Pointwise Inequalities in Variable Sobolev Spaces and Applications

Abstract

Pointwise estimates for variable exponent Sobolev functions are derived to obtain several results on Sobolev spaces with variable exponent. Hypersingular operators acting in these spaces are considered and the corresponding boundedness and pointwise statements are given over bounded open sets with Lipschitz boundary. Moreover, classical Sobolev embeddings into Hölder spaces are generalized to the variable exponent setting.