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 Wϑ1,p(.)(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_\vartheta ^{1,p(.)}({\mathbb{R}^n})$$\end{document}. 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.