On Some Properties of Relative Capacity and Thinness in Weighted Variable Exponent Sobolev Spaces.

We define the weighted relative p(.)-capacity and discuss its properties in the space W ϑ 1 , p (.) ( R n) . Also, we investigate some properties of the weighted variable Sobolev capacity. It is shown that there is a relation between these two capacities. Moreover, we introduce the notion of thinness related to this newly defined relative capacity and prove an equivalence statement for this thinness. [ABSTRACT FROM AUTHOR]