On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator

TL;DR

This paper completes the characterization of weights satisfying weak-type inequalities for the maximal and fractional maximal operators, resolving an open problem for all p > 1 by establishing sufficiency of previously known conditions.

Contribution

It proves the sufficiency of Muckenhoupt and Wheeden's conditions for the maximal operator when p > 1 and extends results to fractional maximal operators, completing the characterization.

Findings

Established sufficiency of conditions for the maximal operator when p > 1.

Extended results to fractional maximal operators.

Resolved the open problem posed by Muckenhoupt and Wheeden for p > 1.

Abstract

In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood maximal operator and the Hilbert transform for weights in the class $A_p$, while also deriving necessary conditions to characterize the weights for which these estimates hold. In this paper, we establish the sufficiency of these conditions for the maximal operator when $p > 1$ and present corresponding results for the fractional maximal operators. This completes the characterization and resolves the open problem posed by Muckenhoupt and Wheeden for $p > 1$.