On the Grothendieck duality for the space of holomorphic Sobolev functions

We describe the strong dual space ( Oˢ (D))^* for the space Oˢ (D) = Hˢ (D) ∩ O (D) of holomorphic functions from the Sobolev space Hˢ(D), s ∈ Z, over a bounded simply connected plane domain D with infinitely differential boundary ∂ D. We identify the dual space with the space of holomorhic functions on Cⁿ D that belong to H¹⁻ˢ (G D) 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 Oˢ (D), s ∈ Z). In this way we extend the classical Grothendieck-K{\"o}the-Sebasti\~{a}o e Silva duality for the space of holomorphic functions.