Compactness of Hankel operators with continuous symbols on convex domains

TL;DR

This paper characterizes the compactness of Hankel operators with continuous symbols on convex domains in complex spaces, linking it to the holomorphicity of symbols along boundary varieties and providing partial converses under certain boundary conditions.

Contribution

It establishes a criterion for the compactness of Hankel operators based on boundary holomorphicity and proves a partial converse for domains with finitely many boundary varieties.

Findings

Hankel operator compactness implies boundary holomorphicity of symbols.

Symbols holomorphic along boundary varieties lead to compact Hankel operators under certain conditions.

Partial converse holds for domains with finitely many boundary varieties.

Abstract

Let $\Omega$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $\phi\in C(\bar{\Omega})$. If the Hankel operator $H^{q-1}_{\phi}$ on $(0,q-1)$--forms with symbol $\phi$ is compact, then $\phi$ is holomorphic along $q$--dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, $1\leq q\leq n$, and $\phi\in C(\bar{\Omega})$ is analytic along the ones of dimension $q$ (or higher), then $H^{q-1}_{\phi}$ is compact.