Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 32324, 18 pages
doi:10.1155/2007/32324
Research Article

Hölder Quasicontinuity in Variable Exponent Sobolev Spaces

Petteri Harjulehto,1 Juha Kinnunen,2 and Katja Tuhkanen2

 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

Recommended by H. Bevan Thompson

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.