Astala's Conjecture on Distortion of Hausdorff Measures under Quasiconformal Maps in the Plane

TL;DR

This paper refines inequalities on how quasiconformal maps distort Hausdorff measures in the plane, confirming Astala's conjecture that relates measure zero sets under such maps.

Contribution

It provides a precise formula for the distortion of Hausdorff measures under quasiconformal maps, confirming a conjecture by Astala.

Findings

Established a formula for measure distortion under quasiconformal maps.

Confirmed Astala's conjecture on measure zero set distortion.

Refined previous inequalities on Hausdorff dimension distortion.

Abstract

Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then H^t (E) = 0 implies H^{t'} (g E) = 0, for t'=[2Kt]/[2+(K-1)t]. This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in his celebrated paper on area distortion for quasiconformal maps and answers in the positive a Conjecture of K. Astala in op. cit.