Quantitative Sobolev regularity of quasiregular maps

Francesco Di Plinio¹ ; A. Walton Green² ; Brett D. Wick²

¹Università di Napoli, Dipartimento di Matematica e Applicazioni
²Washington University in Saint Louis, Department of Mathematics

Annales Fennici Mathematici, Tome 50 (2025) no. 1, p. 3–28

Résumé

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 $(I-\mu S_\Omega)^{-1}$ of the Beltrami equation on a sufficiently regular domain $\Omega$, with $\mu\in W^{1,p}(\Omega)$. Here, $S_\Omega$ is the compression of $S$ to a domain $\Omega$. 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.

Détail

DOI : 10.54330/afm.155498

Keywords: Beltrami equation, quasiregular, quasiconformal, Sobolev regularity, compression of singular integrals, T1-theorems, weighted bounds, Beurling–Ahlfors transform