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Characterization of Orlicz–Sobolev space. (English) Zbl 1166.46308
Summary: We give a new characterization of the Orlicz–Sobolev space \(W ^{1,\Psi}(\mathbb{R}^n)\) in terms of a pointwise inequality connected to the Young function \(\Psi\). We also study different Poincaré inequalities in the metric measure space.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
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