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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: March 29, 2012
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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}$.


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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: http://dx.doi.org/10.1090/S0002-9939-2012-11688-0
PII: S 0002-9939(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.