Price's law and precise late-time asymptotics for subextremal Reissner-Nordström black holes
PPrice’s law and precise late-time asymptotics for subextremalReissner–Nordstr¨om black holes
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 CB30WB, UK Department of Mathematics, Radboud University, 6525 AJ Nijmegen, The Netherlands
February 23, 2021
Abstract
In this paper, we prove precise late-time asymptotics for solutions to the waveequation supported on angular frequencies greater or equal to (cid:96) on the domain of outercommunications of subextremal Reissner–Nordstr¨om spacetimes up to and includingthe event horizon. Our asymptotics yield, in particular, sharp upper and lower decayrates which are consistent with Price’s law on such backgrounds. We present a theoryfor inverting the time operator and derive an explicit representation of the leading-order asymptotic coefficient in terms of the Newman–Penrose charges at null infinityassociated with the time integrals. Our method is based on purely physical spacetechniques. For each angular frequency (cid:96) we establish a sharp hierarchy of r -weightedradially commuted estimates with length 2 (cid:96) + 5. We complement this hierarchy witha novel hierarchy of weighted elliptic estimates of length (cid:96) + 1. Contents ∗ [email protected] † [email protected] ‡ [email protected], [email protected] a r X i v : . [ g r- q c ] F e b Higher-order radiation fields and Newman–Penrose charges 165 Hierarchies of r p -weighted estimates 19 r p -weighted estimates for Φ ( (cid:96) ) . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 r p -weighted estimates for T m Φ ( (cid:96) ) . . . . . . . . . . . . . . . . . . . . . . . . 22 ( (cid:96) ) . . . . . . . . . . . . . . . . 449.2 Almost-sharp decay for P ≥ (cid:96) ψ and its derivatives . . . . . . . . . . . . . . . 45
10 Precise late-time asymptotics when I (cid:96) (cid:54) = 0 P (cid:96) ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11 Construction of time integrals 5712 Precise late-time asymptotics when I (cid:96) = 0
59A Basic inequalities 61
Price [Pri72] predicted in 1972 that if ψ solves the wave equation (cid:3) g ψ = 0 , (1.1)on a Schwarzschild spacetime and is supported on a fixed angular frequency (cid:96) (in this casewe denote the linear wave by ψ (cid:96) ) then ψ (cid:96) ∼ t (cid:96) +3 (1.2)asymptotically in time ( t → ∞ ) along constant { r = r } hypersurfaces. In this paperwe provide the first rigorous derivation and proof of the precise late-time asymptotics forsolutions to the wave equation (1.1) on the domain of outer communications of subextremalReissner–Nordstr¨om spacetimes for all angular frequencies (cid:96) ≥ (cid:96) + 3 decay rate isproved for static initial data supported on the (cid:96) frequency (it is worth pointing out thatfor such data 2 (cid:96) + 4 is the expected optimal decay rate). Hintz [Hin20] recently derived the2 (cid:96) + 3 upper bound for general initial data supported on the (cid:96) frequency on Schwarzschildbackgrounds. Ma [Ma20] also derived almost sharp decay rates for (cid:96) = 1 angular modes ofMaxwell fields.The first rigorous work that derived precise late-time asymptotics is [AAG18c, AAG18a,AAG19]. This work obtained asymptotics for general solutions ψ (without any assumptionson the angular frequency) to the wave equation on sub-extremal Reissner–Nordstr¨om space-times. Precise asymptotics were obtained in [Hin20] for some general asymptotically flatspacetimes that include as a special case the subextremal Kerr family of black hole space-times, and in [MZ20] for the Dirac equation on Schwarzschild backgrounds. On the otherhand, [AAG20b] derived precise asymptotics for general solution on extremal Reissner–Nordstr¨om spacetimes. The terms appearing in these asymptotics decay much slower thanin the sub-extremal case in view of the horizon instability [Are11a, Are11b, Are15, Are13]and the presence of conserved charges at the event horizon [Are17]. An application ofthe precise asymptotics in the extremal case was presented in [AAG18b] where it wasshown that the horizon charges can be computed using only the knowledge of the radiationfield at null infinity. On the other hand, proving late-time asymptotics for higher angularfrequencies on extremal Reissner–Nordstr¨om remains an open problem. In this section, we provide a brief summary of our main results. We consider appropriatehypersurfaces Σ τ and S τ in subextremal Reissner–Nordstr¨om that cross the event horizonand terminate at future null infinity as depicted in the figure belowFigure 1: The hypersurfaces Σ τ .We derive the following precise late-time asymptotics for each angular frequency ψ (cid:96) .In fact our asymptotics are summable, in the sense that they hold for solutions ψ ≥ (cid:96) to thewave equation which are supported on angular frequencies greater or equal to (cid:96) . ψ ≥ (cid:96) ( τ, r, θ, ϕ ) ∼ A (1) (cid:96) · r (cid:96) · I (1) (cid:96) [ ψ ]( θ, ϕ ) · τ (cid:96) +3 (1.3)along the hypersurfaces r = r for any r ≥ r + . Here A (1) (cid:96) are numerical constants that3epend on (cid:96) and I (1) (cid:96) [ ψ ] denotes what we call the (cid:96) th time-inverted Newman–Penrose chargeof ψ . This charge is equal to the Newman–Penrose charge (see Section 4) of the time integral(see Section 11) of ψ (cid:96) . In a region close to infinity, we prove that ψ ≥ (cid:96) ( u, v, θ, ϕ ) ∼ ¯ A (1) (cid:96) · r (cid:96) · I (1) (cid:96) [ ψ ]( θ, ϕ ) · u (cid:96) +2 · v (cid:96) +1 (1.4)for a numerical constant ¯ A (1) (cid:96) depending on (cid:96) . In particular the asymptotics for the radia-tion field rψ along null infinity are as follows: rψ ≥ (cid:96) ( τ, ∞ , θ, ϕ ) ∼ (cid:96) I (1) (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · · · · · ( (cid:96) + 2) · τ (cid:96) +2 . (1.5)We also derive asymptotics for higher-order T and ∂ ρ derivatives of ψ , where T is thestationary Killing field and ∂ ρ is the radial vector field tangential to Σ τ . See Section 2 forthe rigorous statements of the main theorems. In this section we provide a summary of the main ideas of the proof of the asymptoticspresented in the previous section. To make our methods clearer, we will present a schematicversion of the main estimates in this section, omitting terms that do not play an importantrole for the structure of the arguments. We also omit difficulties such as capturing theredshift and trapping effects which have been extensively addressed in the literature. Wenote that all the integrals are taken with respect to the volume form corresponding to theinduced metric of the integrating region.
Recall that the energy-momentum tensor corresponding to a linear wave is given by: T µν [ ψ ] . = ∂ µ ψ∂ ν ψ − g µν ∂ α ψ∂ α ψ. The energy current J V [ ψ ] for a vector field V is given by J Vµ [ ψ ] . = T µν [ ψ ] · V ν . We will first show how to obtain almost sharp decay for the standard energy fluxthrough Σ τ : (cid:90) Σ τ J N [ ψ ≥ (cid:96) ] · n τ dµ Σ τ . (1.6)Here N is a globally timelike vector field such that N = T away from the event horizon, dµ Σ τ is the volume form corresponding to the hypersurface Σ τ , and n τ the normal to Σ τ (note that we will also use n to denote the normal to other hypersurfaces in analogoussituations without specifying it). First of all, the Dafermos–Rodnianski hierachy [DR10]schematically reads as: (cid:90) τ (cid:90) Σ τ r p − r · J N [ φ ≥ (cid:96) ] · n τ dµ Σ τ dτ (cid:46) (cid:90) Σ τ r p r · J N [ φ ≥ (cid:96) ] · n τ dµ Σ τ (1.7)4here φ ≥ (cid:96) := rψ ≥ (cid:96) and 0 < p ≤ , where A (cid:46) B (for function A and B ) denotes A ≤ CB where C is a constant, and where r is the radial variable. This hierarchy (combined withan integrated local energy decay estimate), applied with p = 1 and p = 2, yields τ − decayfor the energy flux (1.6). In order to obtain faster decay for the energy flux we need toobtain higher-order versions of (1.7). For this reason we introduce the following weightedderivatives (cid:101) Φ ( k ) = ( − k · (cid:0) r ∂ r (cid:1) k φ ≥ (cid:96) . (1.8)with k ≥
0. Here ∂ r denotes the outgoing null vector field such that ∂ r r = 1. Assuming that ψ is supported on angular frequencies ≥ (cid:96) and its initial data are decaying sufficiently fast(for example, are compactly supported) then we obtain the following schematic hierarchies: (cid:90) τ (cid:90) Σ τ r p − r · J N (cid:104)(cid:101) Φ ( k ) (cid:105) · n τ dµ Σ τ dτ (cid:46) (cid:90) Σ τ r p r · J N (cid:104)(cid:101) Φ ( k ) (cid:105) · n τ dµ Σ τ (1.9)for 0 ≤ k ≤ (cid:96), < p ≤ . The hierarchy (1.9) was previously presented in [AAG18d] for k = 1 and (cid:96) ≥
1. Here wepresent the full range of k and (cid:96) and show that the maximum number of commutationswith r ∂ r is exactly (cid:96) and which allows us to derive precisely (cid:96) + 1 hierarchies.In order to further extend the top order hierarchy (i.e. k = (cid:96) ), we define the modified weighted derivatives Φ ( k ) in an iterative way as follows:Φ ( k ) = (cid:101) Φ ( k ) + k (cid:88) m =1 α k,m Φ ( k − m ) , (1.10)where α k,m denote numerical constants that depend on k and m . These quantities proveuseful for extending the r p -weighted hierarchies to their almost sharp range, and are im-portant in the definition of the so-called Newman–Penrose charges (that will be definedlater). For the top order hierarchy with k = (cid:96) we show the following improved range for p :0 ≤ p < . The above hierarchies can be connected to each other via the following Hardy inequality: (cid:90) τ (cid:90) Σ τ r p − r · J N (cid:104)(cid:101) Φ ( k ) (cid:105) · n τ dµ Σ τ dτ (cid:46) (cid:90) τ (cid:90) Σ τ r p − r · J N (cid:104)(cid:101) Φ ( k +1) (cid:105) · n τ dµ Σ τ (1.11)that holds for all p (cid:54) = − ψ ≥ (cid:96) with T m ψ ≥ (cid:96) then we get additional estimates, which can be thought of asextending the range of p to 0 < p < m in (1.9) for 0 ≤ k ≤ (cid:96) , and 0 < p < m We should note that the r p -hierarchy as stated here is not quite right in the case of p = 2, as in thiscase there is no term involving angular derivatives. For the sake of the schematic exposition we will ignorethis, one can refer to Section 5 for the precise statements.
