A universal $\overline{\partial }$ solution operator on nonsmooth strongly pseudoconvex domains

Abstract

We construct a universal family of $\overline{\partial }$ homotopy operators on bounded $C^{1,1}$ domains that are locally strongly $\mathbb{C}$-linearly convexifiable. In particular this includes $C^2$ strongly pseudoconvex domains and $C^{1,1}$ strongly $\mathbb{C}$-linearly convex domains. The operators satisfy Sobolev estimates ${\mathcal{H}}_q:H^{s,p}\to H^{s+1/2,p}$ and Hölder-Zygmund estimates ${\mathcal{H}}_q:{\mathcal{C}}^s\to {\mathcal{C}}^{s+1/2}$ simultaneously for all $s\in \mathbb{R}$ and $1<p<\infty$. The construction uses a modification of the Leray-Koppelman formula and a refined directional decomposition of the commutator $[\overline{\partial },\mathcal{E}]$.

arXiv comment39 pages, including 12 pages of appendix. Improves the result to locally strongly $\mathbb{C}$-linearly convexifiable domains. Add Appendix B for a counterexample for Hardy estimates on p<1