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On distortion of Hausdorff measures under quasiconformal mappings
Author(s):
István
Prause
Journal:
Conform. Geom. Dyn.
11
(2007),
219-223.
MSC (2000):
Primary 30C62
Posted:
October 31, 2007
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Abstract:
Astala (Acta Math. 173 (1994), no. 1, 37-60) gave optimal bounds for the distortion of Hausdorff dimension under planar quasiconformal maps. The corresponding estimates on the level of Hausdorff measures remain open. We show that these techniques allow for establishing absolute continuity for some weaker Hausdorff measures.
References:
-
- 1.
- K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173 (1994), no. 1, 37-60. MR 1294669 (95m:30028b)
- 2.
- K. Astala and V. Nesi, Composites and quasiconformal mappings: new optimal bounds in two dimensions, Calc. Var. Partial Differential Equations, 18 (2003), no. 4, 335-355. MR 2020365 (2005d:74035)
- 3.
- K. Astala, A. Clop, J. Mateu, J. Orobitg and I. Uriarte-Tuero, Distortion of Hausdorff measures and improved Painlevé removability for bounded quasiregular mappings, Duke Math. J., to appear, preprint version: arXiv:math.CV/0609327.
- 4.
- C.J. Bishop, Distortion of disks by conformal maps, preprint:
http://www.math.sunysb.edu/~bishop/papers/papers.html - 5.
- I. Prause, Distortion of dimension under quasiconformal mappings, Preprint 462, Reports in Mathematics, University of Helsinki, 2007.
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Additional Information:
István
Prause
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland
Address at time of publication:
Institute of Mathematics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email:
istvan.prause@helsinki.fi
DOI:
10.1090/S1088-4173-07-00171-3
PII:
S 1088-4173(07)00171-3
Keywords:
Quasiconformal,
Hausdorff measure
Received by editor(s):
August 10, 2007
Posted:
October 31, 2007
Additional Notes:
The author was supported by the Finnish Academy of Science and Letters through Vilho, Yrjö ja Kalle Väisälän rahasto.
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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