Precise asymptotics of the spin $+2$ Teukolsky field in the Kerr black hole interior
Sebastian Gurriaran




§ ABSTRACT 

    Using a purely physical-space analysis, we prove the precise oscillatory blow-up asymptotics of the spin +2 Teukolsky field in the interior of a subextremal Kerr black hole. In particular, this work gives a new proof of the blueshift instability of the Kerr Cauchy horizon against linearized gravitational perturbations that was first shown by Sbierski <cit.>. In that sense, this work supports the Strong Cosmic Censorship conjecture in Kerr spacetimes. The proof is an extension to the Teukolsky equation of the work <cit.> by Ma and Zhang that treats the scalar wave equation in the interior of Kerr. The analysis relies on the generic polynomial decay on the event horizon of solutions of the Teukolsky equation that arise from compactly supported initial data, as recently proved by Ma and Zhang <cit.> and Millet <cit.> in subextremal Kerr.



Independence Constrained Disentangled Representation Learning from Epistemological Perspective
    Ruoyu Wang 1 Lina Yao 1,2
    September 9, 2024
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§ INTRODUCTION



 §.§ The Kerr black hole interior

We begin with a review of the main features of the interior of Kerr black holes. The Kerr metric describes the spacetime around and inside a rotating non-charged black hole. It is a two-parameter family of stationary and axisymmetric solutions of the Einstein vacuum equations that write, denoting 𝐑𝐢𝐜[𝐠] the Ricci tensor associated to a Lorentzian metric 𝐠,

    𝐑𝐢𝐜[𝐠]=0.

The two parameters are the mass M and the angular momentum per unit mass a of the black hole. In this work, we only consider subextremal Kerr with non-zero angular momentum, i.e. such that 0<|a|<M. The Kerr metric is given in Boyer-Lindquist coordinates (t,r,θ,ϕ)∈ℝ^2×𝕊^2 by 

    𝐠_a,M=-(Δ-a^2sin^2θ)/q^2 t^2-4aMr/q^2sin^2θ t ϕ+q^2/Δ r^2+q^2θ^2+Σ^2/q^2sin^2θϕ^2 ,

where Δ:=r^2-2rM+a^2, q^2:=r^2+a^2cos^2θ, Σ^2:=(r^2+a^2)^2-a^2sin^2θΔ. We define

    μ:=Δ/r^2+a^2,   r_±:=M±√(M^2-a^2),

where r_± are the roots of Δ. The singularities {r=r_±} of the metric are coordinate singularities that vanish when considering Eddington-Finkelstein coordinates

    u:=r^*-t,  :=r^*+t,  ϕ_±:=ϕ± r_mod  mod 2π,

where r^*/ r=μ^-1, r_mod/ r=a/Δ, see Section <ref>. The event horizon {r=r_+} and the Cauchy horizon {r=r_-} can then be properly attached to the Lorentzian manifold (r_-,r_+)××𝕊^2 equipped with the metric 𝐠_a,M, see <cit.> for more details. We will mainly be interested in a region containing the right event horizon and the right Cauchy horizon that are respectively defined by

    _+:={r=r_+}∩{u=-∞},  :={r=r_-}∩{=+∞}.

Note that a result similar to our main theorem can be deduced in a region containing the left event and Cauchy horizons, that are defined by

    _+':={r=r_+}∩{=-∞},  𝒞ℋ'_+:={r=r_-}∩{u=+∞}.

We denote ^2_:='_+∩_+ and ^2_𝒞ℋ:=𝒞ℋ'_+∩ the bifurcations spheres, and i_+, ℐ_+ (resp. i_+', ℐ_+') the right (resp. left) timelike and null infinities.

In this work, by 'Kerr interior' we mean the resulting Lorentzian manifold (,𝐠_a,M) which is ((r_-,r_+)××𝕊^2,𝐠_a,M) to which we attach its boundaries, the event and Cauchy horizons. We will be interested in the asymptotics at  of solutions to the Teukolsky equation, that arise from compactly supported initial data on a spacelike hypersurface Σ_0. See Figure <ref> for the Penrose diagram of Kerr interior and an illustration of the hypersurface Σ_0.



 §.§ Teukolsky equations

Teukolsky <cit.> found that, when linearising a gravitational perturbation of a Kerr black hole in the Newman-Penrose formalism, two curvature components decouple from the linearised gravity system and satisfy wave equations, called the Teukolsky equations. More precisely, we first define the pair of null vector fields given in Boyer-Lindquist coordinates by

    e_3:=1/2(-∂_r+r^2+a^2/Δ∂_t+a/Δ∂_ϕ),     e_4:=1/2(Δ/r^2+a^2∂_r+∂_t+a/r^2+a^2∂_ϕ),

which is aligned with the principal null directions of the Kerr spacetime, and which is regular on _+. We then define the rescaled null pair 

    :=(-μ) e_3,  :=(-μ)^-1 e_4,

which is regular on . We also define the slightly modified null pair

    n:=-(r^2+a^2)/q^2=[(r^2+a^2)∂_t-Δ∂_r+a∂_ϕ]/(2q^2),   l:=-2=r^2+a^2/Δ∂_t+∂_r+a/Δ∂_ϕ,

and the complex vector field

    m:=1/√(2)(r+iacosθ)(iasinθ∂_t+∂_θ+i/sinθ∂_ϕ).

Then, denoting 𝐑̇ the linearized curvature tensor, the scalars

    :=𝐑̇_lmlm,  :=(r-iacosθ)^4𝐑̇_nmnm,

are called respectively the spin +2 and spin -2 Teukolsky scalars. They satisfy the Teukolsky equations that write, for s=± 2,

    -[(r^2+a^2)^2/Δ-a^2 sin ^2 θ] ∂_t^2_s -4 M a r/Δ∂_t∂_ϕ_s  -[a^2/Δ-1/sin ^2 θ] ∂_ϕ^2_s 
        +Δ^-s∂_r(Δ^s+1∂_r _s)+1/sinθ∂_θ(sinθ∂_θ_s)+2 s[a(r-M)/Δ+i cosθ/sin ^2 θ] ∂_ϕ_s  
        +2 s[M(r^2-a^2)/Δ-r-i a cosθ]∂_t_s -[s^2 cos ^2 θ/sin ^2 θ-s]_s=0.


The rescaled scalars 

    :=Δ^2,  :=Δ^-2,

satisfy a rescaled version of the Teukolsky equation, see Section <ref> for the precise definition of the Teukolsky wave operators. Notice that  and  are projections of the linearized curvature on a frame that is regular on _+, while ,  are projections of the linearized curvature on a frame that is regular on . Thus the main result of this work, namely the blow-up asymptotics of  at , is a linear curvature instability statement for the Kerr Cauchy horizon.

The Teukolsky equations were originally introduced to study the stability of the exterior of Kerr black holes. For a review of the litterature concerning the Teukolsky equations, see the introduction of <cit.>, where the decay estimates for the nonlinear analog of the Teukolsky equations derived in <cit.> are used to prove the nonlinear stability of the exterior of slowly rotating Kerr black holes in <cit.>.



 §.§ Blueshift effect and Strong Cosmic Censorship conjecture



  §.§.§ Blueshift effect


The Kerr spacetime is globally hyperbolic only up to the Cauchy horizon ∪𝒞ℋ'_+. Thus the part of the spacetime that is beyond the Cauchy horizon, in the region {r≤ r_-}, can be thought as unphysical as it is not determined by the data on any spacelike hypersurface inside {r≥ r_-}. In other words, the Kerr spacetime can be extended across the Cauchy horizon as a regular solution of the Einstein vacuum equations (<ref>) in infinitely many ways, which goes against determinism in general relativity.

However, it is expected that this unphysical feature is just an artefact of the ideal Kerr spacetime. Indeed, realistic astrophysical black holes are perturbations of Kerr, i.e. they are the maximal globally hyperbolic developpement of initial data close to the one of Kerr. It is expected that these perturbations kill the non-physical features of ideal Kerr. In the linear setting, this is the so-called blueshift effect, introduced by Simpson and Penrose in 1972 <cit.>. It is a heuristic argument according to which the geometry of Kerr interior (they initially wrote the argument for Reissner–Nordström) 
forces propagating waves to blow-up in some way at the Cauchy horizon. This effect is illustrated on Figure <ref>, and is linked to the Strong Cosmic Censorship conjecture.




  §.§.§ Strong Cosmic Censorship conjecture

The Strong Cosmic Censorship (SCC) conjecture was formulated by Penrose in <cit.>, and, in its rough version, states the following :

    The maximal globally hyperbolic developpement (MGHD) of generic initial data for the Einstein equations is inextendible.

In other words, this conjecture states that the unphysical, non-deterministic behavior of spacetimes with non-empty Cauchy horizons (for example Kerr and Reissner-Nordström spacetimes) is non-generic, and thus vanishes upon small perturbations.  See <cit.> for more modern versions of the SCC conjecture. 

A fundamental question in SCC is the regularity for which the MGHD of initial data should be inextendible. The C^0 formulation of SCC was disproved in Kerr by Dafermos and Luk <cit.>. They showed that generic perturbations of the interior of Kerr still present a Cauchy horizon across which the metric is continuously extendible. They also argued that the perturbed Cauchy horizon may be a so-called weak null singularity, which is a singularity weaker than a spacelike curvature singularity as in Schwarzschild. For references on weak null singularities, see <cit.>, <cit.>. See <cit.> for a link between weak null singularities and the C^0,1_loc formulation of SCC. 

In spherical symmetry, the C^2 instability of the Cauchy horizon for the model of the Einstein-Maxwell-scalar field system was proven in <cit.> and <cit.>, extending the results of <cit.>. See also <cit.> for analog results for the Einstein-Maxwell-Klein-Gordon equations. In Kerr, which is only axisymmetric, the full nonlinear problem for the Einstein equations is still open, and we focus in this paper on the model of linearized gravity, where the Teukolsky scalar  represents a specific component of the perturbed curvature tensor. Our main result, namely the blow-up of  on , thus supports the SCC conjecture in the linearized setting.




 §.§ Black hole interior perturbations



  §.§.§ Results related to Price's law

The starting point to prove the instability of solutions of the Teukolsky equation in Kerr interior is Price's law for Teukolsky, i.e. the polynomial lower bound on the event horizon for solutions of the Teukolsky equation arising from compactly supported initial data, see <cit.> for the original works on Price's law. A version of Price's law for the Teukolsky equations in Kerr was heuristically found by Barack and Ori in <cit.>. 

In this paper we use the precise Price's law asymptotics on _+ given by Ma and Zhang in <cit.> for the Teukolsky equations[The Price's law in <cit.> holds for |a|≪ M, and for |a|<M conditionally on an energy-Morawetz bound. This energy-Morawetz estimate has since been proved for |a|<M by Teixeira da Costa and Shlapentokh-Rothman in <cit.>, so that the Price's law in <cit.> holds for the full range |a|<M.]. For another proof of the polynomial lower bound in the full subextremal range |a|<M for solutions of the Teukolsky equation, see the work of Millet <cit.>, that uses spectral methods. For a complete account of results related to Price's law, see <cit.>.



  §.§.§ Previous results on black hole interior perturbations


The first works on the linear instability of the Cauchy horizons in Kerr and Reissner-Nordström black holes consisted in finding explicit solutions that become unbounded in some way at the Cauchy horizon, see for example <cit.>. In <cit.>, a heuristic power tail asymptotic for scalar waves in the interior of Kerr black holes was obtained. Regarding the Teukolsky equations, the oscillatory blow-up asymptotic of our main result in the interior of Kerr black holes, see (<ref>), was first predicted heuristically by Ori <cit.>, writing the azimuthal m-mode of the solution as a late-time expansion ansatz of the form

    ∑_kψ_k(r,θ)t^-k.

The asymptotic behavior (<ref>) was also confirmed in a numerical simulation <cit.>.

A rigorous boundedness statement for solutions of the scalar wave equation inside the spherically symmetric Reissner-Nordström spacetime was proven in <cit.>. Still for the scalar wave equation in the interior of Reissner-Nordström black holes, the blow-up of the energy of generic scalar waves was obtained in <cit.>.  A scattering approach to Cauchy horizon instability in Reissner-Nordström, as well as an application to mass inflation, was presented in <cit.>, on top of the non-linear instability results <cit.> already mentionned in Section <ref>.

In Kerr, for the scalar wave equation, a generic blow-up result for the energy of solutions on the Cauchy horizon was obtained in <cit.>, while the boundedness of solutions at the Cauchy horizon was proven in <cit.> in the slowly rotating case. The boundedness result was then extended to the full subextremal range in <cit.>. A construction of solutions that remain bounded but have infinite energy at the Cauchy horizon was presented in <cit.>. Finally, the precise asymptotics of the scalar field in the interior of a Kerr black hole was proven in <cit.> using a purely physical-space analysis.

Concerning the Teukolsky equations in Kerr interior, the method of proof of <cit.> was extended to the spin +2 Teukolsky equation in the work <cit.>, that proved the blow-up of a weighted L^2 norm on a hypersurface transverse to the Cauchy horizon, relying on frequency analysis. 
 
The goal of the present paper is to rigorously prove the oscillatory blow-up asymptotics of the spin +2 Teukolsky field in the Kerr black hole interior, by extending the physical-space approach of <cit.> to Teukolsky equations, thus providing a new proof of the blow-up results of <cit.>.

We discussed here the references on black hole interior perturbations that are the most relevant to this work. For a more complete account of the results related to black hole interior perturbations, for example in Schwarschild interior or in the cosmological setting, see the introduction in <cit.>.




 §.§ Rough version of the main theorem


This paper rigorously proves the  blueshift instability on the Kerr Cauchy horizon for solutions of the spin +2 Teukolsky equations, by finding the precise oscillatory blow-up asymptotics of the spin +2 Teukolsky scalar. The rough version of the main result of this paper is the following, see Theorem <ref> for the precise formulation.

 Assume that the Price's law
polynomial lower bounds on the event horizon for solutions of the Teukolsky equations hold, as proven in <cit.>. Denote  the spin +2 Teukolsky scalar obtained in a principal null frame regular on the Cauchy horizon. Then  blows up at the Cauchy horizon, exponentially in the Eddington-Finkelstein coordinate , and oscillates at a frequency that blows up at the Cauchy horizon. More precisely,
    
    
  * the amplitude || is proportionnal to Δ^-2(u,)/^7, 
    
  * the oscillation frequency blows-up like log(r-r_-).


    


 We remark the following :

    
  * Anticipating some of the notations that will be introduced later on, we actually show the following precise asymptotic behavior near  :

    ∼Δ^-2(u,)/^7∑_|m|≤ 2A_m(r_-)Q_m,2e^2imr_mod(u,)Y_m,2^+2(cosθ)e^imϕ_-,

        where the constants Q_m,2 depend on the initial data on Σ_0 and are generically non-zero, the constants A_m(r_-) are non-zero for m≠ 0 (see Remark <ref>), ϕ_- is an angular coordinate that is regular on , the functions Y_m,2^+2(cosθ) are the s=+2 spin-weighted spherical harmonics, and r_mod∼log(r-r_-) near , see Sections <ref> and <ref> for more details.
        
  * The blueshift instability at  for  was first proven recently by Sbierski <cit.> who showed the blow-up of a weighted L^2 norm along a hypersurface transverse to . This result suggests a blow-up that is exponential in the Eddington-Finkelstein coordinate , the Cauchy horizon corresponding to =+∞. We prove in this paper a pointwise exponential blow-up, along with an oscillatory behavior, which were both heuristically predicted by Ori <cit.>.
        
  * We will use the version of Price's law proven in <cit.>. It holds true in subextremal Kerr conditionally on the existence of an energy and Morawetz estimate for the Teukolsky equation in the whole subextremal range |a|<M, which was recently proven in <cit.>.









 §.§ Structure of the proof

Although our main result will be about the precise asymptotics of  at , we will actually also obtain precise estimates for  near _+, and we will use the Teukolsky-Starobinsky identities (see Section <ref>) to link  and  there. 

For s=± 2, we denote _s the differential operator on the left-hand side of (<ref>), called the Teukolsky operator, and _s=Δ^s_s(Δ^-s · ) the rescaled Teukolsky operator, such that 
    _s_s=0,  _sψ_s=0.

The analysis is done entirely in physical space, using energy estimates to prove upper bounds on 'error' quantities, in the hope that this method of proof is robust enough to be applied in a more general setting. The first step is to notice that the energy estimates to get polynomial upper bounds done in <cit.> for the scalar wave equation _gψ=0 in the Kerr interior can be extended to the Teukolsky equations _sψ_s=0, but only for negative spin close to _+, and for positive spin close to . This is because we need a fixed sign for the scalar s(r-M) that appears at crucial places in the energy estimates, where s=± 2 is the spin, and because r_+-M>0 while r_–M<0.



We denote  the region containing _+,  and  the regions close to  where Δ has exponential decay in , and  the intermediate region between  and ∪. See Figure <ref> and Section <ref> for the precise definitions of the regions. The assumptions on _+ that we will use are the ones given by <cit.>, i.e. that the error quantities

    :=-,
       :=-,

are bounded by ^-7-δ on _+∩{≥ 1} where δ>0, see Sections <ref> and <ref> for the definitions of Q_m,2, A_m(r), Y_m,2^± 2(cosθ), ϕ_+. The main steps of the proof then go as follows :
    
  * First, we propagate Price's law lower bound for  in . To do this, we propagate the O(^-7-δ) upper bound of  from _+ to  using a redshift energy estimate. This is only possible in region  that contains the event horizon.
    
  * We then use the Teukolsky-Starobinsky identity (<ref>) to propagate in  the O(^-7-δ) Price's law upper bound on _+ for .
    
  * Next, similarly as in  we propagate the lower bound for  in  using an effective blueshift energy estimate for , which is only possible for r close to r_-. This step in region  is necessary to propagate the lower bound up to  where the analysis becomes more delicate (namely, where we can't anymore get sharp decay of the energy from an energy estimate because of the geometry of the Kerr black hole interior near the Cauchy horizon).
    
  * We then
    obtain a non-sharp L^∞ bound for  in  using an energy estimate.
    
  * To get the blow-up asymptotics in , we rewrite the Teukolsky equation _+2=0 as a 1+1 wave equation 
    
    ((r^2+a^2)Δ^2e_4)=O(Δ),

        where we use the previous L^∞ bound to control the right-hand-side. This is where we can effectively see the blueshift heuristic in action. The geometry of Kerr interior is such that  is regular on , and reporting that in the Teukolsky equation gives a factor Δ on the source term that becomes negligeable in . Integrating (<ref>) from Γ directly gives the announced blow-up asymptotics for . This is where we use the definition of region  : Δ decays exponentially in  in , which easily bounds the error terms. 
        
  * We cannot integrate the 1+1 wave equation in  because the slices {u=cst} starting inside  do not cross Γ. Instead, commuting e_3 and the Teukolsky operator, we estimate e_3 in , which allows us to propagate the blow-up of  from  to  by integration.

Here are some further remarks on the analysis :


    
  * To control the derivatives of the error terms, we commute the Teukolsky equation with operators that have good commutations properties, namely ∂_t, ∂_ϕ, e_3, and the Carter operator.
    
