On mappings of finite distortion that are quasiconformal in the unit disk

TL;DR

This paper investigates quasiconformal mappings of the unit disk with controlled distortion, establishing bounds on inverse map continuity, improving bounds for generalized quasidisks, and analyzing regularity near cusp singularities.

Contribution

It provides new bounds for inverse map continuity, improves three point property bounds for generalized quasidisks, and determines optimal regularity near cusp singularities.

Findings

Bound on the modulus of continuity of the inverse map.

Improved bounds for the three point property of generalized quasidisks.

Optimal regularity results for maps with cusp singularities.

Abstract

We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.