Separately analytic functions, generalizations of Hartogs' theorem, and envelopes of holomorphy

V. P. Zaharyuta

Sbornik. Mathematics, Tome 30 (1976) no. 1, pp. 51-67

Résumé

Let $\mathscr D$ and $\mathcal G$ be arbitrary Stein manifolds, $E\subset\mathscr D$ and $F\subset\mathscr G$ compact sets, and $X=(E\times\mathscr G)\cup(\mathscr D\times F)$. Under certain general hypotheses it is proved that a function $f$ on $X$ which is separately analytic, i.e. for which $f(z,w)$ is analytic in $z$ in $\mathscr D$ for any fixed $w\in F$ and analytic in $w$ in $\mathscr G$ for any fixed $z\in E$, extends to an analytic function in some open neighborhood $\widetilde X$ of $X$ which is the envelope of holomorphy of $X$. The envelope of holomorphy of $X$ is studied in those cases in which $X$ has no open envelope of holomorphy. Bibliography: 26 titles.