5n (1.11) (for both cases this is not quite correct, the hierarchies can be extended only aftercommuting m times with ∂ r , and then exchanging the ∂ r derivatives with T derivatives).For all hierarchies with 0 ≤ k ≤ (cid:96) − p = 1 , k = (cid:96) we have 5 estimates. Hence, our r p -hierarchy yields the followingdecay rates for the energy flux and the conformal flux: (cid:90) Σ τ J N [ ψ ≥ (cid:96) ] · n τ dµ Σ τ (cid:46) τ (cid:96) +5 − (cid:15) , (cid:90) Σ τ r · J N [ ψ ≥ (cid:96) ] · n τ dµ Σ τ (cid:46) τ (cid:96) +3 − (cid:15) , for any (cid:15) > We can derive almost sharp decay for the radiation field rψ ≥ (cid:96) | I + using the fundamentaltheorem of calculus and a Hardy inequality: rψ ≥ (cid:96) (cid:46) (cid:115)(cid:90) Σ τ J N [ ψ ≥ (cid:96) ] · n τ dµ Σ τ · (cid:90) Σ τ r · J N [ ψ ≥ (cid:96) ] · n τ dµ Σ τ (cid:46) (cid:114) τ (cid:96) +5 · τ (cid:96) +3 = 1 τ (cid:96) +2 − (cid:15) . This is the optimal rate for rψ ≥ (cid:96) | I + , however it is far from optimal for ψ ≥ (cid:96) | r = r . For ψ ≥ (cid:96) itself we make use of − ∂ ρ ( r − (cid:96) ψ ≥ (cid:96) ) = 2 (cid:96)r − (cid:96) − ψ ≥ (cid:96) + 2 r − (cid:96) ψ ≥ (cid:96) ∂ ρ ψ ≥ (cid:96) ≤ (2 (cid:96) + 1) r − (cid:96) − ψ ≥ (cid:96) + r − (cid:96) +1 ( ∂ ρ ψ ≥ (cid:96) ) ∼ r − (cid:96) − r J N [ ψ ≥ (cid:96) ] · n τ to conclude that r − (cid:96) + (cid:15) ψ ≥ (cid:96) (cid:46) (cid:90) Σ τ r − (cid:96) − (cid:15) J N [ ψ ≥ (cid:96) ] · n τ dµ Σ τ . (1.12)Hence, in order to get the (almost) sharp decay for rψ ≥ (cid:96) | I + it suffices to obtain the sharpdecay for r − (cid:96) + (cid:15) ψ ≥ (cid:96) and hence of the weighted energy flux with decreasing weights r p with p <
0. For this we present a new hierarchy of elliptic estimates.
Let us define the following weighted derivative˜ ∂ ρ = r∂ ρ , where ∂ ρ is the radial tangential vector field on the asymptotically hyperboloidal hyper-surfaces S τ .The hierarchy of elliptic estimates that we derive schematically take the following form: (cid:90) S τ r p (cid:104) D · J N [ ˜ ∂ m +1 ρ ψ ≥ (cid:96) ] + J N [ ˜ ∂ mρ ψ ≥ (cid:96) ] (cid:105) · n τ dµ S τ (cid:46) (cid:90) S τ r p − m (cid:88) s =0 J N [ T ˜ ∂ sρ ψ ≥ (cid:96) ] · n τ dµ S τ (1.13)with − < p < (cid:96) + 1 , ≤ m ≤ (cid:96), D given by (3.1), and where S τ are asymptotically hyperboloidalhypersurfaces (see Figure 1). Dropping the degenerate term, the uncommuted version( m = 0) reads: (cid:90) S τ r p J N [ ψ ≥ (cid:96) ] · n τ dµ S τ (cid:46) (cid:90) S τ r p − J N [ T ψ ≥ (cid:96) ] · n τ dµ S τ (1.14)with − < p < (cid:96) + 1 . The elliptic estimates show that we can add a T derivative at the expense of increasingthe power of the weight r by 2. If ψ is supported on angular frequences ≥ (cid:96) then we cancommute (cid:96) times with ˜ ∂ ρ and in this way we can then replace ψ with T ψ in the weightedenergy fluxes at the expense of an additional r factor.The proof relies on the fact that we can integrate by the equation ∂ ρ (( Dr ) ∂ ρ ψ ≥ (cid:96) ) + ( Dr ) (cid:48) ∂ ρ ψ ≥ (cid:96) + 2 ∂ ρ ψ ≥ (cid:96) = ∂ ρ ¯ F T , where F T involves T derivatives of ψ ≥ (cid:96) . We note that this is possible for the proof of ourestimates due to the fact that the right hand side of the last equation defines an ellipticoperator. This should be compared and contrasted to the Kerr case where something likethat is not possible, see section 7 of [AAG21]. r weights We will use the above elliptic estimates to derive decay for the weighted flux (cid:90) S τ r (cid:96) +1 − (cid:15) J N [ ψ ≥ (cid:96) ] · n τ dµ S τ , whereWe apply (1.14) successively for p = 2 (cid:96) + 1 − (cid:15), (cid:96) − − (cid:15), ..., − (cid:15), − (cid:15) and we get (cid:90) S τ r (cid:96) +1 − (cid:15) J N [ ψ ≥ (cid:96) ] · n τ dµ S τ (cid:46) (cid:90) S τ r (cid:96) − − (cid:15) J N [ T ψ ≥ (cid:96) ] · n τ dµ S τ (cid:46) (cid:90) S τ r (cid:96) − − (cid:15) J N [ T ψ ≥ (cid:96) ] · n τ dµ S τ · · · (cid:46) (cid:90) S τ r − (cid:15) J N [ T (cid:96) − ψ ≥ (cid:96) ] · n τ dµ S τ (cid:46) (cid:90) S τ r − (cid:15) J N [ T (cid:96) ψ ≥ (cid:96) ] · n τ dµ S τ (cid:46) (cid:90) S τ r (cid:15) J N [ T (cid:96) +1 ψ ≥ (cid:96) ] · n τ dµ S τ The top estimate follows by taking p = 2 (cid:96) + 1 − (cid:15) and the bottom estimate by taking p = 1 − (cid:15) which is the admissible range for p . Since (cid:90) S r (cid:15) J N [ ψ ≥ (cid:96) ] · n τ dµ S τ decays like1 τ (cid:96) +4 − (cid:15) we have that (cid:90) S r (cid:15) J N [ T (cid:96) +1 ψ ≥ (cid:96) ] · n τ dµ S τ decays like 1 τ (cid:96) +2+2( (cid:96) +1) − (cid:15) = 1 τ (cid:96) +3) − (cid:15) which via (1.12) yields the required result for ψ ≥ (cid:96) .7 .3.5 Decay of higher order radial derivatives Using the commuted elliptic estimates (1.13) and the above idea we also obtain:1 r (cid:96) − k − (cid:15) ∂ kρ ψ ≥ (cid:96) (cid:46) τ (cid:96) +3 − (cid:15) (1.15)for all 0 ≤ k ≤ (cid:96) . Moreover, we have (cid:101) Φ ( k ) (cid:46) τ ( (cid:96) − k )+ (cid:15) (1.16)for all 0 ≤ k ≤ (cid:96) where (cid:101) Φ ( k ) is as defined in (1.8). Commuting with the time derivative T m increases the decay rate by 2 m with m ≥ In the remaining sections of the introduction, we will summarize our method that allows usto go beyond almost-sharp time-decay estimates and obtain the precise late-time asymp-totics for angular modes ψ (cid:96) . For this reason we will assume that ψ = ψ (cid:96) in the remainderof this introduction.We first need to introduce the Newman–Penrose charges and constants. The Newman–Penrose constants have been presented previously in the context of the Maxwell and theEinstein equations (see [NP65, NP68] for the case of Maxwell and Einstein equations onMinkowski, and also the recent [Ma20] for the Maxwell equations on Schwarzschild), butin this paper we derive them rigorously in the context of the wave equation for all angularfrequencies on subextremal Reissner–Nordstr¨om.Recall that the modified derivative Φ ( (cid:96) +1) was defined in (1.10). Then the limitingfunction on null infinity I (cid:96) [ ψ ]( θ, ϕ ) . = lim r →∞ Φ ( (cid:96) +1) ( u, r, θ, ϕ ) , (1.17)is independent of u . By further decomposing the function I (cid:96) [ ψ ]( θ, ϕ ) relative to sphericalharmonics we obtain I (cid:96) [ ψ ]( θ, ϕ ) . = (cid:96) (cid:88) m = − (cid:96) I m,(cid:96) [ ψ ] Y m,(cid:96) ( θ, ϕ ) , (1.18)and define the constants I m,(cid:96) to be the Newman–Penrose constants of ψ . I (cid:96) (cid:54) = 0We first derive asymptotics in the case where the initial data satisfies I (cid:96) (cid:54) = 0; such dataare certainly not compactly supported, nonetheless our methods still provide almost sharpdecay in this case.Let ( u, v ) denote the standard null coordinates covering the black hole exterior. Work-ing in the near-infinity region B γ α that lies to the right of the curve γ α = { v − u = v α } forsome α ∈ (0 ,
