Sobolev embeddings with variable exponent

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