Late-time tails and mode coupling of linear waves on Kerr spacetimes
LLate-time tails and mode coupling of linear waves on Kerrspacetimes
Yannis Angelopoulos ∗ , Stefanos Aretakis † , and Dejan Gajic ‡ The Division of Physics, Mathematics and Astronomy, Caltech, 1200 E California Blvd, Pasadena CA 91125, USA Department of Mathematics, University of Toronto, 40 St George Street, Toronto, ON, Canada Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK Department of Mathematics, Radboud University, 6525 AJ Nijmegen, The Netherlands
February 23, 2021
Abstract
We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalarwave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angularfrequencies (cid:96) ≥ (cid:96) ≥
2. The (cid:96) -dependent asymptotics on Kerr spacetimes differ significantly fromthe non-rotating Schwarzschild setting (“Price’s law”). The main differences with Schwarzschild areslower decay rates for higher angular frequencies and oscillations along the null generators of the eventhorizon. We introduce a physical space-based method that resolves the following two main difficultiesfor establishing (cid:96) -dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2)a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conservedcharges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates inthe full black hole exterior. This framework is suitable for resolving the conflicting numerology in Kerrlate-time asymptotics that appears in the numerics literature.
Contents ∗ [email protected] † [email protected] ‡ [email protected], [email protected] a r X i v : . [ g r- q c ] F e b Preliminaries: wave equation 26 r p -weighted energy estimates 35 r p -weighted energy estimates for φ ( n ) . . . . . . . . . . . . . . . . . . . . . . . 355.2 Hierarchies of r p -weighted energy estimates for P , P and P . . . . . . . . . . . . . . . . . . 385.3 Additional hierarchies for higher-order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 40 r − k -weighted elliptic estimates 54 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669.2 Decay towards I + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.3 Construction of time integral data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739.4 Time-inverted Newman–Penrose charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10 Late-time polynomial tails: the (cid:96) = 0 projection 78 A γ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7910.2 Asymptotics in R \ A γ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8110.3 Asymptotics with vanishing Newman–Penrose charges . . . . . . . . . . . . . . . . . . . . . . 82
11 Late-time polynomial tails: the (cid:96) = 1 projection 82 A γ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8311.2 Asymptotics in R \ A γ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.3 Asymptotics with vanishing Newman–Penrose charges . . . . . . . . . . . . . . . . . . . . . . 87
12 Late-time polynomial tails: the (cid:96) = 2 projection 88 A γ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8912.2 Asymptotics in R \ A γ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9412.3 Asymptotics with vanishing Newman–Penrose charges . . . . . . . . . . . . . . . . . . . . . . 94 A Weighted pointwise estimates 95B A basic interpolation inequality 97
The Kerr spacetimes ( M M,a , g
M,a ) constitute a 2-parameter family of solutions to the Einstein vacuumequations Ric[ g ] = 0that are expected to describe all possible final states of a wide variety of gravitational collapse scenarios inan astrophysical setting [Pen82]. A preliminary step to describing the intricate dynamical properties of thegravitational radiation that is emitted when black hole exteriors settle down to Kerr solutions is to verify that2hey indeed settle down to Kerr solutions. In the context of the evolution of dynamical black holes arisingfrom “small perturbations” of Kerr initial data to the Einstein equations, this is the black hole stabilityproblem. There has been significant recent progress towards addressing the question of linear and nonlinearstability; see for example [DHR19b, KS20, Joh19, ABBM19, HHV21, SRdC20] and references therein.In this paper, we develop mathematical tools necessary for going beyond the question of stability, byaddressing the precise late-time behaviour of gravitational radiation emitted in the evolution of perturbationsof Kerr initial data in the context of the model problem of the linear scalar wave equation on a fixedsubextremal ( | a | < M ) Kerr spacetime background ( M M,a , g
M,a ): (cid:3) g M,a ψ = 0 . (1.1)The evolution of the wave ψ models the evolution of key dynamical quantities for the Einstein equations.Addressing the precise late-time dynamics of gravitational radiation is motivated by the exterior as well asthe interior of black holes:A) The decay rates and leading-order coefficients of gravitational radiation measured by observers “at in-finity” are expected to encode information about the initial perturbation and the final Kerr solution.Deducing information purely from gravitational radiation is important since it can be detected and ana-lysed experimentally at gravitational wave observatories and forms a key signature of astrophysical blackhole processes, see also [AAG18a].B) The precise late-time behaviour of gravitational radiation crossing the event horizon, the boundarybetween black hole interior and exterior, plays a direct role in uncovering the nature and strength ofsingularities that may be present in black hole interiors. As such, it is important for addressing the strong cosmic censorship conjecture ; see the introduction of [DL17] for a comprehensive overview of theblack hole interior and the strong cosmic censorship conjecture. In this paper, we determine the precise leading-order late-time behaviour of ψ and itshigher angular frequencies ψ (cid:96) ≥ and ψ (cid:96) ≥ on Kerr black hole exteriors. We develop a physicalspace mechanism that exploits the existence of conserved charges together with a time inversion theory, inorder to derive the presence of “tails” in the late-time dynamics. Our mechanism deals with new difficultiesin deriving late-time asymptotics that are caused by the rotation of the black hole background: a coupling ofdifferent angular frequencies and a loss of ellipticity of operators relevant for the time inversion theory. Wederive new phenomena that arise due to rotation: slower decay rates at the level of higher angular frequenciesand oscillations along the event horizon.We provide below an outline of the remainder of the introductory section of the paper. • In Section 1.1, we present the main theorems of the paper. • In Section 1.2, we discuss previous work on late-time asymptotics for waves on black holes. • In Section 1.3, we sketch an analogue of an angular frequency-dependent Price’s law in the Kerr setting. • In Section 1.4, we outline the main new ideas and methods that appear in the proofs of the theorems.
In this section, we present the main results obtained in this paper and provide some additional remarks.We first introduce briefly the notation appearing in the statements of the theorems below. • Let Σ τ denote appropriate spacelike, asymptotically hyperboloidal hypersurfaces in the Kerr manifold M M,a that intersect the future event horizon H + . The spacetime region of interest is then the union (cid:83) τ ∈ [0 , ∞ ) Σ τ which constitutes the main region of interest in M M,a . See Figure 1 for a pictorialrepresentation. • The hypersurfaces Σ τ are foliated by Boyer–Lindquist spheres S τ,r , which we equip with angularcoordinates ( θ, ϕ ∗ ). The label r denotes the radial Boyer–Lindquist coordinate, which takes the values r ∈ [ r + , ∞ ) along Σ τ , with r = r + at H + . 3ote that the components of the rescaled induced metric r − g S τ,r approach the components of themetric on the unit round sphere in standard spherical coordinates ( θ, ϕ ∗ ) as r → ∞ . In this sense, theBoyer–Lindquist spheres are asymptotically round. • The notation ψ (cid:96) and ψ ≥ (cid:96) indicates a projection of the function ψ to standard spherical harmonicmodes on the spheres S τ,r equipped with the unit round metric, with angular frequencies equal to (cid:96) , and greater or equal to (cid:96) , respectively. The spherical harmonics here are defined with respect tothe angular coordinates ( θ, ϕ ∗ ) and are denoted by Y (cid:96),m ( θ, ϕ ∗ ). We denote with π (cid:96) the operator thatprojects a function on S τ,r to the spherical harmonics with angular frequency (cid:96) .See Section 2 for a precise introduction of the spacetime geometry, the Boyer–Lindquist coordinates andother notational conventions. We state here the main theorems proved in the paper.Figure 1: A 2-dimensional representation of the spacetime M M,a , with the hypersurfaces Σ τ and the shadedregion depicting (cid:83) τ ∈ [0 , ∞ ) Σ τ . Each point in the picture represents a Boyer–Lindquist sphere S τ,r and thehypersurface I + , which represents the points ( τ, ∞ , θ, ϕ ∗ ) is depicted at a finite distance. Theorem 1.1.
Let ψ be a solution arising from smooth and compactly supported initial data for (1.1) . Then ψ satisfies the following asymptotic behaviour along { r = r } , for each r ≥ r + : ψ ( τ, r , θ, ϕ ∗ ) = − I (1)0 (1 + τ ) − + O r ( τ − − η ) ,r − ψ ( τ, r , θ, ϕ ∗ ) = − I (1)1 ( θ, ϕ ∗ )(1 + τ ) − + O r ( τ − − η ) ,ψ ≥ ( τ, r , θ, ϕ ∗ ) = − (cid:114) π a I (1)0 Y , ( θ )(1 + τ ) − + O r ( τ − − η ) , where η > , I (1) i are functions on S and denote the time-inverted Newman–Penrose charges of ψ (seeSection 1.1.2), and O r ( τ − p ) denotes schematically terms that can be bounded uniformly in τ by weightedinitial data L -norms multiplied by (1 + τ ) − p , with constants that may depend on r .When r → ∞ the “radiation field” rψ has the following asymptotic behaviour: rψ ( τ, ∞ , θ, ϕ ∗ ) = − I (1)0 (1 + τ ) − + O ( τ − − η ) ,rψ ( τ, ∞ , θ, ϕ ∗ ) = − I (1)1 ( θ, ϕ ∗ )(1 + τ ) − + O ( τ − − η ) ,rψ ≥ ( τ, ∞ , θ, ϕ ∗ ) = (cid:20) − I (1)2 ( θ, ϕ ∗ ) + 83 (cid:114) π a I (1)0 Y , ( θ ) (cid:21) (1 + τ ) − + O ( τ − − η ) , here O ( τ − p ) denote schematically terms that can be bounded uniformly in τ by weighted initial data L -norms multiplied by (1 + τ ) − p . A more detailed version of Theorem 1.1 follows directly by combining Propositions 10.7, 11.7 and 12.7.In these propositions we moreover obtain global analogues of the expressions in Theorem 1.1 with additionalderivatives ∂ kτ on both sides, for arbitrary k ∈ N . Theorem 1.2.
Let ψ be a solution arising from initial data for (1.1) on Σ that are smooth with respect tothe differentiable structure on the conformal compactification of Σ , with non-zero Newman–Penrose charges I (cid:96) , (cid:96) = 0 , , (see Section 1.1.2). Then the rates and coefficients in the late-time expansions presented inTheorem 1.1 and Corollary 1.3 are modified according to the expressions displayed in Table 1. See Section 2.3 for a precise definition of the conformal compactification of Σ . We directly obtain amore detailed version of Theorem 1.1 by combining Corollary 10.5 and Propositions 10.6, 11.5, 11.6, 12.5and 12.6. Proj r = r ≥ r + I + ψ I · τ I · τ ψ r I ( θ, ϕ ∗ ) · τ I ( θ, ϕ ∗ ) · τ ψ ≥ (cid:112) π a I · Y , ( θ ) · τ (cid:2) I ( θ, ϕ ∗ ) − (cid:112) π a I · Y , ( θ ) (cid:3) · τ Table 1: Late-time asymptotics of ψ , ψ , ψ ≥ for non-zero Newman–Penrose charges I (cid:96) Remark 1.1 (Logarithmic next-to-leading order terms) . One can apply the arguments in [AAG19] to thesetting of Kerr spacetimes, with minimal modifications, to obtain the following higher-order extension of thelate asymptotics of rψ at infinity in Theorem 1.1: rψ ( τ, ∞ , θ, ϕ ∗ ) = − I (1)0 (1 + τ ) − + 8 M I (1)0 log(1 + τ )(1 + τ ) − + O ( τ − ) . See also [BVW18] for additional results pertaining to polyhomogeneity in late-time expansions on asymptot-ically Minkowksi spacetimes.
Remark 1.2 (Initial data regularity) . The initial data in Theorem 1.1 are assumed to be compactly supported.This assumption is made purely to simplify the expressions of the time-inverted Newman–Penrose charges I (1) (cid:96) in terms of initial data for ψ ; see Section 9.4. The assumption can be weakened by merely assuming the datahave sufficient, finite regularity with respect to the differentiable structure on the conformal compactificationof the hypersurface Σ , with moreover vanishing Newman–Penrose charges I (cid:96) . In fact, one can considereven rougher data by modifying the powers of r appearing in the definitions of the Newman–Penrose charges,which will affect the decay rates; see also the upcoming [Keh21] for an example of late-time asymptoticsarising from rougher initial data that naturally come up in scattering problems. Remark 1.3 (Blow-up in the black hole interior) . The results in Theorems 1.1 and 1.2 can be applied tojustify the assumptions made along the event horizon in [LS16] in order to derive blow-up of the H normof ψ at the inner horizon, the future causal boundary of the Kerr black hole interior for a (cid:54) = 0 . Remark 1.4 (Linearized Einstein equations) . The methods that are developed in this paper to prove Theo-rems 1.1 and 1.2 could be extended naturally to the setting of the spin-2 Teukolsky equations, when combinedwith the integrated energy estimates in [DHR19a, SRdC20]. As the Teukolsky equations dictate the dynam-ical behaviour of perturbations of Kerr initial data in the context of the linearized Einstein equations, thesemethods provide a clear strategy for deriving late-time tails in the evolution of metric perturbations.
The
Newman–Penrose charges I (cid:96) are the values of asymptotic quantities that are defined along thespheres foliating future null infinity and are conserved in time. See point 3 of Section 1.4.1 for an illustration5f this conservation property and see Section 4 for the precise definition of the Newman–Penrose charges.They were originally introduced in [NP68] on Minkowski background spacetimes and more generally in thecontext of the nonlinear Einstein equations.In the Minkowski spacetime, the Newman–Penrose charges can be expressed as the following weightedderivatives of the radiation field: I (cid:96) [ ψ ] = (2 r L ) (cid:96) ( rψ (cid:96) )( τ, r = ∞ , θ, ϕ ∗ ) , and they are independent of τ . Here, L denotes a standard outgoing null vector field.In Schwarzschild spacetimes, or Kerr spacetimes with mass M and angular parameter a = 0, the abovequantity is still conserved when (cid:96) = 0, but for (cid:96) ≥
1, it has to be modified as follows: I (cid:96) [ ψ ] = r L (2 r w (cid:96) L ( . . . (2 r w L (2 r L ( rψ (cid:96) ))) . . . )( τ, r = ∞ , θ, ϕ ∗ ) , with w i denoting polynomials in r − of degree (cid:96) + 1 − i , such that lim r →∞ w i ( r ) = 1, which depend onthe mass M . See also [AAG21] for a more precise inductive definition of I (cid:96) on Schwarzschild. Analogousmodified expressions can also be shown to hold in more general spherically symmetric, asymptotical flatspacetimes.In order to construct I (cid:96) on general Kerr spacetimes, one has to deal with the coupling of two difficulties:1. A modification of the weighted vector fields r L due to the presence of mass M ,2. The appearance of the spherical harmonic modes ψ (cid:96) − and ψ (cid:96) +2 in the expression for I (cid:96) , due to thepresence of angular momentum a (cid:54) = 0 and the use of Boyer–Lindquist spheres.Note that difficulty 2 is already present in Minkowski when considering a foliation by oblate spheroidsthat are asymptotically round, rather than exactly round spheres, i.e. by taking M = 0 in the Kerr metric,which leads to a spacetime that is isometric to Minkowski. For example, the charge I then takes the form: I [ ψ ] = (cid:20) r L ( rψ ) − a π (sin θT ( rψ )) (cid:21) ( τ, r = ∞ , θ, ϕ ∗ ) , with π and ψ now denoting projections with respect to ellipsoids, rather than round spheres. The presenceof the term π (sin θT ( rψ )) above may be viewed as an effect of angular mode coupling in Boyer–Lindquistcoordinates at infinity. See Section 1.1.4 for more details.In Kerr, I takes the same form as in Minkowski with respect to oblate spheroidal coordinates, but I (cid:96) with (cid:96) ≥ M . We derive the Newman–Penrose charges I (cid:96) in Kerr, with (cid:96) = 0 , , I (cid:96) .The time-inverted Newman–Penrose charges , which are the quantities I (1) (cid:96) appearing in the leading-order terms in Theorem 1.1 can be interpreted as the Newman–Penrose charges of the function: ∂ − τ ψ ( τ, r, θ, ϕ ) := − (cid:90) ∞ τ ψ ( τ (cid:48) , r, θ, ϕ ) dτ (cid:48) , the time integral of ψ . We moreover show that I (1) (cid:96) can be expressed solely in terms of initial data for ψ onΣ , for example, for (cid:96) = 0: I (1)0 = 14 π M ( r + a ) (cid:90) Σ ∩H + ψ dω + 14 π M (cid:90) Σ n ( ψ ) dµ , with n the normal vector field with respect to Σ and dµ the natural volume form with respect to theinduced metric on Σ . See Section 9.4 for the precise integral expressions of I (1) (cid:96) . The eccentricity of the spheroids is then given by | a | √ r + a . .1.3 Horizon oscillations As an immediate corollary of the late-time asymptotics presented in Theorem 1.1, we derive an oscillatoryand decaying behaviour of the (cid:96) = 1 angular mode of ψ when measured along the null generators of theevent horizon. Corollary 1.3.
Let γ θ,ϕ H + ( τ ) denote a null generator of the event horizon H + emanating from the point ( θ, ϕ H + ) on the 2-sphere S ,r + = Σ ∩ H + , with time parameter τ . Then: ψ | γ ( θ,ϕ ∗ ) ( τ ) = − I (1)0 (1 + τ ) − + O ( τ − − η ) ,r − ψ | γ ( θ,ϕ ∗ ) ( τ ) = − (cid:88) m = − I (1)1 m Y ,m ( θ, ϕ H + ) e im ω + τ (1 + τ ) − + O ( τ − − η ) ,ψ ≥ | γ ( θ,ϕ ∗ ) ( τ ) = − (cid:114) π a I (2)0 Y , ( θ )(1 + τ ) − + O ( τ − − η ) , where ω + = ar + a is the angular velocity of the black hole and I (1) (cid:96)m denotes the projection of I (1) (cid:96) to the m -th azimuthal mode. We therefore see that the leading-order behaviour of the (cid:96) = 1 mode in time encodes the angular velocity ofthe black hole in the form of an oscillatory factor e im ω + τ . Corollary 1.3 provides the first rigorous derivationof horizon oscillations in Kerr spacetimes, which were originally suggested by Barack–Ori in [BO99b].The presence of horizon oscillations illustrates the different roles played by the Killing vector fields T and K . Here, T denotes a suitably normalized Killing vector field that is timelike for large values of r , butspacelike near the event horizon, whereas K denotes the Killing vector field that is tangential to the nullgenerators of the horizon and is timelike close to the horizon, but spacelike far away from the horizon. Hence, T provides a “natural” choice of time direction far away from the horizon and K provides a natural choiceof time direction close to the horizon. Note that in the coordinate chart ( τ, r, θ, ϕ ∗ ) that we have chosen, wecan simply express T = ∂ τ . The two Killing vector fields can be related as follows: K = T + ω + Φ , where Φ denotes the Killing vector field that generates the axisymmetry of the Kerr spacetime, and K = T when a = 0.Horizon oscillations follow from the fact that when a (cid:54) = 0 the time derivative T ψ decays one powerfaster than ψ to leading order, whereas the time derivative Kψ does not . In fact, using the aboverelation between T and K , we can see that the non-axisymmetric part of Kψ must agree precisely with ω + Φ ψ to leading order in time, which explains the oscillatory behaviour in the (cid:96) = 1 mode.We note that it has been suggested that horizon oscillations will similarly appear in the leading-order late-time asymptotics of general solutions to non-zero integer spin Teukolsky equations, see [BO99a] and Remark1.4. They are therefore expected to play a leading-order role in the general late-time horizon dynamics inthe setting of linearized gravity and the Maxwell equations on Kerr spacetimes. The spherical harmonic modes in Theorems 1.1 and 1.2 are coupled in their evolution, in contrast with theSchwarzschild case ( a = 0) where they evolve independently.Indeed, when defined with respect to the metric on the unit round sphere on Boyer–Lindquist spheres,the angular mode coupling takes the following form: ρ (cid:3) g M,a ψ (cid:96) = a T ( π (cid:96) (sin θψ ) − sin θπ (cid:96) ψ ) , (1.2)where we can schematically write the right-hand side as: a T ( π (cid:96) (sin θψ ) − sin θπ (cid:96) ψ ) ≈ a ( c − T ψ (cid:96) − + c T ψ (cid:96) + c + T ψ (cid:96) +2 ) , (1.3)7ith c , c − , c + constants, see Lemma 2.4 for a precise expression.If the different angular modes were uncoupled, we would be able to show that modes supported on higher (cid:96) decay faster . This is precisely the case when a = 0, see [AAG21]. In view of (1.2) and (1.3), however, wecan see heuristically that the decay of ψ (cid:96) is limited by the decay of T ψ (cid:96) − . We would therefore have thefollowing schematic relation for the late-time asymptotics of ψ (cid:96) : ψ (cid:96) ∼ T ψ (cid:96) − ∼ . . . ∼ T (cid:96) ψ for even (cid:96) ≥ ,ψ (cid:96) ∼ T ψ (cid:96) − ∼ . . . ∼ T (cid:96) − ψ for odd (cid:96) ≥ . So while the decay rate of ψ (cid:96) is limited by the lower angular modes T (cid:96) ψ and T (cid:96) − ψ , we can use that T -derivatives decay faster, as mentioned in Section 1.1.3, together with the above heuristics to predict thathigher angular frequencies ψ (cid:96) do in fact decay faster than lower angular frequencies, but they decay slowercompared to the uncoupled setting.In this paper, we give a rigorous validation of the above heuristics in the case of the late-time asymptoticsof (cid:96) = 2. We also discuss the general (cid:96) case in Section 1.3.We conclude that while there is no a priori natural choice of 2-spheres to foliate M M,a , Theorems1.1 and 1.2 imply that the choice of Boyer–Lindquist spheres is natural from the point of view of late-timeasymptotics, as it has the favourable property that it results in a “late-time decoupling” of the correspondingspherical harmonic modes ψ (cid:96) , i.e. the terms in the expansion of ψ in τ − at any fixed r up to order N ∈ N will be supported only on spherical harmonic modes with angular frequencies (cid:96) ≤ L ( N ) < ∞ , where weexpect L ( N ) = max { N − , } . In this section we give an overview of some relevant previous results in the literature on late-time tails andsharp decay estimates for waves on black hole spacetimes.
The first discussion on inverse polynomial late-time tails for wave equations on black hole backgroundappeared in a paper of Price [Pri72], who provided a heuristic argument for the presence of a τ − (cid:96) − tail inSchwarzschild. Late-time tails on spherically symmetric backgrounds have since been discussed frequentlyin the physics literature, using both heuristic and numerical arguments; see the introduction of [AAG20] foran overview.The first numerical discussion on late-time tails in the context of subextremal Kerr spacetimes appearedin [KLPA97], which was followed by heuristic analyses in [Hod99] and [BO99b]. The latter work suggestedthe power laws for generic, compactly supported data that are stated in Section 1.3.1. Subsequent numericalwork on late-time tails focused on the case of initial data supported on a single harmonic mode and hasproduced conflicting numerology, caused by the difficulty of characterising the evolutionary coupling andexcitation of spherical harmonic modes (numerically). We refer to [ZKB14, BK14] for the latest numericalresults and corrected predictions of late-time tails and also to the references therein for a more completehistory of the problem. As discussed in Section 1.3.2, the methods developed in the present paper allow for afinal resolution of the differences in the numerology of late-time tails suggested in the literature, by relatingthe numerology to the vanishing and non-vanishing of time-inverted Newman–Penrose charges, which weexpect to align with the numerical results of [ZKB14, BK14]. The first mathematically rigorous derivation of the leading-order late-time behaviour of waves on black holebackground was obtained in [AAG18c, AAG18b] for a variety of spherically symmetric backgrounds. Thepresence of (cid:96) -dependent late-time tails on Schwarzschild and more general subextremal Reissner–Nordstr¨ombackgrounds is derived in [AAG21]. These results appeal to important ideas and results that appeared inprevious literature on (integrated) energy decay estimates, most notably [DR09, DR10], see also [DR13]. Themethods of [AAG18c, AAG18b] have also been adapted to the setting of the Dirac equation on Schwarzschildbackgrounds [MZ20]. 8 proof of the existence of τ − tails on Kerr spacetimes was recently established in [Hin20], using methodsbased in Fourier space.We refer also to [Tat13, MTT12, Hin20] for sharp decay estimates for ψ in a linear setting and [DR05]in a spherically symmetric nonlinear setting. See the also the results in [DSS11, DSS12], investigating the (cid:96) -dependence in decay estimates and [Ma20] for some sharp decay estimates in the context of the Maxwellequations and a characterisation of the Newman–Penrose charges on Schwarzschild in the Maxwell setting.Finally, the results in the present paper appeal to the energy boundedness and integrated energy decayestimates that have already been established in Kerr spacetimes in [DRSR16], which capture the key geomet-ric obstructions to wave decay on Kerr caused by the presence of trapped null geodesics and an ergoregion.We refer to the introduction of [DRSR16] for a complete overview of the history of integrated energy decayestimates on Kerr spacetimes. The leading-order behaviour of waves on extremal black hole backgrounds features additional interestinggeometric phenomena. The first mathematically rigorous proof of the existence and form of late-time tailson extremal Reissner–Nordstr¨om spacetimes was obtained in [AAG20] and follows previous numerical andheuristic results, see the introduction of [AAG20] for a comprehensive overview. The late-time tails in thissetting involve an additional conserved charge that occurs along the event horizon of extremal black holespacetimes and was first discovered in [Are11]. It is connected to the absence of the redshift effect alongextremal black hole horizons and may be related to the Newman–Penrose charges, see [LMRT13, BF13]. Seealso [AAG18a] and the subsequent numerical work [BKS19, BKS21] for discussions on the signature of thisconserved horizon charge at infinity.While decay estimates have also been obtained in the setting of rotating extremal black hole spacetimes,for the special case of axisymmetric solutions [Are12], the late-time properties of non-axisymmetric linearwaves on extremal Kerr ( | a | = M ) remain open. In this setting, only mode stability has been established ina mathematically rigorous setting in [dC20], but heuristics have also been provided on the rates of late-timetails for fixed azimuthal modes in [CGZ16]. One can apply the methods developed in this paper to extend the results of Theorem 1.1 (and Theorem 1.2)beyond (cid:96) = 2 and obtain the following late-time asymptotics for higher spherical harmonic modes: ψ (cid:96) =2 k ( τ, r , θ, ϕ ∗ ) = c (cid:96), a k I (1)0 Y (cid:96), ( θ )(1 + τ ) − (cid:96) − + O r ( τ − (cid:96) − − η ) ,r − ψ (cid:96) =2 k +1 ( τ, r , θ, ϕ ∗ ) = (cid:88) m = − c (cid:96),m a k I (1)1 m Y (cid:96),m ( θ, ϕ ∗ )(1 + τ ) − (cid:96) − + O r ( τ − (cid:96) − − η ) ,rψ (cid:96) =2 k ( τ, ∞ , θ, ϕ ∗ ) = (cid:96) (cid:88) k =0 2 k (cid:88) m = − k c (cid:96),k,m a (cid:96) − k I (1)2 k m Y (cid:96),m ( θ, ϕ ∗ )(1 + τ ) − (cid:96) − + O ( τ − (cid:96) − − η ) ,rψ (cid:96) =2 k +1 ( τ, ∞ , θ, ϕ ∗ ) = (cid:96) − (cid:88) k =0 2 k +1 (cid:88) m = − (2 k +1) c (cid:96),k,m a (cid:96) − (2 k +1) I (1)2 k +1 m Y (cid:96),m ( θ, ϕ ∗ )(1 + τ ) − (cid:96) − + O ( τ − (cid:96) − − η ) , where c (cid:96),m and c (cid:96),k,m are dimensionless constants. We do not provide a proof of the above expressions inthe present paper, but we refer to the heuristics in Section 1.1.4 and leave them for the interested reader toverify, equipped with the techniques and estimates that are developed in the paper.The above behaviour suggests in particular the following version of “Price’s law” at fixed radius for a (cid:54) = 0: ψ (cid:96) ∼ τ − (cid:96) − when (cid:96) is odd ,ψ (cid:96) ∼ τ − (cid:96) − when (cid:96) is even , a = 0: ψ (cid:96) ∼ τ − (cid:96) − , see [AAG21]. We note that the above decay ratesare consistent with the heuristics in [BO99b]. Rather than considering generic initial data supported on all angular frequencies, one can further investigatethe mode-coupling mechanism by considering restricted data, supported on angular frequencies ≥ (cid:96) (cid:48) and theninvestigate the late-time tails of ψ (cid:96) for each (cid:96) ∈ N . This may be viewed as a study of “mode excitation” toleading order in time.For | (cid:96) − (cid:96) (cid:48) | >
2, it follows easily that I (1) (cid:96) = 0. In this case, one would have to apply the time inversionprocedure k times, for suitable values of k , to obtain a non-vanishing higher-order time-inverted charge I ( k ) (cid:96) [ ψ ] := I (cid:96) [ ∂ − kτ ψ ], which would then be relevant for the late-time behaviour. It is straightforward to showvia the equation for the time inverse (9.30) that if the (smooth and compactly supported) initial data aresupported on the angular frequency (cid:96) (cid:48) , with (cid:96) (cid:48) / ∈ { , } , the first (generically) non-zero time-inverted chargefor (cid:96) (cid:54) = (cid:96) (cid:48) would be I ( | (cid:96) (cid:48) − (cid:96) |− (cid:96) [ ψ ] = I (cid:96) [ ∂ −| (cid:96) (cid:48) − (cid:96) | +1 τ ψ ] , if (cid:96) and (cid:96) (cid:48) are both either even or odd (note that I ( k ) (cid:96) vanishes for all k if (cid:96) and (cid:96) (cid:48) have opposite parity).Moreover, I (1) (cid:96) (cid:48) = I (cid:96) (cid:48) [ ∂ − τ ψ ] (cid:54) = 0, generically. See also Section 9.2 of [AAG18b] for a discussion on higher-ordertime inversions in the spherically symmetric context.In light of the non-vanishing of the above higher-order time inverted charges, we expect that, analogouslyto what is described in Section 1.3, one could apply the methods in the present paper to obtain schematicallythe following late-time tails for initial data supported on angular frequency (cid:96) (cid:48) : let j = 0 when (cid:96) (cid:48) is even and j = 1 when (cid:96) (cid:48) is odd, then ψ (cid:96) | r = r ∼ (cid:88) ≤ k ≤(cid:98) max { (cid:96),(cid:96) (cid:48)− } (cid:99) I ( | (cid:96) (cid:48) − k |− k + j τ − (cid:96) − (cid:96) (cid:48) − when (cid:96) (cid:48) / ∈ { , } , (1.4) ψ (cid:96) | r = r ∼ I (1) (cid:96) (cid:48) τ − (cid:96) − (cid:96) (cid:48) − when (cid:96) (cid:48) ∈ { , } , (1.5) rψ (cid:96) | I + ∼ I ( (cid:96) (cid:48) − (cid:96) − (cid:96) τ − (cid:96) (cid:48) when (cid:96) ≤ (cid:96) (cid:48) − , (1.6) rψ (cid:96) | I + ∼ (cid:88) − ≤ k ≤ min { , (cid:96) − (cid:96) (cid:48) } I (1) (cid:96) (cid:48) +2 k τ − (cid:96) − when (cid:96) ≥ (cid:96) (cid:48) , (1.7)with (cid:96) (cid:48) and (cid:96) both either even or odd. Note in particular that (1.5)–(1.7) follow directly from Theorem 1.1and the expressions in Section 1.3.1, whereas (1.4) requires a minor further extension of the methods in thepresent paper.The above power laws agree with an extrapolation of the numerical results of [ZKB14], but deviate fromearlier suggestions in the literature. See also Section 1.2. In this section, we introduce the main new ideas and techniques involved in proving Theorems 1.1 and 1.2.We make use of the following additional notation in this section: • We cover Σ with the coordinates ( ρ , θ, ϕ ∗ ), where ρ = r | Σ and employ the chart ( τ, ρ , θ, ϕ ∗ ) to coverthe spacetime region (cid:83) τ ∈ [0 , ∞ ) Σ τ . Furthermore, we denote the standard volume form on the unitround 2-sphere S by dω and the natural volume form corresponding to the induced metric on Σ τ by dµ τ . • We use the notation T = ∂ τ and X = ∂ ρ , and we use L and L to denote the principal outgoing andingoing null vector fields in Kerr. We also use / ∇ S to denote the covariant derivative on S . • We the following rescaling of ψ : φ = (cid:112) r + a ψ, φ ( τ, ∞ , θ, ϕ ∗ ) is known as the Friedlander radiation field . This is a natural quantity toconsider for scattering problems; see for example [DRSR18]. • We use T to denote the stress energy momentum tensor corresponding to (1.1): T ( V, W ) =
V ψW ψ − g M,a ( V, W )( g − M,a ) αβ ∂ α ψ∂ β ψ. We refer to Section 2 for a more precise introduction to all the above notation. r p -weighted estimates The first step towards determining the leading-order behaviour of ψ is to obtain time decay estimates forenergy norms along the leaves of a suitable spacetime foliation.We consider a foliation by asymptotically hyperboloidal hypersurfaces Σ τ intersecting the event horizon,see Figure 1. We obtain energy decay by using extensions of the original Dafermos–Rodnianski r p -weightedenergy method [DR10]. See [DRSR18] for an application of the Dafermos–Rodnianski r p -weighted energymethod in Kerr and [Mos16] for the case of more general spacetimes.1. ( Dafermos–Rodnianski hierarchy ) Consider first the Dafermos–Rodnianski (D–R) hierarchy, whichtakes the schematic form (cid:90) τ τ (cid:90) Σ τ r p − ( Lφ ) dωd ρ dτ (cid:46) (cid:90) Σ τ r p ( Lφ ) dωd ρ + . . . , (1.8)for 0 < p ≤
2, where we have omitted additional angular and T -derivatives appearing on the left-handside and . . . denotes terms arising from an application of integrated energy (Morawetz) and energyboundedness estimates, which we ignore in the current discussion.Taking p = 1 and applying the mean-value theorem in τ gives: (cid:90) Σ τ ( Lφ ) dωd ρ (cid:46) (1 + τ ) − (cid:90) Σ τ r ( Lφ ) dωd ρ + . . . . Taking p = 2 and applying the mean value theorem once again (along an appropriate sequence oftimes) allows us to estimate the right-hand side further and obtain the following uniform energy decayestimates: (cid:90) Σ τ ( Lφ ) dωd ρ (cid:46) (1 + τ ) − (cid:90) Σ τ r ( Lφ ) dωd ρ + . . . .
