Boundedness and Decay for the Teukolsky Equation on Kerr Spacetimes I: The Case |a|≪M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a|\ll M$$\end{document}

We prove boundedness and polynomial decay statements for solutions of the spin ±2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm \,2$$\end{document} Teukolsky equation on a Kerr exterior background with parameters satisfying |a|≪M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a|\ll M$$\end{document}. The bounds are obtained by introducing generalisations of the higher order quantities P and P̲\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline{P}}$$\end{document} used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters |a|<M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a|<M$$\end{document}. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.


Introduction
The stability of the celebrated Schwarzschild [100] and Kerr metrics [72] remains one of the most important open problems of classical general relativity and has generated a large number of studies over the years since the pioneering paper of Regge-Wheeler [98]. See [42,43] and the introduction of [31] for recent surveys of the problem.
The ultimate question is that of nonlinear stability, that is to say, the dynamic stability of the Kerr family (M, g a,M ) (including the Schwarzschild case a = 0), without symmetry assumptions, as solutions to the Einstein vacuum equations in analogy to the nonlinear stability of Minkowski space, first proven in the monumental [26]. A necessary step to understand nonlinear stability is of course proving suitable versions of linear stability, i.e. boundedness and decay statements for the linearisation of (1) around the Schwarzschild and Kerr solutions. This requires first imposing a gauge in which the equations (1) become well-posed. A complete study of the linear stability of Schwarzschild in a double null gauge has been obtained in our recent [31]. A key step in [31] was proving boundedness and decay for the so-called Teukolsky equation, to be discussed below in Sect. 1.1, which can be thought to suitably control the "gauge invariant" part of the perturbations. See already equation (2). These decay results were then used in [31] to recover appropriate estimates for the full linearisation of (1). The purpose of the present paper is to extend the boundedness and decay results of [31] concerning the Teukolsky equation (2) from the Schwarzschild a = 0 case to the very slowly rotating Kerr case, corresponding to parameters |a| M. We give a rough statement of the main result already in Sect. 1.2 below.
In part II of this series, we shall obtain an analogue of our main theorem for the case of general subextremal Kerr parameters |a| < M. The extremal case |a| = M is exceptional; see Sect. 1.3 for remarks on this and other related problems. In a separate paper, following our previous work on Schwarzschild [31], we will use the above result to show the full linear stability of the Kerr solution in an appropriate gauge.
We end this introduction in Sect. 1.4 with an outline of the paper.

The Teukolsky Equation for General Spin
The original approach to linear stability in the Schwarzschild case centred on socalled metric perturbations, leading to the decoupled equations of Regge-Wheeler [98] and Zerilli [113]. The Regge-Wheeler equation will in fact appear below as formula (7). This approach does not, however, appear to easily generalise to Kerr. Thus, it was a fundamental advance when Teukolsky [107] identified two gauge invariant quantities which decouple from the full linearisation of (1) in the general Kerr case. The quantities, corresponding to the extremal curvature components in the Newman-Penrose formalism [93], can each be expressed by complex scalars α [±2] which satisfy a wave equation, now known as the Teukolsky equation: with s = + 2 and − 2 respectively. The scalars are more properly thought of as spin ± 2 weighted quantities. This generalised an analogous property in the Schwarzschild case identified by Bardeen and Press [14]. These quantities govern the "gauge invariant" part of the perturbations in the sense that an admissible solution of the linearised Einstein equations whose corresponding α [±2] both vanish must be a combination of a linearised Kerr solution and a pure gauge solution [110]. Note that equation (2) can be considered for arbitrary values of s ∈ 1 2 Z. For s = 0, (2) reduces to the covariant wave equation g ψ = 0, while for s = ± 1, (2) arises as an equation satisfied by the extreme components of the Maxwell equations in a null frame [22].

Separability and the Mode Stability of Whiting and Shlapentokh-Rothman
An additional remarkable property of the Teukolsky equation (2) is that it can be formally separated, in analogy with Carter's separation [19] of the wave equation (i.e. the case of s = 0). The separation of the θ -dependence is surprising in the case a = 0 for all s because the Kerr metric only admits ∂ φ and ∂ t as Killing fields. It turns out that considering the ansatz where S [s] m (ν, cos θ) denote spin-weighted oblate spheroidal harmonics, one can derive from (2) an ordinary differential equation for α, which in rescaled form (see (151)) can be written as u + V [s] (ω, m, , r )u = 0 where for s = 0, the potential V [s] is complex valued. (Here denotes differentiation with respect to r * . See Sect. 2.1.) See already (153). The separation (3) was subsequently understood to be related to the presence of an additional Killing tensor [73]. Of course, the problem of decomposing general, initially finite-energy solutions of (2) as appropriate superpositions of (3) is intimately tied with the validity of boundedness and decay results, in view of the necessity of taking the Fourier transform in time. A preliminary question that can be addressed already solely at the level of (4) is that of "mode stability". Mode stability is the statement that there are no initially finite-energy solutions of the form (3) with Im(ω) > 0. This reduces to showing the non-existence of solutions of (4) with Im(ω) > 0 and exponentially decaying boundary conditions both as r * → ∞ and r * → −∞.
In the case a = 0, s = 0, then mode stability can be immediately inferred by applying the physical space energy estimate associated to the Killing vector field ∂ t to a solution of the form (3). The question is highly nontrivial for a = 0, already in the case s = 0, in view of the phenomenon of superradiance, connected to the presence of the so-called ergoregion where ∂ t is spacelike. For s = ± 2, the question is non-trivial even in the case a = 0, as there does not exist an obvious conserved energy current. (In separated form (4), this is related to the fact that the potential V [s] is now complex valued.) In a remarkable paper, Whiting [111] nonetheless succeeded in proving mode stability for (2) for all s in the general subextremal range of parameters |a| < M by cleverly exploiting certain algebraic transformations of the ode (4).
Mode stability has been extended to exclude "resonances" on the real axis, i.e. solutions u of (4) with ω ∈ R with appropriate boundary conditions, by Shlapentokh-Rothman [104] in the case s = 0, who had the insight that the transformations applied in [111] could be extended to the real axis using the theory of oscillatory integrals. Together with a continuity argument in a, [104] can be used to reprove the original [111], and this leads to certain simplifications. The argument generalises to s = ± 2. See also [6] where the techniques of [104] are combined with an alternative complex analytic treatment.
We emphasise that mode stability is a remarkable property tied to the specific form of the equation (2) and to the specific Kerr background, even for s = 0. Indeed, mode stability fails for a = 0 when an arbitrarily small Klein-Gordon mass is added, as was first suggested by [28,112] and proven recently in [103]. Even more surprisingly, mode stability fails when a well-chosen positive compactly supported potential is added to (2), or when the Kerr metric is itself sufficiently deformed, keeping however all its symmetries and separation properties, in a spatially compact region which can be taken arbitrarily far from the ergoregion [89].

Previous Work on Boundedness and Decay
The quantitative study of the Cauchy problem for (2) with s = 0, beyond statements for fixed modes, has become an active field in recent years. The study for higher spin is still less developed beyond the Schwarzschild case. We review some relevant previous work below.
The case s = 0, |a| < M. An early result [75] obtained boundedness for solutions to the Cauchy problem for the scalar wave equation on Schwarzschild (i.e. the case s = 0 and a = 0 of (2)) with regular, localised initial data. Even this involved nontrivial considerations on the event horizon, which can now be understood in a more robust way using the red-shift energy identity [39,43]. Following intense activity in the last decade (e.g. [3,16,18,36,[39][40][41]43,108]) there are now complete boundedness and decay results for (2) with s = 0 in the full subextremal range of Kerr parameters |a| < M [45].
The main difficulties in passing from a = 0 to a = 0 arise from superradiance, mentioned already in the context of mode stability, and the fact that trapped null geodesics no longer approach a unique value of r in physical space. The latter is relevant because integrated local energy decay estimates, an important step in the proof of quantitative decay, must necessarily degenerate at trapping. 1 One way of dealing with the latter difficulty is employing the separation (3) as a method of frequency localising integrated local energy decay estimates. See [40,43]. Once such an estimate is obtained, the difficulty of superradiance can easily be overcome in the |a| M case as the error terms in the ergoregion are small and can be absorbed. For alternative approaches, see [3,108].
The |a| < M case appears a priori to be much more complicated. It turns out, however, that the Schwarzschild-like structure of trapping survives, when appropriately viewed in phase space. Moreover, in the high frequency regime, one can quantify superradiance with the help of the fact that, quite fortuitously, superradiant frequencies happen not to be trapped. See [45]. These good high frequency properties, together with Shlapentokh-Rothman's real mode stability [104] and a continuity argument in a, allow one to extend the exact same boundedness and integrated local energy decay results originally obtained on Schwarzschild to the whole sub-extremal range |a| < M of Kerr parameters. Suitable polynomial decay then follows from an application of the method of r p weighted energy estimates [38,87]. See [45]. For comments on the extremal case |a| = M, see Sect. 1.3.4.
The case s = ± 2, a = 0. As we remarked already above, the Teukolsky equation with s = ± 2, a = 0 has been studied in our previous [31] as part of our complete study of the linear stability of Schwarzschild.
The main difficulty of the s = ± 2 case as opposed to the case s = 0, is that, as discussed already in the context of mode stability, there does not exist an obvious analogue of the conserved energy associated to the Killing field ∂ t . Thus, proving even just boundedness for a = 0 is non-trivial, even just far away from the event horizon.
The key to understanding (2) for s = ± 2, a = 0 in [31] was associating quantities P [±2] to α [±2] satisfying (2). These are physical space versions of transformations first considered by Chandrasekhar [22] and are defined by the expressions 2 Here where/ [±2] denotes the spin-2-weighted Laplacian on the unit sphere Remarkably, (7) is precisely the same equation which appeared as one of the equations governing the "metric perturbations" approach discussed at the beginning of Sect. 1.1! Unlike (2) with s = ± 2, the above equation (7) can be estimated on Schwarzschild just as for the wave equation s = 0, since (7) is indeed endowed with the usual structure of energy estimates. In particular, both boundedness and integrated local energy decay can be obtained. (For analysis of (7), see the previous [17,63] as well as the self-contained treatment in [31].) Estimates for α [±2] could then be recovered directly by integrating (5) as transport equations from initial data. Such integration on its own would lead, however, to "loss of derivatives" in the resulting estimates for α [±2] . The Teukolsky equation itself (2) can be viewed as a further elliptic relation which allows one to gain back these derivatives, leading finally to boundedness results without loss of derivative, as well as integrated local energy decay and pointwise decay.
We remark that, beyond (2), in the context of the full proof of linear stability in [31], further transport equations and elliptic equations could then be used to appropriately estimate the remaining gauge dependent quantities.
Other spins. We note that the scheme of [31] has recently been applied also to the s = ± 1 case by Pasqualotto [94]. This gives an alternative proof of boundedness and polynomial decay for the Maxwell equations on Schwarzschild, proven originally by Blue [12]. See also [105]. Decay for Maxwell in the case |a| M was obtained in 3 See Sects. 2.4 and 3.3 for the precise relation between the tensorial Regge-Wheeler equation defined in [31] and equation (7). Note in particular that [+2] and [−2] satisfy the same equation, which explains the appearance of a single Regge-Wheeler equation in [31]. We also note that both the operators/  [4]. For a direct treatment of (2) for s = ± 1 in the case |a| M, generalising some of the results of [94], there is the recent [81]. (This has more recently been followed by [Ma18b]; see Sect. 1.3.9.) For the cases s = ± 1/2 and s = ± 3/2 see [106]. See [49] for the related massive Dirac equation not covered by (2). We note also the papers [52,53].

The Main Result and First Comments on the Proof
The aim of the present paper is to extend the analysis of (2) for s = ± 2 from the Schwarzschild a = 0 case considered in [31] to the very slowly rotating Kerr case with parameters |a| M. A rough version of our main result is the following: to the spin s = ± 2 Teukolsky equation (2) on Kerr exterior spacetimes (M, g a,M ) arising from regular localised initial data on a Cauchy hypersurface 0 remain uniformly bounded and satisfy an r p -weighted energy hierarchy and polynomial decay.
The precise statements embodying the above will be given as Theorem 4.1. See also immediately Corollary 4.1 and (38) for the pointwise decay statements obtained.
The proof of our Theorem combines the use of the quantities P [±2] introduced in our previous [31] with a simplified version of the framework introduced in [40,45] for frequency localised energy estimates, which as discussed in Sect. 1.1.2 are useful to capture the obstruction to decay associated with trapped null geodesics. (In the special case of axisymmetric solutions, this frequency localisation can be avoided and our proof can be expressed entirely in physical space. See already Sect. 1.2.5.) The crucial observation which allows this technique to work is the following: In the scheme introduced in [31], it is not in fact absolutely necessary that the quantities P [±2] each satisfy a completely decoupled equation (7). It would have been permissible if the equation (7) for P [±2] was somehow still coupled to α [±2] on the right hand side, provided that this coupling was at a suitable "lower order", in the sense that these lower order terms can indeed be recovered by the transport (and elliptic equations) which were used in [31] to estimate α [±2] .
It turns out, remarkably, that when analogues of the quantities P [±2] are defined for Kerr, even though the exact decoupling from α [±2] , respectively, breaks down, the resulting equations indeed only couple to α [±2] in the "weak" sense described above.
We explain below this structure in more detail, and how it is implemented in our proof (where we will in fact be able to circumvent use of elliptic estimates).

