Quantitative Sobolev regularity of quasiregular maps

Abstract

We quantify the Sobolev space norm of the Beltrami resolvent $(I-\mu S{)}^{-1}$, where $S$ is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation $\mu$ in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in $W^{1,p}$, $p\ge 2$. Our proof strategy is then adapted to yield quantitative estimates for the resolvent $(I-\mu S_{\mathrm{\Omega }}{)}^{-1}$ of the Beltrami equation on a sufficiently regular domain $\mathrm{\Omega }$, with $\mu \in W^{1,p}(\mathrm{\Omega })$. Here, $S_{\mathrm{\Omega }}$ is the compression of $S$ to a domain $\mathrm{\Omega }$. Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities.