AI 中文总结
本文提出广义Whiting变换，将次极端Kerr时空中的自旋波动方程映射到双端渐近平坦流形，并利用几何性质通过渐近傅里叶分析得到事件视界上的流估计，为波动方程有界性与衰减研究提供关键工具。
AI 中文摘要
所有近期关于次极端Kerr时空$\KK(a,m)$（具有非小角动量）上波动方程有界性与衰减的进展（见\cite{DRS}，\cite{S-Rita1}，\cite{S-Rita2}，\cite{Millet}，\cite{MaS}，\cite{MaS2}），本质上都依赖于Whiting的开创性发现，即一个强大的积分-微分变换，借助该变换他在\cite{W}中排除了指数增长模式（另见\cite{Yacov}，\cite{AMPW}和\cite{Rita}的进一步发展）。我们给出了该变换的一个纯物理空间上的广义版本，它将$\KK=\KK(a,m)$中自旋-$\sk$波动方程的一般解，映射到新度量流形$\KKt=\KKt(a,m)$上的一个次级波动方程的解，该流形具有两个平稳、渐近平坦的末端。此外，对于某个子域$\DDt\subset\DDt_-$（可被识别为黑洞区域的外部），$\KKt(a, m)$的新度量$\gt$是洛伦兹的，且时间平移$\T$是类时的。利用这两个末端的几何性质，通过经典的渐近傅里叶分析技术，可以推导出$\psi$在未来事件视界处的流的期望界。该估计在我们的姊妹论文\cite{He-K1}中起到了关键作用。
英文摘要
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 $ψ$ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.
Comments67 pages and 2 figures. Typo corrected and text size changed
