Compactness of Hankel operators with symbols continuous on the closure of pseudoconvex domains

TL;DR

This paper investigates the conditions under which Hankel operators with continuous symbols on the closure of certain pseudoconvex domains are compact, linking this to the holomorphicity of compositions with boundary maps.

Contribution

It establishes a characterization of compact Hankel operators with continuous symbols on specific pseudoconvex domains in terms of boundary holomorphicity.

Findings

Hankel operator compactness implies boundary holomorphicity of symbol compositions

Results apply to Lipschitz boundary pseudoconvex domains and convex domains

Provides a new criterion for compactness in terms of boundary behavior

Abstract

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$.