  * Note that, at any point in the analysis, in any region , , , , any estimate that we get can be turned into an estimate in the symmetric region {u≥ 1} by replacing  with u, ψ_s with _-s, e_3 with  and e_4 with . As in <cit.>, this will be useful in region . 


Here are some differences from the previous works in the method of proof :


    
  * Our physical-space proof differs substantially from <cit.> that relies in a crucial way on frequency analysis.
    
  * In <cit.> for the scalar wave equation, the proof is made by decomposing the scalar field ψ onto its ℓ=0 and ℓ≥ 1 modes : ψ=ψ_ℓ=0+ψ_ℓ≥ 1. Using the decay of both quantities on the event horizon, it is shown, commuting the wave equation with the projection on the modes, that the ^-3 lower bound for ψ_ℓ=0 and the O(^-4-δ) better decay of ψ_ℓ≥ 1 propagate. This is different from our proof, where there is no projection on the (spin-weighted) spherical harmonics. We do not need to assume that the modes ℓ≥ 3 of ψ_± 2 decay better than ^-7 on _+. Moreover, in the blueshift region (i.e. for r close to r_-) we will use easier estimates than <cit.> that introduces log multipliers, taking advantage of the fact that we deal with a non-zero spin.







 §.§ Overview of the paper

In Section <ref>, we introduce the geometric background, the operators, the coordinates that we will use in the analysis, as well as the system of equations that we consider. In Section <ref>, we recall the decay assumptions on the event horizon, the so-called Price's law, and we write the precise version of the main theorem. In Section <ref>, we obtain the redshift energy estimates in  for  and use the Teukolsky-Starobinsky identity to get the lower bound for  in . Section <ref> deals with the effective blueshift energy estimates to propagate the lower bound for  in . In Section <ref>, we get the non-sharp L^∞ bound for  in  and we compute and integrate the 1+1 wave equation that will eventually give the precise oscillatory blow-up asymptotics for  in , before propagating this blow-up to region .



 §.§ Acknowledgments

The author would like to thank Jérémie Szeftel for his support and many helpful discussions, and Siyuan Ma for helpful suggestions. This work was partially supported by ERC-2023 AdG 101141855 BlaHSt.



§ PRELIMINARIES

We first introduce some notations. By 'LHS' and 'RHS' we mean respectively 'left-hand side' and 'right-hand side'. If P is an operator acting on a (spin-weighted) scalar ψ, for an integer k≥ 1 and for any norm ·, we use the notation 

    P^≤ kψ:=∑_j=0^kP^jψ.

If f,g are two non-negative scalars, we write f≲ g whenever there is a constant C>0 that depends only on the black hole parameters a,M, on the initial data, and on the smallness constants , γ, such that f≤ Cg on the region considered. We write f=O(g) when |f|≲ |g|. We write f∼ g when f≲ g and g≲ f.




 §.§ Geometry of the interior of subextremal Kerr spacetimes

First, we fix a notation for the Boyer-Lindquist (B-L) coordinate Killing vector fields : 
 
    T:=∂_t,  Φ:=∂_ϕ.

Note that when we write ∂_r, ∂_θ, we mean the B-L coordinate vector fields. We recall the definition of the null pair that we use :

    e_3=1/2(-∂_r+r^2+a^2/ΔT+a/ΔΦ),     e_4=1/2(Δ/r^2+a^2∂_r+T+a/r^2+a^2Φ).

This pair satisfies

    𝐠_a,M(e_3,e_3)=𝐠_a,M(e_4,e_4)=0,  𝐠_a,M(e_3,e_4)=-q^2/2(r^2+a^2),

and is regular on _+, as can be seen by expressing e_3 and e_4 with the ingoing Eddington-Finkelstein coordinate vector fields, see below. We also define the rescaled null pair

    =(-μ)e_3,  =(-μ)^-1e_4,

that is regular on .

As recalled in the introduction, the B-L coordinates are not adapted to the geometry of Kerr, because they are singular at both the event and Cauchy horizons. In this paper, we will use both the Eddington-Finkelstein coordinates and double null-like coordinates (we borrow the terminology from <cit.>) introduced in Section <ref>. Notice that the scalar function 

    μ=Δ/r^2+a^2

vanishes only on the horizons, and that unlike in the Kerr black hole exterior region, μ≤ 0 on . Define the tortoise coordinate r^* by 

    r^*/ r=μ^-1,   r^*(M)=0.

Notice that r^*→ -∞ as r→ r_+ and r^*→ +∞ as r→ r_-. More precisely, defining κ_+ and κ_-  the surface gravities of the event and Cauchy horizons :

    κ_+:=r_+-r_-/4Mr_+,  κ_-:=r_–r_+/4Mr_-,

then the asymptotics of r^* at the horizons are given by

    r^*=1/2κ_+ln(r_+-r)+h_+(r)=1/2κ_-ln(r-r_-)+h_-(r),

where h_±(r) has a finite limit as r→ r_±. Notice that (<ref>) implies that, for r close to r_+, 

    -Δ∼exp(2κ_+r^*),

while for r close to r_-, 

    -Δ∼exp(-2|κ_-|r^*).

Next, we define the coordinates

    :=r^*+t,   u:=r^*-t.

The range of the coordinates u,,r^* is indicated on Figure <ref>. Recall that the right event horizon corresponds to {u=-∞}, while the right Cauchy horizon corresponds to {=+∞}.


As in <cit.>, define also the function r_mod such that 

    r_mod/ r=a/Δ,   r_mod(M)=0.

Then define the ingoing and outgoing angular coordinates

    ϕ_+:=ϕ+r_mod  mod 2π,  ϕ_-:=ϕ-r_mod  mod 2π.

The coordinates  and ϕ_+ are regular on _+={u=-∞} and 𝒞ℋ'_+={u=+∞}, while u and ϕ_- are regular on '_+={=-∞} and ={=+∞}.


  §.§.§ Ingoing Eddington-Finkelstein coordinates

The ingoing Eddington-Finkelstein coordinates are (_in=, r_in=r, θ_in=θ, ϕ_+). It is a set of coordinates that is regular on _+ and 𝒞ℋ'_+. The coordinate vector fields are 

    ∂__in=T,  ∂_r_in=-2e_3,  ∂_θ_in=∂_θ,  ∂_ϕ_+=Φ.



  §.§.§ Outgoing Eddington-Finkelstein coordinates

The outgoing Eddington-Finkelstein coordinates are (u_out=u, r_out=r, θ_out=θ, ϕ_-). It is a set of coordinates that is regular on '_+ and . The coordinate vector fields are 

    ∂_u_out=T,  ∂_r_out=-2,  ∂_θ_out=∂_θ,  ∂_ϕ_-=Φ.



  §.§.§ Double null-like coordinate systems

As in <cit.>, we also use the ingoing double null-like coordinates (,u,θ,ϕ_+) that are regular at _+ (where u=-∞), with coordinate vector fields

    ∂_=e_4-a/r^2+a^2Φ,  ∂_u=-μ e_3,   ∂_θ=Θ,  ∂_ϕ_+=Φ,

where we denote Θ the B-L coordinate vector field ∂_θ to avoid confusion. The equivalent outgoing double null-like coordinates (,u,θ,ϕ_-) are regular at  (where =+∞), with coordinate vector fields

    ∂_=e_4,  ∂_u=-μ e_3+a/r^2+a^2Φ,   ∂_θ=Θ,  ∂_ϕ_+=Φ.

Note that we will only use the ingoing double null-like coordinate system in the redshift region , and the outgoing double null-like coordinate system in the blueshift region ∪∪ so we use the same notations for the ingoing and outgoing double null-like coordinate vector fields ∂_u, ∂_, as there is no danger of confusion.


  §.§.§ Constant  and w spacelike hypersurfaces

We need a family of spacelike hypersurfaces, for which we will apply the energy estimates and get decay in . Note that the constant , u hypersurfaces are not spacelike. Indeed we have 

    𝐠_a,M(∇,∇)=𝐠_a,M(∇ u,∇ u)=a^2sin^2θ/q^2

so that the constant , u hypersurfaces are null at the poles and timelike away from the poles. We define, as in <cit.>, 

    :=-r+r_+,   w:=u-r+r_-,

such that the constant  and w hypersurfaces are spacelike. Indeed, we have <cit.> 

    𝐠_a,M(∇,∇)=𝐠_a,M(∇ w,∇ w)=-r^2+2Mr+a^2cos^2θ/q^2<-1.





 §.§ Spin-weighted scalars



  §.§.§ Spin-weighted scalars and spin-weighted spherical operators

In this section, we consider the round sphere ^2 equipped with its volume element, written in coordinates (θ,ϕ)∈(0,π)×[0,2π) :  

    ν := sinθθϕ.


Let s be an integer. Note that in this work we only consider s=± 2. A spin s scalar is a scalar function that has zero boost weight and proper spin weight, as defined by Geroch, Held and Penrose <cit.>. See <cit.> for a rigorous presentation of the spaces of spin-weighted scalars on the sphere ^2 and on the Kerr interior, and a proof that the Teukosky scalar obtained in the linearisation of a gravitational perturbation in the Newman-Penrose formalism is a spin-weighted scalar on spacetime. See also <cit.> for a precise review of the geometric background of the Teukolsky equation. By 'spin-weighted operator' we mean an operator that takes a spin-weighted scalar into a spin-weighted scalar. Note that a spin 0 scalar is a scalar function.

In Kerr spacetime, the volume element induced on the topological spheres 

    S(u,):={r=r(u,)}∩{t=t(u,)}

by the metric 𝐠_a,M is 

    ν(u,):=Σsinθθϕ

where Σ=√(r^4+2r^2a^2cos^2θ+a^4cos^2θ+2Mra^2sin^2θ)∼ 1 in the Kerr interior. Thus, although they are not round, we still rely on the round volume element ν on the Kerr spheres S(u,) to define L^2(S(u,)) norms. 

    For ψ a spin-weighted scalar on , we define

    ψ_L^2(S(u,)):=(∫_S(u,)|ψ^2|ν)^1/2.


Notice that on a region {r_0≤ r≤ r_+} with r_0∈(r_-,r_+), in the ingoing Eddington-Finkelstein coordinates, as ϕ_+-ϕ=r_mod is constant on S(u,), the definition of the L^2(S(u,)) norm gives 

    ψ_L^2(S(u,))^2=∫_0^π∫_0^2π|ψ|^2(u,,θ,ϕ_+)sinθθϕ_+,

which is a regular norm up to the event horizon {u=-∞}∪{=-∞}. For the same reason, on a region {r_-≤ r≤ r_0} with r_0∈(r_-,r_+), we have 

    ψ_L^2(S(u,))^2=∫_0^π∫_0^2π|ψ|^2(u,,θ,ϕ_-)sinθθϕ_-,

which is a regular norm up to the Cauchy horizon {=+∞}∪{u=+∞}.

    We recall the definition of the following standard spin-weighted differential operators, called the spherical eth operators :
    
    _s   :=∂_θ+i/cosθ∂_ϕ-sθ,
    '_s   :=∂_θ-i/cosθ∂_ϕ+sθ.



    The spherical eth operators modify the spin when applied to a spin-weighted scalar. More precisely,  increases the spin by 1 while ' decreases the spin by 1. See in Section <ref> their effect on spin-weighted spherical harmonics. Note that the spin of the scalar to which we apply the eth operators will be clear in the context, so we drop the subscript s in what follows. 


    We define the spin-weighted Laplacian as
    
    '=1/sinθ∂_θ(sinθ∂_θ)+1/sin^2θ∂_ϕ^2+2iscosθ/sin^2θ∂_ϕ-(s^2^2θ-s).



    We also have the expression
        
    '=1/sinθ∂_θ(sinθ∂_θ)+1/sin^2θ∂_ϕ^2+2iscosθ/sin^2θ∂_ϕ-(s^2^2θ+s)='-2s.




  §.§.§ Spin-weighted spherical harmonics

Let s be a fixed spin. The spin-weighted spherical harmonics are the eigenfunctions of the spin-weighted Laplacian, that is self-adjoint on L^2(^2). They are given by the following family

    Y_m,ℓ^s(cosθ)e^imϕ,  ℓ≥ |s|, -ℓ≤ m≤ℓ

of spin s scalars on the sphere ^2. They form a complete orthonormal basis of the space of spin s scalars on ^2, for the L^2(^2) scalar product. When considering the spheres of Kerr interior, as the B-L coordinate ϕ is singular at the horizons, we need a slightly modified family. Let r_0∈(r_-,r_+). Then for (u,) such that r(u,)∈ [r_0,r_+], the spin s scalars 

    Y_m,ℓ^s(cosθ)e^imϕ_+,  ℓ≥ |s|, -ℓ≤ m≤ℓ

form a complete orthonormal basis of the space of spin s scalars on S(u,), for the L^2(S(u,)) scalar product. Similarly, if r(u,)∈ [r_-,r_0], the spin s scalars 

    Y_m,ℓ^s(cosθ)e^imϕ_-,  ℓ≥ |s|, -ℓ≤ m≤ℓ

form a complete orthonormal basis of the space of spin s scalars on S(u,), for the L^2(S(u,)) scalar product.


    In this work, we will never project equations or spin-weighted scalars on some ℓ or ≥ℓ modes, unlike in <cit.> and <cit.>. Instead, we will simply propagate the dominant term of  on _+ up to . This dominant term is a linear combination of spin-weighted spherical harmonics, and satisfies key properties (see Proposition <ref>) that rely on their algebraic properties.

The following facts are standards. We have :

    '(Y_m,ℓ^s(cosθ)e^imϕ_±)   =-(ℓ-s)(ℓ+s+1)Y_m,ℓ^s(cosθ)e^imϕ_±,
    '(Y_m,ℓ^s(cosθ)e^imϕ_±)   =-(ℓ+s)(ℓ-s+1)Y_m,ℓ^s(cosθ)e^imϕ_±,
    (Y_m,ℓ^s(cosθ)e^imϕ_±)   =-√((ℓ-s)(ℓ+s+1))Y_m,ℓ^s+1(cosθ)e^imϕ_±,
    '(Y_m,ℓ^s(cosθ)e^imϕ_±)   =√((ℓ+s)(ℓ-s+1))Y_m,ℓ^s-1(cosθ)e^imϕ_±.




 §.§ Functional inequalities


We start this section by recalling <cit.> with α=0, which will be used to deduce decay of the energy from an energy estimate.

    Let p>1 and let f:[1,+∞) →[0,+∞) be a continuous function. Assume that there are constants C_0>0, C_1>0, C_2 ≥ 0, C_3 ≥ 0 such that for 1 ≤ x_1<x_2,

    f(x_2)+C_1 ∫_x_1^x_2  f(x) d x ≤ C_0 f(x_1)+C_2 ∫_x_1^x_2 x^-p d x+C_3 x_1^-p .


Then for any x_1 ≥ 1,

    f(x_1) ≤ C x_1^-p

where C is a constant that depends only on f(1), C_0, C_1, C_2, C_3, and p.


    See <cit.>.



  §.§.§ _s and  spin-weighted operators in Kerr spacetime


We define the spin-weighted Carter operator 

    _s:=a^2sin^2θ T^2-2iascosθ T +'_s+1_s.


The main property of the spin-weighted Carter operator is that it commutes with the Teukolsky operator (see (<ref>), (<ref>)). We will use it to bound the angular derivates of solutions to the Teukolsky equation. The following operator is (up to a bounded potential) the spin-weighted scalar equivalent of the tensor covariant derivative ∇_2, where 

    e_2:=1/sinθΦ+asinθ T,

and will be useful to get a Poincaré-type inequality to absorb the zero order terms of the Teukolsky operator when doing energy estimates.
We define the spin-weighted operator

    :=1/sinθΦ+asinθ T+isθq^2/r^2+a^2.


Note that  is not regular at the poles, but for a spin-weighted scalar ψ on spacetime we still have ψ∈ L^∞(S(u,)), see for example <cit.>. We will need the following integration by parts lemma for .

    Let ψ, φ be spin s scalars on . Then 
    
    ∫_S(u,)φ ψ ν=T(∫_S(u,)asinθφψ ν)-∫_S(u,)φψ ν.



Define the real valued function

    f(r,θ):=θq^2/r^2+a^2.

We have

    ∫_S(u,)φ ψ ν   =∫_S(u,)φ(1/sinθΦ+asinθ T)ψ ν+∫_S(u,)φisf(r,θ)ψ ν
       =T(∫_S(u,)asinθφψ ν)-∫_S(u,)(1/sinθΦ+asinθ T)φψ ν
             -∫_S(u,)isf(r,θ)φψ ν
       =T(∫_S(u,)asinθφψ ν)-∫_S(u,)φψ ν,

as stated.


    Notice that we have the expression
    
    T   =r^2+a^2/q^2(μ e_3+e_4)-asinθ/q^2+iascosθ/r^2+a^2,
    Φ   =sinθ(r^2+a^2)/q^2(-asinθ e_4-asinθμ e_3)-iscosθ.

    In particular, T=O(μ)e_3+O(1)e_4+O(1)+O(1), and Φ=O(μ)e_3+O(1)e_4+O(1)+O(1).



  §.§.§ Poincaré and Sobolev inequalities for spin-weighted scalars

We first recall the standard Poincaré inequality for spin-weighted scalars, see for example <cit.>.

    Let ψ be a spin ± 2 scalar on ^2. Then 
    
    2∫_^2|ψ|^2ν   ≤∫_^2(|∂_θψ|^2+|1/sinθ∂_ϕψ+isθψ|^2)ν.


Next, we define the energy for wich we will show decay for solutions of the Teukolsky equation. 

    Let ψ be a spin-weighted scalar on . We define its energy density and degenerate energy density as the scalars 

    𝐞[ψ]   :=|e_3ψ|^2+|e_4ψ|^2+|ψ|^2+|∂_θψ|^2,
    𝐞_deg[ψ]   :=μ^2|e_3ψ|^2+|e_4ψ|^2+|ψ|^2+|∂_θψ|^2.


The following result is a re-writing of the standard Poincaré inequality (<ref>), that will be useful in the energy estimates.

Let ψ be a spin ±2 scalar on . For any (u,) we have the Poincaré inequality

    ψ^2_L^2(S(u,))≲∫_S(u,)𝐞_deg[ψ]ν.



By the standard Poincaré inequality (<ref>), we have 

    ψ^2_L^2(S(u,))≲(|∂_θψ|^2+|1/sinθ∂_ϕψ+isθψ|^2)ν.

Moreover, we have in view of (<ref>),

    1/sinθ∂_ϕ+isθ=r^2+a^2/q^2(-asinθ e_4-asinθμ e_3),

and reinjecting in (<ref>) concludes the proof.

We now recall the standard Sobolev embedding for spin-weighted scalars, see for example Lemma 4.27 and Lemma 4.24 of <cit.>.

    Let ψ be a spin ± 2 scalar on ^2. We have 

    ψ^2_L^∞(^2)≲∫_^2(|ψ|^2+|'ψ|^2)ν.


We will need the following reformulation of the standard Sobolev embedding : 

    Let ψ be a spin s=± 2 scalar on . For any (u,) we have

    ψ^2_L^∞(S(u,))a,M≲∫_S(u,)(|T^≤ 2ψ|^2+|_sψ|^2) ν.



    We begin the proof with the standard spherical Sobolev estimate (<ref>)

    ψ^2_L^∞(S(u,))≲(|ψ|^2+|'ψ|^2)ν

and it remains to express ' with the Carter operator using (<ref>).



