## Acta Mathematica

### Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane

#### Abstract

Let $E \subset \mathbb{C}$ be a compact set, $g:\mathbb{C} \to \mathbb{C}$ be a K-quasiconformal map, and let 0 < t < 2. Let ${\mathcal{H}^t}$ denote t-dimensional Hausdorff measure. Then ${\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}.$

This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in [2] and answers in the positive a conjecture of K. Astala in op. cit.

#### Note

M.T. Lacey was supported in part by a grant from the NSF.

E. T. Sawyer was supported in part by a grant from the NSERC.

#### Article information

Source
Acta Math., Volume 204, Number 2 (2010), 273-292.

Dates
Received: 11 June 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892470

Digital Object Identifier
doi:10.1007/s11511-010-0048-5

Mathematical Reviews number (MathSciNet)
MR2653055

Zentralblatt MATH identifier
1211.30036

Rights
2010 © Institut Mittag-Leffler

#### Citation

Lacey, Michael T.; Sawyer, Eric T.; Uriarte-Tuero, Ignacio. Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane. Acta Math. 204 (2010), no. 2, 273--292. doi:10.1007/s11511-010-0048-5. https://projecteuclid.org/euclid.acta/1485892470

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