1) (where α = α ( (cid:96) ) is a constant depending on the angular frequency (cid:96) ) asdepicted below. 8igure 2: The curve γ α and the region B γ α .By successively integrating in v we obtain asymptotics for radial derivatives of ψ in B γ α in the following order: v ∂ ρ Φ ( (cid:96) ) → ∂ ρ Φ ( (cid:96) ) → Φ ( (cid:96) ) → (cid:101) Φ ( (cid:96) − → (cid:101) Φ ( (cid:96) − → · · · → (cid:101) Φ (1) → φ This also yields precise asymptotics for the radial derivatives ∂ kr ψr (cid:96) − k for 0 ≤ k ≤ (cid:96) in thesame region: ∂ kρ ψ ∼ (cid:101) A (cid:96),k r (cid:96) − k · τ (cid:96) +3 (1.19)for 0 ≤ k ≤ (cid:96) and numerical constants (cid:101) A (cid:96),k . ∂ (cid:96) +1 ρ ψ and global asymptotics One way to propagate the asymptotics for a quantity, say Q , from B γ α to the rest of theblack hole exterior region, all the way up to the event horizon, is to derive faster decay forthe radial derivative ∂ ρ Q and the use the fundamental theorem of calculus for Q . However,as is evident from (1.19), the (sharp) decay rate for all the radial derivatives up to the (cid:96) th -order is the same. For this reason we turn to the radial derivative ∂ (cid:96) +1 ρ ψ .Using the wave equation we have for all k ≥ ∂ k +1 ρ ψ + 1 r k · ∂ kρ ψ = 1 r [ − ∆ S − k ( k + 1)] ∂ k − ρ ψ + ... (1.20)The omitted terms are ∂ ρ derivatives of T ψ , they decay sufficiently fast and can be thoughtof as lower-order terms.If we consider ψ = ψ (cid:96) we have that coefficient of the term ∂ k − ρ ψ does not vanish for k ≤ l , and so no result can be obtained for ∂ k +1 ρ ψ since it is non-trivially coupled with thelower order derivatives. On the other hand when k = (cid:96) + 1, the coefficient of ∂ k − ρ ψ = ∂ (cid:96)ρ ψ vanishes , and this yields ∂ (cid:96) +2 ρ ψ + 1 r (cid:96) + 1) · ∂ (cid:96) +1 ρ ψ = ... (1.21)Multiplying the above equation with r (cid:96) +1) yields r (cid:96) +1) ∂ (cid:96) +2 ρ ψ + 2( (cid:96) + 1) · r (cid:96) +1 ∂ (cid:96) +1 ρ ψ = ∂ ρ (cid:16) r (cid:96) +1) · ∂ (cid:96) +1 ρ ψ (cid:17) = ... (1.22)Integrating in r the above yields decay for ∂ (cid:96) +1 ρ ψ faster than τ − (cid:96) − which can then beused to propagate the precise asymptotics of ∂ (cid:96)ρ ψ everywhere in the exterior region. We9an then inductively propagate the asymptotics of ∂ kρ ψ everywhere from B α in the blackhole exterior region for all k = (cid:96) − , (cid:96) − , ..., , , Dr ) (cid:48)(cid:48) = 2. If we assume that for an angular mode ψ = ψ (cid:96) the Newman–Penrose charge I (cid:96) [ ψ ] vanishesthen we can construct the time integral of ψ , namely a suitably regular solution ˜ ψ to thewave equation such that T ˜ ψ = ψ . We define the time-inverted Newman–Penrose charge of ψ to be I (1) (cid:96) [ ψ ] = I (cid:96) [ ˜ ψ ] . We note that the construction of ˜ ψ requires repeated integration in r (the number of inte-grations depending on (cid:96) ), however one can still obtain explicit, yet complicated, formulasfor I (1) (cid:96) [ ψ ] in terms of the initial data of ψ .We also note that in our case we can still construct the time-inverse by solving an ODE,similarly to the spherically symmetric case, and in contrast to the Kerr case where suchan approach is not possible and where this step involves the inversion of an operator thatis elliptic outside the ergoregion, see section 9 of [AAG21] for details. I (cid:96) = 0We now have all the tools needed to prove the precise asymptotics (1.3), (1.4) and (1.5) inthe case where I (cid:96) [ ψ ] = 0 (again for an angular mode ψ = ψ (cid:96) ). First we construct the timeintegral ˜ ψ of ψ and show that generically its Newman–Penrose charge is non-vanishing.We can then apply the methods of Sections 1.3.8 and 1.3.7 to obtain the asymptotics for˜ ψ . Finally, commuting the derived estimates with T yields the asymptotics for ψ .Moreover, we note that if the time-inverted (cid:96) -th Newman–Penrose charge happens tovanish as well, we can repeat the process, and prove asymptotics that decay one powerfaster. Using our results, one can also try to recover expressions similar to the ones givenfor I (1)0 in [AAG18a], for compactly supproted data. In this case, for time-symmetric data(i.e. T ψ | Σ = 0) that also vanish at the event horizon, the time-inverted (cid:96) -th Newman–Penrose charge will vanish as well, hence in that situation we can show the polynomiallate-time law of τ − (cid:96) − (this should be contrasted with the suboptimal rates obtained forsuch data in [DSS12] and [DSS11]).As a final remark, let us also note that one can use the results of the present paper andthe methods developed in [AAG19] to obtain second order asymptotics. The second author (S.A.) acknowledges support through the NSERC grant 502581 and theOntario Early Researcher Award.
In this section we state the main results that we obtain in this paper.10e state first the precise late-time asymptotics obtained for a linear wave restrictedto angular frequencies ≥ (cid:96) for some (cid:96) ≥ P ≥ (cid:96) ,and by P (cid:96) we denote the spectral projection to frequency (cid:96) ) with vanishing (cid:96) -th Newman–Penrose charge, but non-vanishing time-inverted (cid:96) -th Newman–Penrose charge (such dataare considered to be the most naturally physical ones and include ). Theorem 2.1.
Let ψ be a solution of the wave equation (1.1) in the domain of outercommunications of a subextremal Reissner–Nordstr¨om spacetime up to and including thehorizon. We assume that our data are smooth and compactly supported, which implies thatits Newman–Penrose charges are vanishing. We additionally assume that for some (cid:96) ≥ its time-inverted (cid:96) -th Newman–Penrose charge is non-vanishing, i.e. I (1) (cid:96) [ ψ ] (cid:54) = 0 . Let m ∈ N . For Φ ( k ) the quantities that are given in Proposition 4.1, we assume that E (1) . = (cid:88) k ≤ (cid:96) +1 (cid:16) (cid:90) N u r − δ ( L ( P (cid:96) T m + k Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P (cid:96) +1 T m + k Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P ≥ (cid:96) +2 T m + k Φ ( (cid:96) ) )) dωdv (cid:17) + (cid:96) − (cid:88) s =0 (cid:88) | α |≤ m + (cid:96) +1 ,k ≤ m +3 (cid:96) +1 (cid:90) N u r − δ | / ∇ α S ( L ( P ≥ (cid:96) +2 T k Φ ( s ) )) | dωdv + (cid:88) ≤ k + l ≤ (cid:96) + m +3 (cid:90) Σ u J N [ N k T l ψ ] · n u dµ Σ u < ∞ . Then we have for ( u, v, θ, ϕ ) ∈ { r ≥ R } for some R > r + that (cid:12)(cid:12)(cid:12)(cid:12) P ≥ (cid:96) ( T m ψ )( u, v, θ, ϕ ) r (cid:96) − A (1) (cid:96),m I (1) (cid:96) [ ψ ]( θ, ϕ ) T m (cid:18) u (cid:96) +2 v (cid:96) +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CE (1) u − (cid:96) − − m − η (cid:48) v − (cid:96) − , (2.1) for some η (cid:48) > , for C = C ( D, R, m, (cid:96), η (cid:48) ) > , and for a numerical constant A (1) (cid:96),m thatdepends on (cid:96) and m .For ( τ, r, θ, ϕ ) ∈ { r ≤ R } we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ≥ (cid:96) ( T m ψ )( τ, r, θ, ϕ ) r (cid:96) − A (1) (cid:96),m I (1) (cid:96) [ ψ ]( θ, ϕ ) τ (cid:96) +3+ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CE (1) τ − (cid:96) − − m − η (cid:48) , (2.2) for some η (cid:48) > and for C = C ( D, R, m, (cid:96), η (cid:48) ) > . We now state the precise asymptotics obtained for a linear wave localized at angularfrequencies ≥ (cid:96) for some (cid:96) ≥ (cid:96) -th Newman–Penrose constant. Theorem 2.2.
