A Hartogs type theorem for separately Cr functions

作者：Marigo Raffaele^*

来源：Israel Journal of Mathematics, 2010, 178(1): 107-112.

DOI：10.1007/s11856-010-0059-0

The main step in the proof of Hartogs' theorem on separate analyticity (see [3], [4], [5]) consists in showing that if a function f defined in Delta x Delta is holomorphic for vertical bar z(2)vertical bar < epsilon and separately holomorphic in z(2) when z(1) is kept fixed, then it is jointly holomorphic; the normal convergence of the Taylor series of f is obtained through the celebrated Hartogs' lemma on subharmonic functions.
This result has been generalized in various directions and following different approaches: recently, in [2], E. M. Chirka proved that it is still true if the function is separately analytic along a general smooth foliation of complex curves instead of coordinate lines.
In our work we consider the case where f is separately holomorphic along the complex lines, issued from a real curve gamma, which foliate a real hypersurface M: if f is holomorphic in a neighborhood of gamma, then it is holomorphic in a neighborhood of M. This generalization of Hartogs' lemma also offers a geometric interpretation of a theorem by J. Siciak ([6]) about separate real analyticity (see also L. Baracco and G. Zampieri, [1]).

• 出版日期2010-9