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

Abstract

We construct homotopy formulae $f=\overline{\partial }{\mathcal{H}}_{q}f+{\mathcal{H}}_{q+1}\overline{\partial }f$ for $(0,q)$ forms on the product domain ${\Omega }_1\times \cdots \times {\Omega }_m$, 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 $1<p<\infty$.

16 pages, including 4 pages of appendix