Open Access

Hölder Quasicontinuity in Variable Exponent Sobolev Spaces

Journal of Inequalities and Applications20072007:032324

DOI: 10.1155/2007/32324

Received: 28 May 2006

Accepted: 25 December 2006

Published: 14 February 2007

Abstract

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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Authors’ Affiliations

(1)
Department of Mathematics and Statistics, University of Helsinki
(2)
Department of Mathematical Sciences, University of Oulu

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Copyright

© Petteri Harjulehto et al. 2007

This article is published under license to BioMed Central Ltd. 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.