On a Capacity for Modular Spaces
Abstract
Description
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 $W^{1,p}(Ω)$, the classical Orlicz-Sobolev space $W^{1,Φ}(Ω)$, the Hajłasz-Sobolev space $M^{1,p}(Ω)$, 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 $W^{1,p}(Ω)\cap C_c(\overlineΩ)$ in $W^{1,p}(Ω)$. In this case every function $u\in V$ (a priori defined only on $Ω$) has a trace on the boundary $\partialΩ$ which is unique up to a $\Cap_{p,Ω}$-polar set.