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

Tuesday, November 29, 2016 - 2:00pm
Math Sci 111
Lenka Slavikova
(MU Math)

We will discuss the problem of 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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