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

Abstract

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\"odinger settings. In addition, we also establish the weighted vector-valued inequalities for Schr\"odinger type maximal operators by using weights belonging to $ A_p^{\rho,\tz}$ which includes $A_p$. As their applications, we establish the weighted vector-valued inequalities for some Sch\"odinger type operators and pseudo-differential operators.