Malgrange–Zerner Theorem

In mathematics, Malgrange–Zerner theorem (named for Bernard Malgrange and Martin Zerner) shows that a function on $\mathbb^n$ allowing holomorphic extension in each variable separately can be extended, under certain conditions, to a function holomorphic in all variables jointly. This theorem can be seen as a generalization of Bochner's tube theorem to functions defined on tube-like domains whose base is not an open set.

Theorem Let

:$X=\bigcup_^n \mathbb^\times P \times \mathbb^, \textP=\mathbb+i [0,1),$
and let $W=$ convex hull of $X$. Let $f: X\to \mathbb$ be a locally bounded function such that $f \in C^\infty(X)$ and that for any fixed point $(x_1,\ldots, x_,x_,\ldots,x_n)\in \mathbb^$ the function $f(x_1,\ldots, x_,z,x_,\ldots,x_n)$ is holomorphic in $z$ in the interior of $P$ for each $k=1,\ldots,n$. Then the function $f$ can be uniquely extended to a function holomorphic in the interior of $W$.

History

According to Henry Epstein, this theorem was proved first by Malgrange in 1961 (unpublished), then by Zerner Zerner M. (1961), mimeographed notes of a seminar given in Marseilles (as cited in ), and commmunicated to him privately. Epstein's lectures contain the first published proof (attributed there to Broz, Epstein and Glaser). The assumption $f \in C^\infty(X)$ was later relaxed to $f, _\in C^3$ (see Ref. in ) and finally to $f, _\in C$.

References

{{DEFAULTSORT:Malgrange-Zerner theorem
Several complex variables