Two-weight extrapolation on function spaces and applications

Abstract

This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including $A_1$, $A_p$, and $A_\infty$ extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman--Fefferman inequalities that can be used to show some known sharp $A_1$ inequalities, Muckenhoupt--Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calder\'{o}n--Zygmund operators. Finally, we obtain two-weight inequalities for Littlewood--Paley operators and Fourier integral operators on weighted Banach function spaces.