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A note to ``Mappings of finite distortion: formation of cusps II''


Authors: Pekka Koskela and Juhani Takkinen
Journal: Conform. Geom. Dyn. 14 (2010), 184-189
MSC (2010): Primary 30C62, 30C65
DOI: https://doi.org/10.1090/S1088-4173-2010-00211-0
Published electronically: July 15, 2010
MathSciNet review: 2670509
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Abstract: We consider planar homeomorphisms $ f\colon\mathbb{R}^2\to\mathbb{R}^2$ that are of finite distortion and map the unit disk onto a specific cusp domain $ \Omega_s$. We study the relation between the degree $ s$ of the cusp and the integrability of the distortion function $ K_f$ by sharpening a previous result where $ K_f$ is assumed to be locally exponentially integrable.


References [Enhancements On Off] (What's this?)

  • 1. L.V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301. MR 0154978 (27:4921)
  • 2. P. Haïssinsky, Chirurgie parabolique, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 2, 195-198. MR 1645124 (99i:58127)
  • 3. P. Koskela and J. Onninen, Mappings of finite distortion: Capacity and modulus inequalities, J. Reine Angew. Math. 599 (2006), 1-26. MR 2279096 (2007k:30035)
  • 4. P. Koskela and J. Takkinen, Mappings of finite distortion: formation of cusps, Publ. Mat. 51 (2007), no. 1, 223-242. MR 2307153 (2008e:30026)
  • 5. -, Mappings of finite distortion: Formation of cusps III, Acta. Math. Sin. (English Series) 26 (2010), no. 5, 817-824.
  • 6. O. Lehto and K.I. Virtanen, Quasiconformal mappings in the plane, second ed., Springer-Verlag, New York, 1973. MR 0344463 (49:9202)
  • 7. J. Onninen and X. Zhong, A note on mappings of finite distortion: the sharp modulus of continuity, Michigan Math. J. 53 (2005), no. 2, 329-335. MR 2152704 (2006c:30025)
  • 8. Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
  • 9. J. Takkinen, Mappings of finite distortion: Formation of cusps II, Conform. Geom. Dyn. 11 (2007), 207-218. MR 2354095 (2009f:30055)
  • 10. J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathematics, vol. 229, Springer-Verlag, New York, 1971. MR 0454009 (56:12260)

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Additional Information

Pekka Koskela
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland
Email: pekka.j.koskela@jyu.fi

Juhani Takkinen
Affiliation: Linnantie 8 C 21, 40800 Vaajakoski, Finland
Email: juhani.takkinen@kolumbus.fi

DOI: https://doi.org/10.1090/S1088-4173-2010-00211-0
Keywords: Cusp, homeomorphism, mapping of finite distortion
Received by editor(s): April 17, 2010
Published electronically: July 15, 2010
Additional Notes: The first author was partially supported by the Academy of Finland grants nos. 120927 and 131477
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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