A boundary cross theorem for separately holomorphic functions
Volume 84 / 2004
Annales Polonici Mathematici 84 (2004), 237-271
MSC: Primary 32D15, 32D10.
DOI: 10.4064/ap84-3-6
Abstract
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.
