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

Abstract

We construct homotopy formulae $f=\overline{\partial }{\mathcal{H}}_{q}f+{\mathcal{H}}_{q+1}\overline{\partial }f$ on a bounded domain which is either $C^2$ strongly pseudoconvex or $C^{1,1}$ strongly $\mathbb{C}$-linearly convex. Such operators exhibit 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 $1<p<\infty$. In particular this provides the existence and $\frac{1}{2}$ estimate for solution operator on Sobolev space of negative index these domains. The construction uses a new decomposition for the commutator $[\overline{\partial },\mathcal{E}]$.

32 pages, including 8 pages of appendix