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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10 Citations (Scopus)


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 α-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).
Original languageEnglish
Pages (from-to)539-571
JournalDuke Mathematical Journal
Publication statusPublished - 15 Feb 2008


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