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

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 H\"older-Zygmund estimates $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/2}$ simultaneously for all $s\in\mathbb R$ and $1<p<\infty$. In particular this provides the existence and $\frac12$ estimate for solution operator on Sobolev space of negative index these domains. The construction uses a new decomposition for the commutator $[\overline\partial,\mathcal E]$.