For all sub-extremal Kerr spacetimes |a|<M, angular elliptic estimates control the full linearised curvature by the extremal Teukolsky components up to lower-order terms.
Benomio , The wave equation on black rings and the linear stability of slowly rotating Kerr spacetimes , PhD thesis , Imperial College London, 2020
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Elliptic curvature estimates for linearised gravitational perturbations of Kerr in the full sub-extremal range $|a|<M$
For all sub-extremal Kerr spacetimes |a|<M, angular elliptic estimates control the full linearised curvature by the extremal Teukolsky components up to lower-order terms.