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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:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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