Dual realizations of Bergman spaces on strongly convex domains

TL;DR

This paper explores the relationship between Bergman spaces and analytic functionals on strongly convex domains in higher dimensions, extending planar results and highlighting limitations in generalizing to all convex domains.

Contribution

It extends duality results of Fantappi extquoteright e and Laplace transforms from planar to higher-dimensional strongly convex domains, providing new insights and counterexamples.

Findings

Established duality results for strongly convex domains in higher dimensions

Provided examples showing planar results do not generalize to all convex domains

Extended the understanding of transforms linking analytic functionals and holomorphic functions

Abstract

The Fantappi\`e and Laplace transforms realize isomorphisms between analytic functionals supported on a convex compact set $K\subset{\mathbb C}^n$ and certain spaces of holomorphic functions associated with $K$. Viewing the Bergman space of a bounded domain in ${\mathbb C}^n$ as a subspace of the space of analytic functionals supported on its closure, the images of the restrictions of these transforms have been studied in the planar setting. For the Fantappi\`e transform, this was done for simply connected domains (Napalkov Jr--Yulumukhamtov, 1995), and for the Laplace transform, this was done for convex domains (Napalkov Jr--Yulumukhamtov, 2004). In this paper, we study this problem in higher dimensions for strongly convex domains, and establish duality results analogous to the planar case. We also produce examples to show that the planar results cannot be generalized to all convex domains in higher dimensions.