The Generalisation of P [±2] to Kerr
Our physical-space definition for P [+2] , generalising (5), is given as where R [+2] is a second order operator defined on Kerr generalising the Regge-Wheeler operator appearing on the left hand side of (7), which has good divergence properties and thus admits energy currents. Consistent with the total decoupling in the Schwarzschild case, the coefficient functions c i (r ) above are all O(|a|). Provided that α [+2] can indeed be viewed as being of two degrees lower in differentiability than [+2] , then the right hand side is "zero'th order" in [+2] . Let us note, however, that if we use only the transport relation (9), then the right hand side of (10) can only be viewed as "first order" in [+2] , as integration of (9) does not improve differentiability. Thus, to exploit fully this structure, one must also invoke in general elliptic relations connecting α [+2] and [+2] that can be derived by revisiting equation (2) itself. As we shall see below, it turns out, however, that we shall be able to avoid invoking this by exploiting more carefully the special structure and the non-degeneration of the derivative ∂ r * [±2] . We describe in Sects. 1.2.2-1.2.3 how these terms can be controlled.
We emphasise already that the above structure of the terms appearing on the right hand side of (10) is surprising. Upon perturbing (7) one would expect higher order terms in [±2] to appear which cannot be incorporated in the definition of R [+2] so as to preserve its good divergence properties. We note already that in the axisymmetric case, the right hand side of (10) is of even lower order, as the ∂ φ derivatives vanish. The deeper reason why these terms cancel is not at all clear. See also the remarks in Sect. 1.2.6 below.

Estimates Away from Trapping
Away from trapping, it suffices to treat the right hand side of (10) as if it were at the level of a general "first order" perturbation in [+2] .
To see this, let us note first that suitably away from r = 3M, the f and y-multiplier estimate of [31] leads in the Schwarzschild case to a coercive spacetime integral containing all first derivatives of [+2] (with suitable weights towards the horizon and infinity). This coercivity property away from trapping is manifestly preserved to perturbations to Kerr for |a| < a 0 M sufficiently small. We may add also a small multiple of the r η -current for an η > 0 to generate extra useful weights near infinity. Moreover, we may add a suitable multiple of the energy estimate associated to a vector field ∂ t + χ ω + ∂ φ which connects the Hawking vector field on the horizon with the stationary vector field ∂ t . This ensures positive boundary terms on suitable spacelike and null boundaries, at the expense of generating an O(|a|) bulk term supported where χ = 0, which is chosen to be away from trapping. Thus, this bulk term can again be absorbed by the coercive terms of the f and y-multipliers.
On the other hand, commutation of equation (9) by the Killing fields ∂ t and ∂ φ allows one to estimate all terms involving α and Lα and their derivatives appearing on the right hand side of (10) from the spacetime estimate for [+2] by appropriate transport estimates. (Here, we note that we must make use of the extra r η weight, just as in [31].) Thus, were it not for trapping, one could easily absorb the error terms on the right hand side of (10).

Frequency Localised Analysis of the Coupled System Near Trapping
In view of the above discussion, the terms on the right hand side of (10) are most dangerous near trapping. Let us take a more careful look at the structure of (9)-(10) using our frequency analysis.
At the level of the formally separated solutions (3), the operator L takes the form where r * is a Regge-Wheeler type coordinate, the relation (9) reads where w := (r 2 + a 2 ) 2 (13) and the "Regge-Wheeler" type equation (10) takes the form Here V is a real potential depending smoothly on a which reduces to the separated version of the Regge-Wheeler potential for a = 0 and the c i are bounded functions. Cf. (10) and see Appendix A.3. At the separated level, using a frequency localised version of the current f of [31], chosen to vanish at the (frequency-dependent) maximum of the potential V as in of [40], together with a frequency localised y-current and the frequency-localised energy estimate (multiplication by ω ) one can prove the ODE analogue of a degenerating integrated local energy decay for [±2] , with a right hand side involving the right hand side of (14). Considerations are different in the "trapped frequency range" This should be thought of as a degenerate integrated local energy decay bound for [+2] . Considering the right hand side of (16), we note that naive integration of (12) as a transport equation is not sufficient to control the integral on the right hand side by the left hand side. This is not surprising: In constrast to the considerations away from trapping of Sect. 1.2.2, in general now only terms which can be truly thought of as "zero'th order" in [+2] can manifestly be absorbed by the left hand side of (16), in view of the absence of an ω 2 | [+2] | 2 and | [+2] | 2 coercive term.
One way to try to realise the right hand side of (16) as "zero'th order" in [+2] would be to invoke, in addition to the transport (12), also the elliptic estimates of [31]. It turns out, however, that exploiting the presence of the good first order term |∂ r [+2] | 2 on the left hand side of (16), one can argue in a more elementary manner: Indeed, by commuting (12) with ∂ r * and exploiting the relation (11), one can indeed rewrite the right hand side so as to absorb it into the left hand side.
Let us note finally that for "non-trapped" frequencies (i.e. outside the frequency range (15)), one can arrange the frequency localised multiplier so that terms m 2 | [+2] | 2 and ω 2 | [+2] | 2 appear on the left hand side of (16) without degeneration. One can then easily absorb the right hand side just as in Sect. 1.2.2 treating it essentially as one would a general "first order" term.

Technical Comments
Let us discuss briefly the technical implementation of the above argument.
As in [40], by using the smallness of the Kerr parameter a, the fixed frequency analysis of Sect. 1.2.3, restricted entirely to real frequencies ω ∈ R, can indeed be implemented to general solutions α [±2] of the Cauchy problem for (2), despite the fact that we do not know a priori that solutions are square integrable in time. This requires, however, applying cutoffs to α in order to justify the Fourier transform, and thus one must estimate inhomogeneous versions of (2) and thus also inhomogeneous versions of the resulting ODE (14). These inhomogeneous terms must themselves be bound by the final estimates.
As opposed to the cutoffs of [40,45], we here will only cut off the solution in a region r * ∈ [2 A * 1 , 2 A * 2 ] near trapping. Thus, the resulting inhomogeneous terms will be supported in a fixed region of finite r * . Moreover, the fixed frequency ODE estimates of Sect. 1.2.3 will only be applied in the region r * ∈ [A * 1 , A * 2 ]. They will be combined with physical space estimates of Sect. 1.2.2. These estimates are now coupled however via boundary terms on r = A 1 and r = A 2 . The fixed frequency multipliers applied to [+2] are chosen so as to be frequency independent near A 1 and A 2 and coincide precisely with those used in the physical space estimates in the away region. As a result, after summation over frequencies, the boundary terms in the mutliplier currents exactly cancel. This is similar to a scheme used previously in [7]. There are also boundary terms associated with the transport equations, but these can be absorbed using the smallness of a.
The above argument leads to a degenerate energy boundedness and integrated local energy decay for both [±2] and α [±2] . This preliminary decay bound will be stated as Theorem 9.1. From Theorem 9.1, we can easily improve our estimates at the event horizon, using the red-shift technique of [39], and then we can easily infer polynomial decay using the weighted r p method of [38]-all directly in physical space.

The Axisymmetric Case
We have already remarked that in the axisymmetric case ∂ φ α [±2] = 0, the right hand side of (10) is of lower order. An even more important simplification is that trapped null geodesics all asymptote to a single value of r = r trap which is near 3M, independent of frequency. As a result, there is no need for frequency-localised analysis and the whole argument can be expressed entirely in physical space. This is convenient for non-linear applications. We shall explain how this simplified argument can be explicitly read off from our paper in Sect. 9.6.

Final Remarks
Given the analogue of [104] for s = ± 2, the argument can in principle be applied for the whole subextremal range |a| < M following the continuity argument of [45], but in the present paper we shall only consider the case |a| M, where the lower order terms also have a useful smallness factor bounded by a, and the relevant multiplier currents can thus be constructed as perturbations of Schwarzschild. The general case will be considered in part II of this series, following the more general constructions of [45].
There are other generalisations of P [±2] to Kerr which have been considered previously in the literature, see [21,102] and the recent review [57]. In contrast to our situation, the quantities of [21,102] do indeed satisfy decoupled equations, though the transformations must now be defined in phase space, and the transformed potentials are somewhat non-standard in their frequency dependence. It would be interesting to find an alternative argument using these transformations. We hope to emphasise with our method, however, that exact decoupling is not absolutely necessary for quantities to be useful.

Other Related Results
We collect other related recent results concerning the stability of black holes. The literature has already become vast so the list below is in no way exhaustive. See also the surveys [42,43].

Metric Perturbations
An alternative approach to linear stability in the Schwarzschild case would go through the theory of so-called metric perturbations. See for instance [60,71] for estimates on the additional Zerilli equation which must be understood in that approach. We note the paper [35].

Canonical Energy
As discussed above, one of the difficulties in understanding linearised gravity is the lack of an obvious coercive energy quantity for the full system, even in the a = 0 case. The Lagrangian structure of the Einstein equations (1) does give rise however to a notion of canonical energy, albeit somewhat non-standard in view of diffeomorphism invariance, and this can indeed be used to infer certain weak stability statements. For some recent results which have been obtained using this approach, see [69,97] and the related [64].

Precise Power-Law Asymptotics
Though one expects that the decay bounds obtained here are in principle sufficient for non-linear applications, it is of considerable interest for a wide range of problems to obtain sharp asymptotics of solutions, of the type first suggested by [96]. For upper bounds on decay compatible with some of the asymptotics of [96], see [47,90,91]. Lower bounds were first obtained in [76]. The most satisfying results are the sharp asymptotics recently obtained by [1,2] for the s = 0, a = 0 case. Such results in particular have applications to the interior structure of black holes (see [76]).

Extremality and the Aretakis Instability
Whereas some stability results for s = 0 carry over to the extremal case |a| = M, it turns out that, already in axisymmetry [7], the transversal derivatives along the horizon grow polynomially [7,8]. This phenomenon is now known as the Aretakis instability. The Aretakis instability has been shown to hold also in the case s = ± 2 by [77]. Understanding the non-axisymmetric case is completely open; see [5] for some of the additional new phenomena that arise.

Nonlinear Model Problems and Stability Under Symmetry
Though nonlinear stability of both Schwarzschild and Kerr is still open, various model problems have been considered which address some of the specific technical difficulties expected to occur.
Issues connected to the handling of decay for quadratic nonlinearities in derivatives are addressed in the models considered in [79,80]. The Maxwell-Born-Infeld equations on Schwarzschild were recently considered in [95]. This latter system, of independent interest in the context of high energy physics, can be thought to capture at the same time aspects of both the quasilinear difficulties as well as the tensorial difficulties (at the level of s = ± 1) inherent in (1).
Turning to stability under symmetry, the literature is vast. For the Einstein-scalar field system under spherical symmetry, see [23,36]. For the vacuum equations (1), [62] provides the first result on the non-linear stability of the Schwarzschild solution in symmetry, considering biaxial symmetry in 4 + 1-dimensions. This again reduces to a 1 + 1 problem. Beyond 1 + 1, some aspects of the vacuum stability problem in axisymmetry are captured in a wave-map model problem whose study was initiated by [70]. Very recently, Klainerman-Szeftel [74] have announced a proof of the non-linear stability of Schwarzschild in the polarised, axisymmetric case.

Analogues with = 0
There are analogues of the questions addressed here when the Schwarzschild and Kerr solutions are replaced with the Schwarzschild-(anti) de Sitter metrics and Kerr-(anti) de Sitter metrics, which are solutions of (1) when a cosmological term g μν is added to the right hand side. These solutions are discussed in [20].
In the de Sitter case > 0, the analogous problem is to understand the stability of the spatially compact region bounded by the event and so-called cosmological horizons. Following various linear results [13,37,48,59,109] the full non-linear stability of this region has been obtained in remarkable work of Hintz-Vasy [67]. This de Sitter case is characterized by exponential decay, so many of the usual difficulties of the asymptotically flat case are not present. The stability of the "cosmological region" beyond the event horizon has been considered in [101].
The case of < 0 has been of considerable interest in the context of high energy physics. Already, pure AdS spacetime fails to be globally hyperbolic. In general, asymptotically AdS spacetimes have a timelike boundary at infinity where boundary conditions must be prescribed to obtained well-posed problems.
For reflective boundary conditions, the analogue of equation (2) on pure AdS space admits infinitely many periodic solutions. In view of this lack of decay in the reflective case, it is natural to conjecture instability at the non-linear level [29], once backreaction is taken into account. 4 This nonlinear instability has indeed been seen in the seminal numerical study [15], which moreover sheds light on the relevance of resonant frequencies for calculating a time-scale for growth. Very recently, the full nonlinear instability of pure AdS space has been proven in the simplest model for which the problem can be studied [88], exploiting an alternative physical-space mechanism.
In the case of Kerr-AdS, one has logarithmic decay [65]-but in general no faster [66]-for the analogue of (2) with s = 0, on account of the fact that trapped null geodesics, in contrast with the situation described in Sect. 1.1.2, are now stable. Again, these results may suggest instability at the non-linear level, as this slow rate of decay is in itself insufficient to control backreaction.

