Sobolev embeddings with variable exponent

Let Ω be a bounded open subset of $ℝ^{n}$ with Lipschitz boundary and let $p:\overline{Ω} → [1,∞)$ 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 $f ∈ L^{p(x)}(Ω)$ with first-order distributional derivatives in $L^{p(x)}(Ω)$. It is shown that if 1 ≤ p(x) < n for all x ∈ Ω, then there is a constant c > 0 such that for all $f∈ W^{1,p(x)}(Ω)$, $|f|_{M,Ω} ≤ c|f|_{1,p,Ω}$. Here $|·|_{M,Ω}$ is the norm on an appropriate space of Orlicz-Musielak type and $|·|_{1,p,Ω}$ is the norm on $W^{1, p(x)}(Ω)$. The inequality reduces to the usual Sobolev inequality if $sup_Ω p