Duke Mathematical Journal

Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

K. Astala, A. Clop, J. Mateu, J. Orobitg, and I. Uriarte-Tuero

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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)

Article information

Source
Duke Math. J., Volume 141, Number 3 (2008), 539-571.

Dates
First available in Project Euclid: 15 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1203087637

Digital Object Identifier
doi:10.1215/00127094-2007-005

Mathematical Reviews number (MathSciNet)
MR2387431

Zentralblatt MATH identifier
1140.30009

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

Citation

Astala, K.; Clop, A.; Mateu, J.; Orobitg, J.; Uriarte-Tuero, I. Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings. Duke Math. J. 141 (2008), no. 3, 539--571. doi:10.1215/00127094-2007-005. https://projecteuclid.org/euclid.dmj/1203087637


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