On a Capacity for Modular Spaces

Original · EN

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 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:=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 ₚ,Ω-polar set.