Bochner's tube theorem

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Short description: Theorem about holomorphic functions of several complex variables

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

Theorem Let ωn be a connected open set. Then every function f(z) holomorphic on the tube domain Ω=ω+in can be extended to a function holomorphic on the convex hull ch(Ω).

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 ω be a connected submanifold of n of class-C2. Then every continuous CR function on the tube domain Ω(ω) can be continuously extended to a CR function on Ω(ach(ω)). (Ω(ω)=ω+inn (n2),ach(ω):=ωInt ch(ω)). By "Int ch(S)" we will mean the interior taken in the smallest dimensional space which contains "ch(S)".

References

  1. Bochner, S.; Martin, W.T. (1948). Several Complex Variables. Princeton mathematical series. Princeton University Press. ISBN 978-0-598-34865-4. 
  2. Hounie, J. (2009). "A Proof of Bochner's Tube Theorem". Proceedings of the American Mathematical Society (American Mathematical Society) 137 (12): 4203–4207. doi:10.1090/S0002-9939-09-10057-6. http://www.jstor.org/stable/40590656. Retrieved 2021-06-30. 
  3. Noguchi, Junjiro (2020). "A brief proof of Bochner's tube theorem and a generalized tube". arXiv:2007.04597 [math.CV].
  4. Kazlow, M. (1979). "CR functions and tube manifolds". Transactions of the American Mathematical Society 255: 153. doi:10.1090/S0002-9947-1979-0542875-5. 
  5. Boivin, André; Dwilewicz, Roman (1998). "Extension and Approximation of CR Functions on Tube Manifolds". Transactions of the American Mathematical Society 350 (5): 1945–1956. doi:10.1090/S0002-9947-98-02019-4. .