Almost compact and compact embeddings of variable exponent spaces

Let Omega be an open subset of R-N, and let p, q : Omega -> [1, infinity] be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space L-p(center dot)(Omega) in L-q(center dot)(Omega) to be almost compact. This leads to a condition on Omega, p and q sufficient to ensure that the Sobolev space W-1,W-p(center dot)(Omega) based on L-p(center dot)(Omega) is compactly embedded in L-q(center dot)(Omega); compact embedding results of this type already in the literature are included as special cases.

Automated summary

This article provides necessary and sufficient conditions for embedding variable exponent spaces and presents conditions for compact embedding of Sobolev spaces based on variable exponent spaces. The article also includes previous results as special cases.