On the Grothendieck duality for the space of holomorphic Sobolev functions

TL;DR

This paper characterizes the dual space of holomorphic Sobolev functions on a domain, extending classical duality results by identifying duals with certain boundary and exterior holomorphic functions with Sobolev regularity.

Contribution

It provides a new description of the dual space for holomorphic Sobolev functions and extends classical duality results to functions of finite growth order.

Findings

Dual space characterized by boundary holomorphic functions with Sobolev regularity.

Extension of Grothendieck-K{"o}the-Sebasti ilde{a}o e Silva duality.

Description of duals for functions of finite order of growth.

Abstract

We describe the strong dual space $({\mathcal O}^s (D))^*$ 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 $({\mathcal O}_F (D))^*$ 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{\"o}the-Sebasti\~{a}o e Silva duality for the space of holomorphic functions.