Weighted mixed inequalities for commutators of Schrödinger type operators

We obtain weighted mixed inequalities for the first order commutator of singular integral operators in the Schrödinger setting. Concretely, for $0<\delta \le 1$ we give estimates of commutators of Schrödinger-Calderón-Zygmund operators of $(s,\delta )$ type with $1<s\le \mathrm{\infty }$, and $\text{BMO}(\rho )$ symbols associated to a critical radious function $\rho$. Our results generalizes some previous estimates about mixed inequalities for Schrödinger type operators. We also deal with $A_p^{\rho }$ weights, which can be understood as a perturbation of the $A_p$ Muckenhoupt classes by means of function $\rho$.