Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings
| dc.creator | Astala, Kari | |
| dc.creator | Clop, Albert | |
| dc.creator | Mateu, Joan | |
| dc.creator | Orobitg, Joan | |
| dc.creator | Uriarte-Tuero, Ignacio | |
| dc.date | 2006-09-12 | |
| dc.date.accessioned | 2026-07-07T07:24:45Z | |
| dc.date.available | 2026-07-07T07:24:45Z | |
| dc.description | 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 $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $σ$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ 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 $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$, $0 < t < 2$, we reduce the absolute continuity properties to an open question on conformal mappings. | |
| dc.description | 31 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0609327 | |
| dc.identifier | http://arxiv.org/abs/math/0609327 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/116484 | |
| dc.subject | Complex Variables | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 30C62, 35J15, 35J70 | |
| dc.title | Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings | |
| dc.type | text |