Hölder quasicontinuity of Sobolev functions on metric spaces

Abstract

We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Maly [Ma1] to the Sobolev spaces on metric spaces [H1].