Let ψ be a solution of the wave equation (1.1) in the domain of outercommunications of a subextremal Reissner–Nordstr¨om spacetime up to and including thehorizon. Assume that its (cid:96) -th Newman–Penrose constant is non-vanishing for some fixed (cid:96) ≥ , i.e. I (cid:96) [ ψ ] (cid:54) = 0 . Let m ∈ N . Moreover we assume that our data are smooth and that: sup r (cid:88) s ≤ (cid:88) l ≤ m (cid:18)(cid:90) S | / ∇ s +1 S ( T l Φ ( (cid:96) ) ) | dω (cid:19) ( u , r ) < ∞ , nd sup r (cid:88) s ≤ (cid:88) l ≤ m (cid:88) ≤ k ≤ (cid:96) − (cid:88) ≤ j ≤ (cid:96) − k (cid:18)(cid:90) S | / ∇ s +1 S / ∆ j S ( T l Φ ( k ) ) | dω (cid:19) ( u , r ) < ∞ , for all r , where the Φ ( k ) ’s are given in Proposition 4.1, and that E . = (cid:88) k ≤ (cid:96) +1 (cid:16) (cid:90) N u r − δ ( L ( P (cid:96) T m + k Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P (cid:96) +1 T m + k Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P ≥ (cid:96) +2 T m + k Φ ( (cid:96) ) )) dωdv (cid:17) + (cid:96) − (cid:88) s =0 (cid:88) | α |≤ m + (cid:96) +1 ,k ≤ m +3 (cid:96) +1 (cid:90) N u r − δ | / ∇ α S ( L ( P ≥ (cid:96) +2 T k Φ ( s ) )) | dωdv + (cid:88) ≤ k + l ≤ (cid:96) + m +3 (cid:90) Σ u J N [ N k T l ψ ] · n u dµ Σ u < ∞ . Then we have for ( u, v, θ, ϕ ) ∈ { r ≥ R } for some R > r + that (cid:12)(cid:12)(cid:12)(cid:12) P ≥ (cid:96) ( T m ψ )( u, v, θ, ϕ ) r (cid:96) − A (cid:96),m I (cid:96) [ ψ ]( θ, ϕ ) T m (cid:18) u (cid:96) +1 v (cid:96) +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CEu − (cid:96) − − m − η v − − (cid:96) , (2.3) for some η > , for C = C ( D, R, m, (cid:96), η ) > , and for a quantity A (cid:96),m that depends on (cid:96) and m .For ( τ, r, θ, ϕ ) ∈ { r ≤ R } we have that (cid:12)(cid:12)(cid:12)(cid:12) P ≥ (cid:96) ( T m ψ )( τ, r, θ, ϕ ) r (cid:96) − A (cid:96),m I (cid:96) [ ψ ]( θ, ϕ ) τ (cid:96) +2+ m (cid:12)(cid:12)(cid:12)(cid:12) ≤ CEτ − (cid:96) − − m − η , (2.4) for some η > and for C = C ( D, R, m, (cid:96), η ) > . Finally we state the precise asymptotics obtained for the radiation field of a linear wavelocalized at frequencies ≥ (cid:96) for some (cid:96) ≥ (cid:96) -th Newman–Penrose charge,but non-vanishing time-inverted (cid:96) -th Newman–Penrose charge, at future null infinity I + . Theorem 2.3.
Under the assumptions of Theorem 2.1 we have that along null infinity thefollowing estimate holds true: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T m φ ≥ (cid:96) ( u, ∞ , θ, ϕ ) + 2 (cid:96) I (1) (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · · · · · ( (cid:96) + 2) T m ( u − − (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CE (1) u − − (cid:96) − m − η (1) , (2.5) for some η (1) > , E (1) as in Theorem 2.1, and C = C ( D, R, m, (cid:96), η (1) ) .On the other hand under the assumptions of Theorem 2.2 we have the following estimatealong null infinity: (cid:12)(cid:12)(cid:12)(cid:12) T m φ ≥ (cid:96) ( u, ∞ , θ, ϕ ) − (cid:96) I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · . . . · ( (cid:96) + 1) T m ( u − − (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ CEu − − (cid:96) − m − η , (2.6) for some η > , E as in Theorem 2.2, and C = C ( D, R, m, (cid:96), η ) . Preliminaries
The Reissner–Nordstr¨om spacetimes are the unique spherically symmetric and asymptoti-cally flat 2-parameter family of solutions of the Einstein–Maxwell equations, the two param-eters being the mass
M > e . We consider a subextremalReissner–Nordstr¨om black hole spacetime ( M , g ) where g in ingoing Eddington–Finkelsteincoordinates ( v, r, θ, ϕ ) has the form: g = − Ddv + 2 dvdr + r ( dθ + sin θdϕ ) , with D = 1 − Mr + e r (3.1)where 0 ≤ | e | < M .Note that D has two roots 0 < r − < r + . We will work work in the domain of outercommunications D of M up to and including the future event horizon, that is we will alwayshave r ≥ r + , the future event horizon being the hypersurface H + := { ( v, r, θ, ϕ ) | r = r + } .In Bondi coordinates ( u, r, θ, ϕ ) (that are valid outside the event horizon) g has theform: g = − Ddu − dudr + r ( dθ + sin θdϕ ) . In D we have that u ∈ R , r ∈ [ r + , ∞ ), θ ∈ (0 , π ) and ϕ ∈ (0 , π ). By using the tortoisecoordinate r ∗ ( r ) = r + κ + ln (cid:12)(cid:12)(cid:12) r − r + r + (cid:12)(cid:12)(cid:12) + κ − ln (cid:12)(cid:12)(cid:12) r − r − r − (cid:12)(cid:12)(cid:12) + C , for a constant C (that we will fixshortly), for κ ± = dDdr (cid:12)(cid:12) r = r ± , we define v = u + 2 r ∗ and in double coordinates ( u, v, θ, ϕ )the metric g takes the form g = − Ddudv + r ( dθ + sin θdϕ ) , and in these coordinates we can define future null infinity I + which is the limiting nullhypersurface foliated by 2-spheres where the null hypersurfaces { u = u (cid:48) } terminate as v → ∞ (i.e. it is the limiting hypersurface that is formed by the limit points as r → ∞ of future null geodesics). We additionally note that on r ph = M (cid:18) (cid:113) − e M (cid:19) (wherewe choose C such that r ∗ ( r ph ) = 0) we have the photon sphere which imposes a derivativeloss in the Morawetz estimates that we present in the next section.The vector field T = ∂ u (in ( u, r, θ, ϕ )) is Killing. By / ∇ S we denote the covariantderivative on S with respect to its standard metric, and by / ∆ S the Laplacian on S .Moreover by Ω i , i ∈ { , , } we denote the three Killing vector fields associated to S which can be expressed as: Ω := sin ϕ∂ θ + cot θ cos ϕ∂ ϕ , Ω := − cos ϕ∂ θ + cot θ sin ϕ∂ ϕ , Ω := ∂ ϕ , and using the above we can also define the Ω α vector fields by Ω α := Ω α Ω α Ω α where( α , α , α ) ∈ N and α = (cid:80) i =1 α i . We also consider the vector field ∂ ρ := ∂ r + h ( r ) T, h : [ r + , ∞ ) → R such that1max r ∈ [ r + ,R ] D ( r ) ≤ h ( r ) < D ( r ) for r ≤ R ,0 < D ( r ) − h ( r ) = O ( r − − η ) for r > R , for some η >
0, for some