Scattering Theory
A related problem to that of proving boundedness and decay is developing a scattering theory for (2). Fixed frequency scattering theory for (2) is discussed in [22]. It was in fact the equality of the reflexion and transmission coefficients between the Teukolsky, Regge-Wheeler and Zerilli equations that first suggested the existence of Chandrasekhar's transformations [22]. A definitive physical space scattering theory was developed in the Schwarzschild case in [32,33] for s = 0, see also [92], and was recently extended to the Kerr case in [44] for the full sub-extremal range of parameters |a| < M.
Turning to the fully non-linear theory of (1), a scattering construction of dynamic vacuum spacetimes settling down to Kerr was given in [30]. The free scattering data allowed in the latter were very restricted, however, as the radiation tail was required to decay exponentially in retarded time, and thus the spacetimes produced are measure zero in the set of small perturbations of Kerr relevant for the stability problem.
For scattering for the Maxwell equations, see [9]. For results in the > 0 case, see [56,85] and references therein.

Stability and Instability of the Kerr Black Hole Interior
The conjectured non-linear stability of the Kerr family refers only to the exterior of the black hole region. Considerations in the black hole interior are of a completely different nature. The Schwarzschild case a = 0 terminates at a spacelike singularity, whereas for the rotating Kerr case 0 < |a| < M, the Cauchy development of twoended data can be smoothly extended beyond a Cauchy horizon. The s = 0 case of (2) in the Kerr interior (as well as the simpler Reissner-Nordström case) has been studied in [46,50,51,58,76,78,83,84], and both C 0 -stability but also H 1 -instability have been obtained. See [54,55] for the extremal case. In the full nonlinear theory, it has been proven that if the Kerr exterior stability conjecture is true, then the bifurcate Cauchy horizon is globally C 0 -stable [34]. This implies in particular that the C 0 inextendibility formulation of "strong cosmic censorship" is false. See [24].

Note Added
Very recently, [82] gave a related approach to obtaining integrated local energy decay estimates for the Teukolsky equation in the |a| M case, following the frequency localisation framework of [40] and again based on proving estimates for defined by generalisations of the transformations used in [31].

Outline of the Paper
We end this introduction with an outline of the paper.
We begin in Sect. 2 by recalling the notation from [45] regarding the Kerr metric and presenting the Teukolsky equation in physical space for spin s = ± 2.
In Sect. 4 we shall define various energy quantities which will allow us in particular to formulate our definitive (non-degenerate) boundedness and decay results, stated as Theorem 4.1.
The first step in the proof of Theorem 4.1 is to obtain integrated local energy decay. In Sect. 5, we shall prove a conditional such estimate, using entirely physical space methods, for the coupled system satisfied by [±2] , ψ [±2 , and α [±2] . In view of the way this will be used, we must allow also inhomogeneous terms on the right hand side of the Teukolsky equation. We apply the physical space multiplier estimates and transport estimates and transport estimates directly from [31], except that these estimates must now be coupled. The resulting estimates (see the propositions of Sects. 5.1 and 5.2) contain on their right hand side an additional timelike boundary term on r = A 1 and r = A 2 for A 1 < 3M < A 2 . To control these terms, we will have to frequency localise the estimates in the region r ∈ [A 1 , A 2 ]. We also give certain auxiliary physical space estimates for the homogeneous Teukolsky equation and its derived quantities (Sect. 5.3).
The next three sections will thus concern frequency localisation. Sect. 6 will interpret Teukolsky's separation of (2) for spin s = ± 2 in a framework generalising that introduced in [45] for the s = 0 case. In Sect. 7, we define the frequency localised versions of P [±2] and derive the coupled system of ordinary differential equations satisfied by the P [±2] and α [±2] . In Sect. 8 we then obtain estimates for this coupled system of ODE's in the region r ∈ [A 1 , A 2 ]. The main statement is summarised as Theorem 8.1 and can be thought of as a fixed frequency version of the propositions of Sects. 5.1-5.2, now valid in r ∈ [A 1 , A 2 ]. The estimate is again conditional on controlling boundary terms, but the energy currents will have been chosen so that the most difficult of these, when formally summed, exactly cancel those appearing in the proposition of Sect. 5.1.
In Sect. 9, we shall turn in ernest to the study of the Cauchy problem for (2) to obtain a preliminary degenerate energy boundedness and integrated local energy decay estimate in physical space. This is stated as Theorem 9.1. To obtain this, we cut off our solution of (2) in the future so as to allow for frequency localisation in r ∈ [A 1 , A 2 ]. This allows us to apply Theorem 8.1 and sum over frequencies. We apply also the propositions of Sects. 5.1-5.2 to the cutoff-solution and sum the estimates. The cutoff generates an inhomogeneous term which is however only supported in a compact spacetime region. By revisiting suitable estimates, the cutoff term can then be estimated exploiting the smallness of a, following [40]. (We note that the fact that these cutoffs are here supported in a fixed, finite region of r leads to various simplifications.) We distill a simpler purely physical-space proof for the axisymmetric case in Sect. 9.6.
The final sections will complete the proof of Theorem 4.1 from Theorem 9.1, by first applying red-shift estimates of [39] to obtain non-degenerate control at the horizon (Sect. 10) and then the r p -weighted energy hierarchy of [38] (Sect. 11). This part follows closely the analogous estimates in the Schwarzschild case [31].
Some auxilliary computations are relegated to Appendix A and B.

The Teukolsky Equation on Kerr Exterior Spacetimes
We recall in this section the Teukolsky equation on Kerr spacetimes. We begin in Sect. 2.1 with a review of the Kerr metric. We then present the Teukolsky equation on Kerr in Sect. 2.2, focussing on the case s = ± 2. This will allows us to state a general well-posedness statement in Sect. 2.3. Finally, in Sect. 2.4 we shall recall the relation of the s = ± 2 Teukolsky equation with the system of gravitational perturbations around Kerr.

The Kerr Metric
We review here the Kerr metric and associated structures, following the notation of [45].

Coordinates and Vector Fields
For each |a| < M, recall that the Kerr metric in Boyer-Lindquist coordinates (r , t, θ, φ) takes the form where We recall from [45] the fixed ambient manifold-with-boundary R, diffeomorphic to R + × R × S 2 and the coordinates (r , t * , θ * , φ * ) on R known as Kerr star coordinates.
We recall the relations relating Boyer-Lindquist and Kerr star coordinates. We do not need here the explicit form oft(r ) andφ(r ); see [45], Section 2.1.3 but remark that they both vanish for r ≥ 9/4M. When expressed in Kerr star coordinates, the metric (17) (defined a priori only in the interior of R) extends to a smooth metric on R, i.e. it extends smoothly to the event horizon H + defined as the boundary ∂R = {r = r + }.
It is easy to see that the coordinate vector fields T = ∂ t * and = ∂ φ * of the fixed coordinate system coincide for all a, M with the coordinate vector fields ∂ t and ∂ φ of Boyer-Lindquist coordinates, which are Killing for the metric (17). We recall that T is spacelike in the so-called ergoregion S = { − a 2 sin 2 θ < 0}. Setting we recall that the Killing field is null on the horizon H + and is timelike in An additional important coordinate will be r * defined to be a function r * (r ) such that and centred as in [45] so that r * (3M) = 0. Note that r * → −∞ as r → r + , while r * → ∞ as r → ∞. Given a parameter R thought of as an r -value, we will often denote r * (R) by R * . The vector fields where ∂ r * is defined with respect to (r * , t, θ, φ) coordinates, define principal null directions. We have the normalisation The vector field L extends smoothly to H + to be parallel to the null generator, while L extends smoothly to H + so as to vanish identically. The quantity −1 L has a smooth nontrivial limit on H + . The vector fields L and L are again T -(and -)invariant.

Foliations and the Volume Form
For all values τ ∈ R, we recall that the hypersurfaces τ = {t * = τ } are spacelike (see [45], Section 2.2.5). We will denote the unit future normal of τ by n τ . We recall the notation For polynomial decay following the method of [38,87], we will also require hypersurfaces τ which connect the event horizon and null infinity. For this we fix some 0 < η < 1 and define the coordinatẽ where ξ is a smooth cut-off function equal to zero for r ≤ R η and equal to 1 for r ≥ R η + M. It is straightforward if tedious to show that for R η sufficiently large (and a suitably chosen function ξ ) the hypersurfaces τ defined by are smooth and spacelike everywhere, in fact c η r −η−1 ≤ −g ∇t * , ∇t * ≤ C η r −η−1 indicating that the hypersurfaces become asymptotically null near infinity. We take this R η as fixed from now on. We will in fact use coordinates t * , r , θ, φ * and perform estimates in the spacetime regions See Fig. 1.
We compute the volume form in the different coordinate systems (recalling that the r and θ coordinates are common to all coordinate systems, so ρ 2 = r 2 + a 2 cos 2 θ is unambiguously defined) We will often use the notation Denoting the (timelike) unit normal to the hypersurfaces (22) by n τ we compute in coordinates r ,t * , θ * , φ * g τ g r 2 + a 2 L, n τ = v (r , θ) ρ 2 sin θ and g τ g L, n τ In particular, the volume element on slices of constantt * = τ satisfies For future reference we note that, again in coordinates r ,t * , θ * , φ * , we have on the null hypersurfaces corresponding to the horizon and null infinity respectively the relations where the volume forms are understood to be themselves normalised by L and L, respectively. The above will be the expressions that arise in the context of the divergence theorem. Finally, we note the covariant identities which are most easily checked in Boyer-Lindquist coordinates.

The Very Slowly Rotating Case |a| < a 0 M
In the present paper, we will restrict to the very slowly rotating case. This will allow us to exploit certain simplifications which arise from closeness to Schwarzschild.
Recall that the hypersurface r = 3M in Schwarzschild is known as the photon sphere and corresponds to the set where integrated local energy decay estimates necessarily degenerate. In the case |a| < a 0 M the trapping is localised near r = 3M while the ergoregion S is localised near r = 2M. See the general discussion in [43]. Let us quantify this below by fixing certain parameters.
We will fix parameters A 1 < 3M < A 2 sufficiently close to 3M. We note already that for sufficiently small |a| < a 0 M, then all future trapped null geodesics will asymptote to an r value contained in r ∈ [A 1 , A 2 ]. (We shall not use this fact directly, but rather, a related property concerning the maximum of a potential function associated to the separated wave equation. See already Lemma 8.2.1.) We moreover can choose a 0 small enough so that in addition, R K > A 1 and so that the ergoregion satisfies S ⊂ {r * < 4 A * 1 }. Fixing a cutoff function χ(r * ) which is equal to 1 for r * ≤ 4 A * 1 and 0 for r * ≥ 2 A * 1 we define the vector field T + ω + χ . We note that by our arrangement, this vector field is now timelike for all r > r + , Killing outside {4 A * 1 < r * < 2 A * 1 }, null on H + , and equal to T on {r * ≥ A * 1 }. Finally, let us note that, if A * 1 is sufficiently small, then restricting to small a 0 , we have that t = t * for r * ≥ 2 A * 1 for all |a| < a 0 . We note in particular

Parameters and Conventions
This paper will rely on fixing a number of parameters which will appear in the proof. We have just discussed the parameters η and which have already been fixed. We will also introduce fixed parameters which will be connected to adding multiplier constructions on Schwarzschild, as well as parameters C , c , C delimiting frequency ranges. In particular, eventually, these can be all thought of as fixed in terms of M alone. We will introduce an additional smallness parameter ε associated to the cutoffs in time. (This notation is retained from our [40].) Again, eventually, this will be fixed depending only on M.
Finally, we will exploit the slowly rotating assumption by employing a 0 as a smallness parameter, which will only be fixed at the end of the proof.
We introduce the following conventions regarding inequalities. For non-negative quantities E 1 and E 2 , by We will sometimes use the notation where Q is not necessarily a non-negative quantity. In this context, this will mean that there exist constants c(M), C(M) such that Note that two inequalities of the above form can be added when the terms Q are identical.
Before certain parameters are fixed, say δ 1 , we will use the notation δ 1 to denote the additional dependence on δ 1 of the constant C(M, δ 1 ) appearing in various inequalities. Only when the parameter is definitively fixed in terms of M, can δ 1 be replaced by .
On the other hand, in the context of the restriction to a 0 M, which will appear ubiquitously, the constant implicit in may depend on all parameters yet to be fixed. This will not cause confusion because restriction to smaller a will always be favourable in every estimate.

