Variations on Hartogs and Henkin-Tumanov Theorems

Abstract

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.