Cruz-Uribe, Diening, Fiorenza: A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces

Cruz-Uribe, D. and Diening, L. and Fiorenza, A.:
A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces
Bollettino dell'Unione Matematica Italiana Serie 9 2 (2009), fasc. n.1, p. 151-173, (English)
pdf (322 Kb), djvu (197 Kb). | MR 2493649 | Zbl 1207.42011

Sunto

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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