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

Slavíková, Lenka

doi:10.1090/proc/13355

On the necessity of bump conditions for the two-weighted maximal inequality
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by Lenka Slavíková PDF

Proc. Amer. Math. Soc. 145 (2017), 109-118 Request permission

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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