Logarithmic Holder continuous mappings and Beltrami equation

TL;DR

This paper investigates mappings satisfying the inverse Poletsky inequality, establishing their logarithmic Hölder continuity at inner points and demonstrating the existence of continuous ACL solutions to the Beltrami equation with this regularity.

Contribution

It introduces new regularity results for mappings under the inverse Poletsky inequality and proves the existence of continuous solutions to the Beltrami equation with logarithmic Hölder continuity.

Findings

Mappings are logarithmic Hölder continuous at inner points.

Existence of continuous ACL solutions to the Beltrami equation with this regularity.

Majorant integrability on spheres influences local behavior of mappings.

Abstract

The paper is devoted to the study of mappings satisfying the inverse Poletsky inequality. We study the local behavior of these mappings, moreover, we are most interested in the case when the corresponding majorant is integrable on some set of spheres of positive linear measure. The most important result is a logarithmic H\"{o}lder continuity of such mappings at inner points. As a corollary, we obtained the existence of a continuous $ACL$-solution of the Beltrami equation, which is logarithmic H\"{o}lder continuous.