A note on quasiconformal maps with Holder-continuous dilatation

TL;DR

This paper provides an alternative proof for a condition ensuring quasiconformal maps with smooth domain support are bi-Lipschitz, and extends the result to domains with corners, enhancing understanding of geometric regularity.

Contribution

It offers a new proof of a known condition for bi-Lipschitz quasiconformal maps and extends the theorem to include domains with corners, broadening its applicability.

Findings

Alternative proof of the sufficient condition for bi-Lipschitz quasiconformal maps.

Extension of the theorem to domains with corners.

Broader understanding of boundary regularity effects on quasiconformal maps.

Abstract

Here we give an alternate proof of a sufficient condition due to J. Mateu, J. Orobitg, and J. Verdera for a quasiconformal map of the plane with dilatation supported in a smooth domain to be bi-Lipschitz. We also extend this theorem to cover boundaries with certain types of corners.