Sawyer estimates of mixed type for operators associated to a critical radius function

Abstract

We prove mixed inequalities for the Hardy–Littlewood maximal function Mρ ,σ , where ρ is a critical radius function and σ ≥ 0. We also exhibit and prove an extension of Cruz-Uribe,Martell and Pérez extrapolation result in Cruz-Uribe et al. (JMath IntMath Res Not 2005(30):1849–1871, 2005) to the setting of Muckenhoupt weights associated to a critical radius function ρ. This theorem allows us to give mixed inequalities for Schrödinger–Calderón–Zygmund operators, extending some previous estimates that we have already proved in Berra et al. (Potential Anal 60(1):253–283, 2024). Since we are dealing with u ∈ Aρ 1 and v ∈ Aρ ∞, the proof involves a quite subtle argument related with the original ideas from Sawyer Sawyer (Proc AmMath Soc 93(4):610–614, 1985).