TY - JOUR

T1 - Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

AU - Astala, K.

AU - Clop, A.

AU - Mateu, J.

AU - Orobitg, J.

AU - Uriarte-Tuero, I.

PY - 2008/2/15

Y1 - 2008/2/15

N2 - 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).

AB - 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).

U2 - https://doi.org/10.1215/00127094-2007-005

DO - https://doi.org/10.1215/00127094-2007-005

M3 - Article

VL - 141

SP - 539

EP - 571

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

ER -