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Asymptotics of linear waves and resonances with applications to black holes

We apply the results of arXiv:1301.5633 to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a|<M and M\Lambda a<<1. We prove that if the initial data is localized at frequencies \lambda>>1, then the energy norm of the solution is bounded by O(\lambda^{1/2} exp(-(\nu_min - \epsilon)t/2) + \lambda^(-\infty)), for t<C log\lambda, where \nu_min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(\lambda exp(-(\nu_min -\epsilon)t) + \lambda^(-\infty)) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.

modulo O(λ −∞ ) errors. This splitting is achieved using the Fourier integral operator Π constructed in [Dy13] using the global dynamics of the flow.
For the Λ > 0 case, we furthermore study resonances, or quasi-normal modes, the complex frequencies z of solutions to the wave equation of the form e −izt v(x). Under a pinching condition ν max < 2ν min which is numerically verified to be true for a large range of parameters (see Figure 2(a)), we show existence of a band of quasi-normal modes satisfying a Weyl law -Theorem 3. In particular, this provides a large family of exact high frequency solutions to the wave equation that decay no faster than e −(νmax+ε)t/2 . We finally compare our theoretical prediction on the imaginary parts of resonances in the band with the exact quasi-normal modes for Kerr computed by the authors of [BeCaSt], obtaining remarkable agreement -see Figure 2(b).
Theorems 1-3 are related to the resonance expansion and quantization condition proved for the slowly rotating Kerr-de Sitter in [Dy12]. In this paper, however, we use dynamical assumptions stable under perturbations, rather than complete integrability of geodesic flow on Kerr(-de Sitter), and do not recover the precise results of [Dy12].
Statement of results. The Kerr(-de Sitter) metric, described in detail in §3.1, depends on three parameters, M (mass), a (speed of rotation), and Λ (cosmological constant). We assume that the dimensionless quantities a/M and ΛM 2 lie in a small neighborhood (see Figure 1(a)) of either the Schwarzschild(-de Sitter) case, a = 0, 9ΛM 2 < 1, (1.1) or the subextremal Kerr case Λ = 0, |a| < M. (1.2) Our results apply as long as certain dynamical conditions are satisfied, and likely hold for a larger range of parameters, see the remark following Proposition 3.2. To facilitate the discussion of perturbations, we adopt the abstract framework of §2.2, with the spacetime X 0 = R t × X 0 and a Lorentzian metricg on X 0 which is stationary in the sense that ∂ t is Killing. The space slice X 0 is noncompact because of the spatial infinity and/or event horizon(s); to measure the distance to those, we use a function µ ∈ C ∞ (X 0 ; (0, ∞)), such that X δ := {µ > δ} is compact for each δ > 0. For the exact Kerr(-de Sitter metric), the function µ is defined in (3.6).
We study solutions to the wave equation in X 0 , with f 0 , f 1 ∈ C ∞ 0 (X 0 ) and the time variable shifted so that the metric continues smoothly past the event horizons -see (3.45). To simplify the statements, and because our work focuses on the phenomena driven by the trapped set, we only study the behavior of solutions in X δ 1 for some small δ 1 > 0. Define the energy norm u(t) E := u(t) H 1 (X δ 1 ) + ∂ t u(t) L 2 (X δ 1 ) .
Assume that f 0 (λ), f 1 (λ) ∈ C ∞ 0 (X δ 1 ) are localized at frequency ∼ λ → ∞ in the sense of (1.6). Then the solution u λ to (1.3) with initial data f 0 , f 1 satisfies the bound u λ (t) E ≤ C(λ 1/2 e −(ν min −ε)t/2 + λ −N ) u λ (0) E , 0 ≤ t ≤ T log λ. (1.5) Here we say that f = f (λ) is localized at frequencies ∼ λ, if for each coordinate neighborhood U in X 0 and each χ ∈ C ∞ 0 (U ), the Fourier transforms χf (ξ) in the corresponding coordinate system satisfy for each N , where C U,χ > 0 is a constant independent of λ. For the proof, it is more convenient to use semiclassical rescaling of frequencies ξ → hξ, where h = λ −1 → 0 is the semiclassical parameter, and the notion of h-wavefront set WF h (f ) ⊂ T * X 0 . The requirement that f j is microlocalized at frequencies ∼ h −1 is then equivalent to stating that WF h (f j ) is contained in a fixed compact subset of T * X 0 \ 0, with 0 denoting the zero section; see §2.1 for details.
The main component of the proof of Theorem 1 is the following Theorem 2. Under the assumptions of Theorem 1, for each families f 0 (h), f 1 (h) ∈ C ∞ 0 (X δ 1 ) with WF h (f j ) contained in a fixed compact subset of T * X 0 \ 0 and u(h) the corresponding solution to (1.3), for t 0 large enough there exists a decomposition u(t, x) = u Π (t, x) + u R (t, x), t 0 ≤ t ≤ T log(1/h), x ∈ X δ/2 , such that g u Π (t), g u R (t) are O(h N ) H N h on X δ 1 uniformly in t ∈ [t 0 , T log(1/h)], and we have uniformly in t 0 ≤ t ≤ T log(1/h), (1.10) The decomposition u = u Π +u R is achieved in §2.4 using the Fourier integral operator Π constructed for r-normally hyperbolic trapped sets in [Dy13]. The component u Π enjoys additional microlocal properties, such as localization on the outgoing tail and approximately solving a pseudodifferential equation -see the proof of Theorem 4 in §2.4 and [Dy13,§8.5]. We note that (1.9) gives a lower bound on the rate of decay of the approximate solution u Π , if u Π (t 0 ) E is not too small compared to u(0) E , and the existence of a large family of solutions with the latter property follows from the construction of u Π . We remark that Theorems 1 and 2 are completely independent from the behavior of linear waves at low frequency. In fact, we do not even use the boundedness in time of solutions for the wave equation, assuming merely that they grow at most exponentially (which is trivially true in our case); this suffices since O(h ∞ ) remainders overcome such growth for t = O(log(1/h)). If a boundedness statement is available, then our results can be extended to all times, though the corresponding rate of decay stays fixed for t log(1/h) because of the O(h ∞ ) term.
To formulate the next result, we restrict to the case Λ > 0, or its small stationary perturbation. In this case, the metric has two event horizons and we consider the discrete set Res of resonances, as defined for example in [Va10]. As a direct application of [Dy13, Theorems 1 and 2], we obtain two gaps and a band of resonances in between with a Weyl law (see Figure 1(b)): Theorem 3. Let ( X 0 ,g) be the Kerr-de Sitter metric with M, a, Λ near one of the cases (1.1) or (1.2) and Λ > 0, or its small stationary perturbation as discussed in §3.6.
, where ν R min (l), ν R max (l) are the minimal/maximal imaginary parts of resonances in the first band defined by (1.14). and the corresponding semiclassical scattering resolvent, namely the inverse of the operator (3.56), is bounded by Ch −2 for z in this region.