2. (
Extending the D–R hierarchy via r L derivatives ) The D–R hierarhcy is limited by the restric-tion p ≤
2. To be able to get a decay rate of (1 + τ ) − and higher, we need control over the left-handside of (1.8) for p ≥
3. By an application of a standard Hardy inequality, we can show that: (cid:90) τ τ (cid:90) Σ τ r ( Lφ ) dωd ρ dτ = (cid:90) τ τ (cid:90) Σ τ r − ( r Lφ ) dωd ρ dτ (cid:46) (cid:90) τ τ (cid:90) Σ τ ( L ( r Lφ )) dωd ρ dτ + . . . . In the case where ψ is supported on (cid:96) ≥
1, we can further estimate: for 0 ≤ p ≤ (cid:90) τ τ (cid:90) Σ τ r p − ( L ( r Lφ ≥ )) dωd ρ dτ (cid:46) (cid:90) Σ τ r p ( L ( r Lφ ≥ ) dωd ρ + a (cid:90) τ τ (cid:90) Σ τ r p − ( r LT φ ) dωd ρ dτ . . . . In fact, more generally, for all n ≤ (cid:96) and 0 ≤ p ≤ (cid:90) τ τ (cid:90) Σ τ r p − ( L ( r L ) n φ ≥ (cid:96) ) dωd ρ dτ (cid:46) (cid:90) Σ τ r p ( L ( r L ) n φ ≥ (cid:96) ) dωd ρ + a (cid:90) τ τ (cid:90) Σ τ r p − (( r L ) n T φ ≥ (cid:96) − ) dωd ρ dτ . . . . (1.9)11hen a = 0, we can therefore continue applying the mean-value theorem together with a Hardyinequality and apply (1.9) in order to obtain (at least) (1 + τ ) − − (cid:96) energy decay for φ ≥ (cid:96) . This is donein [AAG21].However, when a (cid:54) = 0, we have to address the boxed term in (1.9), which reflects the fact that wecannot treat φ ≥ (cid:96) independently from φ ≤ (cid:96) − (recall that the spherical harmonic modes are coupled).For example, in the case (cid:96) = 2, we can rewrite the boxed term in (1.9) as follows: a (cid:90) τ τ (cid:90) Σ τ r p − (( r L ) T φ ) dωd ρ dτ (cid:46) a (cid:88) k =0 (cid:90) τ τ (cid:90) Σ τ r p +3 ( L ( rL ) k T φ ) dωd ρ . By applying the wave equation (1.1) we can exchange T derivatives for L and angular derivatives toestimate further: a (cid:88) k =0 (cid:90) τ τ (cid:90) Σ τ r p +3 ( L ( rL ) k T φ ) dωd ρ (cid:46) a (cid:88) ≤ k + k ≤ (cid:90) τ τ (cid:90) Σ τ r p − | / ∇ k S L ( rL ) k φ | dωd ρ . When then show that the term above can be controlled after showing the estimate (1.8) also holds when φ is replaced with / ∇ k S ( rL ) k φ . That is to say, the (extended) r p -weighted hierarchies remain valid aftercommuting arbitrarily many times with rL and / ∇ S . This favourable commutation property also playsan important role in [Mos16, Sch13]. We stress moreover that this property contrasts strongly withcommutation with r L , which, as outlined above, is more delicate as it is only valid after projecting tosuitably higher angular frequencies.We apply the above strategy for (cid:96) = 1 , , τ ) − − (cid:96) energy decay for φ ≥ (cid:96) and moreover(1 + τ ) − − (cid:96) − k energy decay for T k φ by exploiting the above exchange of T -derivatives for rL -derivatives and angular derivatives to extend the hierarchy even further.See Section 5 for a precise discussion of the various r p -weighted hierarchies and Section 6 for thearguments that use these hierarchies to obtain energy decay.3. ( Maximal length hierarchies via conserved charges at infinity ) In order to obtain sharp energydecay rates, we complement the extended hierarchies above with an additional extension that relies onthe existence of conservation charges at future null infinity.We consider first φ . In order to go beyond p = 2 in the Dafermos–Rodnianski hierarchy (1.8), wereplace Lφ in (1.8) with P , which is defined as follows: P := Lφ −
14 ∆( r + a ) a π (sin θT φ ) . Since P satisfies schematically LP = O ( r − )( φ + a rLφ + a T φ + a rLφ )it follows that lim r →∞ r P | Σ τ is conserved in τ and an analogue of (1.8) holds for the extended range0 < p <
3, with Lφ replaced by P . We use the constant I to denote these limits and refer to thisquantity as the Newman–Penrose charge for (cid:96) = 0; see Section 1.1.2.We can similarly replace ( r L ) φ and ( r L ) φ by the quantities P and P (defined precisely inSection 4) to extend (1.9) with n = (cid:96) ∈ { , , } , to p <
3. See also [AAG18c, AAG18b, AAG21] fora similar extension of the hierarchy of r p -weighted energy estimates in the a = 0 case. In this case,the corresponding limits lim r →∞ r P (cid:96) | Σ τ , which are conserved in τ , are functions of ( θ, ϕ ∗ ) and aredenoted by I (cid:96) ( θ, ϕ ∗ ). We refer to them as the Newman–Penrose charges for (cid:96) = 1 and (cid:96) = 2.We therefore conclude the energy decay rates (1 + τ ) − − (cid:96) − k for T k φ (cid:96) , with (cid:96) = 0 , ,
2, which we cancombine with the already established energy decay rate (1 + τ ) − − k for φ ≥ .The application of the quantities P (cid:96) to r p -weighted energy estimates are derived in Section 5.2.12y a straightforward application of the fundamental theorem of calculus in ρ (see Appendix A) togetherwith standard Sobolev estimates on S , we can convert the energy decay estimates above into the followingpointwise estimates: | rT k ψ | (cid:46) (1 + τ ) − (cid:15) , | rT k ψ | (cid:46) (1 + τ ) − − k + (cid:15) , | rT k ψ | (cid:46) (1 + τ ) − − k + (cid:15) , | rT k ψ ≥ | (cid:46) (1 + τ ) − − k + (cid:15) . Additional difficulties:
In the above outline, we have suppressed the role of the geometric phenomenaof redshift , trapped null geodesics and ergoregion towards establishing energy boundedness and decay in Kerrspacetimes. While these play an important role, their effect on integrated energy estimates has already beendealt with in [DRSR16] and we appeal to these estimates in the present paper. We refer to the introductionof [DRSR16] for a comprehensive discussion.Note however that the estimates in [DRSR16] do not distinguish between different spherical harmonicmodes ψ (cid:96) , so we need to modify them slightly to obtain more refined integrated estimates for ψ ≥ (cid:96) to be ableto carry out the arguments described above. We give an overview of the necessary energy boundedness andintegrated energy decay estimates, including the refined integrated estimates for ψ ≥ (cid:96) , in Section 3.3. We subsequently improve the pointwise decay rates for rψ above further by considering the r -rescalings: ψ , ψ , ψ ≥ and and r − ψ , and we apply in addition a hierarchy of elliptic estimates.We first rewrite the wave equation (1.1) as follows; L ψ = F [ T ψ ] , with L defined as the differential operator: L ψ = X (∆ Xψ ) + 2 aX Φ ψ + / ∆ S ψ and F [ T ψ ] containing only T -derivatives of ψ , see (7.2) for the precise expression. In this section, we treat F [ T ψ ] as an inhomogeneity.The operator L is elliptic when T is a timelike vector field. This is the case outside the ergoregion,i.e. when r − M r + a cos θ >
0. The loss of ellipticity inside the ergoregion prevents the use of astandard elliptic estimate to control ψ in terms of F , which would involve integrating by parts the equation( L ψ ) = F on Σ τ . We establish nevertheless for k ∈ ( , (cid:96) + ): (cid:96) (cid:88) n =0 (cid:90) Σ τ r − k (cid:2) ( X ( rX ) n ψ (cid:96) ) + r − | / ∇ S ( rX ) n ψ (cid:96) | (cid:3) r dωd ρ (cid:46) (cid:96) (cid:88) n =0 (cid:90) Σ τ r − k (( rX ) n ( π (cid:96) F )) dωd ρ (1.10)by applying the following estimates:A.) A “spacelike redshift multiplier estimate”, i.e. when (cid:96) = 0, we integrate by parts on Σ τ ∩ { r ≤ R } theproduct: Xψ · L ψ and more generally, X (cid:96) +1 ψ (cid:96) · X (cid:96) ( L ψ (cid:96) ) . In contrast with the standard redshift estimates [DR09, DR13], which involve integrating by parts inspacetime with timelike vector field multipliers, and which are only valid in a neighbourhood of theevent horizon, the above spacelike redshift estimates hold with arbitrarily large radial coordinate r . In the a = 0 case, where ellipticity only fails at the event horizon, this strategy does work and results in elliptic estimatesthat degenerate at the event horizon. See for example [AAG18c, AAG21]. r ≥ R , with R (cid:29) M , which follows from integrating r − k ( X (cid:96) L ψ ) = r − k ( X (cid:96) F ) where we apply A.) to estimate the boundary terms on r = R . The cross terms that arise whenexpanding the square on the left-hand side above are estimated via a Poincar´e inequality and the rangeof allowed values for k depends therefore on the angular frequency (cid:96) .C.) Another spacelike redshift multiplier estimate to control lower-order derivative terms, by integrating X n +1 ψ (cid:96) · X n ( L ψ (cid:96) ) , with n ≤ (cid:96) − F [ T ψ ], we therefore obtain the following elliptic hierarchy of energy estimates: (cid:96) (cid:88) n =0 (cid:90) Σ τ r − k T [( rX ) n ψ (cid:96) ]( T, n τ ) dµ τ (cid:46) (cid:96) (cid:88) n =0 (cid:90) Σ τ r − k +2 T [( rX ) n T ψ (cid:96) ]( T, n τ ) dµ τ + a (cid:90) Σ τ r − k (cid:8) T [( rX ) n T ψ (cid:96) − ]( T, n τ ) + T [( rX ) n T ψ (cid:96) +2 ]( T, n τ ) (cid:9) dµ τ , (1.11)for k ∈ ( , (cid:96) + ).The elliptic hierarchy allows us to exchange negative r weights for additional T derivatives, which decayfaster in time. One may compare this with extra time decay obtained via an extended hierarchy r p -weightedenergy estimates, as described in Section 1.4.1, where additional time decay follows instead by controllingan additional time integral on the left-hand side and applying the mean-value theorem.In the a = 0 case, we can apply (1.11) starting from k = − δ to k = (cid:96) + − δ to obtain a hierarchy ofestimates of length (cid:96) and therefore estimate: (cid:96) (cid:88) n =0 (cid:90) Σ τ r − (cid:96) − − δ T [( rX ) n ψ (cid:96) ]( T, n τ ) dµ τ (cid:46) (cid:96) (cid:88) n =0 (cid:90) Σ τ r − δ T [( rX ) n T (cid:96) ψ (cid:96) ]( T, n τ ) dµ τ , and obtain the sharp decay rate of the weighted energy on the left-hand side, together with the sharp decayrate of r − (cid:96) ψ (cid:96) .When a (cid:54) = 0, the above strategy can be repeated for (cid:96) = 0 and (cid:96) = 1, but it fails for (cid:96) = 2, due to thepresence of the terms involving ψ on the right-hand side of (1.11) when (cid:96) = 0, which limits the decay rateand hence the length of the elliptic hierarchy. We refer to Section 7 for a precise discussion of the abovearguments.When a (cid:54) = 0, we therefore conclude pointwise decay estimates with the following weights: | T k ψ | (cid:46) (1 + τ ) − (cid:15) , | r − T k ψ | (cid:46) (1 + τ ) − − k + (cid:15) , | T k ψ | (cid:46) (1 + τ ) − − k + (cid:15) , | T k ψ ≥ | (cid:46) (1 + τ ) − − k + (cid:15) . Note in particular that the decay rate for the (cid:96) = 1 and (cid:96) = 2 mode is the same, in contrast with the a = 0case. The pointwise decay rates stated in Sections 1.4.1 and 1.4.2 are (almost) sharp in the case of initial data forwhich the conserved Newman–Penrose charges I (cid:96) are non-zero, with (cid:96) = 0 , ,
2. For initial data that decayfaster as r → ∞ , such that I (cid:96) = 0, it is possible to gain one power in the decay rate.14he idea is the following: we can show that ψ can be thought of as being a time derivative of anothersolution to (1.1), i.e. T (cid:101) ψ = ψ, for some suitably regular function (cid:101) ψ such that (cid:101) ψ Σ → r → ∞ and (cid:3) g M,a (cid:101) ψ = 0. We denote T − ψ := (cid:101) ψ and refer to it as the time inverse of ψ .When the charges I (cid:96) [ ψ ] are zero, for example, when the initial data for ψ is compactly supported, it turnsout that the charges corresponding to (cid:101) ψ are finite, i.e. I (cid:96) [ (cid:101) ψ ] < ∞ or (cid:96) = 0 , , I (cid:96) [ (cid:101) ψ ] (cid:54) = 0. Furthermore, we can express I (cid:96) [ (cid:101) ψ ] purely in terms of initial datafor ψ , see Section 1.1.2 and Section 9.4.We construct the initial data leading to T − ψ by solving the equation: L (cid:101) ψ = F [ ψ ] . As explained in Section 1.4.2, the operator L is elliptic outside the ergoregion. Constructing L − requires theuse of spacelike redshift multipliers and is reminiscent of the strategy for constructing resolvent operators in[War15]. Note that in the a = 0 setting, the spherical modes ψ (cid:96) can be treated independently, so obtaining L − for a single mode amounts to solving a standard ODE, see [AAG21]. When a (cid:54) = 0, however, it isneccessary to construct L − acting on the full solution before projecting to a fixed spherical harmonic mode.In Section 9, we construct L − and established r -decay properties T − ψ | Σ required to apply the timedecay arguments sketched in the sections when ψ is replaced with T − ψ .We then obtain in particular the decay estimates: | ψ | = | T (cid:101) ψ | (cid:46) (1 + τ ) − (cid:15) , | rψ | = | rT (cid:101) ψ | (cid:46) (1 + τ ) − (cid:15) | r − ψ | = | r − T (cid:101) ψ | (cid:46) (1 + τ ) − − k + (cid:15) , | rψ | = | rT (cid:101) ψ | (cid:46) (1 + τ ) − − k + (cid:15) | ψ | = | T (cid:101) ψ | (cid:46) (1 + τ ) − − k + (cid:15) , | rψ | = | rT (cid:101) ψ | (cid:46) (1 + τ ) − − k + (cid:15) | ψ ≥ | = | T (cid:101) ψ ≥ | (cid:46) (1 + τ ) − + (cid:15) , | rψ ≥ | = | rT (cid:101) ψ ≥ | (cid:46) (1 + τ ) − + (cid:15) , which are sharp up to arbitrarily small (cid:15) . In order to derive the precise leading order terms of ψ and ψ ≥ (cid:96) with (cid:96) = 1 ,
2, we need to go beyond the upper bound estimates outlined in the previous sections.We extend the method developed in [AAG18b] in a spherically symmetric setting to the setting of Kerrspacetimes. The strategy is to derive the leading-order asymptotics for T k (cid:101) ψ (with arbitrary k ) by: 1)extending the conservation and non-vanishing property of r P (cid:96) from r = to a suitable far-away spacetimeregion to obtain the late-time asymptotics for rψ (cid:96) there, and 2) propagating these asymptotics to the restof the spacetime by using that X (cid:96) (cid:101) ψ (cid:96) and T X (cid:96) (cid:101) ψ (cid:96) both decay faster in time than X (cid:96) (cid:101) ψ (cid:96) when (cid:96) = 0 , X (cid:101) ψ and T (cid:101) ψ decay faster than (cid:101) ψ , so we can simplyintegrate (multiply times) in the X direction. This allows us to prove Theorem 1.2. Theorem 1.1 then followssimply from the observation that ψ = T (cid:101) ψ .We refer to a more expansive outline of a similar strategy in the spherically symmetric setting in [AAG18b,AAG21]. The key new difficulty in Kerr in this step is the coupling between spherical harmonic modes ψ (cid:96) .See Section 1.1.4 for an outline of this phenomenon. More concretely, obtaining late-time asymptotics in thefar-away region for ψ requires deriving first the late-time asymptotics of ψ and plugging those in suitablyin the derivation for ψ . A precise derivation of the late-time asymptotics for the (cid:96) = 0 , , .5 Outline of the paper We provide below an outline of the remaining sections in the paper. • In Section 2 we introduce the Kerr geometry and the relevant vector fields and spacetime foliations.We moreover state a systematic method for integrating by parts, which we apply to derive all theenergy estimates in the paper. Finally, we state and derive some key properties regarding sphericalharmonics. • In Section 3 we provide various different forms of the wave equation 1.1 and state the preliminaryenergy boundedness and integrated energy decay estimates, which we utilise in the rest of the paper. • In Section 4 we define the Newman–Penrose charges for the (cid:96) = 0 , , • In Section 5 we derive the main hierarchies of r p -weighted energy estimates. • In Section 6 we use the r p -weighted energy estimates from Section 5 to derive energy decay estimates. • In Section 7, we derive a hierarchy of weighted elliptic estimates. • In Section 8, we convert the (weighted) energy decay estimates into pointwise decay estimates. • In Section 9, we construct the time inverse T − ψ and establish its decay properties along Σ . Wemoreover define the time-inverted Newman–Penrose charges and derive explicit expressions for themin terms of initial data for ψ on Σ . • Finally, in Sections 10–12, we use the pointwise estimates from Section 8, applied moreover to both ψ and the time inverse T − ψ constructed in Section 9, to derive the precise late-time asymptotics for ψ , ψ and ψ and hence determine the precise form of the corresponding late-time tails. The second author (S.A.) acknowledges support through the NSERC grant 502581 and the Ontario EarlyResearcher Award.
In this section, we introduce the Kerr family of spacetimes and the spacetime foliations of interest. Wemoreover derive a systematic method of integrating by parts to derive energy estimates, and we prove somerelevant properties of spherical harmonic decompositions.
We introduce in this section the 2-parameter family of Kerr black hole exteriors ( M M,a , g
M,a ) in ingoingKerr coordinates .Let M M,a = R v × [ r + , ∞ ) r × ( S ) θ,ϕ ∗ be a manifold-with-boundary equipped with a coordinate chart( v, r, θ, ϕ ∗ ) that is global, excluding the standard degeneration of spherical coordinates ( θ, ϕ ∗ ) on S . Let g M,a denote the Lorentzian metric: g M,a = − ρ − (cid:0) ∆ − a sin θ (cid:1) dv + 2 dvdr − M arρ − sin θdvdϕ ∗ − a sin θdrdϕ ∗ + ρ dθ + ρ − (( r + a ) − a ∆ sin θ ) sin θdϕ ∗ , (2.1)where ∆ = r − M r + a = ( r + − r )( r − r − ) ,r + , − = M ± (cid:112) M − a , = r + a cos θ. We will restrict our considerations to sub-extremal Kerr spacetimes by assuming that | a | < M , with M > We denote the boundary of M M,a with H + := { r = r + } and will refer to H + as the (future) eventhorizon of the spacetime. The level sets S v (cid:48) ,r (cid:48) = { v = v (cid:48) , r = r (cid:48) } are 2-surfaces diffeomorphic to S , whichare known as the Boyer–Lindquist spheres . Note that it is possible to extend the spacetimes ( M M,a , g
M,a )smoothly by attaching a black hole region where r − < r < r + . This extension will not be necessary in thepresent paper.On the manifold M M,a \ H + , one can alternatively consider the standard Boyer–Lindquist coordinates ( t, r, θ, ϕ ), where t = v − r ∗ ,ϕ = ϕ ∗ + (cid:90) ∞ r a ∆ dr (cid:48) mod 2 π, where r ∗ : ( r + , ∞ ) → R is a solution to dr ∗ dr = r + a ∆ , defined uniquely up to a constant.We introduce also the function u : M M,a \ H + → R , with u = v − r ∗ = t − r ∗ . It will be convenient to state moreover the inverse metric g − M,a : g − M,a = a ρ − sin θ∂ v ⊗ ∂ v + ρ − ( r + a )[ ∂ v ⊗ ∂ r + ∂ r ⊗ ∂ v ] + ∆ ρ − ∂ r ⊗ ∂ r + aρ − [( ∂ v + ∂ r ) ⊗ ∂ ϕ ∗ + ∂ ϕ ∗ ⊗ ( ∂ v + ∂ r )] + ρ − [ ∂ θ ⊗ ∂ θ + sin − θ∂ ϕ ∗ ⊗ ∂ ϕ ∗ ] . (2.2) The following vector fields are Killing vector fields with respect to g M,a : T = ∂ v , Φ = ∂ ϕ ∗ ,K = T + ω + Φ , where ω + = ar + a may be interpreted as the “angular velocity of the Kerr black hole”. Furthermore, K istangent to H + .It can easily be verified that the vector field T is timelike for all ( v, r, θ, ϕ ∗ ) satisfying the condition:∆ − a sin θ > . The subset { ∆ − a sin θ < } , where T fails to be causal is called the ergoregion . Note that { ∆ − a sin θ < } ⊂ { r < M } . We will introduce the following additional vector fields, which will play an important role in later analysis: Y = ∂ r ,L = − ∆2( r + a ) Y, Extremal Kerr spacetimes correspond to the subclass of spacetimes satisfying | a | = M . = T + ar + a Φ − L. The vector fields L and L are null and define the principal null directions .Note moreover the following identities L ( v ) = 1 ,L ( u ) = 1 ,g ( L, L ) = − ∆ ρ r + a ) , [ L, L ] = ar ∆( r + a ) Φ ,L ( ϕ ∗ ) = 0 ,L ( ϕ ∗ ) = ar + a . We introduce moreover the horizon azimuthal angle ϕ H + = ϕ ∗ − ω + v, (2.3)which satisfies K ( ϕ H + ) = 0, i.e. it is constant along the null generators of the future event horizon H + .Finally, we introduce some notation for angular derivatives. Consider S equipped with standard sphericalcoordinates ( θ, ϕ ∗ ). We denote with / ∇ S the covariant derivative (Levi–Civita connection) on the unit roundsphere S . The corresponding Laplacian / ∆ S takes the following form in ( θ, ϕ ∗ ) coordinates: / ∆ S ( · ) = 1sin θ ∂ θ (sin θ∂ θ ( · )) + 1sin θ ∂ ϕ ∗ ( · ) . It will be convenient to keep track of decay with respect to the coordinate r via the following notation: let f : [ r + , ∞ ) r → R and k ∈ Z . We write f = O ( r − k ) when there exists a constant C >
0, depending only on M and a such that | f ( r ) | ≤ Cr − k for all r ∈ [ r + , ∞ ). We write f = O N ( r − k ) when moreover d n fdx n = O ( r − k − n )for all 0 ≤ n ≤ N , where f is assumed to be suitably regular. Finally, we write f = O ∞ ( r − k ) if f = O N ( r − k )for all N ∈ N .Consider the following time function on M M,a : τ = v − (cid:90) rr + h ( r (cid:48) ) dr (cid:48) − v , where v ∈ R > and h : [ r + , ∞ ) → R is a smooth, non-negative function, such that2( r + a )∆ − h ( r ) = O ( r − ) , (2.4) h ( r ) (cid:18) r + a )∆ − h ( r ) (cid:19) > a ∆ − sin θ. (2.5)By construction, T ( τ ) = 1.Let ˆ h : [0 , r + ] x → R , withˆ h ( x ) = 2( x − + a ) − h ( x − )( x − − M x − + a ) . Then ˆ h ( x ) = O ( x ) by (2.4), where we take O ( x k ), O N ( x k ) and O ∞ ( x k ) to have the same meaning asabove with x replacing r .We will make the following additional assumptions for the sake of convenience:ˆ h ∈ C ∞ ([0 , r − ]) , (2.6)18 h (0) =: h > . (2.7)Smoothness of ˆ h is not a necessary assumption and the main results below are still valid with a lowerregularity assumption on ˆ h , in particular they hold also when ˆ h ∈ C N ([0 , r − ]), with N ∈ N chosensuitably large.Define the following 1-parameter family of hypersurfaces:Σ τ (cid:48) := { τ = τ (cid:48) } . We denote Σ := Σ and R = J + (Σ) ∩ J − ( H + ).Let R > M be an arbitrarily large radius. Then we introduce the notation: D τ τ := (cid:91) τ ≤ τ ≤ τ Σ τ ∩ { r ≥ R } . We moreover denote N τ = Σ τ ∩ { r ≥ R } .Let ρ = r | Σ . Then ( ρ , θ, ϕ ∗ ) defines a coordinate chart on Σ and ( τ, ρ , θ, ϕ ∗ ) defines a coordinate charton R . We moreover introduce the notation: X := ( ∂ ρ ) Σ τ = Y + hT. Note that we can express the metric g M,a as follows in ( τ, ρ , θ, ϕ ∗ ) coordinates: g M,a = − ρ − (cid:0) ∆ − a sin θ (cid:1) dτ − (cid:0) ρ − h (∆ − a sin θ ) − (cid:1) dτ d ρ + (cid:0) ρ − h (∆ − a sin θ ) (cid:1) ρ − hd ρ − M arρ − sin θdτ dϕ ∗ − a + 2 M arρ − h ) sin θd ρ dϕ ∗ + ρ dθ + ρ − (( r + a ) − a ∆ sin θ ) sin θdϕ ∗ . We further define the function s = v − r ∗ + (cid:90) rr + h ( r (cid:48) ) dr (cid:48) + v = τ − r ∗ + 2 (cid:90) rr + h ( r (cid:48) ) dr (cid:48) + 2 v . and consider the corresponding level sets I s (cid:48) := { s = s (cid:48) } ∩ R . We can equip the level sets I s with the coordinate chart ( τ, θ, ϕ ∗ ), and we express:( ∂ τ ) I s = T −
12 1 h − r + a ∆ X, = − r + a )∆ − hh − r + a ∆ T −
12 ∆ r + a + (∆ h − r + a )) Y. See Figure 2 below for the relevant pictorial representations.19igure 2: A 2-dimensional representation of the spacetime M M,a and the hypersurfaces Σ τ i , I s i , with i = 1 , τ < τ , s < s . Each point in the picture represents a Boyer–Lindquist sphere S v,r and the hypersurface I + is depicted at a finite distance.The lemma below establishes the key causual properties of the hypersurfaces Σ τ , which are also repre-sented pictorially in Figure 2. Lemma 2.1.
The 1-parameter family { Σ τ } τ ≥ and the level sets I s satisfy the following properties:1. Σ τ and I s (if non-empty) are spacelike for all τ, s ≥ ,2. Σ τ is isometric to Σ τ (cid:48) for all τ, τ (cid:48) ≥ ,3. Σ τ ∩ H + = { v = τ + v } ∩ H + = S τ + v ,r + ,4. For v sufficiently large, depending on h , there exists u > such that Σ ⊂ J + ( { u = u } ) .Proof. Property 3. follows directly from the definition of τ . Property 2. follows from the fact that each Σ τ can be obtained from Σ along the flow corresponding to the Killing vector field T .By (2.2), it follows that g − ( dτ, dτ ) = g − ( dv − hdr, dv − hdr ) = a ρ − sin θ − ∆ ρ − h (cid:18) r + a )∆ − h (cid:19) , so by (2.5), Property 1. must also hold for Σ τ . The same conclusion follows for I s : g − ( ds, ds ) = g − ( dv +( h − r + a )∆ − ) dr, dv +( h − r + a )∆ − ) dr ) = a ρ − sin θ − ∆ ρ − h (cid:18) r + a )∆ − h (cid:19) < . In order to infer that property 4. holds, we first observe that u | Σ ( r ) = v | Σ ( r ) − r ∗ ( r ) = v + (cid:90) rr + h ( r (cid:48) ) dr (cid:48) − r ∗ ( r ) . so u | Σ (2 r + ) = v − (cid:82) r + r + h ( r (cid:48) ) dr (cid:48) − r ∗ (2 r + ) > v for v suitably large. Then for r ≥ r + u | Σ ( r ) = u | Σ (2 r + ) + (cid:90) r r + ( h − r (cid:48) + a )∆ )( r (cid:48) ) dr (cid:48) → u ∈ R as r → ∞ , as the integral above is well-defined by the asymptotics in (2.4). For v suitably large, we can additionallyensure positivity of u . 20et n τ denote the future-directed unit normal vector field with respect to Σ τ . We may extend n τ as avector field on R and introduce the vector field N as follows via a smooth cut-off function: N = n τ when r ≤ M , (2.8) N = T when r ≥ M . (2.9)such that N is timelike everywhere. It follows moreover that exists a constant c = c ( M, a ) >
0, such that g ( N, N ) ≤ − c .From Lemma 2.1 it follows that { Σ τ } τ ≥ defines a foliation of R by isometric spacelike hypersurfaces.We can express dµ , the natural volume form with respect to g M,a , as follows: dµ := (cid:112) det g M,a dvdrdθdϕ ∗ = ρ dωdvdr = ρ dωdτ d ρ , with dω = sin θdθdϕ ∗ the natural volume form on S .Various estimates on the induced volume forms on Σ τ and I s that are relevant in the context of divergencetheorems are contained in Section 2.1.2 of [DHR19a]. For the sake of completeness and notational convention,we prove these estimates in Lemma 2.2 below. Lemma 2.2.
The volume forms dµ τ := (cid:112) det g | Σ τ d ρ dθdϕ ∗ and dµ I s := (cid:112) det g | I s dτ dθdϕ ∗ satisfy the fol-lowing properties: (i) We can express dµ τ = m ( ρ , θ ) rdωd ρ ,dµ I s = m ( ρ , θ ) rdωdτ in { r ≥ r + } , with m i : [ r + , ∞ ) × (0 , π ) → R , i = 1 , , smooth functions satisfying: c h ≤ m i ≤ C h , for positive constants c h , C h , depending on M , a and the choice of h . (ii) We can moreover express: − g ( L, n τ ) (cid:112) det g | Σ τ = m L, ( ρ , θ ) sin θ, − g (cid:18) r + a ∆ L, n τ (cid:19) (cid:112) det g | Σ τ = m L, ( ρ , θ ) r sin θ, − g ( L, n I s ) (cid:112) det g | I s = m L, ( ρ , θ ) r sin θ, − g (cid:18) r + a ∆ L, n I s (cid:19) (cid:112) det g | I s = m L, ( ρ , θ ) sin θ, with m L,i , m
L,i : [ r + , ∞ ) × (0 , π ) → R , i = 1 , , smooth functions satisfying: c h ≤ m L,i , m
L,i ≤ C h , for c h , C h > . (iii) The following divergence identities hold: ∇ α (cid:18) ρ − r + a ∆ L α (cid:19) = 0 , ∇ α (cid:18) ρ − r + a ∆ L α (cid:19) = 0 . We used here that g ( T, T ) < r > M , i.e. outside the ergoregion. roof. Property (i) results from the observation that g ( X, X ) ∼ r − and g (( ∂ τ ) I s , ( ∂ τ ) I s ) ∼ r − .In order to obtain property (ii), note first that we can express: g − = g − ( dτ (cid:93) , dτ (cid:93) ) dτ (cid:93) ⊗ dτ (cid:93) + g − | Σ τ , so we have that with respect to ( τ, ρ , θ, ϕ ∗ ) coordinates:det g = det g | Σ τ g − ( dτ (cid:93) , dτ (cid:93) )and we must therefore have that − (cid:112) det g | Σ τ n τ = ρ sin θdτ (cid:93) . (2.10)Furthermore, ρ dτ (cid:93) = [ a sin θ − ( r + a ) h ] T + [ r + a − ∆ h ] Y + a (1 − h )Φ ,ρ ds (cid:93) = [ a sin θ − ( r + a )∆ − ˆ h ( r − )] T + [ r + a − ˆ h ( r − )] Y + a (1 − h )Φ , where ˆ h ( r − ) = 2( r + a ) − ∆ h ( r ), so after a straightforward computation, we obtain g (cid:18) r + a ∆ L, ρ dτ (cid:93) (cid:19) = 12 ( r + a ) h + (1 − h ) a sin θ = r + O ( r ) ,g (cid:0) L, ρ dτ (cid:93) (cid:1) = 12 ˆ h (1 /r ) + O ( r − ) . Similar expressions involving ( ds ) (cid:93) follow after replacing h ( r ) with ∆ − ˆ h ( r − ) and vice versa. Property (ii)then follows from the assumption that ˆ h (0) = h > conformal radial coordinate x , defined as follows in R : x = 1 ρ . We now define the conformally rescaled metric ˆ g M,a in the region R = [0 , ∞ ) τ × (0 , r − ] x × S :ˆ g M,a := r − g M,a = x g M,a . Note thatˆ g M,a = − ( x + O ∞ ( x )) dτ + 2(1 + O ∞ ( x )) dτ dx + O ∞ ( x ) dx + O ∞ ( x ) sin θdτ dϕ ∗ − a + O ∞ ( x )) dxdϕ ∗ + (1 + O ∞ ( x )) dθ + (1 + O ∞ ( x )) sin θdϕ ∗ , and the above metric components are smooth functions with respect to ( τ, x, θ, ϕ ∗ ). Furthermore, ˆ g M,a canbe smoothly extended to the manifold-with-boundary: (cid:98) R = [0 , ∞ ) τ × [0 , r − ] x × S . We refer to the boundary I + = [0 , ∞ ) τ × { } x × S as future null infinity . It follows immediately that I + is a null hypersurface with respect to ˆ g M,a . Further-more, it is straightforward to see that x | I s ( τ ) → s → ∞ .We moreover denote with (cid:98) Σ τ := { τ } τ × [0 , r − ] x × S the extension of Σ τ to (cid:98) R . 22 .4 Additional notation We will denote with H k (Σ τ ), with k ∈ N , the standard Sobelev spaces with respect to the natural volumeform corresponding to the induced metric g M,a | Σ τ . Similarly, will denote with H k ( (cid:98) Σ τ ), with k ∈ N , thestandard Sobelev spaces with respect to the natural volume form corresponding to the induced conformalmetric ˆ g M,a | Σ τ .Similarly, we use H k ( S ) to denote Sobolev spaces of functions on S with respect to the standard volumeform.We will frequently use the notations c and C to indicate constants appearing at the right-hand side ofan inequality. When the notation C or c appears in an inequality, we will make use of the following “algebraof constants”: C + C = CC = C, c + c = cc = c. in order to avoid the introduction of additional notation to denote different constants in an estimate.We will use the notation f ∼ g , with f, g two non-negative definite expressions to mean: cg ≤ f ≤ Cg, with constants 0 < c < C that depend only on a , M and the function h that determines the foliation { Σ τ } . We appeal to the properties of Σ τ and I s established in Lemma 2.2 to derive a systematic method ofintegrating by parts; see also Remark 5.1 of [DHR19a]. Lemma 2.3.
Let F L , F L be smooth functions on M M,a and let F / ∇ be a smooth vector field on S v,r extendedto M M,a , such that the following identity holds: L ( F L ) + L ( F L ) + / div S F / ∇ + Φ( F Φ ) + J = 0 , (2.11) where / div S denotes the divergence operator with respect to the round metric on S . (i) Then div (cid:20)(cid:18) ρ − r + a ∆ F L (cid:19) L + (cid:18) ρ − r + a ∆ F L (cid:19) L (cid:21) + ρ − r + a ∆ J + ρ − r + a ∆ ( / div S F / ∇ + Φ( F Φ )) = 0 . The following integral identity holds: (cid:90) Σ τ ∩{ s ≤ S } ρ − F L (cid:112) det g Σ τ g (cid:18) r + a ∆ L, − n Σ τ (cid:19) + ρ − F L (cid:112) det g Σ τ g (cid:18) r + a ∆ L, − n Σ τ (cid:19) d ρ dθdϕ ∗ + (cid:90) H + ∩{ τ ≤ τ ≤ τ } F L dωdτ + (cid:90) I S ∩{ τ ≤ τ ≤ τ } ρ − F L (cid:112) det g I S g (cid:18) r + a ∆ L, − n I s (cid:19) + ρ − F L (cid:112) det g I S g (cid:18) r + a ∆ L, − n I S (cid:19) dτ dθdϕ ∗ + (cid:90) D τ τ ∩{ s ≤ S } ρ − r + a ∆ J dµ = (cid:90) Σ τ ∩{ s ≤ S } ρ − F L (cid:112) det g Σ τ g (cid:18) r + a ∆ L, − n Σ τ (cid:19) + ρ − F L (cid:112) det g Σ τ g (cid:18) r + a ∆ L, − n Σ τ (cid:19) d ρ dθdϕ ∗ . (ii) If F L , F L , F / ∇ are non-negative definite, then we have that: (cid:90) Σ τ ∩{ s ≤ S } (cid:0) F L + ∆ − F L (cid:1) dωd ρ + (cid:90) I S ∩{ τ ≤ τ ≤ τ } (cid:0) F L + r − F L (cid:1) dωdτ + (cid:90) τ τ (cid:34)(cid:90) Σ τ ∩{ s ≤ S } r + a ∆ J dωd ρ (cid:35) dτ + (cid:90) H + ∩{ τ ≤ τ ≤ τ } F L dωdτ ∼ (cid:90) Σ τ ∩{ s ≤ S } (cid:0) F L + ∆ − F L (cid:1) dωd ρ . roof. Part (i) follows directly from Stokes’ Theorem combined with (iii) of Lemma 2.2. Part (ii) follows byapplying additionally the remaining identities in Lemma 2.2.
Let (cid:96) ∈ N and consider the following projection operators π (cid:96) : L ( S ) → L ( S ) ,π (cid:96) f = f (cid:96) := (cid:96) (cid:88) m = − (cid:96) f (cid:96),m Y (cid:96),m ( θ, ϕ ∗ ) , with f (cid:96),m ∈ C and Y (cid:96),m ( θ, ϕ ∗ ), m = − (cid:96), . . . , (cid:96) spherical harmonics with angular momentum (cid:96) , with respectto the polar angle θ and the azimuthal angle ϕ ∗ .Note that ∞ (cid:88) (cid:96) =0 (cid:90) S f (cid:96) dω = (cid:90) S f dω. The operator π (cid:96) is well-defined on the function space C ∞ (Σ τ ), where we interpret π (cid:96) as acting on therestrictions of functions in C ∞ (Σ τ ) to functions on the Boyer–Lindquist spheres foliating Σ τ , which we coverwith angular coordinates ( θ, ϕ ∗ ). Since π (cid:96) is a bounded linear operator with respect to || · || L (Σ τ ) , thefollowing extension is also well-defined: π (cid:96) : L (Σ τ ) → L (Σ τ ) . We will introduce the following additional notation: let L ∈ N and f ∈ L (Σ τ ), then f ≤ L := L (cid:88) (cid:96) =0 π (cid:96) f,f ≥ L +1 := f − f ≤ L . We will moreover need to investigate how the projection operators π (cid:96) act on product functions of theform sin θf , with f ∈ L ( S ). Lemma 2.4.
Let f ∈ L ( S ) . Then there exist numerical constants N (cid:96),m, − , N (cid:96),m, , N (cid:96),m, +2 , such that π (cid:96) (sin θf ) = (cid:96) (cid:88) m = − (cid:96) ( N (cid:96),m, − f (cid:96) − ,m + N (cid:96),m, f (cid:96),m + N (cid:96),m, +2 f (cid:96) +2 ,m ) Y (cid:96),m ( θ, ϕ ∗ ) , (2.12) with N (cid:96),m, and N (cid:96),m, +2 non-vanishing and N (cid:96),m, − non-vanishing if and only if | m | ≤ (cid:96) − .In particular, π (sin θπ ( f )) = − (cid:114) π π ( f ) Y , ( θ ) . (2.13) Proof.
We can express: π (cid:96) (sin θf ) = (cid:96) (cid:88) m = − (cid:96) ∞ (cid:88) (cid:96) (cid:48) = | m | f (cid:96) (cid:48) ,m (cid:90) S sin θY (cid:96) (cid:48) ,m Y (cid:96),m sin θdθdϕ Y (cid:96),m . Note that for given (cid:96) and (cid:96) (cid:48) , the integral (cid:90) S sin θY (cid:96) (cid:48) ,m Y (cid:96),m sin θdθdϕ = 0if and only if (cid:90) − (1 − x ) P m(cid:96) (cid:48) ( x ) P m(cid:96) ( x ) dx = 0 , (2.14)24here P m(cid:96) and P m(cid:96) (cid:48) denote associated Legendre polynomials. By the definition of Legendre polynomials, itfollows immediately that the above integral vanishes if (cid:96) (cid:48) − (cid:96) is odd and is non-vanishing when (cid:96) = (cid:96) (cid:48) . Itremains to consider the cases when (cid:96) (cid:48) − (cid:96) is even and non-zero. Without loss of generality, we can assumethat 0 ≤ m ≤ (cid:96) (cid:48) and (cid:96) ≥ (cid:96) (cid:48) + 2.We use the following standard recursive relations between associated Legendre polynomials (see forexample Chapter 8 of [AS72]): let l (cid:54) = 0, then (cid:112) − x P ml = − (2 l + 1) − [( l − m + 1)( l − m + 2) P m − l +1 − ( l + m − l + m ) P m − l − ] for m ≥ , (cid:112) − x P ml = − (2 l + 1) − [ P m +1 l +1 − P m +1 l − ] for m ≤ l − . to write(2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(1 − x ) P m(cid:96) (cid:48) P m(cid:96) = (cid:2) ( (cid:96) − m + 1)( (cid:96) − m + 2) P m − (cid:96) +1 − ( (cid:96) + m − (cid:96) + m ) P m − (cid:96) − (cid:3) · (cid:2) ( (cid:96) (cid:48) − m + 1)( (cid:96) (cid:48) − m + 2) P m − (cid:96) (cid:48) +1 − ( (cid:96) (cid:48) + m − (cid:96) (cid:48) + m ) P m − (cid:96) (cid:48) − (cid:3) , when (cid:96) (cid:48) (cid:54) = 0 and m ≥
1. For (cid:96) (cid:48) (cid:54) = 0 and m = 0, we obtain instead(2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(1 − x ) P (cid:96) (cid:48) P (cid:96) = [ P (cid:96) +1 − P (cid:96) − ][ P (cid:96) (cid:48) +1 − P (cid:96) (cid:48) − ] . The integral (2.14) corresponding to the above cases is non-vanishing if and only if (cid:96) = (cid:96) (cid:48) or | (cid:96) − (cid:96) (cid:48) | = 2.It only remains to consider the case: (cid:96) (cid:48) = m = 0 and (cid:96) ≥
2, for which it follows by integrating by partsthat (cid:90) − (1 − x ) P (cid:96) P dx = 2 − (cid:96) ( (cid:96) !) − (cid:90) − (1 − x ) d (cid:96) dx (cid:96) (( x − (cid:96) ) dx is non-vanishing if and only if (cid:96) = 2. The expression (2.12) then follows and (2.13) can easily be computedexplicitly by keeping track of the appropriate normalization constants.In the lemma below, we state various Poincar´e-type inequalities on S . Lemma 2.5.
Let f ∈ H ( S ) . Then: (cid:90) S | / ∇ S f ≥ (cid:96) | dω ≥ (cid:96) ( (cid:96) + 1) (cid:90) S f ≥ (cid:96) dω, (2.15) (cid:90) S | / ∇ S f (cid:96) | dω = (cid:96) ( (cid:96) + 1) (cid:90) S f (cid:96) dω, (2.16) (cid:90) S f (cid:96) + | / ∇ S f (cid:96) | + | / ∇ S f (cid:96) | dω ≤ (1 + (cid:96) ( (cid:96) + 1) + (cid:96) ( (cid:96) + 1) ) (cid:90) S f (cid:96) dω. (2.17) Furthermore, let N ∈ N and assume that f ∈ H max { N, } ( S ) . Then there exists constants < c < C ,depending on N , such that c (cid:90) S | / ∇ s S / ∆ N − s S f | dω ≤ N (cid:88) n =0 (cid:90) S | / ∇ n S f | dω ≤ C (cid:90) S | / ∇ s S / ∆ N − s S f | dω, (2.18) where s = 0 if N is even and s = 1 if N is odd.Proof. The inequalities (2.15)–(2.17) follow by decomposing f ≥ (cid:96) = (cid:80) ∞ (cid:96) (cid:48) = (cid:96) f (cid:96) (cid:48) , using that / ∆ S f (cid:96) = − (cid:96) ( (cid:96) + 1) f (cid:96) and integrating by parts on S .We obtain (2.18) by integrating by parts with respect to covariant derivation on S .25 .7 Hardy inequalities along hyperboloidal hypersurfaces Throughout this paper we will frequently appeal to Hardy inequalities, in the form of the estimates in Lemma2.6 below, to estimate lower-order weighted derivatives in terms of higher-order weighted derivatives.
Lemma 2.6.
Let ≤ a < b and let f ∈ C ([ a, b ]) , then for p ∈ R \ {− } : (cid:90) ba x p f ( x ) dx ≤ p + 1) − (cid:90) ba x p +2 (cid:18) dfdx (cid:19) ( x ) dx + 2( p + 1) − (cid:2) ( x p +1 f )( b ) − ( x p +1 f )( a ) (cid:3) . (2.19) Let r + ≤ r < r and let f ∈ C (Σ τ ) , then for p ∈ R \ {− } : (cid:90) Σ τ ∩{ r ≤ ρ ≤ r } r p f dωd ρ ≤ p + 1) − (cid:90) Σ τ ∩{ r ≤ ρ ≤ r } r p +2 ( Xf ) dωd ρ + 2( p + 1) − (cid:34)(cid:90) S τ,r r p +12 f dω − (cid:90) S τ,r r p +11 f dω (cid:35) . (2.20) And for r + < r < r and f ∈ C ( R ) , there exists a constant C = C ( M, a, r ) > such that (cid:90) Σ τ ∩{ r ≤ ρ ≤ r } r p f dωd ρ ≤ C ( p + 1) − (cid:90) Σ τ ∩{ r ≤ ρ ≤ r } r p +2 ( Lf ) + r p − ( T f ) + r p − (Φ f ) dωd ρ + 2( p + 1) − (cid:34)(cid:90) S τ,r r p +12 f dω − (cid:90) S τ,r r p +11 f dω (cid:35) . (2.21) Proof.
The estimate (2.19) follows from integrating ddx (cid:0) x p +1 f ) (cid:1) and applying the fundamental theorem ofcalculus and then immediately implies (2.20). We conclude (2.21) from (2.20) combined with the followingrelation between L and X : L = ∆2( r + a ) X + O ( r − ) T + O ( r − )Φ . We recall and derive in this section some preliminary estimates for the geometric wave equation on Kerrspacetimes that will form important ingredients for the estimates in subsequent sections.
In the lemma below we represent the wave equation (cid:3) g M,a ψ = 0 in terms of the vector fields T, Y, Φ , X, L and L . Lemma 3.1.
Let ψ ∈ C ∞ ( R → C ) be a solution to (cid:3) g M,a ψ := 1 (cid:112) det g M,a ∂ α (cid:16)(cid:112) det g M,a ( g − M,a ) αβ ∂ β ψ (cid:17) = 0 . (3.1) Then ψ satisfies the following equation: a sin θT ψ + 2( r + a ) T Y ψ + Y (∆ Y ψ ) + 2 aT Φ ψ + 2 aY Φ ψ + 2 rT ψ + / ∆ S ψ. (3.2) We can reformulate (3.2) as follows: X (∆ Xψ ) + 2 aX Φ ψ + / ∆ S ψ = 2[ h ∆ − ( r + a )] XT ψ + [(∆ h ) (cid:48) − r ] T ψ + [2 h ( r + a ) − h ∆ − a sin θ ] T ψ + 2 a ( h − T Φ ψ. (3.3)26 urthermore, the rescaled quantity φ := √ r + a ψ satisfies the equations: a sin θT φ + 2 aT Φ φ + 2 aY Φ φ + 2( r + a ) T Y φ + ( r + a ) Y (cid:18) ∆ r + a Y φ (cid:19) + / ∆ S φ (3.4) − a rr + a Φ φ − (cid:112) r + a ddr ( r ( r + a ) − / ∆) φ, LLφ = a sin θ ∆( r + a ) T φ + 2 a ∆( r + a ) T Φ φ + ∆( r + a ) / ∆ S φ + 2 a r ∆( r + a ) Φ φ (3.5) − r ddr (cid:0) r ∆ ( r + a ) − (cid:1) φ. Proof.
The above identities follow from straightforward computations that use the expressions for g M,a and g − M,a in (2.1) and (2.2), respectively, and the definitions of the vector fields T, Φ , Y, X, L, L in Section 2.2.By rewriting (3.3) in terms of the conformal coordinates ( τ, x, θ, ϕ ∗ ), we obtain an equation for φ (definedin the statement of Lemma 3.1) with smooth coefficients with respect to the differentiable structure on (cid:98) R : Corollary 3.2.
Let ψ ∈ C ∞ ( R → C ) be a solution to (3.1) . Then φ satisfies the following equation in ( τ, x, θ, ϕ ∗ ) coordinates: a x ) ∂ x (cid:18) ∆( r + a ) x ∂ x φ (cid:19) + 2[1 + a x − x ˆ h ( x )] ∂ τ ∂ x φ + (cid:34) − r + a ) − ˆ h ∆ ˆ h + a sin θ (cid:35) ∂ τ φ + / ∆ S φ + 2 a (cid:34) − r + a ) − ˆ h ∆ (cid:35) ∂ τ ∂ ϕ ∗ φ − ax ∂ x ∂ ϕ ∗ φ − (1 + a x ) ddx (cid:32) ˆ hx a x (cid:33) ∂ τ φ − ax a x ∂ ϕ ∗ φ + (1 + a x ) ddx (cid:18) (1 − M x + a )(1 + a x ) (cid:19) φ. (3.6) Remark 3.1.
When viewed as a function on the extended manifold (cid:98) R , the restriction φ | I + is known as the(Friedlander) radiation field . We introduce the following higher-order quantities in { r > r + } : φ ( n ) = (2( r + a ) ∆ − L ) n φ. Observe that we can alternatively express φ ( n ) = ( − n ( ∂ x ) n φ in ( u, x, θ, ϕ ∗ ) coordinates, so φ ( n ) may beconsidered natural higher-order analogues of φ from the point of view of the conformal spacetime ( (cid:98) R , ˆ g M,a ). Proposition 3.3.