The Teukolsky Equation for Spin Weighted Complex Functions
In this section we present the Teukolsky equation on Kerr. We first review in Sect. 2.2.1 the notion of spin s-weighted complex functions and discuss some elementary properties of the spin s-weighted Laplacian in Sect. 2.2.2. We then recall in Sect. 2.2.3 the classical form of the Teukolsky operator for general spin. Finally, specialising to s = ± 2 we derive in Sect. 2.2.4 rescaled quantities which satisfy an equation regular also on the horizon. It is in this form that we will be able to state well-posedness in the section that follows.

Spin s-Weighted Complex Functions on S 2 and R
The Teukolsky equation will concern functions whose (θ, φ) (or equivalently (θ, φ * )) dependence is that of a spin s-weighted function, for s ∈ 1 2 Z. We will always represent such functions as usual functions α(r , t, θ, φ).
Smooth spin s-weighted functions on S 2 naturally arise, in a one-to-one fashion, from complex-valued functions on S 3 (viewed as the Hopf bundle) which transform in a particular way under the group action on the S 1 fibres of S 3 , as will be described now. (Note that this is indeed natural as S 3 can be identified with the bundle of orthonormal frames on S 2 , and the definition of the Teukolsky null curvature components indeed depends on a choice of frame on S 2 . See Sect. 2.4.) Viewing S 3 as the Hopf bundle we have a U (1) action on the S 1 fibres (corresponding to a rotation of the orthonormal frame in the tangent space of S 2 ). Introducing Euler coordinates 5 (θ, φ, ρ) on S 3 we denote this action by e iρ . Now any smooth function F : S 3 → C which transforms as F pe iρ = e −iρs F ( p) for p ∈ S 3 descends to a spin-weighted function on S 2 (by choosing a frame at each point). More precisely, F descends to a section of a complex line bundle over S 2 denoted traditionally by B(R). See [11,27].
Let Z 1 , Z 2 , Z 3 be a basis of right invariant vector fields constituting a global orthonormal frame on S 3 . In Euler coordinates we have the representation F are smooth functions of the Euler coordinates and extend continuously to the poles of the coordinate system at θ = 0 and θ = π . Since spin s-weighted functions on S 2 arise from smooth functions on S 3 as discussed above, there is a natural notion of the space of smooth spin s-weighted The space of smooth spin s-weighted functions on S 2 is denoted S [s] ∞ . Note that considered as usual functions on S 2 , elements of S [s] ∞ are in general not regular at θ = 0.
We define the Sobolev space of smooth spin s-weighted functions on S 2 , denoted [s] H m (sin θ dθ dφ) as the completion of S [s] ∞ with respect to the norm.
. We now define the analogous notions for functions f of the spacetime coordinates (t * , r , θ, φ * ).
We define a smooth complex-valued spin s-weighted function f on R to be a function f : are smooth functions away from the poles and such that e isφ (( 5 f extends continuously to the south (θ = π ) pole. In particular, the restriction of f to fixed values of t * , r is a smooth spin s-weighted function on S 2 . We denote the space of smooth complex-valued spin s-weighted functions on R by We similarly define a smooth complex-valued spin s-weighted function f on a slice τ to be a function f : are smooth functions away from the poles and such that e ±isφ ( in the obvious way. This allows to define the space [s] H m loc ( τ ) as the space of functions on τ such that the restriction to any U τ (meaning that there is a compact set K with We finally note that we can analogously define these spaces for the slices τ , i.e. define the spaces

The Spin s-Weighted Laplacian
Let us note that the operator defined in the introduction, is a smooth operator on The eigenfunctions of/ [s] are again in S [s] ∞ and are known as s-spin weighted spherical harmonics. We shall discuss these (and their twisted analogues) further in Sect. 6.2.1.

An integration by parts yields for
where the right hand side is manifestly non-negative. 6 Introducing the spinorial gradient/ we also have We note that for , Directly from (30) and (32) we deduce the Poincaré inequality Combining (32) and (34) we also deduce

The Teukolsky Operator for General Spin s
Recall that the operator is the traditional representation (see for instance [99]) of the Teukolsky operator with spin s ∈ 1 2 Z. In view of the comments above, this operator is smooth on S [s] ∞ (R\H + ). We will say that such an satisfies the Teukolsky equation if the following holds: The operator (37) is not smooth on S [s] ∞ (R) itself. This is because it has been derived with respect to a choice of frame which degenerates at the horizon. See Sect. 2.4. To obtain a regular equation at the horizon, we must considered rescaled quantities. We turn to this now.

Rescaled Equations
To understand regularity issues at the horizon we must consider rescaled quantities. We will restrict here to s = ± 2. Definẽ Define now the modified Teukolsky operator T [s] by the relation with/ [s] denoting the spin ± 2 weighted Laplacian on the round sphere defined in (29) and with the first order term t [s] given by One sees that (37) for s = + 2 can be rewritten as On the other hand, we observe that T [+2] now is a smooth operator on S [s] ∞ (R) and that its second order part is hyperbolic, in fact, it is exactly equal to − g .
Similarly, we see that (37) for s = −2 can be rewritten as which in turn can be rewritten as where is a first order operator acting smoothly on S [s] ∞ (R). Now we observe that T [−2] − 2 ρ 2 w w L also acts smoothly on S [s] ∞ (R) and that its second order part is exactly equal to − g . This will allow us to state a well-posedness proposition in the section to follow.

Remark 2.1
The weights in (38) for α [+2] will be useful for the global analysis of the equation, whereas the weights for α [−2] will only be useful for the well-posedness below. For this reason, we shall define later (see Sect. 6.2.5) the different rescaled quantities , and deal mostly with the further rescaled quantities u [±2] · w. Note that The first quantity is finite (and generically non-zero) on the horizon H + while the second quantity is finite (and generically non-zero) on null infinity I + which makes them useful in the global considerations below. Note also that both quantities satisfy the simple equations (41) and (42) respectively.

Well-posedness
Standard theory yields that the Teukolsky equation in the form (41), (43) is well-posed , resp. with 0 replacing 0 . We state this as a proposition for reference: Proof cf. Proposition 4.5.1 of [40].

Relation with the System of Gravitational Perturbations
The Teukolsky equation (2) is traditionally derived via the Newman-Penrose formalism [93]. One defines the (complex) null tetrad ( , n, m,m) by which is normalised such that Note that we can obtain an associated real spacetime null frame ( , n, e 1 , e 2 ) by defining (m −m), which then satisfies in particular g (e 1 , e 1 ) = g (e 2 , e 2 ) = 1 and g (e 1 , e 2 ) = 0.
The extremal Newman-Penrose curvature scalars are defined as the following components of the spacetime Weyl tensor 7 Both 0 and 4 vanish for the exact Kerr metric. Remarkably, upon linearising the Einstein vacuum equations (1) (using the above frame) the linearised components 0 and 4 are gauge invariant (with respect to infinitesimal changes of both the frame and the coordinates) and moreover satisfy decoupled equations. Indeed, one may check that α [−2] = (r − ia cos θ ) 4 4 and α [+2] = 0 satisfy precisely the Teukolsky equation (2) for s = −2 and s = 2 respectively. Instead of defining spin s-weighted complex functions 0 , 4 one may (equivalently) define symmetric traceless 2-tensors α and α (living in an appropriate bundle of horizontal tensors) by Using the symmetry and the trace properties of the Weyl tensor we derive the relations which relate the spin 2-weighted complex function and the tensorial version of the curvature components. Of course similar formulae are easily derived for α.
We can now connect directly to our previous [31] where we wrote down the Teukolsky equation for the symmetric traceless tensors α and α in the Schwarzschild spacetime.
As a final remark we note that in the Schwarzschild case considered in [31] the null frame used to define the extremal Weyl components arose directly from a double null foliation of the spacetime. In stark contrast, the algebraically special null frame (l, n, e 1 , e 2 ) in Kerr for a = 0 does not arise from a double null foliation of that spacetime.

Generalised Chandrasekhar Transformations for s = ± 2
In this section, we generalise the physical space reformulations of Chandrasekhar's transformations, given in [31], to Kerr. In accordance with the conventions of our present paper, we will consider complex scalar spin ± 2 weighted quantities α [±2] in place of the tensorial ones of [31]. We begin in Sect. 3 . The latter coupling vanishes in the Schwarzschild case. The precise relation with the tensorial definitions of [31] will be given in Sect. 3.3.
We will typically work with the rescaled functions which are of course again smooth. As in [31], it will be again useful to give a name to the intermediate quantities We can rewrite (46)-(47) as Note that for α [±2] smooth, it is the quantities

The Generalised Inhomogeneous Regge-Wheeler-Type Equation with Error
The importance of the quantities [±2] arises from the following fundamental proposition: then the quantities [±2] satisfy the equation where and Proof Direct calculation. See Appendix.
We will call the operator R [s] defined by (55) the generalised Regge-Wheeler operator. We note that it has smooth coefficients on R 0 and its highest order part is proportional to the wave operator. The equation (54) reduces to the usual Regge-Wheeler equation in the case a = 0: As discussed already in the introduction, we see that (54), although still coupled to α [±2] , retains some of the good structure of (59). The operator (54) has a good divergence structure admitting estimates via integration by parts, i.e. it does not have the problematic first order terms of the Teukolsky operator T [±2] , cf. (39). See already the divergence identities of Sect. 5.1.1. Moreover, the terms J [+2] can be thought of as lower order, from the perspective of [±2] , as they only involve up to second

Relation with the Quantities P and P of [31]
As with the tensorial quantities α and α discussed in Sect. 2.4, in [31] the transformations to the quantities P and P (corresponding to the complex functions P [+2] , P [−2] in this paper) were again given tensorially. In particular, the Regge-Wheeler equation for the symmetric traceless tensor = r 5 P was written tensorially using projected covariant derivatives as (cf. Corollary 7.1 of [31]) where / ∇ 3 and / ∇ 4 are projected (to the spheres of symmetry) covariant derivatives in the null directions, / is the covariant Laplacian associated with the metric on the spheres of symmetry acting on symmetric traceless tensors and 2 = 1 − 2M r . Note that unlike the operator/ [s] considered in this paper, the operator / was defined as a negative operator in [31]. Computing the equation satisfied by the components of in the standard orthonormal frame on the spheres of symmetry one obtains from which one infers that the complex-valued functions [±2] = 11 ∓ i 12 satisfy the Regge-Wheeler equation (59) for s = ± 2. 8

Energy Quantities and Statement of the Main Theorem
We first give certain definitions of weighted energy quantities in Sect. 4.1. This will allow us to give a precise statement of the main theorem of this paper (Theorem 4.1) in

Definitions of Weighted Energies
We will define in this section a number of weighted energies. In addition to those appearing in the statement of Theorem 4.1, we will need to consider various auxiliary quantities.

The Left, Right and Trapped Subregions
We will in particular need to introduce energies localised to various subregions of τ and R(τ 1 , τ 2 ). In anticipation of this, let us define the following subregions Note that For τ , it will be more natural to consider

Weighted Energies for [±2]
The energies in this section will in general be applied to [±2] satisfying the inhomogeneous equation (54).
Let p be a free parameter (which will eventually always take the values 0, η, 1 or 2). We define the following weighted energies on the slices τ We remark that an overbar indicates that the energy has optimised weights near the horizon.
We will also consider the following energy through On the event horizon H + we define the energies On null infinity I + we define the energies In addition to the energy fluxes, we will define the weighted spacetime energies where δ a b is the Kronecker delta symbol and also the degenerate spacetime energies withχ a radial cut-off function equal to 1 in r * ∈ (−∞, Note that

Weighted Energies for
The quantities in this section will in general be applied to α [+2] , ψ [+2] arising from a solutionα [+2] of the inhomogeneous equation (53). We define the following energy densities With these, we define the following weighted energies on the slices τ : Remark 4. 1 We remark already that while these energies contain the T and the derivative only, we can obtain also the L and the L derivative if we control in addition the energy (61) of [+2] . This is because of the relations (107) and (108) and the relation L = −L + 2T + 2a r 2 +a 2 . It will be useful to also consider separately We also use the notation E away τ , p for the sum of the left and the right energies. On (timelike) hypersurfaces of constant r = A > r + we define In the limit r → r + we obtain the energies the event horizon H + which we denote We also define the following weighted spacetime energies , As with the fluxes, it will be useful to also define Finally, we define We note the relations

Weighted Energies for α
The quantities in this section will in general be applied to α [−2] , ψ [−2] arising from a solutionα [−2] of the inhomogeneous equation (53). We define the following weighted energies on the slices τ : 9 We also define the energies by adding to the set in the energies without the overbar the vectorfield r 2 +a 2 L. Hence an overbar again indicates that the energy has been improved near the horizon.
by appropriately restricting the domain in (81)- (82), in analogy with the definitions (70)- (71). 9 Note that in contrast to the [+2]-energies, no p-weights appear. The underlying reason is that the transport estimates for ψ [−2] and α [−2] will always be applied with the same r -weight. Note also in this context that the E-energies for α [−2] and ψ [−2] on the slices τ in (81)-(82) carry the same r -weight as the corresponding spacetime I-energies in (85)- (86). This arises from the fact that the transport for the [−2]quantities happens in the L-direction and the relation (24) between L and the unit normal to the slices τ .
In the limit r → ∞ we define on null infinity I + We also define the following weighted spacetime energies and the energies by adding to the set appearing in the definitions (85)-(86) the vectorfield r 2 +a 2 L. We define again by restricting the domain in (85)- (86), in analogy with (74)- (77). Finally, in analogy with (78)-(79), we define and we note the [−2] version of (80).

