# On a Capacity for Modular Spaces - Mathematics > Functional Analysis

Abstract: The purpose of this article is to define a capacity on certain topologicalmeasure spaces $X$ with respect to certain function spaces $V$ consisting ofmeasurable functions. In this general theory we will not fix the space $V$ butwe emphasize that $V$ can be the classical Sobolev space $W^{1,p}\Omega$, theclassical Orlicz-Sobolev space $W^{1,\Phi}\Omega$, the Haj{\l}asz-Sobolevspace $M^{1,p}\Omega$, the Musielak-Orlicz-Sobolev space or generalizedOrlicz-Sobolev space and many other spaces. Of particular interest is thespace $V:=\tW^{1,p}\Omega$ given as the closure of $W^{1,p}\Omega\capC c\overline\Omega$ in $W^{1,p}\Omega$. In this case every function $u\inV$ 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.

Author: Markus Biegert

Source: https://arxiv.org/