R > r + . We will consider two foliations of D . The first foliation consists of spacelike-null hy-persurfaces Σ τ that are constructed as follows: consider a spacelike and asymptoticallyflat hypersurface Σ that intersects H + on a 2-sphere, a null hypersurface N τ := { ( u = τ, r, θ, ϕ ) | r ≥ R for some R > r + , take some u such that ( Σ ∩ { r = R } ) = ( { u = u } ∩ { r = R } ), and consider Σ u := ( Σ ∩ { r ≤ R } ) ∪ N u . Then define Σ τ := f τ (Σ u ) for f the flow of T . The second foliation consists of hyper-boloidal hypersurfaces S τ , where S := { ( v, r, θ, ϕ ) | v = v − (cid:90) Rr h ( r (cid:48) ) dr (cid:48) , r ≥ r + } , and v is large enough and depends on h and R , and now we can define S τ := f τ ( S ). Finallyin the region { r ≥ R } we define also I τ ( τ , τ ) := { v = τ | u ≤ τ ≤ u ≤ τ } ∩ { r ≥ R } indouble null coordinates.Let us introduce the projection operators P ≥ (cid:96) , P (cid:96) : L ( S ) → L ( S ) by noticing thatany smooth function f can be written as f = ∞ (cid:88) (cid:96) =0 P (cid:96) f, where P (cid:96) f ( u, r, θ, ϕ ) = (cid:96) (cid:88) m = − (cid:96) f m,(cid:96) ( u, r ) Y m,(cid:96) ( θ, ϕ ) , for certain functions f m,(cid:96) and where Y m,(cid:96) are the spherical harmonics which form a completebasis of eigenfunctions of / ∆ S on L ( S ). Note that / ∆ S ( P (cid:96) f ) = − (cid:96) ( (cid:96) + 1) f. We also define P ≥ (cid:96) f = (cid:88) l ≥ (cid:96) P l f. Finally by f = O ( x ) we mean that there exist a constant K such that | f ( x ) | ≤ Kx forall x .We note that the linear wave equation (cid:3) g ψ = 0 , is globally well-posed in D for smooth data ( ψ, n Σ u ψ ) | Σ u ∩{ r ≤ R } and ( ψ, ∂ v ψ ) | N u , orsmooth data ( ψ, n S u ψ ) | S u (see [Ali10]). Note that we can consider data either on Σ u or14 u , as determining data on either hypersurface can always give data on a later S u (cid:48) or Σ u (cid:48) hypersurface respectively by solving a local problem. Note that in the rest of the paper we will always assume that we work witha smooth solution ψ of the linear wave equation on the domain of outer com-munications up to and including the event horizon of a subextremal Reissner–Nordstr¨om black hole spacetime. In particular we will consider smooth data withrespect to the conformal compactification of Σ u , which implies thatsup r (cid:18)(cid:90) S | / ∇ S / ∆ j S ( r ∂ r ) k T m ( rψ ) | dω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Σ u ( r ) < ∞ , (3.2)for any j , k , m ∈ N . Note that we can always assume less (usually we will work with P (cid:96) ψ or P ≥ (cid:96) ψ and we need to assume the previous estimate only for finite j and k , theirrange dependent on (cid:96) ) but we make the aforementioned assumption to slightly simplify thestatements of our results. Let N be a vector field defined as follows (in ingoing ( v, r, θ, ϕ ) coordinates): N := T − ∂ r for r ∈ [ r + , r ], N := T for r ≥ r ,for r + ≤ r < r . Assuming that (cid:90) Σ u J N [ ψ ] · n Σ u dµ Σ u < ∞ , there exists a uniform constant C >
0, such that for all v (cid:90) Σ τ J N [ ψ ] · n τ dµ Σ τ + (cid:90) I v ( u ,τ ) J N [ ψ ] · L r dωdu ≤ C (cid:90) Σ u J N [ ψ ] · n Σ u dµ Σ u . (3.3)In the region { r ≥ R } (where R was chosen in the definition of the Σ hypersurfaces) wehave the following estimate (which is a combination of a Morawetz estimate and a redshift estimate, see [DR13]): there exists a uniform constant C >
0, such that for all u < τ < τ (cid:90) τ τ (cid:32)(cid:90) Σ τ \N τ J N [ ψ ] · n τ dµ Σ τ (cid:33) dτ ≤ C (cid:90) Σ τ ( J N [ ψ ] · n τ + J N [ T ψ ] · n τ ) dµ Σ τ . (3.4)For a proof of the aforementioned estimate see the lecture notes [DR13].Close to the horizon in the region A (that is away from the photon sphere) we have thefollowing local Morawetz estimates ψ that does not lose derivatives: (cid:90) τ τ (cid:32)(cid:90) Σ τ ∩C} J T [ ∂ α ψ ] · n τ dµ Σ τ (cid:33) dτ ≤ C α (cid:88) k ≤| α | (cid:90) Σ τ J N [ T k ψ ] · n τ dµ Σ τ , (3.5) (cid:90) τ τ (cid:32)(cid:90) S τ ∩C} J T [ ∂ α ψ ] · n τ dµ Σ τ (cid:33) dτ ≤ C α (cid:88) k ≤| α | (cid:90) S τ J N [ T k ψ ] · n τ dµ Σ τ , (3.6)for C ∩ ( r ph − δ, r ph + δ ) = ∅ for some δ > C α > C and the choice of α (see [DR09]). 15 Higher-order radiation fields and Newman–Penrose charges
We denote with φ := r · ψ. the zeroth order radiation field corresponding to a solution ψ to (1.1). Using that ψ is asolution to (1.1) it follows that φ satisfies the equation:2 ∂ u ∂ r φ = ∂ r ( D∂ r φ ) − D (cid:48) r − φ + r − / ∆ S φ. (4.1)See for example Appendix of [AAG18d] for a derivation. Setting x = r , in ( u, x, θ, ϕ )coordinates, equation (4.1) turns into:2 ∂ u ∂ x φ + ∂ x ( Dr − ∂ x φ ) + / ∆ S φ − (2 M x − e x ) φ = 0 , (4.2)as ∂ x = − r ∂ r . Note that Dr − = x − M x + e x . Then, if we define (cid:101) Φ ( (cid:96) ) = ∂ (cid:96)x φ (note that (cid:101) Φ ( (cid:96) ) = ( − (cid:96) ( r ∂ r ) (cid:96) φ ), we obtain inductively thefollowing equation :0 = 2 ∂ u ∂ x (cid:101) Φ ( (cid:96) ) + ∂ x ( Dr − ∂ x (cid:101) Φ ( (cid:96) ) ) + 2 (cid:96) ( x − M x + 2 e x ) (cid:101) Φ ( (cid:96) +1) + / ∆ S (cid:101) Φ ( (cid:96) ) + [ (cid:96) ( (cid:96) + 1)(1 − M x + 6 e x ) − (2 M x − e x )] (cid:101) Φ ( (cid:96) ) + [ − ( (cid:96) + 1) (cid:96) ( (cid:96) − M − e x ) − (cid:96) (2 M − e x )] (cid:101) Φ ( (cid:96) − + [( (cid:96) + 1) (cid:96) ( (cid:96) − (cid:96) − e + 2 (cid:96) ( (cid:96) − e ] (cid:101) Φ ( (cid:96) − . (4.3)For a derivation see Section 6.1 of [GW19]. Using the previous equation we show theexistence of conserved quantities at infinity in Proposition 4.1 below. Note that a similarderivation has been presented also by Ma in [Ma20] in the context of the Maxwell equations. Proposition 4.1.
Let (cid:96) ∈ N . Let Φ ( (cid:96) ) satisfy the following inductive definition: Φ (0) := φ, Φ ( (cid:96) ) := (cid:101) Φ ( (cid:96) ) + (cid:96) (cid:88) k =1 α (cid:96),k Φ ( (cid:96) − k ) , for appropriate constants { α (cid:96),k } (cid:96)k =1 . Then ∂ u ∂ x Φ ( (cid:96) ) + ∂ x ( Dr − ∂ x Φ ( (cid:96) ) ) + 2 (cid:96) ( x − M x + 2 e x )Φ ( (cid:96) ) + [ (cid:96) ( (cid:96) + 1) + / ∆ S ]Φ ( (cid:96) ) + + (cid:96) (cid:88) k =0 O ( x ) (cid:101) Φ ( k ) = 0 . (4.4) Proof.
We will establish (4.4) by strong induction. Note that (4.4) holds for n = 0. Nowsuppose (4.4) holds for n ≤ N , N ∈ N . Then by combining (4.3) with n = N + 1 and164.4) with n = N and n = N −
1, we obtain the following equation for Φ ( N +1) :0 = 2 ∂ u ∂ x Φ ( N +1) + ∂ x ( Dr − ∂ x Φ ( N +1) ) + 2( N + 1)( x − M x + 2 e x )Φ ( N +1) + [( N + 1)( N + 2) + / ∆ S ] (cid:101) Φ ( N +1) + N (cid:88) k =0 [ c k + α N +1 ,N +1 − k ( N + 1 − k )] (cid:101) Φ ( k ) + N +1 (cid:88) k =0 O ( x ) (cid:101) Φ ( k ) , where the c k ’s are constants that depend on the constants { α n,k } nk =0 that have alreadybeen determined due to the induction hypothesis. In order to conclude that (4.4) holdsalso for n = N + 1, we must therefore have that c k + α N +1 ,N +1 − k ( N + 1 − k ) = 0 for every k ∈ { , . . . , N } , which of course we can always do as each one of these equations is a linear algebraicequation in α N +1 ,m , m ∈ { , . . . , N } .We will refer to Φ ( n ) and (cid:101) Φ ( n ) as the n -th order radiation fields . In outgoing ( u, r, θ, ϕ )coordinates let L = 12 D∂ r , L = ∂ u − D∂ r , and we have the following equations for the (cid:101) Φ ( (cid:96) ) ’s and the Φ ( (cid:96) ) ’s: Corollary 4.2.
