On the Grothendieck Duality for the Space of Holomorphic Sobolev Functions

We describe the strong dual space (O-s(D))* for the space O-s(D) = H-s(D) boolean AND O(D) of holomorphic functions from the Sobolev space H-s(D), s is an element of Z, over a bounded simply connected plane domain D with infinitely differential boundary 9D . We identify the dual space with the space of holomorhic functions on C-n\D that belong to H1-s(G\(sic)) for any bounded domain G, containing the compact D, and vanish at the infinity. As a corollary, we obtain a description of the strong dual space (OF(D))* for the space OF(D) of holomorphic functions of finite order of growth in D (here, OF(D) is endowed with the inductive limit topology with respect to the family of spaces O-s(D), s is an element of Z). In this way we extend the classical Grothendieck-Kothe-Sebastiao e Silva duality for the space of holomorphic functions.