Let ψ ∈ C ∞ ( R → C ) be a solution to (3.1) . Then φ ( n ) satisfies the following equation LLφ ( n ) = a sin θ ∆( r + a ) T φ ( n ) + 2 a ∆( r + a ) T Φ φ ( n ) + 2(1 + 2 n ) a r ∆( r + a ) Φ φ ( n ) + ∆( r + a ) − / ∆ S φ ( n ) + ∆( r + a ) − [ n ( n + 1) + O ∞ ( r − )] φ ( n ) − [4 nr − + O ∞ ( r − )] Lφ ( n ) + a n − (cid:88) k =0 O ∞ ( r − )Φ φ ( k ) + n n − (cid:88) k =0 O ∞ ( r − ) φ ( k ) . (3.7) Proof.
We will prove (3.7) by induction. First, note that the n = 0 case follows from (3.5). Now suppose(3.7) holds for some n ∈ N . Then4 LLφ ( n +1) = 4 L (cid:16) L (cid:16) − ( r + a ) Lφ ( n ) (cid:17)(cid:17) + 4[ L, L ] φ ( n +1) = 8 L (cid:16) ∆ − ( r + a ) LLφ ( n ) (cid:17) − L (cid:18) ddr (∆ − ( r + a ) )∆( r + a ) − Lφ ( n ) (cid:19) + 4 ar ∆( r + a ) Φ φ ( n +1) .
27y applying (3.7), we obtain8 L (cid:16) ∆ − ( r + a ) LLφ ( n ) (cid:17) = a sin θ ∆( r + a ) T φ ( n +1) + 2 a ∆( r + a ) T Φ φ ( n +1) + ∆( r + a ) − / ∆ S φ ( n +1) + 4(1 − n ) a r ∆( r + a ) Φ φ ( n +1) + ∆( r + a ) − [ n ( n + 1) + O ∞ ( r − )] φ ( n +1) − [4 nr − + O ∞ ( r − )] Lφ ( n +1) + a n (cid:88) k =0 O ∞ ( r − )Φ φ ( k ) + ( n + 1) n (cid:88) k =0 O ∞ ( r − ) φ ( k ) . Furthermore, − L (cid:18) ddr (∆ − ( r + a ) )∆( r + a ) − Lφ ( n ) (cid:19) = [ − r − + O ∞ ( r − )] Lφ ( n +1) + [2 r − + O ∞ ( r − )] φ ( n ) By combining the above equations, we arrive at (3.7) with n replaced by n +1, which concludes the inductionargument. It will be convenient to appeal to a propagation of regularity result for the wave equation (3.1) with respectto the differentiable structure on both R and (cid:98) R . Proposition 3.4.
Let k ∈ N . (i) Let (Ψ , Ψ (cid:48) ) ∈ H k +1loc (Σ ) × H k loc (Σ ) . Then there exists a solution ψ to (3.1) (in the distributional sense),such that for all T ≥ : ψ ∈ C ([0 , T ] , H k +1loc (Σ )) ∩ C ([0 , T ] , H k loc (Σ )) ,ψ | Σ = Ψ ,T ψ | Σ = Ψ (cid:48) . (ii) Assume moreover that ( √ r + a Ψ , √ r + a Ψ (cid:48) ) ∈ H k +1 ( (cid:98) Σ ) × H k ( (cid:98) Σ ) . Then, for all τ ≥ , ( φ | Σ τ , T φ | Σ τ ) ∈ H k +1 ( (cid:98) Σ τ ) × H k ( (cid:98) Σ τ ) . (3.8)(iii) Let (Ψ , Ψ (cid:48) ) ∈ C ∞ (Σ ) × C ∞ (Σ ) . Assume moreover that ( (cid:112) r + a Ψ , (cid:112) r + a Ψ (cid:48) ) ∈ H N +1 ( (cid:98) Σ ) × H N ( (cid:98) Σ ) for some N ∈ N . Then for all τ ≥ and k, m, n ∈ N , such that k + m + n ≤ N , (cid:90) S | / ∇ k S ∂ mτ φ ( n ) | ( τ, ρ , θ, ϕ ) dω attains a finite limit as ρ → ∞ .Proof. Part (i) follows from standard result for linear wave equations, see for example [Sog08]. In order toobtain part (ii), we consider (3.6) and observe that the coefficients in the equation are smooth functionsof ( x, θ ) (note in particular the smoothness in r .) Therefore, (3.8) follows from standard (finite-in-time)higher-order energy estimates with respect to ( τ, x, θ, ϕ ∗ ) coordinates.By the fundamental theorem of calculus it follows that for 0 < x ≤ r − and x ≤ x (cid:90) S | / ∇ k S ∂ k τ ∂ k x φ | ( τ, x, θ, ϕ ) dω = (cid:90) S | / ∇ k S ∂ k τ ∂ k x φ | ( τ, x , θ, ϕ ) dω − (cid:90) x x / ∇ k S ∂ k τ ∂ k x φ · / ∇ k S ∂ k τ ∂ k +1 x φ ( τ, x (cid:48) , θ, ϕ ) dωdx (cid:48) . ψ with respect to thedifferentiable structure on R , which in turn follows from the smoothness of Ψ and Ψ (cid:48) , together with astandard propagation of regularity argument; see again [Sog08].Then, after applying Cauchy–Schwarz to the second term on the right-hand side of the equation above,together with (3.8) and (3.6), it follows that the limit of the left-hand side at x = 0 is well-defined provided k + k ≤ N . Since / ∇ k S ∂ mτ φ ( n ) , with k + m + n ≤ N , can be written as a linear combination of / ∇ k S ∂ k τ ∂ k x φ with k + k + k ≤ N , we obtain part (iii).Throughout the remainder of the paper we will assume for the sake of convenience that the initial datafor (3.1) satisfy: ( (cid:112) r + a Ψ , (cid:112) r + a Ψ (cid:48) ) ∈ C ∞ ( (cid:98) Σ ) × C ∞ ( (cid:98) Σ ) , though this assumption can be significantly weakened by a standard density argument with respect to the(weighted) energy norms in the relevant estimates. Two key ingredients towards deriving the results in the present article are energy boundedness and integratedlocal energy decay . In the context of subextremal Kerr spacetimes with | a | < M , these were both derived in[DRSR16]. We summarize the relevant results as Theorem A below. Theorem A (Dafermos–Rodnianski–Shlapentokh-Rothman, [DRSR16]) . Let ψ be a solution to (3.1) arisingfrom suitably regular initial data with respect to (cid:98) R . Then the following estimates hold: (i) There exists a constant C = C ( M, a ) > , such that for any τ ≥ τ ≥ : (cid:90) Σ τ J N [ ψ ] · n τ r dωd ρ ≤ C (cid:90) Σ τ J N [ ψ ] · n τ r dωd ρ . (3.9)(ii) Let n ∈ N . For R > suitably large and R > , there exists a constant C = C ( n, R , R , M, a ) > ,such that for any τ ≥ τ ≥ : (cid:88) ≤ k + k + k ≤ n (cid:90) τ τ (cid:34)(cid:90) Σ τ ∩{ R ≤ r ≤ R } | / ∇ k S T k Y k ψ | dωd ρ (cid:35) dτ ≤ C (cid:88) m ≤ n (cid:90) Σ τ J N [ T m ψ ] · n τ r dωd ρ . (3.10) Furthermore, for all δ > , there exists C = C ( n, M, a ) > , such that (cid:90) D τ τ r − − δ (cid:2) ( Lφ ) + ( Lφ ) + r − φ + r − | / ∇ S φ | (cid:3) dωdρdτ ≤ C (cid:90) Σ τ J N [ ψ ] · n τ r dωd ρ . (3.11)(iii) Let n ∈ N . For R > arbitrarily large, there exists a constant C = C ( n, R , M, a ) > , such that forany τ ≥ τ ≥ : (cid:88) ≤ k + k + k ≤ n (cid:90) τ τ (cid:34)(cid:90) Σ τ ∩{ r ≤ R } | / ∇ k S T k Y k ψ | dωd ρ (cid:35) dτ ≤ C (cid:88) k + k ≤ n +1 (cid:90) Σ J N [ T k Φ k ψ ] · n τ r dωd ρ . (3.12)We note moreover that by definition of Σ τ in Section 2.3, and the choice h (cid:54) = 0 in particular, we canestimate J N [ ψ ] · n τ ∼ ( Xψ ) + r − ( T ψ ) + r − | / ∇ S ψ | . It follows immediately by combining (3.9) and (3.10) and applying (2.20) with p = 0 that for all 0 ≤ τ ≤ τ : (cid:90) N τ ( Lφ ) + r − ( Lφ ) + r − | / ∇ S φ | dωdρ + (cid:90) I s ∩ D τ τ ( Lφ ) + r − ( Lφ ) + r − | / ∇ S φ | dωdρ ≤ C (cid:90) Σ τ J N [ ψ ] · n τ r dωd ρ . (3.13)In order to obtain more refined energy decay estimates for the projections ψ ≥ (cid:96) , we need additionalintegrated local energy decay estimates for ψ ≥ (cid:96) . 29 roposition 3.5. Let ψ be a solution to (3.1) arising from suitably regular initial data with respect to (cid:98) R .Then the following estimates hold for (cid:96) ∈ N : (i) For R > M suitably large, there exists a constant C = C ( M, a, R ) > , such that for any τ ≥ τ ≥ : (cid:90) Σ τ J N [ ψ ≥ (cid:96) ] · n τ r dωd ρ + (cid:90) ∞ τ (cid:34)(cid:90) Σ τ ∩{ r ≥ R } M δ r − δ [( T ψ ≥ (cid:96) ) + ( Y ψ ≥ (cid:96) ) ] + r − | / ∇ S ψ ≥ (cid:96) | + M δ r − − δ ψ ≥ (cid:96) dωd ρ (cid:35) dτ ≤ Ca (cid:90) ∞ τ (cid:20)(cid:90) Σ τ M − δ r − δ (cid:2) ( T ψ ≥ max { (cid:96) − , } ) + M ( T ψ ≥ max { (cid:96) − , } ) + ( T Φ ψ ≥ max { (cid:96) − , } ) (cid:3) dωd ρ (cid:21) dτ + C (cid:90) Σ τ (cid:2) J N [ ψ ≥ (cid:96) ] · n τ + a J N [ T ψ ≥ max { ,(cid:96) − } ] · n τ (cid:3) r dωd ρ . (3.14)(ii) There exists a constant C = C ( M, a ) > , such that for any τ ≥ τ ≥ : (cid:90) ∞ τ (cid:20)(cid:90) Σ τ M δ r − δ [( T ψ ≥ (cid:96) ) + ( Y ψ ≥ (cid:96) ) ] + r − | / ∇ S ψ ≥ (cid:96) | + M δ r − − δ ψ ≥ (cid:96) dωd ρ (cid:21) dτ ≤ Ca (cid:88) k + k ≤ (cid:90) τ τ (cid:20)(cid:90) Σ τ M − δ r − δ ( T k Φ k ψ ≥ max { (cid:96) − , } ) dωd ρ (cid:21) dτ + C (cid:88) k + k ≤ (cid:90) Σ τ M k J N [ T k Φ k ψ ≥ (cid:96) ] · n τ + a M k J N [ T k Φ k ψ ≥ max { ,(cid:96) − } ] · n τ r dωd ρ . (3.15)(iii) Let n ∈ N . For R > M suitably large and R > R , there exists a constant C = C ( M, a, n, R , R ) > , such that for any τ ≥ : (cid:88) ≤ k + k + k ≤ n (cid:90) ∞ τ (cid:34)(cid:90) Σ τ ∩{ R ≤ r ≤ R } | / ∇ k S T k Y k ψ ≥ (cid:96) | dωd ρ (cid:35) dτ ≤ C n (cid:88) k =0 (cid:90) Σ τ J N [ T k ψ ≥ (cid:96) ] · n τ r dωd ρ + (cid:88) ≤ l ≤(cid:100) (cid:96)/ (cid:101) ≤ l + l ≤ l a (cid:90) Σ τ J N [ T k +2 l T l Φ l ψ ≥ max { (cid:96) − l, } · n τ r dωd ρ . (3.16) Proof.
Let ψ be a solution to (3.1) arising from suitably regular initial data with respect to (cid:98) R . Then ψ ≥ (cid:96) satisfies the following inhomogeneous equation: ρ (cid:3) g ψ ≥ (cid:96) = a T [ π ≥ (cid:96) , sin θ ] ψ =: F ≥ (cid:96) . We can generalize the Morawetz estimate in Proposition 9.1.1 of [DRSR16] to inhomogeneous wave equationswhere the inhomogeneity F ≥ (cid:96) is given by the above expression and we consider a hyperboloidal foliation byΣ τ rather than an asymptotically flat foliation; see also Proposition 9.8.1 of [DRSR16]. We obtain thefollowing estimate after assuming without loss of generality that M = 1: (cid:90) ∞ τ (cid:20)(cid:90) Σ τ r − δ [ ζ ( T ψ ≥ (cid:96) ) + ( Y ψ ≥ (cid:96) ) ] + r − ζ | / ∇ S ψ ≥ (cid:96) | + r − − δ ψ ≥ (cid:96) dωd ρ (cid:21) dτ ≤ C (cid:90) ∞ τ (cid:20)(cid:90) Σ τ O ∞ ( r ) F ≥ (cid:96) · ( T ψ ≥ (cid:96) + Φ ψ ≥ (cid:96) + Y ψ ≥ (cid:96) + r − ψ ≥ (cid:96) ) dωd ρ (cid:21) dτ + C (cid:90) Σ τ J N [ ψ ≥ (cid:96) ] · n r dωd ρ , δ > ζ a smooth cut-off function that vanishes in an interval [ − s , M + 3 M, M + s , M ], with − s , M + 3 M > r + and s , M < ∞ depending on a and M .After integrating by parts in T and taking (cid:15) > (cid:90) ∞ τ (cid:20)(cid:90) Σ τ O ∞ ( r ) F ≥ (cid:96) · ( T ψ ≥ (cid:96) + Φ ψ ≥ (cid:96) + Y ψ + ψ ≥ (cid:96) ) dωd ρ (cid:21) dτ ≤ (cid:15) (cid:90) ∞ τ (cid:20)(cid:90) Σ τ r − δ [ ζ ( T ψ ≥ (cid:96) ) + ( Y ψ ≥ (cid:96) ) ] + r − ζ | / ∇ S ψ ≥ (cid:96) | + r − − δ ψ ≥ (cid:96) dωd ρ (cid:21) dτ + C(cid:15) − (cid:90) ∞ τ (cid:20)(cid:90) Σ τ r − δ F ≥ (cid:96) dωd ρ (cid:21) dτ + C(cid:15) − (cid:90) ∞ τ (cid:34)(cid:90) Σ τ ∩{− s , M +3 M ≤ r ≤ M + s , M } O ∞ ( r )( T F ≥ (cid:96) + Φ F ≥ (cid:96) ) · ψ ≥ (cid:96) dωd ρ (cid:35) dτ + C(cid:15) − (cid:90) Σ τ ∩{− s , M +3 M ≤ r ≤ M + s , M } | F ≥ (cid:96) || ψ ≥ (cid:96) | dωd ρ , with suitable s , M < s , M < (3 M − r + ) and s , M < s , M < ∞ . We have that (cid:90) S F ≥ (cid:96) dω ≤ C (cid:90) S ( T ψ ≥ (cid:96) − ) dω. We obtain (3.14) by combining the above estimates. The estimate (3.15) follows by additionally applyingthe above estimates with ψ replaced by T ψ or Φ ψ (using that T and Φ are Killing vector fields), to removethe degenerate factor ζ on the left-hand side. The estimate (3.16) follows from standard higher-order ellipticestimates in the far-away region 2 M < R < r < R , where T is timelike, together with a repeated applicationof (3.14), using moreover that all estimates apply to T k ψ replacing ψ . In this section we derive conservation laws for weighted derivatives of φ along I + by constructing Newman–Penrose charges in Kerr spacetimes. These Newman–Penrose charges will play a key role in the analysis inthe remainder of the paper, both when deriving sharp decay estimates and precise late-time asymptotics forsolutions to (3.1).We first introduce the following renormalized derivatives of φ :ˇ φ (1) := (cid:2) α + α Φ Φ) r − + ( β + β Φ Φ + β Φ Φ ) r − (cid:3) r + a ) ∆ Lφ, (4.1)ˇ φ (2) := (cid:2) γ + γ Φ Φ) r − (cid:3) r + a ) ∆ L ˇ φ (1) , (4.2)with α = − M, β = − M − a , γ = − M,α Φ = a, β Φ = 0 , β Φ = 13 a γ Φ = 0 . It will be useful to consider a decompositions into azimuthal modes ψ = (cid:80) m ∈ Z ( ψ ) m , where ( ψ ) m can beexpressed via the following spherical harmonic decomposition:( ψ ) m = ∞ (cid:88) (cid:96) = | m | ψ (cid:96),m Y (cid:96),m . Note that Φ( ψ ) m = im ( ψ ) m and the azimuthal modes ( ψ ) m are decoupled , i.e. they independently satisfy(3.1): (cid:3) g M,a ( ψ ) m = 0, so it is possible to consider each m -th azimuthal mode independently.31e can then split φ = π ( φ ) = (cid:88) m = − ( φ ) m and φ = π ( φ ) = (cid:88) m = − ( φ ) m . with Φ( φ j ) m = im ( φ j ) m for j = 1 ,
2. When restricted to fixed azimuthal modes, (4.1) and (4.2) take theform: ( ˇ φ (1) ) m := (cid:2) α + imα Φ ) r − + ( β + imβ Φ − m β Φ ) r − (cid:3) r + a ) ∆ L ( φ ) m , ( ˇ φ (2) ) m := (cid:2) γ + imγ Φ ) r − (cid:3) r + a ) ∆ L ( ˇ φ (1) ) m . In a slight abuse of notation, when restricting to fixed azimuthal modes with azimuthal number m , we willtherefore also use α , β and γ to denote also the complex numbers α + imα Φ , β + imβ Φ + m β Φ and γ + imγ Φ ,respectively. Proposition 4.1.
Let ψ ∈ C ∞ ( R → C ) denote a solution to (3.1) that is supported on a fixed azimuthalmode with azimuthal number m , i.e. it satisfies Φ ψ = imψ . Then: r + a ) ∆ LL ˇ φ (1)1 = 4 ( r + a ) ∆ LLπ ( ˇ φ (1) ) = (cid:104) a π (sin θT ˇ φ (1) ) + 2 aT Φ ˇ φ (1)1 (cid:105) + [ − α − α Φ Φ + O ∞ ( r − )][ a π (sin θT φ ) + 2 aT Φ φ ]+ [ − r + O ∞ (1)] L ˇ φ (1)1 + O ∞ ( r − ) ˇ φ (1)1 + O ∞ ( r − )Φ ˇ φ (1)1 + O ∞ ( r − ) φ (4.3) and r + a ) ∆ LL ˇ φ (2)2 = (cid:104) a π (sin θT ˇ φ (2) ) + 2 aT Φ ˇ φ (2)2 (cid:105) + [ M − γ − a Φ + O ∞ ( r − )][ a π (sin θT ˇ φ (1) ) + 2 aT Φ ˇ φ (1)2 ]+ [2 β + 2 β Φ Φ + 2 β Φ Φ + γ ( a Φ − M ) + O ∞ ( r − )][ a π (sin θT φ ) + 2 aT Φ φ ]+ [ − r + O ∞ ( r )] L ˇ φ (2)2 + O ∞ ( r − ) ˇ φ (2)2 + O ∞ ( r − )Φ ˇ φ (2)2 + O ∞ ( r − )Φ ˇ φ (1)2 + O ∞ ( r − ) ˇ φ (1)2 + O ∞ ( r − )Φ φ + O ∞ ( r − ) φ , (4.4) where we allow the constants in the terms in O ∞ ( r − k ) , k = 0 , , to depend also on m .Proof. We have that ˇ φ (1) = (1 + αr − + βr − ) 2( r + a ) ∆ Lφ, and therefore4 L ˇ φ (1) = =: T (cid:122) (cid:125)(cid:124) (cid:123) L (cid:18) (1 + αr − + βr − ) 2( r + a ) ∆ (cid:19) (1 + αr − + βr − ) − ∆( r + a ) ˇ φ (1) + =: T (cid:122) (cid:125)(cid:124) (cid:123) (1 + αr − + βr − ) 2( r + a ) ∆ 4 LLφ .
We can write T = ∆( r + a ) [ − r + 2( α + 2 M ) + O ∞ ( r − )] ˇ φ (1) , where we allow the terms in O ∞ ( r − ) to depend on α and β .Hence, L ( T ) = ∆( r + a ) [ − r + 2( α + 2 M ) + O ∞ ( r − )] L ˇ φ (1) + ∆( r + a ) (cid:2) − (2 α + 12 M ) r − + O ∞ ( r − ) (cid:3) ˇ φ (1) . LLφ = ∆( r + a ) (cid:20) a sin θT φ + 2 aT Φ φ + / ∆ S φ + 2 a rr + a Φ φ (cid:21) + ∆( r + a ) [ − M r − − a r − + O ∞ ( r − )] φ. Therefore, T = 2(1+ αr − + βr − ) (cid:20) a sin θT φ + 2 aT Φ φ + / ∆ S φ + 2 a rr + a Φ φ + ( − M r − − a r − + O ∞ ( r − )) φ (cid:21) . We then obtain L ( T ) = ∆( r + a ) (cid:20) a sin θT ˇ φ (1) + 2 aT Φ ˇ φ (1) + 2 a rr + a Φ ˇ φ (1) + / ∆ S ˇ φ (1) (cid:21) + ∆( r + a ) [ − M r − − a r − + O ∞ ( r − )] ˇ φ (1) + ∆( r + a ) [ − α − βr − + O ∞ ( r − )][ a sin θT φ + 2 aT Φ φ + / ∆ S φ ]+ ∆( r + a ) [ − a − aαr − + O ∞ ( r − )]Φ φ + ∆( r + a ) (cid:2) M + (2 a + 4 αM ) r − + O ∞ ( r − ) (cid:3) φ Combining the expressions for L ( T ) and L ( T ) and commuting L and L , we obtain:4 ( r + a ) ∆ LL ˇ φ (1) = (cid:20) a sin θT ˇ φ (1) + 2 aT Φ ˇ φ (1) + 6 a rr + a Φ ˇ φ (1) (cid:21) + [ − r + 2( α + 2 M ) + O ∞ ( r − )] L ˇ φ (1) + (cid:2) / ∆ S (cid:3) ˇ φ (1) + [( − α − M ) r − + O ∞ ( r − )] ˇ φ (1) + [ − α − βr − + O ∞ ( r − )][ a sin θT φ + 2 aT Φ φ + / ∆ S φ ]+ [ − a − aαr − + O ∞ ( r − )]Φ φ + (cid:2) M + (2 a + 4 αM ) r − + O ∞ ( r − ) (cid:3) φ (4.5)In order to guarantee that the right-hand side above either contains a T -derivative or vanishes when we actwith π on both sides and take r → ∞ , we use that α = − M + iam. Now let ˇ φ (2) = (1 + γr − ) 2( r + a ) ∆ L ˇ φ (1) , so 4 L ˇ φ (2) = =: T (cid:122) (cid:125)(cid:124) (cid:123) L (cid:18) (1 + γr − ) 2( r + a ) ∆ (cid:19) (1 + γr − ) − ∆( r + a ) ˇ φ (2) + =: T (cid:122) (cid:125)(cid:124) (cid:123) (1 + γr − ) 2( r + a ) ∆ 4 LL ˇ φ (1) . Note that L ( T ) = ∆( r + a ) [ − r + O ∞ ( r )] L ˇ φ (2) + ∆( r + a ) (cid:2) O ∞ ( r − ) (cid:3) ˇ φ (2) , where the terms in O ∞ ( r − ) depend on γ .By (4.5) with α = − M + iam , we have that: T = 2(1 + γr − )[ a sin θT ˇ φ (1) + 2 aT Φ ˇ φ (1) + 6 ar ( r + a ) − Φ ˇ φ (1) ]+ ∆( r + a ) − [ − r + 2( M + iam ) + O ∞ ( r − )] ˇ φ (2) + 2(1 + γr − )[2 + / ∆ S ] ˇ φ (1) + 2(1 + γr − )[( − M − iam ) r − + O ∞ ( r − )] ˇ φ (1) + 2(1 + γr − )[ M − iam − βr − + O ∞ ( r − )][ a sin θT φ + 2 aT Φ φ + / ∆ S φ ]+ 2(1 + γr − )[ − a + (4 M a − ima ) r − + O ∞ ( r − )]Φ φ + 2(1 + γr − )[2 M + (2 a − M + 4 iamM ) r − + O ∞ ( r − )] φ r + a ) ∆ L ( T ) = [ a sin θT ˇ φ (2) + 2 aT Φ ˇ φ (2) + 6 ar ( r + a ) − Φ ˇ φ (2) ]+ [ − r + O ∞ ( r )] L ˇ φ (2) + [2 + / ∆ S + O ∞ ( r − )] ˇ φ (2) − [ γ + O ∞ ( r − )][ a sin θT ˇ φ (1) + 2 aT Φ ˇ φ (1) ] − [6 a + O ∞ ( r − ]Φ ˇ φ (1) + [2 + O ∞ ( r − )] ˇ φ (2) − [ γ + O ∞ ( r − )](2 + / ∆ S ) ˇ φ (1) + [12 M + 2 iam + O ∞ ( r − )] ˇ φ (1) + [ M − iam + O ∞ ( r − )]( a sin θT ˇ φ (1) + 2 aT Φ ˇ φ (1) + / ∆ S ˇ φ (1) )+ [ − a + O ∞ ( r − )]Φ ˇ φ (1) + [2 M + O ∞ ( r − )] ˇ φ (1) + [ − γ ( M − iam ) + 2 β + O ∞ ( r − )]( a sin θT φ + 2 aT Φ φ + / ∆ S φ )+ [2 aγ − M a + 4 ima + O ∞ ( r − )]Φ φ + [ − M γ − (2 a − M + 4 iamM ) + O ∞ ( r − )] φ. Combining the above expression with the expression for L ( T ), using that Φ φ = imφ (and commuting L and L ), we obtain4 ( r + a ) ∆ LL ˇ φ (2) = (cid:20) a sin θT ˇ φ (2) + 2 aT Φ ˇ φ (2) + 10 a rr + a Φ ˇ φ (2) (cid:21) + [ − r + O ∞ ( r )] L ˇ φ (2) + (cid:2) / ∆ S (cid:3) ˇ φ (2) + O ∞ ( r − ) ˇ φ (2) + [ M − aim − γ + O ∞ ( r − )][ a sin θT ˇ φ (1) + 2 aT Φ ˇ φ (1) ]+ [ − γ (2 + / ∆ S ) + 14 M − iam + ( M − iam ) / ∆ S + O ∞ ( r − )] ˇ φ (1) + [2 β − M γ + iamγ + O ∞ ( r − )]( a sin θT φ + 2 aT Φ φ )+ [( − M γ + iamγ + 2 β ) / ∆ S − M γ − a + 4 M − iamM + 2 iamγ − iamM − m a + O ∞ ( r − )] φ (4.6)In order to guarantee that the terms involving ˇ φ (2) and ˇ φ (1) on the right-hand side above either containsa T -derivative or vanishes when we act with π on both sides and take r → ∞ , we use that: γ = − M. With the above value of γ , it follows that if we moreover use that β = − M − a − a m , then all the terms involving φ on the right-hand side above either contain a T -derivative or vanish when weact with π on both sides and take r → ∞ .We define the following special linear combinations of derivatives of ˇ φ ( j ) , j = 0 , , r + a ) ∆ P := ( r + a ) ∆ Lφ − a π (sin θT φ ) , (4.7)( r + a ) ∆ P := ( r + a ) ∆ L ˇ φ (1)1 − (cid:104) a π (sin θT ˇ φ (1) ) + 2 a Φ ˇ φ (1)1 (cid:105) (4.8) −
14 [ − α − α Φ Φ][ a π (sin θT φ ) + 2 a Φ φ ] , ( r + a ) ∆ P := ( r + a ) ∆ L ˇ φ (2)2 − (cid:104) a π (sin θT ˇ φ (2) ) + 2 a Φ ˇ φ (2)2 (cid:105) (4.9) −
14 [ M − γ − a Φ][ a π (sin θT ˇ φ (1) ) + 2 a Φ ˇ φ (1)2 ] −
14 [2 β + 2 β Φ Φ + γ ( a Φ − M )][ a π (sin θT φ ) + 2 a Φ φ ] . orollary 4.2. Let ψ ∈ C ∞ ( R → C ) denote a solution to (3.1) that is supported on a fixed azimuthal modewith azimuthal number m , i.e. it satisfies Φ ψ = imψ . Then: LP = O ∞ ( r − ) φ + a O ∞ ( r − ) T π (sin θφ ) + a O ∞ ( r − ) LT π (sin θφ ) , (4.10)4 LP = [ − r − + O ∞ ( r − )] P + O ∞ ( r − )[ ˇ φ (1)1 + φ ] + a O ∞ ( r − )[ T π (sin θ ˇ φ (1) ) + T π (sin θφ )] (4.11)+ a O ∞ ( r − )[ LT π (sin θ ˇ φ (1) ) + LT π (sin θφ )] , LP = [ − r − + O ∞ ( r − )] P + O ∞ ( r − )[ ˇ φ (2)2 + ˇ φ (1)2 + φ ] (4.12)+ a O ∞ ( r − )[ T π (sin θ ˇ φ (2) ) + T π (sin θ ˇ φ (1) ) + T π (sin θφ )]+ a O ∞ ( r − )[ LT π (sin θ ˇ φ (2) ) + LT π (sin θ ˇ φ (1) ) + LT π (sin θφ )] , where we allow the constants in the terms in O ∞ ( r − k ) , k = 0 , , to depend also on m .Furthermore, if φ ∈ C ( (cid:98) R ) , then lim ρ →∞ L ( r P )( τ, ρ ) = 0 , (4.13)lim ρ →∞ L ( r P )( τ, ρ , θ, ϕ ∗ ) = 0 , (4.14)lim ρ →∞ L ( r P )( τ, ρ , θ, ϕ ∗ ) = 0 . (4.15) Proof.
By the rewriting the equations in Proposition 4.1 in terms of the quantities P i , i = 0 , ,
2, using thedefinitions (4.7)–(4.9), we obtain (4.10)–(4.12).Now suppose φ ∈ C ( (cid:98) R ), then the limits on the left-hand sides of (4.13)–(4.15) are well-defined. Byapplying (4.10)–(4.12), we moreover show that these limits must vanish. Definition 4.1.
Let ψ be a solution to such that φ = √ r + a ψ ∈ C ( (cid:98) R ) . Then we define the Newman–Penrosecharges as the following limits: I [ ψ ] = lim ρ →∞ r P ( τ, ρ ) ,I [ ψ ]( θ, ϕ ∗ ) = lim ρ →∞ r P ( τ, ρ , θ, ϕ ∗ ) ,I [ ψ ]( θ, ϕ ∗ ) = lim ρ →∞ r P ( τ, ρ , θ, ϕ ∗ ) , which are well-defined and conserved in τ by (4.13) – (4.15) . r p -weighted energy estimates In this section, we derive the key weighted hierarchies of estimates that are involved in establishing sharpenergy decay estimates.We will use R to denote the area radius that appears in the definitions of D τ τ and N τ in Section 2.3. Wewill need to take R > r + appropriately large for the estimates in the sections below to hold. r p -weighted energy estimates for φ ( n ) This section is concerned with establishing hierarchies of weighted energy estimates for the quantities φ ( n ) ,which were introduced in Section 3.1. In Proposition 5.1 below, we show that we can derive weighted energyestimates for a larger range of n ∈ N , provided we restrict to φ ( n ) ≥ (cid:96) with (cid:96) appropriately large.35 roposition 5.1. Let n ∈ N , (cid:96) ∈ N , such that (cid:96) ≥ n and − n < p ≤ . For R > r + suitably large, thereexists a constant C = C ( M, a, n, (cid:96), R, p ) > , such that (cid:90) N τ r p ( Lφ ( n ) ≥ (cid:96) ) + r p − [ | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ n ) ] dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( Lφ ( n ) ≥ (cid:96) ) + (2 − p ) r p − (cid:16) | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) + δ n δ (cid:96) a sin θ ( T φ ) (cid:17) dωd ρ (cid:21) dτ ≤ C (cid:90) N τ r p ( Lφ ( n ) ≥ (cid:96) ) + r p − (cid:16) | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) (cid:17) dωd ρ + C ( n + (cid:96) ) (cid:90) τ τ (cid:20)(cid:90) N τ r p − a ( T φ ( n ) ≥ max { (cid:96) − , } ) + r p − a ( T Φ φ ( n ) ≥ (cid:96) ) dωd ρ (cid:21) dτ + Cn (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( L Φ φ ( n − ≥ (cid:96) ) + r p − ( LT φ ( n − ≥ (cid:96) ) dωd ρ (cid:21) dτ + Cn n − (cid:88) m =0 (cid:90) τ τ (cid:20)(cid:90) N τ a r p − (Φ φ ( m ) ≥ (cid:96) ) + r p − ( φ ( m ) ≥ (cid:96) ) dωd ρ (cid:21) dτ + C n (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ≥ (cid:96) ] · n τ r dωd ρ + a (1 − δ n δ (cid:96) ) (cid:88) ≤ l ≤(cid:100) (cid:96)/ (cid:101) ≤ l + l ≤ l (cid:90) Σ τ J N [ T m +2 l T l Φ l ψ ≥ max { (cid:96) − k, } · n τ r dωd ρ (5.1) Furthermore, the estimate (5.1) also holds with the last line replaced by C n (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ] · n τ r dωd ρ . (5.2) Proof.
We apply π ≥ (cid:96) and then multiply both sides of (3.7) by − r p − r + a ) ∆ Lφ ( n ) ≥ (cid:96) to obtain:0 = − r p − Lφ ( n ) ≥ (cid:96) (cid:34) − r + a ) ∆ LLφ ( n ) ≥ (cid:96) + a T π ≥ (cid:96) (sin θφ ( n ) ) + 2 aT Φ φ ( n ) ≥ (cid:96) + 2(1 + 2 n ) a rr + a Φ φ ( n ) ≥ (cid:96) + / ∆ S φ ( n ) ≥ (cid:96) + [ n ( n + 1) + O ∞ ( r − )] φ ( n ) ≥ (cid:96) − [4 nr − + O ∞ ( r − )] ( r + a ) ∆ Lφ ( n ) ≥ (cid:96) + a n − (cid:88) m =0 O ∞ ( r )Φ φ ( m ) ≥ (cid:96) + n n − (cid:88) k =0 O ∞ ( r ) φ ( m ) ≥ (cid:96) (cid:35) =: L ( F r p LL ) + L ( F r p LL ) + / div S F r p L/ ∇ + Φ( F r p L Φ ) + J r pL . We will determine F r p LL , F r p LL and J r pL below so that the second equality above holds.2 r p − r + a ) ∆ Lφ ( n ) ≥ (cid:96) · LLφ ( n ) ≥ (cid:96) :We can write2 r p − ( r + a ) ∆ Lφ ( n ) ≥ (cid:96) · LLφ ( n ) ≥ (cid:96) = L (cid:18) r p − ( r + a ) ∆ ( Lφ ( n ) ≥ (cid:96) ) (cid:19) + (cid:16) p O ∞ ( r − ) (cid:17) r p − ( Lφ ( n ) ≥ (cid:96) ) . - r p − Lφ ( n ) ≥ (cid:96) · ( / ∆ S + n ( n + 1)) φ ( n ) ≥ (cid:96) :We can write − r p − Lφ ( n ) ≥ (cid:96) · ( / ∆ S + n ( n + 1)) φ ( n ) ≥ (cid:96) = 14 L (cid:16) r p − | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1) r p − ( φ ( n ) ≥ (cid:96) ) (cid:17) + 18 (2 − p ) r p − ∆ r + a (cid:104) | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) (cid:105) . r p − Lφ ( n ) ≥ (cid:96) · a T π ≥ (cid:96) (sin θφ ( n ) ):We only rewrite this term if n = 0 and (cid:96) = 0, as π ≥ = id (for n (cid:54) = 0 or (cid:96) (cid:54) = 0 we directly group itwith J r P L . We can write − r p − Lφ · a sin θT φ = − T (cid:18) r p − Lφ · a sin θT φ (cid:19) + 14 r p − a sin θL (( T φ ) )= − ( L + L ) (cid:18) r p − Lφ · a sin θT φ (cid:19) + L (cid:18) r p − a sin θ ( T φ ) (cid:19) + Φ (cid:18) a r + a ) r p − Lφ · a sin θT φ (cid:19) + 18 (2 − p ) ∆ r + a r p − a sin θ ( T φ ) . − r p − Lφ ( n ) ≥ (cid:96) · aT Φ φ ( n ) ≥ (cid:96) :Again, we only rewrite this term if n = 0 and (cid:96) = 0: − ar p − Lφ · T Φ φ = − ar p − Lφ · ( L + L − ar + a Φ)Φ φ = − Φ (cid:18) a r p − ( Lφ ) + a r + a Lφ Φ φ (cid:19) + r p − a r + a ) L ((Φ φ ) ) − L ( ar p − Lφ Φ φ ) + L ( ar p − Lφ )Φ φ = − Φ (cid:18) a r p − ( Lφ ) + a r + a Lφ Φ φ (cid:19) + L (cid:18) r p − a r + a ) (Φ φ ) (cid:19) + ar p − Φ φLLφ + (cid:20)
12 (4 − p ) a r p − + O ∞ ( r p − ) (cid:21) (Φ φ ) + O ∞ ( r p − ) Lφ Φ φ. By using (3.5), we can further write: ar p − Φ φLLφ = 14 a r p − sin θ ∆( r + a ) Φ φT φ + 12 a ∆( r + a ) r p − Φ φT Φ φ + a r + a ) r p − Φ φ / ∆ S φ + 12 a r p − ∆( r + a ) (Φ φ ) + O ∞ ( r p − ) φ Φ φ. We have that14 a r p − sin θ ∆( r + a ) Φ φT φ + 12 a ∆( r + a ) r p − Φ φT Φ φ = T (cid:18) a r p − sin θ ∆( r + a ) Φ φT φ (cid:19) + T (cid:18) a ∆( r + a ) r p − (Φ φ ) (cid:19) + Φ (cid:18) a r p − sin θ ∆( r + a ) ( T φ ) (cid:19) and a r + a ) r p − Φ φ / ∆ S φ = / div S (cid:18) a r + a ) r p − Φ φ / ∇ S φ (cid:19) + Φ (cid:18) a r + a ) r p − | / ∇ S φ | (cid:19) . We combine the above identities to obtain: F r p LL = 14 r p − ( | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) ) + δ n δ (cid:96) (cid:34) a sin θr p − ( T φ ) − r p − a sin θLφT φ + r p − a r + a ) (Φ φ ) + 14 a r p − sin θ ∆( r + a ) Φ φT φ (cid:35) , F r p LL = r p − ( r + a ) ∆ ( Lφ ( n ) ≥ (cid:96) ) + δ n δ (cid:96) (cid:34) − r p − a sin θLφT φ + 14 a ∆( r + a ) r p − (Φ φ )
37 14 a r p − sin θ ∆( r + a ) Φ φT φ (cid:35) , J r p L = (cid:18) p + 4 n O ∞ ( r − ) (cid:19) r p − ( Lφ ( n ) ≥ (cid:96) ) + 18 (2 − p ) r p − ∆ r + a [ | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) ]+ O ∞ ( r p − ) φ ( n ) ≥ (cid:96) Lφ ( n ) ≥ (cid:96) + O ∞ ( r p − )Φ φ ( n ) ≥ (cid:96) Lφ ( n ) ≥ (cid:96) + a (1 − δ n δ (cid:96) ) n − (cid:88) m =0 O ∞ ( r p − )Φ φ ( m ) ≥ (cid:96) Lφ ( n ) ≥ (cid:96) + (1 − δ n δ (cid:96) ) n − (cid:88) m =0 O ∞ ( r p − ) φ ( m ) ≥ (cid:96) Lφ ( n ) ≥ (cid:96) + δ n δ (cid:96) (cid:20)
18 (2 − p ) ∆ r + a r p − a sin θ ( T φ ) + (cid:18)
12 (5 − p ) r p − + O ∞ ( r p − ) (cid:19) (Φ φ ) + O ∞ ( r p − ) φ Φ φ (cid:21) + (1 − δ n δ (cid:96) ) (cid:20) − a r p − Lφ ( n ) ≥ (cid:96) T π ≥ (cid:96) (sin θφ ( n ) ) − ar p − Lφ ( n ) ≥ (cid:96) T Φ φ ( n ) ≥ (cid:96) (cid:21) . We will now apply Lemma 2.3. We deal with the boundary term at r = R by a standard averagingargument: we multiply the F r p L (cid:3) terms with a suitable smooth cut-off function χ : [ r + , ∞ ) →
0, with χ ( ρ ) = 0 for r ≤ R − M and χ ( ρ ) = 1 for r ≥ R , see for example the proof of Proposition 6.5 of [AAG20].Note that when n ≥
1, the flux terms on N τ and I s coming from the terms in F r p LL and F r p LL are non-negative definite for R > − n < p ≤
2. When n = 0, we can easily estimate the fluxterms without a good sign by applying (3.13).The terms without a sign in J r p L are estimated by applying a weighted Young’s inequality. In the n = (cid:96) = 0 case we additionally apply (3.10) and (2.21). For example: we estimate r p − | Φ φ || Lφ ( n ) | + r p − | φ || Lφ ( n ) | ≤ (cid:15)r p − ( Lφ ) + C(cid:15) − r p − (Φ φ ) + C(cid:15) − r p − φ and apply (3.10) to estimate the (Φ φ ) term for p ≤
2, whereas we estimate the φ term using (2.21) (aftermultiplying with the cut-off χ ): (cid:90) τ τ (cid:90) Σ τ ∩{ r ≥ R − M } (cid:15) − r p − χ φ dωd ρ ≤ C (cid:90) τ τ (cid:90) Σ τ ∩{ r ≥ R − M } (cid:15) − r p − χ ( Lφ ) + r p − χ ( T φ ) + r p − χ (Φ φ ) + r p − ( χ (cid:48) ) φ dωd ρ , where we absorb the ( Lφ ) term into the left-hand side and control the remaining terms with (3.11) and(3.10).In the n ≥ (cid:90) τ τ (cid:90) N τ r p − ( φ ( n ) ) dωd ρ dτ ≤ C (cid:90) τ τ (cid:90) N τ r p − ( Lφ ( n ) ≥ (cid:96) ) + r p − ( LT φ ( n − ≥ (cid:96) ) + r p − ( L Φ φ ( n − ≥ (cid:96) ) dωd ρ dτ + . . . , where . . . denotes boundary terms at r = R that can easily be absorbed via an averaging argument and(3.16), as above.The estimate (5.1) then follows. Similarly, we obtain the alternate version of (5.1), with the right-handside (5.2) by applying (3.12) instead of (3.16). r p -weighted energy estimates for P , P and P When restricting to ψ (cid:96) , with (cid:96) = 0 , ,
2, we can obtain weighted hierarchies of energy estimates with largerweights in r compared to those derived in Section 5.1 by considering the quantities P i , i = 0 , , Proposition 5.2.