 §.§ System of equations




  §.§.§ Different expressions of the Teukolsky operators

Recall from (<ref>) the expression of the Teukolsky operator obtained in a frame regular at , originally found by Teukolsky <cit.> :

    _s:=     -[(r^2+a^2)^2/Δ-a^2 sin ^2 θ] T^2 -4 M a r/Δ T Φ -[a^2/Δ-1/sin ^2 θ] Φ^2  
        +Δ^-s∂_r(Δ^s+1∂_r )+1/sinθ∂_θ(sinθ∂_θ)+2 s[a(r-M)/Δ+i cosθ/sin ^2 θ] Φ
        +2 s[M(r^2-a^2)/Δ-r-i a cosθ] T -[s^2 cos ^2 θ/sin ^2 θ-s],

such that for s=± 2, the spin s Teukolsky equation writes

    _s_s=0.

The rescaled Teukolsky operator, obtained in the rescaled frame that is regular on _+ is 
    _s:=Δ^s_s(Δ^-s · ),

such that for s=± 2, recalling ψ_s=Δ^s_s,

    _sψ_s=0.

The expression of _s in B-L coordinates is 

    _s=     -[(r^2+a^2)^2/Δ-a^2 sin ^2 θ] T^2 -4 M a r/Δ T Φ -[a^2/Δ-1/sin ^2 θ] Φ^2  
        +Δ^-s∂_r(Δ^s+1∂_r )+1/sinθ∂_θ(sinθ∂_θ)+2 s[a(r-M)/Δ+i cosθ/sin ^2 θ] Φ
        +2 s[M(r^2-a^2)/Δ-r-i a cosθ] T -[s^2 cos ^2 θ/sin ^2 θ+s] -4 s(r-M) ∂_r.

Notice that 

    _s=_s+2s+4s(r-M)∂_r=_s+2s+4s(r-M)(μ^-1e_4-e_3).

To do the energy estimates, it is convenient to write the Teukolsky operators in terms of e_3, e_4, and .

We have :
    
    _s=   -4(r^2+a^2)e_3e_4+ ^2+1sinθ∂_θ(sinθ∂_θ)-4 iascosθ T+2isa^2sinθcosθ(r^2+a^2)
       +2 r (e_4-μ e_3)+4 s[(r-M)e_3- rT]+2ar r^2+a^2Φ- s-s^2a^4sin^2θcos^2θ(r^2+a^2)^2,

    
    _s=   -4(r^2+a^2)e_3e_4+ ^2+1/sinθ∂_θ(sinθ∂_θ)-4 iascosθ T+2isa^2sinθcosθ/(r^2+a^2)
       +2 r (e_4-μ e_3)+4 s[(r-M)μ^-1e_4- rT]+2ar /r^2+a^2Φ+ s-s^2a^4sin^2θcos^2θ/(r^2+a^2)^2.



    The computation is done in the easiest way by rewriting
    
    ^2+2isa^2cosθsinθ/(r^2+a^2)-s^2a^4sin^2θcos^2θ/(r^2+a^2)^2=^2,  :=1/sinθΦ+asinθ T+isθ,

    and by using (<ref>) to deduce (<ref>) from (<ref>).

The Teukolsky operators can also be written <cit.> as

    _s=-(r^2+a^2)^2-a^2Δsin^2θ/ΔT^2+∂_r(Δ∂_r)-4aMr/ΔTΦ-a^2/ΔΦ^2
             +_s-1'_s-2iascosθ T+4s[(r-M)e_3-rT],
       _s=-(r^2+a^2)^2-a^2Δsin^2θ/ΔT^2+∂_r(Δ∂_r)-4aMr/ΔTΦ-a^2/ΔΦ^2
             +'_s+1_s-2iascosθ T+4s[(r-M)μ^-1e_4-rT].

Note that these expressions show the crucial fact that T, Φ and the Carter operator commute with the Teukolsky equation :

    [_s,T]=[_s,_s]=[_s,Φ]=[_s,T]=[_s,_s]=[_s,Φ]=0.

To compute the commutator of the Teukolsky operator with e_3, we will rewrite _s in terms of e_3, T, and the angular operators.

    We have 

    𝐓_s=4Δ e_3^2+   a^2sin^2θ T^2-4aΦ e_3-4(r^2+a^2)Te_3+2aTΦ
       +_s-1'_s+4(r-M)(s-1)e_3+(2r(1-2s)-2aiscosθ)T.



    
Using (<ref>) and

    ∂_r=-2e_3+a/ΔΦ+r^2+a^2/ΔT,

we get 

    _s   =-(r^2+a^2)^2-a^2Δsin^2θ/ΔT^2+(-2e_3+a/ΔΦ+r^2+a^2/ΔT)(-2Δ e_3+aΦ+(r^2+a^2)T)
             -4aMr/ΔTΦ-a^2/ΔΦ^2+_s-1'_s-2iascosθ T+4s[(r-M)e_3-rT]
       =4Δ e_3^2+a^2sin^2θ T^2-4(r-M)e_3-4aΦ e_3-4(r^2+a^2)Te_3+a^2/ΔΦ^2+2a(r^2+a^2)/ΔTΦ+2r T
             -4aMr/ΔTΦ-a^2/ΔΦ^2+_s-1'_s-2iascosθ T+4s[(r-M)e_3-rT],

simplifying the coefficients gives the result.


    We have 
    
    [_s,e_3]=4(r-M)e_3^2-4rTe_3+2(s-1)e_3+(1-2s)T.



    We simply use the above expression for _s and the fact that [_s-1,e_3]=['_s,e_3]=0.



  §.§.§ Teukolsky-Starobinsky identities

On top of the Teukolsky equations, we also assume that the Teukolsky-Starobinsky identities (TSI) hold. The Teukolsky-Starobinsky identities are a PDE system relating the 4th order angular and radial derivatives of ψ_± 2. Like the Teukolsky equations, they are obtained from the linearisation of a gravitational perturbation of the Einstein vacuum equations around Kerr spacetime. Their differential form was first derived in <cit.> in frequency space, while their covariant form is derived in <cit.>. Recalling from (<ref>), (<ref>) the coordinate vector fields 

    ∂_r_out=∂_r+r^2+a^2/ΔT+a/ΔΦ=-2,  ∂_r_in=∂_r-r^2+a^2/ΔT-a/ΔΦ=-2e_3,

the TSI for the spin ± 2 write <cit.>, in Kerr spacetime,

    ('-iasinθ T)^4-12MT   =Δ^2∂_r_out^4(Δ^2),
    
        (+iasinθ T)^4+12MT   =∂_r_in^4().

As mentionned in <cit.>, (<ref>) and (<ref>) are physical space versions of the frequency space TSI's obtained in <cit.>, and it is also possible to obtain (<ref>) and (<ref>) from the covariant TSI in <cit.>. We will actually only use (<ref>) in this work, close to _+, and not (<ref>).


§ STATEMENT OF THE MAIN THEOREM


Recall that we denote by ,  the spin -2 and +2 scalars that are solutions of the spin ±2 Teukolsky equations :

    _+2=0,  _-2=0.

As before, we denote =Δ^-2 that satisfies _+2=0. In this section, we state our main result on the precise asymptotics of  at . To this end, we first introduce the different subregions of the Kerr interior that we will consider.



 §.§ The different regions of the Kerr black hole interior


Fix r_𝔟∈ (r_-,r_+) close to r_-, γ>0 small, that will both be chosen later in the energy estimates. More precisely,  is chosen in Appendix <ref> and γ is chosen in Section <ref>. We define the following subregions of the Kerr black hole interior, see Figure <ref> :

    𝐈:={≤ r≤ r_+}∩{≥ 1},
       𝐈𝐈:={r_-≤ r≤}∩{2r^*≤^γ}∩{w≤ w_,γ},
       𝐈𝐈𝐈:={r_-≤ r≤}∩{2r^*≥^γ}∩{w≤ w_,γ},
       :={≥_,γ}∩{w≥ w_,γ},

where w_,γ:=2^*-(2^*)^1/γ-+r_- and _,γ:=(2^*)^1/γ are such that {w=w_,γ} and {=_,γ} intersect Γ and {r=}, where the hypersurface Γ is defined by 
    Γ:={2r^*=^γ}.
 

  
    
Region  is the redshift region that contains _+, region ∪ is the blueshift region very close to  where the scalar Δ decays exponentially towards the Cauchy horizon, and region  is an intermediate region, where the blueshift effect is already effective. 


In region  we will obtain redshift energy estimates for the Teukolsky equation _sψ_s=0 for s=-2, while in ∪∪ we will derive effective blueshift estimates for s=+2. For the energy estimates, we use the coordinate system (u,,θ,ϕ_+) in 𝐈, and the coordinate system (u,,θ,ϕ_-) in ∪∪. 

In the next section, we provide the assumptions on _+ on which all the analysis is based. The goal of the analysis will be to successively propagate the polynomial bounds on _+ to regions , ,  and . 


 §.§ Main assumptions on the event horizon

We first define our initial spacelike hypersurface Σ__0 as the union of three spacelike hypersurfaces :

    Σ_0:=Σ_τ_0∪Σ_int∪Σ_τ'_0,

similarly as in <cit.>. More precisely, we 
define Σ_τ_0 as the hypersurface defined in <cit.>, i.e. a constant τ hypersurface, where (τ,r,θ,ϕ_+) is a hyperboloidal coordinate system on the right part of the exterior of Kerr spacetime. Similarly, Σ_τ'_0 is a constant τ' hypersurface, where (τ',r,θ,ϕ_-) is a hyperboloidal coordinate system on the left part of the exterior of Kerr spacetime. We chose Σ_int as any spacelike hypersurface inside the Kerr interior that joins Σ_τ_0 and Σ_τ'_0  such that the union of the three hypersurfaces is spacelike, see Figure <ref>. The Cauchy problem for the Teukolsky equation with initial data on Σ_0 is well-posed on the future maximal Cauchy development ^+(Σ_0) of Σ_0. We will prove the precise asymptotics of the Teukosky field on ∩^+(Σ_0), that is the part of  (thus in the grey shaded area) that is located above Σ_int in Figure <ref>. Without loss of generality we can assume that at the intersection of Σ_0 and _+, we have ≤ 1, and symmetrically that u≤ 1 at Σ_0∩'_+.


The conclusion of the main theorem will be applicable to solutions of the Teukolsky equations that arise from compactly supported initial data on Σ_0, but we will actually only rely on the Price's law results of <cit.>, that we write down now.

We denote by  the set of tangential derivatives on the event horizon. More precisely, we use

    ∇:={T,Φ, , '},

and for k=(k_1,k_2,k_3,k_4)∈ℕ^4, we define |k|=k_1+k_2+k_3+k_4 and ^k=T^k_1Φ^k_2^k_3(')^k_4.

Let N_j^±, N_k^±, (N_j^+)', (N_k^+)'≥ 1 be sufficiently large integers that we will choose later, see Remark <ref> for the precise values. 

We will consider ψ_± 2 such that there is δ>0 such that for j≤ N_j^- and |k|≤ N_k^-, 
    
    |T^j^k(-)|≲^-7-j-δ   on   _+∩{≥ 1},

    and such that for j≤ N_j^+, |k|≤ N_k^+, and l≤ 3, 
    
    |T^j^ke_3^l(-)|≲^-7-j-δ   on   _+∩{≥ 1}.

We also assume the less precise assumption on '_+ : for j≤ (N_j^+)', |k|≤ (N_k^+)', and l≤ 1,

    |T^j^k^l|≲ u^-7-j   on   '_+∩{u≥ 1}.


Assumptions (<ref>), (<ref>), (<ref>) with |k|=l=0 correspond to the so-called Price's law, and were recently shown to hold true in subextremal Kerr by Ma and Zhang <cit.>. More precisely, they have shown that this holds for initially smooth and compactly supported solutions on Σ_0 of the Teukolsky equations _± 2ψ_± 2=0, where :

    
  * The constants Q_m,2 are defined as 2^7ℚ_m,2/5 where the constants ℚ_m,2 are  defined in <cit.>, depend on the values of ψ_± 2 on the initial hypersurface Σ_0, and are generically non-zero.
    
  * The functions A_m(r) are defined as (r^2+a^2)^2𝔣_+2,m(r) where 𝔣_+2,m(r) is precisely defined in <cit.>. The explicit computation of A_m(r) gives
    
    A_m(r)=1/3[3Δ^2   +(r-M)(4(a^2-M^2)+6Δ)iam-(2Δ+6(a^2-M^2)+4(r-M)^2)a^2m^2
       -4(r-M)ia^3m^3+2a^4m^4],
 
see Appendix <ref>.

    Statements (<ref>), (<ref>), (<ref>) with non-zero |k|,l can be deduced[The statements about the tangential derivatives on the event horizon can be obtained easily using the fact that Φ, T and the Carter operator commute with the Teukolsky equation and directly applying the main theorem of <cit.>. Statement (<ref>) for the e_3^≤ 3 derivatives can be deduced as follows. Define  as in (<ref>) and let r_0>r_+. Differentiating <cit.> by e_3^n for n=1,2,3, restricting on {r=r_0}, and using the bounds <cit.> as well as e_3∼μ(r_0)∂_ρ+O(1)T on {r=r_0} gives |e_3^n|≲^-7-δ on {r=r_0}. Then the TSI (<ref>) gives |e_3^4|≲^-7-δ on {r_0≤ r≤ r_+}, and all is left to do is integrate this bound from r=r_0 to r=r_+.] directly from the results in <cit.>, for solutions of the Teukolsky equations arising from smooth and compactly supported initial data on Σ_0.




 §.§ Precise version of the main theorem

We state the main result of this paper.

    Let ψ_± 2 be solutions of the Teukolsky equations _± 2ψ_±2=0 on the interior of Kerr spacetime. Assume that the Teukolsky-Starobinsky identity (<ref>) holds in , as well as the assumptions (<ref>), (<ref>), (<ref>) on the event horizon. Then, denoting =Δ^-2, in region ∪ close to  we have the following asymptotic behaviour :

    (u,,θ,ϕ_-)=Δ^-2(u,)/^7∑_|m|≤ 2A_m(r_-)Q_m,2e^2imr_mod(u,)Y_m,2^+2(cosθ)e^imϕ_-+O(Δ^-2^-7-δ).
 


    A few remarks are in order.
    
        
  * As discussed in Section <ref>,  is the Teukolsky scalar obtained from the linearisation of a gravitational perturbation obtained in the Newman-Penrose tetrad that is regular at . Thus Theorem <ref> should be interpreted as a linear curvature instability statement for the Kerr Cauchy horizon, as Δ^-2/^7 blows up on , exponentially in  in ∪. Moreover, since the function r_mod blows up on  as r_mod∼log(r-r_-), (<ref>) implies large oscillations of  close to , as announced by Ori <cit.>.
        
  * Notice that the expression of A_m(r) gives
        
    A_m(r_-)=2am(am+2i√(M^2-a^2))(M^2+a^2(m^2-1)).

        Thus for any subextremal Kerr black hole parameters (a,M), for m=± 1 and m=± 2, we have A_m(r_-)≠ 0. Note that the identity A_0(r_-)=0 confirms the heuristics arguments of Ori <cit.> according to which the m=0 azimuthal mode of  decays better than Δ^-2^-7. 
        
  * We also prove, see Proposition <ref>, in  intermediate region ,

    (u,,θ,ϕ_-)=Δ^-2(u,)/^7∑_|m|≤ 2A_m(r)Q_m,2e^2imr_mod(u,)Y_m,2^+2(cosθ)e^imϕ_-+O(Δ^-2^-7-δ).
 
        
  * In view of the asymptotic behaviors (<ref>) and  (<ref>), there exists C>1 large enough such that the following uniform lower bound

    _L^2(S(u,))≳Δ^-2(u,)/^7,   in   {≥ C}∩{r_-< r≤},

        holds for generic initial data, in accordance with the results of <cit.>.
        
  * Inspecting the proof, we find that the minimal values of N_j^±, N_k^± and (N_j^+)', (N_k^+)' for which we prove (<ref>) are N_j^+=5, N_k^+=5, N_j^-=15, N_k^-=14, and (N_j^+)'=2, (N_k^+)'=3. We did not try to optimize this loss of derivatives. We mainly lose derivatives when we apply the Sobolev embedding on the spheres (which loses each time two T and angular derivatives) and when we integrate the Teukolsky-Starobinsky identity from _+. Assuming that higher order 
        derivatives decay on the event horizon leads to an asymptotic (<ref>) that holds for higher order T, , and Φ derivatives. 
        
  * By assuming the equivalent statement of assumptions (<ref>), (<ref>) for _± 2 on '_+, we can get an equivalent statement in the symmetric region {u≥ 1} at any point in the analysis, by replacing  with u, ψ_s with _-s, e_3 with  and e_4 with . As in <cit.> for the scalar wave equation, this will not be useful until we try to get the asymptotics on the upper part of , i.e. in region . There, we will need the boundedness of the energy density [e_3^≤ 1] on {u≥ 1}∩{r=}. This is why we also assume (<ref>), which ensures this energy density bound, and which holds for physical initial data compactly supported near the right part of the Kerr exterior. We note that to prove the final result in ∪, that contains the lower part of , the analysis does not require any assumptions on ℋ'_+.
        
  * We also prove a result on the precise asymptotics of ψ_- 2 in redshift region , see Proposition <ref>, and Theorem <ref> for a more general result on the spin -2 Teukolsky equation in . See Proposition <ref> for the precise asymptotics of  in .

    

The rest of the paper is devoted to the proof of Theorem <ref>. In Section <ref> we prove the precise asymptotics of ψ_± 2 in region , see Propositions <ref> and <ref>. In Section <ref>, we prove the precise asymptotics of  in region , see Proposition <ref>. Finally, in Section <ref>, we prove the precise asymptotic behavior (<ref>) of  in regions  and , see Theorems <ref> and <ref>, hence concluding the proof of Theorem <ref>.



§ PRECISE ASYMPTOTICS IN REDSHIFT REGION 𝐈 NEAR _+

We begin the proof of Theorem <ref>, with the description of the precise asymptotics in region . In Section <ref>, we show a redshift energy estimate, that we will eventually apply to  in Section <ref> to propagate the ansatz for  from _+ to . Finally, in Section <ref>, using the Teukolsky-Starobinsky identity (<ref>) we propagate the ansatz for  from _+ to .



 §.§ Energy method for the spin -2 Teukolsky equation in 


We begin this section with the following definition.

    Let V and c be real numbers. We define the following spin-weighted operator :
    
    _s^(c,V):=   -4(r^2+a^2)e_3e_4+ ^2+1sinθ∂_θ(sinθ∂_θ)-4 iascosθ T+2isa^2sinθcosθ(r^2+a^2)
       +2 r (e_4-μ e_3)+4cs[(r-M)e_3- rT]+2ar r^2+a^2Φ- s-s^2a^4sin^2θcos^2θ(r^2+a^2)^2+V.


We remark the following :

    
  * We introduce the modified Teukolsky operator _s^(c,V) in order to do energy estimates for a more general class of operators, which will be useful after commuting the Teukolsky operator with e_3.
    
  * We will use the estimates of this section for a finite number of explicit constants (c,V) thus we still write the bounds that depend on (c,V) with notations O, ≲.
    
