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

• Volume: 2, Issue: 1, page 151-173
• ISSN: 0392-4041

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## Abstract

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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, maps $L^{p(\cdot)}$ into $L^{q(\cdot)}$, where $1/p(\cdot)-1/q(\cdot)=a/n$.

## How to cite

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Cruz-Uribe, D., Diening, L., and Fiorenza, A.. "A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces." Bollettino dell'Unione Matematica Italiana 2.1 (2009): 151-173. <http://eudml.org/doc/290576>.

@article{Cruz2009,
abstract = {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 $1/p(\cdot) - 1/q(\cdot) = a/n$.},
author = {Cruz-Uribe, D., Diening, L., Fiorenza, A.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {151-173},
publisher = {Unione Matematica Italiana},
title = {A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces},
url = {http://eudml.org/doc/290576},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Cruz-Uribe, D.
AU - Diening, L.
AU - Fiorenza, A.
TI - A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/2//
PB - Unione Matematica Italiana
VL - 2
IS - 1
SP - 151
EP - 173
AB - 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 $1/p(\cdot) - 1/q(\cdot) = a/n$.
LA - eng
UR - http://eudml.org/doc/290576
ER -

## References

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