Sobolev regularity of the canonical solutions to $\bar{\partial }$ on product domains

Abstract

Let $\Omega$ be a product domain in ${\mathbb{C}}^n,n\ge 2$, where each slice has smooth boundary. We observe that the canonical solution operator for the $\bar{\partial }$ equation on $\Omega$ is bounded in $W^{k,p}(\Omega )$, $k\in {\mathbb{Z}}^{+},1<p<\infty$. This Sobolev regularity is sharp in view of Kerzman-type examples.

7 pages