Quantitative Sobolev regularity of quasiregular maps

Abstract

We quantify the Sobolev space norm of the Beltrami resolvent $(I-\hat{I}¼\mathcal{B}{)}^{-1}$, where $\mathcal{B}$ is the Beurling-Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation $\hat{I}¼$ 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 $(I-\hat{I}¼{\mathcal{B}}_{\hat{I}}\text{c◯}{)}^{-1}$ of the Beltrami equation on a sufficiently regular domain $\hat{I}\text{c◯}$, with $\hat{I}¼\in W^{1,p}(\hat{I}\text{c◯})$. Here, ${\mathcal{B}}_{\hat{I}}\text{c◯}$ is the compression of $\mathcal{B}$ to a domain $\hat{I}\text{c◯}$. 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.

25 pages. Final version to appear in Ann. Fenn. Math