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.