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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 \(x\in\Omega\).

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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