  * Using (<ref>) we get _s=_s^(1,0). More generally, 
    
    _s^(c,V)=_s+4(c-1)s[(r-M)e_3-rT]+V,

    which proves, using (<ref>),
    
    [_s^(c,V),T]=[_s^(c,V),]=0.




In all this section, we denote ψ a spin -2 scalar such that there are constants 

    V∈ℝ,   c>0,  β>1,
 
such that for 0≤ j≤ N_j^-, 0≤ |k|≤ N_k^-, and 0≤ 2 k_1+k_2≤ N_k^-,

    ∙    T^j^kψ=O(^-β-j)   on   _+∩{≥ 1},
    ∙    _-2^(c,V)T^j_-2^k_1Φ^k_2ψ=O(^-β-j)   in   .

The goal of this section is to propagate the upper bound (<ref>) for ψ on the event horizon to region  using a redshift energy estimate, see Proposition <ref>. Recall the definition (<ref>) of the energy density [ψ]=|e_3ψ|^2+|e_4ψ|^2+|Uψ|^2+|∂_θψ|^2.

    Assume that ψ is a spin -2 scalar that satisfies (<ref>), (<ref>) with c>0, β>1. Then for 0≤ j≤ N_j^-, 0≤ 2 k_1+k_2≤ N_k^–1, and for _1≥ 1,
    
    ∬_{w=w_1}∩𝐈𝐞[T^j_-2^k_1Φ^k_2ψ](-μ) ν  u≲_1^-2β-2j.



    The choice of the negative -2 spin is the right one in the redshift region 𝐈 to be able to obtain a positive bulk term in the energy estimate. Actually, we will see that the sign that matters is the one of s(r-M) where s is the spin, thus we already anticipate that we will only be able to control solutions with positive +2 spin in the blueshift region ∪∪.


In what follows, we denote s=-2. The computations are done for a general spin s, but the bulk term will only be positive for s< 0. In view of the assumptions (<ref>), (<ref>), it suffices to treat the case j=k_1=k_2=0. Recall that in 𝐈, in coordinates (u,,θ,ϕ_+), we have ∂_u=-μ e_3 and ∂_=e_4-a/(r^2+a^2)Φ. Thus multiplying (<ref>) by μ we get the following expressions :

    μ_s^(c,V)   =4(r^2+a^2)∂_ue_4+μ𝒰^2+μsinθ∂_θ(sinθ∂_θ)-4μ iascosθ T+2isμa^2sinθcosθ(r^2+a^2)U
       +2 rμ (e_4+∂_u)-4cs[(r-M)∂_u+μ rT]+2arμr^2+a^2Φ-μ s-s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2+μ V
       =4(r^2+a^2)e_4∂_u+μ𝒰^2+μsinθ∂_θ(sinθ∂_θ)-4μ iascosθ T+2isμa^2sinθcosθ(r^2+a^2)U
         +2 rμ (e_4+∂_u)-4cs[(r-M)∂_u+μ rT]-2arμr^2+a^2Φ-μ s-s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2+μ V,

where we used (<ref>) to get 

    4(r^2+a^2)[∂_u,e_4]=-4arμr^2+a^2Φ.

As in <cit.> we now multiply (<ref>) by μ and by the complex conjugate of 
    X(ψ):=f(r)∂_uψ+g(r)e_4ψ,

where we choose 

    f(r)=(-μ)^-1(r^2+a^2)^p,   g(r)=(r^2+a^2)^p

with a real number p=p(a,M,c,V)≫1 large enough that will be chosen in Appendix <ref>. We then integrate over S(u,) against ν, and take the real part, to get

    ∫_S(u,)(f(r)∂_uψμ_s^(c,V)ψ)+(g(r)e_4ψμ_s^(c,V)ψ)ν=∫_S(u,)μ(X(ψ)O(^-β))ν.


The computation of the left-hand side of (<ref>), done in Appendix <ref> gives : 

    ∂_(∫_S(u,)𝐅_[ψ]ν)+∂_u(∫_S(u,)𝐅_u[ψ]ν)+∫_S(u,)𝐁[ψ]ν=∫_S(u,)μ(X(ψ)O(^-β))ν,

where 
    𝐅_[ψ]:=2(r^2+a^2)f(r)|∂_uψ|^2-12μ g(r)(|∂_θψ|^2+|Uψ|^2)+asinθ f(r)μℜ(∂_uψUψ)+asinθ g(r)μℜ(e_4ψUψ),


    𝐅_u[ψ]:=2(r^2+a^2)g(r)|e_4ψ|^2-12μ f(r)(|∂_θψ|^2+|Uψ|^2)-asinθ f(r)μℜ(∂_uψUψ)-asinθ g(r)μℜ(e_4ψUψ),

and the bulk term is

    𝐁[ψ]:   =2(rμ g(r)-∂_u((r^2+a^2)g(r)))|e_4ψ|^2+2(rμ f(r)-e_4((r^2+a^2)f(r)))|∂_uψ|^2
          +12(∂_u(μ f(r))+e_4(μ g(r)))(|∂_θψ|^2+|Uψ|^2)
         +μ g(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)-4cs(r-M)f(r)|∂_uψ|^2
         +4μ g(r)ascosθℑ(e_4ψTψ)-4srg(r)μℜ(e_4ψTψ)+g(r)[2rμ-4cs(r-M)]ℜ(e_4ψ∂_uψ)
         +2g(r)arμr^2+a^2ℜ(e_4ψΦψ)-g(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(e_4ψψ)
         -2sg(r)μa^2sinθcosθ(r^2+a^2)ℑ(e_4ψUψ)+μ f(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)
         +4μ f(r)ascosθℑ(∂_uψTψ)-4srf(r)μℜ(∂_uψTψ)+2rμ f(r)ℜ(∂_uψe_4ψ)
         -2f(r)arμr^2+a^2ℜ(∂_uψΦψ)-f(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(∂_uψψ)
         -2sf(r)μa^2sinθcosθ(r^2+a^2)ℑ(∂_uψUψ).



    Setting s=0 in the LHS of (<ref>), we find the same expression as in <cit.> for the scalar wave.


    See Appendix <ref>.

We follow <cit.> and integrate (<ref>) in 𝐈∩{_1≤≤_2} with respect to u to get 

    ∬_{w=w_2}∩𝐈𝒯_w[ψ] νd u+∬_{r=r_𝔟}∩{w_1 ≤w≤w_2}𝒯_r[ψ] νdu+∭_{w_1 ≤w≤w_2}∩𝐈𝐁[ψ] νd u du
    
    =    ∬_{w=w_1}∩𝐈𝒯_w[ψ] νd u+∬_ℋ_+∩{w_1 ≤w≤w_2}𝒯_u[ψ] νdu+∭_{w_1 ≤w≤w_2}∩𝐈μ(X(ψ)O(^-β)) νd u du,

where 
    𝒯_w[ψ]=(1-μ/2)𝐅_u[ψ]-12μ𝐅_u[ψ],  𝒯_r[ψ]=-12μ(𝐅_u[ψ]+𝐅_u[ψ]),   and  𝒯_u[ψ]=𝐅_u[ψ].

We now estimate the different quantities using the choice of f and g.

Control of the bulk terms. First, we prove a lower bound for the bulk term 𝐁[ψ], which is a manifestation of the redshift effect.

    For p=p(a,M,c,V) chosen large enough and for s=-2, we have in 
    
    ∫_S(u,)𝐁[ψ]ν≳ (-μ)∫_S(u,)[ψ]ν.



    See Appendix <ref>.

To control the other bulk term on the RHS of (<ref>), we write, for ε>0,

    |∭   _{w_1 ≤w≤w_2}∩𝐈μ(X(ψ)O(^-β)) νd u du|≤
       ε∭_{w_1 ≤w≤w_2}∩𝐈 |X(ψ)|^2(-μ)νd u du+ε^-1∭_{w_1 ≤w≤w_2}∩𝐈^-2β (-μ)νd u du.

Moreover, changing variables[Note that on =cst, we have μ u=(2-μ) r.] from u to r, and the fact that r is bounded, we get

    ∭_{w_1 ≤w≤w_2}∩𝐈^-2β (-μ)νd u du≲∬_{w_1 ≤w≤w_2}∩𝐈^-2βd r du≲∫__1^_2^-2β.

We also have |X(§ψ)|^2≲ |e_3ψ|^2+|e_4ψ|^2≤[ψ]. Thus, choosing ε>0 small enough such that the first term on the RHS of (<ref>) is absorbed in the left-hand side of (<ref>), we get 

    ∬_{w=w_2}∩𝐈𝒯_w[ψ] νd u+∬_{r=r_𝔟}∩{w_1 ≤w≤w_2}𝒯_r[ψ] νdu+∭_{w_1 ≤w≤w_2}∩𝐈𝐞[ψ] (-μ)νd u du
    ≲   ∬_{w=w_1}∩𝐈𝒯_w[ψ] νd u+∬_ℋ_+∩{w_1 ≤w≤w_2}𝒯_u[ψ] νdu+∫__1^_2^-2β.

Control of the boundary terms. We first deal with the boundary terms on =cst : 

    𝒯_w[ψ]   =(1-μ/2)𝐅_u[ψ]-12μ𝐅_u[ψ]
       =2(r^2+a^2)f(r)|∂_uψ|^2-12μ g(r)(|∂_θψ|^2+|ψ|^2)+asinθμℜ(X(ψ)ψ)
           -μ/2[2(r^2+a^2)f(r)|∂_uψ|^2+2(r^2+a^2)g(r)|e_4ψ|^2-12μ (f(r)+g(r))(|∂_θψ|^2+|ψ|^2)]
       ∼ (-μ)[ψ],

where we absorbed the term asinθμℜ(X(ψ)ψ) using 

    |asinθμ f(r) ℜ(∂_uψψ)|≤ a^2f(r)|∂_uψ|^2+1/4μ^2f(r)|ψ|^2

and

    |asinθμ g(r) ℜ(e_4ψψ)|≤ -3/4μ a^2g(r)|e_4ψ|^2-1/3μ g(r)|ψ|^2.

Then, for the term on {r=}, we have 

    𝒯_r[ψ]=-12μ(𝐅_u[ψ]+𝐅_u[ψ])∼[ψ],   on   {r=}.

To control the term on the event horizon, we use[Notice that e_4=-μ e_3+O(1)T+O(1)Φ.]  𝒯_u[ψ]∼ |e_4ψ|^2≲ |Tψ|^2+|Φψ|^2 on _+. Thus, using (<ref>)  we get on _+∩{≥ 1},

    ∫_S(-∞,)𝒯_u[ψ]ν≲∫_S(-∞,)|Tψ|^2+|Φψ|^2ν≲^-2β.

This concludes the estimates of the boundary terms.  Together with (<ref>), this yields 

    ∬   _{w=w_2}∩𝐈𝐞[ψ](-μ) νd u+∭_{w_1 ≤w≤w_2}∩𝐈𝐞[ψ](-μ) νd u du
       +∬_{w_1 ≤w≤w_2}∩{r=}𝐞[ψ]νdu≲∬_{w=w_1}∩𝐈𝐞[ψ](-μ) νd u+∫_w_1^w_2w^-2βdw.

Thus using Lemma <ref> concludes the proof, dropping the term on {r=} which is non-negative, and using the fact that for any _0≥ 1, the initial energy 

    ∬_{w=w_0}∩𝐈𝐞[ψ](-μ) νd u≲ 1

is finite[In the notations of Lemma <ref>, this guaranties that f(1) is finite.]. This is clear by standard existence results for linear wave equations with smooth initial data, and because {w=w_0}∩𝐈 is a compact region inside a globally hyperbolic spacetime.

The following result will be useful to deduce decay in L^2(S(u,)) norm from energy decay.

    Let ψ be a ± 2 spin-weighted scalar on . We define the scalar function 
    f(,r):=ψ_L^2(S(u(r,),(r,))).

    Then for ≥ 1 and r_2≤ r_1,
    
    f(,r_2)≲ f(,r_1)+(∫_r_2^r_1∫_S(,r')[ψ]ν r')^1/2.



    In coordinates (,r,θ,ϕ_+) we have ∂_r_in=-2e_3, ∂_=T.
    Thus for a fixed ,
    
    r(f(,r))= r(f(=+r-r_+,r))=(∂_r_in+∂_)f(,r).

    Moreover, by a Cauchy-Schwarz inequality,[Notice that even in {r_-≤ r≤}, we can rewrite the L^2(S(u,)) norm as an integration with respect to ϕ_+, doing the change of variables ϕ_+=ϕ_-+2r_mod, as shown in Section <ref>.]
    
    |(∂_r_in+∂_)(f^2)|   =|(∂_r_in+∂_)(∫_0^π∫_0^2π|ψ|^2(,r,θ,ϕ_+)sinθθϕ_+)|
       =|2ℜ(∫_0^π∫_0^2π(ψ (∂_r_in+∂_)ψ)(,r,θ,ϕ_+)sinθθϕ_+)|
       ≲ 2f(∂_r_in+∂_)ψ_L^2(S(,r))=2f(T-2e_3)ψ_L^2(S(,r)),

    which yields
    
    |(∂_r_in+∂_)f|=1/2f|(∂_r_in+∂_)(f^2)|≲(T-2e_3)ψ_L^2(S(,r)).

    Integrating (<ref>), together with the bound (<ref>) gives
    
    f(,r_2)≲ f(,r_1)+∫_r^r_+(∫_S(,r')|Tψ|^2+|e_3ψ|^2ν)^1/2 r',

    and we conclude the proof of Lemma <ref> using the decomposition (<ref>) that gives T=O(1)e_3+O(1)e_4+O(1)+O(1), the Poincaré inequality (<ref>), and a Cauchy-Schwarz inequality.


    Assume that ψ is a spin -2 scalar that satisfies (<ref>), (<ref>) with c>0, β>1. Then we have in 𝐈, for 0≤ j≤ N_j^- and 0≤ 2 k_1+k_2≤ N_k^–1,
    
    T^j^k_1Φ^k_2ψ_L^2(S(u,))≲^-β-j.



As T, Φ, and  commute with _-2^(c,V), it suffices to prove the case j=k_1=k_2=0. Let (u,)∈ and denote _1=(u,), r=r(u,). Using Lemma <ref> with r_1=r_+ and r_2=r we get 

    ψ_L^2(S(u,))≲ψ_L^2(S(-∞,_1))+(∫_r^r_+∫_S(,r')[ψ]ν r')^1/2.

Changing variables from r to u, taking into account Footnote <ref>, we get using Proposition <ref> :

    (∫_r^r_+∫_S(,r')[ψ]ν r')^1/2≲(∬_{w=w_1}∩𝐈𝐞[ψ](-μ) ν  u)^1/2≲_1^-β.

Moreover, ψ_L^2(S(-∞,_1))≲_1^-β by (<ref>), thus using _1^-β≲^-β we get

    ψ_L^2(S(u,))≲_1^-β≲^-β.


The previous energy and L^2(S(u,)) decay estimates, combined with the Sobolev embedding (<ref>) give the following polynomial bound propagation result.

    Assume that ψ is a spin -2 scalar that satisfies (<ref>), (<ref>) with c>0, β>1. Then we have in , for 0≤ j≤ N_j^–2 and 0≤ 2 k_1+k_2≤ N_k^–3,
    
    |T^j_-2^k_1Φ^k_2ψ|≲^-β-j.



It suffices to treat the case j=k=0. Using the Sobolev embedding (<ref>) and Proposition <ref>,  we get pointwisely,

    |ψ|^2≲∫_S(u,)|T^≤ 2ψ|^2ν+∫_S(u,)|_-2ψ|^2ν≲^-2β,

which is the stated estimate for j=k=0.


We conclude this section with the following result, that proves decay of e_3ψ under additionnal assumptions on _+. We also prove a precise energy bound on {r=}. 
 Assume that ψ is a spin -2 scalar that satisfies, for 0≤ j≤ N_j^-, 0≤ |k|≤ N_k^-, and 0≤ 2k_1+k_2≤ N_k^-,

    ∙    e_3^≤ 1T^j^kψ=O(^-β-j)   on   _+∩{≥ 1},
    ∙    e_3^≤ 1_-2T^j_-2^k_1Φ^k_2ψ=O(^-β-j)   in   ,

 where β>1. Then for j≤ N_j^–2 and 2k_1+k_2≤ N_k^–3, we have in ,

    |e_3^≤ 1T^j^k_1Φ^k_2ψ|≲^-β-j,

    and for 1≤_1≤_2 we have the enery bound on {r=}
    
    ∬_{w_1 ≤w≤w_2}∩{r=}𝐞[T^j^k_1Φ^k_2e_3^≤ 1ψ]νdu≲_1^-2β-2j+∫_w_1^w_2w^-2β-2jdw.



    We already proved (<ref>) in the case without the e_3 derivative in Proposition <ref>. Thus it remains to prove the bound (<ref>) with the e_3 derivative, as well as (<ref>). The proof is based on the energy estimates done in Section <ref>, and on a commutation of the Teukolsky operator with e_3. Using Proposition <ref>, we find that e_3ψ satisfies

    (_-2-4[(r-M)e_3-rT]+6)[e_3ψ]=5Tψ+e_3_-2ψ.

Commuting with T^j^k_1Φ^k_2 and using (<ref>), (<ref>) without the e_3 derivative, and (<ref>) yields[Note that the bound for the RHS holds for j≤ N_j-3 but after integration on the spheres (which is the only bound that we need in the energy estimates) it holds for j≤ N_j-2 by Proposition <ref>.]

    _-2^(3/2,6)[T^j^k_1Φ^k_2e_3ψ]=O(^-β-j),   in   ,

Thus using assumption (<ref>) on _+ with the e_3 derivative, by Proposition <ref> with the parameters β>1, c=3/2>0 and V=6, we get for j≤ N_j^–2 and 2k_1+k_2≤ N_k^–3,

    |T^j^k_1Φ^k_2e_3ψ|≲^-β-j,   in   ,

as stated. To get the energy bound (<ref>), notice that dropping the non-negative first and second terms on the LHS of (<ref>) gives in this context, for _2≥_1,

    ∬_{w_1 ≤w≤w_2}∩{r=}   𝐞[T^j^k_1Φ^k_2e_3^≤ 1ψ]νdu≲
       ∬_{w=w_1}∩𝐈𝐞[T^j^k_1Φ^k_2e_3^≤ 1ψ](-μ) νd u+∫_w_1^w_2w^-2β-2jdw,

where β=7+δ. Using Proposition <ref> for e_3^≤ 1ψ yields 

    ∬_{w_1 ≤w≤w_2}∩{r=}𝐞[T^j^k_1Φ^k_2e_3^≤ 1ψ]νdu≲_1^-2β-2j+∫_w_1^w_2w^-2β-2jdw,

as stated.


    Actually, further commutations with e_3 only improve the redshift effect. More precisely, for k≥ 0, e_3^kψ satisfies
    
    _-2^(c_k,V_k)[e_3^kψ]=e_3^k_-2ψ+T^≤ 1e_3^≤ k-1ψ,

    where c_k=(k+2)/2. Thus, assuming decay of e_3^k_-2ψ in  and of e_3^kψ on _+ allows one to use Proposition <ref> to successively control all the derivatives e_3^kψ, k≥ 0, in .




