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)
"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