A Characterization of ${\mathcal{M}}_{A_1}({\mathbb{R}}^n)$

Abstract

We characterize the set of all measurable functions on $\RR^n$ possessing an $A_1$ majorant, denoted as $\cM_{A_1}(\RR^n)$, by certain Banach function spaces. We prove that a function has an $A_1$ majorant if and only if it belongs to some Banach function space for which the Hardy-Littlewood maximal operator is bounded. This answers the question posted by G. Knese, J. M${}^c$Carthy, and K. Moen.