Under the pinching condition
and for ε small enough so that ν max + ε < 2(ν min − ε), we have the Weyl law as h → 0, where c K is the symplectic volume of a certain part of the trapped set K, see (2.16) and (3.42).
The pinching condition (1.12) is true for the non-rotating case a = 0, since ν min = ν max there (see Proposition 3.8). However, it is violated for the nearly extremal case M − |a| M , at least for Λ small enough; in fact, as |a|/M → 1, ν max stays bounded away from zero, while ν min converges to zero -see Proposition 3.9 and Figure 2(a). Note that (ν min − ε)/2 is the size of the resonance free strip and thus gives the minimal rate of exponential decay of linear waves on Kerr-de Sitter, modulo terms coming from finitely many resonances, by means of a resonance expansion -see for example [Va10,Lemma 3.1].
To demonstrate the sharpness of the size of the band of resonances {Im ω ∈ 1 2 [−ν max − ε, −ν min + ε]}, we use the exact quasi-normal modes for the Kerr metric computed (formally, since one cannot meromorphically continue the resolvent in the Λ = 0 case; however, one could consider the case of a very small positive Λ) by Berti-Cardoso-Starinets [BeCaSt]. Similarly to the quantization condition of [Dy12], these resonances ω mlk are indexed by three integer parameters m ≥ 0 (depth), l ≥ 0 (angular energy), and k ∈ [−l, l] (angular momentum). The parameter l roughly corresponds to the real part of the resonance and the parameter m, to its imaginary part. We define l] (− Im ω 0lk ). (1.14) We compare ν R min (l), ν R max (l) with ν min /2, ν max /2 and plot the supremum of the relative error over a/M ∈ [0, 0.95] for different values of l; the error decays like O(l −1 ) -see Figure 2(b).
Previous work. We give an overview of results on decay and non-decay on black hole backgrounds; for a more detailed discussion of previous results on normally hyperbolic trapped sets and resonance asymptotics, see the introduction to [Dy13].
The study of boundedness of solutions to the wave equation for the Schwarzschild (Λ = a = 0) black hole was initiated in [Wa, KaWa] and decay results for this case have been proved in [BlSt,DaRo09,MMTT,Lu10a]. The slowly rotating Kerr case (Λ = 0, |a| M ) was considered in [AnBl,DaRo11,DaRo08,Ta,TaTo,To,MeTaTo,Lu10b], and the full subextremal Kerr case (Λ = 0, |a| < M ) in [FKSY,FKSYErr,DaRo10,DaRo12,Sh1,Sh2] -see [DaRo12] for a more detailed overview. In either case the decay is polynomial in time, with the optimal decay rate O(t −3 ). A decay rate of O(t −2l−3 ), known as Price's Law, was proved in [DoScSo11,DoScSo12] for linear waves on the Schwarzschild black hole for solutions living on the l-th spherical harmonic; the constant in the O(·) depends on l. Our Theorem 1 improves on these decay rates in the high frequency regime l = λ 1, for times O(log λ).
The extremal Kerr case (Λ = 0, |a| = M ) was recently studied for axisymmetric solutions in [Ar12], with a weaker upper bound due to the degeneracy of the event horizon. The earlier work [Ar11a,Ar11b] suggests that one cannot expect the O(t −3 ) decay to hold in the extremal case. In the high frequency regime studied here, we do not expect to get exponential decay due to the presence of slowly damped geodesics near the event horizon, see Figure 2(a) above.
The Schwarzschild-de Sitter case (Λ > 0, a = 0) was considered in [SBZw,BoHä,DaRo07,MeSBVa], proving an exponential decay rate at all frequencies, a quantization condition for resonances, and a resonance expansion, all relying on separation of variables techniques. In [Dy11a,Dy11b,Dy12], a same flavor of results was proved for the slowly rotating Kerr-de Sitter (Λ > 0, |a| M ). The problem was then studied from a more geometric perspective, aiming for results that do not depend on symmetries and apply to perturbations of the metric -the resonance free strip of [WuZw] for normally hyperbolic trapping, the gluing method of [DaVa], and the analysis of the event horizons and low frequencies of [Va10] together give an exponential decay rate which is stable under perturbations, for Λ > 0, |a| < √ 3 2 M , provided that there are no resonances in the upper half-plane except for the resonance at zero. Our Theorem 3 provides detailed information on the behavior of resonances below the resonance free strip of [WuZw], without relying on the symmetries of the problem.
Finally, we mention the the Kerr-AdS case (Λ < 0). The metric in this case exhibits strong (elliptic) trapping, which suggests that the decay of linear waves is very slow because of the high frequency contributions. A logarithmic upper bound was proved in [HoSm11], and existence of resonances exponentially close to the real axis and a logarithmic lower bound were established in [Ga,HoSm13].
Quasi-normal modes (QNMs) of black holes have a rich history of study in the physics literature, see [KoSc]. The exact QNMs of Kerr black holes were computed in [BeCaSt], which we use for Figure 2(b). The high-frequency approximation for QNMs, using separation of variables and WKB techniques, has been obtained in [YNZZZC, YZZNBC, Ho]. In particular, for the nearly extremal Kerr case their size of the resonance free strip agrees with Proposition 3.9; moreover, they find a large number of QNMs with small imaginary parts, which correspond to a positive proportion of the Liouville tori on the trapped set lying close to the event horizon. See [YNZZZC] for an overview of the recent physics literature on the topic. We finally remark that the speed of rotation of an astrophysical black hole (NGC 1365) has recently been accurately measured in [RHMWBCCGHNSZ], yielding a high speed of rotation: a/M ≥ 0.84 at 90% confidence.
Structure of the paper. In §2, we study semiclassical properties of solutions to the wave equation on stationary Lorentzian metrics with noncompact space slices. We operate under the geometric and dynamical assumptions of §2.2; while these assumptions are motivated by Kerr(-de Sitter) metrics and their stationary perturbations, no explicit mention of these metrics is made. The analysis of §2 works in a fixed compact subset of the space slice, and the results apply under microlocal assumptions in this compact subset (namely, outgoing property of solutions to the wave equation for Theorems 1-2 and meromorphic continuation of the scattering resolvent with an outgoing parametrix for Theorem 3) which are verified for our specific applications in §3.4 and §3.5. In §2.3, we reduce the problem to the space slice via the stationary d'Alembert-Beltrami operator and show that some of the assumptions of [Dy13,§ §4.1,5.1] are satisfied. In §2.4, we use the methods of [Dy13] to prove asymptotics of outgoing solutions to the wave equation.
Next, §3 contains the applications of [Dy13] and §2 to the Kerr(-de Sitter) metrics and their perturbations. In §3.1, we define the metrics and establish their basic properties, verifying in particular the geometric assumptions of §2.2. In §3.2, we show that the trapping is r-normally hyperbolic, verifying the dynamical assumptions of §2.2.
