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Sobolev embeddings with variable exponent. (English) Zbl 0974.46040

Let ${\Omega }$ be a bounded open subset of ${ℝ}^{n}$ with Lipschitz boundary and let $p:\overline{{\Omega }}\to \left[1,\infty \right)$ be Lipschitz-continuous. The authors consider the generalised Lebesgue space ${L}^{p\left(x\right)}\left({\Omega }\right)$ and the corresponding Sobolev space ${W}^{1,p\left(x\right)}\left({\Omega }\right)$, consisting of all $f\in {L}^{p\left(x\right)}\left({\Omega }\right)$ with first-order distributional derivatives in ${L}^{p\left(x\right)}\left({\Omega }\right)$.

It is shown that if $1\le p\left(x\right) for all $x\in {\Omega }$, then there is a constant $c>0$ such that for all $f\in {W}^{1,p\left(x\right)}\left({\Omega }\right)$,

${\parallel f\parallel }_{M,{\Omega }}\le c{\parallel f\parallel }_{1,p,{\Omega }}·$

Here ${\parallel ·\parallel }_{M,{\Omega }}$ is the norm on an appropriate space of Orlicz-Musielak type and ${\parallel ·\parallel }_{1,p,{\Omega }}$ is the norm on ${W}^{1,p\left(x\right)}\left({\Omega }\right)$. The inequality reduces to the usual Sobolev inequality if ${sup}_{{\Omega }}p. Corresponding results are proved for the case in which $p\left(x\right)>n$ for all $x\in {\Omega }$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives, differential and integral operators