Let < p < . Then the following estimates hold: for R > r + suitably large, there existsa constant C ( M, a, R, p ) > such that: or (cid:96) = 0 : (cid:90) N τ r p ( P ) dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( P + ( Lφ ) ) dωd ρ (cid:21) dτ ≤ C (cid:90) N τ r p ( P ) dωd ρ + C (cid:90) τ τ (cid:20)(cid:90) N τ a r p − ( LT φ ) + r p − ( T φ ) dωd ρ (cid:21) dτ + C (cid:90) Σ τ J T [ ψ ] · n τ r dωd ρ , for (cid:96) = 1 : (cid:90) N τ r p ( P ) dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( P + ( L ˇ φ (1)1 ) ) dωd ρ (cid:21) dτ ≤ C (cid:90) N τ r p ( P ) dωd ρ + Ca (cid:88) i =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( LT φ ( i ) ≥ ) + r p − ( T φ ( i ) ≥ ) dωd ρ (cid:21) dτ + C (cid:88) j =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( L ( rL ) j T φ ) + r p − ( T φ ) dωd ρ (cid:21) dτ + C (cid:88) m =0 (cid:90) Σ τ J T [ T m ψ ] · n τ r dωd ρ , for (cid:96) = 2 : (cid:90) N τ r p ( P ) dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( P + ( L ˇ φ (2)2 ) ) dωd ρ (cid:21) dτ ≤ C (cid:90) N τ r p ( P ) dωd ρ + Ca (cid:88) i =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( LT φ ( i ) ≥ ) + r p − ( T φ ( i ) ≥ ) dωd ρ (cid:21) dτ + C (cid:88) i =0 1 (cid:88) j =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( L ( rL ) j T φ ( i )2 ) + r p − ( T φ ) dωd ρ (cid:21) dτ + Ca (cid:88) j =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p +1 ( L ( rL ) j T φ ) + r p − ( T φ ) dωd ρ (cid:21) dτ + C (cid:88) m =0 (cid:90) Σ τ J T [ T m ψ ] · n τ r dωd ρ . Proof.
Note first of all that for any function f ∈ L ( S ), we can decompose f = (cid:80) m ∈ Z f m , with Φ f m = imf m such that (cid:90) S | f | dω = (cid:88) m ∈ Z (cid:90) S | f m | dω. We take the azimuthal modes ψ m to satisfy the above equality and multiply both sides of (4.10)–(4.12)with − P j , j = 0 , ,
2, sum over m = − j, . . . , j , respectively, and apply Lemma 2.4 to obtain the following(schematic) identities:0 = L ( r p P ) + (cid:104) p O ∞ ( r − ) (cid:105) r p − P + O ∞ ( r p − ) P (cid:2) φ + a rLT φ + a T φ + a rLT φ + a T φ (cid:3) , L ( r p P ) + (cid:20) p + 42 + O ∞ ( r − ) (cid:21) r p − P + O ∞ ( r p − ) P (cid:104) ˇ φ (1)1 + φ + a rLT ˇ φ (1)1 , + a rLT φ , + a T ˇ φ (1)1 , + a T φ , (cid:105) , L ( r p P ) + (cid:20) p + 82 + O ∞ ( r − ) (cid:21) r p − P + O ∞ ( r p − ) P (cid:88) i =1 (cid:104) ˇ φ ( i )2 + φ + a rLT ˇ φ (2)0 , , + a rLT φ , , + a T ˇ φ ( i )0 , , + a T φ , , (cid:105) . Note that we can estimate (suppressing Φ derivatives):( Lφ ) ≤ CP + Cr − ( T φ ) , L ˇ φ (1)1 ) ≤ CP + C (cid:88) i =0 r − ( φ ( i )1 ) + r − ( T φ ( i )3 ) + r − ( T φ ( i )1 ) ≤ CP + C (cid:88) i =0 r − ( φ ( i )1 ) + r − ( T φ ( i )3 ) + r − ( T φ ) + ( LT φ ) , ( L ˇ φ (2)2 ) ≤ CP + C (cid:88) i =0 r − ( φ ( i )2 ) + r − ( T φ ( i )0 ) + r − ( T φ ( i )2 ) + r − ( T φ ( i )4 ) ≤ CP + C [ r − ( ˇ φ (2)2 ) + r − ( T φ (2)4 ) ]+ C (cid:88) i =0 r − ( φ ( i )2 ) + r − ( T φ ( i )2 , ) + ( LT φ (1)2 ) + [ r ( L T φ ) + r ( LT φ ) + r − ( T φ ) ] . Furthermore, we apply (2.21), to estimate for suitably large
R > (cid:90) N τ r p − φ dωd ρ ≤ C (cid:90) N τ r p − ( Lφ ) + r p − ( T φ ) dωd ρ + . . . , (cid:90) N τ r p − [( ˇ φ (1)1 ) + φ ] dωd ρ ≤ C (cid:90) N τ r p − ( L ˇ φ (1)1 ) + r p − ( LT φ ) + r p − ( T φ ) dωd ρ + . . . , (cid:90) N τ r p − [( ˇ φ (2)2 ) + ( φ (1)2 ) + φ ] dωd ρ ≤ C (cid:90) N τ r p − ( L ˇ φ (2)2 ) + r p − ( LT φ (1)2 ) + r p − ( LT φ ) + r p − ( T φ ) dωd ρ + . . . , where . . . denotes omitted terms arising from averaging the boundary terms at r = R via a cut-off function.We conclude the proof by applying Lemma 2.3 together with the above identities and estimates. Remark 5.1.
Observe that the integrals of r p − ( T φ ) on the right-hand sides of of the estimates in Propo-sition 5.2 can be further estimated for p < using the energy boundedness property in (3.11) . In this section we extend the hierarchies of r p -weighted estimates established in Sections 5.1 and 5.2 tohigher-order quantities of the form / ∇ α S ( rL ) k T m φ ( n ) and ( rL ) k T m P j by commuting the wave operator (cid:3) g with the vector fields rL , T and the angular derivative operator / ∇ S .Note that T is a Killing vector field, so T k ψ is a solution to (3.1) for any k ∈ N and the estimatesderived in Sections 5.1 and 5.2 immediately apply when φ is replaced by T k φ .The operator (cid:3) g , however, does not commute with rL and / ∇ S , so the aim of the this section is to showthat nevertheless analogues of the r p -weighted energy estimates derived in Sections 5.1 and 5.2 still hold for / ∇ α S ( rL ) k φ ( n ) .The higher-order hierarchies of r p -weighted estimates derived in this section are essential for dealing withthe coupling of spherical harmonic modes when proving improved energy decay estimates for the projections ψ ≥ (cid:96) , with (cid:96) ≥ ( rL ) k We will show in this section that r p -weighted energy estimates that are analogous to those derived in Sections5.1 and 5.2 still hold when φ is replaced by the higher-order quantities ( rL ) k φ , with k ∈ N . Lemma 5.3.
Let ψ ∈ C ∞ ( R ) be a solution to (3.1) . Let n, k ∈ N . Then ( rL ) k φ ( n ) satisfies the following quation LL ( rL ) k φ ( n ) = a sin θ ∆( r + a ) T ( rL ) k φ ( n ) + 2 a ∆( r + a ) T Φ( rL ) k φ ( n ) + ∆( r + a ) − / ∆ S ( rL ) k φ ( n ) + ( − kr − + O ∞ ( r − ))[ a sin θT ( rL ) k − φ ( n ) + 2 aT Φ( rL ) k − φ ( n ) + / ∆ S ( rL ) k − φ ( n ) ] − (cid:20)(cid:18) n + 12 k (cid:19) r − + O ∞ ( r − ) (cid:21) L ( rL ) k φ ( n ) + r − (cid:20) n ( n + 1) + (cid:18) n + 12 (cid:19) k + O ∞ ( r − ) (cid:21) ( rL ) k φ ( n ) + k − (cid:88) j =1 O ∞ ( r − )( rL ) j φ ( n ) + k (cid:88) j =0 O ∞ ( r − )Φ( rL ) j φ ( n ) + k ( k − k − (cid:88) j =0 O ∞ ( r − )[ a sin θT ( rL ) j φ ( n ) + 2 aT Φ( rL ) j φ ( n ) + / ∆ S ( rL ) j φ ( n ) ]+ a n − (cid:88) m =0 k (cid:88) j =0 O ∞ ( r − )Φ( rL ) j φ ( m ) + n n − (cid:88) m =0 k (cid:88) j =0 O ∞ ( r − )( rL ) j φ ( m ) . (5.3) Furthermore, for ψ supported on a fixed azimuthal mode, we have that, schematically, L ( rL ) k P = − k (cid:20) r − + O ∞ ( r − ) (cid:21) ( rL ) k P + k k − (cid:88) j =0 O ∞ ( r − )( rL ) j P + k (cid:88) j =0 O ∞ ( r − )( rL ) j φ (5.4)+ k +1 (cid:88) j =0 O ∞ ( r − ) T ( rL ) j π (sin θφ ) , L ( rL ) k P = (cid:20) − (cid:18) k (cid:19) r − + O ∞ ( r − ) (cid:21) ( rL ) k P + k k − (cid:88) j =0 O ∞ ( r − )( rL ) j P + k (cid:88) j =0 O ∞ ( r − )[( rL ) j ˇ φ (1)1 + ( rL ) j φ ](5.5)+ k +1 (cid:88) j =0 O ∞ ( r − )[( rL ) j T π (sin θ ˇ φ (1) ) + ( rL ) j T π (sin θφ )] , L ( rL ) k P = (cid:20) − (cid:18) k (cid:19) r − + O ∞ ( r − ) (cid:21) ( rL ) k P + k k − (cid:88) j =0 O ∞ ( r − )( rL ) j P (5.6)+ k (cid:88) j =0 O ∞ ( r − )[( rL ) j ˇ φ (2)2 + ( rL ) j ˇ φ (1)2 + ( rL ) j φ ]+ k +1 (cid:88) j =0 O ∞ ( r − )[( rL ) j T π (sin θ ˇ φ (2) ) + ( rL ) j T π (sin θ ˇ φ (2) ) + ( rL ) j T π (sin θφ )] . Proof.
We have that LL ( rL ) k φ ( n ) = L ( rLL ( rL ) k − φ ( n ) ) −
12 ∆ r + a r − L ( rL ) k φ − L (cid:18)
12 ∆ r + a r − (cid:19) ( rL ) k φ + [ L, L ]( rL ) k φ = L ( rLL ( rL ) k − φ ( n ) ) −
12 ∆ r + a r − L ( rL ) k φ + (cid:20) kr − + O ∞ ( r − ) (cid:21) ( rL ) k φ + O ∞ ( r − )( rL ) k Φ φ. Furthermore, L ( rL ) k P j = rL ( L ( rL ) k − P j ) −
12 ∆ r + a r − ( rL ) k P j + r [ L, L ]( rL ) k − P j = rL ( L ( rL ) k − P j ) −
12 ∆ r + a r − ( rL ) k P j + O ∞ ( r − )( rL ) k − Φ P j Proposition 5.4.
Let n, (cid:96) ∈ N , with (cid:96) ≥ n , K ∈ N and − n < p ≤ . For R > r + suitably large, thereexists a constant C = C ( M, a, n, (cid:96), K, R, p ) > , such that (cid:90) N τ r p ( Lφ ( n ) ≥ (cid:96) ) + r p − (cid:16) | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) (cid:17) dωd ρ + K K (cid:88) k =1 (cid:90) N τ r p ( L ( rL ) k φ ( n ) ≥ (cid:96) ) + r p − | / ∇ S ( rL ) k φ ( n ) ≥ (cid:96) | dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( Lφ ( n ) ≥ (cid:96) ) + (2 − p ) r p − (cid:16) | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) (cid:17) dωd ρ (cid:21) dτ + K K (cid:88) k =1 (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( L ( rL ) k φ ( n ) ≥ (cid:96) ) + (2 − p ) r p − | / ∇ S ( rL ) k φ ( n ) ≥ (cid:96) | dωd ρ (cid:21) dτ ≤ C (cid:90) N τ r p ( Lφ ( n ) ≥ (cid:96) ) + r p − (cid:16) | / ∇ S φ ( n ) ≥ (cid:96) | − n ( n + 1)( φ ( n ) ≥ (cid:96) ) (cid:17) dωd ρ + CK K (cid:88) k =1 (cid:90) N τ r p ( L ( rL ) k φ ( n ) ≥ (cid:96) ) + r p − | / ∇ S ( rL ) k φ ( n ) ≥ (cid:96) | dωd ρ + C ( n + (cid:96) ) (cid:90) τ τ (cid:20)(cid:90) N τ r p − a (( rL ) K T φ ( n ) ≥ max { (cid:96) − , } ) + r p − a (( rL ) K T Φ φ ( n ) ≥ (cid:96) ) dωd ρ (cid:21) dτ + CK K − (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − a (( rL ) k T φ ( n ) ≥ max { (cid:96) − , } ) + r p − a (( rL ) k T Φ φ ( n ) ≥ (cid:96) ) dωd ρ (cid:21) dτ + Cn (cid:90) τ τ (cid:20)(cid:90) N τ r p − ( L Φ φ ( n − ≥ (cid:96) ) + r p − ( LT φ ( n − ≥ (cid:96) ) dωd ρ (cid:21) dτ + Cn K (cid:88) k =0 n − (cid:88) m =0 (cid:90) τ τ (cid:20)(cid:90) N τ a r p − (( rL ) k Φ φ ( m ) ≥ (cid:96) ) + r p − (( rL ) k φ ( m ) ≥ (cid:96) ) dωd ρ (cid:21) dτ + Cδ (cid:96) δ n K (cid:88) i =1 K − (cid:88) k =0 (cid:90) N τi a r p − ( L ( rL ) k T φ ) dωd ρ + C n + K (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ≥ (cid:96) ] · n τ r dωd ρ + a (1 − δ (cid:96) δ n ) (cid:88) ≤ l ≤(cid:100) (cid:96)/ (cid:101) ≤ l + l ≤ l (cid:90) Σ τ J N [ T m +2 l T l Φ l ψ ≥ max { (cid:96) − l, } · n τ r dωd ρ . (5.7) Furthermore, the estimate (5.7) also holds with the last line replaced by C n + K (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ] · n τ r dωd ρ . Proof.
We will prove the proposition via induction. The K = 0 case follows from Proposition 5.1. Nowsuppose (5.7) holds for some K ∈ N .We then proceed analogously to the proof of Proposition 5.1 by considering the multiplier − r p − r + a ) ∆ L ( rL ) K +1 φ ( n ) ≥ (cid:96) k = K + 1:0 = − r p − L ( rL ) K +1 φ ( n ) ≥ (cid:96) (cid:34) − r + a ) ∆ LL ( rL ) K +1 φ ( n ) ≥ (cid:96) + a sin θ ∆( r + a ) T ( rL ) K +1 φ ( n ) + 2 a ∆( r + a ) T Φ( rL ) K +1 φ ( n ) + ∆( r + a ) − / ∆ S ( rL ) K +1 φ ( n ) + ( − ( K + 1) r − + O ∞ ( r − ))[ a sin θT ( rL ) K φ ( n ) + 2 aT Φ( rL ) K φ ( n ) + / ∆ S ( rL ) K φ ( n ) ] − (cid:20)(cid:18) n + 12 ( K + 1) (cid:19) r − + O ∞ ( r − ) (cid:21) L ( rL ) K +1 φ ( n ) + r − (cid:20) n ( n + 1) + (cid:18) n + 12 (cid:19) ( K + 1) + O ∞ ( r − ) (cid:21) ( rL ) K +1 φ ( n ) + K (cid:88) j =1 O ∞ ( r − )( rL ) j φ ( n ) + K +1 (cid:88) j =0 O ∞ ( r − )Φ( rL ) j φ ( n ) + K ( K + 1) k − (cid:88) j =0 O ∞ ( r − )[ a sin θT ( rL ) j φ ( n ) + 2 aT Φ( rL ) j φ ( n ) + / ∆ S ( rL ) j φ ( n ) ]+ a n − (cid:88) m =0 K +1 (cid:88) j =0 O ∞ ( r − )Φ( rL ) j φ ( m ) + n n − (cid:88) m =0 K +1 (cid:88) j =0 O ∞ ( r − )( rL ) j φ ( m ) (cid:35) =: L ( F r p LL ) + L ( F r p LL ) + / div S F r p L/ ∇ + Φ( F r p L Φ ) + J r pL . The rest of the proof proceeds analogously to the proof of Proposition 5.1, applying in addition the inductionhypothesis (5.7) where needed, with the only term that requires an additional, new, argument being of theform: (cid:20)
12 ( K + 1) r p − + O ∞ ( r p − ) (cid:21) / ∆ S ( rL ) K φ ( n ) ≥ (cid:96) · L ( rL ) K +1 φ ( n ) ≥ (cid:96) . We start by splitting: / ∆ S ( rL ) K φ ( n ) ≥ (cid:96) = − (cid:96) ( (cid:96) + 1)( rL ) K φ ( n ) (cid:96) + / ∆ S ( rL ) K φ ( n ) ≥ (cid:96) +1 . Then we apply a Young’sinequality to obtain: r p − | ( rL ) K φ ( n ) (cid:96) || L ( rL ) K +1 φ ( n ) ≥ (cid:96) | ≤ (cid:15)r p − ( L ( rL ) K +1 φ ( n ) ≥ (cid:96) )) + C (cid:15) r p − (( rL ) K φ ( n ) (cid:96) ) . We can immediately absorb the first term on the right-hand side. We can also control the second term, usingthe induction hypothesis (5.7) if K ≥
1. If K = 0, we instead apply (2.21) to estimate (cid:90) N τ r p − ( φ ( n ) (cid:96) ) dωd ρ ≤ C (cid:90) N τi r p − ( Lφ ( n ) ≥ (cid:96) ) + r p − ( T φ ( n ) ≥ (cid:96) ) + r p − (Φ φ ( n ) ≥ (cid:96) ) dωd ρ + ( . . . ) , with ( . . . ) here denoting boundary terms that can be estimated via the use of a smooth cut-off function.We can similarly split ( rL ) K +1 φ ( n ) ≥ (cid:96) = ( rL ) K φ ( n ) (cid:96) + ( rL ) K φ ( n ) ≥ (cid:96) +1 and repeat the argument above.We apply a Leibniz rule in the L and angular directions to estimate the remaining terms: (cid:20)
12 ( K + 1) r p − + O ∞ ( r p − ) (cid:21) / ∆ S ( rL ) K φ ( n ) ≥ (cid:96) +1 · L ( rL ) K +1 φ ( n ) ≥ (cid:96) +1 = L (cid:20) − (cid:18)
12 ( K + 1) r p − + O ∞ ( r p − ) (cid:19) / ∇ S ( rL ) K φ ( n ) ≥ (cid:96) +1 · / ∇ S ( rL ) K +1 φ ( n ) ≥ (cid:96) +1 (cid:21) + (cid:20)
12 ( K + 1) r p − + O ∞ ( r p − ) (cid:21) | / ∇ S ( rL ) k +1 φ ( n ) ≥ (cid:96) +1 | −
12 (2 − p )[ r p − + O ∞ ( r p − )] / ∇ S ( rL ) K φ ( n ) ≥ (cid:96) +1 · / ∇ S ( rL ) K +1 φ ( n ) ≥ (cid:96) +1 + / div S ( . . . ) . Note moreover that the second term on the right-hand side above has a good sign that we use to control thethird term after applying Young’s inequality. The first term on the right-hand side above contributes to the43erm F r p LL . Note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18)
12 ( K + 1) r p − + O ∞ ( r p − ) (cid:19) / ∇ S ( rL ) K φ ( n ) ≥ (cid:96) +1 · / ∇ S ( rL ) K +1 φ ( n ) ≥ (cid:96) +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15)r p − | / ∇ S ( rL ) K +1 φ ( n ) ≥ (cid:96) +1 | + C (cid:15) r p − | / ∇ S ( rL ) K φ ( n ) ≥ (cid:96) +1 | . We can absorb the terms in the right-hand side above into the remaining terms in F r p LL together with theterms on the left-hand side of the induction hypothesis (5.7).Finally, note that when n = (cid:96) = 0, we cannot no longer energy boundedness (3.13) to control the followingintegrals along N τ i that show up when repeating the arguments of the proof of Proposition 5.1: (cid:88) i =1 (cid:90) N τi a r p − (( rL ) K +1 T φ ) dωd ρ . Proposition 5.5.
Let K ∈ N and < p < . Then the following estimates hold: for R > r + suitably large,there exists a constant C ( M, a, R, p ) > such that:for (cid:96) = 0 : K (cid:88) k =0 (cid:90) N τ r p (( rL ) k P ) dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − (( rL ) k P ) + r p − ( L ( rL ) k φ ) dωd ρ (cid:21) dτ ≤ C K (cid:88) k =0 (cid:90) N τ r p (( rL ) k P ) dωd ρ + C K +1 (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − (( rL ) k T φ ) dωd ρ (cid:21) dτ + C K (cid:88) m =0 (cid:90) Σ τ J T [ T m ψ ] · n τ r dωd ρ , for (cid:96) = 1 : K (cid:88) k =0 (cid:90) N τ r p (( rL ) k P ) dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − (( rL ) k P ) + r p − ( L ( rL ) k ˇ φ (1)1 ) dωd ρ (cid:21) dτ ≤ C K (cid:88) k =0 (cid:90) N τ r p (( rL ) k P ) dωd ρ + C K +1 (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − (( rL ) k T φ ≥ ) + r p − (( rL ) k T φ (1) ≥ ) + r p − ( L ( rL ) k T φ ≥ ) dωd ρ (cid:21) dτ + C K +1 (cid:88) m =0 (cid:90) Σ τ J T [ T m ψ ] · n τ r dωd ρ , or (cid:96) = 2 : K (cid:88) k =0 (cid:90) N τ r p (( rL ) k P ) dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − (( rL ) k P ) + r p − ( L ( rL ) k ˇ φ (2)2 ) dωd ρ (cid:21) dτ ≤ C K (cid:88) k =0 (cid:90) N τ r p (( rL ) k P ) dωd ρ + Ca K +1 (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p +1 ( L ( rL ) k T φ ) + r p − (( rL ) k T φ ) , dωd ρ (cid:21) dτ + C K +1 (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − (( rL ) k T φ ≥ ) + r p − (( rL ) k T φ (1) ≥ ) + r p − (( rL ) k T φ (2) ≥ ) + r p − ( L ( rL ) k T φ (1) ≥ ) dωd ρ (cid:21) dτ + C K +2 (cid:88) m =0 (cid:90) Σ τ J T [ T m ψ ] · n τ r dωd ρ . Proof.
We repeat the proof of Proposition 5.2, using equations (5.4)–(5.6) instead of (4.10)–(4.12), with therole of P i , i = 0 , ,
2, replaced by ( rL ) k P i , with 0 ≤ k ≤ K . rL , / ∇ S and T In this section we show that we can also obtain r p -weighted energy estimates when φ is replaced with / ∇ α S φ , α ∈ N , by making use of the estimates derived in Section 5.3.1.For the sake of convenience, will consider separately φ ( n ) with n = 0 (Proposition 5.6) and n ≥ Proposition 5.6.
Let
K, J ∈ N and < p < . For R > r + suitably large, there exists a constant C = C ( M, a, K, J, R, p ) > , such that (cid:88) j + α ≤ J K (cid:88) k =0 (cid:90) N τ r p | / ∇ α S T j L ( rL ) k φ | + r p − | / ∇ α +1 S T j ( rL ) k φ | dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − | / ∇ α S T j L ( rL ) k φ | + (2 − p ) r p − (cid:16) | / ∇ α +1 S ( rL ) k T j φ | + a sin θ | / ∇ α S T j +1 φ | (cid:17) dωd ρ (cid:21) dτ ≤ C (cid:88) j + α ≤ J (cid:40) K (cid:88) k =0 (cid:90) N τ r p | / ∇ α S T j L ( rL ) k φ | + r p − | / ∇ α +1 S T j ( rL ) k φ | dωd ρ + K K − (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − a | / ∇ α S ( rL ) k T j φ | + r p − a | / ∇ α S ( rL ) k T j Φ φ | dωd ρ (cid:21) dτ + K (cid:88) i =1 K − (cid:88) k =0 (cid:90) N τi a r p − | / ∇ α S L ( rL ) k T j +1 φ ) dωd ρ (cid:41) + C n + K + J (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ] · n τ r dωd ρ . (5.8) Proof.
Note that the J = 0 case follows from Proposition 5.4. We then carry out an inductive argument in J by proceeding analogously to the proof of Proposition 5.4 with n = 0, but considering multipliers of the form − ( − α r p − r + a ) ∆ / ∆ α S L ( rL ) k T j φ . Note that the estimates in the proof of Proposition 5.4 automaticallyhold when φ is replaced by T j φ , since T commutes with the differential operators in (5.3) by the Killingproperty of T j . Furthermore, we integrate by parts an additional α times by parts in the angular directionin order to deal with the / ∆ α S derivative. For this step, we moreover use that / ∆ S commutes with most ofthe terms in (5.3).We will see, for example, the term:2( − α r p − ( r + a ) ∆ / ∆ α S L ( rL ) k φ · LL ( rL ) k φ α ∈ N , we integrate by parts on S to obtain:2( − α r p − ( r + a ) ∆ / ∆ α S L ( rL ) k φ · LL ( rL ) k φ = L (cid:20) r p − ( r + a ) ∆ ( / ∆ α S L ( rL ) k φ ) (cid:21) + (cid:16) p O ∞ ( r − ) (cid:17) r p − ( / ∆ α S L ( rL ) k φ ) + . . . , where . . . denotes terms that vanish after integration over S .If α ∈ N + 1, we instead obtain:2( − α r p − ( r + a ) ∆ / ∆ α S L ( rL ) k φ · LL ( rL ) k φ = L (cid:20) r p − ( r + a ) ∆ | / ∇ S / ∆ α − S L ( rL ) k φ | (cid:21) + (cid:16) p O ∞ ( r − ) (cid:17) r p − | / ∇ S / ∆ α − S L ( rL ) k φ | + . . . . In both cases, control over the desired covariant derivatives on S follows from (2.18). Other terms in(5.3) that commute with / ∆ S can be treated similarly.Hence, the only term that requires an additional argument when K = 0 is the following: − ( − α r p − ( r + a ) / ∆ α S LT j φ · a sin θT j φ, because [ / ∆ S , sin θ ] (cid:54) = 0.After integrating by parts in the angular directions, we obtain − ( − α r p − ( r + a ) / ∆ α S LT j φ · a sin θT j φ = − r p − ( r + a ) / ∇ s S / ∆ α − s S LT j φ · a / ∇ s S / ∆ α − s S (sin θT j φ )+ . . . , with s = 0 if α is even and s = 1 if α is odd.By using the following estimates: | / ∇ S (sin θ ) | ≤ N sin θ, | / ∇ s S / ∆ k S (sin θ ) | ≤ N k , for all k ∈ N and s = 0 ,
1, with N , N k numerical constants together with (2.18), we can estimate (cid:90) S − r p − ( r + a ) / ∇ s S / ∆ α − s S LT j φ · a / ∇ s S / ∆ α − s S (sin θT j φ ) dω ≥ (cid:90) S − r p − ( r + a ) / ∇ s S / ∆ α − s S LT j φ · a sin θ / ∇ s S / ∆ α − s S T j φ − (cid:15)r p − | / ∇ s S / ∆ α − s S LT j φ | − C (cid:15) r p − sin θ ( / ∆ α − s S T j φ ) − C (cid:15) ( α − r p − sin θ α − (cid:88) β =0 | / ∇ β S T j φ | dω, with (cid:15) > α = 1, we can estimate theintegral of the term on the right-hand side above with the factor C (cid:15) by applying (3.11), using that p <
2. If α ≥
2, we instead appeal to the induction step to estimate both terms with the factor C (cid:15) , using again that p < − r p − ( r + a ) / ∇ s S / ∆ α − s S LT j φ · a sin θ / ∇ s S / ∆ α − s S T j φ as in the proof of Proposition 5.1. Remark 5.2.
In contrast with Proposition 5.4, the Proposition 5.6 requires p < rather than p ≤ . Analternative method for controlling higher-order angular derivatives, which would include the case p = 2 , wouldbe to use that (cid:3) g commutes with the Carter operator Q = / ∆ S + a sin θT − Φ , which is an additionalsymmetry property of the Kerr metric. Since the p = 2 estimate is not relevant for the energy decay estimatesin this paper, we have commuted instead with / ∆ S in the proof of Proposition 5.6, as this is less reliant onthe symmetry properties of the background metric. roposition 5.7. Let n ≥ and (cid:96) ≥ n , or n = 0 and (cid:96) ≥ , K, J ∈ N and < p < . For R > r + suitablylarge, there exists a constant C = C ( M, a, K, J, n, (cid:96), R, p ) > , such that (cid:88) α + j ≤ J (cid:40) (cid:90) N τ r p | / ∇ α LT j φ ( n ) ≥ (cid:96) | + r p − (cid:16) | / ∇ α +1 S T j φ ( n ) ≥ (cid:96) | − n ( n + 1) | / ∇ α T j φ ( n ) ≥ (cid:96) | (cid:17) dωd ρ + K K (cid:88) k =1 (cid:90) N τ r p | / ∇ α L ( rL ) k T j φ ( n ) ≥ (cid:96) | + r p − | / ∇ α +1 S ( rL ) k T j φ ( n ) ≥ (cid:96) | dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − | / ∇ α LT j φ ( n ) ≥ (cid:96) | + (2 − p ) r p − (cid:16) | / ∇ α +1 S T j φ ( n ) ≥ (cid:96) | − n ( n + 1) | / ∇ α T j φ ( n ) ≥ (cid:96) | (cid:17) dωd ρ (cid:21) dτ + K K (cid:88) k =1 (cid:90) τ τ (cid:20)(cid:90) N τ r p − | / ∇ α L ( rL ) k T j φ ( n ) ≥ (cid:96) ) + (2 − p ) r p − [ | / ∇ α +1 S ( rL ) k T j φ ( n ) ≥ (cid:96) | dωd ρ (cid:21) dτ (cid:41) ≤ C (cid:88) α + j ≤ J (cid:40) (cid:90) N τ r p | / ∇ α LT j φ ( n ) ≥ (cid:96) | + r p − (cid:16) | / ∇ α +1 S T j φ ( n ) ≥ (cid:96) | − n ( n + 1) | / ∇ α T j φ ( n ) ≥ (cid:96) | (cid:17) dωd ρ + K K (cid:88) k =1 (cid:90) N τ r p | / ∇ α L ( rL ) k T j φ ( n ) ≥ (cid:96) | + r p − | / ∇ α +1 S ( rL ) k T j φ ( n ) ≥ (cid:96) | dωd ρ + (cid:90) τ τ (cid:20)(cid:90) N τ r p − a | / ∇ α ( rL ) K T j φ ( n ) ≥ max { (cid:96) − , } | + r p − a | / ∇ α ( rL ) K T j Φ φ ( n ) ≥ (cid:96) | dωd ρ (cid:21) dτ + K K − (cid:88) k =0 (cid:90) τ τ (cid:20)(cid:90) N τ r p − a | / ∇ α ( rL ) k T j φ ( n ) ≥ max { (cid:96) − , } | + r p − a | / ∇ α ( rL ) k T j Φ φ ( n ) ≥ (cid:96) | dωd ρ (cid:21) dτ + (cid:90) τ τ (cid:20)(cid:90) N τ a r p − | / ∇ α L Φ T j φ ( n − ≥ (cid:96) | + r p − | / ∇ α LT j φ ( n − ≥ (cid:96) | dωd ρ (cid:21) dτ + K (cid:88) k =0 n − (cid:88) m =0 (cid:90) τ τ (cid:20)(cid:90) N τ a r p − | / ∇ α ( rL ) k Φ T j φ ( m ) ≥ (cid:96) | + r p − | / ∇ α ( rL ) k T j φ ( m ) ≥ (cid:96) | dωd ρ (cid:21) dτ (cid:41) + C n + K + J (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ≥ (cid:96) ] · n τ r dωd ρ + a (cid:88) ≤ l ≤(cid:100) (cid:96)/ (cid:101) ≤ l + l ≤ l (cid:90) Σ τ J N [ T m +2 l T l Φ l ψ ≥ max { (cid:96) − l, } · n τ r dωd ρ . (5.9) Furthermore, the estimate (5.7) also holds with the last line replaced by C n + K + J (cid:88) m =0 (cid:90) Σ τ J N [ T m ψ ] · n τ r dωd ρ . Proof.
We then carry out an inductive argument in J by repeating the arguments in the proof of Propo-sition 5.4, but considering multipliers of the form − ( − α r p − r + a ) ∆ / ∆ α S L ( rL ) k T j φ ( n ) ≤ (cid:96) and additionallyintegrating by parts in the angular directions. In contrast with the proof of Proposition 5.6, the fact that[ / ∇ S , sin θ ] (cid:54) = 0 does not affect the argument, as all terms involving factors of sin θ are estimated using astraightforward Young’s inequality. T -derivatives We can obtain control over additional r p -weighted hierarchies when considering T φ rather than φ , by usingthe favourable commutation properties of rL and / ∇ S that follow from Propositions 5.6, 5.7 and 5.5. Thesewill be important for proving better energy decay rates for additional T -derivatives.The relevant key lemma is the following: 47 emma 5.8. Let n, K, J ∈ N and (cid:96) ≥ n . Then there exists a constant C ( M, a, R, p, n, K, J, (cid:96) ) > suchthat for all p ∈ R : K (cid:88) k =0 J (cid:88) α =0 (cid:90) S r p +1 | / ∇ α S L ( rL ) k T φ ( n ) ≥ (cid:96) | dω ≤ CK J (cid:88) α =0 K (cid:88) k =0 (cid:88) γ + β ≤ (cid:90) S r p − | / ∇ α + γ S L ( rL ) k + β φ ( n ) ≥ (cid:96) | + r p − | / ∇ γ + α +1 S ( rL ) k + β φ ( n ) ≥ (cid:96) | + r p − | / ∇ γ + α S ( rL ) k + β φ ( n ) ≥ (cid:96) | dω + C J (cid:88) α =0 K (cid:88) k =0 (cid:90) S a r p − | / ∇ α S ( rL ) k T φ ( n ) ≥ max { (cid:96) − , } | + a r p − | / ∇ α S ( rL ) k T Φ φ ( n ) ≥ (cid:96) | dω + C J (cid:88) α =0 K (cid:88) k =0 n − (cid:88) m =0 (cid:90) S a r p − | / ∇ α S ( rL ) k Φ φ ( m ) ≥ (cid:96) | + r p − | / ∇ α S ( rL ) k φ ( m ) ≥ (cid:96) | dω. (5.10) Furthermore, K (cid:88) k =0 (cid:90) S r p +1 (( rL ) k T P j ) dω ≤ C K +1 (cid:88) k =0 (cid:90) S r p − (( rL ) k P j ) + j (cid:88) i =0 r p − (( rL ) k ˇ φ ( i ) j ) dω + C K +1 (cid:88) k =0 j (cid:88) i =0 (cid:90) S a r p − [(( rL ) k T φ ( i ) ≥ j ) + (( rL ) k T φ ( i ) ≤ j − ) ] dω. (5.11) Proof.
We first split:
T L ( rL ) k φ ( n ) = LL ( rL ) k φ ( n ) + LL ( rL ) k φ ( n ) − ar + a Φ L ( rL ) k φ ( n ) ,T ( rL ) k P j = L ( rL ) k P j + r − ( rL ) k +1 P j − ar + a Φ( rL ) k P j and then we square both sides of the equations together with Lemma 5.3 to estimate ( LL ( rL ) k φ ( n ) ) and( L ( rL ) k P j ) and obtain (5.10) with J = 0.Now take J ≥
1. Then we repeat the above procedure, but rather than squaring both sides of Lemma5.3, we multiply the right-hand side of Lemma 5.3 with ( − α / ∆ α S acting on the right-hand side of Lemma5.3 and integrate by parts in the angular directions, using moreover (2.18). In this section, we convert the hierarchy of r p -weighted estimates established in Propositions 5.6, 5.7 and5.5 into energy decay estimates. We outline below the strategy for obtaining both sharp energy decay ratesfor ψ (cid:96) with (cid:96) = 0 , , ψ ≥ :A) We first obtain a preliminary energy decay estimate for the full solution ψ in Proposition 6.1 ( Proposition6.2 ).B) We then consider the restriction ψ ≥ and obtain a corresponding improved energy decay estimate usingthe preliminary energy decay established in Proposition 6.1 to deal with the contributions of the mode ψ that play a role in the estimate. We then consider the further restrictions ψ ≥ and ψ ≥ and keepimproving the energy decay estimates successively. ( Proposition 6.2 ).C) Subsequently, we restrict to the single spherical harmonic modes ψ , ψ and ψ and improve the energydecay estimate even further. The corresponding equations are coupled with the remaining sphericalharmonic modes and the terms that appear in the estimates due to this coupling are controlled usingthe energy decay estimates already established in Proposition 6.2. We arrive at energy estimates thatare sharp when considering initial data with non-vanishing Newman–Penrose charges I (cid:96) , (cid:96) = 0 , , Proposition 6.3 ) 48y applying the time-inversion theory from Section 9 below, we will perform an additional step:D) Applying the energy decay estimates Proposition 6.3 to the time-integral T − ψ (Proposition 9.9) allows usto obtain sharp energy decay estimates when considering initial data with vanishing Newman–Penrosecharges, in particular, compactly supported initial data (
Corollary 9.10 ). Proposition 6.1.
Let K ∈ N and δ > be arbitrarily small. Then there exists a constant C = C ( M, a, K, R, δ ) > , (cid:90) Σ τ J N [ T K ψ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k + α ≤ K (1 + τ ) − m − K − k ) (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ | dωd ρ ≤ C (1 + τ ) − δ − K E ≥ ,K,δ [ ψ ] (6.1) with E ≥ ,K,δ [ ψ ] := (cid:88) ≤ j = j + j ≤ (cid:88) k + k + α ≤ K (cid:90) N r − δ − j | / ∇ α S L ( rL ) k T j +2 k Φ j φ | + r − − δ − j | / ∇ α S ( rL ) k T j +2 k Φ j φ | dωd ρ + (cid:88) ≤ k + k ≤ K (cid:90) Σ J N [ T k Φ k ψ ] · n r dωd ρ . Proof.
We consider first the case K = 0. We obtain energy decay by applying Proposition 5.1 with n = (cid:96) = 0in combination with the following ingredients:1) the mean value theorem along dyadic time intervals (the “pigeonhole principle”),2) an interpolation inequality (Lemma B.1) ,3) energy boundedness estimate in the form (3.9), (3.13) and (3.11),4) an integrated local energy decay estimate (3.12).Let { τ i } be a dyadic sequence, then Proposition 5.1 with p = 1 together with (3.11) implies that: (cid:90) τ i +1 τ i (cid:90) N τ ( Lφ ) + r − ( Lφ ) + r − | / ∇ S φ | dωd ρ dτ ≤ C (cid:90) N τi r ( Lφ ) + r − | / ∇ S φ | dωd ρ + C (cid:90) Σ τi J N [ ψ ] · n τ i r dωd ρ . (6.2)By combining (6.2) with (3.10), applying the mean value theorem and then applying (3.9), we obtain τ − decay for the N -energy, with a loss of T and Φ derivatives on the right-hand side. We can improve thisenergy decay rate by considering the spacetime integral of the right-hand side of (6.2), applying Proposition5.1 with p = 2 − δ together with (3.11) and repeating the arguments above together with the interpolationinequality in Lemma B.1 to obtain (6.1) with K = 0. Note that the decay rate corresponds to the totallength of the hierarchy of r p -weighted estimates applied minus δ .Now consider K = 1. The above arguments hold automatically for ψ replaced by T ψ , but we can improvethe energy decay even further by considering p = 3 − δ in the following way: we apply Lemma 5.8 with p = 1 − δ and J = K = 0, and instead of Proposition 5.1, we appeal to the r p -estimates in Proposition 5.6with p = 1, J = 1, and K = 1. When then obtain τ − δ decay. Subsequently, we consider Proposition 5.6with p = 2 − δ . As the length of our hierarchy is now 4, we are left τ − δ decay of the N -energy. In thisprocess, we automatically obtain control over additional r -weighted quantities with slower decay rates.We can treat the general K case inductively, by repeatedly applying Lemma 5.8 with n = (cid:96) = 0 and p = 1 − δ and suitably high values of J and K , together with Proposition 5.6 with p = 1 and p = 2 − δ andsuitably high values of J and K . We obtain in this way a hierarchy of length 2 + 2 K and hence, τ − − K decay for the N -energy. 49 roposition 6.2. Let K ∈ N and δ > be arbitrarily small. Then there exists a constant C = C ( M, a, K, R, δ ) > , (cid:90) Σ τ J N [ T K ψ ≥ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k + α ≤ K (1 + τ ) − m − K − k )+ δ (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ ≥ | dωd ρ (6.3)+ (cid:88) m =0 (cid:88) k + k + α ≤ K (1 + τ ) − − m − K − k )+ δ (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ (1) ≥ | dωd ρ ≤ C (1 + τ ) − δ − K E ≥ ,K,δ [ ψ ] , (cid:90) Σ τ J N [ T K ψ ≥ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k + α ≤ K (1 + τ ) − m − K − k )+ δ (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ ≥ | dωd ρ (6.4)+ (cid:88) m =0 2 (cid:88) j =1 (cid:88) k + k + α ≤ K (1 + τ ) − j − m − K − k )+ δ (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ ( j ) ≥ | dωd ρ ≤ C (1 + τ ) − δ − K E ≥ ,K,δ [ ψ ] , (cid:90) Σ τ J N [ T K ψ ≥ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k + α ≤ K (1 + τ ) − m − K − k )+ δ (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ ≥ | dωd ρ (6.5)+ (cid:88) m =0 3 (cid:88) j =1 (cid:88) k + k + α ≤ K (1 + τ ) − j − m − K − k )+ δ (cid:90) N τ r m − δ | / ∇ α S L ( rL ) k T k φ ( j ) ≥ | dωd ρ ≤ C (1 + τ ) − δ − K E ≥ ,K,δ [ ψ ] , with E ≥ (cid:96),K,δ [ ψ ] := (cid:88) m ≤ M T,(cid:96) ,m ≤ M Φ ,(cid:96) (cid:40) (cid:96) (cid:88) j =0 (cid:88) k + α ≤ K +3 − j (cid:90) N r − δ | / ∇ α S L ( rL ) k T m Φ m φ ( j ) ≥ j | + r − − δ | / ∇ α S ( rL ) k T m Φ m φ ( j ) ≥ j | dωd ρ + (cid:90) Σ J T [ T m Φ m ψ ] · n r dωd ρ (cid:41) , where (cid:96) = 1 , , and M T,(cid:96) and M Φ ,(cid:96) , are suitably large integers.Proof. We consider first (6.3). Let K = 0. Note that the estimates of Proposition 6.1 still hold when wereplace φ in the left-hand side by φ ≥ , by orthogonality of the spherical harmonic modes. When considering φ ≥ , we can further estimate via (2.21): (cid:90) τ i +1 τ i (cid:90) N τ r − δ [( Lφ ≥ ) + r − | / ∇ S φ ≥ | ] dωd ρ dτ ≤ C (cid:88) m + m ≤ (cid:90) τ i +1 τ i (cid:90) N τ r − δ (cid:104) ( Lφ (1) ≥ ) + ( LT m Φ m φ ≥ ) + r − | / ∇ S T m Φ m φ ≥ | (cid:105) dωd ρ dτ.
