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Abstract

The goal of this paper is to unify the theory of weights beyond the setting of weighted Lebesgue spaces in the general setting of quasi-Banach function spaces. We prove new characterizations for the boundedness of singular integrals, pose several conjectures, and prove partial results related to the duality of the Hardy-Littlewood maximal operator. Furthermore, we give an overview of the theory applied to weighted variable Lebesgue, Morrey, and Musielak-Orlicz spaces.