The Muckenhoupt condition

TL;DR

This paper extends the Muckenhoupt weight theory to quasi-Banach function spaces, providing new characterizations for singular integral boundedness and exploring duality in maximal operators across various function spaces.

Contribution

It introduces a unified framework for weights beyond Lebesgue spaces, with new characterizations and partial results on duality and boundedness in diverse quasi-Banach spaces.

Findings

New characterizations for singular integral boundedness

Partial results on Hardy-Littlewood maximal operator duality

Overview of applications to variable Lebesgue, Morrey, and Musielak-Orlicz spaces

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.