# On some properties of relative capacity and thinness in weighted variable exponent Sobolev spaces

## TL;DR

This paper explores properties of weighted relative capacity and thinness in variable exponent Sobolev spaces, establishing relations and equivalences between these concepts in the context of weighted capacities.

## Contribution

It introduces and analyzes weighted relative p(.)-capacity and thinness, establishing their properties and interrelations in weighted variable exponent Sobolev spaces.

## Key findings

- Relation between weighted relative capacity and Sobolev capacity
- Introduction of thinness concept in this context
- Equivalence of thinness with respect to the new capacity

## Abstract

In this paper, we define weighted relative $p(.)$-capacity and discuss properties of capacity in the space $W_{\vartheta }^{1,p(.)}(\mathbb{R}^{n}).$ Also, we investigate some properties of weighted variable Sobolev capacity. It is shown that there is a relation between these two capacities. Moreover, we introduce a thinness in sense to this new defined relative capacity and prove an equivalence statement for this thinness.