Sharp Pointwise Contraction of Mappings with Integrable Distortion Which Are Quasiconformal in a Disk

TL;DR

This paper investigates quasiconformal mappings with integrable distortion in a disk, establishing a sharp bound for the inverse's modulus of continuity and analyzing their multifractal spectra.

Contribution

It provides the first sharp bounds for the inverse modulus of continuity and explores the compression multifractal spectra of these mappings.

Findings

Established a sharp bound for the inverse mapping's modulus of continuity.

Derived bounds for the compression multifractal spectra.

Demonstrated the sharpness of the continuity bound.

Abstract

We consider quasiconformal mappings of the unit disk that have a planar extension which have $p$-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this bound. Furthermore, we also obtain bounds for the compression multifractal spectra of such mappings.