Hölder quasicontinuity in variable exponent sobolev spaces

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.