  We prove mixed inequalities for the Hardy-Littlewood maximal function
$M^{\rho,\sigma}$, where $\rho$ is a critical radius function and $\sigma\geq
0$. We also exhibit and prove an extension of Cruz-Uribe, Martell and P\'erez
extrapolation result in \cite{CruzUribe-Martell-Perez} to the setting of
Muckenhoupt weights associated to a critical radius function $\rho$. 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 \cite{BPQ}. Since we are dealing with unrelated
weights, the proof involves a quite subtle argument related with the original
ideas from Sawyer in \cite{Sawyer}.
