A boundary cross theorem for separately holomorphic functions

Peter Pflug¹ ; Viêt-Anh Nguyên²

¹ Fachbereich Mathematik Carl von Ossietzky Universität Oldenburg Postfach 2503 D-26111 Oldenburg, Germany
² Fachbereich Mathematik Carl von Ossietzky Universität Oldenburg Postfach 2503, D–26111 Oldenburg, Germany

Annales Polonici Mathematici, Tome 84 (2004) no. 3, pp. 237-271.

Résumé

Let $D\subset \mathbb C^n$ and $G\subset \mathbb C^m$ be pseudoconvex domains, let $A$ (resp. $B$) be an open subset of the boundary $\partial D$ (resp. $\partial G$) and let $X$ be the $2$-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that the domain $D$ (resp. $G$) is locally $\mathcal{C}^2$ smooth on $A$ (resp. $B$). We shall determine the “envelope of holomorphy" $\widehat{X}$ of $X$ in the sense that any function continuous on $X$ and separately holomorphic on $(A\times G) \cup (D\times B)$ extends to a function continuous on $\widehat{X}$ and holomorphic on the interior of $\widehat{X}.$ A generalization of this result to $N$-fold crosses is also given.

Détail

DOI : 10.4064/ap84-3-6

Keywords: subset mathbb subset mathbb pseudoconvex domains resp subset boundary partial resp partial fold cross cup times cup times cup suppose addition domain resp locally mathcal smooth resp shall determine envelope holomorphy widehat sense function continuous separately holomorphic times cup times extends function continuous widehat holomorphic interior widehat generalization result n fold crosses given