In §3.3, we study in greater detail trapping in the Schwarzschild(-de Sitter) case a = 0 and in the nearly extremal Kerr case Λ = 0, a = M − , in particular showing that the pinching condition (1.12) is violated for the latter case; we also study numerically some properties of the trapping for the general Kerr case, and give a formula for the constant in the Weyl law. In §3.4, we study solutions to the wave equation on Kerr(de Sitter), using the results of §2.4 to prove Theorems 1 and 2. In §3.5, we use the results of [Dy13] and [Va10] to prove Theorem 3 for Kerr-de Sitter. Finally, in §3.6, we explain why our results apply to small smooth stationary perturbations of Kerr(-de Sitter) metrics.
2. General framework for linear waves 2.1. Semiclassical preliminaries. We start by briefly reviewing some notions of semiclassical analysis, following [Dy13,§3]. For a detailed introduction to the subject, the reader is directed to [Zw].
Let X be an n-dimensional manifold without boundary. Following [Dy13, §3.1], we consider the class Ψ k (X) of all semiclassical pseudodifferential operators with classical symbols of order k. If X is noncompact, we impose no restrictions on how fast the corresponding symbols can grow at spatial infinity. The microsupport of a pseudodifferential operator A ∈ Ψ k (X), also known as its h-wavefront set WF h (A), is a closed subset of the fiber-radially compactified cotangent bundle T * X. We denote by Ψ comp (X) the class of all pseudodifferential operators whose wavefront set is a compact subset of T * X (and in particular lies away from the fiber infinity). Finally, we say that notions apply to tempered distributions and operators below.
Using pseudodifferential operators, we can study microlocalization of h-tempered distributions, namely families of distributions u(h) ∈ D (X) having a polynomial in h bound in some Sobolev norms on compact sets, by means of the wavefront set WF h (u) ⊂ T * X. Using Schwartz kernels, we can furthermore study h-tempered op- Besides pseudodifferential operators (whose wavefront set is this framework is the image under the diagonal embedding T * X → T * (X × X) of the wavefront set used in the previous paragraph) we will use the class I comp (Λ) of compactly supported and compactly microlocalized Fourier integral operators associated to some canonical relation The h-wavefront set of an h-tempered family of distributions u(h) can be characterized using the semiclassical Fourier transform We have (x, ξ) ∈ WF h (u) if and only if there exists a coordinate neighborhood U x of x in X, a function χ ∈ C ∞ 0 (U x ) with χ(x) = 0, and a neighborhood U ξ of ξ in T * x X such that if we consider χu as a function on R n using the corresponding coordinate system, then for each N , (2.1) The proof is done analogously to [HöIII,Theorem 18.1.27].
One additional concept that we need is microlocalization of distributions depending on the time variable that varies in a set whose size can grow with h. Assume that u(t; h) is a family of distributions on (−ε, T (h) + ε) × X, where ε > 0 is fixed and where τ is the momentum corresponding to t and T * X × R τ is the fiber-radial compactification of the vector bundle T * X × R τ , with R τ part of the fiber, as follows: (x, ξ, τ ) does not lie in WF h (u) if and only if there exists a neighborhood U of (x, ξ, τ ) in T * X × R τ such that is simply the closure of the projection of WF h (u) onto the (x, ξ, τ ) variables. The notion of WF h makes it possible to talk about u being microlocalized inside, or being O(h ∞ ), on subsets of T * ((−ε, T (h) + ε) × X) independent of t.
We now discuss restrictions to space slices. Assume that u(h) ∈ D ((−ε, T (h) + ε) × X) is h-tempered uniformly in t and moreover, WF h (u) does not intersect the spatial fiber infinity {ξ = 0, τ = ∞}. Then u (as well as all its derivatives in t) is a smooth function of t with values in D (X), u(t) is h-tempered uniformly in t ∈ [0, T (h)], and One can see this using (2.1) and the formula for the Fourier transform of the restriction w of v ∈ S (R n+1 ) to the hypersurface {t = 0}:

General assumptions.
In this section, we study Lorentzian metrics whose space slice is noncompact, and define r-normal hyperbolicity and the dynamical quantities ν min , ν max in this case.
Geometric assumptions. We assume that: (1) ( X 0 ,g) is an n + 1 dimensional Lorentzian manifold of signature (1, n), and X 0 = R t × X 0 , where X 0 , the space slice, is a manifold without boundary; (2) the metricg is stationary in the sense that its coefficients do not depend on t, or equivalently, ∂ t is a Killing field; (3) the space slices {t = const} are spacelike, or equivalently, the covector dt is timelike with respect to the dual metricg −1 on T * X 0 ; The (nonsemiclassical) principal symbol of the d'Alembert-Beltrami operator g (without the negative sign), denoted byp(x,ξ), is The Hamiltonian flow ofp is the (rescaled) geodesic flow on T * x X 0 ; we are in particular interested in nontrivial lightlike geodesics, i.e. the flow lines of Hp on the set {p = 0} \ 0, where 0 denotes the zero section.
Note that we do not assume that the vector field ∂ t is timelike, since this is false inside the ergoregion for rotating black holes. Because of this, the intersections of the sets {τ = const}, invariant under the geodesic flow, with the energy surface {p = 0} need not be compact in the ξ direction, and it is possible that ξ will blow up in finite time along a flow line of Hp, while x stays in a compact subset of X 0 . 1 We consider instead the rescaled flowφ s := exp(sHp/∂ τp ) on {p = 0} \ 0. (2.3) Here ∂ τp (x,ξ) = −2g −1 x (ξ, dt) never vanishes on {p = 0} \ 0 by assumption (3). Since Hpt = ∂ τp , the variable t grows linearly with unit rate along the flowφ s . The flow lines of (2.3) exist for all s as long as x stays in a compact subset of X 0 . The flow is homogeneous, which makes it possible to define it on the cosphere bundle S * X 0 , which is the quotient of T * X 0 \ 0 by the action of dilations. Finally, the flow preserves the restriction of the symplectic form to the tangent bundle of {p = 0}.
We next split the light cone {p = 0} \ 0 into the sets C + and C − of positively and negatively time oriented covectors: (2.6) Since ∂ τp never vanishes on {p = 0}\0 by assumption (3) We fix the sign of τ on the trapped set, in particular requiring that K ⊂ {τ = 0}: Dynamical assumptions. We now formulate the assumptions on the dynamical structure of the flow (2.3). They are analogous to the assumptions of [Dy13, §5.1] and related to them in §2.3 below. We start by requiring that Γ ± are regular: (6) for a large constant r, Γ ± are codimension 1 orientable C r submanifolds of {p = 0} \ 0; (7) Γ ± intersect transversely inside {p = 0}\0, and the intersections K∩{t = const} are symplectic submanifolds of T * X 0 .
