Capacity and Hausdorff measure in Musielak-Orlicz-Sobolev spaces

Abstract

In this paper, we show that sets with zero Sobolev $p(\cdot)$-capacity have generalized Hausdorff $h(\cdot)$-measure zero, for some gauge function $h(\cdot).$ We also prove that sets with zero Musielak-Orlicz-Sobolev $\Phi(\cdot,\cdot)$-capacity, for a particular class of functions $\Phi(\cdot,\cdot),$ have generalized Hausdorff $h(\cdot)$-measure zero, for a suitable gauge function $h(\cdot).$