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

RESUMO

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 $\KK(a,m)$, 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 \textit{physical space}, generalized version, of the transformation which takes general solutions of {spin-$\sk$} wave equations in $\KK=\KK(a,m)$ to solutions of a secondary wave equation on a new metric manifold $\KKt=\KKt(a,m)$ with two stationary, asymptotically flat, ends. Moreover, for a certain subdomain $\DDt\subset\DDt_-$, which can be identified as the exterior of a black hole region, the new metric $\gt$ of $\KKt(a, m)$ is Lorentzian and the time translation $\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}.