We next define a natural invariant decomposition of the tangent space to {p = 0} at K. Let (T Γ ± ) ⊥ be the symplectic complement of the tangent space to Γ ± . Since Γ ± has codimension 2 and is contained in {p = 0}, (T Γ ± ) ⊥ is a two-dimensional vector subbundle of T (T * X 0 ) containing Hp. Since Hpt = 0 on {p = 0} \ 0, we can define the one-dimensional vector subbundles of T (T * X 0 ) Since Γ ± is a codimension 1 submanifold of {p = 0} and Hp is tangent to Γ ± , we see that Γ ± is coisotropic and then V ± are one-dimensional subbundles of T Γ ± ; moreover, Since the flowφ s from (2.3) maps the space slice {t = t 0 } to {t = t 0 + s} and Hp is tangent to T Γ ± , we see that the splittings (2.8) are invariant underφ s .
We now formulate the dynamical assumptions on the linearization of the flowφ s with respect to the splitting (2.8). Define the minimal expansion rate in the transverse direction ν min as the supremum of all ν for which there exists a constant C such that with · denoting any smooth t-independent norm on the fibers of T (T * X 0 ), homogeneous of degree zero with respect to dilations on T * X 0 . Similarly, define ν max as the infimum of all ν for which these exists a constant c > 0 such that We now formulate the dynamical assumption of r-normal hyperbolicity: (8) ν min > rµ max , where µ max is the maximal expansion rate of the flow along K, defined as the infimum of all ν for which there exists a constant C such that (2.11) The large constant r determines how many terms we need to obtain in semiclassical expansions, and how many derivatives of these terms need to exist -see [Dy13]. Theorem 3 simply needs r to be large (in principle, depending on the dimension), while Theorems 1 and 2 require r to be large enough depending on N, T . For exact Kerr(-de Sitter) metrics, our assumptions are satisfied for all r, but a small perturbation will satisfy them for some fixed large r depending on the size of the perturbation.
2.3. Reduction to the space slice. We now put a Lorentzian manifold ( X 0 ,g) satisfying assumptions of §2.2 into the framework of [Dy13]. Consider the stationary d'Alembert-Beltrami operator Pg(ω), ω ∈ C, the second order semiclassical differential operator on the space slice X 0 obtained by replacing hD t by −ω in the semiclassical d'Alembert-Beltrami operator h 2 g . The principal symbol of Pg(ω) is given by wherep is defined in (2.2) and the right-hand side does not depend on t. We will show that the operator Pg(ω) satisfies a subset of the assumptions of [Dy13, § §4.1, 5.1].
First of all, we need to understand the solutions in ω to the equation p = 0. Let be the unique real solution ω to the equation p(x, ξ; ω) = 0 such that (t, x, −ω, ξ) ∈ C + , with the positive time oriented light cone C + defined in (2.6). The existence and uniqueness of such solution follows from assumption (3) in §2.2, and we also find from the definition of C + that (2.12) We can write C + as the graph of p: The level sets of p are not compact if ∂ t is not timelike. To avoid dealing with the fiber infinity, we use assumption (5) in §2.2 to identify a bounded region in T * X 0 invariant under the flow and containing the trapped set: There exists an open conic subset W ⊂ C + , independent of t, such that Proof. Consider a conic neighborhood W 0 of K ∩ C + in C + independent of t and such that the closure of W 0 is contained in {µ > δ 0 /2} ∩ {τ < 0}; this is possible by assumption (5) and since K is contained in {µ ≥ δ 0 }. Let W ⊂ C + be the union of all maximally extended flow lines of (2.3) passing through W 0 . Then W is an open conic subset of C + containing K ∩ C + and invariant under the flow (2.3). It remains to show that each point (x,ξ) ∈ C + ∩ {τ ≥ 0} has a neighborhood that does not intersect W.
To see this, note that the corresponding trajectory γ(s) of (2.3) does not lie in Γ + ∪ Γ − (as otherwise, the projection of γ(s) onto the cosphere bundle would converge to K as s → +∞ or s → −∞, by [Dy13,Lemma 4.1]; it remains to use assumption (5) and the fact that τ is constant on γ(s)). We then see that γ(s) escapes for both s → +∞ and s → −∞ and does not intersect the closure of W 0 and same is true for nearby trajectories; therefore, a neighborhood of (x,ξ) does not intersect W.

Arguing similarly (using an open conic subset
we construct an open conic subset W of C + independent of t and such that and W, W are invariant under the flow (2.3). Now, take small δ 1 > 0 and define  (5)), and the closure of U is contained in U . Moreover, the projections of U, U onto the (x, τ, ξ) variables are bounded because W, W are conic and W, W ⊂ {τ = 0}.
and similarly for U . Note that U ⊂ {|p − 1| < δ 1 } and U ⊂ {|p − 1| < 2δ 1 }. Since U is bounded, and by (2.12), for δ 1 > 0 small enough and ( We now study the Hamiltonian flow of p. Since and for each t, we see that the flow of H p is the projection of the rescaled geodesic flow (2.3) on C + : We now verify some of the assumptions of [Dy13, §4.1]. We let X be an n-dimensional manifold containing X 0 (for the Kerr-de Sitter metric it is constructed in §3.5) and consider the volume form d Vol on X 0 related to the volume form d Vol on X 0 generated byg by the formula d Vol = dt∧d Vol. The operator Pg(ω) is a semiclassical pseudodifferential operator depending holomorphically on ω ∈ Ω := {|ω − 1| < 2δ 1 } and p is its semiclassical principal symbol. We do not specify the spaces H 1 , H 2 here and do not establish any mapping or Fredholm properties of Pg(ω); for our specific applications it is done in §3.5. Except for these mapping properties, the assumptions (1), (2), and (5)-(9) of [Dy13, §4.1] are satisfied, with U, U defined above, [α 0 , α 1 ] := [1−δ 1 /2, 1+δ 1 /2], and the incoming/outgoing tails Γ ± on the space slice given by (for each t) and similarly for the trapped set Finally, the dynamical assumptions of [Dy13, §5.1] are also satisfied, as follows directly from (2.14) and the dynamical assumptions of §2.2. Note that the subbundles , and the expansion rates ν min , ν max , µ max defined in (2.9)-(2.11) coincide with those defined in [Dy13,(5 To relate the constants for the Weyl laws in Theorem 3 and [Dy13, Theorem 2], we note that for [a, b] ⊂ (1 − δ 1 /2, 1 + δ 1 /2), Here Vol σ and Volσ stand for symplectic volume forms of order 2n − 2 on T * X 0 and T * X 0 , respectively. The constant c K from Theorem 3 is then given by ( 2.16) 2.4. Applications to linear waves. In this section, we apply the results of [Dy13] to understand the decay properties of linear waves; Theorem 4 below forms the base for the proofs of Theorems 1 and 2 in §3.4.

