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15 February 2008 Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings
K. Astala, A. Clop, J. Mateu, J. Orobitg, I. Uriarte-Tuero
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Duke Math. J. 141(3): 539-571 (15 February 2008). DOI: 10.1215/00127094-2007-005

Abstract

The classical Painlevé theorem tells us 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 2/(K+1). We show that when K>1, unexpectedly one has improved removability. More precisely, we prove that sets E of σ-finite Hausdorff (2/(K+1))-measure are removable for bounded K-quasiregular mappings. On the other hand, dim(E)=2/(K+1) is not enough to guarantee this property.

We also study absolute continuity properties of pullbacks of Hausdorff measures under K-quasiconformal mappings: in particular, at the relevant dimensions 1 and 2/(K+1). For general Hausdorff measures Ht, 0<t<2, we reduce the absolute continuity properties to an open question on conformal mappings (see Conjecture 2.3)

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K. Astala. A. Clop. J. Mateu. J. Orobitg. I. Uriarte-Tuero. "Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings." Duke Math. J. 141 (3) 539 - 571, 15 February 2008. https://doi.org/10.1215/00127094-2007-005

Information

Published: 15 February 2008
First available in Project Euclid: 15 February 2008

zbMATH: 1140.30009
MathSciNet: MR2387431
Digital Object Identifier: 10.1215/00127094-2007-005

Subjects:
Primary: 30C62
Secondary: 35J15, 35J70

Rights: Copyright © 2008 Duke University Press

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Vol.141 • No. 3 • 15 February 2008
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