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Maximal function on generalized Lebesgue spaces \(L^{p(\cdot)}\). (English) Zbl 1071.42014

The author proves that the Hardy-Littlewood maximal operator \(M\) is bounded on a Lebesgue space \(L^{p(\cdot)}(\mathbb R^ d)\) with variable exponent (introduced in [O. Kováčik and J. Rákosník, Czech. Math. J. 41(116), No. 4, 592–618 (1991; Zbl 0784.46029)]) under a certain continuity assumption on the exponent function \(p\). Moreover, he deduces continuity of mollifying sequences and density of \(C^\infty(\overline{\Omega})\) in the Sobolev space \(W^{1,p(\cdot)}(\Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb R^ d\) with Lipschitz boundary.
Reviewer: Petr Gurka (Praha)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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

Citations:

Zbl 0784.46029
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