50e then apply Proposition 5.1 with n = (cid:96) = 1 and p = 1 − δ to obtain: (cid:90) τ i +1 τ i (cid:90) N τ r − δ [( Lφ ≥ ) + r − | / ∇ S φ ≥ | ] dωd ρ dτ ≤ C (cid:90) N τi r − δ ( Lφ (1) ≥ ) + r − − δ [ | / ∇ S φ (1) ≥ | − φ (1) ≥ ) ] dωd ρ + C (cid:88) m + m ≤ (cid:90) τ i +1 τ i (cid:90) N τ r − δ ( LT T m Φ m φ ) dωd ρ dτ + C (cid:90) τ i +1 τ i (cid:90) N τ r − − δ [(Φ φ ≥ ) + ( φ ≥ ) ] dωd ρ dτ + C (cid:88) m + m ≤ E , ,δ [ T m Φ m ψ ] + C (cid:88) m =0 (cid:90) Σ τi J N [ T m ψ ] · n i r dωd ρ . (6.6)We apply (2.21) to estimate further the third integral on the right-hand side of (6.6): (cid:90) τ i +1 τ i (cid:90) N τ r − − δ [(Φ φ ≥ ) + ( φ ≥ ) ] dωd ρ dτ ≤ C (cid:90) τ i +1 τ i (cid:90) N τ r − δ [( L Φ φ ≥ ) + ( Lφ ≥ ) ] dωd ρ dτ + C (cid:88) m + m =1 (cid:90) τ i +1 τ i (cid:90) N τ r − − δ [(Φ T m Φ m Φ φ ≥ ) + ( T m Φ m φ ≥ ) ] dωd ρ dτ. Both terms can be controlled by applying the estimates in the proof of Proposition 6.1.Note that the second integral on the right-hand side of (6.6) does not involve φ ≥ , but rather the fullfunction φ , so we cannot appeal to Proposition 5.1 with n = 1. However, as it involves T φ rather than φ ,we can apply Lemma 5.8 with p = 1 and J = K = 0 to control it using Proposition 6.1 with K = 1. We arethen left with τ − δ energy decay for the N -energy of ψ ≥ .In order to obtain τ − δ decay, we apply Proposition 5.1 with n = (cid:96) = 1 and p = 2 − δ to controlthe right-hand side of (6.6). Note that the energy boundedness estimate (3.9) and local integrated energydecay estimates (3.10) and (3.12) have N -energies of ψ on the right-hand side, for which we have merelyestablished τ − δ decay, so after applying the mean-value theorem along dyadic intervals, we are limited to τ − δ energy decay for ψ ≥ .We therefore appeal instead to the refined energy boundedness estimate (3.14) and the refined integratedlocal integrated energy decay estimates (3.15) and (3.16) for ψ ≥ , together with Proposition 5.1 with n = 0and (cid:96) = 1, so that we do not see the N -energy of the full solution ψ , but rather the N -energy of T -derivativesof ψ , which decay faster in time.With the above observations, taking into account the additional loss of T and Φ derivatives due to the useof the refined energy boundedness and local integrated energy decay estimates, we obtain (6.3) with K = 0.Note that we have extended the total hierarchy of r p -weighted estimates by 2, which results in τ − δ decayfor the N -energy of ψ ≥ , compared to τ − δ -decay for ψ ≥ .Proposition 5.1 with n = (cid:96) = 1 and p = 2 − δ , p = 1 − δ and p = − δ provides moreover boundedness ofthe higher-order weighted energies: (cid:90) N τ (1 + τ ) − m r m − δ ( Lφ (1) ) dωd ρ with m = 0 , , K ≥ n = (cid:96) = 1and p = 1 − δ and suitably high values of J and K , together with Proposition 5.7 with p = 1 − δ and p = 2 − δ and suitably high values of J and K .Now consider (6.4). The estimates for ψ ≥ still apply to ψ ≥ . As in the (cid:96) ≥ (cid:90) τ i +1 τ i (cid:90) N τ r − δ [( Lφ (1) ≥ ) + r − | / ∇ S φ (1) ≥ | ] dωd ρ dτ ≤ C (cid:88) m + m ≤ (cid:90) τ i +1 τ i (cid:90) N τ r − δ (cid:104) ( Lφ (2) ≥ ) + ( LT m Φ m φ (1) ≥ ) + r − | / ∇ S T m Φ m φ (1) ≥ | (cid:105) dωd ρ dτ.
51e then apply Proposition 5.1 with n = (cid:96) = 2 and p = 1 − δ and p = 2 − δ to extend the hierarchy of r p -weighted estimates by 2 (in the K = 0 case). Note that we see in particular on the right-hand side theterm: (cid:90) τ i +1 τ i (cid:90) N τ r p − a ( T φ (2) ) dωd ρ ≤ C (cid:88) k =0 (cid:90) τ i +1 τ i (cid:90) N τ r p +3 a ( L ( rL ) k T φ ) dωd ρ By applying Lemma 5.8 twice, it follows that we can estimate the resulting integral with the left-hand sideof (6.1) with K = 2 and additional commutations with T and Φ.We avoid N -energies of the full solution ψ appearing on the right-hand side of Proposition 5.1 with n = 0 , (cid:96) = 2, at the expense of losing additional T and Φ derivatives on the right-hand side. Similarly,we apply (3.14), (3.15) and (3.16) to ensure we only see N -energies of sufficiently many T -derivatives of thefull solution ψ on the right-hand side, which decay suitably fast. The rest of the argument proceeds in asimilar manner.Finally, we prove (6.5) by naturally following the strategy outlined above. Note that we exploit here thefact that the equation for ψ ≥ is only coupled with ψ ≥ , and the terms involving T φ ≥ when K = 0 can beestimated via (6.3) with K = 2. The estimates for higher K then follow inductively as above. Remark 6.1.
In principle, one can keep precise track of the number of additional T and Φ derivatives onthe right-hand sides of the inequalities in Proposition 6.2, but we do not expect this number to be sharp.In order to arrive at a sharp number of T and Φ derivatives, one can instead decompose ψ into a boundedfrequency part with respect to a time frequency ω and azimuthal mode m , for which there will not be a lossof T and Φ derivatives, and a high-frequency part (supported on suitably large frequencies ω and m ) that isdecoupled with lower (cid:96) modes, as these are supported on lower values of m . These two parts can be treatedseparately. We do not carry out this procedure in the present paper. Proposition 6.3.
Let K ∈ N and δ > be arbitrarily small. Then there exists a constant C = C ( M, a, K, R, δ ) > , such that (cid:90) Σ τ J N [ T K ψ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k ≤ K (1 + τ ) − m − K − k )+ δ (cid:90) N τ r m − δ ( L ( rL ) k T k φ ) dωd ρ (6.7) ≤ C (1 + τ ) − δ − K E ,K,δ [ ψ ] , (cid:90) Σ τ J N [ T K ψ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k ≤ K (1 + τ ) − m − K − k )+ δ (cid:90) N τ r m − δ ( L ( rL ) k T k φ ) dωd ρ (6.8)+ (cid:88) m =0 (cid:88) k + k ≤ K (1 + τ ) − − m − K − k )+ δ (cid:90) N τ r m − δ ( L ( rL ) k T k φ (1)1 ) dωd ρ , ≤ C (1 + τ ) − δ − K E ,K,δ [ ψ ] , (cid:90) Σ τ J N [ T K ψ ] · n τ r dωd ρ + (cid:88) m =1 (cid:88) k + k ≤ K (1 + τ ) − m − K − k )+ δ (cid:90) N τ r m − δ ( L ( rL ) k T k φ ) dωd ρ (6.9)+ (cid:88) j =0 2 (cid:88) m =0 (cid:88) k + k + α ≤ K (1 + τ ) − j − m − K − k )+ δ (cid:90) N τ r m − δ ( L ( rL ) k T k φ ( j )2 ) dωd ρ ≤ C (1 + τ ) − δ − K E ,K,δ [ ψ ] . with E (cid:96),K [ ψ ] := (cid:88) m ≤ M (cid:96),T (cid:40) (cid:96) (cid:88) j =0 (cid:88) k + α ≤ K + (cid:96) − j (cid:90) N r − δ | / ∇ α S L ( rL ) k T m φ ( j ) ≥ j | + r − − δ | / ∇ α S ( rL ) k T m φ ( j ) ≥ j | + r − δ | / ∇ α S ( rL ) k T m P (cid:96) | dωd ρ + (cid:90) Σ J N [ T m ψ ] · n r dωd ρ (cid:41) . where M (cid:96),T are suitably large integers. roof. We consider first ψ with K = 0. Then, we can extend the length of the hierarchy of r p -estimates inthe proof of Proposition 6.1 by 2, via a consideration of the quantity P and an application of the estimatesin Proposition 5.2 for (cid:96) = 0 with p = 3 − δ and p = 4 − δ .Note that the full φ appears on the right-hand side of the relevant estimate of Proposition 5.2, but itcan be easily estimated using Proposition 5.2. The lengths of the hierarchies in the proof of Proposition 6.1that we appeal to when K ≥ r p -estimates applies also whenconsidering ψ and ψ . We consider P j with j = 1 , φ , we apply Proposition 6.2. Remark 6.2.
Note that the estimates for each ψ (cid:96) in Proposition 6.3 can be derived independently of eachother, i.e. they are not coupled. The mode coupling that occurs in these estimates can instead be dealt withby applying the estimates that we already established in Propositions 6.1 and 6.2. It will be useful to establish also energy decay estimates for the commuted quantities of the form ( rX ) j ψ (cid:96) ,with (cid:96) = 0 , , Corollary 6.4.
Let (cid:96) = 0 , , . Let δ > be arbitrarily small and K, J ∈ N . Then there exists a constant C = C ( M, a, R, K, J, (cid:96), δ ) > , such that J (cid:88) j =0 (cid:90) Σ τ J N [( rX ) j T K ψ (cid:96) ] · n τ r dωd ρ ≤ C (1 + τ ) − − (cid:96) + δ − K E (cid:96),J + K,δ [ ψ ] + J (cid:88) j =0 E (cid:96),K,δ [ N j ψ ] (6.10) Furthermore, for (cid:96) = 3 , , there exists a constant C = C ( M, a, R, K, J, (cid:96), δ ) > , such that J (cid:88) j =0 (cid:90) Σ τ J N [( rX ) j T K ψ (cid:96) ] · n τ r dωd ρ ≤ C (1 + τ ) − δ − K E ≥ ,J + K,δ [ ψ ] + J (cid:88) j =0 E ≥ ,K,δ [ N j ψ ] . (6.11) Proof.
Let (cid:96) = 0 , , K = 0. We will show that (6.10) and (6.10) hold by induction in J . Then J = 0 case follows from Proposition 6.3 for the (cid:96) = 0 , , (cid:96) = 3 , (cid:88) ≤ j + j ≤ J (cid:90) N τ r − δ ( L ( rL ) j T j φ (cid:96) ) dωd ρ ≤ CE (cid:96),J,δ [ ψ ](1 + τ ) − − (cid:96) + δ . (6.12)Furthermore, by applying the arguments in the proof of Lemma 5.8, we obtain: (cid:90) N τ r − δ ( L ( rL ) J +1 φ (cid:96) ) dωd ρ ≤ C J (cid:88) j =0 (cid:90) N τ r − δ ( L ( rL ) j T φ (cid:96) ) dωd ρ + C J (cid:88) j =0 (cid:90) N τ r − δ ( L ( rL ) j φ (cid:96) ) dωd ρ + C J (cid:88) j =0 (cid:90) N τ r − − δ (( rL ) j T φ ) + r − − δ (( rL ) j φ (cid:96) ) dωd ρ . By Proposition 6.3, we have that J (cid:88) j =0 (cid:90) N τ r − δ ( L ( rL ) j φ (cid:96) ) dωd ρ ≤ CE (cid:96),J,δ [ ψ ](1 + τ ) − − (cid:96) , (6.13) J (cid:88) j =0 (cid:90) N τ r − δ ( L ( rL ) j φ (cid:96) ) dωd ρ ≤ CE (cid:96),J,δ [ ψ ](1 + τ ) − (cid:96) . (6.14)53sing (6.14) together with the fact that (6.12) also holds if we replace ψ with T ψ , we therefore obtain(6.12) with J replaced by J + 1.In order to remover the r − δ degeneracy in (6.12), we interpolate between (6.12) and (6.13) using LemmaB.1 to obtain J (cid:88) j =0 (cid:90) N τ J N [( rL ) j ψ (cid:96) ] · n τ r dωd ρ ≤ CE (cid:96),J,δ [ ψ ](1 + τ ) − − (cid:96) +2 δ . The K ≥ X in terms of L and T and using that (3.1) commuteswith T , we obtain the desired estimates on N τ = Σ τ ∩ { r ≥ R } .We obtain the desired estimate on Σ τ ∩ { r + < r ≤ r ≤ R } , with r arbitrarily close to r + , via standardelliptic estimates and commutation of (3.1) with T and Φ: in particular, in the K = 0 we have that J (cid:88) j =0 (cid:90) Σ τ ∩{ r +
We multiply both sides of (7.1) by Xψ , to obtain XψF = ∆
XψX ψ + ∆ (cid:48) ( Xψ ) + 2 aXψX Φ ψ + Xψ / ∆ S ψ = X (cid:18)
12 ∆( Xψ ) − | / ∇ S ψ | (cid:19) + 12 ∆ (cid:48) ( Xψ ) + Φ( a ( Xψ ) ) + / div S ( Xψ / ∇ S ψ ) . Integrating along Σ τ and using that lim ρ →∞ / ∇ S ψ ( τ, ρ , θ, ϕ ) = 0, we obtain (7.3).In order to obtain (7.4), we first apply Young’s inequality to the terms in r − F to obtain r − F (cid:46) r ( XT φ ) + r − ( T φ ) + r − (Φ T φ ) + r − ( T φ ) , absorbing the integral of r − ( T φ ) into the remaining terms, using (2.20), and then relating with J T [ T ψ ] · n τ by applying (2.20) once more.In the proposition below, we obtain local, higher-order versions of the estimate (7.3) by projecting tofixed spherical harmonic modes. These provide control inside the ergoregion and will be combined with theelliptic estimates outside the ergoregion derived in Section 7.2. Proposition 7.2.
Let ψ be a solution to (7.1) that is suitably regular, then: (i) For arbitrary r > r + , there exists a constant C = C ( M, a, r , (cid:96) ) > such that (cid:90) Σ τ ∩{ r ≤ r } ( X (∆ X (cid:96) +1 ψ (cid:96) )) + ( X (cid:96) +1 ψ (cid:96) ) dωd ρ ≤ C (cid:90) Σ τ ∩{ r ≤ r } ( π (cid:96) X (cid:96) F ) dωd ρ . (7.6)(ii) For all r ≥ r + , there exists a constant C = C ( M, a ) > , such that furthermore: || X (cid:96) +1 ψ (cid:96) || L ( S τ,r (cid:48) ) ≤ C sup r + ≤ r ≤ r (cid:48) || r − π (cid:96) X (cid:96) F || L ( S τ,r ) . (7.7)(iii) There exists a constant C = C ( M, a, (cid:96) ) > such that || X (cid:96) +1 ψ (cid:96) || L ( S τ,r (cid:48) ) ≤ C (cid:96) +1 (cid:88) k =0 sup r + ≤ r ≤ r (cid:48) || r − (cid:96) ( rX ) k T ψ (cid:96) || L ( S τ,r ) + C (cid:96) (cid:88) k =0 sup r + ≤ r ≤ r (cid:48) || r − − (cid:96) ( rX ) k T ψ (cid:96) || L ( S τ,r ) + Ca sup r + ≤ r ≤ r (cid:48) || r − X (cid:96) T ψ (cid:96) − || L ( S τ,r ) + Ca sup r + ≤ r ≤ r (cid:48) || r − X (cid:96) T ψ (cid:96) +2 || L ( S τ,r ) . (7.8)(iv) There exists a constant C = C ( M, a, (cid:96) ) > such that for (cid:96) ≥ : || r X (cid:96) +1 ψ (cid:96) || L ( S τ,r (cid:48) ) ≤ C (cid:96) +1 (cid:88) k =0 sup r + ≤ r ≤ r (cid:48) || r − (cid:96) ( rX ) k T ψ (cid:96) || L ( S τ,r ) + C (cid:96) (cid:88) k =0 sup r + ≤ r ≤ r (cid:48) || r − (cid:96) ( rX ) k T ψ (cid:96) || L ( S τ,r ) + Ca sup r + ≤ r ≤ r (cid:48) || X (cid:96) T ψ (cid:96) − || L ( S τ,r ) + Ca sup r + ≤ r ≤ r (cid:48) || X (cid:96) T ψ (cid:96) +2 || L ( S τ,r ) . (7.9)55 roof. It follows easily by induction that for all n ≥ X n +1 (∆ Xψ ) = X (∆ X n +1 ψ ) + n ∆ (cid:48) X n +1 ψ + n ( n + 1) X n ψ. Hence, X n F = X (∆ X n +1 ψ ) + n ∆ (cid:48) X n +1 ψ + n ( n + 1) X n ψ + 2 aX Φ X n ψ + / ∆ S ( X n ψ ) . (7.10)A cancellation occurs in (7.10) when n = (cid:96) after projecting onto the (cid:96) -th spherical harmonic mode: X (cid:96) π (cid:96) F = X (∆ X (cid:96) +1 ψ (cid:96) ) + (cid:96) ∆ (cid:48) X (cid:96) +1 ψ (cid:96) + 2 a Φ X (cid:96) +1 ψ (cid:96) (7.11)Now consider the multiplier X (cid:96) +1 ψ (cid:96) to obtain: X (cid:96) +1 ψ (cid:96) · X (cid:96) π (cid:96) F =∆ X (cid:96) +1 ψ (cid:96) X (cid:96) +2 ψ (cid:96) + ( (cid:96) + 1)∆ (cid:48) ( X (cid:96) +1 ψ (cid:96) ) + 2 aX (cid:96) +1 ψ (cid:96) X (cid:96) +1 Φ ψ (cid:96) = X (cid:18)
12 ∆( X (cid:96) +1 ψ (cid:96) ) (cid:19) + Φ( a ( X (cid:96) +1 ψ (cid:96) ) ) + (cid:18) (cid:96) + 12 (cid:19) ∆ (cid:48) ( X (cid:96) +1 ψ (cid:96) ) . Hence, we can integrate the above inequality to obtain for all r ≥ r + : (cid:90) S
12 ∆( X (cid:96) +1 ψ (cid:96) ) | r = r ,τ = τ (cid:48) dω + (cid:18) (cid:96) + 12 (cid:19) (cid:90) Σ τ (cid:48) ∩{ r ≤ r } ∆ (cid:48) ( X (cid:96) +1 ψ (cid:96) ) dωd ρ ≤ (cid:90) Σ τ (cid:48) ∩{ r ≤ r } | X (cid:96) +1 ψ (cid:96) · X (cid:96) π (cid:96) F | dωd ρ ≤ (cid:90) Σ τ (cid:48) ∩{ r ≤ r } (cid:15) ∆ (cid:48) ( X (cid:96) +1 ψ (cid:96) ) + 12 (cid:15) (cid:48) ( X (cid:96) π (cid:96) F ) dωd ρ . (7.12)Then, (7.6) immediately follows after absorbing the ( X (cid:96) +1 ψ (cid:96) ) term on the very right-hand side of (7.12)into the left-hand side (taking (cid:15) > X (∆ X (cid:96) +1 ψ (cid:96) ) term by using:( X (∆ X (cid:96) +1 ψ (cid:96) ) ≤ C (∆ (cid:48) X (cid:96) +1 ψ (cid:96) ) + Ca (Φ X (cid:96) +1 ψ (cid:96) ) + ( X (cid:96) π (cid:96) F ) . It moreover follows that for r (cid:48) ∈ [ r + , r ]: (cid:90) S ∆( r (cid:48) )( X (cid:96) +1 ψ (cid:96) ) | r = r (cid:48) ,τ = τ (cid:48) dω ≤ C (cid:90) Σ τ (cid:48) ∩{ r ≤ r (cid:48) } r − ( X (cid:96) π (cid:96) F ) dωdρ ≤ C ∆( r (cid:48) ) r (cid:48)− (cid:90) S sup r + ≤ r ≤ r (cid:48) r − ( X (cid:96) π (cid:96) F ) | τ = τ (cid:48) dω and hence we obtain (7.7) for the range r (cid:48) ∈ [ r + , r ], with r > r + arbitrarily large, with a constant C depending moreover on r .In order to prove (7.7) for r (cid:48) ∈ ( r , ∞ ), we appeal again to (7.12) and Young’s inequality to estimate: (cid:90) S
12 sup r ≤ ˜ r ≤ r (cid:48) ∆( X (cid:96) +1 ψ (cid:96) ) | r =˜ r,τ = τ (cid:48) dω ≤ (cid:90) S
12 ∆( X (cid:96) +1 ψ (cid:96) ) | r = r dω + (cid:90) Σ τ (cid:48) ∩{ r ≤ ˜ r ≤ r (cid:48) } | X (cid:96) +1 ψ (cid:96) · X (cid:96) π (cid:96) F | dωd ρ ≤ (cid:90) S
12 ∆( X (cid:96) +1 ψ (cid:96) ) | r = r ,τ = τ (cid:48) dω + 14( r (cid:48) − r ) (cid:90) Σ τ (cid:48) ∩{ r ≤ ˜ r ≤ r (cid:48) } ∆( X (cid:96) +1 ψ (cid:96) ) dωd ρ + ( r (cid:48) − r ) (cid:90) Σ τ (cid:48) ∩{ r ≤ ˜ r ≤ r (cid:48) } ∆ − ( X (cid:96) π (cid:96) F ) dωd ρ ≤ (cid:90) S
12 ∆( X (cid:96) +1 ψ (cid:96) ) | r = r ,τ = τ (cid:48) dω + 14 (cid:90) S sup r ≤ ˜ r ≤ r (cid:48) ∆( X (cid:96) +1 ψ (cid:96) ) | r =˜ r,τ = τ (cid:48) dω + ( r (cid:48) − r ) (cid:90) S sup r ≤ ˜ r ≤ r (cid:48) ∆ − ( X (cid:96) π (cid:96) F ) | r = r (cid:48) ,τ = τ (cid:48) dω. Hence, for all r (cid:48) ∈ [ r , ∞ ) (cid:90) S ∆( X (cid:96) +1 ψ (cid:96) ) | r = r (cid:48) ,τ = τ (cid:48) dω ≤ r (cid:48) − r ) (cid:90) S sup r ≤ ˜ r ≤ r (cid:48) ∆ − ( X (cid:96) π (cid:96) F ) | r = r (cid:48) ,τ = τ (cid:48) dω + C ∆( r ) (cid:90) S sup r + ≤ r ≤ r r − ( X (cid:96) π (cid:96) F ) | τ = τ (cid:48) dω. F the right-hand side of (7.2).Finally, we observe that we can rearrange the terms in (7.11) and multiply by ∆ (cid:96) as follows:∆ (cid:96) X (cid:96) π (cid:96) F = X (∆ (cid:96) +1 X (cid:96) +1 ψ (cid:96) ) + 2 a ∆ (cid:96) Φ X (cid:96) +1 ψ (cid:96) . Suppose now that (cid:96) ≥
1. Then we integrate to obtain for ρ ≥ r : | ∆ (cid:96) +1 X (cid:96) +1 ψ (cid:96) | ( τ, ρ , θ, ϕ ∗ ) ≤ Cr (cid:96) +1 || X (cid:96) +1 ψ (cid:96) || L ∞ (Σ τ ) + C ∆ (cid:96) || rX (cid:96) π (cid:96) F || L ∞ (Σ τ ) , and therefore || rX (cid:96) +1 ψ (cid:96) || L ∞ (Σ τ ) ≤ C || X (cid:96) +1 ψ (cid:96) || L ∞ (Σ τ ) + || rX (cid:96) π (cid:96) F || L ∞ (Σ τ ) . (7.13)Repeating the above step, using now (7.13), we obtain for (cid:96) ≥ | ∆ (cid:96) +1 X (cid:96) +1 ψ (cid:96) | ( τ, ρ , θ, ϕ ∗ ) ≤ Cr (cid:96) ∆ (cid:96) || rX (cid:96) +1 ψ (cid:96) || L ∞ (Σ τ ) + ∆ (cid:96) || rX (cid:96) π (cid:96) F || L ∞ (Σ τ ) and (7.9) follows by applying a standard Sobolev inequality on S . In this section, we derive elliptic estimates outside the ergoregion, making use of the uniform ellipticity of L when restricted to a region where r is sufficiently large. In order to couple these estimates with the spacelikeredshift estimates from Section 7.1, we restrict to fixed spherical harmonic modes. Proposition 7.3.
Let (cid:96) ∈ N and k ∈ R , with k < (cid:96) + . Let ψ be a solution to (7.1) , such that rψ ∈ C ∞ ( (cid:98) Σ τ ) .Then, for R > M ≥ r + suitably large, there exists a constant C = C ( M, a, R , (cid:96), k ) > , such that (cid:90) Σ τ ∩{ r ≥ R } r k +2 ( X (cid:96) +1 ψ (cid:96) ) dωd ρ + r k ( X (∆ X (cid:96) +1 ψ (cid:96) )) dωd ρ ≤ C (cid:90) Σ τ r k ( X (cid:96) π (cid:96) F ) dωd ρ . (7.14) Proof.
Denote G n := X n F − a Φ X n +1 ψ. Then we can write G n = X (∆ X n +1 ψ ) + n ∆ (cid:48) X n +1 ψ + n ( n + 1) X n ψ + / ∆ S ( X n ψ ) , so that X (∆ X (cid:96) +1 ψ (cid:96) ) + (cid:96) ∆ (cid:48) X (cid:96) +1 ψ = π (cid:96) G (cid:96) . Let χ : [ r + , ∞ ) be a cut-off function such that χ ( r ) = 1 for r ≥ R and χ ( r ) = 0 for r ≤ R − M , with R > r + + M to be chosen suitably large. Then for any k ∈ N χ r k ( X (∆ X (cid:96) +1 ψ (cid:96) )) + χ (cid:96) (∆ (cid:48) ) r k ( X (cid:96) +1 ψ (cid:96) ) + χ (cid:96) ∆ (cid:48) r k X (cid:96) +1 ψX (∆ X (cid:96) +1 ψ (cid:96) ) = χr k ( π (cid:96) G (cid:96) ) . We can further estimate χ (cid:96) ∆ (cid:48) r k X (cid:96) +1 ψ (cid:96) X (∆ X (cid:96) +1 ψ (cid:96) ) = 2 (cid:96)χ (∆ (cid:48) ) r k ( X (cid:96) +1 ψ (cid:96) ) + (cid:96)χ ∆ (cid:48) ∆ r k X (( X (cid:96) +1 ψ (cid:96) ) )= (cid:96) (cid:2) (cid:48) ) r k − (∆ (cid:48) ∆ r k ) (cid:48) (cid:3) χ ( X (cid:96) +1 ψ (cid:96) ) − (cid:96)χ (cid:48) χ ∆ (cid:48) ∆( X (cid:96) +1 ψ (cid:96) ) + X ( (cid:96)χ ∆ (cid:48) ∆ r k ( X (cid:96) +1 ψ (cid:96) ) )= − (cid:96) [4 k − O ( r − )] r k +2 ( X (cid:96) +1 ψ (cid:96) ) − (cid:96)χ (cid:48) χ ∆ (cid:48) ∆( X (cid:96) +1 ψ (cid:96) ) + X ( (cid:96)χ ∆ (cid:48) r k ( X (cid:96) +1 ψ (cid:96) ) ) . Note that (cid:90) Σ τ X ( (cid:96)χ ∆ (cid:48) ∆ r k ( X (cid:96) +1 ψ (cid:96) ) ) dωd ρ = 0if (cid:96) = 0 or k < (cid:96) + . 57y (2.20) we can moreover estimate for k < (cid:96) + and (cid:15) > k − − (cid:15) ) (cid:90) Σ τ r k − ∆ χ ( X (cid:96) +1 ψ (cid:96) ) dωd ρ ≤ (1 − (cid:15) ) (cid:90) Σ τ χ r k ( X (∆ X (cid:96) +1 ψ (cid:96) )) dωd ρ + C (cid:15) (cid:90) Σ τ ( χ (cid:48) ) ∆ ( X (cid:96) +1 ψ (cid:96) ) dωd ρ . Finally, observe that r k ( π (cid:96) G (cid:96) ) ≤ Cr k ( X (cid:96) π (cid:96) F ) + a r k (Φ X (cid:96) +1 ψ (cid:96) ) . We then combine the above estimates to obtain the following integral inequality: (cid:90) Σ τ χ(cid:15)r k ( X (∆ X (cid:96) +1 ψ (cid:96) )) + (cid:20) (cid:96) + (1 − k )2 (cid:96) + 14 (2 k − − (cid:15) (2 k − O ∞ ( r − ) (cid:21) χ r k +2 ( X (cid:96) +1 ψ (cid:96) ) dωd ρ ≤ C (cid:15),R ,(cid:96) (cid:90) Σ τ ∩{ R − M ≤ r ≤ R } ( X (cid:96) +1 ψ (cid:96) ) dωd ρ + C (cid:90) Σ τ r k ( X (cid:96) π (cid:96) F ) dωd ρ . We can choose R suitably large and (cid:15) > < (2 (cid:96) ) + (1 − k )2 (cid:96) + 14 (2 k − = (cid:18) k − (cid:96) − (cid:19) , which follows in particular from the condition k < (cid:96) + .We conclude (7.14) by applying additionally (7.6).In the proposition below, we obtain additional control over arbitrarily many X -derivatives of ψ (cid:96) , startingfrom the estimates established in Proposition 7.3. Proposition 7.4.
Let
J, (cid:96) ∈ N . Restrict − < k < (cid:96) + when (cid:96) ≥ and k > − when (cid:96) = 0 . Let ψ be asolution to (7.1) , such that rψ ∈ C ∞ ( (cid:98) Σ τ ) . Then, for R > M ≥ r + suitably large, there exists a constant C = C ( M, a, R , (cid:96), k, J ) > , such that (cid:96) + J (cid:88) n =0 (cid:90) Σ τ r − k +2 ( X ( rX ) n ψ (cid:96) ) + (cid:96) r − k (( rX ) n ψ (cid:96) ) dωd ρ ≤ C (cid:96) + J (cid:88) n =0 (cid:90) Σ τ r − k (( rX ) n ( π (cid:96) F )) dωd ρ . (7.15) If ψ is a solution to (3.1) , then (cid:96) + J (cid:88) n =0 (cid:90) Σ τ r − k +2 ( X ( rX ) n ψ (cid:96) ) + (cid:96) r − k (( rX ) n ψ (cid:96) ) dωd ρ ≤ C (cid:96) + J (cid:88) n =0 (cid:90) Σ τ r − k +4 ( XT ( rX ) n ψ (cid:96) ) + r − k +2 ( T ( rX ) n ψ (cid:96) ) + a r − k ( T ( rX ) n ψ (cid:96) − ) + a r − k ( T ( rX ) n ψ (cid:96) +2 ) dωd ρ . (7.16) Proof.
Let n ∈ N . By (7.10) it follows that r m X n +1 ψ (cid:96) X n π (cid:96) F = ∆ r m X n +1 ψ (cid:96) X n +2 ψ (cid:96) + ( n + 1)∆ (cid:48) r m ( X n +1 ψ (cid:96) ) − (cid:2) (cid:96) ( (cid:96) + 1) − n ( n + 1) (cid:3) r m X n +1 ψ (cid:96) X n ψ (cid:96) + 2 ar m X n +1 ψ (cid:96) Φ X n +1 ψ (cid:96) = ∆ r m X n +1 ψ (cid:96) X n +2 ψ (cid:96) + ( n + 1)∆ (cid:48) r m ( X n +1 ψ (cid:96) ) − X (cid:18)
12 [ (cid:96) ( (cid:96) + 1) − n ( n + 1)] r m ( X n ψ (cid:96) ) (cid:19) + 12 m [ (cid:96) ( (cid:96) + 1) − n ( n + 1)] r m − ( X n ψ (cid:96) ) + Φ( ar m ( X n +1 ψ (cid:96) ) ) . (7.17)We integrate over Σ τ ∩ { r ≥ R } , where R ≥ r + will be chosen appropriately large: − (cid:90) Σ τ ∩{ r ≥ R } X (cid:18)
12 ( (cid:96) ( (cid:96) + 1) − n ( n + 1)) r m ( X n ψ (cid:96) ) (cid:19) dωd ρ = (cid:90) S τ,R
12 ( (cid:96) ( (cid:96) + 1) − n ( n + 1)) r m ( X n ψ (cid:96) ) dω m < n + 2. J = 0:We consider first the case J = 0. If (cid:96) = 0, then (7.15) follows immediately from (7.14). Suppose (cid:96) ≥ n ≤ (cid:96) −
1. Then we estimate in { r ≥ R } : | ∆ r m X n +1 ψ (cid:96) X n +2 ψ (cid:96) | ≤ (cid:15) ∆ r m − ( X n +1 ψ (cid:96) ) + 12 (cid:15) ∆ r m +1 ( X n +2 ψ (cid:96) ) , | r m X n +1 ψ (cid:96) X n π (cid:96) F | ≤ (cid:15)r m +1 ( X n +1 ψ (cid:96) ) + 12 (cid:15) r m − ( X n π (cid:96) F ) . In the case when m ≤ n = (cid:96) −
1, we need to additionally apply (7.10) in { r ≥ R } to furtherestimate: − m [ (cid:96) ( (cid:96) + 1) − ( (cid:96) − (cid:96) ] r m − ( X (cid:96) − ψ (cid:96) ) = − (cid:96)mr m − ( X (cid:96) − ψ (cid:96) ) ≤ − m (1 + (cid:15) ) (cid:96) [ r + O ( r )] r m − ( X (cid:96) ψ (cid:96) ) + C (cid:15) r m +2 ( X (cid:96) +1 ψ (cid:96) ) + C (cid:15) r m − ( X (cid:96) − π (cid:96) F ) ≤ − m (1 + (cid:15) ) (cid:96) [1 + O ( r − )]∆ (cid:48) r m ( X (cid:96) ψ (cid:96) ) + C (cid:15) r m +1 ( X (cid:96) +1 ψ (cid:96) ) + C (cid:15) r m − ( X (cid:96) − π (cid:96) F ) . Hence, for m > − R appropriately large and (cid:15) > X (cid:96) ψ (cid:96) ) = ( X n +1 ψ (cid:96) ) term on the very right-hand side of (7.17)with n = (cid:96) − − < m < (cid:96) −
1) + 2, we therefore obtain: (cid:90) Σ τ ∩{ r ≥ R } r m +1 ( X (cid:96) ψ (cid:96) ) + r m − ( X (cid:96) − ψ (cid:96) ) dωd ρ ≤ C (cid:90) Σ τ ∩{ r ≥ R } r m +3 ( X (cid:96) +1 ψ (cid:96) ) dωd ρ + C (cid:90) Σ τ ∩{ r ≥ R } r m − ( X (cid:96) − π (cid:96) F ) dωd ρ . We can now apply (7.14) to estimate the right-hand side further and obtain (cid:90) Σ τ ∩{ r ≥ R } r m +1 ( X (cid:96) ψ (cid:96) ) + r m − ( X (cid:96) − ψ (cid:96) ) dωd ρ ≤ C (cid:90) Σ τ r m − ( X (cid:96) − π (cid:96) F ) + r m +1 ( X (cid:96) π (cid:96) F ) dωd ρ . We can easily remove the restriction to r ≥ R above by applying additionally (2.20) together with (7.6).Suppose (cid:96) ≥
2. By (7.10), we can moreover estimate, for all n ≤ (cid:96) − X n ψ (cid:96) ) ≤ Cr ( X n +1 ψ (cid:96) ) + Cr ( X n +2 ψ (cid:96) ) + ( X n π (cid:96) F ) , so we can in fact control all lower-order derivatives: (cid:96) (cid:88) l =0 (cid:90) Σ τ r m +1 − l ( X (cid:96) − l ψ (cid:96) ) dωd ρ ≤ C (cid:96) (cid:88) l =0 (cid:90) Σ τ r m +1 − l ( X (cid:96) − l π (cid:96) F ) dωd ρ , for − < m < (cid:96) . By rearranging terms, taking k := (cid:96) − m +12 and expanding the terms in ( rX ) n π (cid:96) F , weconclude that (7.15) must hold for J = 0. J ≥ J ≥
1. We will carry out an induction argument in J . First of all, we have ob-tained above the J = 0 case. Suppose (7.15) holds for some J ∈ N . We will show that it also holds for J replaced with J + 1. Consider (7.17) with n = (cid:96) + J + 1. Then we write∆ r m X n +1 ψ (cid:96) X n +2 ψ (cid:96) + ( n + 1)∆ (cid:48) r m ( X n +1 ψ (cid:96) ) = (cid:20)(cid:18) (cid:96) + J + 32 (cid:19) r ∆ (cid:48) − m (cid:21) r m − ( X (cid:96) + J +2 ψ (cid:96) ) + X (cid:18)
12 ∆ r m ( X (cid:96) + J +2 ψ (cid:96) ) (cid:19) . m < (cid:96) + J + 1) + 2. We can therefore integrate the right-hand side of (7.17) over Σ τ toobtain for m < (cid:96) + J + 1) + 2: (cid:90) Σ τ r m +1 ( X (cid:96) + J +2 ψ (cid:96) ) dωd ρ ≤ C (cid:90) Σ τ r m − ( X (cid:96) + J +1 ψ (cid:96) ) dωd ρ + C (cid:90) S τ,r + ( X (cid:96) + J +1 ψ (cid:96) ) dω + C (cid:90) Σ τ r m − ( X (cid:96) + J +1 π (cid:96) F ) dωd ρ . We then arrive at (7.15) with J replaced by J + 1 by taking k = (cid:96) − m +12 and applying the inductionassumption.We finally obtain (7.16) by expanding out the terms in ( rX ) n π (cid:96) F . We establish below improved energy decay for energies restricted to ψ , ψ and ψ containing additional r -weights with negative powers, via an application of an r − k -weighted hierarchy of elliptic estimates fromProposition 7.4. These are important for establishing almost-sharp pointwise decay in regions of bounded r . Proposition 7.5.
Let δ > be arbitrarily small and K, J ∈ N . Then there exists a constant C = C ( M, a, R, K, J, (cid:96), δ ) > , such that J (cid:88) n =0 (cid:90) Σ τ r − J N [( rX ) n T K ψ ] · n τ r dωd ρ (7.18) ≤ C (cid:88) (cid:96) ∈{ , } (1 + τ ) − δ − K E (cid:96),J + K +1 ,δ [ ψ ] + J (cid:88) j =0 E (cid:96),K +1 ,δ [ N j ψ ] , J (cid:88) n =0 (cid:90) Σ τ r − η J N [( rX ) n T K ψ ] · n τ r dωd ρ (7.19) ≤ C (1 + τ ) − η + δ − K E ,J + K +3 ,δ [ ψ ] + J (cid:88) j =0 E ,K +2 ,δ [ N j ψ ] , J (cid:88) n =0 (cid:90) Σ τ r − − η J N [( rX ) n T K ψ ] · n τ r dωd ρ (7.20) ≤ C (1 + τ ) − η + δ − K E ,J + K +5 ,δ [ ψ ] + E ,J + K +5 ,δ [ ψ ] + J (cid:88) j =0 E ,K +3 ,δ [ N j ψ ] + J (cid:88) j =0 E ,K +3 ,δ [ N j ψ ] , (cid:90) Σ τ r − J N [ T K ψ ≥ ] · n τ r dωd ρ ≤ C (1 + τ ) − η + δ − K ( E ≥ ,K +1 ,δ [ ψ ] + E ,K +2 ,δ [ ψ ] + E ,K +2 ,δ [ ψ ]) . (7.21) Proof.
We start by considering ψ . By (7.16) with k = 1, together with (2.20) we have that J (cid:88) n =0 (cid:90) Σ τ r − J N [( rX ) n ψ ] · n τ r dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ ( X ( rX ) n ψ ) + r − ( T ( rX ) n ψ ) dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ r ( XT ( rX ) n ψ ) + ( T ( rX ) n ψ ) + r − ( T ( rX ) n ψ ) dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ ( J N [( rX ) n T ψ ] + J N [( rX ) n T ψ ]) · n τ r dωd ρ . (7.22)60e conclude that (7.18) must hold by applying (6.10) for (cid:96) = 0 and (cid:96) = 2.We apply (7.16) with k = − η and then k = − η , with η > (cid:96) = 1 to obtain: J (cid:88) n =0 (cid:90) Σ τ r − η J N [( rX ) n ψ ] · n τ r dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ r − η ( X ( rX ) n ψ ) + r − η (( rX ) n ψ ) + r − η ( T ( rX ) n ψ ) dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ r η ( X ( rX ) n T ψ ) + r − η (( rX ) n T ψ ) + r − η ( T ( rX ) n ψ ) + r − η ( T ( rX ) n ψ ) dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ r η ( X ( rX ) n T ψ ) + r η (( rX ) n T ψ ) + r − η ( T ( rX ) n ψ ) dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ (cid:0) r η J N [( rX ) n T ψ ] + J N [( rX ) n T ψ ] (cid:1) · n τ r dωd ρ . By combining the energy decay estimates with r − δ weights in (6.5) and (6.8) with the energy decay estimatesfrom Corollary 6.4 via the interpolation inequality in Lemma B.1, we therefore obtain: J (cid:88) n =0 (cid:90) Σ τ r − η J N [( rX ) n T K ψ ] · n τ r dωd ρ ≤ C (1+ τ ) − − (cid:96) + δ + η − K E ,J + K +3 ,δ [ ψ ] + J (cid:88) j =0 E ,K +2 ,δ [ N j ψ ] . We can similarly apply (7.16) with k = + δ for (cid:96) = 2 to obtain: J (cid:88) n =0 (cid:90) Σ τ r − − δ J N [( rX ) n ψ ] · n τ r dωd ρ ≤ C J (cid:88) n =0 (cid:90) Σ τ r − δ ( X ( rX ) n T ψ ) + r − − δ (( rX ) n T ψ ) + r − − δ ( T ( rX ) n ψ ) + r − − η ( T ( rX ) n ψ ) + r − − δ ( T ( rX ) n ψ ) ωd ρ ≤ J (cid:88) n =0 (cid:90) Σ τ r − δ J N [( rX ) n T ψ ] · n τ + J N [( rX ) n T ψ ] · n τ r dωd ρ + J (cid:88) n =0 (cid:90) Σ τ r − − δ ( T ( rX ) n ψ ) dωd ρ . Note that by (A.4) with h = ψ , we can further estimate: J (cid:88) n =0 (cid:90) Σ τ r − − δ ( T ( rX ) n ψ ) dωd ρ ≤ C J (cid:88) n =0 || T ( rX ) n ψ || L ∞ (Σ τ ) ≤ C (cid:115)(cid:90) Σ τ r − J N [ T ψ ] · n τ r dωd ρ (cid:115)(cid:90) Σ τ J N [ T ψ ] · n τ r dωd ρ . Hence, after applying (6.10), (6.11) and (7.18), we conclude that J (cid:88) n =0 (cid:90) Σ τ r − − δ J N [( rX ) n T K ψ ] · n τ r dωd ρ ≤ C (1 + τ ) − − (cid:96) − K E ,J + K +5 ,δ [ ψ ] + E ,J + K +5 ,δ [ ψ ] + J (cid:88) j =0 E ,K +3 ,δ [ N j ψ ] + J (cid:88) j =0 E ,K +3 ,δ [ N j ψ ] . ψ replaced by T K ψ , we can estimate (cid:90) Σ τ r − J N [ T K ψ ≥ ] · n τ r dωd ρ ≤ C (cid:90) Σ τ J N [ T K +1 ψ ≥ ] · n τ r dωd ρ + C (cid:90) N τ r ( LT K +1 φ ≥ ) · n τ r dωd ρ + C (cid:90) Σ τ ( J N [ T K +2 ψ ] + J N [ T K +2 ψ ]) · n τ r dωd ρ . Hence, (7.21) follows by combining the above equation with the energy decay estimates (6.5), (6.8) and(6.9).
Remark 7.1.
In the proof of Proposition 7.5, we applied (7.16) with k = + δ in the (cid:96) = 2 case. Notethat (7.16) in fact applies with − η ≤ k ≤ − η when (cid:96) = 2 . However, as we already applied an almost-sharp decay estimate for T ψ , which does not further improve by considering additional weights in r − ,we cannot exploit the full hierarchy of elliptic estimates to improve the above decay rate when considering r − η J N [( rX ) n ψ ] instead of r − − η J N [( rX ) n ψ ] . This is a manifestation of angular mode couplinglimiting the sharp decay rate of the (cid:96) = 2 mode.
We apply the energy decay estimates of Section 6 and 7.3 to obtain L ∞ estimates for ψ , rψ and varioushigher-order quantities of the form ( rX ) J T K ψ . These pointwise estimates will be used in subsequent sectionsto determine the precise late-time behaviour of ψ ≥ (cid:96) with (cid:96) = 0 , , Proposition 8.1.
Let δ > be arbitrarily small and K, J ∈ N . Then there exists a constant C = C ( M, a, R, δ, K, J ) > , such that || T K φ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.1) || ( rX ) J T K φ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) (cid:96) ∈{ , } E (cid:96),K + J,δ [ ψ ] + J (cid:88) j =0 E (cid:96),K,δ [ N j ψ ] , (8.2) ||√ rT K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.3) || T K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:115) (cid:88) (cid:96) ∈{ , } E (cid:96),K +1 ,δ [ ψ ] , (8.4) || ( rX ) T K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) (cid:96) ∈{ , } E (cid:96),K +2 ,δ [ ψ ] + (cid:88) j =0 E (cid:96),K +1 ,δ [ N j ψ ] , (8.5) Proof.
We obtain (8.1) and (8.3) by applying (A.1) and (A.3) with f = φ and h = ψ together with theenergy decay estimate (6.7).The estimate (8.4) follows by applying (A.4) with h = ψ in combination with (6.7) and (7.18). In orderto obtain (8.2) and (8.5), we instead apply the higher-order energy decay estimate (6.10) in combinationwith (A.4). Proposition 8.2.
Let δ > be arbitrarily small and K, J ∈ N . Then there exists a constant C = C ( M, a, R, K, J, δ ) > , such that for m = 0 , : || T K φ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.6) ||√ rT K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.7) || T K φ (1)1 || L ∞ ( N τ ) + (1 + τ ) (1+ δ ) || r − (1+ δ ) T K φ (1)1 || L ∞ ( N τ ) (8.8) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , | ( rX ) J T K φ ( m )1 || L ∞ ( N τ ) + (1 + τ ) (1+ δ ) || r − (1+ δ ) ( rX ) J T K φ ( m )1 || L ∞ ( N τ ) (8.9) ≤ C (1 + τ ) − j − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) E ,K + J,δ [ ψ ] + J (cid:88) j =0 E ,K,δ [ N j ψ ] ,r − || T K ψ || L ∞ (Σ τ ) + || XT K ψ || L ∞ (Σ τ ) (8.10) ≤ C (1 + τ ) − − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) E ,K +3 ,δ [ ψ ] + (cid:88) j =0 E ,K +2 ,δ [ N j ψ ] , || rX T K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) E ,K +4 ,δ [ ψ ] + (cid:88) j =0 E ,K +2 ,δ [ N j ψ ] . (8.11) Proof.
We note first of all that by a standard Sobolev inequality on S together with (2.17), the L ∞ ( S )norm of ψ and can be uniformly bounded by its L ( S ) norm.We obtain (8.6) and (8.7) by applying (A.1) and (A.3) with f = φ and h = ψ together with the energydecay estimate (6.8). Similarly, (8.8) follows from (A.1) and (A.2) with f = φ (1) and (6.8).The estimate (8.10) follows by applying (A.5) with k = 1, h = ψ and h = rXψ , together with theenergy decay estimates (6.8) and (7.19) and Lemma B.1. The estimate (8.11) follows by additionally taking h = ( rX ) ψ .Finally, (8.9) follows as (8.8) but via the higher-order energy decay estimates (7.19). Proposition 8.3.
Let δ > be arbitrarily small and K, J ∈ N . Then there exists a constant C = C ( M, a, R, K, J, δ ) > , such that for m = 0 , , : || rT K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.12) ||√ rT K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.13) || T K φ ( m )2 || L ∞ ( N τ ) +(1 + τ ) + δ || r − (1+ δ ) T K φ ( m )2 || L ∞ ( N τ ) ≤ C (1 + τ ) − − (2 − m ) − K +2 δ (cid:113) E ,K,δ [ ψ ] , (8.14) || ( rX ) J T K φ ( m )2 || L ∞ ( N τ ) +(1 + τ ) + δ || r − (1+ δ ) ( rX ) J T K φ ( m )2 || L ∞ ( N τ ) (8.15) ≤ C (1 + τ ) − − (2 − m ) − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) E ,K + J,δ [ ψ ] + J (cid:88) j =0 E ,K,δ [ N j ψ ] , J +2 (cid:88) n =0 || ( rX ) n T K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:32)(cid:118)(cid:117)(cid:117)(cid:116) E ,K + J +5 ,δ [ ψ ] + (cid:88) j =0 E ,K + J +3 ,δ [ N j ψ ] (8.16)+ (cid:118)(cid:117)(cid:117)(cid:116) E ,K + J +5 ,δ [ ψ ] + (cid:88) j =0 E ,K + J +3 ,δ [ N j ψ ] (cid:33) . Proof.
Note that as in the (cid:96) = 1 case the L ∞ ( S ) norm of ψ and can be uniformly bounded by its L ( S )norm.We obtain (8.12) and (8.13) by applying (A.1) and (A.3) with f = φ and h = ψ together with theenergy decay estimate (6.9). Similarly, (8.14) follows from (A.1) and (A.2) with f = φ ( m ) , m = 1 ,
2, and(6.8).By applying (A.5) for k = 0, together with (7.20), (6.10) and Lemma B.1, we obtain (8.16).Finally, (8.15) follows as (8.14), but via the higher-order energy decay estimates (7.19). Proposition 8.4.
Let δ > be arbitrarily small and K, J ∈ N . Let Q denote the Carter operator: Q = / ∆ S + a sin θT − Φ . hen there exists a constant C = C ( M, a, R, K, J, δ ) > , such that for m = 0 , , , : || T K φ ≥ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:115) (cid:88) j + j + j ≤ E ≥ ,K,δ [ Q j T j Φ j ψ ] , (8.17) ||√ rT K ψ ≥ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:115) (cid:88) j + j + j ≤ E ≥ ,K,δ [ Q j T j Φ j ψ ] , (8.18) || T K ψ ≥ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:34) (cid:88) j + j + j ≤ E ≥ ,K +1 ,δ [ Q j T j Φ j ψ ] + E ,K +2 ,δ [ Q j T j Φ j ψ ](8.19)+ E ,K +2 ,δ [ Q j T j Φ j ψ ] (cid:35) . We also have the following additional estimates for (cid:96) = 3 , : || ( rX ) J T K φ ( m ) (cid:96) || L ∞ ( N τ ) +(1 + τ ) + δ || r − (1+ δ ) ( rX ) J T K φ ( m ) (cid:96) || L ∞ ( N τ ) (8.20) ≤ C (1 + τ ) − + m − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) E ≥ ,K + J,δ [ ψ ] + J (cid:88) j =0 E ≥ ,K,δ [ N j ψ ] , || r XT K ψ (cid:96) || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:34)(cid:113) E ≥ ,K +1 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K,δ [ ψ ] (cid:35) . (8.21) Proof.
In order to bound the L ∞ ( S ) norm of ψ ≥ we consider the Carter operator Q and note that: || ψ ≥ || L ∞ ( S ) ≤ C (cid:90) S ψ ≥ + | / ∇ S ψ ≥ | + | / ∇ S ψ ≥ | dω ≤ C (cid:90) S ψ ≥ + ( / ∆ S ψ ≥ ) dω ≤ C (cid:90) S ψ ≥ + ( Qψ ≥ ) + ( T ψ ≥ ) + (Φ ψ ≥ ) dω. Furthermore, since [ T, (cid:3) g ] = [Φ , (cid:3) g ] = [ Q, (cid:3) g ] = 0, all estimates derived for ψ automatically hold for ψ replaced by Qψ , T ψ or Φ ψ . It therefore remains only to establish control of the L ( S )-norm of ψ ≥ .We obtain (8.17) and (8.18) by applying (A.1) and (A.3) with f = φ ≥ and h = ψ ≥ together with theenergy decay estimate (6.5). The estimate (8.21) follows similarly by taking h = rXψ or h = rXψ andapplying the energy decay estimate (6.10) with (cid:96) = 3 ,
4, respectively.Finally, (8.19) and (8.20) follow from (A.5) with k = 0 and h = ψ ≥ , and (A.1), (A.2), respectively,combined with (7.21).In the following proposition we establish improved decay rates for certain X -derivatives of ψ (cid:96) comparedto the decay of ψ (cid:96) itself. These estimates will be crucial for propagating asymptotics of ψ (cid:96) from a regionnear infinity to the rest of the spacetime. Proposition 8.5.
Let δ > be arbitrarily small and K ∈ N . Then there exists a constant C = C ( M, a, R, K, δ ) > , such that: || XT K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:88) (cid:96) ∈{ , } (cid:113) E (cid:96),K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E (cid:96),K +1 ,δ [ N j ψ ] , (8.22) || X T K ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:34)(cid:113) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +3 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +2 ,δ [ ψ ] (8.23)64 (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) , || r X T K ψ || L ∞ (Σ τ ) + || rX T K ψ || L ∞ (Σ τ ) (8.24) ≤ C (1 + τ ) − − K +2 δ (cid:34)(cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ]+ (cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ]+ (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) . Proof.
Recall from (7.8) that || X (cid:96) +1 ψ (cid:96) || L ( S τ,r (cid:48) ) ≤ C (cid:96) +1 (cid:88) k =0 sup r + ≤ r ≤ r (cid:48) || r − (cid:96) ( rX ) k T ψ (cid:96) || L ( S τ,r ) + C (cid:96) (cid:88) k =0 sup r + ≤ r ≤ r (cid:48) || r − − (cid:96) ( rX ) k T ψ (cid:96) || L ( S τ,r ) + Ca sup r + ≤ r ≤ r (cid:48) || r − X (cid:96) T ψ (cid:96) − || L ( S τ,r ) + Ca sup r + ≤ r ≤ r (cid:48) || r − X (cid:96) T ψ (cid:96) +2 || L ( S τ,r ) . Hence, || XT K ψ || L ∞ (Σ τ ) ≤ C (cid:88) k =0 || ( rX ) k T ψ || L ∞ (Σ τ ) + C || r − T ψ || L ∞ (Σ τ ) + Ca || r − T ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) (cid:96) ∈{ , } E (cid:96),K +2 ,δ [ ψ ] + (cid:88) j =0 E (cid:96),K +1 ,δ [ N j ψ ] . Furthermore, by (8.10) and (8.11). || X T K ψ || L ∞ (Σ τ ) ≤ C (cid:88) k =0 || r − ( rX ) k T ψ || L ∞ (Σ τ ) + C (cid:88) k =0 || r − ( rX ) k T ψ || L ∞ (Σ τ ) + Ca || r − XT ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:34)(cid:113) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +3 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +2 ,δ [ ψ ]+ (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) . We apply (7.9) and (8.16) to obtain: || r X T K ψ || L ∞ (Σ τ ) ≤ C (cid:88) k =0 || ( rX ) k T ψ || L ∞ (Σ τ ) + C (cid:88) k =0 || r − ( rX ) k T ψ || L ∞ (Σ τ ) + Ca || XT ψ || L ∞ (Σ τ ) + Ca || XT ψ || L ∞ (Σ τ ) ≤ C (1 + τ ) − − K +2 δ (cid:34)(cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ]+ (cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ]+ (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) . We obtain an estimate for || rX T K ψ || L ∞ (Σ τ ) by integrating from ρ = ∞ .65 Elliptic theory of time inversion
The aim of this section is to construct a solution (cid:101) ψ to (3.1), such that T (cid:101) ψ = ψ, with ψ another solution to (3.1) arising from smooth, compactly supported initial data on Σ . We willdenote T − ψ = (cid:101) ψ . This propositions in this section are self-contained and independent from the estimatesin Sections 5–8.The construction of (cid:101) ψ relies on the invertibility of the differential operator L : L f = X (∆ Xf ) + 2 aX Φ f + / ∆ S f, which was introduced in Section 7, acting on suitable Hilbert spaces.Let H k denote the completion of the space C ∞ rad (Σ ) := { f ∈ C ∞ (Σ ) | ( r + a ) f ∈ C ∞ ( (cid:98) Σ ) } , with respect to the norm || · || k , defined as follows: || f || k := (cid:88) k + k ≤ k (cid:90) Σ | / ∇ k S ( rX ) k f | dωd ρ . Let D k ( L ) denote the closure of C ∞ rad (Σ ) under the norm: || f || k + ||L f || k . Then L : D k ( L ) → H k is a densely defined, closed, linear operator. L In this section, we will establish invertibility and regularity properties of L . The strategy for obtaininginvertibility of L can be compared to the strategy of obtaining invertibility of the resolvent operators con-sidered in Section 4 of [War15]. In particular, as in [War15], the use of (elliptic) redshift estimates will beimportant.The following proposition contains the key estimates that are relevant for invertibility: Proposition 9.1.
Let ( r + a ) f ∈ C ∞ ( (cid:98) Σ ) . Then (cid:90) Σ f + r ( Xf ) + | / ∇ S f | dωd ρ ≤ C (cid:90) Σ ( L f ) dωd ρ . (9.1) More generally, || f || k + || / ∇ S f || k + || rXf || k ≤ C ||L f || k . (9.2) Furthermore, H k +1 ⊂ D k ( L ) and the equation (9.2) holds also for all f ∈ D k ( L ) .Proof. Let q ∈ R . After applying the Leibniz rule, we obtain the following identity: r q Xf L f = 12 X ( r q ∆( Xf ) ) + 12 (cid:0) ∆ (cid:48) − q ∆ r − (cid:1) r q ( Xf ) + a Φ( r q ( Xf ) ) + div S ( r q Xf / ∇ S f ) − X ( r q | / ∇ S f | ) + 12 qr q − | / ∇ S f | . (9.3)Integrating the above identity with q = 1 over Σ and using that ( r + a ) f ∈ C ∞ ( (cid:98) Σ ), we therefore obtain: (cid:90) Σ
12 ( r − a )( Xf ) + 12 | / ∇ S f | dωd ρ + 12 (cid:90) Σ ∩{ ρ = r + } r | / ∇ S f | dω ≤ (cid:90) Σ r | Xf ||L f | dωd ρ
66y applying a weighted Young’s inequality to right-hand side and absorbing the resulting ( Xf ) term intothe left-hand side, we are left with: (cid:90) Σ r ( Xf ) + | / ∇ S f | dωd ρ + (cid:90) Σ ∩{ ρ = r + } | / ∇ S f | dω ≤ C (cid:90) Σ ( L f ) dωd ρ . (9.4)By applying (2.20) we obtain (9.1).In order to obtain (9.2), we first observe that: X k ( L f − k ( k + 1) f ) = X (∆ X k +1 f ) + k ∆ (cid:48) X k +1 f + 2 a Φ X k +1 f + / ∆ S X k f. We then multiply both sides by ( − k / ∆ k S r k +1 X k +1 f and apply the Leibniz rule multiple times to obtain: r k +1 / ∇ s S / ∆ k − s S X k +1 f · / ∇ s S / ∆ k − s S X k ( L f − k ( k + 1) f ) = (cid:18) k + 12 (cid:19) [∆ (cid:48) − ∆ r − ] r k +1 | / ∇ s S / ∆ k − s S X k +1 f | + (cid:18) k + 12 (cid:19) r k | / ∇ − s S / ∆ k s S X k f | − X ( r k +1 | / ∇ − s S / ∆ k s S X k f | ) + . . . where s = 0 if k is even and s = 1 if k is odd, and the terms in . . . on the right-hand side are totalderivatives that vanish after integrating over Σ . We therefore obtain: (cid:90) Σ r k +2 | / ∇ s S / ∆ k − s S X k +1 f | + r k | / ∇ − s S / ∆ k s S X k f | dωd ρ ≤ C (cid:90) Σ r k | / ∇ s S / ∆ k − s S X k ( L f − k ( k + 1) f ) | dωd ρ . (9.5)We then obtain (9.2) via (2.18) and a straightforward induction argument in k .Now suppose f ∈ D k ( L ). By definition, there exists a sequence { f j } such that ( r + a ) f k ∈ C ∞ ( (cid:98) Σ )and f j → f , L f k → L f with respect to || · || k . Applying the estimates above to the difference f j − f i , j, i ∈ N , we can conclude that { f j } is Cauchy with respect to || · || k +1 and hence the limit f ∈ H k +1 . Itfollows moreover that (9.2) holds for f ∈ D k ( L ).Let L ∗ : D ( L ∗ ) → H denote the Hilbert space adjoint operator of L with respect to the standard L norm on [ r + , ∞ ) × S , i.e. v ∈ D ( L ∗ ) if and only if: there exists a f ∈ H such that for all u ∈ C ∞ c (Σ ) (cid:104)L u, v (cid:105) L ([ r + , ∞ ) × S ) = (cid:104) u, f (cid:105) L ([ r + , ∞ ) × S ) and L ∗ v := f .We moreover define the operator L † : D ( L † ) → H k , with L † u = L u and where D ( L † ) is the closure of C ∞ rad , ∗ (Σ ) = { f ∈ C ∞ rad (Σ ) | f | r = r + = 0 } under the norm || u || + ||L † u || . Lemma 9.2.
The following identity holds: L † = L ∗ , with D ( L † ) = D ( L ∗ ) .Proof. We will establish the equivalent statement: ( L † ) ∗ = L with D (( L † ) ∗ ) = D ( L ).By definition of the adjoint ( L † ) ∗ , we have that for all u ∈ C ∞ rad , ∗ (Σ ) and v ∈ D (( L † ) ∗ ): (cid:104)L † u, v (cid:105) L = (cid:104) u, ( L † ) ∗ v (cid:105) L . (9.6)Now, let u ∈ C ∞ rad , ∗ (Σ ) and v ∈ C ∞ rad (Σ ). Then we can integrate by parts and use that u | r = r + = 0 and u, v are compactly supported to obtain: (cid:104)L † u, v (cid:105) L = (cid:104) u, L v (cid:105) L . (9.7)Since L and L † are closed operators, by construction, we have that (9.7) must also hold for u ∈ D ( L † )and v ∈ D ( L ). Applying both (9.6) and (9.7), we can therefore infer that: D ( L ) ⊆ D (( L † ) ∗ )67nd ( L † ) ∗ | D ( L ) = L .In order to conclude that D ( L ) = D (( L † ) ∗ ), it remains to show that for any element v ∈ D (( L † ) ∗ ),there exists a sequence ( v k ), with v k ∈ C ∞ rad (Σ ) such that || v − v k || L ([ r + , ∞ ) × S ) + ||L ( v − v k ) || L ([ r + , ∞ ) × S ) → k → ∞ . (9.8)First, observe that if v ∈ D (( L † ) ∗ ), then by (9.6) combined with (9.7), L v ∈ H exists in a weak sense.Then, we can take v k to be a convolution of v with a suitable mollifier to conclude that (9.8) must hold. Proposition 9.3.
The operator L : D k ( L ) → H k is invertible and the inverse L − : H k → D k ( L ) satisfies ||L − ( F ) || k + || rX L − ( F ) || k + || / ∇ S L − ( F ) || k ≤ C || F || k , (9.9) for any F ∈ H k .Proof. We first consider k = 0. By Proposition 9.1, we have that L is injective, i.e. ker L = { } . We willconclude that L is bijective by showing that Ran L = H = L ([ r + , ∞ ) × S ).Let v ∈ (Ran L ) ⊥ and u ∈ C ∞ rad (Σ ). Then0 = (cid:104) v, L u (cid:105) L = (cid:104)L ∗ v, u (cid:105) L , so v ∈ D ( L ∗ ) and moreover v ∈ ker L ∗ . Hence, (Ran L ) ⊥ ⊆ ker L ∗ . Furthermore, if v ∈ ker L ∗ , then we cansimilarly conclude that v ∈ (Ran L ) ⊥ and therefore (Ran L ) ⊥ = ker L ∗ .By Lemma 9.2, we have that L ∗ = L † . By definition of L † , we moreover have that L † | C ∞ rad , ∗ (Σ ) = L| C ∞ rad , ∗ (Σ ) , so we can apply Proposition 9.1 again to obtain ker L ∗ = { } .Note, by Proposition 9.1 together with the fact that L is closed, it follows that Ran L is closed and henceRan L = (Ran L ) ⊥⊥ = (ker L ∗ ) ⊥ = H . We conclude that L : D ( L ) → H is bijective and we denote its inverse by L − . The estimate (9.9) with k = 0 follows immediately by taking f = L − ( F ) in (9.2).We can obtain the k > L − ( F ) ∈ H K and (9.9) with k = K for some K ∈ N . Then we can repeat the argument above where in the proof ofLemma 9.2 we replace L by the operator / ∇ k S X k ( L − k ( k + 1)), with k + k = 1, applying (9.5) insteadof (9.1) to conclude that in fact / ∇ k S ( rX ) k L − ( F ), with k + k = 1 lies in H K and (9.9) holds for L − ( F )replaced by / ∇ k S X k L − ( F ). Hence, L − ( F ) ∈ H K +1 and (9.9) holds for k = K + 1.By Proposition 9.3 together with a standard Sobolev inequality, we immediately obtain: Corollary 9.4.
Let F ∈ C ∞ rad (Σ ) . Then L − ( F ) ∈ (cid:84) k ∈ N H k and in particular L − ( F ) ∈ C ∞ (Σ ) . I + In this section, we will establish additional r -decay estimates of L − ( F ) (with suitably decaying F ) towards I + . We will derive elliptic analogues of the r p -weighted estimates from Section 5.We first define the following higher-order quantities: f ( n ) := (( r + a ) X ) n f, ˇ f (0) := f, ˇ f (1) := [1 + ( α + α Φ Φ) r − + ( β + β Φ Φ ) r − ]( r + a ) Xf, ˇ f (2) := [1 + γr − ]( r + a ) X ˇ f (1) , where the coefficients α, α Φ , β, β Φ , γ, γ Φ are chosen as in Section 4.We moreover observe that [ L , π (cid:96) ] = 0 ,
68o we can independently derive additional decay estimates for L − ( F (cid:96) ) = ( L − ( F )) (cid:96) .We define the following auxiliary operators: (cid:98) L := ( r + a ) L ( r + a ) − , (cid:98) L ( n ) := ( r + a )( X ( r + a )) n ( r + a ) − (cid:98) L , ˇ L (0) := ( r + a ) L ( r + a ) − , ˇ L (1) := ( r + a ) X ( r + a ) (cid:2) α + α Φ Φ) r − + ( β + β Φ Φ ) r − (cid:3) ( r + a ) − (cid:98) L , ˇ L (2) := ( r + a ) X ( r + a )[1 + γr − ]( r + a ) − ˇ L (1) . Lemma 9.5.
Let n ∈ N , then ( r + a ) − (cid:98) L ( n ) f := X (cid:16) ∆( r + a ) − Xf ( n ) (cid:17) − n [1 + O ∞ ( r − )] Xf ( n ) + 2 a ( r + a ) − X Φ f ( n ) + ( r + a ) − [ / ∆ S − n ( n + 1)] f ( n ) + O ∞ ( r − )[ f ( n ) + Φ f ( n ) ]+ n n − (cid:88) m =0 O ∞ ( r − )[ f ( m ) + Φ f ( m ) ] . (9.10) Furthermore, for n = 0 , , : ( r + a ) − ˇ L ( n ) f n := X (cid:16) ∆( r + a ) − Xf ( n ) n (cid:17) − n [1 + O ∞ ( r − )] Xf ( n ) n + 2 a ( r + a ) − X Φ f ( n ) n + n (cid:88) m =0 O ∞ ( r − )[ f ( m ) n + Φ f ( m ) n ] . (9.11) Proof.
We can write L (( r + a ) − f ) = X (cid:16) ∆ X (( r + a ) − / f ) (cid:17) + 2 aX ( r + a ) − / Φ f ) + ( r + a ) − / / ∆ S f = X (cid:16) ( r + a ) / ∆( r + a ) − Xf (cid:17) − X (cid:16) ∆ r ( r + a ) − / f (cid:17) + 2 a ( r + a ) − / X Φ f + ( r + a ) − / / ∆ S f − ar ( r + a ) − / Φ f = ( r + a ) / X (cid:0) ∆( r + a ) − Xf (cid:1) + 2 a ( r + a ) − / X Φ f − ar ( r + a ) − / Φ f + ( r + a ) − / / ∆ S f − (cid:16) ∆ r ( r + a ) − / (cid:17) (cid:48) f. (9.12)Hence, (9.10) follows for n = 0. The n > f ( n ) as in the proof of Proposition 4.1.The following proposition contains the elliptic analogues of r p -weighted energy estimates: Proposition 9.6.
Let n ∈ { , , } , then for f ∈ C ∞ ( (cid:98) Σ ) and δ ∈ (0 , , there exists a constant C δ ( M, a ) > , such that (cid:90) Σ r − δ ( Xf ( n ) ≥ n +1 ) + r − − δ | / ∇ S f ( n ) ≥ n +1 | + r − − δ ( f ( n ) ≥ n +1 ) dωd ρ ≤ C δ (cid:90) Σ r − − δ (cid:32) (cid:98) L ( n ) f ≥ n +1 − n n − (cid:88) m =0 O ∞ ( r )[ f ( m ) ≥ n +1 + Φ f ( m ) ≥ n +1 ] (cid:33) dωd ρ + C δ n (cid:88) m =0 (cid:90) Σ (( rX ) m L (( r + a ) − f ≥ n +1 )) dωdρ (9.13)69 nd (cid:90) Σ r − δ ( X ˇ f ( n ) n ) + r − − δ ( ˇ f ( n ) n ) dωd ρ ≤ C (cid:90) Σ r − − δ (cid:32) ˇ L ( n ) f − n n − (cid:88) m =0 O ∞ ( r )[ f ( m ) n + Φ f ( m ) n ] (cid:33) dω + C n (cid:88) m =0 (cid:90) Σ (( rX ) m L (( r + a ) − f n )) dωdρ. (9.14) Proof.
Let χ be a smooth cut-off function such that χ ( r ) = 1 for r ≥ R and χ ( r ) = 0 for r ≤ R − M , where R > r + + M will be chosen suitably large. Consider the multiplier − ( r + a ) p Xf . Then by (9.10) we obtain − ( r + a ) p − Xf · (cid:98) L f = −
12 ∆ − ( r + a ) p +22 X (cid:0) ∆ ( r + a ) − ( Xf ) (cid:1) − a ( r + a ) p − Φ(( Xf ) )+ / div S ( . . . ) −
12 ( r + a ) p − X ( | / ∇ S f | ) + O ( r p − ) χ Φ f Xf + O ( r p − ) f Xf = (cid:104) p r p − + O ( r p − ) (cid:105) ( Xf ) + 12 ( p − r p − + O ( r p − )] | / ∇ S f | + O ( r p − )Φ f Xf + O ( r p − ) f Xf + 12 ( r + a ) p − | / ∇ S f | | H + − X (cid:16) ∆( r + a ) p − ( Xf ) (cid:17) − a ( r + a ) p − Φ(( Xf ) ) + / div S ( . . . ) . We integrate over Σ , applying (2.20) and (9.1) to control terms in bounded r regions, to obtain: for1 ≤ p ≤ (cid:90) Σ r p − ( Xf ) + (2 − p ) r p − | / ∇ S f | + r − f dωd ρ ≤ C (cid:90) Σ r p − ( (cid:98) L f ) + ( L (( r + a ) − f ) dωd ρ . (9.15)Note that when replacing f with f = π f , we can in fact take p = 4 − δ to obtain: (cid:90) Σ r − δ ( Xf ) + r − − δ f dωd ρ ≤ C (cid:90) Σ r − − δ ( ˇ L f ) + ( L (( r + a ) − f ) dωd ρ . (9.16)For n > − ( r + a ) p Xf ( n ) together with ( r + a ) − (cid:98) L ( n ) f , and we proceedanalogously to above to obtain: (cid:90) Σ r p − ( Xf ( n ) ) + (2 − p ) r p − [ | / ∇ S f ( n ) | − n ( n + 1)( f ( n ) ) ] dωd ρ ≤ C (cid:90) Σ r p − (cid:32) (cid:98) L ( n ) f − n − (cid:88) m =0 O ∞ ( r )[ f ( m ) + Φ f ( m ) ] (cid:33) + n (cid:88) m =0 (( rX ) m L (( r + a ) − f )) dωd ρ , (9.17)for 0 < p < n = 1. Then for f ≥ , the left-hand side of (9.17) is non-negative definite by (2.15). Furthermore,if we consider (9.11) with the multiplier − ( r + a ) p χX ˇ f (1)1 , we can take p = 4 − δ to obtain: (cid:90) Σ r − δ ( X ˇ f (1)1 ) + r − − δ ( ˇ f (1)1 ) dωd ρ ≤ C (cid:90) Σ r − − δ (cid:16) ˇ L (1) f − O ∞ ( r )[ f + Φ f ] (cid:17) dω + C (cid:88) m =0 (cid:90) Σ (( rX ) m L (( r + a ) − f )) dωdρ. (9.18)Similarly, for n = 2 and f ≥ , (9.17) has a non-negative definite left-hand side. For f , we consider (9.11)with the multiplier − ( r + a ) p χX ˇ f (2)2 with p = 4 − δ to obtain: (cid:90) Σ r − δ ( X ˇ f (2)2 ) + r − − δ ( ˇ f (2)2 ) dωd ρ ≤ C (cid:90) Σ r − − δ (cid:32) ˇ L (2) f − O ∞ ( r ) (cid:88) m =0 [ f ( m )2 + Φ f ( m )2 ] (cid:33) dω + C (cid:88) m =0 (cid:90) Σ (( rX ) m L (( r + a ) − f )) dωdρ. (9.19)70e can moreover obtain r p -weighted energy estimates for higher-order derivatives with respect to / ∇ S and rX : Corollary 9.7.
Let n ∈ { , , } and k ∈ N , then for f ∈ C ∞ ( (cid:98) Σ ) and δ ∈ (0 , , there exists a constant C δ ( M, a, k ) > , such that (cid:88) k ≤ k + k ≤ k +1 (cid:90) Σ r − − δ | / ∇ k S ( rX ) k f ( n ) ≥ n +1 | dωd ρ ≤ C δ (cid:88) k + k = k (cid:40) (cid:90) Σ r − − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / ∇ k S ( rX ) k (cid:34) (cid:98) L ( n ) f ≥ n +1 − n − (cid:88) m =0 O ∞ ( r )( f ( m ) ≥ n +1 + Φ f ( m ) ≥ n +1 ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω + C δ n (cid:88) m =0 (cid:90) Σ | / ∇ k S ( rX ) m + k L (( r + a ) − f ≥ n +1 ) | dωdρ (cid:41) . (9.20) and (cid:90) Σ r − δ ( X ( rX ) k ˇ f ( n ) n ) + r − − δ (( rX ) k ˇ f ( n ) n ) dωd ρ ≤ C (cid:90) Σ r − − δ (cid:32) ( rX ) k ˇ L ( n ) f − nO ∞ ( r ) n − (cid:88) m =0 [( rX ) k f ( m ) n + Φ( rX ) k f ( m ) n ] (cid:33) dω + C n (cid:88) m =0 (cid:90) Σ (( rX ) m + k L (( r + a ) − f n )) dωdρ. (9.21) Proof.
We consider ( rX ) k (cid:98) L ( n ) f with the multiplier − ( − k ( r + a ) p X / ∆ k S ( rX ) k f , and ( rX ) k ˇ L ( n ) f n with the multiplier − ( r + a ) p X ( rX ) k f n and proceed to integrate by parts over S , as in the proof ofProposition 9.1, and apply the arguments of Proposition 9.6.In order to establish stronger decay properties of L − ( F ), we introduce the following norms: let 0 < δ < || f || k, , ,δ := (cid:88) k ≤ k (cid:90) Σ r − − δ (( rX ) k f ) dωd ρ , || f || k, ,m,δ := (cid:88) k ≤ k m (cid:88) n =0 (cid:90) Σ r − − δ (( rX ) k f ( n )1 ) dωd ρ ≤ m ≤ , || f || k, ,m,δ := (cid:88) k ≤ k m (cid:88) n =0 (cid:90) Σ r − − δ (( rX ) k f ( n )2 ) dωd ρ ≤ m ≤ , || f || k, ,m,δ := (cid:88) k + k ≤ k m (cid:88) n =0 (cid:90) Σ r − − δ | / ∇ k S ( rX ) k f ( n ) ≥ | dωd ρ ≤ m ≤ . We denote with H k,n,m,δ the completions of: π n C ∞ ( (cid:98) Σ ) for n ∈ { , , } and π ≥ C ∞ ( (cid:98) Σ) for n = 3 underthe norms || · || k, ,m,δ defined above.Let D k,n,m,δ , with n ∈ { , , } , denote the completion of π n C ∞ ( (cid:98) Σ ) under the graph norm || · || k,n,m,δ with respect to the operator A , where A = ˇ L (0) for n = 0, and when n ∈ { , } : A = (cid:98) L ( m ) for m ≤ n − A = ˇ L ( n ) , for m = n .Similarly, let D k, ,m,δ , denote the completion of π ≥ C ∞ ( (cid:98) Σ ) under the graph norm || · || k, ,m,δ , withrespect to the operator A , where A = (cid:98) L ( m ) . 71hen A : D k,n,m,δ → H k,n,m,δ is a densely defined, closed, linear operator. Proposition 9.8.
Let K ∈ N and let ( r + a ) − F ∈ H K . Then there exists a unique f ∈ ( r + a ) H K ,such that (cid:98) L f = F. If moreover, ( r + a ) ˇ F ∈ H K, , ,δ ,F ∈ H K, , ,δ and ( r + a ) ˇ F (1)1 ∈ H K, , ,δ ,F ∈ H K, , ,δ and ( r + a ) ˇ F (2)2 ∈ H K, , ,δ ,F ≥ ∈ H K, , ,δ , then we can estimate || f || K, , ,δ + || r Xf || K, , ,δ ≤ C || ( r + a ) ˇ F || K, , ,δ , (9.22) || f || K, , ,δ + || r Xf || K, , ,δ ≤ C (cid:104) || F || K, , ,δ + || ( r + a ) ˇ F || K, , ,δ (cid:105) , (9.23) || f || K, , ,δ + || r Xf || K, , ,δ ≤ C (cid:104) || F || K, , ,δ + || ( r + a ) ˇ F || K, , ,δ (cid:105) , (9.24) || f ≥ || K +1 , , ,δ ≤ C || F ≥ || K, , ,δ . (9.25) Proof.
The existence and uniqueness of f ∈ ( r + a ) H K follows directly from Proposition 9.3. Theimproved estimates (9.22)–(9.25) would follow from Corollary 9.7, if we knew a priori that f and its appro-priately weighted derivatives decayed sufficiently fast towards I + . Since this decay does not follow from thefact that f ∈ ( r + a ) H K , we proceed by deriving invertibility of the operators A : D k,m,δ → H k,m,δ , asdefined above.We consider first (9.22) with K = 0. Consider the restricted inverse operator: (cid:98) L − : H , , ,δ → L − ( H , , ,δ ) ⊆ ( r + a ) H , Let F ∈ H , , ,δ and write f := (cid:98) L − ( F ). Let { ( f ) j } be a sequence in C ∞ ( (cid:99) Σ ) defined as follows:( f ) j = f ∗ η j , with η j a standard mollifier defined on an extension of (cid:98) Σ (where we view (cid:98) Σ as a compact subset of R × S ).Note that { ( r + a ) − ( f ) j } converges to ( r + a ) − f with respect to || · || .By linearity of L , we also have that ( F ) j := L ( f ) j = F ∗ η j , so by F ∈ H , , ,δ , it is straightforwardto show that: || F − ( F ) j || , , ,δ → j → ∞ . Hence, { ( F ) j } is Cauchy with respect to || · || , , ,δ and we can apply (9.14) to the difference ( f ) j − ( f ) i , i, j ∈ N , to obtain (in particular) the Cauchy property of { ( f ) j } with respect to || · || , , ,δ . It follows that f ∈ H , , ,δ and we can conclude that (9.22) holds with K = 0.In order to derive (9.23) with K = 0, we first establish invertibility of (cid:98) L : D , , ,δ → H , , ,δ as outlinedabove and then consider the operator ˇ L (1) : D , , ,δ → H , , ,δ . Similarly, we obtain (9.24) by first considering (cid:98) L : D , , ,δ → H , , ,δ , then (cid:98) L (1) : D , , ,δ → H , , ,δ , and finally ˇ L (2) : D , , ,δ → H , , ,δ . In order toderive (9.25), we analogously establish successively invertibility of (cid:98) L ( n ) for n = 0 , , , K > .3 Construction of time integral data If ψ is a solution to (3.1), then it follows from (3.3) that the restriction ψ | Σ satisfies the inhomogeneousequation (cid:98) L ψ | Σ = F [ T ψ | Σ ] , with( r + a ) − F [ f ] := 2[ h ∆ − ( r + a )] Xf + [(∆ h ) (cid:48) − r ] f + [2 h ( r + a ) − h ∆ − a sin θ ] T f + 2 a ( h − f. Proposition 9.9.
