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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ L\log L$ results for the maximal operator in variable $ L^p$ spaces


Authors: D. Cruz-Uribe SFO and A. Fiorenza
Journal: Trans. Amer. Math. Soc. 361 (2009), 2631-2647
MSC (2000): Primary 42B25, 42B35
Published electronically: November 19, 2008
MathSciNet review: 2471932
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Abstract: We generalize the classical $ L\log L$ inequalities of Wiener and Stein for the Hardy-Littlewood maximal operator to variable $ L^p$ spaces where the exponent function $ p(\cdot)$ approaches $ 1$ in value. We prove a modular inequality with no assumptions on the exponent function, and a strong norm inequality if we assume the exponent function is log-Hölder continuous. As an application of our approach we give another proof of a related endpoint result due to Hästö.


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Additional Information

D. Cruz-Uribe SFO
Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106-3100
Email: david.cruzuribe@trincoll.edu

A. Fiorenza
Affiliation: Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli, Via Monteoliveto, 3, I-80134 Napoli, Italy – and – Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino, 111, I-80131 Napoli, Italy
Email: fiorenza@unina.it

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04608-4
PII: S 0002-9947(08)04608-4
Keywords: Variable Lebesgue space, maximal operators
Received by editor(s): November 30, 2006
Received by editor(s) in revised form: July 23, 2007
Published electronically: November 19, 2008
Additional Notes: The first author was partially supported by the Stewart-Dorwart faculty development fund of Trinity College. Both authors would like to thank the anonymous referee for the close reading of the original version of this paper.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.