 §.§ Precise asymptotics of ψ_-2 in region 𝐈

We state the main result of this section.

    Assume that  satisfies (<ref>). Then, we have in 𝐈,
    
    =+,

where for j≤ N_j^–2 and 2k_1+k_2≤ N_k^–3,

    |T^j_-2^k_1Φ^k_2|≲^-7-j-δ,   in .

If, additionnally, we assume that for j≤ N_j^- and |k|≤ N_k^-,

    |e_3T^j^k|≲^-7-j-δ   on   _+∩{≥ 1},

then for j≤ N_j^–2 and 2k_1+k_2≤ N_k^–3,

    |e_3T^j_-2^k_1Φ^k_2|≲^-7-j-δ,   in .




    Let 
    
    :=-.

    Using _-2ψ_-2=0, as well as [T,_-2]=[_-2,_-2]=[Φ,_-2]=0, we get in 

    e_3^≤ 1_-T^j_-2^k_1Φ^k_2=-e_3^≤ 1T^j_-2^k_1Φ^k_2_-2().

Notice that e_3(θ)=e_3()=e_3(ϕ_+)=0 so that 

    e_3()=0.

Thus using the expression (<ref>) of the Teukolsky operator, we get 

    _-2   ()
       =(a^2sin^2θ T^2-4(r^2+a^2)Te_3+2aTΦ-(6r+4iacosθ)T)[],

since we also have ' Y_m,2^-2(cosθ)=0 by (<ref>). The explicit computation of (<ref>) yields 

    e_3^≤ 1T^j_-2^k_1Φ^k_2_-2()=O(^-8-j).

Thus  satisfies (<ref>) and (<ref>) with β=7+δ>1, c=1, and V=0 thanks to the assumption (<ref>) on _+. Using Proposition <ref>, we get in 

    |T^j_-2^k_1Φ^k_2|≲^-7-j-δ,

as stated. The result (<ref>) with the e_3 derivative is a direct application of Theorem <ref>.



Notice that (<ref>) implies in particular the energy bound on {r=}∩{≥ 1} :
    
    ∬_{r=}∩{≥ 1}[T^j_-2^k_1Φ^k_2e_3^≤ 1]ν≲ 1.

We will use the symmetric version of this bound for , in the region {u≥ 1}. This is where we will use the initial assumption (<ref>) on '_+. Recall that it will only be used to deduce the precise asymptotics of  on the upper part of the Cauchy horizon, i.e. in , see Remark <ref>. The symmetric argument in the region {u≥ 1}, together with assumption (<ref>) gives the following analog of (<ref>) : 

    ∬_{r=}∩{u≥ 1}[T^j^k_1Φ^k_2^≤ 1]ν u≲ 1.




 §.§ Precise asymptotics of ψ_+2 in region 𝐈

We have the ansatz for  on the event horizon, given by (<ref>). We show that this ansatz propagates to region , using the asymptotics for  in  derived in Section <ref>, and the Teukolsky-Starobinsky identity (TSI) (<ref>). The idea is that the TSI can be rewritten as a relation between  and , with error terms that can be bounded using the fact that T derivatives of  gain powers of ^-1. This will imply that the O(^-7-δ) bound for  in  also holds for .

    Assume that  satisfies (<ref>). Then we have in ,
    
    =+,

where for j≤min(N_j^--10,N_j^+) and 2k_1+k_2≤min(N_k^--9,N_k^+), 

    |e_3^≤ 3T^j_+2^k_1Φ^k_2|≲^-7-j-δ.



Let 

    :=-.

To highlight the important points of the argument, we begin with the case k_1=0. We commute with T^jΦ^k_2 and develop the LHS of TSI (<ref>) to get 

    ()^4T^jΦ^k_2+∑_1≤ p≤ 4,0≤ q≤ 3O(1)()^qT^p+jΦ^k_2=∂_r_in^4T^jΦ^k_2.

Next, we substract from both sides of (<ref>) the quantity

    ()^4T^jΦ^k_2()   =24T^jΦ^k_2(∑_|m|≤ 2Q_m,2Y_m,2^+2(cosθ)e^imϕ_+)
       =∂_r_in^4T^jΦ^k_2(),

where we used (<ref>) four times, and the expression (<ref>) of A_m(r) to get ∂_r_in^4(A_m(r))=A_m^(4)(r)=24. This gives 

    ()^4T^jΦ^k_2+∑_1≤ p≤ 4,0≤ q≤ 3O(1)()^qT^p+jΦ^k_2=∂_r_in^4T^jΦ^k_2.

The crucial point is that the second term on the LHS of (<ref>) contains at least one extra T derivative[Also notice that we lose a lot of derivatives to control this term.] compared to the other terms in (<ref>). In order to estimate this term, we use the Sobolev embedding (<ref>) and (<ref>) to obtain in  

    |∑_1≤ p≤ 4,0≤ q≤ 3O(1)()^qT^p+jΦ^k_2|   ≲∑_1≤ p≤ 4,0≤ q≤ 3(')^≤1()^qT^p+jΦ^k_2_L^2(S(u,))
       ≲∑_p=1^4(')^≤ 3T^j+pΦ^k_2_L^2(S(u,)).

Using the definition (<ref>) of the Carter operator we developp 

    (')^≤ 3T^j+pΦ^k_2=∑_0≤ n≤ 3,  0≤ m≤ 6 O(1)_-2^nT^mT^j+pΦ^k_2.

Thus using Proposition <ref> we get[Notice that the bound |(')^≤ 3T^j+pΦ^k_2|≲^-8-j holds only for j≤ N_j-12 by Proposition <ref>, but using instead Proposition <ref> we get (')^≤ 3T^j+pΦ^k_2_L^2(S(u,))≲^-8-j for j≤ N_j-10.] for p≥ 1, |(')^≤ 3T^j+pΦ^k_2|≲^-8-j, and hence 

    |∑_1≤ p≤ 4,0≤ q≤ 3O(1)()^qT^p+jΦ^k_2|≲^-8-j.

Next, we use the Sobolev embedding again to get 
    |()^4T^jΦ^k_2|≲(')^≤3T^jΦ^k_2_L^2(S(u,))
 and we can developp again the RHS to get 

    (')^≤3T^jΦ^k_2=∑_0≤ n≤ 3, 0≤ m≤ 6O(1)_-2^nT^mT^jΦ^k_2

which gives, using Proposition <ref>, 

    |()^4T^jΦ^k_2|≲^-7-j-δ.

Combining (<ref>), (<ref>) and (<ref>) yields 

    |∂_r_in^4T^jΦ^k_2|≲^-7-j-δ   in    .

Now, in the Eddington-Finkelstein coordinates (,r,θ,ϕ_+), and using the initial condition (<ref>) on _+, we easily integrate (<ref>) four times from r_+ to r on =cst to get |e_3^≤ 3T^jΦ^k_2|≲^-7-j-δ in , which concludes the proof of (<ref>) in the case k_1=0.

Finally, to treat the case k_1≠ 0, first notice that we can developp again

    T^j _+2^k_1Φ^k_2=∑_0≤ n≤ k,0≤ m≤ 2kO(1)(')^nT^m+jΦ^k_2

so we only need to show the O(^-7-j-δ) decay of (')^nT^jΦ^k_2 for any j,n. Differentiating (<ref>) by (')^n gives

    (')^n()^4T^jΦ^k_2+∑_1≤ p≤ 4,0≤ q≤ 3O(1)(')^n()^qT^p+jΦ^k_2=∂_r_in^4(')^nT^jΦ^k_2,

and applying the exact same techniques as in the case k_1=0, controlling any angular derivative ,' using the Carter operator, gives 

    |∂_r_in^4T^j _+2^k_1Φ^k_2|≲^-7-j-δ,

and thus (<ref>) by integrating from _+ on 
=cst as in the case k_1=0.


Using the fact that the integral is taken on {r=}, the control (<ref>) of T^j^k_1Φ^k_2 in  as well as the relation =e_3+O(1)T+O(1)Φ on {r=}, we can rewrite the energy bound (<ref>) as

    ∬_{r=}∩{u≥ 1}[T^j^ke_3^≤ 1]ν u≲ 1.


We continue this section by proving an energy bound for  on {r=}∩{≥ 1}. This will be useful information on the initial data when doing the energy estimates for the spin +2 Teukolsky equation in region . The following result is a corollary of Proposition <ref>.
 
     Assume that  satisfies (<ref>). Then we have, for j≤min(N_j^–12,N_j^+-2) and 2k_1+k_2≤min(N_k^–11,N_k^+-2), 
     
    ∫_S(u,)[T^j_+2^k_1Φ^k_2e_3^≤ 1]ν=O(^-14-2δ-2j)   on   {r=}∩{≥ 1}.
 
 
 
Using (<ref>) we get 

    |e_3^≤ 2T^j _+2^k_1Φ^k_2|≲^-7-j-δ,    on   {r=}.

Next, we write 

    e_4=-μ e_3+T+a/r^2+a^2Φ.
 
Using Proposition <ref>, we can bound

    Φ T^j _+2^k_1Φ^k_2e_3^≤ 1_L^2(S(u,))≲^-7-j-δ

on {r=}, as well as |TT^j _+2^k_1Φ^k_2e_3^≤ 1|≲^-8-j. Thus we get 

    |e_4T^j _+2^k_1Φ^k_2e_3^≤ 1|ν≲^-14-2δ-2j,    on   {r=}.

Finally, denoting 

    ψ:=T^j _+2^k_1Φ^k_2e_3^≤ 1,

to bound |∂_θψ|^2+|𝒰ψ|^2 we first write

    |∂_θψ|^2+|𝒰ψ|^2≲ |∂_θψ|^2+1/sin^2θ|Φψ+iscosθψ|^2+|ψ|^2.

Then we use an integration by parts formula (see for example <cit.>) to get

    ∫_S(u,)|∂_θψ|^2+1/sin^2θ|Φψ+2icosθψ|^2ν   =∫_S(u,)('+2)ψ·ψν
       ≲ψ^2_L^2(S(u,))+(')ψ^2_L^2(S(u,)).

This gives, reinjecting the definition of the Carter operator (<ref>), and using Proposition <ref> ,

    ∫_S(u,)|∂_θ T^j _+2^k_1Φ^k_2e_3^≤ 1|^2+|𝒰T^j    _+2^k_1Φ^k_2e_3^≤ 1|^2ν
       ≲T^≤ 2_+2^≤ k+1T^je_3^≤ 1^2_L^2(S(u,))≲^-14-2δ-2j,

which concludes the proof of Corollary <ref>.
 
 We continue with an energy boundedness result on {r=} for  that will be used only at the end of the paper to get the precise asymptotics in region .

    Assume that  satisfies (<ref>). Then we have for j≤min(N_j^–12,N_j^+-2) and 2k_1+k_2≤min(N_k^–11,N_k^+-2),
    
    ∬_{r=}[T^j^k_1Φ^k_2e_3^≤ 1]≲ 1.



    The proof is made by combining Corollary <ref> on {r=}∩{≥ 1} and the energy bound (<ref>) on {r=}∩{u≥ 1}.




§ PRECISE ASYMPTOTICS IN BLUESHIFT INTERMEDIATE REGION 𝐈𝐈




 §.§ Energy method for the spin +2 Teukolsky equation in 

In this section, we consider ψ a spin +2 scalar such that there are constants

    V∈ℝ,   c>1/4,  β>1,

such that for 0≤ j≤ N_j', and 0≤ 2k_1+k_2≤ N_k',

    ∙    T^j_+2^k_1Φ^k_2ψ=O(^-β-j)   on   {r=r_𝔟}∩{≥ 1}
        ,
    ∙    [T^j_+2^k_1Φ^k_2ψ]ν=O(^-2β-2j)   on   {r=r_𝔟}∩{≥ 1},
    ∙    _+2^(c,V)T^j_+2^k_1Φ^k_2ψ=O(^-β-j)   in   .

Recall that we have

    e_3=-μ e_3,  e_4=(-μ)^-1e_4.

The goal of this section is to propagate the polynomial decay (<ref>) on the spacelike hypersurface {r=} to region , see Proposition <ref>. In this region, we consider the positive +2 spin because it provides a positive bulk term in the energy estimate for the Teukolsky equation. We begin by proving the following energy estimate, that holds only for c>1/4 (more precisely, see (<ref>)).

    Assume that ψ is a spin +2 scalar satisfying (<ref>) and (<ref>) with c>1/4, β>1. Then for γ>0 small enough, we have, for j≤ N_j', 2k_1+k_2≤ N_k', and _1≥ 1,
    
    ∬_{=_1}∩𝐈𝐈[T^j_+2^k_1Φ^k_2ψ](-μ) ν d u≲_1^-2β-2j.



    From now on, we assume that γ>0 is small enough such that Proposition <ref> holds.


In what follows, we denote s=+2. Similarly as in , the computations work for a general spin s, but the bulk term will only be positive for s>0. Once again, it suffices to treat the case j=k_1=k_2=0. Recall that in the coordinates (u,,θ,ϕ_-), we have ∂_u=-μ e_3+a/(r^2+a^2)Φ, ∂_=e_4. Thus we compute, in region 𝐈𝐈 :

    μ_s^(c,V)   =4(r^2+a^2)e_3∂_+ μ𝒰^2+μ/sinθ∂_θ(sinθ∂_θ)-4 iasμcosθ T+2isμa^2sinθcosθ/(r^2+a^2)U
           +2 rμ (e_3+∂_)-4 s[(r-M)e_3+ rμ T]+2arμ/r^2+a^2Φ+ sμ -s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2,
       =4(r^2+a^2)∂_e_3+ μ𝒰^2+μ/sinθ∂_θ(sinθ∂_θ)-4 iasμcosθ T+2isμa^2sinθcosθ/(r^2+a^2)U
           +2 rμ (e_3+∂_)-4 s[(r-M)e_3+ rμ T]-2arμ/r^2+a^2Φ+ sμ -s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2,

where we used 

    4(r^2+a^2)[∂_,]=4arμ/r^2+a^2Φ.

Next, similarly as in , we multiply (<ref>) by μ and by the complex conjugate of 

    X(ψ):=f(r)e_3ψ+g(r)∂_ψ

where 
    f(r):=(r^2+a^2)^p(-μ)^-1,   g(r)=(r^2+a^2)^p,
 we take the real part, and we integrate on S(u,) with respect to ν. We get 

    (f(r)e_3ψμ_s^(c,V)ψ)ν+(g(r)∂_ψμ_s^(c,V)ψ)ν=μ(X(ψ)O(^-β))ν.

Making the substitutions[Recall that our convention for the derivatives ∂_u, ∂_ is that we use the ingoing double null like coordinates in {r≥} and the outgoing double null like coordinates in {r≤}.] 

    ∂_u→,   e_4→∂_,

the same computation as in the proof of (<ref>) in Appendix <ref> for the energy method in the redshift region gives

    ∂_(∫_S(u,)𝐅_[ψ]ν)+∂_u(∫_S(u,)𝐅_u[ψ]ν)+∫_S(u,)𝐁[ψ]ν=μ(X(ψ)O(^-β))ν,

where 

    𝐅_[ψ]=2(r^2+a^2)f(r)|e_3ψ|^2-1/2μ g(r)(|∂_θψ|^2+|Uψ|^2)+asinθμℜ(X(ψ)Uψ),


    𝐅_u[ψ]=2(r^2+a^2)g(r)|∂_ψ|^2-1/2μ f(r)(|∂_θψ|^2+|Uψ|^2)-asinθμℜ(X(ψ)Uψ)),

and

    𝐁[ψ]   =2(rμ f(r)-∂_((r^2+a^2)f(r)))|e_3ψ|^2+2(rμ g(r)-∂_u((r^2+a^2)g(r)))|∂_ψ|^2
          +1/2(∂_u(μ f(r))+∂_(μ g(r)))(|∂_θψ|^2+|Uψ|^2)
         +μ g(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)
         +4μ g(r)ascosθℑ(∂_ψTψ)-4srg(r)μℜ(∂_ψTψ)
         +2g(r)arμ/r^2+a^2ℜ(∂_ψΦψ)-g(r)(μ s+s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2-μ V)ℜ(∂_ψψ)
         -2sg(r)μa^2sinθcosθ/(r^2+a^2)ℑ(∂_ψUψ)+g(r)(2rμ-4cs(r-M))(∂_ψe_3ψ)
         +μ f(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)
         +4μ f(r)ascosθℑ(e_3ψTψ)-4srμℜ(e_3ψTψ)+2 rμ f(r)ℜ(e_3ψ∂_ψ)
         -2f(r)arμ/r^2+a^2ℜ(e_3ψΦψ)-f(r)(μ s+s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2-μ V)ℜ(e_3ψψ)
         -2sf(r)μa^2sinθcosθ/(r^2+a^2)ℑ(e_3ψUψ)-4cs(r-M)f(r)|e_3ψ|^2.



Integrating (<ref>) on {_1≤≤_2}∩ with respect to u gives 

    ∬_{=_2}∩𝐈𝐈𝒯_[ψ] ν d u+∬_Γ∩{_1 ≤≤_2}𝒯_Γ[ψ] ν d  u+∭_{_1 ≤≤_2}∩𝐈𝐈𝐁[ψ] ν  u
    
    =    ∬_{=_1}∩𝐈𝐈𝒯_[ψ] ν d u+∬_{r=r_𝔟}∩{_1 ≤≤_2}𝒯_r[ψ] ν d  u+∭_{w_1 ≤w≤w_2}∩𝐈μ(X(ψ)O(^-β)) νd u du,

where  
    𝒯_[ψ]=(1-1/2μ) 𝐅_[ψ]-1/2μ𝐅_u[ψ],  𝒯_r[ψ]=-1/2μ(𝐅_u[ψ]+𝐅_u[ψ]),

and 
    𝒯_Γ[ψ]=𝐅_u[ψ]+(1-γ^γ-1)𝐅_[ψ].
Now we estimate the different quantities involved, with the previous choice of f and g.

Control of the bulk terms. We first prove a lower bound for the bulk term, which holds thanks to an effective blueshift effect, taking place for strictly positive spins.

    For p=p(a,M)≫ 1 large enough, and =(a,M) sufficiently close to r_-, we have in {r_-≤ r≤},
    
    𝐁[ψ]ν≳(-μ)[ψ]ν.



    See Appendix <ref>, this is where we use the assumption c>1/4.

To control the other bulk term on the RHS of (<ref>), we write, for ε>0, similarly as in ,

    |∭_{_1 ≤≤_2}∩𝐈𝐈μ(X(ψ)   O(^-β))ν  u|≲
       ε∭_{_1 ≤≤_2}∩𝐈𝐈[ψ](-μ)ν  u+ε^-1∫__1^_2^-2β.

Thus, choosing ε>0 small enough such that the first term on the RHS of (<ref>) is absorbed in the LHS of (<ref>), we get 

    ∬_{=_2}∩𝐈𝐈𝒯_[ψ] ν d u+∬_Γ∩{_1 ≤≤_2}𝒯_Γ[ψ] ν d  u+∭_{_1 ≤≤_2}∩𝐈𝐈[ψ] ν  u
    
    =    ∬_{=_1}∩𝐈𝐈𝒯_[ψ] ν d u+∬_{r=r_𝔟}∩{_1 ≤≤_2}𝒯_r[ψ] ν d  u+∫__1^_2^-2β.

