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

Michael T. Lacey School of Mathematics, Georgia Institute of Technology Eric T. Sawyer Department of Mathematics & Statistics, McMaster University Ignacio Uriarte-Tuero Mathematics Department, University of Missouri

TBD mathscidoc:1701.332015

Acta Mathematica, 204, (2), 273-292, 2008.6
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 $$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $$
Quasiconformal; Hausdorff measure; Removability
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@inproceedings{michael2008astala’s,
  title={Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane},
  author={Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203355038737724},
  booktitle={Acta Mathematica},
  volume={204},
  number={2},
  pages={273-292},
  year={2008},
}
Michael T. Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero. Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane. 2008. Vol. 204. In Acta Mathematica. pp.273-292. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203355038737724.
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