Une classe de domaines pseudoconvexes. (A class of pseudoconvex domains) (Q1090813)

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.]