Consider a family of approximate solutions
Here h 1 is the semiclassical parameter and T (h) > 0 depends on h (for our particular application, T (h) = T log(1/h) for some constant T ). We assume that u is h-tempered uniformly in t, as defined in §2.1. Then by the elliptic estimate (see for instance [Dy13,Proposition 3.2]), u is microlocalized on the light cone: The second of these inequalities is trivial; the first one is done by applying the standard energy estimate for the wave equation to the function χ(t − t 0 )u, with χ ∈ C ∞ 0 (− , ) equal to 1 near 0 and > 0 small depending on δ 1 , δ 2 .
We furthermore restrict ourselves to the following class of outgoing solutions, see Figure 3(a): The main result of this section is Theorem 4. Fix T, N, ε > 0 and let the assumptions of §2.2 hold, including r-normal hyperbolicity with r large depending on T, N . Assume that u is an outgoing solution to Then for t 0 large enough and independent of h, we can write For the proof, we assume that the metric is r-normally hyperbolic for all r, and prove the bounds for all T, N (so that O(h N ) becomes O(h ∞ )); since semiclassical arguments require finitely many derivatives to work, the statement will be true for r large depending on T and N .
We first recall the factorization of [Dy13,Lemma 4.3]: where S(ω) is a family of pseudodifferential operators elliptic near U, and such that S(ω) * = S(ω) for ω ∈ R, and P is a self-adjoint pseudodifferential operator, moreover we assume that it is compactly supported and compactly microlocalized. If we define the self-adjoint pseudodifferential operator S on X 0 by replacing ω by −hD t in S(ω), then we get note that u(t) and its t-derivatives are bounded uniformly in t with values in htempered distributions on X 0 by the discussion of restrictions to space slices in §2.1 and by (2.18). We have by (2.17), (2.20), (2.21), and (2.28), We next use the construction of [Dy13, Lemma 5.1], which (combined with the homogeneity of the flow) gives functions ϕ ± defined in a conic neighborhood of K in T * X 0 , such that Γ ± = {ϕ ± = 0} in this neighborhood, ϕ ± are homogeneous of degree zero, and where c ± are some smooth functions on the domain of ϕ ± . Then for small δ > 0, We now fix δ small enough so that [Dy13, Theorem 3 in §7.1 and Proposition 7.1] apply, giving a Fourier integral operator Π ∈ I comp (Λ • ) which satisfies the equations [Dy13,(5.12)]. Also, we define We now derive certain conditions on the microlocalization of u for large enough times, see Figure 3 . Then by propagation of singularities (see for example [Dy13,Proposition 3.4]) for the equation (2.29), we see that either there exists s 0 ∈ [−t 1 , 0] such that γ(s 0 ) ∈ {δ 1 ≤ µ ≤ 2δ 1 } ∩ WF h (u(t + s 0 )), or γ(s) ∈ X 2δ 1 for all s ∈ [−t 1 , 0]. However, in the first of these two cases, by (2.31) we have γ(s 0 ) ∈ {µ ≤ 2δ 1 } ∩ {H p µ ≤ 0}, which implies that γ(0) ∈ {µ ≤ 2δ 1 } by assumption (4) in §2.2, a contradiction. Therefore, It remains to note that for t 1 large enough, (2.38) By [Dy13, Proposition 6.1 and §6.2], we have (2.39) We now use the methods of [Dy13,§8] to prove a microlocal version of Theorem 4 near the trapped set: Proposition 2.5. There exist compactly supported A 0 , A 1 ∈ Ψ comp (X 0 ) microlocalized inside W , elliptic on W , and such that for t ∈ [t 1 , T log(1/h)], Proof. We will use the operators Θ ± , Ξ constructed in [Dy13,Proposition 7.1]. The microlocalization statements we make will be uniform in t ∈ [t 1 , T log(1/h)].
Then similarly to [Dy13,(8.14)], we find By (2.29) and (2.38), Similarly to [Dy13,Proposition 8.3], we use the commutation relation [P, Similarly to [Dy13,(8.18 , and similarly to [Dy13,(8.19)] we have and the inequality is strict on W . Similarly to [Dy13,Lemma 8.4], by sharp Gårding inequality we get for an appropriate choice of A 1 and some A 0 ∈ Ψ comp (X 0 ) microlocalized inside W . Also similarly to [Dy13,Lemma 8.4], by propagation of singularities for the equation (2.44) we get for t 1 large enough, for an appropriate choice of A 0 . By (2.45) and (2.46), we see that Breaking the second integral on the right-hand side in two pieces and estimating each of them separately by (2.47) and (2.48), we get for an appropriate choice of A 0 , We can moreover assume that X − has the form A * 1 A 1 + X * 1 X 1 + O(h ∞ ) for some pseudodifferential operator X 1 ; this can be arranged since σ(X − ) > 0 on WF h (A 1 ) and the argument of [Dy13,§8.2] only depends on the principal symbol of X − , which can be taken to be the square of a smooth function. Then (2.49) To prove (2.42), it remains to note that (1 − Π)u(t) = v(t) + O(h ∞ ) microlocally near W and v(t 1 ) L 2 ≤ Ch −1 u(t 1 ) L 2 by part 1 of [Dy13,Proposition 6.13].
To prove (2.43), we argue similarly to (2.45), but use the equation (2.29) instead of (2.44). We get and the inequality is strict on W . The remainder of the proof of (2.43) proceeds exactly as the proof of (2.49).
(2.54) Therefore, Next, by (2.21) each backwards flow line of e tHp starting in X 2δ 1 either stays forever in X 2δ 1 or reaches the complement of WF h (u(t)) -see the proof of Proposition 2.4. By propagation of singularities for the equation (2.29), we find Also, for t 1 large enough, each flow line γ(t), t ∈ [−t 1 , 0], of H p such that γ(0) ∈ X δ 1 either satisfies γ(−t 1 ) ∈ W and γ([−t 1 , 0]) ⊂ X δ 1 , or there exists s ∈ [−t 1 , 0] such that γ(s) ∈ WF h (u(t)) for t ∈ [t 1 , T log(1/h)] and γ([s, 0]) ⊂ X δ 1 . This is true since if γ(s) ∈ WF h (u(t)), then γ([s − t 1 , s]) ⊂ X δ 1 , see the proof of Proposition 2.4. By propagation of singularities for (2.29), we get (2.55) By (2.29) and (2.51), we have (hD t + P )( Su R (t)) = O(h ∞ ) on X δ 1 . Using propagation of singularities for this equation in a manner similar to (2.55), we obtain by (2.54) Combining these estimates with (2.39)-(2.43), we arrive to holding uniformly in t ∈ [t 0 − 1, T log(1/h)]. To obtain (2.22)-(2.26) from here, we need to remove the operator S from the estimates; for that, we can use the fact that S is bounded uniformly in h on L 2 t,x together with the equivalency of the norms h · L ∞ t Ex and · L 2 t,x for solutions of the wave equation (2.17) given by (2.19) and the functions of interest being microlocalized at frequencies ∼ h −1 .
