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Linear dilatation and differentiability of homeomorphisms of $ \mathbb{R}^n$


Author: Bruce Hanson
Journal: Proc. Amer. Math. Soc. 140 (2012), 3541-3547
MSC (2010): Primary 30C65; Secondary 26B05
DOI: https://doi.org/10.1090/S0002-9939-2012-11688-0
Published electronically: March 29, 2012
MathSciNet review: 2929022
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Abstract: According to a classical result, if $ \Omega $ is a domain in $ \mathbb{R}^d$, where $ d>1$, $ f: \Omega \rightarrow \mathbb{R}^d$ is a homeomorphism and the lim-sup dilatation $ H_f$ of $ f$ is finite almost everywhere on $ \Omega $, then $ f$ is differentiable almost everywhere on $ \Omega $. We show that this theorem fails if $ H_f$ is replaced by the lim-inf dilatation $ h_f$. Our example demonstrates the sharpness of recent results of Kallunki and Koskela concerning the $ h_f$ function and also of Balogh and Csörnyei involving the lower-scaled oscillation of continuous functions $ f: \Omega \rightarrow \mathbb{R}$.


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

  • [BC] Balogh, Z., and M. Csörnyei, Scaled oscillation and regularity, Proc. Amer. Math. Soc. 134 9 (2006), 2667-2675. MR 2213746 (2007b:46052)
  • [G] Gehring, F.W., Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 0139735 (25:3166)
  • [HK] Heinonen, J., and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), 61-79. MR 1323982 (96e:30051)
  • [KK] Kallunki, S., and P. Koskela, Exceptional sets for the definition of quasiconformality, American Journal of Mathematics 122 (2000), 735-743. MR 1771571 (2001h:37095)
  • [KM] Kallunki, S., and O. Martio, ACL homeomorphisms and linear dilatation, Proc. Amer. Math. Soc. 130 (2002), 1073-1078. MR 1873781 (2002i:30022)
  • [V] Väisälä, J., Lectures on $ n$-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, No. 229, Springer-Verlag, New York, 1971. MR 0454009 (56:12260)

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Additional Information

Bruce Hanson
Affiliation: Department of Mathematics, Statistics and Computer Science, St. Olaf College, Northfield, Minnesota 55057
Email: hansonb@stolaf.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11688-0
Received by editor(s): April 12, 2011
Published electronically: March 29, 2012
Dedicated: In memory of Juha Heinonen
Communicated by: Mario Bonk
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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