Fix (cid:96) ≥ . Then (cid:101) Φ ( (cid:96) ) satisfies: ∂ r ∂ u (cid:101) Φ ( (cid:96) ) = ∂ r ( D∂ r (cid:101) Φ ( (cid:96) ) ) + r − / ∆ S (cid:101) Φ ( (cid:96) ) + [ − (cid:96)r − + O ( r − )] ∂ r (cid:101) Φ ( (cid:96) ) + (cid:2) (cid:96) ( (cid:96) + 1) r − + O ( r − ) (cid:3) (cid:101) Φ ( (cid:96) ) + (cid:96) − (cid:88) k =0 O ( r − ) (cid:101) Φ ( k ) (4.5) or equivalently, LL (cid:101) Φ ( (cid:96) ) = Dr − / ∆ S (cid:101) Φ ( (cid:96) ) + [ − (cid:96)r − + O ( r − )] L (cid:101) Φ ( (cid:96) ) + D (cid:2) (cid:96) ( (cid:96) + 1) r − + O ( r − ) (cid:3) (cid:101) Φ ( (cid:96) ) + (cid:96) − (cid:88) k =0 O ( r − ) (cid:101) Φ ( k ) . (4.6) Furthermore, for Φ ( (cid:96) ) we have that P (cid:96) Φ ( (cid:96) ) satisfies: ∂ r ∂ u ( P (cid:96) Φ ( (cid:96) ) ) = ∂ r ( D∂ r ( P (cid:96) Φ ( (cid:96) ) )) + ( − (cid:96)r − + O ( r − )) ∂ r ( P (cid:96) Φ ( (cid:96) ) )+ O ( r − ) P (cid:96) Φ ( (cid:96) ) + (cid:96) − (cid:88) k =0 O ( r − ) P (cid:96) Φ ( k ) , (4.7)17r equivalently,4 LL ( P (cid:96) Φ ( (cid:96) ) ) = [ − (cid:96)r − + O ( r − )] L ( P (cid:96) Φ ( (cid:96) ) ) + (cid:96) (cid:88) k =0 O ( r − ) P (cid:96) Φ ( k ) . (4.8)Finally we note that from equation (4.8) it follows that the (cid:96) -th Newman–Penrosequantities are finite along I + . Proposition 4.3.
Fix (cid:96) ≥ , and assume that (cid:90) Σ J T [ ψ ] · n Σ dµ Σ < ∞ . For any u ≥ u if lim r →∞ / ∆ j S (cid:101) Φ ( k ) ( u , r, θ, ϕ ) < ∞ , with ≤ j ≤ (cid:96) − k and ≤ k ≤ (cid:96) , then lim r →∞ (cid:101) Φ ( (cid:96) ) ( u, r, θ, ϕ ) < ∞ . The proof of the above Proposition can be done inductively in the same as the proofof Proposition 6.2 of [AAG20b] (see also section 3.2 of [AAG18d]).Using the aforementioned Proposition 4.3 and Proposition 4.1 we get the followingimportant result.
Theorem 4.4.
Fix (cid:96) ≥ and assume that lim r →∞ / ∆ j S (cid:101) Φ ( k ) ( u , r, θ, ϕ ) < ∞ , for all ≤ j ≤ (cid:96) − k and ≤ k ≤ (cid:96) . Then ∂ x ( P (cid:96) Φ (cid:96) ) is conserved along I + . Definition 4.1.
Let I (cid:96) [ ψ ]( θ, ϕ ) . = lim r →∞ ( − (cid:96) +1 ∂ x ( P (cid:96) Φ ( (cid:96) ) )( u, r, θ, ϕ ) , (4.9) which is independent of u as shown in Proposition 4.1. We will refer to the quantities I (cid:96) [ ψ ] (usually omitting the dependence on θ and ϕ ) as the (cid:96) -th Newman–Penrose charges of alinear wave ψ . Note that we can also have a spherical harmonics decomposition of eachNewman–Penrose charge as I (cid:96) [ ψ ]( θ, ϕ ) . = (cid:96) (cid:88) m = − (cid:96) I m,(cid:96) [ ψ ] Y m,(cid:96) ( θ, ϕ ) , (4.10) and the Newman–Penrose constants I m,(cid:96) determine the value of the Newman–Penrosecharge.We will also refer to the quantities ∂ x Φ ( (cid:96) ) as the (cid:96) -th Newman–Penrose quantities (notethat there is no angular frequency localization in the latest quantities). Hierarchies of r p -weighted estimates r p -weighted estimates for Φ ( (cid:96) ) Proposition 5.1.
Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96), and for all u ≤ u ≤ u : (cid:90) N u r p ( L ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) dωdv + 12 (cid:90) u u (cid:90) N u ( p + 4 (cid:96) ) r p − ( L ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) dωdvdu + 18 (cid:90) u u (cid:90) N u (2 − p ) r p − D (cid:16) | / ∇ S ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) ) | − (cid:96) ( (cid:96) + 1)( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) ) (cid:17) dωdvdu ≤ C (cid:90) N u r p ( L ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ (cid:96) (cid:90) Σ u J T [ T k P ≥ (cid:96) ψ ] · n u dµ Σ u , where C = C ( D, R, (cid:96) ) > and we can take R = ( p + 4 (cid:96) ) − R ( (cid:96), D ) > . The proof of the aforementioned Proposition follows along the same lines as the proofof Proposition 6.5 of [AAG20b] by using equation (10.7).Now we present an extended hierarchy for a linear wave localized at frequency (cid:96) withnon-zero (cid:96) -th Newman–Penrose charge.
Proposition 5.2.
Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96), and for all u ≤ u ≤ u : (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv + 12 (cid:90) u u (cid:90) N u ( p + 4 (cid:96) ) r p − ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu ≤ C (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ (cid:96) (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u , where C = C ( D, R, (cid:96) ) > and we can take R = max { ( p + 4 (cid:96) ) − , ( p − − } R ( (cid:96), D ) > .Proof. Let us use ψ to denote P (cid:96) ψ , for convenience. We then proceed as in the proof ofProposition 5.1, using instead (4.7), and then we use that for P (cid:96) ψ the Poincar´e inequality becomes an equality : (cid:90) S | / ∇ S Φ ( (cid:96) ) | dω = (cid:96) ( (cid:96) + 1) (cid:90) S Φ (cid:96) ) dω, which allows us to write (cid:90) N u r p ( L ( χ Φ ( (cid:96) ) )) dωdv + 12 (cid:90) u u (cid:90) N u [( p + 4 (cid:96) ) r p − + O ( r p − )]( L ( χ Φ ( (cid:96) ) )) dωdvdu = (cid:90) N u r p ( L ( χ Φ ( (cid:96) ) )) dωdv + J + (cid:88) | α |≤ (cid:90) u u (cid:90) N u r p − L ( χ Φ ( (cid:96) ) ) · R χ [ ∂ α Φ ( (cid:96) ) ] dωdvdu, (5.1)with J := (cid:96) (cid:88) k =0 (cid:90) u u (cid:90) N u O ( r p − ) χ Φ ( k ) · L ( χ Φ ( (cid:96) ) ) dωdudv,
19e apply a weighted Young’s inequality to estimate | J | ≤ (cid:90) u u (cid:90) N u (cid:15) ( p +4 (cid:96) ) r p − ( L ( χ Φ ( (cid:96) ) )) dωdudv + C (cid:15) ( p +4 (cid:96) ) − (cid:96) (cid:88) k =0 (cid:90) u u (cid:90) N u r p − χ Φ k ) dωdudv. As in the proof of Proposition 5.1, the right-hand side of the above equation can be absorbedinto the left-hand side of (5.1) if p < R χ terms that appear can be estimated using (3.4).Next we show how the above hierarchy for a wave localized at frequency (cid:96) can beextended even further by assuming that its (cid:96) -th Newman–Penrose constant vanishes. Proposition 5.3.
Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96), and for all u ≤ u ≤ u we have that: (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv + 12 (cid:90) u u (cid:90) N u ( p + 4 (cid:96) ) r p − ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu ≤ C (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ (cid:96) (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u , where C = C ( D, R, (cid:96) ) > and we can take R = max { ( p + 4 (cid:96) ) − , ( p − − } R ( (cid:96), D ) > .Furthermore there exists R > sufficiently large, such that for p ∈ ( − (cid:96), and for all u ≤ u ≤ u we have that: (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv + 12 (cid:90) u u (cid:90) N u ( p + 4 (cid:96) ) r p − ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu ≤ C (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ (cid:96) (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u + C ( p − − E aux ,(cid:96) , where C = C ( D, R, (cid:96) ) > and we can take R = ( p + 4 (cid:96) ) − R ( (cid:96), D ) > and E aux ,(cid:96) = (cid:90) N r − (cid:15) ( L ( P (cid:96) Φ ( k ) )) dωdv + (cid:96) (cid:88) j =0 E [ T j P (cid:96) ψ ] , where E [ ψ ] := (cid:88) m ≤ (cid:90) Σ J N [ T m ψ ] · n dµ Σ + (cid:88) m ≤ (cid:90) N r ( LT m φ ) + r ( LT m +1 φ ) dωdv + (cid:90) N r − (cid:15) ( L (cid:101) Φ (1) ) + r − (cid:15) ( LT (cid:101) Φ (1) ) dωdv. Proof.
