On a Capacity for Modular Spaces

TL;DR

This paper introduces a capacity concept for various topological measure spaces with respect to different function spaces, including Sobolev and Orlicz-Sobolev spaces, emphasizing boundary trace properties.

Contribution

It generalizes the notion of capacity to a broad class of function spaces and analyzes boundary trace properties in this unified framework.

Findings

Capacity defined for classical and generalized Sobolev spaces

Functions in the space have unique boundary traces up to polar sets

Framework applicable to many function spaces beyond classical Sobolev

Abstract

The purpose of this article is to define a capacity on certain topological measure spaces $X$ with respect to certain function spaces $V$ consisting of measurable functions. In this general theory we will not fix the space $V$ but we emphasize that $V$ can be the classical Sobolev space $W^{1,p}(\Omega)$, the classical Orlicz-Sobolev space $W^{1,\Phi}(\Omega)$, the Haj{\l}asz-Sobolev space $M^{1,p}(\Omega)$, the Musielak-Orlicz-Sobolev space (or generalized Orlicz-Sobolev space) and many other spaces. Of particular interest is the space $V:=\tW^{1,p}(\Omega)$ given as the closure of $W^{1,p}(\Omega)\cap C_c(\overline\Omega)$ in $W^{1,p}(\Omega)$. In this case every function $u\in V$ (a priori defined only on $\Omega$) has a trace on the boundary $\partial\Omega$ which is unique up to a $\Cap_{p,\Omega}$-polar set.