Precise Statement of the Main Theorem: Theorem 4.1
We are now ready to give a precise version of the main theorem stated in Sect. 1.2: be as in the well-posedness Proposition 2.3.1, and let α [±2] , P [±2] , [±2] , ψ [±2] be as defined by (38), (46), (47), (48), (49) and (50). Then the following estimates hold: For each statement, the Sobolev exponent j in the initial data norm is assumed large enough so that the quantities on the right hand sides of the corresponding estimates above are well defined.
Let us note that we can easily deduce from the above an alternative version where initial data is posed (and weighted norms given) on 0 instead of 0 . We suffice here with the remark that smooth, compactly supported initial data on 0 trivially give rise to initial data on 0 satisfying the assumptions of the above theorem.
As an example of the pointwise estimates which follow immediately from the above theorem, let us note the following pointwise corollary (recall that 0 < η < 1 was fixed in Sect. 2.1.2): ) be smooth and of compact support. Then the solutioñ α satisfies where C depends on an appropriate higher Sobolev weighted norm.
The above decay rates can be improved following [87].

Remark 4.4
Note that, in view of Remark 4.3, one sees that the decay in r provided for 0 by Corollary 4.1 is weaker than peeling, consistent with the fact that, just as in [26], our weighted energies do not in fact impose initially the validity of peeling. This is important since it has been shown that peeling does not hold for generic physically interesting data [25].

The Logic of the Proof
The remainder of the paper concerns the proof of Theorem 4.1.
Sections 5-8 are preliminary: Section 5 will prove an integrated energy estimate for and α [±2] arising from a solution of the homogeneous equation (37). Sects. 6-8 will concern so-called [A 1 , A 2 ]-admissible solutions and will provide frequency-localised estimates in the region [A 1 , A 2 ], again with boundary terms on r = A i .
The proof proper of Theorem 4.1 commences in Sect. 9 where the degenerate integrated local energy decay and boundedness statements are proven (statement 1.), using the results of Sects. 5-9, applied to a particular solution α [±2] Q of the inhomogeneous equation (53) which arises by cutting off a solution α of the homogeneous equation so that, when restricted to the r -range [A 1 , Q is compactly supported in t * ∈ [0, τ final ]. The estimate of statement 1. follows by appropriately summing the estimates of Sects. 5 and 8 applied to α Q . We note already that when summing, the most dangerous boundary terms on r = A i have been arranged to precisely cancel, while the error term arising from the inhomogeneous term on the right hand side of the equation of α Q can easily be absorbed in view of its support properties and the auxiliary estimates of Sect. 5.3. Finally, in Sect. 9.6, we will distill from our argument a simpler, purely physical space proof of statement 1. for the axisymmetric case.
The degenerate boundedness and integrated local energy decay are combined with redshift estimates in Sect. 10 to obtain statement 2.
Finally, the weighted r p estimates are obtained in Sect. 11, giving statements 3.-4.

Conditional Physical Space Estimates
In this section, we will derive certain physical space estimates for can be controlled from initial data provided that boundary terms on r = A i can be controlled. (To understand the latter boundary term, this estimate must be combined with that obtained in Sect. 8.) Finally, we shall need some auxiliary physical space estimates (applied throughout R) for [±2] , ψ [±2] and α [±2] arising from a solution of the homogeneous Teukolsky equation (37). These will be given in Sect. 5.3.

Let us note that we may always assume in what follows that any α [±2] referred to is in S [±2]
∞ ( R 0 ).
We note already that the boundary terms (τ 1 , τ 2 ) appearing above formally coincide with those of the fixed frequency identity to be obtained in Sect. 8.2. Thus these terms will cancel when all identities are summed in Sect. 9.
In what follows, our multiplier constructions will be identical for [+2] and [−2] . We will thus denote these simply as . The spin weight will be explicitly denoted however for the terms arising from the right hand side of (54).

Multiplier Identities
The proof of Proposition 5.1.1 will rely on various multiplier identities for (54). These are analogous for standard multiplier estimates proven for solutions of the scalar wave equation and in particular generalise specific estimates which have been proven for the Regge-Wheeler equation (60) on Schwarzschild in [31].
Multiplying (54) where ≡ denotes equality after integration with respect to the measure sin θ dθ dφ and The y identity. Multiplying (54) by y L − L for a smooth radial function y and taking the real part produces (use the formulae of Appendix B.4) where ≡ denotes equality after integration with respect to the measure sin θ dθ dφ and The h identity. Multiplying (54) by h for a smooth radial function h and taking real parts leads to (use the formulae of Appendix B.2) where ≡ denotes equality after integration with respect to the measure sin θ dθ dφ and The f identity. Adding the y-identity with y = f and the h-identity with h = f for f a smooth radial function yields the identity (recall (289)) where ≡ denotes equality after integration with respect to the measure sin θ dθ dφ and The r p -weighted identity. We multiply (54) by r p β 4 ξ L with β 4 = 1 + 4 M r and ξ a smooth radial cut-off satisfying ξ = 0 for r ≤ R and ξ = 1 for r ≥ R + M with R is chosen directly below (99) depending only on M. After taking the real parts of the resulting identity we obtain (use the formulae of Appendix B.6) where ≡ denotes equality after integration with respect to the measure sin θ dθ dφ and It is easy to see that we can choose R in the cut-off function such that the coefficients of |L | 2 , |/ ∇ [s] | 2 and | | 2 in (99) are all non-negative in r ≥ R + M for p ∈ [0, 2] and we henceforth make that choice.

Remark 5.1 (Conversion into divergence identities)
To convert the identities derived in this section into proper spacetime divergence identities (from which the boundary contributions, etc., are most easily assessed) we recall the identities (26). Since the left hand side of any multiplier identity above has the schematic form with E sin θ dθ dφ = 0, we can use (26) to convert them into the divergence form This is easily integrated using Stokes' theorem and making use of the formulae (24) and (25) for the normals to the spacelike hypersurfaces (and the horizon and null infinity). Therefore it is the above identity which provides the precise sense in which the F's in the identities indeed correspond to boundary terms. Note the term involving E disappears after integration with respect to the spacetime volume form (23).

Proof of Proposition 5.1.1
We define (cf. [31]) and whereχ is a cutoff function such thatχ = 0 for r ≤ 9M andχ = 1 for r ≥ 10M. We note the following Schwarzschild proposition Proposition 5.1.2 [31] In the Schwarzschild case a = 0, then As a consequence, for δ 1 and δ 2 sufficiently small and arbitrary E we have Note that in view of Remark 5.1, upon application of the divergence theorem, the left hand side leads to a term which controls the integrand of I deg η .
We have: 1. Given any E > 1, and sufficiently small δ 1 , δ 2 , then for |a| < a 0 M sufficiently small, the resulting bulk term is nonnegative and in fact satisfies the coercitivity estimate R away (τ 1 ,τ 2 ) This follows from (a) Proposition 5.1.2, (b) smooth dependence on a to infer coercivity away from the horizon and away from infinity, (c) the fact that for all a, the term I r η manifestly controls the integrand of I 5. The remaining flux terms on r = A 1 and r = A 2 produce exactly the expression where (recalling (89), (91) and (95)) This again follows from Remark 5.1: In (t, r * , θ, φ)-coordinates we have that 1 √ g r * r * ∂ r * is the unit normal to constant r * hypersurfaces and ρ 2 1 √ g r * r * r 2 +a 2 sin θ dθ dφdt is the induced volume element. Using that 2∂ r * = L − L and that ∂ r * is orthogonal to L + L the result follows. Observe that there is no contribution from F r η in (103) because that multiplier is supported away from A 2 . 6. The inhomogeneous term involving G [±2] on the right hand side of (54) generates the term (104) where (recall again Remark 5.1) 7. By Cauchy-Schwarz, the term generated by the inhomogeneous term involving J [±2] on the right hand side of (54) can be bounded (with a constant depending on E) by the expression with the subindex [η] = η in case of +2, and [η] being dropped entirely in case of s = − 2. Note that the last term can be absorbed in view of (102), for sufficiently small |a| < a 0 M.
Thus, for E sufficiently large, and δ 1 , δ 2 sufficiently small, one obtains immediately the statement of Proposition 5.1.

1.
In what follows, we will now consider E as fixed in terms of M, and thus incorporate the E dependence into the , etc. We will further constrain δ 1 and δ 2 in Sect. 8.2 and thus we will continue to denote explicitly dependence of constants on δ 1 , δ 2 .

Transport
and the following estimate in R left (τ 1 , τ 2 ): Proof We recall from (49) and (51) the relations From (107) we derive for n ≥ 0 ∇ a r n 1 ρ 2 and hence ∇ a ⎛ ⎝ r n 1 ρ 2 Moreover, the same estimate (110)  ) on the right since the relation (107) trivially commutes with the Killing fields T and respectively. We will refer to those estimates as the "T -commuted and -commuted (110)" below. Similarly from (108), ) on the right. We again refer to the latter as the "T -commuted and -commuted (111)" below. Let us first obtain the estimate in R right (τ 1 , τ 2 ). The case in R left (τ 1 , τ 2 ) is analogous but easier since weights in r do not play a role. We add • (111) with n ∈ {η, 1, 2 − η} • the -commuted (111) with n ∈ {η, 1, 2 − η} • the T -commuted (111) with n = 2 − η integrated over R right (τ 1 , τ 2 ). Combining the above we conclude for p ∈ {η, 1, 2} the estimate Turning to the estimate (110) we add Combining the above we conclude for p ∈ {η, 1, 2} (note that for p = 2 there is an η-loss in the definition of the densities (67), (68), ensuring that we can indeed set p = 2) Combining (113) and (112) yields the desired estimate to the right of trapping. As remarked above, the estimate in the "left region" R left (τ 1 , τ 2 ) is easier and left to the reader.  (47), (50). Then we have the following estimate in R right (τ 1 , τ 2 ):

Transport
and the following estimate in R left (τ 1 , τ 2 ):

Remark 5.2
As the proof will show, these estimates also hold replacing I Proof We recall the relations From (116) we derive (recall ρ 2 = r 2 + a 2 cos 2 θ ) for any n, η ∈ R ∇ a r 2 + a 2 −n−1+4 and hence, choosing n = 3, we have for any η > 0 the estimate Moreover, the same estimate holds replacing 1 + 1 r η by 1 r η on the left and r 1+η by r 1−η on the right (cf. Remark 5.2). Note that the estimate (118) also holds replacing α From (117) we derive Moreover, the same estimate holds replacing 1 + 1 r η by 1 r η on the left and r 1+η by r 1−η on the right (cf. Remark 5.2). Note that the estimate (119) also holds replacing ψ We are now ready to prove the estimate in R left (τ 1 , τ 2 ).

Integrating (119) and the T -commuted and -commuted (119) over
Integrating (118) and the T -commuted and -commuted (118) over R left (τ 1 , τ 2 ) produces Combining the last two estimates produces the desired estimate in R left (τ 1 , τ 2 ). The estimate in R right (τ 1 , τ 2 ) is proven entirely analogously and is again left to the reader. The only important observation is that the good ψ-spacetime term generated from (119) is stronger (in terms of r -weight) than what is needed on the left hand side of (118).

Auxiliary Estimates
We collect a number of auxiliary estimates we shall require.

The Homogeneous
Here the subindex [η] is equal to η in case of s = +2 and it is dropped entirely in case s = −2.
Proof The inequality (122) follows from integrating the identity (88) associated with the multiplier T + ω + χ over the region R (τ 1 , τ 2 ) using Remark 5.1. The details are as follows. Note that G [s] = 0 and that for the boundary terms one has while for the spacetime term clearly It remains to estimate the term In view of the fact that the support of χ is away from the degeneration of I deg we can easily control the ω + χ -part by the right hand side of (122) using the Cauchy-Schwarz inequality. For the remaining term Re T [±2] J [±2] we restrict the proof to the s = +2 case, the s = −2 case being completely analogous. We recall from Proposition 3.2.1 that where |c 1 (r )| 1, |c 2 (r )| r −1 , |c 3 (r )| r −1 and |c 4 (r )| 1. Note that unless we are in the region near trapping all of these terms feeding into (124) are easily controlled by the right hand side of (122) using the Cauchy-Schwarz inequality. We can also assume without loss of generality τ 2 > τ 1 + 2 in (124) as otherwise we can again apply Cauchy-Schwarz and estimate the spacetime integral of T [+2] by the supremum of the energy through each slice τ and absorb the term on the left using that a is small.

