Weighted estimates for Beltrami equations

TL;DR

This paper establishes weighted a priori estimates for generalized Beltrami equations with VMO coefficients and applies these results to enhance the regularity of Jacobians in quasiconformal mappings.

Contribution

It introduces new weighted estimates for Beltrami equations with VMO coefficients, advancing understanding of regularity in quasiconformal mappings.

Findings

Weighted estimates in L^p(ω) for Beltrami equations.

Improved regularity results for Jacobians of quasiconformal maps.

Applicability to coefficients in VMO with ellipticity and weights in A_p.

Abstract

We obtain a priori estimates in $L^p(\omega)$ for the generalized Beltrami equation, provided that the coefficients are compactly supported $VMO$ functions with the expected ellipticity condition, and the weight $\omega$ lies in the Muckenhoupt class $A_p$. As an application, we obtain improved regularity for the jacobian of certain quasiconformal mappings.