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Bochner's tube theorem

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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 C n {\displaystyle \mathbb {C} ^{n}} can be extended to the convex hull of this domain.

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

A classic reference is (Theorem 9). See also for other proofs.

Generalizations

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

Theorem Let ω {\displaystyle \omega } be a connected submanifold of R n {\displaystyle \mathbb {R} ^{n}} of class- C 2 {\displaystyle C^{2}} . Then every continuous CR function on the tube domain Ω ( ω ) {\displaystyle \Omega (\omega )} can be continuously extended to a CR function on Ω ( ach ( ω ) ) .   ( Ω ( ω ) = ω + i R n ⊂ C n   ( n ≥ 2 ) , ach ( ω ) := ω ∪ Int   ch ( ω ) ) {\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)".

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. 137 (12). American Mathematical Society: 4203–4207. doi:10.1090/S0002-9939-09-10057-6. JSTOR 40590656.
  3. Noguchi, Junjiro (2020). "A brief proof of Bochner's tube theorem and a generalized tube". arXiv:2007.04597 .
  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. JSTOR 117646..
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