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On the spaces $$L^{p(x)}(\Omega)$$ and $$W^{m,p(x)}(\Omega)$$. (English) Zbl 1028.46041
A measurable function $$u:\Omega\to \mathbb{R}$$ 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.

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
$$L^p$$-space; Sobolev space
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