The metric is originally defined on here θ ∈ [0, π] and ϕ ∈ S 1 = R/(2πZ) are the spherical coordinates on S 2 . The numbers r − < r + are the roots of ∆ r defined below; in particular, ∆ r > 0 on (r − , r + ) and ±∂ r ∆ r (r ± ) < 0. The metric becomes singular on the surfaces {r = r ± }, known as the event horizons; however, this can be fixed by a change of coordinates, see §3.4.
The Kerr(-de Sitter) family admits the scaling M → sM, Λ → s −2 Λ, a → sa, r → sr, t → st for s > 0; therefore, we often consider the parameters a/M and ΛM 2 invariant under this scaling. We assume that a/M, ΛM 2 lie in a neighborhood of the Schwarzschild(-de Sitter) case (1.1) or the Kerr case (1.2). Then for Λ > 0, ∆ r is a degree 4 polynomial with real roots r 1 < r 2 < r − < r + , with r − > M . For Λ = 0, ∆ r is a degree 2 polynomial with real roots r 1 < M < r − ; we put r + = ∞. The general set of Λ and a for which ∆ r has all real roots as above was studied numerically in [AkMa,§3], and is pictured on Figure 1(a) in the introduction. Note that in [AkMa], the roots are labeled r −− < r − < r + < r C ; we do not adopt this (perhaps more standard) convention in favor of the notation of [Dy11a,Dy12,Va10], and since the roots r 1 , r 2 are irrelevant in our analysis.
We now assume that r ≤ 14M and µ < δ 0 . Then (here the constants do not depend on δ 0 and are locally uniform in M, Λ, a) Then for δ 0 small enough, by (3.13) the expression (3.14) has the same sign as 3.2. Structure of the trapped set. We now study the structure of trapping for Kerr(-de Sitter) metrics, summarized in the following Proposition 3.2. For (ΛM 2 , a/M ) in a neighborhood of the union of (1.1) and (1.2), assumptions (5)-(8) of §2.2 are satisfied, with µ max = 0 (see (2.11)) and the trapped set (see Definition 2.1) given by Remark. The assumptions on M, Λ, a can quite possibly be relaxed. The only parts of the proof that need us to be in a neighborhood of (1.1) or (1.2) are (3.18) and (3.19). Several other statements require that α is small (in particular, (3.27) requires α < √ 2−1 √ 2+1 ), but this is true for the full admissible range of parameters depicted on Figure 1(a) in the introduction. However, if Λ, a are large enough so that r ∈ (r − , r + ), ∂ r ∆ r (r) = 0 =⇒ ∆ r (r) = a 2 , (3.16) then the trapped set contains points with τ = 0 (and also ξ r = ξ θ = 0, θ = π/2, ξ ϕ = 0), which prevent us from having a meromorphic continuation of the resolvent and violate the required assumption (5) in §2.2 -see the discussion preceding [Va10,(6.13)]. The set of values of Λ, a satisfying (3.16) is pictured as the dashed line on Figure 1(a).
We next claim that ∂ 2 r G < 0 on K.
To study the behavior of K in the angular variables, we introduce the equatorial set This is a closed conic subset of K invariant under the flow (2.3) (which is proved similarly to the invariance of K). We have Indeed, (3.24) Putting θ = π/2 into (3.24), we solve for ∆ r and substitute it into (3.23), obtaining At the poles {θ = 0, π}, we have This follows immediately from (3.3), as ξ 1 = ξ 2 = 0 would imply G θ = 0, which is impossible given that ξ r = 0, G = 0, andξ = 0.
We now consider the following two cases: Case 1: |Φ 0 (r)| + |∂ r Φ 0 (r)| > 0 for all r ∈ (r − , r + ). In this case, the set of solutions to (3.30) is a closed one-dimensional submanifold of T * (r − , r + ) and the Hamiltonian field of H 0 is nonvanishing on this manifold. This manifold has no compact connected components, as the function Φ 0 (r) cannot achieve a local maximum on it by (3.31). It follows that the geodesic (x(t),ξ(t)) escapes in both time directions.
We also see from (3.19) and the fact that ∂ τp = 0 on {p = 0}\0 (as follows from (3.5)) that the matrix of Poisson brackets of functions G, ∂ r G, ξ r , t on K is nondegenerate, which implies that the intersections K ∩ {t = const} are symplectic submanifolds of T * X 0 . Together with Proposition 3.5, this verifies assumptions (6) and (7) of §2.2.
It remains to verify r-normal hyperbolicity of the flowφ s defined in (2.3). We start by showing that the maximal expansion rate in the directions of the trapped set µ max , defined in (2.11), is equal to zero: Proposition 3.6. For each ε > 0, there exists a constant C such that for each v ∈ T K, |dφ s v| ≤ Ce ε|s| |v|.
Here | · | denotes any fixed smooth homogeneous norm on the fibers of T K.
Proof. Using the group property of the flow, it suffices to show that for each ε > 0 there exists T > 0 such that for each v ∈ T K, |dφ T v| < e εT |v|. (3.32) Since K is a closed conic set, and K ∩ {τ = 1} ∩ {t = 0} is compact, it suffices to show that for each flow line γ(s) of (2.3) on K, there exists T such that (3.32) holds If γ(s) is a trajectory of (2.3) on T K, then r, ξ r = 0, τ are constant on γ(s) and the generator of the flow does not depend on the variable t; therefore, it suffices to show (3.32) for the restriction of the matrix of dφ T to the ∂ θ , ∂ ϕ , ∂ ξ θ , ∂ ξϕ variables. This is equivalent to considering the Hamiltonian flow of G in the θ, ϕ, ξ θ , ξ ϕ variables only, on T * S 2 . Recall that the equatorial set K e = K ∩ {θ = π/2, ξ θ = 0} defined in (3.21) is invariant underφ s . We then consider two cases: Case 1: γ(s) ∈ K e for all s. Then the differentials of G and ξ ϕ are linearly independent by (3.26) and (3.27). Since {G, ξ ϕ } = 0, by Arnold-Liouville theorem (see for example [Dy13, Proposition 2.8]), there is a local symplectomorphism from a neighborhood of γ(s) in T * S 2 to T * T 2 , where T 2 is the two-dimensional torus, which conjugates G to some function f (η 1 , η 2 ); here (y 1 , y 2 , η 1 , η 2 ) are the canonical coordinates on T * T 2 . The corresponding evolution of tangent vectors is given by ∂ s v y (s) = ∇ 2 f (η(s))v η (s), ∂ s v η (s) = 0, and (3.32) follows.
