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

##### MSC:
 4.6e+31 Spaces of measurable functions 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems