On $\bar{\mathrm{\partial }}$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates

We construct homotopy formulae $f=\bar{\mathrm{\partial }}{\mathcal{H}}_{q}f+{\mathcal{H}}_{q+1}\bar{\mathrm{\partial }}f$ for $(0,q)$ forms on the product domain ${\mathrm{\Omega }}_1\times \cdots \times {\mathrm{\Omega }}_m$, where each ${\mathrm{\Omega }}_j$ 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 $\bar{\mathrm{\partial }}$ equation with optimal Sobolev regularity $W^{k,p}\to W^{k,p}$ simultaneously for all $k\in \mathbb{Z}$ and $1<p<\mathrm{\infty }$.