On the grothendieck duality for the space of holomorphic Sobolev functions

Arkadii B. Levskii, Alexander A. Shlapunov

Siberian Federal University, Krasnoyarsk, Russian Federation

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–Köthe–Sebastião e Silva duality for the space of holomorphic functions.

Keywords: duality theorems, holomorphic functions of finite order of growth.

UDC: 517.53

Received: 10.03.2024
Received in revised form: 02.04.2024
Accepted: 10.05.2024

Language: English