Quasiconformal distortion of Hausdorff measures

TL;DR

This paper investigates how planar K-quasiconformal maps distort Hausdorff measures, establishing a relation between measures of sets before and after the transformation with explicit bounds.

Contribution

It provides a new quantitative relationship describing how quasiconformal maps distort Hausdorff measures in the plane, extending previous understanding.

Findings

Sets of finite (t')-Hausdorff measure are mapped to sets of finite t-Hausdorff measure.

Derived an explicit inequality relating H^t(E) and H^{t'}(f(E)) with a constant depending on K.

Established a formula connecting t, t', and K for measure distortion.

Abstract

In this paper we prove that if f is a planar K-quasiconformal map and 0<t<2, t' = 2t/(2K-Kt+t), then f transforms sets of finite (t')-Hausdorff measure into sets of finite t-Hausdorff measure. We also prove the following more quantitative statement: If E is a planar set, then H^t(E) \leq C(K) H^{t'}(f(E))^{t/(t'K)}, where H^s stands for the s-Hausdorff measure.