---
title: Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings
url: https://www.emergentmind.com/papers/0609327
type: paper
arxiv_id: '0609327'
arxiv_url: https://arxiv.org/abs/0609327
published: '2006-09-12'
categories:
- math.CV
- math.AP
---

# Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

## Abstract

The classical Painlev\'e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $\sigma$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity properties of pull-backs of Hausdorff measures under $K$-quasiconformal mappings, in particular at the relevant dimensions 1 and $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$, $0 < t < 2$, we reduce the absolute continuity properties to an open question on conformal mappings.