Sobolev embeddings with variable exponent

Let Ω be a bounded open subset of ℝⁿ with Lipschitz boundary and let p:Ω̄ → [1,∞) be Lipschitz-continuous. We consider the generalised Lebesgue space Lᵖ⁽ˣ⁾(Ω) and the corresponding Sobolev space W1,p(x)(Ω), consisting of all $f