Geodesic
A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 151-173.

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We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot)}$ whenever the exponent function $p(\cdot)$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot)$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_{a}$, $0 a n$, maps $L^{p(\cdot)}$ into $L^{q(\cdot)}$, where $1/p(\cdot) - 1/q(\cdot) = a/n$. 
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Cruz-Uribe, D.; Diening, L.; Fiorenza, A. A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 151-173. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a7/

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