Compactness of Hankel operators with continuous symbols

TL;DR

This paper characterizes the compactness of Hankel operators with continuous symbols on convex Reinhardt domains in terms of the holomorphicity of the symbol along boundary analytic discs.

Contribution

It provides a precise criterion linking the compactness of Hankel operators to the boundary behavior of the symbol on convex Reinhardt domains.

Findings

Hankel operator $H_{}$ is compact iff $$ is holomorphic along all boundary analytic discs.

The result applies specifically to bounded convex Reinhardt domains in $C^2$.

Establishes a boundary regularity condition for compactness of Hankel operators.

Abstract

Let $\Omega$ be a bounded convex Reinhardt domain in $\mathbb{C}^2$ and $\phi\in C(\bar{\Omega})$. We show that the Hankel operator $H_{\phi}$ is compact if and only if $\phi$ is holomorphic along every non-trivial analytic disc in the boundary of $\Omega$.