Acta Mathematica

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

Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero

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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.


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane 35J15: Second-order elliptic equations 35J70: Degenerate elliptic equations

Quasiconformal Hausdorff measure Removability

2010 © Institut Mittag-Leffler


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.

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