Sobolev embeddings with variable exponent

Let Ω be a bounded open subset of ℝ^n with Lipschitz boundary and let p: Ω → [1, ∞) be Lipschitz-continuous. We consider the generalised Lebesgue space L^p (x) (Ω) and the corresponding Sobolev space W^1, p (x) (Ω), consisting of all f