Consider initial data ( ψ | Σ , T ψ Σ ) for (3.1) , with ( φ | Σ , T φ Σ ) ∈ ( C ∞ ( (cid:98) Σ)) . (i) Then there exists a unique solution to (3.1) , denoted T − ψ , such that T − ψ ∈ C ∞ ( R ) , T ( T − ψ ) = ψ and r − T − ψ | Σ ∈ L (Σ ) and T − ψ | Σ ∈ ˙ H (Σ ) . (ii) If we moreover assume that ( I [ ψ ] , I [ ψ ] , I [ ψ ]) = (0 , , , (9.26) then the energy norms appearing on the right-hand sides of the estimates of Section 8 are finite for all K ∈ N and δ > if we replace ψ with T − ψ .Proof. We can rewrite F [ ψ | Σ ] as follows in terms of φ | Σ : F [ ψ | Σ ] = 2[ h ∆ − ( r + a )] Xφ | Σ + [(∆ h ) (cid:48) − h ∆) r ( r + a ) − ] φ | Σ + [2 h ( r + a ) − h ∆] T φ | Σ − a sin θπ ( T φ | Σ )= 2 r P + O ∞ ( r − )[ φ (1)0 | Σ + T φ | Σ + φ | Σ ] . (9.27)By combining the equations in Proposition 4.1 with the expressions in Lemma 9.5, we obtain moreover:ˇ F (1)1 [ ψ | Σ ] = 2[ r + O ∞ ( r )] P | Σ + O ∞ ( r − ) (cid:88) m =0 1 (cid:88) j =0 φ (2)1 + T j φ (1)2 m +1 + T j φ m +1 , (9.28)ˇ F (2)2 [ ψ | Σ ] = 2[ r + O ∞ ( r )] P | Σ + O ∞ ( r − ) (cid:88) m =0 1 (cid:88) j =0 φ (3)2 + T j φ (2)2 m + T j φ (1)2 m + φ m , (9.29)Denote: T − ψ | Σ := L − ( F [ ψ | Σ ]) . Then by the regularity established in Corollary 9.4, T − ψ | Σ ∈ C ∞ (Σ ) and we denote with T − ψ thesolution to (3.1) with initial data ( T − ψ | Σ , ψ Σ ). We then have that T ( T − ψ ) = ψ. Furthermore, by the injectivity properties of L following from Proposition 9.3, T − ψ must the unique solutionto (3.1) satisfying the conditions: T ( T − ψ ) = ψ , r − T − ψ | Σ ∈ L (Σ ) and T − ψ | Σ ∈ ˙ H (Σ ).Furthermore, by the above expressions ( r + a ) F , ( r + a ) ˇ F , ( r + a ) ˇ F ∈ C ∞ ( (cid:98) Σ ), if (9.26)holds, so we can apply Proposition 9.8 to conclude that for T − φ := ( r + a ) T − ψ :( T − φ ) | Σ , r X ( T − φ ) | Σ ∈ H K, , ,δ , ( T − φ ) | Σ , r X ( T − φ ) | Σ ∈ H K, , ,δ , ( T − φ ) | Σ , r X ( T − φ ) | Σ ∈ H K, , ,δ , ( T − φ ) ≥ | Σ , rX ( T − φ ) ≥ | Σ ∈ H K, , ,δ . for all K ≥ δ > ψ with T − ψ . 73 orollary 9.10. Consider initial data ( ψ | Σ , T ψ Σ ) for (3.1) , with ( φ | Σ , T φ Σ ) ∈ ( C ∞ ( (cid:98) Σ)) , such that (9.26) holds.Then the energy decay estimates in Section 6 and the pointwise decay estimates in Section 8 hold when ψ is replaced with T − ψ and we can moreover express: T − ψ ( τ, ρ , θ, ϕ ∗ ) = − (cid:90) ∞ τ ψ ( τ (cid:48) , ρ , θ, ϕ ∗ ) , dτ (cid:48) . . In this section, we will express the Newman–Penrose charges I [ T − ψ ], I m [ T − ψ ], with | m | ≤ I m [ T − ψ ], with | m | ≤
2, which are defined in Section 4, in terms of integrals over F [ ψ | Σ ], where L ( T − ψ | Σ ) = ( r + a ) − F [ ψ | Σ ]= 2[ h ∆ − ( r + a )] Xψ | Σ + [(∆ h ) (cid:48) − r ] ψ | Σ + [2 h ( r + a ) − h ∆ − a sin θ ] T ψ | Σ + 2 a ( h − ψ | Σ . (9.30) Definition 9.1.
We define the time-inverted Newman–Penrose charges I (1) (cid:96) [ ψ ] , with (cid:96) = 0 , , as follows: I (1) (cid:96) [ ψ ] := I (cid:96) [ T − ψ ] . Proposition 9.11.
Consider initial data ( ψ | Σ , T ψ | Σ ) for (3.1) , with ( φ | Σ , T φ Σ ) ∈ ( C ∞ ( (cid:98) Σ)) , such thatmoreover I [ ψ ] = 0 . Then we can express: I (1)0 [ ψ ] = I [ T − ψ ] = M (cid:90) ∞ r + ( r + a ) − π F ( ρ (cid:48) ) d ρ (cid:48) −
12 lim ρ (cid:48) →∞ ρ (cid:48) π F ( ρ (cid:48) ) − lim ρ (cid:48) →∞ [∆ h − r + a )] φ ( ρ (cid:48) ) − a lim ρ (cid:48) →∞ π (sin θφ )( ρ (cid:48) ) . In particular, if ( φ | Σ , T φ Σ ) ∈ ( C ∞ c (Σ)) , then I (1)0 [ ψ ] = M (cid:90) ∞ r + ( r + a ) − F [ ψ | Σ ]( ρ (cid:48) ) d ρ . Furthermore, if ( φ | Σ , T φ Σ ) ∈ ( C ∞ c (Σ)) , then I (1)1 m [ ψ ] = I m [ T − ψ ] = 12 M (cid:90) ∞ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m [ ψ | Σ ])( ρ (cid:48) ) d ρ (cid:48) ,I (1)2 m [ ψ ] = I m [ T − ψ ] = 15 M (cid:90) ∞ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m [ ψ | Σ ])( ρ (cid:48) ) d ρ (cid:48) . Proof.
We will suppress in the notation below the restriction | Σ . We first consider the projection onto (cid:96) = 0.The quantity ( T − ψ ) satisfies: X (∆ X ( T − ψ ) ) = ( r + a ) − F and hence, by integrating from ρ = r + , we obtain X ( T − ψ ) ( ρ ) = ∆ − (cid:90) ρ r + ( r + a ) − F ( ρ (cid:48) ) d ρ (cid:48) . Integrating the above expression again, starting from ρ = ∞ and using that lim ρ →∞ ( T − ψ ) ( ρ ) = 0 by theregularity properties of ( T − ψ ) following from Proposition 9.9, we obtain( T − ψ ) ( ρ ) = − (cid:90) ∞ ρ ∆ − ( ρ ) (cid:90) ρ r + ( r + a ) − F ( ρ ) d ρ d ρ .
74t will be convenient to denote Q = (cid:90) ∞ r + ( r + a ) − F ( ρ (cid:48) ) d ρ (cid:48) ,G ( ρ ) = − (cid:90) ∞ ρ ( r + a ) − F ( ρ (cid:48) ) d ρ . It follows that G ( ρ ) = − ρ − lim ρ (cid:48) →∞ ρ (cid:48) F ( ρ (cid:48) ) + O ∞ ( ρ − ) , with lim ρ (cid:48) →∞ ρ (cid:48) F ( ρ (cid:48) ) well-defined by Proposition 9.9. Then we obtain for ρ ≥ R , with R > r + :( T − φ ) ( ρ ) = − ( r + a ) (cid:90) ∞ ρ [ ρ − + 2 M ρ − + O ∞ ( ρ − )][ Q − ρ − lim ρ (cid:48) →∞ ρ (cid:48) F ( ρ (cid:48) ) + O ∞ ( ρ − )] d ρ = − Q (1 + M ρ − ) + 12 lim ρ (cid:48) →∞ ρ (cid:48) F ( ρ (cid:48) ) ρ − + O ∞ ( ρ − )( r XT − φ ) ( ρ ) = M Q −
12 lim ρ (cid:48) →∞ ρ (cid:48) F ( ρ (cid:48) ) + O ∞ ( ρ − ) . In order to determine I [ T − ψ ], we need the following relation between X , L and T : X = 2( r + a )∆ − L + ( h − r + a )∆ − ) T − a ∆ − Φ . We conclude that I [ T − ψ ] = M Q −
12 lim ρ (cid:48) →∞ ρ (cid:48) F ( ρ (cid:48) ) − lim ρ (cid:48) →∞ (∆ h − r + a )) φ ( ρ (cid:48) ) − a lim ρ (cid:48) →∞ π (sin θφ )( ρ (cid:48) ) . By rearranging the terms in (7.11), we obtain the following expression for general (cid:96) : X ( e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ (cid:96) +1 X (cid:96) +1 ( T − ψ ) (cid:96)m ) = e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ (cid:96) X (cid:96) F (cid:96)m . (9.31)Now consider the projection onto (cid:96) = 1 and m ∈ {− , , } and integrate (9.31) to obtain: X ( T − ψ ) m ( ρ ) = ∆ − e iam (cid:82) ∞ r ∆ − dr (cid:48) (cid:90) ρ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ (cid:48) . Integrating the above expression again, starting from ρ = ∞ and using that lim ρ →∞ X k T − ψ ( ρ ) = 0 bythe regularity properties of T − ψ following from Proposition 9.9, we obtain X ( T − ψ ) m ( ρ ) = − (cid:90) ∞ ρ ∆ − e iam (cid:82) ∞ r ∆ − dr (cid:48) (cid:90) ρ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ d ρ , ( T − ψ ) m ( ρ ) = (cid:90) ∞ ρ (cid:90) ∞ ρ ∆ − e iam (cid:82) ∞ r ∆ − dr (cid:48) (cid:90) ρ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ d ρ d ρ By assumption of compact support of φ | Σ , the following expressions are well-defined Q m = (cid:90) ∞ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ (cid:48) < ∞ ,G m ( ρ ) = − (cid:90) ∞ ρ e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ (cid:48) and it follows that G m has compact support, so in particular: G m ( ρ ) = O ∞ ( ρ − ) . (cid:16) − r + r (cid:17) e − iam (cid:82) ∞ r ∆ − dr (cid:48) = 1 + O ∞ ( ρ − ) . We can therefore express for ρ ≥ R :( T − ψ ) m ( ρ ) = (cid:90) ∞ ρ (cid:90) ∞ ρ [ ρ − + 4 M ρ − + O ∞ ( ρ − )] (cid:2) Q m + O ∞ ( ρ − ) (cid:3) d ρ d ρ , and hence ( T − φ ) m ( ρ ) =( r + a ) (cid:90) ∞ ρ (cid:90) ∞ ρ [ Q m ρ − + 4 M Q m ρ − + O ∞ ( ρ − ) d ρ d ρ = 16 Q m ρ − + 13 M Q m ρ − + O ∞ ( ρ − ) . We then obtain: r X ( T − φ ) m ( ρ ) = − Q m − M Q m ρ − + O ∞ ( ρ − ) ,r X ( r X ( T − φ ) m )( ρ ) = 23 M Q m + O ∞ ( ρ − ) . By using the compactness of the support of ( φ, T φ ), together with (9.31) and lim r →∞ ( T − φ ) = 0, weobtain: I m [ T − ψ ] = lim r →∞ r X ( T − ˇ φ (1) ) m + ima ( T − ˇ φ (1) ) m = lim r →∞ r X ( r X ( T − φ ) m ) − ( α + imα Φ ) r X ( T − φ ) m + imar X ( T − φ ) m = 12 M Q m . Finally, we consider the projection onto (cid:96) = 2 and m ∈ {− , , . . . , } and integrate (9.31) to obtain: X ( T − ψ ) m ( ρ ) = ∆ − e iam (cid:82) ∞ r ∆ − dr (cid:48) (cid:90) ρ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ (cid:48) . Integrating the above expression multiple times, starting from ρ = ∞ and using that lim ρ →∞ X k ( T − ψ ) ( ρ ) =0 by the regularity properties of ( T − ψ ) following from Proposition 9.9, we obtain( T − ψ ) m ( ρ ) = − (cid:90) ∞ ρ (cid:90) ∞ ρ (cid:90) ∞ ρ ∆ − e iam (cid:82) ∞ r ∆ − dr (cid:48) (cid:90) ρ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ . . . d ρ . By assumption of compact support of φ | Σ , the following expressions are well-defined Q m = (cid:90) ∞ r + e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ (cid:48) < ∞ ,G m ( ρ ) = − (cid:90) ∞ ρ e − iam (cid:82) ∞ r ∆ − dr (cid:48) ∆ X (( r + a ) − F m )( ρ (cid:48) ) d ρ (cid:48) and it follows that G m is compactly supported, so in particular: G m ( ρ ) = O ∞ ( ρ − ) . We can therefore express( T − φ ) m ( ρ ) = − ( r + a ) (cid:90) ∞ ρ (cid:90) ∞ ρ (cid:90) ∞ ρ Q m ρ − + 6 M Q m ρ − + O ∞ ( ρ − ) d ρ d ρ d ρ = − Q m ρ − − M Q m ρ − + O ∞ ( ρ − ) .
76e then obtain: r X ( T − φ ) m ( ρ ) = 130 Q m ρ − + 320 M Q m ρ − + O ∞ ( ρ − ) , ( r X ) ( T − φ ) m ( ρ ) = − Q m − M Q m ρ − + O ∞ ( ρ − ) , ( r X ) ( T − φ ) m ( ρ ) = 310 M Q m + O ∞ ( ρ − ) . We use that lim r →∞ ( T − φ ) = lim r →∞ r X ( T − φ ) = 0 together with compactness of the support of( φ, T φ ) to express: I m [ T − ψ ] = lim r →∞ r X ( T − ˇ φ (2) ) m + ima ( T − ˇ φ (2) ) m = lim r →∞ ( r X ) T − φ m − ( γ + α + imα Φ )( r X ) T − φ m + ima ( r X ) ( T − φ ) m = 15 M Q m . Remark 9.1.
One can generalize the argument in the proof of Proposition 9.11 in order to define I (cid:96)m [ T − ψ ] for (cid:96) = 1 , when the initial data ( ψ | Σ , T ψ | Σ ) is not compactly supported, but satisfies ( φ | Σ , T φ Σ ) ∈ ( C N ( (cid:98) Σ)) , for some suitably large N , together with the conditions: I (cid:96) [ ψ ] = 0 , as in the (cid:96) = 0 case. In thissetting, the expressions for I (cid:96)m [ T − ψ ] will be considerably more complicated than in the case of compactlysupported initial data, so we do not pursue this generalization here. In the corollary below, we obtain several simplified expressions for the time-inverted Newman–Penrosecharges I (1) (cid:96)m [ ψ ] with (cid:96) = 0 , , Corollary 9.12.
Let ( φ | Σ , T φ Σ ) ∈ ( C ∞ c (Σ)) . (i) We can express: I [ T − ψ ] = = 14 π M ( r + a ) (cid:90) Σ ∩H + ψ dω + 14 π M (cid:90) Σ n ( ψ ) dµ . (ii) We alternatively express I i [ T − ψ ] , with i = 1 , , as integrals along I + : I (1)0 [ ψ ] = M (cid:90) ∞ φ | I + ( τ ) dτI (1)1 [ ψ ]( θ, ϕ ∗ ) = 3 M (cid:90) ∞ r Xφ | I + ( τ, θ, ϕ ∗ ) dτ,I (1)2 [ ψ ]( θ, ϕ ∗ ) = 6 M (cid:90) ∞ ( r X ) φ | I + ( τ, θ, ϕ ∗ ) dτ. Proof.
In order to obtain (i), we first observe that ρ dτ (cid:93) = ( r + a − h ∆) X + [ a sin θ + h ∆ − h ( r + a )] T + a (1 − h )Φ . We can therefore write( r + a ) − F [ ψ | Σ ] = − ρ dτ (cid:93) ( ψ ) | Σ + X (cid:2) ( h ∆ − ( r + a )) ψ (cid:3) − a (1 − h )Φ( ψ ) . Recall from (2.10) that − ρ sin θdτ (cid:93) ( ψ ) | Σ = n ( ψ ) (cid:112) det g | Σ . The expression in (i) now immediately follows. 77e consider now (ii). Note that the quantities Q and Q im , with i = 1 ,
2, which are defined in the proofof Proposition 9.11, can be expressed as limits of projected radiation fields of T − ψ :( T − φ ) | Σ ( ∞ ) = − Q ,r X ( T − φ ) m | Σ ( ∞ ) = − Q m , ( r X ) ( T − φ ) m | Σ ( ∞ ) = − Q m . Using that ( T − φ ) | I + , r X ( T − φ ) m | I + , ( r X ) ( T − φ ) m | I + → τ → ∞ , by the decay estimates established in Propositions 8.1–8.4 applied to T − ψ , we can integrate φ | I + , r Xφ m | I + , ( r X ) φ m | I + to obtain (ii).
10 Late-time polynomial tails: the (cid:96) = 0 projection
We derive in this section the precise leading-order behaviour in time of ψ ≥ (cid:96) , with (cid:96) = 0 , ,
2, which will takethe form of inverse polynomial tails.We introduce the following additional initial data quantities: for 0 < β ≤ R > r + arbitrarilylarge, we define D ,β,K [ ψ ] := max ≤ k ≤ K (cid:12)(cid:12)(cid:12)(cid:12) v β + k L k (cid:0) P − I [ ψ ] v − (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ (Σ ∩{ r ≥ R } ) . We moreover introduce the following auxiliary norm on Σ τ : for 0 < δ <
1, we define S ,δ,K [ ψ ] := sup τ ≥ (cid:34) (1 + τ ) − δ (cid:88) k + k ≤ K (1 + τ ) k || ( rL ) k T k φ || L ∞ (Σ τ ) + a (1 + τ ) − δ (cid:88) (cid:96) ∈{ , } (cid:88) k + k ≤ K +1 ,k ≤ K (1 + τ ) k || ( rL ) k T k +1 φ (cid:96) || L ∞ (Σ τ ) (cid:35) . The lemma below establishes boundedness of S ,δ,K [ ψ ] in terms of the weighted initial data energy normsdefined in Section 6. Lemma 10.1.
There exists a constant C = C ( M, a, δ, K ) > such that: S ,δ,K [ ψ ] ≤ C (cid:115) (cid:88) k + j ≤ K +2 E ,k, δ [ N j ψ ] + E ,k, δ [ N j ψ ] . Proof.
The estimate follows immediately by applying the estimates in Proposition 8.1 and 8.3.We will consider in this section the timelike hypersurfaces γ α = { v − u = v α } , with α < { r ≥ R } , with R > r + suitably large: A γ α = { ( u, v, θ, ϕ ∗ ) ∈ R ∩ { r ≥ R } | u ≤ v − v α } . Let us moreover introduce the notation ( u, v γ α ( u ) , θ, ϕ ∗ ) and ( u γ α ( v ) , v, θ, ϕ ∗ ) for points on the curve γ α .78igure 3: A 2-dimensional representation of the spacetime region A γ α . A γ α The following relation between r and the variables u and v will be important: Lemma 10.2.
Let R > r + . Then there exists constants c = c ( M, a, R ) and C = C ( M, a, R ) , such thatin the region r ≥ R : cM log (cid:18) | v − u | M (cid:19) ≤ (cid:12)(cid:12)(cid:12)(cid:12) r − v − u (cid:12)(cid:12)(cid:12)(cid:12) ≤ CM log (cid:18) | v − u | M (cid:19) . (10.1) Proof.
We recall that v − u = 2 r ∗ and dr ∗ dr = r + a ∆ . Solutions r ∗ have the following form: r ∗ ( r ) = r + 2 M log r + O ∞ ( r ) . The estimates in (10.1) then follows immediately.
Proposition 10.3.
Let α K > K K . Then there exists δ ( α K ) > suitably small, such that (cid:12)(cid:12) L K P ( u, v ) − I [ ψ ] L k ( v − ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ] v − (3+ K − δ ) α K + CD ,β,K [ ψ ] v − − K − β . (10.2) Proof.
Let K ∈ N . By acting with L K on both sides of (4.10), we obtain:4 LL K P = K (cid:88) k =0 O ∞ ( r − − K )( rL ) k φ + K +1 (cid:88) k =0 O ∞ ( r − − K )[( rL ) k T φ , + ( rL ) k T φ ] . (10.3)We can estimate || LL K P || L ∞ (Σ τ ) ≤ C (1 + τ ) − δ r − − K S ,δ,K [ ψ ] . (10.4)Integrating from Σ in the L direction, applying (10.1) together with (10.4), we obtain (cid:12)(cid:12) L K P ( u, v ) − L K P ( u Σ ( v ) , v ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ] (cid:90) uu Σ0 ( v ) (1 + τ ) − δ ( u (cid:48) , v ) r − − K ( u (cid:48) , v ) du (cid:48) ≤ CS ,δ,K [ ψ ] (cid:90) uu Σ0 ( v ) u (cid:48)− δ ( v − u (cid:48) ) − − K du (cid:48) ≤ CS ,δ,K [ ψ ] v − (3+ K − δ ) α K (cid:90) uu Σ0 ( v ) u (cid:48)− − δ du (cid:48) ≤ CS ,δ,K [ ψ ] v − (3+ K − δ ) α K . D ,β,K [ ψ ]: (cid:12)(cid:12) L K P ( u Σ ( v ) , v ) − I [ ψ ] L K ( v − ) (cid:12)(cid:12) ≤ Cv − − K − β D ,β,K [ ψ ] . By combining the above, we obtain (10.4).
Corollary 10.4.
Let α K > K K . Then there exists δ ( α K ) > suitably small, such that (cid:12)(cid:12) LT K φ ( u, v ) − I [ ψ ] L K ( v − ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ] (cid:32) v − (3+ K − δ ) α K + u − K + δ ( v − u ) − + u − δ ( v − u ) − − K + u − − K + δ ( v − u ) − (cid:33) + CD ,β,K [ ψ ] v − − K − β . (10.5) Proof.
It follows from a straightforward inductive argument that we can express T K P = L K P + K − (cid:88) k =0 LL k T K − − k P . Furthermore, we have that K − (cid:88) k =0 | LL k T K − − k P | ( u, v ) ≤ C K − (cid:88) k =0 u − ( K − k )+ δ r − − k S ,δ,K [ ψ ] ≤ S ,δ,K [ ψ ][ u − K + δ r − + u − δ r − − K ] . Now, recall that P = Lφ − a ∆( r + a ) π (sin θT φ ) , so we can estimate | LT K φ ( u, v ) − T K P ( u, v ) | ≤ CS ,δ,K [ ψ ]( v − u ) − u − − K + δ . The estimate (10.5) follows by combining the above and applying (10.1).
Corollary 10.5.
For α (cid:48) K suitably close to 1, δ > suitably small and < β ≤ suitably large, there exists ν > and a constant C = C ( M, a, α (cid:48) K , ν, δ ) > such that (cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( u, v ) − I [ ψ ] T K (cid:18) uv (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | (cid:19) v − u − − K − ν , (10.6) for all ( u, v, θ, ϕ ) ∈ A γ α (cid:48) K .In particular, (cid:12)(cid:12) T K φ | I + − I [ ψ ] T K (cid:0) u − (cid:1)(cid:12)(cid:12) ≤ C (cid:18) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:19) u − − K − ν , (10.7) (cid:12)(cid:12)(cid:12) T K ψ | γ α (cid:48) K − I [ ψ ] T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12) ≤ C (cid:18) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:19) (1 + τ ) − − K − ν . (10.8) Proof.
Observe first of all that for K ≥ (cid:90) vv γαK ( u ) L K ( v − ) dv (cid:48) = L K − ( v − ) − L K − ( v (cid:48)− ) | v (cid:48) = v γαK ( u ) | v − − Kγ αK ( u ) − u − − K | ≤ Cu − − K + α k and L K − ( v − ) − L K − ( v (cid:48)− ) | v (cid:48) = u = T K (cid:18) u − v (cid:19) = ( v − u ) T K (cid:18) uv (cid:19) . Using the above together with (10.5) and taking α K suitably close to 1, we can integrate LT K φ to obtain (cid:12)(cid:12)(cid:12)(cid:12) T K φ ( u, v ) − T K φ ( u, v γ αK ( u ) ) − I [ ψ ]( v − u ) T K (cid:18) uv (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | ) u − − K − ν , for some ν > δ (cid:48) > C > | T K φ | ( u, v γ αK ( u ) ) ≤ Cr ( u, v γ αK ( u ) ) u − − K +2 δ (cid:48) (cid:113) E ,K,δ (cid:48) [ ψ ] ≤ C (cid:113) E ,K [ ψ ] u − + α k − K + δ (cid:48) , so that (cid:12)(cid:12)(cid:12)(cid:12) T K φ ( u, v ) − I [ ψ ] T K (cid:18) v − uuv (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:19) u − − K − ν , for δ > r , we moreover obtain the following estimate in the smaller region A γ α (cid:48) K , with α (cid:48) K > α K suitably close to 1, then there exists a constant ν > (cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( u, v ) − I [ ψ ] T K (cid:18) uv (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:19) v − u − − K − ν . R \ A γ α In this section, we propagate the late-time asymptotics derived in Corollary 10.5 to the rest of the spacetime.
Proposition 10.6.
Let K ∈ N and let < α < be sufficiently close to . Then there exists a constant C > such that (cid:12)(cid:12) T K ψ − I [ ψ ] T K ((1 + τ ) − ) (cid:12)(cid:12) ( τ, ρ ) ≤ C (1 + τ ) − − K − ν S ,δ,K [ ψ ] + D ,β,K [ ψ ] + | I [ ψ ] | + (cid:88) (cid:96) ∈{ , } (cid:113) E (cid:96),K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E (cid:96),K +1 ,δ [ N j ψ ] (10.9) for all ( τ, ρ , θ, ϕ ∗ ) ∈ R \ A γ α .Proof. We apply the fundamental theorem of calculus, integrating XT K ψ between ρ = ρ (cid:48) and ρ = ρ γ α ( τ ) .We use (10.8) to estimate the boundary term on ρ = ρ γ α ( τ ) and (8.22) to estimate the contribution of theintegral. Remark 10.1.
One can insert the expression (10.6) when integrating both sides of (10.3) to refine theasymptotics of φ along I + and obtain the next-to-leading order logarithmic asymptotics; see Remark 1.1and also [AAG19] for an application of this argument in the case a = 0 . We do not pursue this refinementin the present paper. The estimates in Section 10.1 and 10.2 provide the late-time asymptotics for ψ arising from initial data with I [ ψ ] (cid:54) = 0. Using the time-integral construction from Section 9, we can moreover apply these estimates to thesetting when I [ ψ ] = 0, and in particular, to the setting where the initial data of ψ is smooth and compactlysupported. Proposition 10.7.
Consider initial data ( ψ | Σ , T ψ | Σ ) for (3.1) , with ( φ | Σ , T φ Σ ) ∈ ( C ∞ ( (cid:98) Σ)) , such thatmoreover I [ ψ ] = 0 . Let r > r + . Then there exists a ν > and a constant C = C ( M, a, Σ , r , ν ) > , suchthat (cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( u, v ) − I [ T − ψ ] T K +1 (cid:18) uv (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cv − u − − K − ν (cid:18)(cid:113) E ,K +1 ,δ [ T − ψ ] + S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + | I [ T − ψ ] | (cid:19) in { r ≥ r } , whereas in { r ≤ r } , we can express: (cid:12)(cid:12) T K ψ + 8 I [ T − ψ ] T K ((1 + τ ) − ) (cid:12)(cid:12) ( τ, ρ ) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + | I [ T − ψ ] | + (cid:88) (cid:96) ∈{ , } (cid:113) E (cid:96),K +3 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E (cid:96),K +2 ,δ [ N j T − ψ ] (cid:33) . Proof.
We apply Propositions 9.9 and 9.11 to conclude that T − φ has sufficiently high regularity to concludethat all the relevant initial data energy norms for T − ψ are finite and I [ T − ψ ] is well-defined. From this itfollows that Corollary 10.5 and Proposition 10.6 immediately apply when ψ is replaced by T − ψ .
11 Late-time polynomial tails: the (cid:96) = 1 projection
We derive in this section the leading-order late-time asymptotics of the restriction ψ of ψ .It will be useful for the estimates in this section to define the following quantities: let β > < δ < D ,β,K [ ψ ] := max ≤ k ≤ K || v β + k L k (cid:0) r − P − I [ ψ ] v − (cid:1) || L ∞ (Σ ∩{ r ≥ R } ) ,S ,δ,K [ ψ ] := sup τ ≥ (1 + τ ) − δ (cid:88) j =0 (cid:88) ≤ k + k ≤ K +1 k ≤ K (1 + τ ) k || ( rL ) k T k ˇ φ ( j )1 || L ∞ (Σ τ ) + sup τ ≥ (1 + τ ) − δ (cid:88) (cid:96) ∈{ , } (cid:88) j =0 (cid:88) ≤ k + k ≤ K +1 k ≤ K (1 + τ ) k || ( rL ) k T k +1 ˇ φ ( j ) (cid:96) || L ∞ (Σ τ ) . Lemma 11.1.
There exists a constant C = C ( M, a, δ, K ) > such that: S ,δ,K [ ψ ] ≤ C (cid:115) (cid:88) k + j ≤ K +2 E ,k, δ [ N j ψ ] + E ≥ ,k, δ [ N j ψ ] . Proof.
The estimate follows immediately by applying the estimates in Proposition 8.2 and 8.4.82 A γ α Proposition 11.2.
Let α K > K K . Then there exists δ ( α K ) > suitably small, such that (cid:12)(cid:12) L K ( r − P )( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ ) L K ( v − ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ] v − (5+ K − δ ) α K + C ( D ,β,K [ ψ ]+ I [ ψ ]) v − (4+ K + β ) . (11.1) Proof.
Let K ∈ N . By acting with r − and then with L K on both sides of (4.11), we obtain:4 LL K ( r − P ) = K (cid:88) k =0 O ∞ ( r − − K )( rL ) k P + (cid:88) j =0 K (cid:88) k =0 O ∞ ( r − − K )( rL ) k φ ( j )1 + (cid:88) j =0 K +1 (cid:88) k =0 O ∞ ( r − − K )( rL ) k T π (sin θφ ( j ) ) . (11.2)We can further estimate for all k ∈ N : | ( rL ) k P − r − ( rL ) k +1 ˇ φ (1)1 | ( u, v, θ, ϕ ∗ ) ≤ Cr − ( u, v ) (cid:88) j =0 a || ( rL ) k T ˇ φ ( j )3 || L ∞ (Σ τ ) + a || ( rL ) k ˇ φ ( j )1 || L ∞ (Σ τ ) (11.3)and hence obtain: | LL K ( r − P ) | ( u, v, θ, ϕ ∗ ) ≤ Cu − δ r − − K S ,δ,K [ ψ ] . (11.4)Integrating from Σ in the L direction, applying (10.1) together with (10.4), we obtain (cid:12)(cid:12) L K ( r − P )( u, v, θ, ϕ ∗ ) − L K ( r − P )( u Σ ( v ) , v, θ, ϕ ∗ ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ] (cid:90) uu Σ0 ( v ) (1 + τ ) − δ ( u (cid:48) , v ) r − − K ( u (cid:48) , v ) du (cid:48) ≤ CS ,δ,K [ ψ ] (cid:90) uu Σ0 ( v ) u (cid:48)− δ ( v − u (cid:48) ) − − K du (cid:48) ≤ CS ,δ,K [ ψ ] v − (5+ K − δ ) α K (cid:90) uu Σ0 ( v ) u (cid:48)− − δ du (cid:48) ≤ CS ,δ,K [ ψ ] v − (5+ K − δ ) α K . Furthermore, we have that (cid:12)(cid:12) L K ( r − P )( u Σ ( v ) , v, θ, ϕ ∗ ) − I [ ψ ] L K ( v − ) (cid:12)(cid:12) ≤ Cv − − K − β D ,β,K [ ψ ] . so we can conclude that (cid:12)(cid:12) L K ( r − P )( u, v, θ, ϕ ∗ ) − I [ ψ ] L K ( v − ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ] v − (5+ K − δ ) α K + CD ,β,K [ ψ ] v − − K − β . Corollary 11.3.
Let α K > K K . Then there exists δ ( α K ) > suitably small, such that (cid:12)(cid:12)(cid:12) LT K ˇ φ (1)1 ( u, v, θ, ϕ ∗ ) − I [ ψ ]( v − u ) L K ( v − ) (cid:12)(cid:12)(cid:12) ≤ CS ,δ,K [ ψ ][ a ( v − u ) − u − − K + δ + ( v − u ) v − (5+ K − δ ) α K + u − K + δ ( v − u ) − + u − δ ( v − u ) − − K ]+ CD ,β,K [ ψ ]( v − u ) v − − K − β . (11.5) Proof.
By induction, it follows that we can express r − T K P = L K ( r − P ) + K − (cid:88) k =0 LL k ( r − T K − − k P ) . P replaced by T K − − k P , we obtain: K − (cid:88) k =0 | LL k T K − − k P | ( u, v ) ≤ C K − (cid:88) k =0 u − ( K − k )+ δ r − − k S ,δ,K [ ψ ] ≤ S ,δ,K [ ψ ][ u − K + δ r − + u − δ r − − K ] . Hence, it follows from (11.1) that (cid:12)(cid:12) T K P ( u, v, θ, ϕ ∗ ) − I [ ψ ] r L K ( v − ) (cid:12)(cid:12) ≤ CS ,δ,K [ ψ ][ r v − (5+ K − δ ) α K + u − K + δ r − + u − δ r − − K ]+ CD ,β,K [ ψ ] r v − − K − β . By (11.3) with k = 0 and φ replaced by T K φ , we obtain: (cid:12)(cid:12)(cid:12) LT K ˇ φ (1)1 ( u, v, θ, ϕ ∗ ) − T K P (cid:12)(cid:12)(cid:12) ≤ CaS ,δ,K [ ψ ] r − u − − K + δ . By combining the above and applying (10.1), we conclude that (11.5) holds.The lemma below provides some integral identities that will be useful for integrating (multiple times) inthe L -direction. Lemma 11.4.
Let K ∈ N and m ∈ R , with m > . Then: (cid:90) v v ( v − u ) m − L K ( v − m ) dv = ( m − − (cid:20) ( v − u ) m − T K (cid:18) uv m − (cid:19) | v = v − ( v − u ) m − T K (cid:18) uv m − (cid:19) | v = v (cid:21) . (11.6) Proof.
Note that (cid:90) v v ( v − u ) m − L K ( v − m ) dv = (cid:90) v v T K (( v − u ) m − v − m ) dv = (cid:90) v v LT K (cid:18) ( m − − ( v − u ) m − uv m − (cid:19) dv =( m − − (cid:20) ( v − u ) m − T K (cid:18) uv m − (cid:19) | v = v − ( v − u ) m − T K (cid:18) uv m − (cid:19) | v = v (cid:21) . Proposition 11.5.
For α (cid:48)(cid:48) K suitably close to 1, δ > suitably small and < β ≤ suitably large, thereexists ν > and a constant C = C ( M, a, α (cid:48)(cid:48) K , ν, δ ) > such that (cid:12)(cid:12)(cid:12) T K ˇ φ (1)1 ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ )( v − u ) T K ( u − v − ) (cid:12)(cid:12)(cid:12) (11.7) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) u − − K − ν , (11.8) (cid:12)(cid:12)(cid:12) r − T K ψ ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ ) T K ( u − v − ) (cid:12)(cid:12)(cid:12) (11.9) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) v − u − − K − ν , (11.10) (cid:12)(cid:12)(cid:12) LT K ψ ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ ) T K ( u − v − ) (cid:12)(cid:12)(cid:12) (11.11) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) v − u − − K − ν (11.12)84 n A γ α (cid:48)(cid:48) K .In particular, we have that (cid:12)(cid:12)(cid:12)(cid:12) T K ˇ φ (1)1 | I + − I [ ψ ] T K ( u − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) u − − K − ν , (11.13) (cid:12)(cid:12)(cid:12)(cid:12) T K φ | I + − I [ ψ ] T K ( u − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) u − − K − ν (11.14) and (cid:12)(cid:12)(cid:12)(cid:12) r − T K ψ | γ α (cid:48) K − I [ ψ ] T K ( τ − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) τ − − K − ν , (11.15) (cid:12)(cid:12)(cid:12)(cid:12) XT K ψ | γ α (cid:48) K − I [ ψ ] T K ( τ − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) τ − − K − ν . (11.16) Proof.
Let ( u, v, θ, ϕ ∗ ) be a point in A γ αK . Let us denote with γ L the integral curve tangent to L connectingthe point ( u, v, θ, ϕ ∗ ) with the curve γ α K , where ˙ γ L = L . Then ( u γ L ( v (cid:48) ) , θ γ L ( v (cid:48) )) = ( u, θ ) for all v γ αK ( u ) ≤ v (cid:48) ≤ v and L ( ϕ ∗ ) = ar + a with ( ϕ ∗ ) γ L ( v ) = ϕ ∗ .We can therefore obtain for all v γ αK ( u ) ≤ v (cid:48) ≤ v : | ϕ ∗ − ( ϕ ∗ ) γ L ( v (cid:48) ) | ≤ (cid:90) vv γαK | a | ( r + a ) − dv (cid:48) ≤ Cu − α K . (11.17)Hence, for all v γ αK ( u ) ≤ v (cid:48) ≤ v : | I ( θ, ( ϕ ∗ ) γ L ( v (cid:48) )) − I ( θ, ϕ ∗ ) | ≤ C (cid:88) m = − | I m | u − α K . (11.18)By combining the above estimate with Lemma 11.4 for m = 4, we have that: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) vv γαK ( u ) I [ ψ ] | γ L ( v (cid:48) − u ) L K ( v (cid:48)− ) dv (cid:48) − I [ ψ ]( θ, ϕ ∗ )( v − u ) T K ( u − v − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:88) m = − | I m | u − − K +3 α K . By integrating (11.5), and applying (8.8), we obtain | T K ˇ φ (1)1 | ( u, v γ αK ( u ) , θ, ϕ ∗ ) ≤ Cr (1+ δ ) ( u, v γ αK ( u ) ) u − − K +2 δ (cid:113) E ,K,δ [ ψ ] ≤ C (cid:113) E ,K,δ [ ψ ] u − + α K − K +2 δ , we then obtain (11.7) and (11.13) for some ν > T K ˇ φ (1)1 = 2( r + O ( r )) Lφ + aKO ( r − ) T K − φ , we have that: (cid:12)(cid:12)(cid:12)(cid:12) LT K φ ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ )( v − u ) T K ( u − v − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m | (cid:33) ( v − u ) − u − − K − ν . (11.19)85et α (cid:48) K > α K , then we can estimate in A γ α (cid:48) K using (8.7): | T K φ | ( u, v γ α (cid:48) K ( u ) , θ, ϕ ) ≤ C (cid:113) E ,K,δ [ ψ ] u − + α (cid:48) K − K + δ , so by integrating (11.19) and applying (11.18) and Lemma 11.4 again, taking α (cid:48) K > α K suitably large, weobtain the following estimate in A γ α (cid:48) K : (cid:12)(cid:12)(cid:12)(cid:12) T K φ ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ )( v − u ) T K ( u − v − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:32)(cid:113) E ,K,δ [ ψ ] + S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m | (cid:33) u − − K − ν . (11.20)Hence, (11.14) follows.We conclude that (11.9) must hold in the smaller region A γ α (cid:48)(cid:48) K , with α (cid:48)(cid:48) K > α K suitably large, bymultiplying both sides of (11.20) with r − .Finally, we combine (11.20) with (11.19) and use that Lψ = − ∆ r r + a ) r − ψ + 1 √ r + a Lφ to obtain (11.11). The estimates (11.15) and (11.15) then follows immediately. R \ A γ α In this section, we will extend the late-time asymptotics in Proposition 11.5 to the rest of the spacetime.In particular, we will demonstrate the presence of oscillations in the late-time behaviour along the nullgenerators of the event horizon H + . Proposition 11.6.
Let K ∈ N and let α > be arbitrarily large. Then there exists a constant C > suchthat (cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( τ, ρ , θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ ) ρ T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12) ( τ, ρ , θ, ϕ ∗ ) (11.21) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | + (cid:113) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +3 ,δ [ N j ψ ]+ (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:33) , (cid:12)(cid:12)(cid:12)(cid:12) XT K ψ ( τ, ρ , θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ ) T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12) ( τ, ρ , θ, ϕ ∗ ) (11.22) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m | + (cid:113) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +3 ,δ [ N j ψ ]+ (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:33) , for all ( τ, ρ , θ, ϕ ∗ ) ∈ R \ A γ α . n particular, the following oscillatory asymptotics hold along the null generators of H + : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ | H + ( v, θ, ϕ H + ) − r + 1 (cid:88) m = − I m [ ψ ] Y ,m ( θ, ϕ H + ) e im ω + v T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m | + (cid:113) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +3 ,δ [ N j ψ ]+ (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:33) , (11.23) where we use Y (cid:96),m ( θ, ϕ H + ) denote spherical harmonics with respect to the polar angle θ and the azimuthalangle ϕ H + .Proof. In order to obtain (11.22), we apply the fundamental theorem of calculus, integrating X T K ψ between ρ = ρ (cid:48) and ρ = ρ γ α ( τ ) . We use (11.16) to estimate the boundary term on ρ = ρ γ α ( τ ) and (8.23) toestimate the contribution of the integral.The estimate (11.21) then follows by applying the fundamental theorem of calculus again, integratingnow XT K ψ between ρ = ρ (cid:48) and ρ = ρ γ α ( τ ) and using the estimates (11.22) and (11.15).Finally, (11.23) follows from (11.21) by using the relation (2.3) between ϕ H + and ϕ ∗ to conclude that e imϕ ∗ = e imϕ H + e im ω + v , and hence Y (cid:96),m ( θ, ϕ ∗ ) = Y (cid:96),m ( θ, ϕ H + ) e im ω + v . We can now apply the time-integral construction from Section 9 to obtain late-time asymptotics for ψ arising from initial data that are smooth and compactly supported. Proposition 11.7.