 
 
 
Control of the boundary terms. As in region , we have

    𝒯_[ψ]∼(-μ)[ψ],    in   ,

and

    𝒯_r[ψ]∼[ψ],   on   {r=}.

We also have, as in <cit.>

    𝒯_Γ[ψ]=𝐅_u[ψ]+(1-γ^γ-1)𝐅_[ψ]≥ f(r)|ψ|^2+g(r)|e_4ψ|^2-μ(f(r)+g(r))()≥ 0

for γ>0 small enough. Thus combining these boundary terms estimates with (<ref>) yields

    ∬_{=_2}∩𝐈𝐈[ψ]  (-μ)ν d u+∭_{_1 ≤≤_2}∩𝐈𝐈[ψ] ν  u
       ≲∬_{=_1}∩𝐈𝐈[ψ] (-μ)ν d u+∬_{r=r_𝔟}∩{_1 ≤≤_2}[ψ] ν du+∫__1^_2^-2β.

Using the energy assumption (<ref>) on {r=}, we get 

    ∬_{=_2}∩𝐈𝐈[ψ]  (-μ)ν d u+∭_{_1 ≤≤_2}∩𝐈𝐈[ψ] ν  u
       ≲∬_{=_1}∩𝐈𝐈[ψ] (-μ)ν d u+∫__1^_2^-2β,

 and we conclude the proof of Proposition <ref> using Lemma <ref>, using the fact that for any _0≥ 1, the initial energy 
 
    ∬_{=_0}∩𝐈𝐈[ψ](-μ) ν d u≲ 1

 is finite, as in the end of the proof of Proposition <ref>.



    Assume that ψ is a spin +2 scalar satisfying (<ref>), (<ref>), and (<ref>) with c>1/4, β>1. Then we have in , for j≤ N_j' and 2k_1+k_2≤ N_k',
    
    T^j_+2^k_1Φ^k_2ψ_L^2(S(u,))≲^-β-j.



The proof is exactly the same as the one of Proposition <ref>, i.e. we use Lemma <ref> and use the energy decay given by Proposition <ref> to bound the integrated term.


    Assume that ψ is a spin +2 scalar satisfying (<ref>), (<ref>), and (<ref>) with c>1/4, β>1. Then we have in , for j≤ N_j'-2 and 2k_1+k_2≤ N_k'-2,
    
    |T^j_+2^k_1Φ^k_2ψ|≲^-β-j.



It suffices to treat the case j=k_1=k_2=0. Using the Sobolev embedding (<ref>) and Proposition <ref>,  we get pointwisely,

    |ψ|^2≲∫_S(u,)|T^≤ 2ψ|^2ν+∫_S(u,)|_-2ψ|^2ν≲^-2β,

which concludes the case j=k_1=k_2=0.




 §.§ Precise asymptotics of ψ_+2 in region 𝐈𝐈

We use the energy method of Section <ref> to get the following results.

    Assume that  satisfies (<ref>). Then we have in , 
    
    ψ_+2=+,

    where for j≤min(N_j^–14,N_j^+-4) and 2k_1+k_2≤min(N_k^–13,N_k^+-4), 
    
    |T^j_+2^k_1Φ^k_2|≲^-7-δ-j.


The proof of Proposition <ref> requires the following lemma. 

    In , for j,k≥ 0 we have

    e_3^≤ 1_+T^j_+2^k_1Φ^k_2(1/^7∑_|m|≤2A_m(r)Q_m,Y_m,^+(cosθ)e^imϕ_+)=O(^-8-j).



    The computation requires the precise expression of A_m(r), and the expression (<ref>) of the Teukolsky operator _+2, see Appendix <ref> for the complete proof.


By Proposition <ref>, we get that  satisfies (<ref>) with β=7+δ. Moreover, by Corollary <ref>,  satisfies (<ref>). In order to use Proposition <ref>, it remains to check that  satisfies (<ref>). Using Lemma <ref> and _+2=0, we get, in , 

    _+T^j_+2^k_1Φ^k_2=O(^-8-j).

This proves that  satisfies (<ref>) with β=7+δ>1, c=1>1/4 and V=0, thus by Proposition <ref> we get in 

    |T^j_+2^k_1Φ^k_2|≲^-7-δ-j,

as stated.

The rest of this section is devoted to obtaining O(^-7-δ-j) pointwise decay for e_3 T^j_+2^k_1Φ^k_2 and e_4^k_1Φ^k_2T^j in . This will be used in Section <ref> as initial data on Γ to integrate a 1+1 wave equation that will eventually lead to the blow-up of  on . Unlike in region  where we already controled  and could use TSI to get decay of e_3, the proof in  is done by commuting the Teukolsky equation with e_3 and applying the energy method of Section <ref>.


    Assume that  satisfies (<ref>). Then we have, in , for j≤min(N_j^–14,N_j^+-4) and 2k_1+k_2≤min(N_k^–13,N_k^+-4),

    |T^j_+2^k_1Φ^k_2e_3|≲^-7-j-δ.



By Proposition <ref>, we get that e_3 satisfies (<ref>) with β=7+δ. Moreover, by Corollary <ref>, e_3 satisfies (<ref>). In order to use Proposition <ref>, it remains to check that e_3 satisfies (<ref>). Using the commutator between _+2 and e_3 given by Proposition <ref>, and Lemma <ref> to write

    T^j^k_1Φ^k_2e_3_+2()=O(^-8-j),

we get

    (_+2-4[(r-M)e_3   +rT]-2)[T^j^k_1Φ^k_2e_3]
       =T^j^k_1Φ^k_2(-3T-e_3_+2())
       =O(^-8-j),

where we also used[Once again, the reader interested in the count of the loss of derivatives will notice that the bound for the RHS holds only for j≤ N_j-15, but also for j≤ N_j-14 after integration on the sphere, which is the only bound that we use in the energy estimates.] Proposition <ref>. Using (<ref>) gives

    _+2^(1/2,-2)T^j^k_1Φ^k_2e_3=O(^-8-j).

This proves that e_3 satisfies (<ref>) with β=7+δ>1, c=1/2>1/4 and V=-2, thus by Proposition <ref> we get in 

    |T^j_+2^k_1Φ^k_2e_3|≲^-7-j-δ,

as stated.


    Assume that  satisfies (<ref>). Then we have, in , for j≤min(N_j^–15,N_j^+-5) and 2k_1+k_2≤min(N_k^–14,N_k^+-5),

    |e_4 T^j_+2^k_1Φ^k_2|≲^-7-δ-j.



    We have e_4=-μ e_3+O(1)T+O(1)Φ, which gives the stated bound by Propositions <ref> and <ref>. 



§ PRECISE ASYMPTOTICS IN BLUESHIFT REGION 𝐈𝐈𝐈∪ NEAR 

In all this section, we assume that ψ_± 2 satisfy the assumptions of Theorem <ref>.


 §.§ Energy estimate and pointwise bounds for  in 


We have, for (w_1,_1)∈, and for j≤min(N_j^–12,N_j^+-2) and 2k_1+k_2≤min(N_k^–11,N_k^+-2), 

    ∬_{=_1, w≤ w_1}∩(∪)[T^j_+2^k_1Φ^k_2]ν r≲ 1.



As T and _+2 commute with the Teukolsky equation, it suffices to treat the case j=k=0. We implement the same energy method as in region  (see Section <ref>), noticing that the computations of the energy method work also in , and we integrate (<ref>) with[In this case, _+2^(c,V)ψ=_+2=0, so the RHS in (<ref>) is exactly zero.] ψ= on {w≤ w_1, ≤_1}∩(∪). We get

    ∬_{w≤ w_1, =_1}∩(∪)𝒯_[] ν d u+∬_{w= w_1, ≤_1}∩(∪)𝒯_w[] ν d u
       +∭_{w≤ w_1, ≤_1}∩(∪)𝐁[] ν  u= ∬_{r=}∩{_(w_1)≤≤_1}𝒯_r[] ν d u,

where the boundary term on {w=w_1} is

    𝒯_w[]   :=(1-μ/2)𝐅_u[]-(μ/2)𝐅_[]
       =2(r^2+a^2)g(r)|∂_ψ|^2-12μ f(r)(|∂_θψ|^2+|ψ|^2)-asinθμℜ(X(ψ)ψ)
           -μ/2[2(r^2+a^2)f(r)|ψ|^2+2(r^2+a^2)g(r)|∂_ψ|^2-12μ (f(r)+g(r))(|∂_θψ|^2+|ψ|^2)]
       ∼ (-μ)[ψ],

where we absorbed the term asinθμℜ(X(ψ)ψ) using 

    |asinθμ g(r) ℜ(∂_ψψ)|≤ a^2g(r)|∂_ψ|^2+1/4μ^2g(r)|ψ|^2

and

    |asinθμ f(r) ℜ(ψψ)|≤ -3/4μ a^2f(r)|ψ|^2-1/3μ f(r)|ψ|^2.

Using (<ref>) and (<ref>), as well as Lemma <ref>, this yields

    ∬_{w≤ w_1, =_1}∩(∪)[](-μ)ν u+∬_{≤_1, w=w_1}∩(∪)[](-μ)ν u
       +∭_{w≤ w_1, ≤_1}∩(∪)[](-μ)ν u≲∬_{r=}∩{_(w_1)≤≤_1}[]ν.


Using Corollary <ref>, and the trivial bound

    []≲^-14,

we get 

    ∬_{r=}∩{_(w_1)≤≤_1}[]ν≲∫_{_(w_1)≤≤_1}^-14≲ 1.

Since we have by changing variable :

    ∬_{w≤ w_1, =_1}∩(∪)[](-μ)ν u∼∬_{w≤ w_1, =_1}∩(∪)[]ν r,

the conclusion of the proof of Proposition <ref> follows from (<ref>) and (<ref>).


We have in , for j≤min(N_j^–14,N_j^+-4) and 2k_1+k_2≤min(N_k^–13,N_k^+-4), 

    |T^j_+2^k_1Φ^k_2|≲ 1.



As in the proof of Proposition <ref>, we use Lemma <ref> and use the energy decay given by Proposition <ref> to bound the integrated term, which yields

    T^j_+2^k_1Φ^k_2_L^2(S(u,))≲ 1,   in   .

Using the Sobolev embedding (<ref>), we infer 

    |T^j^k_1Φ^k_2|^2≲∫_S(u,)|T^≤ j+2^k_1Φ^k_2|^2ν+∫_S(u,)|T^j^k+1|^2ν≲ 1,

as stated.



 §.§ The coupling of _s to ψ_s through a 1+1 wave equation

The goal of this section is to reformulate the Teukolsky equation _+2=0 as a 1+1 wave equation for  with a right-hand-side that we can control, so that we can solve explicitely for  by integrating twice the equation.

    We have, for j,k_1,k_2≥ 0 and s=± 2,

    ((r^2+a^2)Δ^s e_4T^j_s^k_1Φ^k_2_s)=1/4(-μ)(_s+2aTΦ-2(1+2s)rT)[T^j_s^k_1Φ^k_2ψ_s].



    Notice that we control the right-hand-side of (<ref>) thanks to the L^∞ bounds of Proposition <ref>. The blow-up of  on  will come from the integration of (<ref>), as the inverse of the factor Δ^ blows up exponentially on .


    We have 
    
    ((r^2+a^2)Δ^se_4_s)   =rμΔ^s e_4_s+sΔ^s(r-M)e_4_s+(r^2+a^2)Δ^s  e_4_s
       =1/4μΔ^s(-4(r^2+a^2)e_3e_4+4s(r-M)μ^-1e_4+4re_4)[_s].

Using e_4-μ e_3=μ∂_r=2e_4-T-a/(r^2+a^2)Φ we get
    
    ((r^2+a^2)   Δ^s e_4_s)=
       1/4μΔ^s(-4(r^2+a^2)e_3e_4+4s(r-M)μ^-1e_4+2r(e_4-μ e_3)+2r T+2ra/r^2+a^2Φ)[_s].

Next, using the expression (<ref>) we get
    
    _s=   -4(r^2+a^2)e_3e_4+ U^2+1/sinθ∂_θ(sinθ∂_θ)-4iascosθ T
       +2 r (e_4-μ e_3)+4s[(r-M)μ^-1e_4- rT]+2ar /r^2+a^2Φ+ s,

where 

    U^2=1/sin^2θΦ^2+a^2sin^2 T^2+2a TΦ+2aiscosθ T+2iscosθ/sin^2θΦ-s^2^2θ.

We infer, using the definition of the Carter operator (<ref>),
    
    _s=   -4(r^2+a^2)e_3e_4+2 r (e_4-μ e_3)+4s(r-M)μ^-1e_4+2ar /r^2+a^2Φ
           +'+a^2sin^2 T^2+2aTΦ-2aiscosθ T-4srT
       =-4(r^2+a^2)e_3e_4+2 r (e_4-μ e_3)+4s(r-M)μ^-1e_4+2ar /r^2+a^2Φ
           +_s+2aTΦ-4srT.

Thus combining this with the Teukolsky equation _s_s=0 gives (<ref>) with j=k_1=k_2=0, where we use the fact that Δ^s commutes with _s, T and Φ. We then extend this to general non zero j,k_1,k_2 by commuting with T^j_s^k_1Φ^k_2.




 §.§ Precise asymptotics of  in  and blow-up at ∩

In this section we will use the crucial fact that 2r^*≥^γ in , thus by (<ref>),

    -Δ∼exp(-2|κ_-|r^*)≤exp(-|κ_-|^γ)   in .


    We have in ,
    
    (u,,θ,ϕ_-)=Δ^-2(u,)/^7∑_|m|≤2A_m(r_-)e^2imr_mod(u,)Q_m,Y_m,^+(cosθ)e^imϕ_-
        +Err[],

where for j≤min(N_j^–15,N_j^+-5) and 2k_1+k_2≤min(N_k^–14,N_k^+-5),

    |T^j^k_1Φ^k_2Err[]|≲Δ^-2^-7-j-δ   in   .



The proof is basically done by integrating the 1+1 wave equation (<ref>). A bit of work is necessary at the end of the proof to get rid of the dependance in u that comes from the boundary terms on Γ, i.e. to prove that the upper bound for Err[] is uniform in u. Recall that we have the ansatz  given by Proposition <ref>:

    =1/^7∑_|m|≤2A_m(r)Q_m,Y_m,^+(cosθ)e^imϕ_++,     on   Γ

where 
    |e_4^≤ 1T^j^k_1Φ^k_2|≲^-7-j-δ    on  Γ⊆,

using Proposition <ref> and Corollary <ref>. This implies 

    e_4=   μ∂_r(Δ^-2)/2^7∑_|m|≤2A_m(r)Q_m,Y_m,^+(cosθ)e^imϕ_++Δ^-2/2^7∑_|m|≤2μ A_m'(r)Q_m,Y_m,^+(cosθ)e^imϕ_+
       +aΔ^-2/(r^2+a^2)^7∑_|m|≤2imA_m(r)Q_m,Y_m,^+(cosθ)e^imϕ_++Err[e_4],

with 

    T^j^k_1Φ^k_2Err[e_4]=O(Δ^-2^-7-j-δ)     on   Γ.

Also, notice that the term 

    S:=Δ^-2μ/2^7∑_|m|≤2 A_m'(r)Q_m,Y_m,^+(cosθ)e^imϕ_+

has an extra factor μ that decays exponentially on Γ, such that |^k_1Φ^k_2T^jS|≲-Δ^-1^-7-j. We can thus add S to the error term while still satisfying (<ref>), to get on Γ,

    Δ^2e_4=1/^7∑_|m|≤2A_m(r)(ima-2(r-M)/r^2+a^2)Q_m,Y_m,^+(cosθ)e^imϕ_++Δ^2Err[e_4].

Next, we define 

    Z(r,θ,ϕ_+):=∑_|m|≤2A_m(r)(ima-2(r-M)/r^2+a^2)Q_m,Y_m,^+(cosθ)e^imϕ_+.

Then we integrate the 1+1 wave equation (<ref>) on a curve from Γ to (u,,θ,ϕ_+) with  constant ,θ,ϕ_+, using =∂_u in these coordinates. Using Proposition <ref> to bound the RHS, and denoting u_Γ()=^γ-, we get in 

    |(r^2+a^2)Δ^2T^j^k_1Φ^k_2e_4-T^j^k_1Φ^k_2(r_Γ()^2+a^2)/^7   Z(r_Γ(),θ,ϕ_+)+O(^-7-δ-j)|
       ≲∫_u_Γ()^u(-μ)(u',) u'
       ≲(u-u_Γ())e^-|κ_-|^γ≲^-7-δ-j,

where we used the definition of  to write the exponential decay of μ with  in , and the fact that u-u_Γ()=u+-^γ≲ in , as u≲ 1 in . Moreover, 

    |^k_1Φ^k_2(r_Γ()^2+a^2/r^2+a^2)Z(r_Γ(),θ,ϕ_+)-^k_1Φ^k_2∑_|m|≤2A_m(r_-)ima-2(r-M)/r^2+a^2Q_m,Y_m,^+(cosθ)e^imϕ_+|
             ≲∑_|m|≤2|A_m(r_Γ())(ima-2(r_Γ()-M))-A_m(r_-)(ima-2(r-M))|
             ≲ r_Γ()-r_-≲|μ(r_Γ())|=O(exp(-|κ_-|^γ)),

in , which together with (<ref>) yields

    T^j^k_1Φ^k_2e_4(u,,θ,ϕ_+)=T^j^k_1Φ^k_2   Δ^-2(u,)/^7∑_|m|≤2A_m(r_-)ima-2(r-M)/r^2+a^2Q_m,Y_m,^+(cosθ)e^imϕ_+
       +O(Δ^-2^-7-j-δ),   in   .

We finally integrate (<ref>) on a curve from Γ to (u,,θ,ϕ_-) with  constant u,θ,ϕ_-, using e_4=∂_ in these coordinates, and we get, using Proposition <ref>,

    T^j^k_1Φ^k_2(u,,θ,ϕ_-)
       =T^j^k_1Φ^k_2Δ^-2(r_Γ(u))/_Γ(u)^7∑_|m|≤2A_m(r_Γ(u))Q_m,Y_m,^+(cosθ)e^imϕ_-+2imr_mod(r_Γ(u))+O(Δ^-2(r_Γ(u))/_Γ(u)^7+j+δ)
       +∫__Γ(u)^ T^j^k_1Φ^k_2Δ^-2(u,')/(')^7∑_|m|≤2[A_m(r_-)ima-2(r'-M)/(r')^2+a^2Q_m,Y_m,^+(cosθ)e^imϕ_-+2imr_mod(u,')]'
       + ∫__Γ(u)^ O(Δ^-2(u,')(')^-7-j-δ)'.

