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Title: Variations on Hartogs and Henkin-Tumanov Theorems
Authors: Mascolo, Raffaella
Keywords: Hartogs TheoremCR FunctionsSeparate Analyticity
Issue Date: 2007
Publisher: EUT Edizioni Università di Trieste
Source: Raffaella Mascolo, "Variations on Hartogs and Henkin-Tumanov Theorems”, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 39 (2007), pp. 395–406.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
39 (2007)
There are equivalent characterizations for holomorphic
functions defined on open sets of $\mathbb C^n$; first of all, they can be represented locally as sums of convergent power series. It is obvious
that a holomorphic function of several complex variables is separately holomorphic in each variable. Just separating variables, a
lot of the well-known properties of holomorphic functions of one
complex variable, as the integral Cauchy formula, have a corresponding version in several complex variables; for separation of
variables, we need the function to be continuous. Surprisingly, a
function which is separately holomorphic, is indeed C^0 and even
C^1 and therefore holomorphic (Hartogs Theorem, 1906).
This short note deals with the problem of separate analyticity
and extends the discussion to the case of separately CR functions
defined on CR manifolds. We present our result of [5] and explain
how it is related to the former literature. In particular, we explain
its link with former results by Henkin and Tumanov of 1983 and
by Hanges and Treves of 1983.
Type: Article
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.39 (2007)

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