On a Capacity for Modular Spaces

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 Hajłasz-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 $V:=\tW^{1,p}(Ω)$ 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 ${⋒}_{p,\Omega }$-polar set.