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

 

On some questions related to the maximal operator on variable $ L^p$ spaces


Author: Andrei K. Lerner
Journal: Trans. Amer. Math. Soc. 362 (2010), 4229-4242
MSC (2000): Primary 42B25, 46E30
Published electronically: March 26, 2010
MathSciNet review: 2608404
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Abstract: Let $ \mathcal{P}(\mathbb{R}^n)$ be the class of all exponents $ p$ for which the Hardy-Littlewood maximal operator $ M$ is bounded on $ L^{p(\cdot)}({\mathbb{R}}^n)$. A recent result by T. Kopaliani provides a characterization of $ \mathcal{P}$ in terms of the Muckenhoupt-type condition $ A$ under some restrictions on the behavior of $ p$ at infinity. We give a different proof of a slightly extended version of this result. Then we characterize a weak type $ \big(p(\cdot),p(\cdot)\big)$ property of $ M$ in terms of $ A$ for radially decreasing $ p$. Finally, we construct an example showing that $ p\in\mathcal{P}(\mathbb{R}^n)$ does not imply $ p(\cdot)-\alpha\in \mathcal{P}(\mathbb{R}^n)$ for all $ \alpha< p_--1$. Similarly, $ p\in\mathcal{P}(\mathbb{R}^n)$ does not imply $ \alpha p(\cdot)\in \mathcal{P}(\mathbb{R}^n)$ for all $ \alpha>1/p_-$.


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Additional Information

Andrei K. Lerner
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: aklerner@netvision.net.il

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05066-X
PII: S 0002-9947(10)05066-X
Keywords: Maximal operator, variable $L^p$ spaces.
Received by editor(s): June 8, 2008
Published electronically: March 26, 2010
Additional Notes: This work was supported by the Spanish Ministry of Education under the program “Programa Ramón y Cajal, 2006”.
Article copyright: © Copyright 2010 American Mathematical Society