Generalized Whiting transform and Flux Estimates for Wave Equations in Subextremal Kerr
Abstract
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⊂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 ψ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.
Related
- Dual modes in Kerr spacetimes and the Whiting transform: Mode stability revisited
- Optimal distributional estimates of the multiple Hilbert transform
- Reverse Translation and Convolution Estimates in the Framework of the Mehler-Fock Transform
- Moduli of smoothness and growth of the Laguerre transform: two-sided estimates
- Dimension free weak-type endpoint estimates for the vectors of the Dunkl--Riesz transform
BibTeX
@misc{he2026generalized,
title = {{Generalized Whiting transform and Flux Estimates for Wave Equations in Subextremal Kerr}},
author = {Lili He and Sergiu Klainerman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06344},
eprint = {2609.06344},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06344},
}
RIS
TY - UNPB
TI - Generalized Whiting transform and Flux Estimates for Wave Equations in Subextremal Kerr
AU - Lili He
AU - Sergiu Klainerman
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.06344
UR - https://arxiv.org/abs/2609.06344
AB - 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⊂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 ψ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.
ER -