Remark 5.3
A frequency localised version of this proof can be found in the proof of Proposition 8.4.1.  (46), (47), (49), (50). Then for any τ step > 0 there exists an a 0 M such that for |a| < a 0 we have for any τ 1 > 0

Local in Time Estimates
where C = C(M) (and the implicit constant in is independent of both τ step and τ 1 , according to our general conventions). Here the subindex [η] is equal to η in case of s = + 2 and it is dropped entirely in case s = − 2.
Proof We first note that sup τ 1 ≤τ ≤τ 1 +τ step This follows easily by the estimates of the previous sections. We now apply (122) with τ 2 taken in τ 1 ≤ τ 2 ≤ τ 1 + τ step , noting that the first three terms on the right hand side can be bounded by |a|τ step times the right hand side of (132). Restricting a 0 so that in particular |a|τ step e Cτ step < 1 we obtain (130).
We note that we can repeat the transport estimates of Sect. 5.2, now for the homogeneous equations, and applied globally in R(τ 1 , τ 1 + τ step ), obtaining Note that the term is I η and not I deg η . In view of (note the η on the left but the 0 on the right hand side), we obtain (131) for sufficiently small a.
Remark 5. 4 We note that a more careful examination of the Schwarzschild case and Cauchy stability yields that the inequality (131) can be proven without the τ step factor on the first term of right hand side, provided I 0 is replaced by I deg 0 . We shall not however require this here.

The Admissible Class and Teukolsky's Separation
In this section we will implement Teukolsky's separation [107] of (36) for s = ± 2.
To make sense a priori of the formal separation of [107], one must in particular work in a class of functions for which one can indeed take the Fourier transform in time. This requires applying the analysis to functions which satisfy certain timeintegrability properties. A useful such class is the "sufficiently integrable, outgoing" class defined in [44,45] for the s = 0 case.
In the present paper, it turns out that we shall only require Fourier analysis in the region r ∈ [A 1 , A 2 ]. We may thus consider the more elementary setting of what we shall call the [A 1 , A 2 ]-admissible class where time square integrability is only required for r ∈ [A 1 , A 2 ]. (We will in fact assume compact support in t * in this r -range.) This leads to a number of useful simplifications. In particular, we need not refer to the asymptotic analysis of the ODE's as r * → ±∞, as was done in [44,45], in order to infer boundary behaviour.
The section is organised as follows: We will define our elementary notion of [A 1 , A 2 ]-admissible class in Sect. 6.1. We will then implement Teukolsky's separation in Sect. 6.2, deriving the radial ODE, valid for r ∈ [A 1 , (We note already that, in practice, the results of this section will be applied to solutions of the inhomogeneous Teukolsky equation which arises from applying a suitable cutoff to solutions of (37). The restriction of Fourier analysis to the range r * ∈ [A * 1 , A * 2 ] will allow us to use a cutoff whose derivatives are supported in a region of finite r * ∈ [2 A * 1 , 2 A * 2 ], leading to additional simplifications with respect to [45]. We will only turn to this in Sect. 9.)

The [A 1 , A 2 ]-admissible Class
We define an admissible class of functions for our frequency analysis. This is to be compared with the class of sufficiently integrable functions from [44,45]. Since we will only apply frequency localisation in a neighbourhood of trapping, we only consider the behaviour in the fixed r -region [A 1 , A 2 ] with r + < A 1 < A 2 < ∞ defined in Sect. 2.1.3. (Recall in this region that t = t * =t * .) On the other hand, for convenience, we will assume compact support in t for these r -values, as this is what we shall indeed obtain after applying cutoffs.
Definition 6.1 Let a 0 < M, |a| < a 0 and let g = g a,M . We say that a smooth complex valued spin± 2 weighted functionα : Remark 6.1 One could work with the weaker condition that (cf. [45]) for all j ≥ 1, the following holds with the only caveat that in the frequency analysis we would have to restrict to generic frequency ω for the ODE to be satisfied in the classical sense.

Teukolsky's Separation
We will now implement Teukolsky's formal separation of the operator (36) in the context of [A 1 , A 2 ]-admissible spin-s weighted functions α [s] for s = ± 2. We begin in Sect. 6.2.1 with a review of the basic properties of spin-weighted oblate spheroidal harmonics and their associated eigenvalues λ [s] m (ν). We will then turn immediately in Sect. 6.2.2 to some elementary estimates for the eigenvalues λ [s] m (ν) which will be useful later in the paper. Next, we shall apply these oblate spheroidals together with the Fourier transform in time in Sect. 6.2.3 to define coefficients α . We then give Proposition 6.2.1 in Sect. 6.2.4, stating that these coefficients satisfy an ordinary differential equation with respect to r * ; this is the content of Teukolsky's remarkable separation of (36).

Spin-Weighted Oblate Spheroidal Harmonics
Let ν ∈ R, s = 0, ±2 and consider the self-adjoint operator/ [s] (ν) defined bẙ ∞ , which we recall is a dense subset of L 2 (sin θ dθ dφ). This has a complete collection of eigenfunctions with eigenvalues λ [s] m ∈ R, indexed by m ∈ Z, ≥ max(|m|, |s|). These are known as the spin-weighted oblate 10 The eigenfuctions themselves satisfy We note the following familiar special cases: 1. For s = 0 one obtains the oblate spheroidal harmonics familiar from the angular part of the separation equation of the scalar wave equation on Kerr [45]. The case s = 0 and ν = 0 recovers the standard spherical harmonics S [0] m (0, cos θ)e imφ = Y m with eigenvalues ( + 1).

For ν = 0, then/
[s] (0) is the spin-s-weighted Laplacian and one obtains the spin-weighted spherical harmonics, whose eigenvalues can also be determined explicitly where the last inequality follows from the relation | | ≥ |s|. For future reference we note the relation We finally remark also the general relation allowing us to restrict to s = + 2 without loss of generality when obtaining estimates on the λ [s] m (ν). For various asymptotics concerning the behaviour of λ [s] m see [10].

The Coefficients α
Setting ν = aω, for each ω ∈ R we may decomposê We obtain then the representation where these relations are to be interpreted in L 2 t L 2 S 2 . We also recall as in [40,45] the following Plancherel relations

The Radial ODE
We here state a proposition that implements Teukolsky's formal separation of (36)  can be defined.
Let us first introduce the following shorthand notation We have the following In view of our definitions, the proof is immediate from the usual formal derivation of (150). See [68]. The s = 0 case corresponds precisely to Proposition 5.2.1 of [45].
Note the difference between (149) and our in (141). It is only the latter quantity which will appear in the estimates of this paper. We have retained (149) to faciliate comparison with the literature.

The Rescaled Coefficients
For s = 0, this reduces to the form of the separated wave equation used in [45].

The Frequency-Localised Transformations
In this section, we will define frequency localised versions of the quantities P [±2] , [±2] , ψ [±2] of Sect. 3 and the Regge-Wheeler type equation (54). We begin in Sect. 7.1 with the definitions of the frequency localised version of the null frame L, L. We then derive in Sect. 7.2 the frequency localised expression for [±2] followed in Sect. 7.3 with the frequency localised form of (54).

The Separated Null Frame
Note that (following the conventions in [45]) we have the following formal analogues: We define the separated frame operators (corresponding to the principal null directions (20)) by We have retained the notation of (20) without fear of confusion. Also note that (138) implies the following formal analogue:

The Frequency Localised Coefficients
[+2],(aω) m = r 2 + a 2 3/2 P and the inhomogeneous terms by Remark 7. 2 We may rewrite the potential Here we see the dependence in the spin is entirely contained in the definition of [±2] .   [45], the results of this section can be understood as results about ODE's, independently of the particular framework of Sect. 6. We have thus tried to give as self-contained a statement as possible.

Statement of Theorem 8.1: The Main Fixed Frequency Estimates
In the present section we consider the coupled system of ODEs satisfied by u, ψ and and state a fixed frequency analogue of local integrated energy decay, in the region r ∈ [A 1 , A 2 ] near trapping.

Frequency Localised Norms
Before formulating the theorem, we define certain energy norms.
In view of Remark 7.3, the natural setting of the theorem refers only to an admissible frequency triple (ω, m, ) (cf. Definition 6.2) and associated solutions u  Given the above, let us define the quantities as well as the boundary energies for i = 1, 2: In the above, r trap is a parameter depending on M, a and the frequency triple (ω, m, ) to be determined later. For "trapped" frequencies, we will have r * trap ∈ [A * 1 /4, A * 2 /4], but it will be important that in various high frequency but untrapped frequency ranges, we can take r trap = 0.
Note that since this is a region of fixed finite r , bounded away from infinity and the horizon, no r -weights or -factors need appear in the above norms.
Finally, it will be convenient if we introduce the alternate notation which will be useful when referring to boundary terms in contexts where the choice of term depends on the spin.

Statement of the Theorem
Theorem 8.1 Given 0 ≤ a 0 M sufficiently small, then the following is true. Let 0 ≤ a ≤ a 0 and let (ω, m, ) be an admissible frequency triple. Let E > 1 be the parameter fixed after Proposition 5.1.1. Given a parameter δ 1 < 1, let f 0 , y 0 be defined by (100) and (100) as in the proof of Proposition 5.1.1.
Then one can choose sufficiently small δ 1 < 1 depending only on M, and functions f , y and an r -value r trap , depending on the parameters a, M and the frequency triple (ω, m, ) but satisfying the uniform bounds such that, for all smooth solutions u [±2] of (153) on [A 1 , A 2 ] and associated ψ [±2] and [±2] , then where and Q is given by (172).

Multiplier Estimates for [±2]
We begin in this section with frequency localised bounds for [±2] . Frequency localisation is necessary to capture trapping, in the style of our previous [40]. The multipliers will be frequency independent at r = A 1 and r = A 2 and will in fact match exactly those applied in Sect. 5.1. This is ensured by (166). As a result, in the setting of Sect. 9, the boundary terms Q(A i ) which will appear below, after summation over frequencies, will exactly cancel the terms Q(A i ) appearing in Proposition 5.
where K [±2] is defined by and Q is given by (172).

Proof
The estimate (170) will be proven by using multiplier identities. The relevant frequency-localised current templates, corresponding precisely to the physical space multiplier identities used in Sect. 5.1, will be defined in Sect. 8.2.1 below. For a specific combination of these currents, the bulk term will control the integrand of the left hand side of (170) whereas the boundary terms (after summation over frequencies) will correspond precisely to the boundary terms of Proposition 5.1.1. This coercivity is stated as Proposition 8.2.2 in Sect. 8.2.2. The precise choice of the functions f and y will be frequency dependent and is carried out separately for the frequency ranges G 1 and G 2 in Sects. 8

.2.3 and 8.2.4 respectively.
In the rest of this subsection, we will always write in the place of [±2] , as the choice of the multipliers will not depend on the spin. We will write V in place of V [±2] , and for [±2] , remembering that the dependence of V [±2] on the spin in the context of the separation is completely contained in the different definition of [±2] ; see formula (163). We will only refer explicitly to s = ± 2 when discussing the inhomogeneous terms on the right hand side of (160).

The Frequency-Localised Multiplier Current Templates
Let us define the frequency localised multiplier currents which correspond to the physical space multipliers of Sect. 5.1: If satisfies for an admissible frequency triple (ω, m, ), then, since V is real, we have Let us remark already that if α is an [A 1 , A 2 ]-admissible solution of the inhomogeneous Teukolsky equation (53), such that the restriction of α to r ∈ [A 1 , A 2 ] is supported in t = t * =t * ∈ (τ 1 , τ 2 ), then the identity corresponding to applying resp. with Q y , Q T , yields precisely the identities of Sect. 5.1.1 applied in the region R trap (τ 1 , τ 2 ). (Note that by our choices from Sect. 2.1.3, we have T = T + ω + χ in this region, and note moreover that the boundary terms ont * = τ i vanish by the restriction on the support.)

The Total Current Q and Its Coercivity Properties
For all frequencies, we will apply the identity corresponding to a current of the form for appropriate choices of functions f , y. The coercivity statement is given by the following: Proof See Sects. 8.2.3 and 8.2.4 .

Lemma 8.2.1
Let 0 < c < 1 and C > 1 be arbitrary. For sufficiently small |a| < a 0 M, then for all frequency triples in the range G 1 , the potential V 0 of (161) has a unique maximum r max satisfying property 2. and . If m = 0, then r max is manifestly independent of ω and .
Proof This is an easy computation in view of (163). For the region G 1 \{ +ω 2 +m 2 ≤ C }, one uses the bound and the smallness of a. For the region { +ω 2 +m 2 ≤ C } it suffices to use the general bound ≥ 1 and the smallness of a. Notice that according to our conventions, the constant in the indeed only depends on M, since smallness of a can be used to absorb the c and C dependence.
Let χ(r * ) be a cutoff function such that We define now This function obviously satisfies property 1. and is easily seen to satisfy property 3. It remains to show property 4. By (175) and the definition of f we have On the other hand, for |a| a 0 < M sufficiently small, we have that | f 0 − f | ≤ c(a), and thus Finally, we note that V = V 0 + V 1 + V 2 , and we have It follows readily that property 4. indeed holds for frequencies in G 1 . Now, given a parameter δ 1 < 1, we define the function Note that this function satisfies (166). We compute where we are using also that | f | ≤ 1 implies that | f 3 | ≤ | f |. Note on the other hand that for sufficiently small |a| < a 0 M, we have for all frequencies in G 1 , in view of the general bound and the bound which holds in G 1 . Thus It follows that we may choose δ 1 sufficiently small so as for Henceforth, δ 1 will be fixed. In particular, according to our conventions, we may replace the δ 1 factor by 1 on the right hand side of (180). In view of (180) and (178), examining the identities of Sect. 8.2.1, we have obtained the degenerate coercivity of (Q f + Q y 1 ) .
We would like to improve this coercivity in the "angular-dominated" subrange of G 1 . Let us now introduce a new parameter C 1 and consider the range Noting that we have in G 1 , it follows that for C sufficiently large, we have in (181). Henceforth, C will be fixed. We may now define a new small parameter δ 3 > 0 and define a function where χ is the cutoff from above. We have that for frequency triples in (181), . In particular, we may choose δ 3 sufficiently small, with the smallness requirement depending only on M, so that, defining we have in (181). Henceforth, δ 3 will be fixed.
We are ready now for our final definitions. In the range (181), we define y by (182). Since δ 3 is now fixed we may now write We thus can set r trap = 0.
For the remaining frequencies in G 1 , i.e. for frequencies in G 1 ∩ { < C ω 2 }, we define simply y = y 1 and r trap = r max .
Finally, we consider the current

EQ T
for E the parameter fixed in Sect. 5.1. Thus, applying the identity corresponding to (172) in view of (178), (180) and (183), we obtain that Proposition 8.2.2 holds for all frequencies in G 1 .

The G 2 Range
We define this frequency range to be the complement of G 1 , i.e.
These are the "time-dominated" large frequencies.
We may choose c sufficiently small, and C sufficiently large, so that for sufficiently small |a| < a 0 M, we have Henceforth, c and C will be fixed by the above restriction. We note that it is certainly the case that C ≥ c .
Consider the function f 0 of the previous section. We define simply f = f 0 for frequencies in G 2 .
Given the parameter δ 1 fixed in Sect. 8.2.3, we define now y = δ 1 f . It follows from (185) that in the range G 2 we have We may define thus the parameter r trap = 0 for the frequency range G 2 . Finally, we may again add

EQ T
for E the parameter fixed in Sect. 5.1. Thus again applying the identity to (172) with the above definitions we obtain that Proposition 8.2.2 holds for all frequencies in G 2 .
Since G 1 ∪G 2 contains all admissible frequencies, the results of this section together with Section 8.2.3 imply that Proposition 8.2.2, and thus (170), indeed holds.
The proof of Proposition 8.2.1 is now complete. Let us recall that in the course of the above proof, we have fixed the parameter δ 1 . This allows us to fix also δ 2 of Proposition 5.1.1. Since E has been fixed previously, it follows that all dependences on parameters can be removed from the in the statement of Proposition 5.1.1.

Transport Estimates for ψ [±2] and u [±2]
In this section we will prove frequency-localised versions for the transport estimates of [31] to obtain estimates for u [+2] and ψ [+2] from [+2] as well as for u [−2] and The main result of the section is: With the assumptions of Theorem 8.1, we have the following estimates: Proof We consider first the case +2 of (186). Adding the identity arising from multiplying (157) by r √ ψ [+2] and its complex conjugate by r √ ψ [+2] leads after integration and applying Cauchy-Schwarz on the right hand side to the estimate Similarly, adding the identity arising from multiplying (156) by r u [+2] w and its complex conjugate by ru [+2] w leads after integration and applying Cauchy-Schwarz on the right hand side to the estimate Combining (187) and (188) yields (186) without the m 2 and ω 2 terms in the norms on the left.
To obtain the estimate with the m 2 and ω 2 terms we define the frequency ranges where C is the constant of Sect. 8.2.3. In view of the general bound (179) which holds for all admissible frequencies, it follows that in the frequency range F , we have and thus F is contained in the frequency range (181). It follows that r trap = 0 for F , i.e. these frequencies are not "trapped". Suppose first that (ω, m) lie in the frequency range F . Since r trap = 0, we have Multiplying thus (157) Multiplying this by r √ ψ [+2] and adding the complex conjugate multiplied by r √ ψ [+2] we find, upon integration and using Cauchy-Schwarz on the right hand side, the estimate Using the pointwise relation (157) In the range F , restricting to sufficiently small |a| < a 0 M, we have that It follows that in the inequality (192), we can replace the factor in round bracket on the left hand side simply by ω 2 and absorb the second term on the right by the left hand side. This establishes (186) for the ψ [+2] -norm on the left. We can now multiply (188) by m 2 and ω 2 and use the estimate just obtained for ψ [+2] to establish the estimate (186) also for the u [+2] w-term. The proof of (186) is now complete.
To prove (186) for s = −2 one follows the identical argument but choosing the multiplier 1 r instead of r .
appearing in Proposition 8.2.1 satisfies Proof Since f , f and y are all uniformly bounded we have by Cauchy-Schwarz: For the last remaining term, , we observe that we only need to estimate and where c (r ) denotes a generic bounded real-valued function with uniformly bounded derivative in A * 1 , A * 2 (whose explicit form may change in the estimates below). This is because the other terms appearing in J [±2] are again easily controlled via Cauchy-Schwarz and satisfy the estimate (194). We show how to estimate these terms for s = + 2, the case s = − 2 being completely analogous.
For the first term of (195) we have where we have used the (frequency localised) relation between [+2] and ψ [+2] twice. Now the first three terms on the right hand side are again easily controlled using Cauchy-Schwarz (as well as the simple fact that for i = 1, 2 . For the term in the last line we integrate the first summand by parts while the second is already manifestly controlled by dψ [±2] 2 . This leads immediately to (193). For the second term of (195), write The second term on the right is already manifestly controlled by dψ [±2] 2 and for the first we integrate by parts

Back to Physical Space: Energy Boundedness and Integrated Local Energy Decay
We now turn in this section in ernest to the study of the Cauchy problem for (37) for s = ± 2. The main result of this section will be a uniform (degenerate) energy boundedness and integrated energy decay statement. This will be stated as Theorem 9.1 of Sect. 9.1. This corresponds to statement 1. of the main result of the paper, Theorem 4.1.
The remainder of the section will then be devoted to the proof of Theorem 9.1. We first define in Sect. 9.2 a cutoff version α [±2] Q of our solution α [±2] of (37) such that α [±2] Q satisfies an inhomogeneous equation (53), whose inhomogeneous term F [±2 Q is localised in time to be supported only "near"t * = 0 and "near"t * = τ final and in space to be supported only in The cutoff is such that restricted to Q is compactly supported int * ∈ [0, τ final ]. This allows us in Sect. 9.3 to then apply the results of Sect. 8 to such α [±2] Q , summing the resulting estimate over frequencies. In Sect. 9.4 we shall combine this estimate with the conditional estimates of Sect. 5, using also the auxiliary estimates of Sect. 5.3 to obtain a global integrated energy decay statement, with an error term, however, on the right side arising from the cutoff. Finally, we shall bound this latter error terms associated to the cutoff in Sect. 9.5, again using the auxiliary estimates of Sect. 5.3, allowing us to infer the statement of Theorem 9.1.
As remarked in Sect. 1.2.5, in the axisymmetric case, one can directly distill from the calculations of this paper an alternative, simpler proof of Theorem 9.1 expressed entirely in physical space. We do this in Sect. 9.6.

Statement of Degenerate Boundedness and Integrated Energy Decay
• the η-weighted energy boundedness estimate Similarly, for s = −2 , we have • the basic degenerate Morawetz estimate • the η-weighted energy boundedness estimate In the proof of the theorem, we may assume for convenience that the data ) are smooth. It follows that all associated appropriately rescaled quantities [±2] , etc., are smooth in R 0 . To ease notation we define the data quantities Here the subindex [η] is equal to η in case of s = +2 and it is dropped entirely in case s = −2.

Remark 9.2
As the proof shows and is already clear from the support of the cut-offs, only the spacetime integrals in the overlap region are needed on the right hand side of (209).
Proof We first prove (208). Note that the support of G is manifestly contained in the support (207) ofF [±2] Q . Moreover, one easily sees that one obtains sum of terms containing with r and horizon weights which are uniformly bounded in view of the support. From the conditions (204) defining , it follows that for any k 1 + k 2 + k 3 ≥ 1, where we have used also that t = t * =t * in this region by our choices in Sect. 2.1.3. It follows that all terms in the expression for G pick up an ε factor. The inequality (208) now follows from Cauchy-Schwarz, the definition of the norms and Remarks 4.1 and 4.2 , where in addition we have appealed to the coarea formula and size oft * -support for the term involving [±2] . The proof is the same for (209), except that the nontrivial r dependence of given by (204) means that ε on the right hand side of (210) must now be replaced by 1 outside of r ∈ [A 1 , A 2 ], and thus the ε 2 factor of (208) is no longer present in the right hand side of the final estimate.
We will in fact not use the bound (209) directly, but similar bounds for physical space terms that arise from multiplying G and G∂ r .

The Summed Relation
In view of the support ofα [±2] Q and the smoothness of (206), it follows that α [±2] Q manifestly satisfies the [A 1 , A 2 ]-admissibility condition of Definition 6.1. In a slight abuse of notation, we will denote the coefficients of α [±2] Q , [±2] Q , etc., without the Q subscript. 12 We where ).

Global Physical Space Estimates
Let us first combine the above estimates with the conditional physical space estimates proven in Sect. 5. Q as in (205), (38) and (46)- (50). Then we have the estimates Q with τ 1 = 0, τ 2 = τ final and with p = η) to a suitable constant times the estimate (212) of Proposition 9.3.1. The constant ensures that the terms E r =A 2 on the left hand side of (105) is sufficient to absorb the analogous term on the right hand side of (212). Finally, we now add to the previous combination a suitable constant times the second estimate (106) of Proposition 5.2.1, again so that the boundary terms on E r =A 1 are now absorbed. We obtain thus We fix now a sufficiently small parameter e depending only on M. It follows that, restricting to a 0 e, we may sum e× (215) with (217) to absorb both the first term on the right hand side of (215) and the middle two terms on the right hand side of (217). The desired (213) follows.
In the rest of this subsection, we proceed to remove the Q from the quantities on the left hand side of (213).
Putting together the local-in-time Proposition 5.3.2 and the (T + χ ω + )-energy estimate Proposition 5.3.1 we obtain first the following: Proposition 9.4.2 With the notation of Proposition 9.4.1, we have the additional estimates Here the subindex [η] is equal to η in case of s = +2 and it is dropped entirely in case s = −2.
Finally, to obtain (221), we argue as follows. Revisiting the transport estimates of Sect. 5.2, we can estimate the left hand side from initial data, I deg η [ [±2] ](0, τ final − ε −1 ), the left hand side of (219) and the left hand side of (220).
Using once again the auxiliary estimates of Sect. 5.3, we can now improve this to: Proof Appealing to Proposition 5.3.2 with τ 1 = τ final − ε −1 and with τ step = ε −1 , and using (221), we obtain Finally, we apply (220) to absorb the last term on the right hand side, obtaining thus (222). Repeating now the proof of (220) we obtain (223).

Controlling the Term
and consists of quadratic terms one of which always contains a [±2] -term. Thus, by Cauchy-Schwarz this can easily be bounded by the first three lines of the right hand side of (225), where an ε −1 factor is introduced on the term, compensated by an ε on the other terms. (The extra ε factor in ε −2 arises from estimating a spacetime integral by the supremum. Cf. the proof of (209).) For H 1 , by the exact Plancherel formulae of Sect. 6.2.3, the integral (226) transforms into a physical space integral supported in which similarly to before, is obviously estimable from the first three lines of the right hand side of (225). (In fact, one could replace the factor ε −2 with 1, since, just as in the proof of (208),t * derivatives of the cutoff always generate extra ε factors; we will use this idea below for estimating the remaining term.) For H 2 , we first apply Cauchy-Schwarz, introducing a ε −1 , where we have used (165) to bound the f , f and y factors uniformly over frequencies.
We now apply Plancherel. We note that by Proposition 9.2.1, the first term on the right hand side is bounded by ε −1 × the right hand side of (208) while the second term is manifestly bounded by We obtain (225) for H 2 , finishing the proof.

