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

TL;DR

This paper investigates whether bump conditions, which are sufficient but not known to be necessary, are actually required for the boundedness of the Hardy-Littlewood maximal operator in two-weight inequalities.

Contribution

The study demonstrates that bump conditions are generally not necessary for the two-weighted maximal inequality, challenging previous assumptions about their role.

Findings

Bump conditions are sufficient but not necessary for boundedness.

The paper provides counterexamples to the necessity of bump conditions.

Results clarify the limitations of bump conditions in two-weight inequalities.

Abstract

We study the necessity of bump conditions for the boundedness of the Hardy-Littlewood maximal operator $M$ from $L^p(v)$ into $L^p(w)$, where $1<p<\infty$. The conditions in question are obtained by replacing the average of $\sigma=v^{-\frac{1}{p-1}}$ in the Muckenhoupt $A_p$-condition by an average with respect to certain Banach function space, and are known to be sufficient for the two-weighted maximal inequality. We show that these conditions are in general not necessary for the boundedness of $M$ from $L^p(v)$ into $L^p(w)$.