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 ( 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 : = W ˜ 1 , p ( Ω ) given as the closure of W 1 , p ( Ω ) ∩ C c ( Ω ¯ ) in W 1 , 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 Cap p , Ω -polar set.