Bochner's tube theorem

In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in ${\displaystyle \mathbb {C} ^{n}}$ can be extended to the convex hull of this domain.

Theorem Let ${\displaystyle \omega \subset \mathbb {R} ^{n}}$ be a connected open set. Then every function ${\displaystyle f(z)}$ holomorphic on the tube domain ${\displaystyle \Omega =\omega +i\mathbb {R} ^{n}}$ can be extended to a function holomorphic on the convex hull ${\displaystyle \operatorname {ch} (\Omega )}$.

A classic reference is ^[1] (Theorem 9). See also ^[2][3] for other proofs.

Generalizations

The generalized version of this theorem was first proved by Kazlow (1979),^[4] also proved by Boivin and Dwilewicz (1998)^[5] under more less complicated hypothese.

Theorem Let ${\displaystyle \omega }$ be a connected submanifold of ${\displaystyle \mathbb {R} ^{n}}$ of class-${\displaystyle C^{2}}$. Then every continuous CR function on the tube domain ${\displaystyle \Omega (\omega )}$ can be continuously extended to a CR function on ${\displaystyle \Omega ({\text{ach}}(\omega )).\ \left(\Omega (\omega )=\omega +i\mathbb {R} ^{n}\subset \mathbb {C} ^{n}\ \left(n\geq 2\right),{\text{ach}}(\omega ):=\omega \cup {\text{Int}}\ {\text{ch}}(\omega )\right)}$. By "Int ch(S)" we will mean the interior taken in the smallest dimensional space which contains "ch(S)".