. 41, 39-49 (1984; Zbl 0578.32031)]. It is also proved that if \(\partial D\) is of class \(C^ 3\) and B-regular, then D is uniformly H-convex in the sense of Cirka, so that in particular \(\bar D\) has a basis of Stein neighbourhoods; all known examples of pseudo-convex domains with this property seem to be covered by this result. Some approximation theorems are also proved. [Remark. Theorem 4.3 is stated for B-regular compact sets with empty interior, though it seems obvious from the definition that B-regularity implies empty interior.]
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32U05
zbMATH DE Number: 4008809
zbMATH Keywords: \({\bar \partial }\)-Neumann problem
zbMATH Keywords: global regularity
zbMATH Keywords: pseudoconvex domain
zbMATH Keywords: approximation
reviewed by: R. R. Simha
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4008809