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The maximal operator on weighted variable Lebesgue spaces. (English) Zbl 1273.42018

Summary: We study the boundedness of the maximal operator on the weighted variable exponent Lebesgue spaces \(L^{p(\cdot)}_\omega (\Omega)\). For a given log-Hölder continuous exponent \(p\) with \(1<\inf p\leqslant \sup p<\infty\) we present a necessary and sufficient condition on the weight \(\omega\) for the boundedness of \(M\). This condition is a generalization of the classical Muckenhoupt condition.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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