In mathematics, Malgrange–Zerner theorem (named for Bernard Malgrange and Martin Zerner) shows that a function on
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[1][2] Let and let
convex hull of
be a locally bounded function such that
and that for any fixed point
the function
is holomorphic in
in the interior of
Then the function
can be uniquely extended to a function holomorphic in the interior of
According to Henry Epstein,[1][3] this theorem was proved first by Malgrange in 1961 (unpublished), then by Zerner [4] (as cited in [1]), and communicated to him privately.
Epstein's lectures [1] contain the first published proof (attributed there to Broz, Epstein and Glaser).
The assumption
was later relaxed to
(see Ref.
[1] in [2]) and finally to