We prove mixed inequalities for the Hardy–Littlewood maximal function \(M^{\rho ,\sigma }\) , where \(\rho \) is a critical radius function and \(\sigma \ge 0\) . We also exhibit and prove an extension of Cruz-Uribe, Martell and Pérez extrapolation result in Cruz-Uribe et al. (J Math Int Math Res Not 2005(30):1849–1871, 2005) to the setting of Muckenhoupt weights associated to a critical radius function \(\rho \) . 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 \(u\in A_1^\rho \) and \(v\in A_\infty ^\rho \) , the proof involves a quite subtle argument related with the original ideas from Sawyer Sawyer (Proc Am Math Soc 93(4):610–614, 1985).
