Some extensions of Cartan–Behnke–Stein's theorem

Oka Q15] proved that any domain of holomorphy in C is a Cousin-I domain, that is, a domain in which any additive Cousin's distribution has a solution. Conversely Cartan pT] had stated that any Cousin-I domain in C is a domain of holomorphy and Behnke-Stein £2] gave its proof. From Cartan £6] C{(0, 0, 0)} is a Cousin-I domain which is not a domain of holomorphy and from Thullen [18] C-{(0, 0)} is not a domain of holomorphy but a Cousin-II domain, that is, a domain in which any multiple Cousin's distribution has a solution. These two facts suggest that Cartan-Behnke-Stein's theorem can not be generalized for Cousin-I domain in C(^^3) and for Cousin-II domain in C"(»^2) directly. The main purpose of the present paper is to extend Cartan-Behnke-Stein's theorem in the following form: Let L be an abelian complex Lie group and 31^ be the sheaf of all germs of holomorphic mappings in L. A domain (D, qf) over C with H^D, 2Ii)=0 is a domain of holomorphy. If D is a domain in C with continuous boundary such that TV-^DftP, ^L) = 0 for any simply connected and relatively compact polycylinder P in C, D is a domain of holomorphy. Moreover these two results hold also for L = GL(p, C).