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On the necessity of bump conditions for the two-weighted maximal inequality


Author: Lenka Slavíková
Journal: Proc. Amer. Math. Soc. 145 (2017), 109-118
MSC (2010): Primary 42B25; Secondary 42B35
DOI: https://doi.org/10.1090/proc/13355
Published electronically: September 30, 2016
MathSciNet review: 3565364
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Abstract: We study the necessity of bump conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted $ L^p$ spaces with different weights. The conditions in question are obtained by replacing the $ L^{p'}$-average of $ \sigma ^{\frac {1}{p'}}$ in the Muckenhoupt $ A_p$-condition by an average with respect to a stronger Banach function norm, and are known to be sufficient for the two-weighted maximal inequality. We show that these conditions are in general not necessary for such an inequality to be true.


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Additional Information

Lenka Slavíková
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Address at time of publication: Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA
Email: slavikoval@missouri.edu

DOI: https://doi.org/10.1090/proc/13355
Keywords: Bump condition, two-weighted inequality, Hardy-Littlewood maximal operator
Received by editor(s): September 29, 2015
Received by editor(s) in revised form: February 4, 2016
Published electronically: September 30, 2016
Additional Notes: This research was partly supported by the grant P201-13-14743S of the Grant Agency of the Czech Republic.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society