Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 32324, 18 pages
Hölder Quasicontinuity in Variable Exponent Sobolev Spaces
1Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2b), Helsinki 00014, Finland
2Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland
Received 28 May 2006; Revised 6 November 2006; Accepted 25 December 2006
Academic Editor: H. Bevan Thompson
Copyright © 2007 Petteri 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
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.