Local energy estimates and Strichartz estimates on Schwarzschild and Kerr black hole backgrounds

In this thesis, we prove decay for solutions of the linear wave equation in Schwarzschild and Kerr geometries. This is considered to be the first step in proving the stability of the Kerr space-time under small perturbations of the initial data set.;We begin by presenting the corresponding decay results in the case of the Minkowski metric (flat space), the easiest example of a spacetime satisfying the Einstein equations. We obtain the optimal L2 decay rates for both Minkowski and small perturbations by commuting the wave operator with a suitably chosen vector field in the spirit of Morawetz. We will use these results later, to deal with the regions near infinity for Schwarzschild and Kerr.;We continue by obtaining decay results in the case of the Schwarzschild metric (non-rotating black hole). We first obtain a weaker local energy result by commuting the wave operator with a smooth vector field X that can be extended across the event horizon. There will be a polynomial loss at r = 3M due to the existence of trapped geodesics. We then localize around the trapped set, and prove a stronger estimate which only has a logarithmic loss. Finally, we construct a parametrix that satisfies the Strichartz estimates and which produces an error that can be controlled by the local energy estimates discussed above. Due to the logarithmic loss, this will work for all nonsharp Strichartz exponents.;Finally, we obtain decay results in the case of the Kerr metric (rotating black hole). Due to the more complicated nature of the trapped set, we cannot obtain a positive commutator from using vector fields, as was recently proved by Alinhac. Instead for small angular momentum a we can look at the Kerr geometry as a small perturbation of Schwarzschild and commute the wave operator with a suitable pseudodifferential operator which will be very close ( O(a)) to X. There will be a polynomial loss along the trapped geodesics close to r = 3M Similarly to the Schwarzschild case, we improve the estimate to a logarithmic loss and use the parametrix construction and the local energy estimates to obtain global-in-time Strichartz estimates for all nonsharp Strichartz exponents.