  We quantify the Sobolev space norm of the Beltrami resolvent $(I- \mu
\mathcal S)^{-1}$, where $\mathcal S$ is the Beurling-Ahlfors transform, in
terms of the corresponding Sobolev space norm of the dilatation $\mu$ in the
critical and supercrticial 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 $(I-\mu \mathcal
S_\Omega)^{-1}$ of the Beltrami equation on a sufficiently regular domain
$\Omega$, with $\mu\in W^{1,p}(\Omega)$. Here, $\mathcal S_\Omega$ is the
compression of $\mathcal S$ 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
Calder\'on-Zygmund operators to domains, recently obtained by the authors, to
extend the Astala-Iwaniec-Saksman technique to higher regularities.
