Finely quasiconformal mappings

TL;DR

This paper introduces a relaxed, more natural definition of quasiconformality suitable for low-regularity mappings and demonstrates that these 'finely quasiconformal' mappings are indeed quasiconformal, with implications for Sobolev regularity.

Contribution

It proposes a new relaxed metric definition of quasiconformality applicable to low-regularity mappings and proves their equivalence to classical quasiconformal mappings in the plane.

Findings

The relaxed definition applies to mappings in $W_{ ext{loc}}^{1,1}$.

Finely quasiconformal mappings are quasiconformal in the plane.

These mappings exhibit Sobolev regularity.

Abstract

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}(\mathbb{R}^n;\mathbb{R}^n)$-mappings. Then we show on the plane that this relaxed definition can be used to prove Sobolev regularity, and that these ``finely quasiconformal'' mappings are in fact quasiconformal.