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

Let $\mathrm{\Omega }$ 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 $\bar{\mathrm{\partial }}$ equation on $\mathrm{\Omega }$ is bounded in $W^{k,p}(\mathrm{\Omega })$, $k\in {\mathbb{Z}}^{+},1<p<\mathrm{\infty }$. This Sobolev regularity is sharp in view of Kerzman-type examples.