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 -quasiregular mappings in planar domains, the corresponding critical dimension is . We show that when , unexpectedly one has improved removability. More precisely, we prove that sets of -finite Hausdorff -measure are removable for bounded -quasiregular mappings. On the other hand, is not enough to guarantee this property.
We also study absolute continuity properties of pullbacks of Hausdorff measures under -quasiconformal mappings: in particular, at the relevant dimensions and . For general Hausdorff measures , , we reduce the absolute continuity properties to an open question on conformal mappings (see Conjecture 2.3)
Citation
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