Generalized Whiting transform and Flux Estimates for Wave Equations in Subextremal Kerr

All recent advances \cite{DRS}, \cite{S-Rita1}, \cite{S-Rita2}, \cite{Millet}, \cite{MaS}, \cite{MaS2} on the boundedness and decay for wave equations on subextremal Kerr spacetimes $\text{KK}(a,m)$, with non-small angular rotation, rely in an essential way on Whiting’s groundbreaking discovery of a powerful integro-differential transformation with the help of which he ruled out exponentially growing modes in \cite{W} (see also \cite{Yacov}, \cite{AMPW} and \cite{Rita} for further developments). We give a purely $\mathit{\text{physical}}\mathit{\text{space}}$, generalized version, of the transformation which takes general solutions of {spin-$\text{sk}$} wave equations in $\text{KK}=\text{KK}(a,m)$ to solutions of a secondary wave equation on a new metric manifold $\text{KKt}=\text{KKt}(a,m)$ with two stationary, asymptotically flat, ends. Moreover, for a certain subdomain $\text{DDt}\subset {\text{DDt}}_{-}$, which can be identified as the exterior of a black hole region, the new metric $>$ of $\text{KKt}(a,m)$ is Lorentzian and the time translation $\text{T}$ is timelike. Using the geometric properties of the two ends one can then deduce, by classical asymptotic Fourier analysis techniques, the desired bound for the flux of $\psi$ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.