A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces

Cruz-Uribe, D. ; Diening, L. ; Fiorenza, A.

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 151-173

Résumé

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 $1/p(\cdot) - 1/q(\cdot) = a/n$.