We quantify the Sobolev space norm of the Beltrami resolvent $(I- \mu \mathcal{ B})^{ -1}$, where $\mathcal B$ 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 $W^{ 1,p}$, $p \geq 2$. Our proof strategy is then adapted to yield quantitative estimates for the resolvent $(I-\mu { \mathcal B}_\Omega)^{ -1}$ of the Beltrami equation on a sufficiently regular domain $\Omega$, with $\mu\in W^{ 1,p}(\Omega)$. Here, ${ \mathcal B}_\Omega$ is the compression of ${ \mathcal B}$ 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 Calder\'on-Zygmund operators to domains, recently obtained by the authors, to extend the Astala-Iwaniec-Saksman technique to higher regularities.
Categories: math.AP math.CA math.CV