On a capacity for modular spaces

Abstract

AbstractThe 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 W1,p(Ω), the classical Orlicz–Sobolev space W1,Φ(Ω), the Hajłasz–Sobolev space M1,p(Ω), the Musielak–Orlicz–Sobolev space (or generalized Orlicz–Sobolev space) and many other spaces. Of particular interest is the space V:=W˜1,p(Ω) given as the closure of W1,p(Ω)∩Cc(Ω¯) in W1,p(Ω). In this case every function u∈V (a priori defined only on Ω) has a trace on the boundary ∂Ω which is unique up to a Capp,Ω-polar set