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

We prove mixed inequalities for the Hardy-Littlewood maximal function M^, , where is a critical radius function and 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. 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.