Teukolsky equations in perturbations of Kerr

The Kerr stability conjecture has been proved in the slowly rotating case, i.e., $\mathrm{\mid }a\mathrm{\mid }\ll m$, in the sequence of works \cite{KS-GCM1} \cite{KS-GCM2} \cite{KS:Kerr} by Sergiu Klainerman and the author, \cite{GKS22} by Elena Giorgi, Sergiu Klainerman and the author, and \cite{Shen} by Dawei Shen, and extended in \cite{Sze}, by the author, to the full subextremal range $\mathrm{\mid }a\mathrm{\mid }<m$, thereby completing the proof of the Kerr stability conjecture. The proof in \cite{Sze} crucially relies on the two companion papers \cite{MaSz24} \cite{MaSz26} by Siyuan Ma and the author, in which we prove energy-Morawetz estimates respectively for the scalar wave equation and for Teukolsky equations on perturbations of Kerr with $\mathrm{\mid }a\mathrm{\mid }<m$. In addition, two results in \cite{Sze}, concerning the derivation of the Teukolsky wave-transport system in perturbations of Kerr and energy-Morawetz estimates for Teukolsky in the setting of \cite{Sze}, are stated without proofs. The goal of the present paper companion paper to \cite{Sze} is to provide the proof of these two results.