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

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.