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Domaines pseudoconvexes d’ordre général et fonctions pseudoconvexes d’ordre général. (Pseudoconvex domains and pseudoconvex functions of general order). (French) Zbl 0728.32009

An open subset \(D\subset {\mathbb{C}}^ n\) is called pseudoconvex of order k, \(0\leq k\leq n-1\), if its complement \({\mathbb{C}}^ n\setminus D\) has the same continuity as an analytic set of pure dimension k. An open set \(D\subset {\mathbb{C}}^ n\) is pseudoconvex in the usual sense iff it is pseudoconvex of order n-1. Also it is easy to see that a q-pseudoconvex open set \(D\subset {\mathbb{C}}^ n\) (in the sense of Andreotti and Grauert) is pseudoconvex of order n-q but in general the converse is not valid. Using the notion of subpluriharmonic function the author introduces also the definition of pseudoconvex function of order k.
The main result of the paper may be stated as follows:
If \(D\subset {\mathbb{C}}^ n\) is an open subset, then the following conditions are equivalent:
1) D is pseudoconvex of order k,
2) -log d is a pseudoconvex function of order k on D, where d denotes the Euclidean distance at the boundary of D,
3) there exists an exhaustion function on D which is pseudoconvex of order k.
This theorem generalizes a well-known result due to Oka for \(k=n-1\).

MSC:

32E10 Stein spaces
32F10 \(q\)-convexity, \(q\)-concavity
32T99 Pseudoconvex domains
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