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Proceedings of the American Mathematical Society

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A remark on the maximal operator for radial measures


Author: Adrián Infante
Journal: Proc. Amer. Math. Soc. 139 (2011), 2899-2902
MSC (2000): Primary 42B25
DOI: https://doi.org/10.1090/S0002-9939-2011-10727-5
Published electronically: January 14, 2011
MathSciNet review: 2801630
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Abstract: The purpose of this paper is to prove that there exist measures $ d\mu(x)=\gamma(x)dx$, with $ \gamma(x)=\gamma_{0}(\vert x\vert)$ and $ \gamma_{0}$ being a decreasing and positive function, such that the Hardy-Littlewood maximal operator, $ \mathcal{M}_{\mu}$, associated to the measure $ \mu$ does not map $ L^{p}_{\mu}(\mathbb{R}^{n})$ into weak $ L^{p}_{\mu}(\mathbb{R}^{n})$, for every $ p<\infty$. This result answers an open question of P. Sjögren and F. Soria.


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Additional Information

Adrián Infante
Affiliation: Department of Mathematics, Universidad Simón Bolívar, Caracas, Venezuela
Email: ainfante@usb.ve

DOI: https://doi.org/10.1090/S0002-9939-2011-10727-5
Keywords: Rotation invariant measure, maximal operator, is of weak type
Received by editor(s): April 6, 2010
Received by editor(s) in revised form: August 4, 2010
Published electronically: January 14, 2011
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2011 American Mathematical Society