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On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$. (English) Zbl 1028.46041
A measurable function $u:\Omega\to \bbfR$ belongs to $L^{p(x)} (\Omega)$, by definition, if $\lim_{\lambda \downarrow 0}\int_\Omega |\lambda u(x)|^{p(x)} dx=0$. The authors study the properties of the space $L^{p(x)} (\Omega)$, equipped with some kind of Luxemburg norm. They also consider a parallel construction for the Sobolev space $W^{m,p(x)} (\Omega)$. Such constructions are motivated by certain elliptic or variational problems.

46E30Spaces of measurable functions
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
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