On the Grothendieck duality for the space of holomorphic Sobolev functions

Abstract

We describe the strong dual space $({\mathcal{O}}^s(D){)}^{\ast }$ for the space ${\mathcal{O}}^s(D)=H^s(D)\cap \mathcal{O}(D)$ of holomorphic functions from the Sobolev space $H^s(D)$, $s\in \mathbb{Z}$, over a bounded simply connected plane domain $D$ with infinitely differential boundary $\partial D$. We identify the dual space with the space of holomorhic functions on ${\mathbb{C}}^n\setminus \overline{D}$ that belong to $H^{1-s}(G\setminus \overline{D})$ for any bounded domain $G$, containing the compact $\overline{D}$, and vanish at the infinity. As a corollary, we obtain a description of the strong dual space $({\mathcal{O}}_F(D){)}^{\ast }$ for the space ${\mathcal{O}}_F(D)$ of holomorphic functions of finite order of growth in $D$ (here, ${\mathcal{O}}_F(D)$ is endowed with the inductive limit topology with respect to the family of spaces ${\mathcal{O}}^s(D)$, $s\in \mathbb{Z}$). In this way we extend the classical Grothendieck-K{ö}the-Sebastião e Silva duality for the space of holomorphic functions.