2006/09/12 by Astala, Kari, Clop, Albert, Mateu, Joan +2
#30C62 #35J15 #35J70 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0609327
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, dim(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 \cal Ht, 0 < t < 2, we reduce the absolute continuity properties to an open question on conformal mappings.