Let V ± be the one-dimensional subbundles of T Γ ± defined in (2.7), invariant under the flowφ s . Since dφ ∓ vanishes on T K and is not identically zero on T K Γ ± , we can fix a basis v ± of V ± | K by requiring that dφ ∓ · v ± = 1.
Denote by V = H G /∂ τ G the generator of the flowφ s . The Lie derivative L V v ± is a multiple of v ± ; to compute it, we use the identity Since (3.33) holds on Γ + ∪ Γ − , we get on vectors tangent to Γ ± , which implies immediately Proposition 3.7. The expansion rates defined in (2.9) and (2.10) are given by whereν > 0 is the function on K defined in (3.34) and • ϕ s ds.
Together, Proposition 3.6 and 3.7 verify assumption (8) of §2.2 and finish the proof of Proposition 3.2.
3.3. Trapping in special cases. We now establish some properties of the trapped set K and the local expansion rateν, defined in (3.34), in two special cases. We start with the Schwarzschild(-de Sitter) case (1.1), when everything can be described explicitly: Proposition 3.8. For a = 0, we have (3.36) Proof. We recall from (3.17) that K is given by the equations G = 0, ξ r = 0, Ψ = 0, where Ψ is computed using (3.20): Ψ(r) = 2τ r 2 (r − 3M ).
We next consider the case when Λ = 0 and a approaches the maximal rotation speed M from below, calculating the expansion rates on two equators to show that the pinching condition (1.12) is violated: Proposition 3.9. Fix M and assume that Λ = 0, a = M − , 0 < 1.
Then K e , defined in (3.21), is the union of two conical sets where R + ( ), F + ( ) are smooth functions of , R − ( ), F − ( ) are smooth functions of √ , and (see Figure 4) Finally, the expansion ratesν defined in (3.34) are given by (see also Figure 2(a) in the introduction) Proof. The set K e is defined by equations ξ r = ξ θ = 0, θ = π/2, and (see (3.17)) ((r 2 + a 2 )τ + aξ ϕ ) 2 = ∆ r (r)(aτ + ξ ϕ ) 2 , 4rτ ∆ r (r) = ((r 2 + a 2 )τ + aξ ϕ )∂ r ∆ r (r). Recall that ∆ r (r) = r 2 + a 2 − 2M r. Putting A = (r 2 + a 2 )τ + aξ ϕ and B = aτ + ξ ϕ , we rewrite these as The second equation can be written as (r 2 + 2a 2 − 3M r)A = 2a∆ r (r)B. Solving for B and substituting into the first equation, we get This is a fourth order polynomial equation in r with coefficients depending on and with a root at r = 0; we will study the behavior of the other three roots as → 0. We write (3.40) as By the implicit function theorem, for small enough, the equation (3.40) has a solution R = R + ( ) = 4M + O( ). We next identify the two roots lying near r = M ; they are solutions to the equations The solution with the negative sign lies to the left of r − > M , therefore we ignore it. The solution with the positive sign, which we denote by R − ( ), exists for small enough by the implicit function theorem and we find R − ( ) = M + 8 M/3 + O( ).
We start by formulating a well-posed problem for the wave equation on the Kerrde Sitter background. For that, we in particular need to shift the time variable, see [Dy11a,§1] and [Dy12,§1.1]. Let µ be the defining function of the event horizons and/or spatial infinity defined in (3.6) and fix a small constant δ 1 , used in Theorem 4 as well as in (2.13). To continue the metric smoothly past the event horizons, we make the change of variables where F t , F ϕ are smooth real-valued functions on (r − , r + ) such that • F t (r) = ± 1+α ∆r(r) (r 2 + a 2 ) + f ± (r) and F ϕ (r) = ± 1+α ∆r(r) a near r = r ± , where f ± are smooth functions (for the Kerr case Λ = 0, we only require this at r = r − ) • F t (r) = F ϕ (r) = 0 near {µ ≥ δ 1 /10} (and also for r large enough in the Kerr case Λ = 0); • the covector dt * is timelike everywhere; equivalently, the level surfaces of t * are spacelike.
Consider the initial-value problem for the wave equation (here s ≥ 0 is integer) This problem is well-posed, based on standard methods for hyperbolic equations [Tay,§6.5] and the following crude energy estimate: if we consider functions on X −δ 1 as functions of t * with values in functions on X −δ 1 , then for t ≥ 0, (3.46) To prove (3.46) for s = 0, we use the standard energy estimate on Ω = X −δ 1 ∩ {0 ≤ t * ≤ t } for hyperbolic equations (see [Tay,§2.8], [Dy11a,Proposition 1.1], or [Dy11b, §1.1]), with the timelike vector field N equal to ∂ t (a Killing field) for large r (in the case Λ = 0) and tog −1 (dt * ) close to the event horizon(s); by (3.44), the boundary ∂Ω is spacelike and N points inside of Ω on {t * = 0} and outside of it elsewhere on ∂Ω.
The higher order estimates are obtained from here as in [Tay,(6.5.14)], commuting with differential operators in the scattering calculus.
We now assume that Let u be the corresponding solution to (3.45) and assume that it is extended to small negative times (which can be done by taking a smaller δ 1 and using the local existence result backwards in time). By (3.46), we see that u is h-tempered uniformly for t * ∈ [0, T log(1/h)]. Similarly to (2.18), WF h (u) ⊂ {p * = 0}. Moreover, using standard microlocal analysis for hyperbolic equations, we get a pseudodifferential one-to-one correspondence between (f 0 , f 1 ) and (u + (0), u − (0)), where u ± are the components of u microlocalized on C ± , the positive and negative parts of the light cone, each solving an equation of the form (hD t + P ± )u ± = O(h ∞ ) for some spatial pseudodifferential operators P ± (similarly to (2.28)). This gives By the same correspondence, if WF h (u) ∩ {t * = 0} is compact and covered by finitely many open subsets of T * X −δ 1 \ 0, then we can apply the associated pseudodifferential partition of unity to f 0 , f 1 to split u into several solutions to the wave equation such that the wavefront set of each solution at t * = 0 is contained in one of the covering sets. The resulting solutions can then each be analysed separately.
We next assume that supp f 0 ∪ supp f 1 ⊂ X δ 1 .
We obtain some restrictions on the microlocalization of u for large times. For that, we need to consider the dynamics of the geodesic flow on the extended spacetime X δ 1 . Define the flowφ s similarly to (2.3), rescaling the geodesic flow so that the variable t * is growing with speed 1. Since t = t * , ϕ = ϕ * on X δ 1 /10 , the flow lines ofφ s and ϕ s coincide on X δ 1 /10 . If γ(s) is a flow line ofφ s such that γ(0) ∈ X δ 1 /10 and γ is not trapped for positive times according to Definition 2.1, then either γ(s) escapes to the Euclidean infinity (for Λ = 0) or γ(s) crosses one of the event horizons at some fixed positive time s 0 , and µ(γ(s)) < 0 is strictly decreasing for s > s 0 (see the discussion following [Va10,(6.22)], verifying [Va10,(2.8)]); in the latter case, we say that γ escapes through the event horizons.
