Compactness of Hankel operators with continuous symbols on convex domains

Abstract

Let $\Omega$ be a bounded convex domain in ${\mathbb{C}}^n$, $n\ge 2$, $1\le q\le (n-1)$, and $\varphi \in C(\bar{\Omega })$. If the Hankel operator $H_{\varphi }^{q-1}$ on $(0,q-1)$--forms with symbol $\varphi$ is compact, then $\varphi$ 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, $1\le q\le n$, and $\varphi \in C(\bar{\Omega })$ is analytic along the ones of dimension $q$ (or higher), then $H_{\varphi }^{q-1}$ is compact.

minor changes, to appear in Houston J. Math