Beltrami equation with coefficient in Sobolev and Besov spaces

TL;DR

This paper investigates the regularity of quasiregular solutions to the Beltrami equation when the coefficient belongs to specific Sobolev and Besov spaces, establishing conditions for derivatives to also lie in these spaces.

Contribution

It introduces new function spaces on the complex plane where solutions' derivatives inherit the regularity of the Beltrami coefficient.

Findings

Solutions have derivatives in the same function space as the coefficient.

Provides conditions linking the Beltrami coefficient's space to the solution's regularity.

Extends regularity results for quasiregular solutions in Sobolev and Besov spaces.

Abstract

Our goal in this work is to present some function spaces on the complex plane $\C$, $X(\C)$, for which the quasiregular solutions of the Beltrami equation, $\bar\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\C)$.