Weighted Variable Sobolev Spaces and Capacity

Abstract

We define weighted variable Sobolev capacity and discuss properties of capacity in the space 𝑊1,𝑝(⋅)(ℝ𝑛,𝑤). We investigate the role of capacity in the pointwise definition of functions in this space if the Hardy-Littlewood maximal operator is bounded on the space 𝑊1,𝑝(⋅)(ℝ𝑛,𝑤). Also it is shown the relation between the Sobolev capacity and Bessel capacity.