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Sobolev embeddings with variable exponent. (English) Zbl 0974.46040
Let $$\Omega$$ be a bounded open subset of $$\mathbb R^n$$ with Lipschitz boundary and let $$p:\overline{\Omega}\to [1,\infty)$$ be Lipschitz-continuous. The authors consider the generalised Lebesgue space $$L^{p(x)}(\Omega)$$ and the corresponding Sobolev space $$W^{1,p(x)}(\Omega)$$, consisting of all $$f\in L^{p(x)}(\Omega)$$ with first-order distributional derivatives in $$L^{p(x)}(\Omega)$$.
It is shown that if $$1\leq p(x)<n$$ for all $$x\in\Omega$$, then there is a constant $$c>0$$ such that for all $$f\in W^{1,p(x)}(\Omega)$$, $\|f\|_{M,\Omega}\leq c \|f\|_{1,p,\Omega}.$ Here $$\|\cdot \|_{M,\Omega}$$ is the norm on an appropriate space of Orlicz-Musielak type and $$\|\cdot \|_{1,p,\Omega}$$ is the norm on $$W^{1,p(x)}(\Omega)$$. The inequality reduces to the usual Sobolev inequality if $$\sup _{\Omega}p<n$$. Corresponding results are proved for the case in which $$p(x)>n$$ for all $$x\in\Omega$$.

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