Variations on Hartogs and Henkin-Tumanov Theorems

There are equivalent characterizations for holomorphic \nfunctions 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 \nthat a holomorphic function of several complex variables is separately holomorphic in each variable. Just separating variables, a \nlot of the well-known properties of holomorphic functions of one \ncomplex variable, as the integral Cauchy formula, have a corresponding version in several complex variables; for separation of \nvariables, we need the function to be continuous. Surprisingly, a \nfunction which is separately holomorphic, is indeed C^0 and even \nC^1 and therefore holomorphic (Hartogs Theorem, 1906). \nThis short note deals with the problem of separate analyticity \nand extends the discussion to the case of separately CR functions \ndefined on CR manifolds. We present our result of [5] and explain \nhow it is related to the former literature. In particular, we explain \nits link with former results by Henkin and Tumanov of 1983 and \nby Hanges and Treves of 1983.