Note on the Axisymmetric Case: A Pure Physical-Space Proof
We note that in the axisymmetric case ∂ φ α [±2] = 0, the physical space multiplier and transport estimates of Sect. 5 can be applied directly globally in the region R(τ 1 , τ 2 ), i.e. without the restriction to R away (τ 1 , τ 2 ). This leads already to a much shorter proof of Theorem 9.1 which can be expressed entirely with physical space methods. We explain how this physical-space proof can be distilled directly from the more general calculations of Sect. 8 done at fixed frequency.
Given |a| < a 0 M sufficiently small, let r trap be the unique value given by Lemma 8.2.1 and define f by (176) and y by δ 1 ((1 − χ) where χ is the cutoff appearing in (177) andχ is the cutoff appearing in (101). The calculation of Sect. 8.2.3 now shows that the coercivity property of the physical space current I f + I y holds globally in R trap (τ 1 , τ 2 ) and thus (102) holds when integrated globally in R(τ 1 , τ 2 ), i.e. without restriction to the "away" region and with I away replaced by I deg η . One also produces an estimate for the future boundary term: in view of property 3. of the proof of Proposition 5.1.1. We apply this estimate then in the region R(0, τ 2 ) directly to [±2] arising from a solution α [±2] of the homogeneous Teukolsky equation (37).
We must estimate the error term arising from the coupling J [±2] . For this we turn first to global transport estimates.
Note that in the axisymmetric case, the simple estimate applied in Sect. 8.3 for frequencies in the range F applies now for all frequencies (since F = ∅ if m = 0) and corresponds to commuting the transport equations by ∂ r * and integrating by parts. This physical space procedure, say in the [+2] case, allows one to obtain the estimate Note that I trap (0, τ 2 ) is degenerate and thus controlled by I deg η (0, τ 2 ). Summing (232) with the estimates obtained from (105) and (106), as in the proof of Proposition 9.4.1, allows one to estimate finally With this we estimate the new contribution to J ±2] coming from the region R trap (τ 1 , τ 2 ). The only difficult term is the one arising from the T multiplier. In the fixed frequency estimate of Sect. 8.4, this corresponded to passing an ω from ψ to before applying Cauchy-Schwarz. In physical space, this corresponds simply to integration by parts in t. By this physical space estimate, we obtain that the resulting term is bounded by where the future boundary terms arise from this integration by parts. (Note that all other terms in J [±2] are estimated by the first line of (234) alone.) Combining with the original statement of Proposition 9.4.1, this yields In view of (233), for sufficiently small |a| < a 0 M, one can absorb the terms (234) on the right hand side of (235) into the left hand side. The remaining statements of Theorem 9.1 follow immediately.

The Redshift Effect and Its Associated Morawetz Estimate
In this section we will obtain statement 2. of Theorem 4.1 concerning the boundedness and integrated local energy decay of the so-called red-shifted energy. The required statement is contained in Theorem 10.1 below.

Statement of Red-Shifted Boundedness and Integrated Decay
• the basic non-degenerate Morawetz estimate • the η-weighted energy boundedness estimate For s = −2 • the basic degenerate Morawetz estimate • the basic non-degenerate Morawetz estimate • the η-weighted energy boundedness estimate

Proof of Theorem 10.1
We only prove the s = + 2 case. The s = − 2 case is completely analogous and slightly easier because the term J [−2] has stronger degeneration near the event horizon. Note that in Sect. 9 we have already proven the estimates (236) and (238) provided we drop all overbars from the energies that appear. The estimate (237), which does not degenerate in a neighbourhood of r = 3M but loses a derivative, is a simple corollary of (236) and (238) again provided we drop all overbars from the energies. Hence the only task left is to improve the L derivative in the energies that appear. This is achieved using the redshift multiplier of [39,43]: The redshift identity. Recall the notational conventions of Sect. 5.1.1. Multiplying (54) by Y = 1 w ξ L (with ξ a smooth radial cut-off function equal to 1 for r ∈ r + , r + + 1 4 M and equal to zero for r ≥ r + + 1 2 M) and taking the real parts yields (use the formulae of Appendix B.5) We apply the identity (242) to the equation satisfied by [+2] . In particular, G [s] = 0 because α [+2] satisfies the homogeneous Teukolsky equation. Upon integration over R (τ 1 , τ 2 ) (recalling Remark 5.1) we obtain (236) and (238) after making the following observations: • The first term in F Y L and the first term in F Y L are manifestly non-negative and produce precisely the desired improvement in the L derivative and the missing angular derivative in the horizon term respectively. All other terms appearing as boundary terms can now be controlled using Cauchy-Schwarz and (35) by the energies without the overbar (sometimes borrowing an from the just obtained good L-derivative term and the good angular term respectively is required).
• Examining (245), the term 1 2 ξ w w 2 |L | 2 is manifestly positive and produces precisely the desired improvement of the |L | 2 in the spacetime energy without the overbar. All other terms can be controlled by the spacetime energy without the overbar, sometimes borrowing an from the improved |L | 2 term.
• The error term is controlled using Cauchy's inequality with and the energies Finally, the estimate (237) follows from its un-overbarred version by adding the just established (236).

The r p -Weighted Hierarchy and the Main Decay Result
To complete the proof of Theorem 4.1, it remains to obtain statements 3. and 4. concerning the r p -weighted hierarchy and polynomial decay. The required statement is contained in Theorem 11.1 below.

Statement of the Decay Theorem
for the initial data energy and for the initial data energy

Proof of Theorem 11.1 for s = +2
The s = +2 case of Theorem 11.1 will be proven in Sect. 11 and the non-degenerate estimate Proof Given α [+2] we apply the multiplier identity (96) to [+2] . To the identity that is being produced after integration over R(τ 1 , τ 2 ), we can add a large constant B and the estimate

Completing the Proof of Theorem 11.1
Combining the estimate of Proposition 11.2.1 with that of Proposition 11.2.2 we deduce for p ∈ {η, 1, 2} (first for K = 0 and then by trivial commutation with the Killing field T for any K ∈ N) the estimate and also Let us denote the right hand side of the second estimate on the initial data slice 0 (i.e. for τ 1 = τ 0 ) by Applying (255) for K = 1 and p = 2 implies (after using a standard argument involving dyadic sequences) along a dyadic sequence τ n ∼ 2 n τ 0 the estimate Using the above and applying (254) for p = 1, K = 1 between the time τ 1 = τ n and any τ 2 ∈ (τ n , τ n+1 yields the previous estimate for any τ , not only the members of the dyadic sequence. Turning back to (255) now with K = 0 we use the previous estimate and a similar dyadic argument to produce along a dyadic sequence the estimate Using the above and applying (254) with p = η and K = 0 now yields the estimate (246) of Theorem 11.1.

Proof of Theorem 11.1 for s = −2
The s = − 2 case of Theorem 11.1 will be proven in Sect. 11 and the estimate The same estimates hold with an overbar on all terms.
Proof Multiply (119) by a cut-off function ξ which is equal to 1 for r ≥ 9M and equal to zero for r ≤ 8M. Bringing ξ inside the first bracket produces an error-term supported in [8M, 9M], which (upon integration) is for any n controlled by the basic estimate (239). Upon integration of the resulting identity we deduce (260) for being the identity in the energies appearing. Now we observe that the same estimate holds for the T and commuted equations (note that (117) commutes trivially with the Killing fields T and so the estimate (111) trivially holds for the commuted variables). The estimate (259) is proven completely analogously except that here no cut-off is required in view of the non-degenerate norm of ψ [−2] appearing on the right hand side: One first applies (118) and the same estimate for the T and commuted variables.
To obtain the estimates with an overbar one first commutes (117) and (116) with L and notes that the analogue of (119) and (118) can now be applied with the error from the commutator L, L ∼ a r 3 being controlled by the previous step. Secondly, one commutes (117) and (116) with the vectorfield r 2 +a 2 L which extends regularly to the horizon and observes that the additional commutator term leads to a good sign (near the horizon) in the estimates (119) and (118).

Completing the Proof of Theorem 11.1 for s = −2
Combining the estimate of Proposition 11.3.1 with that of Proposition 11.3.2 we deduce for p ∈ {η, 1, 2} (first for K = 0 and then by trivial commutation with the Killing field T for any K ∈ N) the estimate and also K k=0 Let us denote the right hand side of the second estimate on the initial data slice 0 (i.e. for τ 1 = τ 0 ) by D K +1, p [−2] , ψ [−2] , α [−2] (τ 0 ). Applying the estimate (262) for K = 1 and p = 2 implies (after using a standard argument involving dyadic sequences) along a dyadic sequence τ n ∼ 2 n τ 0 the estimate Using the above and applying (261) for p = 1, K = 1 between the time τ 1 = τ n and any τ 2 ∈ (τ n , τ n+1 yields the previous estimate for any τ , not only the members of the dyadic sequence. Turning back to (262) now with K = 0 we use the previous estimate and a similar dyadic argument to produce along a dyadic sequence the estimate Using the above and applying (261) with p = η and K = 0 now yields (247) of Theorem 11.1. To obtain the second estimate, one simply repeats the above proof using that the estimate of Proposition 11.3.2 also holds for the energies with an overbar. and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

A.1 The Teukolsky Equation
Recall from (13) the definition w = (r 2 +a 2 ) 2 . Using (39) as well as recalling the definitions

A.2 Derivation of the [s] Equation in Physical Space
We derive the equation for s = +2 in physical space. Observing the commutation relation L, L = 4ra r 2 + a 2 w · , we obtain after applying L to the Teukolsky equation (268) which can be written more succinctly as The terms in the second line of (274) simplify to w −8a · −r 2 + a 2 r 2 + a 2 + 20a 2 r 3 − 3Mr 2 + ra 2 + Ma 2 r 2 + a 2 2 Finally, for the double commutator term in the third line of (274) we obtain +12a 3 w r r 2 + a 2 − 3a 2 w Combining the previous equations we have therefore established the formula of Proposition 3.2.1 for s = +2. The formula for s = −2 is proven entirely analogously exchanging the roles of L and L. For completeness, we nevertheless explicitly derive the (equivalent) separated form of the equation for s = −2 in the next section.

A.3 Derivation of the [s] Equation in Separated Form
As mentioned at the end of the previous section and mainly to illustrate the equivalence between deriving equations in the physical space and the separated picture, we now derive the equation for [−2] in separated form from (271). Recall that L and L are now given by the separated frame operators (155) and (154). Observing the commutation relation L, L = 4ra r 2 + a 2 w · im , we obtain after applying L to the separated Teukolsky equation (271) which can be written more succinctly as The terms in the second line simplify to w +8aim · −r 2 + a 2 r 2 + a 2 + 20a 2 r 3 − 3Mr 2 + ra 2 + Ma 2 r 2 + a 2 2 Finally, for the third line of (274) excluding the inhomogeneous term we obtain −12a 3 w r r 2 + a 2 im − 3a 2 w r 4 − a 4 + 10Mr 3 − 6Ma 2 r (r 2 + a 2 ) 2 u [−2] w . (283) In summary, we have established the following formula for s = −2: where the right hand is side given by J [s] = ρ 2 r 2 + a 2 2 −4s r 2 − a 2 r 2 + a 2 aim − 20a 2 r 3 − 3Mr 2 + ra 2 + Ma 2 r 2 + a 2 2 √ ψ [s] + a 2 ρ 2 r 2 + a 2 2 −6s r r 2 + a 2 aim + 3 We have therefore proven Proposition 7.3.1 for s = −2. The s = +2 case is proven entirely analogously or can be easily deduced directly from the physical space formula of Proposition 3.

B Auxilliary Calculations for Physical Space Multipliers
We first recall the relations L + L = 2T + 2 a r 2 + a 2 , L − L = 2∂ r * .
We will consider the identities generated by the following four multipliers (the smooth radial cut-offs χ , ξ and the smooth radial functions f , h, y are chosen appropriately in the body of the paper) 1.
The T -energy: T 2. The Lagragian multiplier: h 3. The -multiplier: ω + χ (χ a radial cut-off) 4. The y-multiplier: f L − L 5. The redshift multiplier: 1 w ξ L (ξ a radial cut-off near the horizon) 6. The r p weighed multiplier: r p β k ξ L with β k = 1 + k M r (ξ a radial cut-off near infinity) each acting on the second order terms in the equation (54), namely (recall w = (r 2 +a 2 ) 2 ) I. 1 2 L L + L L II. w/ The point is that the 0 th order terms in (54) are easy to handle while for the (only) first order term in (54), 2iswa cos θ T , we observe that for X any real vectorfield commuting with T we have 2iswa cos θ T X = T 2iswa cos θ X − X 2iswa cos θ T and hence Re 2iswa cos θ T X = T iswa cos θ X − X iswa cos θ T + X (iswa cos θ ) T = −T swa cos θ Im X + X swa cos θ Im T − X (swa cos θ ) Im T .
In particular for X = T the right hand side is zero while for X = ω + χ only the first two terms survive (and only the first after integration in φ).