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In this paper, we generalize the $A_\fz$ extrapolation theorem in \cite{cmp}and the $A_p$ extrapolation theorem of Rubio de Francia to Schr\"odingersettings. In addition, we also establish the weighted vector-valuedinequalities for Schr\"odinger type maximal operators by using weightsbelonging to $ A_p^{\rho,\tz}$ which includes $A_p$. As their applications, weestablish the weighted vector-valued inequalities for some Sch\"odinger typeoperators and pseudo-differential operators.

Extrapolation from $A_\fz^{\rho,\fz}$, vector-valued inequalities and applications in the Schr\"odinger settings