Theory of capacities in fracional Sobolev spaces with variable exponents

TL;DR

This paper develops a capacities theory for fractional Sobolev spaces with variable exponents, establishing fundamental properties and showing that these capacities are Choquet capacities with all Borel sets being capacitable.

Contribution

It introduces a new capacities framework for fractional Sobolev spaces with variable exponents, including proofs of key properties and capacity measurability.

Findings

Both capacities are Choquet capacities

All Borel sets are capacitable

Fundamental properties like monotonicity and outer capacity are established

Abstract

In this paper we develop a capacities theory connected with the fractional Sobolev spaces with variable exponents. Two kinds of capacities are studied: Sobolev capacity and relative capacity. Basic properties of capacities, including monotonicity, outer capacity and several results, are studies. We prove that both capacities is a Choquet capacity and all borel sets are capacitable.