In recent joint work with Vasy, we analyze the low energy behavior of differential form-valued waves on black hole spacetimes. In order to deduce asymptotics and decay from this, one in addition needs high energy estimates for the wave operator acting on sections of the form bundle. The present paper provides these on perturbations of Schwarzschild-de Sitter spaces in all spacetime dimensions @math . In fact, we prove exponential decay, up to a finite-dimensional space of resonances, of waves valued in any finite rank subbundle of the tensor bundle, which in particular includes differential forms and symmetric tensors. As the main technical tool for working on vector bundles that do not have a natural positive definite inner product, we introduce pseudodifferential inner products, which are inner products depending on the position in phase space.
@cite_18: 
@cite_14: We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming outgoing tails. An important application is wave decay on Kerr and Kerr-de Sitter black holes. We recover the optimal size of the strip and give an @math resolvent bound there. We next show existence of deeper resonance free strips under the @math -normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the ne-sided cutoff resolvent on the real line.
@cite_26: We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an ( ) derivative loss in the angular directions.
@cite_7: We establish a Bohr–Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild–de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.
@cite_8: 
@cite_9: We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recently, roughly by extending the problem across the boundary of the compactification of the asymptotically hyperbolic space in a suitable manner. The main novelty is that the non-scalar nature of the operator is dealt with by relating it to a problem on an asymptotically Minkowski space to motivate the choice of the extension across the conformal boundary.
@cite_27: We prove an asymptotic formula for the number of scattering resonances in a strip near the real axis when the trapped set is r-normally hyperbolic with r large and a pinching condition on the normal expansion rates holds. Our dynamical assumptions are stable under smooth perturbations and motivated by the setting of black holes. The key tool is a Fourier integral operator which microlocally projects onto the resonant states in the strip. In addition to Weyl law, this operator provides new information about microlocal concentration of resonant states.
@cite_12: We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schrodinger propagator as well as local energy decay results for the wave equation.
The framework in which we describe resonances was introduced by Vasy @cite_8 . In the scalar setting, this can directly be combined with estimates at normally hyperbolic trapping @cite_27 @cite_14 @cite_12 to obtain resonance expansions for scalar waves. On exact Kerr-de Sitter space, Dyatlov proved a significant strengthening of this in @cite_7 , obtaining a full resonance expansion for scalar waves, improving on the result of Bony and H "afner @cite_26 in the Schwarzschild-de Sitter setting, which in turn followed S 'a Barreto and Zworski @cite_18 . Vasy @cite_9 proved the meromorphic continuation of the resolvent of the Laplacian on differential forms on asymptotically hyperbolic spaces, and the fact that the underlying analysis of @cite_8 works on sections of vector bundles just as it does on functions is fundamental for the present paper.