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Open Access Research Article

Hölder Quasicontinuity in Variable Exponent Sobolev Spaces

Petteri Harjulehto1*, Juha Kinnunen2 and Katja Tuhkanen2

Author Affiliations

1 Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2b), Helsinki 00014, Finland

2 Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland

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Journal of Inequalities and Applications 2007, 2007:032324  doi:10.1155/2007/32324

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2007/1/032324

Received:28 May 2006
Revisions received:6 November 2006
Accepted:25 December 2006
Published:14 February 2007

© 2007 Harjulehto et al.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We show that a function in the variable exponent Sobolev spaces coincides with a Hölder continuous Sobolev function outside a small exceptional set. This gives us a method to approximate a Sobolev function with Hölder continuous functions in the Sobolev norm. Our argument is based on a Whitney-type extension and maximal function estimates. The size of the exceptional set is estimated in terms of Lebesgue measure and a capacity. In these estimates, we use the fractional maximal function as a test function for the capacity.


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