Consider initial data ( ψ | Σ , T ψ | Σ ) for (3.1) , with ( φ | Σ , T φ Σ ) ∈ ( C ∞ c (Σ )) . Let r > r + . Then there exists a ν > and a constant C = C ( M, a, Σ , r , ν ) > , such that (cid:12)(cid:12)(cid:12) r − T K ψ ( u, v, θ, ϕ ∗ ) − I [ T − ψ ]( θ, ϕ ∗ ) T K +1 ( u − v − ) (cid:12)(cid:12)(cid:12) ≤ Cv − u − − K − ν (cid:32) S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ ψ ] + (cid:88) m = − | I m [ T − ψ ] | + (cid:113) E ,K +6 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j T − ψ ] + (cid:113) E ≥ ,K +3 ,δ [ T − ψ ]+ (cid:88) j =0 (cid:113) E ≥ ,K +3 ,δ [ N j T − ψ ] (cid:33) in { r ≥ r } , whereas in { r ≤ r } , we can express: (cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( τ, ρ , θ, ϕ ∗ ) + 323 I [ T − ψ ]( θ, ϕ ∗ ) ρ T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12) ( τ, ρ , θ, ϕ ∗ ) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + (cid:88) m = − | I m [ T − ψ ] | + (cid:113) E ,K +6 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j T − ψ ] + (cid:113) E ≥ ,K +3 ,δ [ T − ψ ]+ (cid:88) j =0 (cid:113) E ≥ ,K +3 ,δ [ N j T − ψ ] (cid:33) . n particular, the following oscillatory asymptotics hold along the null generators of H + : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ | H + ( v, θ, ϕ H + ) + 323 r + 1 (cid:88) m = − I m [ T − ψ ] Y ,m ( θ, ϕ H + ) e im ω + v T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + (cid:88) m = − | I m [ T − ψ ] | + (cid:113) E ,K +6 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j T − ψ ] + (cid:113) E ≥ ,K +3 ,δ [ T − ψ ]+ (cid:88) j =0 (cid:113) E ≥ ,K +3 ,δ [ N j T − ψ ] (cid:33) . Proof.
We apply Propositions 9.9 and 9.11 to conclude that T − φ has sufficiently high regularity to concludethat all the relevant energies for T − ψ are finite and I [ T − ψ ] is well-defined. Then the estimates followsimmediately by applying the estimates in Propositions 11.5 and 11.6 to T − ψ instead of ψ .
12 Late-time polynomial tails: the (cid:96) = 2 projection
In this section we derive the late-time asymptotics for the projection ψ . In contrast with the (cid:96) = 0 , the leading-order late-time behaviour of ψ is coupled with the leading-order late-timebehaviour of ψ and will therefore not just depend on the Newman–Penrose charge I [ ψ ], but also on I [ ψ ].As will be shown below, this coupling mechanism will limit the decay rate of ψ in regions of bounded r .It will be useful for the estimates in this section to define the following quantities: let β > < δ < D ,β,K [ ψ ] := max ≤ k ≤ K || v β + k L k (cid:0) r − P − I [ ψ ] v − (cid:1) || L ∞ (Σ ∩{ r ≥ R } ) ,S ,δ,K [ ψ ] := sup τ ≥ (1 + τ ) − δ (cid:88) j =0 (cid:88) ≤ k + k ≤ K +1 k ≤ K (1 + τ ) k || ( rL ) k T k ˇ φ ( j )2 || L ∞ (Σ τ ) + sup τ ≥ (1 + τ ) − δ (cid:88) (cid:96) ∈{ , } (cid:88) j =0 (cid:88) ≤ k + k ≤ K +1 k ≤ K (1 + τ ) k || ( rL ) k T k +1 ˇ φ ( j ) (cid:96) || L ∞ (Σ τ ) + sup τ ≥ (1 + τ ) − δ (cid:88) j =0 (cid:88) ≤ k + k ≤ K +1 k ≤ K (1 + τ ) k || ( rL ) k T k +1 ˇ φ ( j )0 || L ∞ (Σ τ ) + S ,δ, K [ ψ ] + | I [ ψ ] | + D , − δ, K [ ψ ] . Lemma 12.1.
There exists a constant C = C ( M, a, δ, K ) > such that: S ,δ,K [ ψ ] ≤ C (cid:88) k + j ≤ K +2 (cid:113) E ,k, δ [ N j ψ ] + (cid:113) E ≥ ,k, δ [ N j ψ ] + (cid:113) E ,k +3 , δ [ N j ψ ] + C | I [ ψ ] | + CD , − δ, K [ ψ ] . Proof.
The estimate follows immediately by applying the estimates in Proposition 8.3 and 8.3, together withLemma 10.1 and also (12.6) below to estimate the terms involving ( rL ) k T k +1 ˇ φ (1)0 .88 A γ α We recall here (4.12), but with the dependence on π (sin θ ˇ φ (2)0 ) stated more precisely:4 LP = [ − r − + O ∞ ( r − )] P − [2 r − + O ∞ ( r − )] a ∆( r + a ) π (sin θT ˇ φ (2)0 ) + L ( a ∆( r + a ) π (sin θT ˇ φ (2)0 ))+ O ∞ ( r − )[ ˇ φ (2)2 + ˇ φ (1)2 + φ ] + O ∞ ( r − )[ T π (sin θ ˇ φ (2)4 ) + T π (sin θ ˇ φ (1) ) + T π (sin θφ )]+ O ∞ ( r − )[ LT π (sin θ ˇ φ (2)4 ) + LT π (sin θ ˇ φ (1) ) + LT π (sin θφ )]Due to the slower decay properties of ˇ φ (2)0 , it will be convenient to put some of the terms involving ˇ φ (2)0 onthe left-hand side of the above equation by introducing the modified quantity: (cid:101) P = P − r L (cid:18) r − a ∆( r + a ) π (sin θT ˇ φ (2)0 ) (cid:19) . Then:4 L ( r − (cid:101) P ) = L (cid:18) r − a ∆( r + a ) π (sin θ ˇ φ (2)0 ) (cid:19) + O ∞ ( r − ) P + O ∞ ( r − )[ ˇ φ (2)2 + ˇ φ (1)2 + φ ] + O ∞ ( r − )[ T π (sin θ ˇ φ (2)4 ) + T π (sin θ ˇ φ (1) ) + T π (sin θφ )]+ O ∞ ( r − )[ rLT π (sin θ ˇ φ (2)4 ) + rLT π (sin θ ˇ φ (1) ) + rLT π (sin θφ )] . (12.1)In order to determine the late-time asymptotics of ψ , it will be necessary to appeal to the late-timeasymptotics of ψ obtained in Section 10. We derive in the lemma below the necessary estimates for ψ . Lemma 12.2.
Let K ∈ N and α K > K K . Then there exists δ ( α K ) > suitably small, such that for all j, m ∈ N with j + m = K , we have that in A γ αK : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L j +1 (cid:20) r − a ∆( r + a ) T m ˇ φ (2)0 (cid:21) − a I [ ψ ] L K T m (cid:32) v − u ) − v − − v − u ) − v − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12.2) ≤ C ( r − − j u − − m + δ + r − − K u − δ + v − (7 − K +2 δ ) α K − ) S ,δ,K − [ ψ ] , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L j +2 (cid:20) r − a ∆( r + a ) T m ˇ φ (2)0 (cid:21) − a I [ ψ ] LL K T m (cid:32) − v − u ) − v − − v − u ) − v − + 16( v − u ) − v − + 24( v − u ) − v − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12.3) ≤ C ( r − − j u − − m + δ + r − − K u − δ + v − (8 − K +2 δ ) α K ) S ,δ,K [ ψ ] , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ∆( r + a ) T K +1 ˇ φ (2)0 − a I [ ψ ] L K (cid:18)
32 ( v − u ) v − − ( v − u ) v − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12.4) ≤ C ( r − − K u − δ + r − u − − K + δ + v − (4 − K +2 δ ) α K − ) S ,δ,K − [ ψ ] . Furthermore, for all k ≤ K , we can estimate r − | ( rL ) k ( r − T ˇ φ (2)0 ) | ≤ CS ,δ,K [ ψ ][( r − v − + r − u − δ + r − u − δ ) (12.5) Proof.
We can write14 L (cid:18) r − a ∆( r + a ) T m ˇ φ (2)0 (cid:19) = 12 L (cid:16) a (cid:2) r − + O ∞ ( r − ) (cid:3) LT m ˇ φ (1)0 (cid:17) = L (cid:0) a (cid:2) r − + O ∞ ( r − ) (cid:3) L T m φ (cid:1) + L (cid:0) a (cid:2) r − + O ∞ ( r − ) (cid:3) LT m φ (cid:1) = a ( r − + O ∞ ( r − )) L T m φ − a ( r − + O ∞ ( r − )) sin θLT m φ . L K (cid:20) L (cid:18) r − a ∆( r + a ) ˇ φ (2)0 (cid:19)(cid:21) = a K (cid:88) k =0 K ! k !( K − k )! L k ( r − + O ∞ ( r − )) L K − k T m φ − a K (cid:88) k =0 K ! k !( K − k )! L k ( r − + O ∞ ( r − )) L K − k T m φ By (10.2) together with the arguments in the proof of Corollary 10.4, we moreover have that for all j ≥ β ≥ (cid:12)(cid:12) L j T m φ ( u, v ) + 2 I [ ψ ] L j + m ( v − ) (cid:12)(cid:12) ≤ CS ,δ,j + m − [ ψ ] (cid:104) r − − j u − − m + δ + u − δ r − − j − m + v − (2+ j + m − δ ) α j + m − (cid:105) + CD ,β,K [ ψ ] v − − j − m − β (12.6)in A γ αj + m − . By combining the above, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L j +1 (cid:20) r − a ∆( r + a ) T m ˇ φ (2)0 (cid:21) − a I [ ψ ] T m L j (cid:32) v − u ) − v − − v − u ) − v − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( r − − j u − − m + δ + r − − K u − δ + v − (7 − K +2 δ ) α K − ) S ,δ,K − [ ψ ] . and hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L j +2 (cid:20) r − a ∆( r + a ) T m ˇ φ (2)0 (cid:21) − a I [ ψ ] L j T m (cid:32) v − u ) − v − − v − u ) − v − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( r − − j u − − m + δ + r − − K u − δ + v − (8 − K +2 δ ) α K ) S ,δ,K [ ψ ] . We obtain (12.4) and (12.5) in an analogous manner, appealing again to the arguments in the proof ofCorollary 10.4.Let f be a suitably regular function on S . Then recall that by (2.13), we can write: π (sin θπ ( f )) = − (cid:114) π π ( f ) Y , ( θ ) . Proposition 12.3.
Let α K > K K . Then there exists δ ( α K ) > suitably small, such that in A γ αK : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L K ( r − (cid:101) P ) − I [ ψ ] L K ( v − ) + 83 (cid:114) π a I [ ψ ] Y , L K ( v − )+ 83 (cid:114) π a I [ ψ ] Y , L K (cid:32) − v − u ) − v − − v − u ) − v − + 16( v − u ) − v − + 24( v − u ) − v − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cv − (8 − K +2 δ ) α K +1 S ,δ,K [ ψ ] + CD ,β,K [ ψ ] v − − K − β . (12.7) Proof.
Let K ∈ N . By multiplying with r − and then acting with L K on both sides of (4.12), we obtain:4 LL K ( r − (cid:101) P ) = L K (cid:18) r − a ∆( r + a ) π (sin θ ˇ φ (2)0 ) (cid:19) + K (cid:88) k =0 O ∞ ( r − − K )( rL ) k P + (cid:88) j =0 K (cid:88) k =0 O ∞ ( r − − K )( rL ) k ˇ φ ( j )2 + (cid:88) j =0 K +1 (cid:88) k =0 O ∞ ( r − − K )( rL ) k T π (sin θφ ( j )4 )+ (cid:88) j =0 K +1 (cid:88) k =0 O ∞ ( r − − K )( rL ) k T π (sin θ ˇ φ ( j )0 ) . (12.8)90e can further estimate for all k ∈ N : | ( rL ) k P − r − ( rL ) k +1 ˇ φ (2)2 | ( u, v, θ, ϕ ∗ ) ≤ Ca | ( rL ) k ( r − T ˇ φ (2)0 ) | + Car − ( u, v ) (cid:88) j =0 ( a || ( rL ) k T ˇ φ ( j )0 || L ∞ (Σ τ ) + || ( rL ) k ˇ φ ( j )2 || L ∞ (Σ τ ) + a || ( rL ) k T ˇ φ ( j )4 || L ∞ (Σ τ ) + a || ( rL ) k T ˇ φ (2)4 || L ∞ (Σ τ ) ) . (12.9)and hence obtain after applying (12.3) and (12.5): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (cid:34) L K ( r − (cid:101) P ) + 83 (cid:114) π a I [ ψ ] Y , L K (cid:32) − v − u ) − v − − v − u ) − v − + 16( v − u ) − v − + 24( v − u ) − v − (cid:33)(cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( r − − K u − δ + v − (8 − K +2 δ ) α K ) S ,δ,K [ ψ ] . (12.10)Integrating from Σ in the L direction, applying (10.1) together with (10.4), we obtain (cid:90) uu Σ0 ( v ) u (cid:48)− δ r − − K ( u (cid:48) , v ) du (cid:48) ≤ C (cid:90) uu Σ0 ( v ) u (cid:48)− δ ( v − u (cid:48) ) − − K du (cid:48) ≤ Cv − (7+ K − δ ) α K (cid:90) uu Σ0 ( v ) u (cid:48)− − δ du (cid:48) ≤ Cv − (7+ K − δ ) α K . Hence, (12.7) follows after integrating the above inequalities.
Corollary 12.4.
Let K ∈ N and α K > K K . Then there exists δ ( α K ) > suitably small, such that in A γ αK : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) LT K ˇ φ (2)2 − I [ ψ ]( v − u ) L K ( v − ) + 403 (cid:114) π a I [ ψ ] Y , ( v − u ) L K ( v − )+ 83 (cid:114) π a Y , I [ ψ ] T K (cid:18) − v − u ) v − + 32 ( v − u ) v − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( u − δ r − − K + u − − K + δ r − + r v − (8 K +2 δ ) α K +1 ) S ,δ,K [ ψ ] + CD ,β,K [ ψ ] r v − − K − β . (12.11) Proof.
In light of the estimate (12.7), it is convenient to introduce a second modification of P : (cid:102)(cid:101) P := (cid:101) P + 83 (cid:114) π a I [ ψ ] Y , (cid:32) − v − u ) − v − − v − u ) − v − + 16( v − u ) − v − + 24( v − u ) − v − (cid:33) By (12.10), we have that | L ( r − L K (cid:102)(cid:101) P ) | ≤ C ( r − − K u − δ + v − (8 − K +2 δ ) α K ) S ,δ,K [ ψ ]and by induction, it follows that we can express r − T K (cid:102)(cid:101) P = L K ( r − (cid:102)(cid:101) P ) + K − (cid:88) k =0 LL k ( r − T K − − k (cid:102)(cid:101) P ) . Furthermore, by repeating the arguments leading to (12.10) for P replaced by T K − − k P and applying(12.3) and (12.5), we obtain K − (cid:88) k =0 | LL k ( r − T K − − k (cid:102) P ) | ( u, v, θ, ϕ ∗ ) ≤ C ( u − δ r − − K + u − − K + δ r − + v − (7 − K +2 δ ) α K ) S ,δ,K [ ψ ] . LT K ˇ φ (2)2 = T K (cid:102)(cid:101) P + 83 (cid:114) π a I [ ψ ] Y , T K (cid:32) − v − u ) v − −
32 ( v − u ) v − + ( v − u ) v − + 32 v − (cid:33) + 14 r T K L (cid:18) r − a ∆( r + a ) π (sin θ ˇ φ (2)0 ) (cid:19) + 14 a ∆( r + a ) π (sin θT K +1 ˇ φ (2)0 ) , so after applying Lemma 12.2 we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) LT K ˇ φ (2)2 − I [ ψ ]( v − u ) L K ( v − ) + 403 (cid:114) π a I [ ψ ] Y , ( v − u ) L K ( v − )+ 83 (cid:114) π a Y , I [ ψ ] T K (cid:18) − v − u ) v − + 32 ( v − u ) v − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( u − δ r − − K + u − − K + δ r − + r v − (8 K +2 δ ) α K +1 ) S ,δ,K [ ψ ] + CD ,β,K [ ψ ] r v − − K − β . Proposition 12.5.
For α (cid:48)(cid:48)(cid:48) K suitably close to 1, δ > suitably small and < β ≤ suitably large, thereexists ν > and a constant C = C ( M, a, α (cid:48)(cid:48)(cid:48) K , ν, δ ) > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ )( v − u ) T K ( u − v − )+ 643 (cid:114) π a I [ ψ ] Y , ( θ ) T K (cid:34) u − v − ( v − u ) − u − v − ( v − u ) + 14 u − v − (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cu − − K − ν v − ( S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | ) + C (cid:113) E ,K,δ [ ψ ] u − − K − ν (12.12) in A γ α (cid:48)(cid:48)(cid:48) K and along I + : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K φ | I + − I [ ψ ] T K ( u − ) + 89 (cid:114) π a I [ ψ ] Y , ( θ ) T K ( u − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12.13) ≤ Cu − − K − ν (cid:16) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:17) , (12.14) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ˇ φ (1)2 | I + − I [ ψ ] T K ( u − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12.15) ≤ Cu − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:33) , (12.16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ˇ φ (2)2 | I + − I [ ψ ] T K ( u − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12.17) ≤ Cu − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:33) . (12.18)In particular, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ | γ α (cid:48)(cid:48) K + 163 (cid:114) π a I [ ψ ] Y , ( θ ) T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | + (cid:113) E ,K,δ [ ψ ] (cid:33) . (12.19)92 roof. Note first of all that (11.18) holds also with I [ ψ ] replaced by I [ ψ ].We then integrate (12.11) in the L direction, starting from γ α K and use (8.15) to obtain: | T K ˇ φ (2)2 | ( u, v γ αK ( u ) , θ, ϕ ∗ ) ≤ Cr (1+ δ (cid:48) ) ( u, v γ αK ( u ) ) u − − K +2 δ (cid:48) (cid:113) E ,K,δ (cid:48) [ ψ ] ≤ C (cid:113) E ,K,δ (cid:48) [ ψ ] u − + α K − K +2 δ (cid:48) , for δ (cid:48) > < α K < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ˇ φ (2)2 ( u, v, θ, ϕ ∗ ) − I [ ψ ]( θ, ϕ ∗ )( v − u ) T K ( u − v − )+ 83 (cid:114) π a I [ ψ ] Y , ( θ ) T K (cid:34) u − v − ( v − u ) − u − v − ( v − u ) + 12 u − v − ( v − u ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cu − − K − ν ( v − u ) v − ( S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | ) + C (cid:113) E ,K,δ [ ψ ] u − − K − ν , for some ν >
0, with δ > T K ˇ φ (2)2 = 2( r + O ( r )) LT K ˇ φ (1)2 + KO ( r − ) T K − ˇ φ (1)2 , we can integrate the above equation oncemore, starting from γ α (cid:48) K , with α (cid:48) K > α K suitably large (depending on ν above), using (8.15) again toestimate: | T K ˇ φ (1)2 | ( u, v γ αK ( u ) , θ, ϕ ) ≤ C (cid:113) E ,K,δ (cid:48) [ ψ ] u − + α K − K +2 δ (cid:48) , for δ (cid:48) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ˇ φ (1)2 − I [ ψ ]( v − u ) T K ( u − v − )+ 163 (cid:114) π a I [ ψ ] Y , T K (cid:34) u − v − ( v − u ) − u − v − ( v − u ) + 14 u − v − ( v − u ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cu − − K − ν ( v − u ) v − ( S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | ) + C (cid:113) E ,K,δ [ ψ ] u − − K − ν in A γ α (cid:48) K with δ > T K ˇ φ (1)2 = 2( r + O ( r )) Lφ + KO ( r − ) T K − φ and for α (cid:48) K < α (cid:48)(cid:48) K < | T K φ | ( u, v γ α (cid:48)(cid:48) K ( u ) , θ, ϕ ) ≤ C (cid:113) E ,K,δ (cid:48) [ ψ ] u − + α (cid:48)(cid:48) K − K +2 δ (cid:48) , we integrate a final time starting from γ α (cid:48)(cid:48) K , with α (cid:48)(cid:48) K > α (cid:48)(cid:48) K suitably large and δ (cid:48) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K φ − I [ ψ ]( v − u ) T K ( u − v − ) + 323 (cid:114) π a I [ ψ ] Y , T K (cid:34) u − v − ( v − u ) − u − v − ( v − u ) + 14 u − v − ( v − u ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cu − − K − ν ( v − u ) v − ( S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | ) + C (cid:113) E ,K,δ [ ψ ] u − − K − ν in A γ α (cid:48)(cid:48) K . We obtain in particular (12.13).The estimate (12.12) then follows by restricting to a smaller region A γ α (cid:48)(cid:48)(cid:48) K , with α (cid:48)(cid:48)(cid:48) K > α (cid:48)(cid:48) K suitably largedepending on ν and dividing the equation above by r . The estimate (12.19) follows directly.93 R \ A γ α We extend now the late-time asymptotics derived in the region A γ α in Proposition 12.5 to the rest of thespacetime. Proposition 12.6.
Let K ∈ N and let α > be arbitrarily large. Then there exists a constant C = C ( M, a, K, δ ) > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ − (cid:114) π a I [ ψ ] Y , ( θ ) T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( τ, ρ , θ, ϕ ∗ ) ≤ C (1 + τ ) − − K − ν (cid:32) S ,δ,K [ ψ ] + D ,β,K [ ψ ] + (cid:88) m = − | I m [ ψ ] | (cid:33) + C (1 + τ ) − − K − ν (cid:34)(cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ] + (cid:113) E ,K +7 ,δ [ ψ ]+ (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) . (12.20) for all ( τ, ρ , θ, ϕ ∗ ) ∈ R \ A γ α and δ > suitably small.Proof. First of all, by (8.16), we have that: | XT K ψ | γ α ( τ ) | ( τ, θ, ϕ ∗ ) ≤ C (1 + τ ) − − α − K +2 δ (cid:32)(cid:118)(cid:117)(cid:117)(cid:116) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 E ,K +3 ,δ [ N j ψ ]+ (cid:118)(cid:117)(cid:117)(cid:116) E ,K +5 ,δ [ ψ ] + (cid:88) j =0 E ,K +3 ,δ [ N j ψ ] (cid:33) . By (8.24), we moreover have that | rX T K ψ | ( τ, ρ , θ, ϕ ∗ ) ≤ C (1 + τ ) − − K +2 δ (cid:34)(cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ] + (cid:113) E ,K +7 ,δ [ ψ ]+ (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) . We apply the fundamental theorem of calculus, integrating X T K ψ between ρ = ρ (cid:48) and ρ = ρ γ α ( τ ) , togetherwith the estimates above to conclude that there exists a ν > | T K Xψ | ( τ, ρ , θ, ϕ ∗ ) ≤ C (1 + τ ) − − K − ν (cid:34)(cid:113) E ,K +7 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ] + (cid:113) E ,K +7 ,δ [ ψ ]+ (cid:88) j =0 (cid:113) E ,K +4 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +2 ,δ [ ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +2 ,δ [ N j ψ ] (cid:35) . (12.21)We conclude the proof by applying the fundamental theorem of calculus again, integrating now XT K ψ between ρ = ρ (cid:48) and ρ = ρ γ α ( τ ) . The corresponding boundary term at ρ = ρ γ α ( τ ) can be estimated by(12.19) and we estimate the integral term with (12.21). Taking α suitably large, we arrive at (12.20). We apply here the time-integral construction from Section 9 to obtain the late-time asymptotics for ψ arising from initial data that is smooth and compactly supported.94 roposition 12.7. Consider initial data ( ψ | Σ , T ψ | Σ ) for (3.1) , with ( φ | Σ , T φ Σ ) ∈ ( C ∞ c (Σ)) . Let r >r + . Then there exists a ν > and a constant C = C ( M, a, Σ , r , ν ) > , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ ( u, v, θ, ϕ ∗ ) − I [ T − ψ ]( θ, ϕ ∗ )( v − u ) T K +1 ( u − v − )+ 643 (cid:114) π a I [ T − ψ ] Y , ( θ ) T K +1 (cid:34) u − v − ( v − u ) − u − v − ( v − u ) + 14 u − v − (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cu − − K − ν v − ( S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + (cid:88) m = − | I m [ T − ψ ] | )+ Cu − − K − ν v − (cid:34)(cid:113) E ,K +8 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ,K +5 ,δ [ N j T − ψ ] + (cid:113) E ,K +8 ,δ [ T − ψ ]+ (cid:88) j =0 (cid:113) E ,K +5 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +3 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +3 ,δ [ N j T − ψ ] (cid:35) in { r ≥ r } .In particular, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K φ | I + ( u, θ, ϕ ∗ ) + 25 I [ T − ψ ]( θ, ϕ ∗ ) T K ( u − ) − (cid:114) π a I [ T − ψ ] Y , ( θ ) T K ( u − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cu − − K − ν (cid:16) S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + (cid:88) m = − | I m [ T − ψ ] | + (cid:113) E ,K +1 ,δ [ T − ψ ] (cid:17) . In { r ≤ r } , we can express: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T K ψ + 163 (cid:114) π a I [ T − ψ ] Y , ( θ ) T K ((1 + τ ) − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( τ, ρ , θ, ϕ ∗ ) ≤ C (1 + τ ) − − K − ν ( S ,δ,K +1 [ T − ψ ] + D ,β,K +1 [ T − ψ ] + (cid:88) m = − | I m [ T − ψ ] | )+ C (1 + τ ) − − K − ν (cid:34)(cid:113) E ,K +8 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ,K +5 ,δ [ N j T − ψ ] + (cid:113) E ,K +8 ,δ [ T − ψ ]+ (cid:88) j =0 (cid:113) E ,K +5 ,δ [ N j ψ ] + (cid:113) E ≥ ,K +3 ,δ [ T − ψ ] + (cid:88) j =0 (cid:113) E ≥ ,K +3 ,δ [ N j T − ψ ] (cid:35) . Proof.
We apply Propositions 9.9 and 9.11 to conclude that T − φ has sufficiently high regularity to concludethat all the relevant energies for T − ψ are finite and I [ T − ψ ] is well-defined. Then the estimates followsimmediately by applying the estimates in Propositions 12.5 and 12.6 to T − ψ instead of ψ . A Weighted pointwise estimates
We derive in this section a lemma which is convenient for turning weighted energy estimates into pointwiseestimates.
Lemma A.1.
Let h, f : R → R be C functions, such that lim ρ →∞ h ( τ, ρ , θ, ϕ ∗ ) = 0 . Let k ∈ R > and > arbitrarily small. Let k ≥ , then there exists C = C ( M, a, R, k, δ ) > , such that (cid:90) S τ (cid:48) , ρ (cid:48) f dω ≤ C (cid:90) N τ (cid:48) r δ ( Lf ) + r − δ [( T f ) + (Φ f ) ] dωd ρ + (cid:90) Σ τ (cid:48) ∩{ R − M ≤ r ≤ R } f dωd ρ if ρ (cid:48) > R, (A.1) (cid:90) S τ (cid:48) , ρ (cid:48) ρ (cid:48)− − δ f dω ≤ C (cid:90) N τ (cid:48) r − δ ( Lf ) + r − − δ [( T f ) + (Φ f ) ] dωd ρ + (cid:90) Σ τ (cid:48) ∩{ R − M ≤ r ≤ R } f dωd ρ if ρ (cid:48) > R, (A.2) (cid:90) S τ (cid:48) , ρ (cid:48) ρ (cid:48) h dω ≤ C (cid:90) Σ τ (cid:48) J N [ h ] · n τ r dωd ρ , (A.3) (cid:90) S τ (cid:48) , ρ (cid:48) h dω ≤ C (cid:115)(cid:90) Σ τ (cid:48) r − J N [ h ] · n τ r dωd ρ · (cid:115)(cid:90) Σ τ (cid:48) J N [ h ] · n τ r dωd ρ , (A.4) (cid:90) S τ (cid:48) , ρ (cid:48) ρ (cid:48)− k h dω ≤ C (cid:90) Σ τ (cid:48) (cid:0) r − k − J N [ h ] · n τ + r − k − h (cid:1) r dωd ρ . (A.5) Proof.
Let R > r + + M and let χ : [ r + , ∞ ) → R be a smooth cut-off function, such that χ ( r ) = 1 for all r ≥ R and χ = 0 for r ≤ R − M .By applying the fundamental theorem of calculus, integrating from ρ = R − M , together with Cauchy–Schwarz and (2.20), we obtain for ρ (cid:48) ≥ R : f ( ρ (cid:48) , θ, ϕ ∗ , τ (cid:48) ) = (cid:90) ρ (cid:48) R − M χf X ( χf ) d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) ≤ (cid:115)(cid:90) ρ (cid:48) R − M r − − δ ( χf ) d ρ · (cid:115)(cid:90) ρ (cid:48) R − M r δ ( X ( χf )) d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) ≤ C (cid:115)(cid:90) ρ (cid:48) R − M r δ ( Xf ) d ρ · (cid:115)(cid:90) ρ (cid:48) R − M r − δ ( Xf ) d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) ≤ C (cid:90) ρ (cid:48) R − M r δ ( Lf ) + r − δ [( T f ) + (Φ f ) ] d ρ + C (cid:90) R R − M f d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) . The estimate (A.1) then follows by integrating over S , choosing R appropriately.We similarly obtain ρ (cid:48)− − δ f ( ρ (cid:48) , θ, ϕ ∗ , τ (cid:48) ) = (cid:90) ρ (cid:48) R − M − (1 + δ ) r − − δ ( χf ) + 2 r − − δ χf X ( χf ) d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) ≤ (cid:90) ρ (cid:48) R − M − (cid:18)
12 + δ (cid:19) r − ( χf ) + 2 r − δ ( X ( χf )) d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) ≤ C (cid:90) ρ (cid:48) R − M r − δ ( Lf ) + r − − δ [( T f ) + (Φ f ) ] d ρ + C (cid:90) R R − M f d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) . We apply the fundamental theorem of calculus again to obtain: h ( ρ (cid:48) , θ, ϕ ∗ , τ (cid:48) ) = (cid:18) − (cid:90) ∞ ρ (cid:48) X ( h ) d ρ (cid:19) (cid:12)(cid:12)(cid:12) τ = τ (cid:48) ≤ ρ (cid:48)− (cid:90) ∞ ρ (cid:48) r ( Xh ) d ρ (cid:12)(cid:12)(cid:12) τ = τ (cid:48) . The estimate (A.3) then follows by integrating over S and applying Cauchy–Schwarz again on S .96y applying the fundamental theorem of calculus, integrating from ρ = ∞ , in combination with Cauchy–Schwarz and (2.20), we obtain h ( ρ (cid:48) , θ, ϕ ∗ , τ (cid:48) ) = 0 − (cid:90) ∞ ρ (cid:48) hXh d ρ (cid:48) ≤ C (cid:115)(cid:90) ∞ ρ (cid:48) h d ρ (cid:115)(cid:90) ∞ ρ (cid:48) ( Xh ) d ρ ≤ C (cid:115)(cid:90) ∞ ρ (cid:48) r ( Xh ) d ρ (cid:115)(cid:90) ρ (cid:48) R − M ( Xh ) d ρ . We obtain (A.4) by integrating over S .Finally, we repeat the above application of the fundamental theorem of calculus, integrating from ρ = ∞ to obtain r − k h ≥ ( ρ , θ, ϕ ∗ , τ (cid:48) ) = 0 − (cid:90) ∞ ρ (cid:48) X ( r − k h ≥ ) d ρ ≤ C (cid:90) ∞ ρ (cid:48) r − k − h ≥ + r − k +1 ( Xh ≥ ) d ρ We then integrate over S and apply (2.15) to obtain (A.4). B A basic interpolation inequality
The lemma below is useful for interpolating between r - and τ -decay. Lemma B.1.
Let f : R → R be a continuous function, such that for q , q ∈ R with ≤ q ≤ q and r + < r < r ≤ ∞ : (cid:90) r r (cid:90) S r q f ( τ, ρ , θ, ϕ ∗ ) dωd ρ ≤ D (1 + τ ) − p , (B.1) (cid:90) r r (cid:90) S r q f ( τ, ρ , θ, ϕ ∗ ) dωd ρ ≤ D (1 + τ ) − p +( q − q ) . (B.2) Then for all q ≤ q ≤ q : (cid:90) r r (cid:90) S r q f ( τ, ρ , θ, ϕ ∗ ) dωd ρ ≤ ( D + D )(1 + τ ) − p +( q − q ) . (B.3) Proof.
We split [ r , r ] = J ≤ + J > , with J ≤ = [ r , r ] × S ∩{ r ≤ (1+ τ ) } and J > = [ r , r ] × S ∩{ r > (1+ τ ) } .Then by applying (B.1) and (B.2), we obtain (cid:90) r r (cid:90) S r q f dωd ρ = (cid:90) J ≤ r q f dωd ρ + (cid:90) J > r q f dωd ρ ≤ (1 + τ ) q − q (cid:90) J ≤ r q f dωd ρ + (1 + τ ) − ( q − q ) (cid:90) J > r q f dωd ρ ≤ D (1 + τ ) − p +( q − q ) + D (1 + τ ) − ( q − q ) − p +( q − q ) , from which (B.3) immediately follows. References [AAG18a] Y. Angelopoulos, S. Aretakis, and D. Gajic. Horizon hair of extremal black holes and measure-ments at null infinity.
Physical Review Letters , 121(13):131102, 2018.97AAG18b] Y. Angelopoulos, S. Aretakis, and D. Gajic. Late-time asymptotics for the wave equation onspherically symmetric, stationary backgrounds.
Advances in Mathematics , 323:529–621, 2018.[AAG18c] Y. Angelopoulos, S. Aretakis, and D. Gajic. A vector field approach to almost-sharp decay forthe wave equation on spherically symmetric, stationary spacetimes.
Annals of PDE , 4(2):15,2018.[AAG19] Y. Angelopoulos, S. Aretakis, and D. Gajic. Logarithmic corrections in the asymptotic expansionfor the radiation field along null infinity.
Journal of Hyperbolic Differential Equations , 16(01):1–34, 2019.[AAG20] Y. Angelopoulos, S. Aretakis, and D. Gajic. Late-time asymptotics for the wave equation onextremal Reissner–Nordstr¨om backgrounds.
Advances in Mathematics , 375, 2020.[AAG21] Y. Angelopoulos, S. Aretakis, and D. Gajic. Price’s law and precise asymptotics for subextremalReissner–Nordstr¨om black holes. preprint , 2021.[ABBM19] L. Andersson, T. B¨ackdahl, P. Blue, and S. Ma. Stability for linearized gravity on the Kerrspacetime. arXiv:1903.03859 , 2019.[Are11] S. Aretakis. Stability and instability of extreme Reissner–Nordstr¨om black hole spacetimes forlinear scalar perturbations I.
Commun. Math. Phys. , 307:17–63, 2011.[Are12] S. Aretakis. Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds.
J. Funct. Analysis , 263:2770–2831, 2012.[AS72] M. Abramowitz and I. A. Stegun.
Handbook of mathematical functions with formulas, graphs,and mathematical tables . US Government printing office, 1972.[BF13] P. Bizon and H. Friedrich. A remark about the wave equations on the extreme Reissner–Nordstr¨om black hole exterior.
Class. Quantum Grav. , 30:065001, 2013.[BK14] L. M. Burko and G. Khanna. Mode coupling mechanism for late-time Kerr tails.
Physical ReviewD , 89(4):044037, 2014.[BKS19] L. M. Burko, G. Khanna, and S. Sabharwal. Transient scalar hair for nearly extreme black holes.
Physical Review Research , 1(3):033106, 2019.[BKS21] L. M. Burko, G. Khanna, and S. Sabharwal. Scalar and gravitational hair for extreme Kerr blackholes.
Physical Review D , 103(2):L021502, 2021.[BO99a] L. Barack and A. Ori. Late-time decay of gravitational and electromagnetic perturbations alongthe event horizon.
Physical Review D , 60(12):124005, 1999.[BO99b] L. Barack and A. Ori. Late-time decay of scalar perturbations outside rotating black holes.
Phys.Rev. Lett. , 82(4388-4391), 1999.[BVW18] D. Baskin, A. Vasy, and J. Wunsch. Asymptotics of scalar waves on long-range asymptoticallyminkowski spaces.
Advances in Mathematics , 328:160–216, 2018.[CGZ16] M. Casals, S. E. Gralla, and P. Zimmerman. Horizon instability of extremal Kerr black holes:Nonaxisymmetric modes and enhanced growth rate.
Phys. Rev. D , 94:064003, 2016.[dC20] R. Teixeira da Costa. Mode stability for the Teukolsky equation on extremal and subextremalKerr spacetimes.
Communications in Mathematical Physics , 378(1):705–781, 2020.[DHR19a] M. Dafermos, G. Holzegel, and I. Rodnianski. Boundedness and Decay for the Teukolsky Equa-tion on Kerr Spacetimes I: The Case | a | (cid:28) M . Annals of PDE , 5(2):1–118, 2019.[DHR19b] M. Dafermos, G. Holzegel, and I. Rodnianski. The linear stability of the Schwarzschild solutionto gravitational perturbations.
Acta Math. , 222:1–214, 2019.98DL17] M. Dafermos and J. Luk. The interior of dynamical vacuum black holes I: The C -stability ofthe Kerr Cauchy horizon. arXiv:1710.01722 , 2017.[DR05] M. Dafermos and I. Rodnianski. A proof of Price’s law for the collapse of a self-gravitating scalarfield. Invent. Math. , 162:381–457, 2005.[DR09] M. Dafermos and I. Rodnianski. The redshift effect and radiation decay on black hole spacetimes.
Comm. Pure Appl. Math. , 62:859–919, arXiv:0512.119, 2009.[DR10] M. Dafermos and I. Rodnianski. A new physical-space approach to decay for the wave equa-tion with applications to black hole spacetimes.
XVIth International Congress on MathematicalPhysics , pages 421–432, 2010.[DR13] M. Dafermos and I. Rodnianski. Lectures on black holes and linear waves. in Evolution equa-tions, Clay Mathematics Proceedings, Vol. 17, Amer. Math. Soc., Providence, RI, , pages 97–205,arXiv:0811.0354, 2013.[DRSR16] M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman. Decay for solutions of the waveequation on Kerr exterior spacetimes III: The full subextremal case | a | < m . Annals of Math ,183:787–913, 2016.[DRSR18] M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman. A scattering theory for the waveequation on Kerr black hole exteriors.
Ann. Sci. ´ec. Norm. Sup´er , 51(2):371–486, 2018.[DSS11] R. Donninger, W. Schlag, and A. Soffer. A proof of Price’s law on Schwarzschild black holemanifolds for all angular momenta.
Adv. Math. , 226:484–540, 2011.[DSS12] R. Donninger, W. Schlag, and A. Soffer. On pointwise decay of linear waves on a Schwarzschildblack hole background.
Comm. Math. Phys. , 309:51–86, 2012.[HHV21] D. H¨afner, P. Hintz, and A. Vasy. Linear stability of slowly rotating Kerr black holes.
Inventionesmathematicae , 223:1227–1406, 2021.[Hin20] P. Hintz. A sharp version of Price’s law for wave decay on asymptotically flat spacetimes. arXiv:2004.01664 , 2020.[Hod99] S. Hod. Mode-coupling in rotating gravitational collapse of a scalar field.
Physical Review D ,61(2):024033, 1999.[Joh19] T. W. Johnson. The linear stability of the Schwarzschild solution to gravitational perturbationsin the generalised wave gauge.
Annals of PDE , 5(2):1–92, 2019.[Keh21] L. M. A. Kehrberger. The Case Against Smooth Null Infinity II: A Logarithmically ModifiedPrice’s Law. to appear , 2021.[KLPA97] W. Krivan, P. Laguna, P. Papadopoulos, and N. Andersson. Dynamics of perturbations ofrotating black holes.
Physical Review D , 56(6):3395, 1997.[KS20] S. Klainerman and J. Szeftel.
Global Nonlinear Stability of Schwarzschild Spacetime under Po-larized Perturbations . Annals of Mathematical Studies. Princeton University Press, 2020.[LMRT13] J. Lucietti, K. Murata, H. S. Reall, and N. Tanahashi. On the horizon instability of an extremeReissner–Nordstr¨om black hole.
JHEP , 1303:035, 2013.[LS16] J. Luk and J. Sbierski. Instability results for the wave equation in the interior of Kerr blackholes.
Journal of Functional Analysis , 271(7):1948 – 1995, 2016.[Ma20] S. Ma. Almost Price’s law in Schwarzschild and decay estimates in Kerr for Maxwell field. arXiv:2005.12492 , 2020. 99Mos16] G. Moschidis. The r p -weighted energy method of Dafermos and Rodnianski in general asymp-totically flat spacetimes and applications. Annals of PDE , 2:6, 2016.[MTT12] J. Metcalfe, D. Tataru, and M. Tohaneanu. Price’s law on nonstationary spacetimes.
Advancesin Mathematics , 230:995–1028, 2012.[MZ20] S. Ma and L. Zhang. Sharp decay estimates for massless Dirac fields on a Schwarzschild back-ground. arXiv:2008.11429 , 2020.[NP68] E. T. Newman and R. Penrose. New conservation laws for zero rest mass fields in asympoticallyflat space-time.
Proc. R. Soc. A , 305:175204, 1968.[Pen82] R. Penrose.
Some unsolved problems in classical general relativity , pages 631–668. PrincetonUniversity Press, 1982.[Pri72] R. Price. Non-spherical perturbations of relativistic gravitational collapse. I. Scalar and gravita-tional perturbations.
Phys. Rev. D , 3:2419–2438, 1972.[Sch13] V. Schlue. Decay of linear waves on higher-dimensional Schwarzschild black holes.
Analysis andPDE , 6(3):515–600, 2013.[Sog08] C. D. Sogge.
Lectures on non-linear wave equations . International Press Boston, 2008.[SRdC20] Y. Shlapentokh-Rothman and R. Teixeira da Costa. Boundedness and decay for the Teukol-sky equation on Kerr in the full subextremal range | a | < M : frequency space analysis. arXiv:2007.07211 , 2020.[Tat13] D. Tataru. Local decay of waves on asymptotically flat stationary space-times. American Journalof Mathematics , 135:361–401, 2013.[War15] C. M. Warnick. On Quasinormal Modes of Asymptotically Anti-de Sitter Black Holes.
Commu-nications in Mathematical Physics , 333(2):959–1035, 2015.[ZKB14] A. Zengino˘glu, G. Khanna, and L. M. Burko. Intermediate behavior of Kerr tails.