Localization of compactness of Hankel operators on pseudoconvex domains

Abstract

We prove the following localization for compactness of Hankel operators on Bergman spaces. Assume that $\Omega $ is a bounded pseudoconvex domain in $\mathbb{C} ^{n}$, $p$ is a boundary point of $\Omega $, and $B(p,r)$ is a ball centered at $p$ with radius $r$ so that $U=\Omega \cap B(p,r)$ is connected. We show that if the Hankel operator $H^{\Omega }_{\phi}$ with symbol $\phi\in C^{1}(\overline{\Omega } )$ is compact on $A^{2}(\Omega )$ then $H^{U}_{R_{U}(\phi)}$ is compact on $A^{2}(U)$ where $R_{U}$ denotes the restriction operator on $U$, and $A^{2}(\Omega )$ and $A^{2}(U)$ denote the Bergman spaces on $\Omega $ and $U$, respectively.