Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

Original · EN

The classical Painlevé theorem tells 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, (E) = 2/K+1 is not enough to guarantee this property. We also study absolute continuity properties of pull-backs 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.