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 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.