We now simplify the terms on the RHS of (<ref>). First, we have 

    ∫__Γ(u)^Δ^-2(u,')(')^-7-j-δ'   =∫__Γ(u)^-r^2+a^2/2(r-M)/(Δ^-2(u,'))(')^-7-j-δ'
       ∼∫__Γ(u)^/(Δ^-2(u,'))(')^-7-j-δ'
       ∼Δ^-2(u,)^-7-j-δ-Δ^-2(u,_Γ(u))_Γ(u)^-7-j-δ
             +(7+j+δ)∫__Γ(u)^Δ^-2(u,')(')^-8-j-δ'

where we used the fact that ∂_Δ^-2(u,)=e_4Δ^-2=(μ/2)∂_rΔ^-2(r). This implies 

    ∫__Γ(u)^Δ^-2(u,')(')^-7-j-δ'∼Δ^-2(u,)^-7-j-δ-Δ^-2(u,_Γ(u))_Γ(u)^-7-j-δ

and thus

    ∫__Γ(u)^ O(Δ^-2(u,')(')^-7-j-δ)'=O(Δ^-2(u,)^-7-j-δ)+O(Δ^-2(u,_Γ(u))_Γ(u)^-7-j-δ).

We now compute the second lign on the RHS of (<ref>). We have 

    ∫__Γ(u)^   T^j^k_1Φ^k_2Δ^-2(u,')/(')^7ima-2(r-M)/r^2+a^2e^2imr_mod(u,')Y_m,^+(cosθ)e^imϕ_-'
       =∫__Γ(u)^/(Δ^-2(u,')e^2imr_mod(u,'))T^j^k_1Φ^k_21/(')^7Y_m,^+(cosθ)e^imϕ_-'
       =T^j(Δ^-2(u,)/^7e^2imr_mod(u,)-Δ^-2(u,_Γ(u))/_Γ(u)^7e^2imr_mod(u,_Γ(u)))^k_1Φ^k_2Y_m,^+(cosθ)e^imϕ_-
             +7∫__Γ(u)^ T^j^k_1Φ^k_2Δ^-2(u,')/(')^8e^2imr_mod(u,')Y_m,^+(cosθ)e^imϕ_-'
       =T^j^k_1Φ^k_2Δ^-2(u,)/^7e^2imr_mod(u,)Y_m,^+(cosθ)e^imϕ_-
             -T^j^k_1Φ^k_2Δ^-2(u,_Γ(u))/_Γ(u)^7e^2imr_mod(r_Γ(u))Y_m,^+(cosθ)e^imϕ_-
             +O(Δ^-2(u,)^-7-j-δ)+O(Δ^-2(u,_Γ(u))_Γ(u)^-7-j-δ),

using the previous computation (<ref>) with δ=1. Next, notice that the first term on the RHS of (<ref>) can be rewritten

    T^j^k_1Φ^k_2Δ^-2(r_Γ(u))/_Γ(u)^7∑_|m|≤2A_m(r_Γ(u))Q_m,Y_m,^+(cosθ)e^imϕ_-+2imr_mod(r_Γ(u))
       =T^j^k_1Φ^k_2Δ^-2(r_Γ(u))/_Γ(u)^7∑_|m|≤2A_m(r_-)Q_m,Y_m,^+(cosθ)e^imϕ_-+2imr_mod(r_Γ(u))+O(Δ^-1(r_Γ(u))_Γ(u)^-7-j)
       =T^j^k_1Φ^k_2Δ^-2(r_Γ(u))/_Γ(u)^7∑_|m|≤2A_m(r_-)Q_m,Y_m,^+(cosθ)e^imϕ_-+2imr_mod(r_Γ(u))+O(Δ^-2(r_Γ(u))_Γ(u)^-7-j-δ).

Thus (<ref>) rewrites, using (<ref>), (<ref>) and (<ref>),

    T^j^k_1Φ^k_2(u,,θ,ϕ_-)   =T^j^k_1Φ^k_2Δ^-2(u,)/^7∑_|m|≤ 2A_m(r_-)Q_m,2e^2imr_mod(u,)Y_m,^+(cosθ)e^imϕ_-
             +O(Δ^-2(u,)^-7-j-δ)+O(Δ^-2(u,_Γ(u))_Γ(u)^-7-j-δ).

We now finish the proof of Theorem <ref> by showing 

    Δ^-2(u,_Γ(u))_Γ(u)^-7-j-δ=O(Δ^-2(u,)^-7-j-δ)

in . This requires some care. Note that in a region {-1≤ u≲ 1} where u is bounded from below and from above, the term that we are trying to bound is O(1) which is controlled by Δ^-2(u,)^-7-j-δ. Thus we restrict the remaining part of the analysis to {u≤ -1}∩, where we want to prove

    Δ^-2(r_Γ(u))/_Γ(u)^7+δ+j≲Δ^-2(u,)/^7+δ+j.

Notice that Δ^-2(r_Γ(u))≤Δ^-2(u,), but that we cannot directly control _Γ(u)^-7-δ-j by ^-7-δ-j in , as a priori we only have _Γ(u)≤. Let γ'∈(γ,1). We write =𝐀∪𝐁 where 

    𝐀:={2r^*≥^γ'}∩,  𝐁:={2r^*≤^γ'}∩.
 

    
  * In 𝐀, we have Δ^-2(r_Γ(u))∼exp(2|κ_-|_Γ(u)^γ)≤exp(2|κ_-|^γ). Moreover by the definition of 𝐀, Δ^-2(u,)≳exp(2|κ_-|^γ'),
    thus 
    
    Δ^-2(r_Γ(u))^7+δ+j/Δ^-2(u,)_Γ(u)^7+δ+j≲exp(2|κ_-|(^γ-^γ'))^7+j+δ≲ 1.

    
  * In 𝐁, we use
    
    Δ^-2(r_Γ(u))/_Γ(u)^7+δ+j≲Δ^-2(u,)/_Γ(u)^7+δ+j,

    thus we only need to show that we can control _Γ(u)^-7-δ-j by ^-7-δ-j there. We recall u+_Γ(u)=_Γ(u)^γ thus _Γ(u)∼ -u as u→-∞. Moreover in 𝐁 we have u+≤^γ' thus ≤^γ'-u and hence 
    
    -u/≥ 1-^γ'-1.

    As we are interested in the asymptotics on , we can restrict the analysis to {≥ 2}, and we get in 𝐁 : 
    
    -u/≥ 1-2^γ'-1>0.

    We finally reinject this back into (<ref>) to get in 𝐁 :
    
    Δ^-2(r_Γ(u))/_Γ(u)^7+δ+j≲Δ^-2(u,)/_Γ(u)^7+δ+j≲Δ^-2(u,)/(-u)^7+δ+j≲Δ^-2(u,)/^7+δ+j.


This concludes the proof of Theorem <ref>.




 §.§ End of the proof of Theorem <ref>.

It remains to prove the asymptotic behavior (<ref>) in region ={≥_,γ}∩{w≥ w_,γ}, where w_,γ=2^*-(2^*)^1/γ-+r_- and _,γ=(2^*)^1/γ. The first step is to notice that we can extend the bounded energy method of Section <ref> to get non-sharp L^∞ bounds for  in .

    We have in , for j≤min(N_j^–12,N_j^+-2) and 2k_1+k_2≤min(N_k^–11,N_k^+-2),
    
    |T^j^k_1Φ^k_2|≲ 1.



    The proof is an extension of the argument in Section <ref> taking into accound the symmetric bounds on {u≥ 1}∩{r=}. Integrating (<ref>) with[Notice that in this case, the RHS is exactly zero.] ψ=T^j^k_1Φ^k_2 on a triangle-shaped region ℛ:={r_-<r<}∩{w≤ w_1}∩{≤_1}, with (w_1,_1)∈, gives similarly as in (<ref>),
    
    ∬_{w≤ w_1, =_1}∩ℛ[T^j^k_1Φ^k_2](-μ)ν u≲∬_∂ℛ∩{r=}[T^j^k_1Φ^k_2]ν≲ 1,

where we bounded the energy term on {r=} using Corollary <ref>. As before, using (<ref>) together with Lemma <ref>, and using the initial ^-7-j bound on {r=} given by Proposition <ref>, we obtain T^j^k_1Φ^k_2_L^2(S(u,))≲ 1 in . We conclude using the Sobolev embedding (<ref>).

The following result, together with Theorem <ref>, concludes the proof of Theorem <ref>.

    We have in ,   
        
    (u,,θ,ϕ_-)=Δ^-2(u,)/^7∑_|m|≤2A_m(r_-)e^2imr_mod(u,)Q_m,Y_m,^+(cosθ)e^imϕ_-
        +Err[],

where for j≤min(N_j^–15,N_j^+-5) and 2k_1+k_2≤min(N_k^–14,N_k^+-5), 
    |T^j^k_1Φ^k_2Err[]|≲Δ^-2^-7-j-δ.




    Note that Theorem <ref> shows that this result holds on {w=  w_,γ}∩{≥_,γ}. We need to prove sufficient decay for the derivative ∂_u (see (<ref>)), that is transverse to the hypersurfaces {w=cst}, to infer the result in ={w≥  w_,γ}∩{≥_,γ} from the result on {w=  w_,γ}∩{≥_,γ} by integration. Using Proposition <ref> 
    we find that e_3 satisfies the PDE 

    (_+2-4[(r-M)e_3-rT]-2)[e_3]=-3T.

Thus using Lemma <ref> and commuting with T^j^k_1Φ^k_2 we obtain 

    _+2^(1/2,-2)T^j^k_1Φ^k_2e_3=O(1),    in  .

This together with the computations of the energy method in ∪, that holds also in , shows that (<ref>) holds in {≥_,γ}∩{r_-<r≤} for ψ=T^j^k_1Φ^k_2e_3 and β=0. Let (u,)∈ and denote the corresponding values (w_1,_1)=(w_1(u,),_1(u,)). Then, integrating (<ref>) on :={r_-<r<}∩{w≤ w_1}∩{≤_1}, with ψ=T^j^k_1Φ^k_2e_3, and with β=0 for the RHS gives :

    ∬_{w≤ w_1, =_1}∩[ψ](-μ)ν u+∬_{≤_1, w=w_1}∩[](-μ)ν u
       +∭_{w≤ w_1, ≤_1}∩[ψ](-μ)ν u≲
       ∬_{r=}∩{_(w_1)≤≤_1}[ψ]ν+∭_{w≤ w_1, ≤_1}∩|X(ψ)|O(-μ)ν u,

where we also used (<ref>) and (<ref>), as well as Lemma <ref>.
Recall that, |X(ψ)|^2≲[ψ] thus using a Cauchy-Schwarz inequality we can bound the last term on the RHS of (<ref>) by

    ε∭_{w≤ w_1, ≤_1}∩[ψ](-μ)ν u+ε^-1∭_{w≤ w_1, ≤_1}∩O(-μ)ν u.

Choosing ε>0 small enough so that the first term of (<ref>) gets absorbed on the LHS of (<ref>). Moreover, 

    ∭_{w≤ w_1, ≤_1}∩(-μ)ν u   =4π∫__(w_1)^_(_1)∫_u_()^u(,w_1)(-μ) u
       ∼4π∫__(w_1)^_(_1)∫_^r(,w_1) r
       ≲_(_1)-_(w_1)
       ≲ w_1+_1+K,

where K(a,M)>0 is a constant, and where we used 

    _(_1)   =_1+-r_+,
    _(w_1)   =2^*-(w_1+-r_-)=-w_1+cst.

Using Corollary <ref> we also have the bound 

    ∬_{r=}∩{_(w_1)≤≤_1}[T^j^k_1Φ^k_2e_3]ν≲ 1.


Thus we have shown

    ∬_{w≤ w_1, =_1}∩[T^j^k_1Φ^k_2e_3](-μ)ν u≲ u+.

Using Lemma <ref> yields[Notice that u+=2r^*>0 in {r_-≤ r≤}.]

    T^j^k_1Φ^k_2e_3_L^2(S^2(u,))≲   T^j^k_1Φ^k_2e_3_L^2(S^2(u(,_1),(,_1)))
       +(∬_{w≤ w_1, =_1}∩[T^j^k_1Φ^k_2e_3](-μ)ν u)^1/2
       ≲√(u+)     in   ,

where we used (<ref>) and (<ref>) to bound the term on {r=}. Using the Sobolev embedding (<ref>) finally gives

    |T^j^k_1Φ^k_2e_3|(u,_1,θ,ϕ_-)≲√(u+)≲ u+,   ().

We will now use the fact that =∂_u in coordinates (u,,θ,ϕ_+), and integrate the previous estimate on =cst from {w= w_,γ} to w(u,) to get information in  from the lower bound that we have on {w= w_,γ} from the Theorem <ref>. We have the estimate 

    | T^j^k_1Φ^k_2|(u,,θ,ϕ_-)≲ -Δ(u+),   ().

Thus integrating on =cst,θ=cst,ϕ_+=cst, we get 

    T^j^k_1Φ^k_2(u,,θ,ϕ_-)=T^j^k_1Φ^k_2(u(w_,γ,   ),,θ,ϕ_-|_w=w_,γ)
       +∫_u(w_,γ,)^uO(-Δ(u',)(u'+)) u',

where ϕ_-|_w=w_,γ=ϕ_-+2r_mod(,u(w_,γ,))-2r_mod(,u). Using Theorem <ref> on {w=  w_,γ}∩{≥_,γ} we obtain 

    T^j^k_1Φ^k_2(u,,θ,ϕ_-)   =T^j^k_1Φ^k_2Δ^-2(u,)/^7∑_|m|≤2A_m(r_-)e^2imr_mod(u,)Q_m,Y_m,^+(cosθ)e^imϕ_-
       +O(Δ^-2(u,)^-7-j-δ)+Δ^-2(u,)∫_u(w_,γ,)^uO(-Δ(u',)(u'+)) u'.

We conclude by proving that in , 

    ∫_u(w_,γ,)^uO(-Δ(u',)(u'+)) u'=O(^-7-j-δ).

We have in , -Δ(u,)∼exp(-|κ_-|(u+)). Thus 

    |∫_u(w_,γ,)^u   O(-Δ(u',)(u'+)) u'|
       ≲exp(-|κ_-|)[∫_u(w_,γ,)^u |u'|exp(-|κ_-|u') u'+∫_u(w_,γ,)^u exp(-|κ_-|u') u']
       ≲exp(-|κ_-|)(C_1+ C_2)
       ≲^-7-j-δ

in , where we defined

    C_1=∫_w_,γ-r_++r_-^+∞ |u'|exp(-|κ_-|u') u',    C_2=∫_w_,γ-r_++r_-^+∞exp(-|κ_-|u') u'.

This concludes the proof of Theorem <ref>.






§ COMPUTATIONS FOR THE ENERGY METHOD



 §.§ Proof of Lemma <ref>


    
In , we compute 

    ∫_S(u,)(f(r)∂_uψμ_s^(c,V)ψ)+(g(r)e_4ψμ_s^(c,V)ψ)ν

using integration by parts on the spheres. We have using (<ref>) :

    ∫_S(u,)(g(r)e_4ψ   μ_s^(c,V)ψ) ν
    
        =   4(r^2+a^2)g(r)∫_S(u,)ℜ(e_4ψ∂_u e_4ψ) ν+μ g(r)∫_S(u,)ℜ(e_4ψ𝒰^2ψ) ν
       +μ g(r)∫_S(u,)ℜ(e_4ψ1sinθ∂_θ(sinθ∂_θψ))ν
       +4μ g(r) as ∫_S(u,)cosθℑ(e_4ψ Tψ)ν
       +2rμ g(r)∫_S(u,)|e_4ψ|^2ν+ g(r) [2rμ -4cs(r-M)]∫_S(u,)ℜ(e_4ψ∂_uψ)ν
       -4srμ g(r)∫_S(u,)ℜ(e_4ψTψ)ν+2g(r)arμr^2+a^2∫_S(u,)ℜ(e_4ψΦψ)ν
       -g(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)∫_S(u,)ℜ(e_4ψψ)ν
       -2sg(r)μa^2sinθcosθ(r^2+a^2)∫_S(u,)ℑ(e_4ψUψ)ν.

We begin with the term 

    4(r^2+a^2)g(r)   ∫_S(u,)ℜ(e_4ψ∂_u e_4ψ) ν=2(r^2+a^2)g(r)∂_u(∫_S(u,) |e_4ψ|^2 ν)
       =∂_u(2(r^2+a^2)g(r)∫_S(u,) |e_4ψ|^2 ν)-2∂_u((r^2+a^2)g(r))∫_S(u,) |e_4ψ|^2 ν.

Using Lemma <ref> and

    [e_4,U]=e_4(isθq^2/r^2+a^2)=isra^2μcosθsinθ/(r^2+a^2)^2,

we get, in view of T=∂_-∂_u,

    μ g(r)∫_S(u,)ℜ(   e_4ψ𝒰^2ψ) ν
       =T(μ g(r)∫_S(u,)asinθℜ(e_4ψUψ) ν)-μ g(r)∫_S(u,)ℜ(Ue_4ψ Uψ) ν
       =(∂_-∂_u)(μ g(r)∫_S(u,)asinθℜ(e_4ψUψ) ν)-12∂_(μ g(r)∫_S(u,)|Uψ|^2 ν)
           +12e_4(μ g(r))∫_S(u,)|Uψ|^2 ν+μ g(r)sra^2μcosθsinθ/(r^2+a^2)^2∫_S(u,)ℑ(ψ Uψ) ν.

We also have, using [e_4,∂_θ]=0,

    μ g(r)∫_S(u,)ℜ(e_4ψ1sinθ∂_θ   (sinθ∂_θψ))ν
       =-μ g(r)∫_S(u,)ℜ(∂_θe_4ψ∂_θψ)ν
       =-12μ g(r)∫_S(u,)e_4|∂_θψ|^2ν
       =∂_(-12μ g(r)∫_S(u,)|∂_θψ|^2ν)+12e_4(μ g(r))∫_S(u,)|∂_θψ|^2ν.

All the remaing terms will be put in the bulk term 𝐁[ψ]. Now we do the same computations for 

    ∫_S(u,)ℜ(f(r)∂_uψ   μ_s^(c,V)ψ) ν
    
        =   4(r^2+a^2)f(r)∫_S(u,)ℜ(∂_uψ e_4∂_uψ) ν+μ f(r)∫_S(u,)ℜ(∂_uψ𝒰^2ψ) ν
       +μ f(r)∫_S(u,)ℜ(∂_uψ1sinθ∂_θ(sinθ∂_θψ))ν
       +4μ f(r) as ∫_S(u,)cosθℑ(∂_uψ Tψ)ν
       +f(r) [2rμ -4cs(r-M)]∫_S(u,)|∂_uψ|^2ν+ 2rμ f(r)∫_S(u,)ℜ(∂_uψe_4ψ)ν
       -4srμ f(r)∫_S(u,)ℜ(∂_uψTψ)ν-2f(r)arμr^2+a^2∫_S(u,)ℜ(∂_uψΦψ)ν
       -f(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)∫_S(u,)ℜ(∂_uψψ)ν
       -2sf(r)μa^2sinθcosθ(r^2+a^2)∫_S(u,)ℑ(∂_uψUψ)ν.

We begin with the term 

    4(r^2+a^2)f(r)   ∫_S(u,)ℜ(∂_uψ e_4∂_uψ) ν=2(r^2+a^2)f(r)e_4(∫_S(u,) |∂_uψ|^2 ν)
       =∂_(2(r^2+a^2)f(r)∫_S(u,) |∂_uψ|^2 ν)-2e_4((r^2+a^2)f(r))∫_S(u,) |∂_uψ|^2 ν.

