We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{ q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $\Omega_1\times\dots\times\Omega_m$, where each $\Omega_j$ is either a bounded Lipschitz domain in $\mathbb C^1$, a bounded strongly pseudoconvex domain with $C^2$ boundary, or a smooth convex domain of finite type. Such homotopy operators $\mathcal H_q$ yield solutions to the $\overline\partial$ equation with optimal Sobolev regularity $W^{ k,p}\to W^{ k,p}$ simultaneously for all $k\in\mathbb Z$ and $1<p<\infty$.
Categories: math.CV