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

preprint2013arXivOpen access

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.

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

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