The next statement makes nontrivial use of the structure of the infinite ends (in particular, using [Me, VaZw, Da] for the asymptotically Euclidean end for Λ = 0) and is the key step for obtaining control on the escaping parts of the solution for long times: Proposition 3.10. Assume that all flow lines ofφ s starting on WF h (u) ∩ {t * = 0} escape, either to the spatial infinity or through the event horizons. Then there exists T 0 > 0 such that uniformly in t * , (3.48) Proof. We first consider the case when WF h (u) ∩ {t * = 0} is contained in a small neighborhood of some (x,ξ) ∈ T * X δ 1 \ 0, and, for γ(s) =φ s (x,ξ), there exists T 0 > 0 such that γ([0, T 0 ]) ⊂ X −3δ 1 /4 and γ(T 0 ) ∈ {µ < −δ 1 /2}. By propagation of singularities for the wave equation (see for example [Dy13,Proposition 3.4]), we see that Then the same bound holds for t * ≥ T 0 in place of T 0 by (3.46) (replacing δ 1 by δ 1 /2).
For the remainder of this proof, we consider the opposite case, when Λ = 0 and each flow line ofφ s starting on WF h (u) ∩ {t * = 0} escapes to the spatial infinity. Fix a large constant R 1 ; we require in particular that X δ 1 ⊂ {r < R 1 }. By propagation of singularities, similarly to the previous paragraph, we may shift the time parameter and assume that WF h (u) ∩ {t * = 0} is contained in a small neighborhood of some (x 0 ,ξ 0 ) ∈ T * X 0 \ 0, where r 0 > R 1 , ∂ s µ(φ s (x 0 ,ξ 0 ))| s=0 < 0. In fact, by (3.46) and finite speed of propagation, we may assume that for t * near 0, the support of u in x is contained in a compact subset of {r > R 1 }. Without loss of generality, we assume that τ 0 < 0. The trajectoryφ s (x 0 ,ξ 0 ) does not intersect {r ≤ R 1 } for s ≥ 0.
To show (3.49), we use the Fourier transform in time, Here ψ 1 (t) is supported in [−δ, ∞) and is equal to 1 on [δ, ∞), for some small fixed δ > 0. The integral converges, as u 1 (t) H 2 (R 3 ) ≤ Ce εt for each ε > 0, as follows from the standard energy estimate for the wave equation (see the paragraph following (3.46)) applied for the timelike Killing vector field ∂ t .
We now apply the results of [Me, VaZw, Da]. For this, note that for any fixed λ, the operator P (λ) lies in Melrose's scattering calculus on the radially compactified R 3 , and for Im λ > 0, the operator P (λ) is elliptic in this calculus in the microlocal sense (that is, elliptic as ξ and/or r go to infinity) -in fact, near r = ∞ the operator P (λ) is close to ∆ 0 − λ 2 , where ∆ 0 is the flat Laplacian on R 3 . Moreover, P (λ) is a symmetric operator when λ ∈ R. This implies that the proofs of [VaZw, Da] apply. Similarly to [Me,Theorem 2], for Im λ > 0, the operator P (λ) is Fredholm H 2 (R 3 ) → L 2 (R 3 ) and invertible for λ outside of a discrete set; we can then fix E > 0 such that P (λ+iE) is invertible for all λ ∈ R.

We can now give
Proofs of Theorems 1 and 2. Without loss of generality (replacing δ 1 by δ 1 /3) we may assume that supp f 0 ∪ supp f 1 ⊂ X 3δ 1 .
3.5. Results for resonances. In this section, we use the results of [Dy13] together with the analysis of § §3.1, 3.2 to prove Theorem 3. As in the statement of this theorem, we consider the Kerr-de Sitter case Λ > 0.
Following [Va10, §6.5], we embed X −δ 1 as an open set into a compact manifold without boundary X, extend P (z) to a second order differential operator on X depending holomorphically on z, and construct a complex absorbing operator Q(z) ∈ Ψ 2 (X), whose Schwartz kernel is supported inside the square of the nonphysical region {µ < 0}. Then [Va10, Theorem 1.1] for Im z ≥ −C 1 and s large enough depending on C 1 , P (z) − iQ(z) is a holomorphic family of Fredholm operators X s → H s−1 (X), where X s = {u ∈ H s (X) | (P (0) − iQ (0) (3.56) We now claim that the operator P(ω) satisfies all the assumptions of [Dy13, § §4.1, 5.1]. Most of these assumptions have already been verified in §2.3, relying on the assumptions of §2.2 which in turn have been verified in § §3.1, 3.2. Given the definition of the spaces H 1 := X s h , H 2 := H s−1 h (X), and the Fredholm property discussed above, it remains to verify assumptions (10) and (11) of [Dy13, §4.1], namely the existence of an outgoing parametrix. This is done by modifying the proof of [Va10,Theorem 2.15] exactly as at the end of [Dy13,§4.4].
Theorem 3 now follows directly by [Dy13, Theorems 1 and 2]; the constant c K is given by (2.16).
3.6. Stability. We finally discuss stability of Theorems 1-3, under perturbations of the metric. We assume that (X 0 ,g) is a Lorentzian manifold which is a small smooth metric perturbation of the exact Kerr(-de Sitter) (as described in §3.1 and with M, Λ, a in a small neighborhood of either (1.1) or (1.2)) and which is moreover stationary (that is, ∂ t is Killing). For perturbations of Kerr (Λ = 0) spacetime, we moreover assume that our perturbation coincides with the exact metric for large r (this assumption can be relaxed; in fact, all we need is for (3.46) and the analysis in Proposition 3.10 to apply, so we may take a small perturbation in the class of scattering metrics). We also assume that the perturbation continues smoothly across the event horizons in the coordinates (3.43). The initial value problem (3.45) is then well-posed, as {µ = −δ 1 } is still spacelike. The results of [Va10] still hold, as discussed in [Va10,§2.7].
It remains to verify that the assumptions of §2.2 still hold for the perturbed metric. Assumptions (1)-(3) are obviously true. Assumption (4) holds with the same function µ, at least for µ(γ(s)) ∈ (δ, δ 0 ), where δ 0 is fixed and δ > 0 is small depending on the size of the perturbation; we take a small enough perturbation so that δ δ 1 , where δ 1 > 0 is the constant used in Theorem 4 in §2.4 and in (2.13). Then the trapped set K for the perturbed metric is close to the original trapped set, which implies assumption (5). Finally, the dynamical assumptions (6)-(8) still hold by the results of [HiPuSh] and the semicontinuity of ν min , ν max , µ max , as discussed in [Dy13,§5.2].