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

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 P\'erez extrapolation result in {CruzUribe-Martell-Perez} to the setting of Muckenhoupt weights associated to a critical radius function ρ. This theorem allows us to give mixed inequalities for Schr\"odinger-Calder\'on-Zygmund operators, extending some previous estimates that we have already proved in {BPQ}. Since we are dealing with unrelated weights, the proof involves a quite subtle argument related with the original ideas from Sawyer in {Sawyer}.