Next we have

    μ f(r)∫_S(u,)ℜ   (∂_uψ𝒰^2ψ) ν
       =T(μ f(r)∫_S(u,)asinθℜ(∂_uψUψ) ν)-μ f(r)∫_S(u,)ℜ(U∂_uψ Uψ) ν
       =(∂_-∂_u)(μ f(r)∫_S(u,)asinθℜ(∂_uψUψ) ν)-12∂_u(μ f(r)∫_S(u,)|Uψ|^2 ν)
           +12∂_u(μ f(r))∫_S(u,)|Uψ|^2 ν+μ f(r)sra^2μcosθsinθ/(r^2+a^2)^2∫_S(u,)ℑ(ψ Uψ) ν,


    μ f(r)∫_S(u,)ℜ(∂_uψ1sinθ∂_θ   (sinθ∂_θψ))ν
       =-μ f(r)∫_S(u,)ℜ(∂_θ∂_uψ∂_θψ)ν
       =-12μ f(r)∫_S(u,)∂_u|∂_θψ|^2ν
       =∂_u(-12μ f(r)∫_S(u,)|∂_θψ|^2ν)+12∂_u(μ f(r))∫_S(u,)|∂_θψ|^2ν.

Combining everything, (<ref>) gives 

    ∂_(∫_S(u,)𝐅_[ψ]ν)+∂_u(∫_S(u,)𝐅_u[ψ]ν)+∫_S(u,)𝐁[ψ]ν=∫_S(u,)μ(X(ψ)O(^-β))ν,

where 

    𝐅_[ψ]=2(r^2+a^2)f(r)|∂_uψ|^2-12μ g(r)(|∂_θψ|^2+|Uψ|^2)+asinθ f(r)μℜ(∂_uψUψ)+asinθ g(r)μℜ(e_4ψUψ),


    𝐅_u[ψ]=2(r^2+a^2)g(r)|e_4ψ|^2-12μ f(r)(|∂_θψ|^2+|Uψ|^2)-asinθ f(r)μℜ(∂_uψUψ)-asinθ g(r)μℜ(e_4ψUψ),

and 

    𝐁[ψ]   =2(rμ g(r)-∂_u((r^2+a^2)g(r)))|e_4ψ|^2+2(rμ f(r)-e_4((r^2+a^2)f(r)))|∂_uψ|^2
          +12(∂_u(μ f(r))+e_4(μ g(r)))(|∂_θψ|^2+|Uψ|^2)
         +μ g(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)-4cs(r-M)f(r)|∂_uψ|^2
         +4μ g(r)ascosθℑ(e_4ψTψ)-4srg(r)μℜ(e_4ψTψ)+g(r)[2rμ-4cs(r-M)]ℜ(e_4ψ∂_uψ)
         +2g(r)arμr^2+a^2ℜ(e_4ψΦψ)-g(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(e_4ψψ)
         -2sg(r)μa^2sinθcosθ(r^2+a^2)ℑ(e_4ψUψ)+μ f(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)
         +4μ f(r)ascosθℑ(∂_uψTψ)-4srf(r)μℜ(∂_uψTψ)+2rμ f(r)ℜ(∂_uψe_4ψ)
         -2f(r)arμr^2+a^2ℜ(∂_uψΦψ)-f(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(∂_uψψ)
         -2sf(r)μa^2sinθcosθ(r^2+a^2)ℑ(∂_uψUψ),

which concludes the proof of Lemma <ref>.



 §.§ Lower bound for the bulk term in  for s=-2


We have 

    r μ g(r)-∂_u((r^2+a^2) g(r))  =-μ p r(r^2+a^2)^p, 
       r μ f(r)-e_4((r^2+a^2) f(r))  =(-μ)^-1[(r-M)-μ(p+1) r](r^2+a^2)^p
                                 ≳ (-μ)^-1(1-μ p)(r^2+a^2)^p,

in  for p large, as r_+-M>0. We also have

    ∂_u(μ f(r))+e_4(μ g(r))  =-μ(r^2+a^2)^p-1(2 M p r^2r^2+a^2+r μ-(r-M))≳ -μ p(r^2+a^2)^p.
 
Denoting the principal bulk term

    𝐁_pr[ψ]:=2(rμ g(   r)-∂_u((r^2+a^2)g(r)))|e_4ψ|^2+2(rμ f(r)-e_4((r^2+a^2)f(r)))|∂_uψ|^2
          +12(∂_u(μ f(r))+e_4(μ g(r)))(|∂_θψ|^2+|Uψ|^2),

we have shown 

    𝐁_pr[ψ]≳ (-μ)(r^2+a^2)^p[p|e_4ψ|^2+(1-μ p)|e_3ψ|^2+p(|∂_θψ|^2+|Uψ|^2)].

The only thing left to prove is that we can take p large enough so that 𝐁[ψ]-𝐁_pr[ψ] can be absorbed in 𝐁_pr[ψ] after integrating on S(u,). This is due to the following mix between weighted Cauchy-Schwarz of the type 

    |ab|≤ε a^2+ε^-1b^2/2

and the Poincaré inequality (<ref>). We have the following bounds : 

    |μ g(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)ν|   ≲ (-μ)(r^2+a^2)^p(_deg[ψ]ν+|Uψ|^2ν)
       ≲(-μ)(r^2+a^2)^p_deg[ψ]ν,


    |4μ g(r)ascosθℑ(e_4ψTψ)-4srg(r)μℜ(e_4ψTψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,

where we used (<ref>) to write

    T=O(μ)e_3+O(1)e_4+O(1)U+O(1).

We continue with 

    |2rμ g(r)ℜ(e_4ψ∂_uψ)|≲(-μ)(r^2+a^2)^p_deg[ψ],


    |2g(r)arμr^2+a^2ℜ(e_4ψΦψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,

where we used (<ref>) to get Φ=O(μ)e_3+O(1)e_4+O(1)U+O(1). Next, 

    |-g(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(e_4ψψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,


    |2sg(r)μa^2sinθcosθ(r^2+a^2)ℑ(e_4ψUψ)|≲(-μ)(r^2+a^2)^p_deg[ψ],


    |μ f(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,


    |4μ f(r)   ascosθℑ(∂_uψTψ)-4srf(r)μℜ(∂_uψTψ)ν|
       ≲(-μ)ε^-1(r^2+a^2)^p|Tψ|^2ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν
       ≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν,


    |2rμ f(r)ℜ(∂_uψe_4ψ)|≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]+(-μ)ε (r^2+a^2)^p|e_3ψ|^2,


    |-2f(r)arμr^2+a^2   ℜ(∂_uψΦψ)ν|≲
       (-μ)ε^-1(r^2+a^2)^p_deg[ψ]ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν,


    |-f(r)(μ s+   s^2μa^4sin^2θcos^2θ/(r^2+a^2)^2-μ V)ℜ(∂_uψψ)ν|≲
       (-μ)ε^-1(r^2+a^2)^p_deg[ψ]ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν,



    |2sf(r)μa^2sinθcosθ(r^2+a^2)ℑ(∂_uψUψ)|≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]+(-μ)ε(r^2+a^2)^p|e_3ψ|^2,


    |-4cs(r-M)g(r)ℜ(e_4ψ∂_uψ)|≲ (-μ)(r^2+a^2)^pε^-1_deg[ψ]+(-μ)(r^2+a^2)^pε|e_3ψ|^2.

Notice that thanks to the μ in front of |e_3ψ|^2 in the definition (<ref>) of _deg[ψ], the integral on S(u,) of _deg[ψ] can be absorbed in the one of (<ref>), for p large enough. Moreover, we chose the value of ε in the weighted Cauchy-Schwarz inequalities above so that all the terms bounded by a constant times (-μ)(r^2+a^2)^pε|e_3ψ|^2 can be absorbed in the term (1-μ p)|e_3ψ|^2≥|e_3ψ|^2 of (<ref>), for ε>0 small enough, after integration on the sphere. The only remaining term in the bulk that we want to absorb is

    -4cs(r-M)f(r)|∂_uψ|^2=-4cs(r-M)(-μ)(r^2+a^2)^p|e_3ψ|^2.
 
As it lacks a factor μ, we can't bound it pointwisely (nor its integral on S(u,)) by _deg[ψ]. But as r_+-M>0, we have r-M≳ 1 for r close to r_+ in , say for r∈[r_+-ε_0,r_+]. As we chose s=-2<0 and c>0, we have

    -4cs(r-M)f(r)|∂_uψ|^2≥ 0,   r∈[r_+-ε_0,r_+].

And for r∈ [,r_+-ε_0], we can absorb -4cs(r-M)f(r)|∂_uψ|^2 in the term 

    -μ p(r^2+a^2)^p|e_3ψ|^2≳ -μ(r_+-ε_0)p(r^2+a^2)^p|e_3ψ|^2

that appears in 𝐁_pr[ψ], for p large enough. This concludes the proof of Lemma <ref>.




 §.§ Lower bound for the bulk term in {r_-<r≤} for s=+2



We have 

    r μ g(r)-∂_u((r^2+a^2) g(r))     =-μ p r(r^2+a^2)^p, 
    
    r μ f(r)-e_4((r^2+a^2) f(r))     =(-μ)^-1[(r-M)-μ(p+1) r](r^2+a^2)^p.

We also have

    ∂_u(μ f(r))+e_4(μ g(r))  =-μ(r^2+a^2)^p-1(2 M p r^2r^2+a^2+r μ-(r-M))≳ -μ p(r^2+a^2)^p.
 
Define the principal bulk

    𝐁_pr[ψ]:=2(rμ g(   r)-∂_u((r^2+a^2)g(r)))|e_4ψ|^2+2(rμ f(r)-e_4((r^2+a^2)f(r)))|ψ|^2
          +12(∂_u(μ f(r))+e_4(μ g(r)))(|∂_θψ|^2+|Uψ|^2)-4cs(r-M)f(r)|ψ|^2.

Note that unlike in the redshift region, we add a term -4cs(r-M)f(r)|ψ|^2 in the principal bulk. The positive spin will help us get a positive simple bulk term, without the need of replacing f and g by more complicated log multipliers, as is needed for the scalar wave equation in <cit.>. We have

    𝐁_pr[ψ]≳ (-μ)p(r^2+a^2)^p   |e_4ψ|^2+2(-μ)^-1[(r-M)(1-2cs)-μ(p+1) r](r^2+a^2)^p|ψ|^2
          +(-μ)p(r^2+a^2)^p(|∂_θψ|^2+|Uψ|^2).

Notice that in , we have r-M≲ -1 thus for s=+2 and c>1/4,

    (r-M)(1-2cs)-μ(p+1) r=(r-M)(1-4c)-μ(p+1)≳ 1-μ p,   r∈(r_-,].

This is where we use the positivity of the spin, to get an effective blueshift effect. We have shown 

    𝐁_pr[ψ]≳ (-μ)(r^2+a^2)^p[p|e_4ψ|^2+(1-μ p)|e_3ψ|^2+p(|∂_θψ|^2+|Uψ|^2)].

The only thing left to prove is that we can take p large enough so that 𝐁[ψ]-𝐁_pr[ψ] can be absorbed in 𝐁_pr[ψ] after integrating on S(u,). This is due to the following mix between weighted Cauchy-Schwarz of the type 

    |ab|≤ε a^2+ε^-1b^2/2

and the Poincaré inequality (<ref>) :

    |μ g(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)ν|   ≲ (-μ)(r^2+a^2)^p(_deg[ψ]ν+|Uψ|^2ν)
       ≲(-μ)(r^2+a^2)^p_deg[ψ]ν,


    |4μ g(r)ascosθℑ(e_4ψTψ)-4srg(r)μℜ(e_4ψTψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,

where we used again (<ref>) to get

    T=O(μ)e_3+O(1)e_4+O(1)U+O(1).

We continue with 

    |2rμ g(r)ℜ(e_4ψψ)|≲(-μ)(r^2+a^2)^p_deg[ψ],


    |-4sc(r-M)g(r)(∂_ψψ)|≲ (-μ)(r^2+a^2)^p(ε|e_3ψ|^2+ε^-1_deg[ψ]),


    |2g(r)arμr^2+a^2ℜ(e_4ψΦψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,

where we used (<ref>) to get Φ=O(μ)e_3+O(1)e_4+O(1)U+O(1). Next, 

    |-g(r)(μ s+s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(e_4ψψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,


    |2sg(r)μa^2sinθcosθ(r^2+a^2)ℑ(e_4ψUψ)|≲(-μ)(r^2+a^2)^p_deg[ψ],


    |μ f(r)sra^2μcosθsinθ/(r^2+a^2)^2ℑ(ψ Uψ)ν|≲(-μ)(r^2+a^2)^p_deg[ψ]ν,


    |4μ f(r)   ascosθℑ(ψTψ)-4srf(r)μℜ(ψTψ)ν|
       ≲(-μ)ε^-1(r^2+a^2)^p|Tψ|^2ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν
       ≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν,


    |2rμ f(r)ℜ(ψe_4ψ)|≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]+(-μ)ε (r^2+a^2)^p|e_3ψ|^2,


    |-2f(r)arμr^2+a^2   ℜ(ψΦψ)ν|
       ≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν,


    |-f(r)(μ s+   s^2μa^4sin^2θcos^2θ(r^2+a^2)^2-μ V)ℜ(ψψ)ν|
       ≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]ν+(-μ)(r^2+a^2)^pε|e_3ψ|^2ν,



    |2sf(r)μa^2sinθcosθ(r^2+a^2)ℑ(ψUψ)|≲(-μ)ε^-1(r^2+a^2)^p_deg[ψ]+(-μ)ε(r^2+a^2)^p|e_3ψ|^2.

Notice that thanks to the μ in front of |e_3ψ|^2 in the definition of _deg[ψ], the integral on S(u,) of _deg[ψ] can be absorbed in the one of (<ref>), for p large enough. Moreover, we choose the value of ε in the weighted Cauchy-Schwarz inequalities above so that all the terms bounded by a constant times (the integral of) (-μ)(r^2+a^2)^pε|e_3ψ|^2 can be absorbed in the term (1-μ p)|e_3ψ|^2≥|e_3ψ|^2 in (<ref>), for ε>0 small enough. This concludes the proof of Lemma <ref>.





§ COMPUTATION OF A_M(R) AND PROOF OF LEMMA <REF>

The polynomial A_m(r)=(r^2+a^2)^2𝔣_m,2(r) is defined in <cit.> by plugging the ansatz 

    

for  into the TSI (<ref>) and requiring the compatibility

    1/24Δ^2∂_r_out^4(Δ^2)=1/^7∑_|m|≤ 2A_m(r)Q_m,2Y_m,2^-2(cosθ)e^imϕ_++O(^-8),

see also <cit.>, where the factor 1/24 corresponds to 1/(2𝔰)! with 𝔰=2. We recall

    ∂_r_out=2μ^-1e_4=∂_r+μ^-1∂_t+a/ΔΦ.

More precisally, the computation done in <cit.> gives

    A_m(r)=1/24e^-imϕ_+Δ^2∂_r_out^4(Δ^2e^imϕ_+),

and then (<ref>) holds, where the O(^-8) term is given by the terms where a ∂_r_out falls on an inverse power of . Let us now compute (<ref>). We have ∂_r_out(e^imϕ_+)=2aim/Δe^imϕ_+ thus 

    ∂_r_out(Δ^2e^imϕ_+)=(∂_rΔ^2+2aimΔ)=2Δ(2(r-M)+aim).

We compute successively

    ∂_r_out^2(Δ^2e^imϕ_+)   =(∂_r+2iam/Δ)(2Δ(2(r-M)+aim))
       =4[2(r-M)^2+3(r-M)aim+Δ-a^2m^2],


    ∂_r_out^3(Δ^2e^imϕ_+)   =(∂_r+2iam/Δ)(4[2(r-M)^2+3(r-M)aim+Δ-a^2m^2])
       =4[4(r-M)^2aim/Δ+6(r-M)-6(r-M)a^2m^2/Δ-2ia^3m^3/Δ+5aim],


    ∂_r_out^4   (Δ^2e^imϕ_+)
       =(∂_r+2iam/Δ)(4[4(r-M)^2aim/Δ+6(r-M)-6(r-M)a^2m^2/Δ-2ia^3m^3/Δ+5aim])
       =4[8(r-M)(a^2-M^2)aim/Δ+6+6a^2m^2/Δ-12a^2m^2(a^2-M^2)/Δ^2+4ia^3m^3(r-M)/Δ^2
           +2aim/Δ(4(r-M)^2aim/Δ+6(r-M)-6(r-M)a^2m^2/Δ-2ia^3m^3/Δ+5aim)].

Thus we get 

    Δ^2∂_r_out^4(Δ^2e^imϕ_+)=8[3Δ^2   +iam(4(a^2-M^2)(r-M)+6Δ(r-M))
       +a^2m^2(3Δ-6(a^2-M^2)-4(r-M)^2-5Δ)
       +ia^3m^3(2(r-M)-6(r-M))+2a^4m^4],

which finally gives

    A_m(r)=1/3[3Δ^2   +(r-M)(4(a^2-M^2)+6Δ)iam-(2Δ+6(a^2-M^2)+4(r-M)^2)a^2m^2
       -4(r-M)ia^3m^3+2a^4m^4].


We use Proposition <ref> to get 

    _+2(   )=
       1/^7∑_|m|≤ 2[Δ A_m”(r)+2(iam-(r-M))A_m'(r)-4 A_m(r)] Q_m,Y_m,2^+2(cosθ)e^imϕ_++Err,

where we used (<ref>) to get '(Y_m,2^+2(cosθ)e^imϕ_+)=-4Y_m,2^+2(cosθ)e^imϕ_+, and where the error term Err is explicit and defined by

    Err:=(a^2sin^2θ T^2-4(r^2+a^2)Te_3+2aTΦ-(6r+4iacosθ)T)[].

It satisfies e_3^≤ 1T^j^kErr=O(^-8-j). It remains only to prove

    Δ A_m”(r)+2(iam-(r-M))A_m'(r)-4 A_m(r)=0.

We have  

    3A_m'(r)=12(r-M)Δ+iam(4(a^2-M^2)+6Δ+12(r-M)^2)-12(r-M)a^2m^2-4ia^3m^3,
       3A_m”(r)=12Δ+24(r-M)^2+36(r-M)iam-12a^2m^2.

Thus we can compute 

    Δ A_m”(r)+2(iam-(r-M))A_m'(r)-4 A_m(r)=1/3[c_0+c_1iam+c_2a^2m^2+c_3ia^3m^3+c_4a^4m^4],

where the coefficients are :

    c_0=12Δ^2+24(r-M)^2Δ-24(r-M)^2Δ-12Δ^2=0,
       c_1=36Δ(r-M)+24(r-M)Δ-2(r-M)(4(a^2-M^2)+6Δ+12(r-M)^2)
           -4(r-M)(4(a^2-M^2)+6Δ)
         =24(r-M)Δ-24(r-M)(a^2-M^2)-24(r-M)^3=24(r-M)(Δ-Δ)=0
       c_2=-12Δ-8(a^2-M^2)-12Δ-24(r-M)^2+24(r-M)^2+8Δ+24(a^2-M^2)+16(r-M)^2
         =-16Δ+16(a^2-M^2)+16(r-M)^2=16(Δ-Δ)=0
        c_3=-24(r-M)+8(r-M)+16(r-M)=0
        c_4=8-8=0,

which concludes the proof of Lemma <ref>.