For the proof of the first estimate in the range ( − (cid:96),
4) we follow the exact sameproof as in Proposition 5.2. Note that the difference in the two cases comes from theassumption on I (cid:96) [ ψ ]. In the current case the right-hand side would be infinite for p ∈ [3 , I (cid:96) [ ψ ] was non-vanishing. 20n the remaining range p ∈ [4 ,
5) we proceed again as in the proof of Proposition 5.2and we rewrite (5.1) by taking a supremum on the right-hand side:sup u ≤ u ≤ u (cid:90) N u r p ( L ( χ Φ ( (cid:96) ) )) dωdv + 12 (cid:90) u u (cid:90) N u ( p + 4 (cid:96) ) r p − ( L ( χ Φ ( (cid:96) ) )) dωdvdu = (cid:90) N u r p ( L ( χ Φ ( (cid:96) ) )) dωdv + J + (cid:88) | α |≤ (cid:90) u u (cid:90) N u r p − L ( χ Φ ( (cid:96) ) ) · R χ [ ∂ α Φ ( (cid:96) ) ] dωdudv, (5.2)Now we will estimate | J | by absorbing it into the (first) flux integral on the left-handside of (5.2) rather than the (second) spacetime integral. We apply a Cauchy–Schwarzinequality with weights in r and u to estimate: r p − χ Φ ( k ) · L ( χ Φ ( (cid:96) ) ) ≤ (cid:15) ( u + 1) − − η r p L ( χ Φ ( (cid:96) ) ) + C (cid:15) ( u + 1) η r p − Φ k ) . We can estimate (cid:15) (cid:90) u u (cid:90) N u ( u + 1) − − η r p L ( χ Φ ( (cid:96) ) ) dωdv ≤ (cid:15) ( u + 1) − η · sup u ≤ u ≤ u (cid:90) N u r p ( L ( χ Φ ( (cid:96) ) )) dωdv, and we can absorb the right-hand side into the left-hand side of (5.2).We note that (cid:90) S ( P (cid:96) Φ ( (cid:96) ) ) ( u, r, θ, ϕ ) dω ≤ Cu − (cid:15) E aux ,(cid:96) , (5.3)in { r ≥ R } as we will show in the next section in Lemma 6.1, and using the above estimatewe have that (cid:90) u u (cid:90) N u ( u + 1) η r p − Φ (cid:96) ) dωdudv ≤ CE aux ,(cid:96) (cid:90) u u (cid:90) ∞ R ( u + 1) − (cid:15) + η r p − drdu The integral on the right-hand side is bounded for p < p − − .It remains to show estimate (5.3) which we do in Lemma 6.1 in section 6. P (cid:96) Φ ( (cid:96) ) In this Section we will show that Proposition 5.2 cannot hold true for p ≥
3, and that thefinal estimate of Proposition 5.3 cannot hold true for p ≥ Proposition 5.4.
Fix some (cid:96) ≥ and consider a solution ψ of (1.1) such that I (cid:96) [ ψ ] (cid:54) = 0 . Then the range of p in Proposition 5.2 is sharp. Specifically for any u ≤ u (cid:48) < ∞ we havethat: (cid:90) ∞ u (cid:48) (cid:90) N u r ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu = ∞ . Proof.
By the definition of the (cid:96) -th Newman–Penrose charge, for some ¯ R = ¯ R ( u (cid:48) ) > R andfor some constant c we have that: r ( ∂ r ( P (cid:96) Φ ( (cid:96) ) )) ≥ CI (cid:96) [ ψ ] for all r ≥ ¯ R . (5.4)21e use the same computation as in Proposition 5.2 for p = 3 but in the region { u ≥ u (cid:48) , r ≥ ¯ R, v ≤ v (cid:96) } for some for some fixed v (cid:96) < ∞ and we have that:12 (cid:90) ∞ u (cid:48) (cid:90) N u ∩{ v ≤ v (cid:96) } [( p + 4 (cid:96) ) r p − + O ( r p − )]( L ( χ Φ ( (cid:96) ) )) dωdvdu − J + (cid:88) | α |≤ (cid:90) ∞ u (cid:48) (cid:90) N u ∩{ v ≤ v (cid:96) } r p − L ( χ Φ ( (cid:96) ) ) · R χ [ ∂ α Φ ( (cid:96) ) ] dωdvdu = (cid:90) N u (cid:48) ∩{ v ≤ v (cid:96) } r ( L ( χ Φ ( (cid:96) ) )) dωdv, where we recall that J := (cid:96) (cid:88) k =0 (cid:90) u u (cid:90) N u O ( r p − ) χ Φ ( k ) · L ( χ Φ ( (cid:96) ) ) dωdudv. By the computations in Proposition 5.2 we note that | J | + (cid:88) | α |≤ (cid:90) ∞ u (cid:48) (cid:90) N u ∩{ v ≤ v (cid:96) } r p − L ( χ Φ ( (cid:96) ) ) · R χ [ ∂ α Φ ( (cid:96) ) ] dωdvdu ≤ C ( ψ, u (cid:48) ) < ∞ , for finite u (cid:48) . Hence we get the desired result noticing that due to (5.4) we have that:lim v (cid:96) →∞ (cid:90) N u (cid:48) ∩{ v ≤ v (cid:96) } r ( L ( χ Φ ( (cid:96) ) )) dωdv = ∞ . In the case I (cid:96) = 0 we can use the time-inversion construction of Section 11 and showthe following: Proposition 5.5.
Fix some (cid:96) ≥ and consider a solution ψ of (1.1) such that I (cid:96) [ ψ ] = 0 . Then the range of p in the final estimate of Proposition 5.2 is sharp. Specifically for any u ≤ u (cid:48) < ∞ we have that: (cid:90) ∞ u (cid:48) (cid:90) N u r ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu = ∞ . The proof is similar to the one of Proposition 5.4 and will hence be omitted. r p -weighted estimates for T m Φ ( (cid:96) ) In this subsection we present an extended hierarchy for the higher order radiation fieldsafter we commute the equation with T . Proposition 5.6.
Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96) + 2 m, m + 2) and for all u ≤ u ≤ u : (cid:90) N u r p ( L m +1 ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) dωdv + (cid:90) u u (cid:90) N u r p − ( L m +1 ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) dωdvdu + (cid:90) u u (cid:90) N u r p − | / ∇ S ( L m ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) | dωdvdu ≤ C (cid:88) l ≤ m (cid:90) N u r p − l ( L m +1 − l ( P ≥ (cid:96) (cid:101) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ m (cid:90) Σ u J T [ T k P ≥ (cid:96) ψ ] · n u dµ Σ u , (5.5)22 here C = C ( D, R, m, (cid:96) ) > . The proof follows along the same lines as the proof of Proposition 7.6 of [AAG20b] andwill not be repeated here.
Proposition 5.7.
Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96) + 2 m, m + 3) and for all u ≤ u ≤ u : (cid:90) N u r p ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdv + (cid:90) u u (cid:90) N u r p − ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu ≤ C (cid:88) l ≤ m (cid:90) N u r p − l ( L m +1 − l ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ m (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u , where C = C ( D, R, m, (cid:96) ) > .Proof. We commute equation (4.8) with L m and we have that2 LL m +1 ( P (cid:96) Φ ( (cid:96) ) ) =[ − (cid:96)r − + O ( r − )] L m +1 ( P (cid:96) Φ ( (cid:96) ) )+ (cid:88) m + m = m,m C(cid:15) (cid:88) m + m = m,m Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96) + 2 m, m + 4) and for all u ≤ u ≤ u : (cid:90) N u r p ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdv + (cid:90) u u (cid:90) N u r p − ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu ≤ C (cid:88) l ≤ m (cid:90) N u r p − l ( L m +1 − l ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ m (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u , and for p ∈ [2 m + 4 , m + 5) and any δ > small enough: (cid:90) N u r p ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdv + (cid:90) u u (cid:90) N u r p − ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdvdu ≤ C (cid:88) l ≤ m (cid:90) N u r p − l ( L m +1 − l ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ m (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u + C E aux,(cid:96),k,s u − δ , where C = C ( D, R, m, (cid:96) ) .Proof. The difference with the proof of Proposition 5.7 is the treatment of J . We applya u -weighted Young’s inequality and for any δ small enough (to be determined later) we25ave that: J ≤ (cid:90) u u (cid:90) N u u δ r p ( L m +1 ( χP (cid:96) Φ ( (cid:96) ) )) dωdvdu + (cid:96) − (cid:88) k =0 (cid:88) m + m = m (cid:90) u u (cid:90) N u u δ r p − − m ( L m ( χP (cid:96) Φ ( k ) )) dωdvdu = (cid:90) u u (cid:90) N u u δ r p ( L m +1 ( χP (cid:96) Φ ( (cid:96) ) )) dωdvdu + (cid:96) − (cid:88) k =0 m (cid:88) s =0 (cid:90) u u (cid:90) N u u δ r p − − m +2 s ( L s ( χP (cid:96) Φ ( k ) )) dωdvdu. (5.9)Now we use (6.6) for m ≥ (cid:96) − (cid:88) k =0 m (cid:88) s =0 (cid:90) u u (cid:90) N u u δ r p − − m +2 s ( L s ( χP (cid:96) Φ ( k ) )) dωdvdu ≤ C (cid:90) u u E aux,(cid:96),m,s u − δ du, where E aux,(cid:96),m,s is given in Lemma 6.2.Finally for this section we show how to use the extended range of the previous twoPropositions in order to obtain additional hierarchies for the solution commuted with T . Proposition 5.9. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then there exists R > sufficiently large, such that for p ∈ ( − (cid:96) + 2 m, m + 3) and for all u ≤ u ≤ u : (cid:90) u u (cid:90) N u r p − ( L m ( P (cid:96) ( T Φ ( (cid:96) ) ))) dωdvdu ≤ C (cid:90) N u r p ( L m +1 ( P (cid:96) Φ ( (cid:96) ) )) dωdv ≤ C (cid:88) l ≤ m (cid:90) N u r p − l ( L m +1 − l ( P (cid:96) Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ m (cid:90) Σ u J T [ T k P (cid:96) ψ ] · n u dµ Σ u , where C = C ( D, R, m, (cid:96) ) > .Under the assumption that I (cid:96) [ ψ ] = 0 , the above estimate holds in the extended range p ∈ ( − (cid:96) + 2 m, m + 5) .Moreover we have that (cid:90) u u (cid:90) N u r p − ( L m ( P ≥ (cid:96) +1 ( T Φ ( (cid:96) ) ))) dωdvdu ≤ C (cid:90) N u r p ( L m +1 ( P ≥ (cid:96) +1 Φ ( (cid:96) ) )) dωdv ≤ ¯ C (cid:88) l ≤ m, | α |≤ (cid:90) N u r p − l ( L m +1 − l Ω α ( P ≥ (cid:96) +1 Φ ( (cid:96) ) )) dωdv + C (cid:88) k ≤ m (cid:90) Σ u J T [ T k P ≥ (cid:96) +1 ψ ] · n u dµ Σ u , for some ¯ C = ¯ C ( D, R, m, (cid:96) ) > , for p ∈ ( − (cid:96) + 2 m, m + 2) . roof. The two estimates can be proven in the same way. The proof follows by usingequation (4.5) after commuting it with L m (something that results in the appearance ofthe angular derivatives in the right-hand side – for fixed (cid:96) see equation (5.6)) and thenmaking use of Propositions 5.6 and 5.7 and that T = L + L . The proof follows the samelines as the proof of Proposition 4.7 of [AAG18d]. In this section we will use the r p -weighted estimates of the previous Section to show decayfor certain energy-type quantities. First we state and prove two auxiliary results that were needed in the proofs of Propositions5.3 and 5.8. Lemma 6.1. Fix (cid:96) ∈ N and consider a solution ψ to (1.1) . Assume that there exists aconstant C > such that (cid:18)(cid:90) S ( P (cid:96) Φ ( k ) ) dω (cid:19) ( u , r ) ≤ C , for ≤ k ≤ (cid:96) .Then there exists C = C ( D, R, (cid:96), (cid:15) ) such that for all ≤ k ≤ (cid:96) (cid:90) N u ( L ( χP (cid:96) Φ ( k ) )) dωdv ≤ Cu − − (cid:96) − k )+ (cid:15) E aux ,(cid:96) , (6.1) (cid:90) N u r ( L ( χP (cid:96) Φ ( k ) )) dωdv ≤ Cu − − (cid:96) − k )+ (cid:15) E aux ,(cid:96) , (6.2) with E aux ,(cid:96) = (cid:96) (cid:88) k =0 (cid:90) N r − (cid:15) ( L ( P (cid:96) Φ ( k ) )) dωdv + (cid:96) (cid:88) j =0 E [ T j P (cid:96) ψ ] , and E [ ψ ] as in Proposition 5.3.In the region { r ≥ R } we can moreover estimate (cid:90) S χ ( P (cid:96) Φ ( k ) ) ( u, r, θ, ϕ ) dω ≤ Cu − − (cid:96) − k )+ (cid:15) E aux ,(cid:96) , for all ≤ k ≤ (cid:96) , with C = C ( D, R, (cid:96), (cid:15) ) > .Proof. From [AAG18d] it follows that there exists a constant C = C ( D, (cid:15), R ) > ψ ≥ that is supported on angular frequencies ≥ 1, we have thefollowing: (cid:90) Σ u J N [ ψ ≥ ] · n u dµ Σ u ≤ Cu − (cid:15) E [ ψ ≥ ] , (6.3)where E [ ψ ≥ ] is as in Proposition 5.3. 27et { u j } be a dyadic sequence. Then we can apply the mean-value theorem to theestimate of Proposition 5.2 with p = 4 − (cid:15) to obtain (cid:90) N u (cid:48) j r − (cid:15) ( L Φ ( (cid:96) ) ) dωdv ≤ Cu − j − (cid:90) N u r − (cid:15) ( L Φ ( (cid:96) ) ) dωdv + u − (cid:15)j − (cid:96) (cid:88) j =0 E [ T j ψ ] , for some u j − ≤ u (cid:48) j ≤ u j − , where we additionally applied (6.3). By the dyadicity of { u j } , { u (cid:48) j +1 } is a dyadic subsequence of { u (cid:48) j } . We can therefore apply the mean-value theoremsuccessively together with the estimate of Proposition 5.2 with p = 3 − (cid:15) ,. . . , p = 1 − (cid:15) anduse the already established decay in the previous step to obtain: (cid:90) N ˜ uj ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ C ˜ u − j E aux ,(cid:96) , (cid:90) N ˜ uj r − (cid:15) ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ C ˜ u − j E aux ,(cid:96) , (cid:90) N ˜ uj r − (cid:15) ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ C ˜ u − j E aux ,(cid:96) , (cid:90) N ˜ uj r − (cid:15) ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ C ˜ u − j E aux ,(cid:96) , for a potentially different dyadic sequence { ˜ u j } . We can use the interpolation estimatesfrom Lemma A.4 to obtain (cid:90) N ˜ uj ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ C ˜ u − (cid:15)j E aux ,(cid:96) , (cid:90) N ˜ uj r ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ C ˜ u − (cid:15)j E aux ,(cid:96) . By an application of the estimates of Proposition 5.2 with p = 0 and p = 2 respectively,we obtain (cid:90) N u ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ Cu − (cid:15) E aux ,(cid:96) , (cid:90) N u r ( L ( χ Φ ( (cid:96) ) )) dωdv ≤ Cu − (cid:15) E aux ,(cid:96) , for all u ≥ u , which gives us, so (6.1) and (6.2) hold for k = (cid:96) .We can moreover apply Hardy’s inequality to estimate (cid:90) N u ( L ( χ Φ ( k ) )) dωdv ≤ C (cid:90) N u r − ( L ( χ Φ ( k +1) )) dωdv + C (cid:90) N u ( χ (cid:48) ) Φ k ) dωdv, (cid:90) N u r ( L ( χ Φ ( k ) )) dωdv ≤ C (cid:90) N u ( L ( χ Φ ( k +1) )) dωdv + C (cid:90) N u ( χ (cid:48) ) Φ k ) dωdv. Hence, we can inductively show that (6.1) and (6.2) hold also for 0 ≤ k < (cid:96) .28e apply the fundamental theorem of calculus, Cauchy–Schwarz and Hardy’s inequality(A.1) to estimate (cid:90) S χ Φ k ) ( u, r, θ, ϕ ) dω = 2 (cid:90) N u χ Φ ( k ) · L ( χ Φ ( k ) ) dωdv ≤ (cid:115)(cid:90) N u r − χ Φ k ) dωdv · (cid:115)(cid:90) N u r ( L ( χ Φ ( k ) )) dωdv ≤ C (cid:115)(cid:90) N u ( L ( χ Φ ( k ) )) dωdv · (cid:115)(cid:90) N u r ( L ( χ Φ ( k ) )) dωdv ≤ u − (cid:96) − k )+ (cid:15) E aux ,(cid:96) . In a similar way using the results of Proposition 5.8 we can show the following. Lemma 6.2. Fix (cid:96), m ∈ N and consider a solution ψ to (1.1) . Assume that there exists aconstant C > such that (cid:88) l ≤ m,k ≤ (cid:96) (cid:18)(cid:90) S ( L l ( P (cid:96) Φ ( k ) )) dω (cid:19) ( u , r ) ≤ C . Then we have the following energy estimates: (cid:90) N u ( L m +1 ( P (cid:96) Φ ( k ) )) dωdv ≤ C E aux,(cid:96),k,l u (cid:96) − k ) − δ , (6.4) and (cid:90) N u r ( L m +1 ( P (cid:96) Φ ( k ) )) dωdv ≤ C E aux,(cid:96),k,l u (cid:96) − k ) − δ , (6.5) for any δ > small enough and k ∈ { , . . . , (cid:96) } , and the following pointwise estimates: (cid:90) S ( L m ( P (cid:96) Φ ( k ) ) dω ≤ C E aux,(cid:96),k, u (cid:96) − k ) − δ , (6.6) for any δ > small enough and k ∈ { , . . . , (cid:96) − } in the region r ≥ R , where C = C ( D, R, (cid:96), δ ) and where E aux,(cid:96),k,l . = (cid:88) j + j ≤ m + (cid:96) (cid:90) N u r m +3 − j − δ ( L m +1 − j T j ( P (cid:96) Φ ( (cid:96) ) )) dωdv + (cid:88) j + j ≤ m + (cid:96) (cid:96) − (cid:88) s = k (cid:90) N u r m +2 − j ( L m +1 − j T j ( P (cid:96) Φ ( s ) )) dωdv + (cid:88) l ≤ (cid:96) + m (cid:90) Σ u J N [ T l P (cid:96) ψ ] · n u dµ Σ u . In the remaining part of the section we now show some decay estimates for the r p -weightedenergies. 29 emma 6.3. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) . Thenwe have that for every u ≥ u that: (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv ≤ C E aux,I (cid:96) (cid:54) =0 ,(cid:96) u − p − δ , (6.7) for p ∈ (0 , , any δ ∈ (0 , small enough, C . = C ( D, R, p, δ, (cid:96) ) , and E aux,I (cid:96) (cid:54) =0 ,m . = (cid:90) N u r − δ ( L ( P (cid:96) Φ ( m ) )) dωdv + m − (cid:88) s =0 (cid:90) N u r ( L ( P (cid:96) Φ ( s ) )) dωdv + (cid:88) l ≤ m (cid:90) Σ u J T [ T l P (cid:96) ψ ] · n u dµ Σ u . If we additionally assume that I (cid:96) [ ψ ] = 0 , we have for every u ≥ u that: (cid:90) N u r p ( L ( P (cid:96) Φ ( (cid:96) ) )) dωdv ≤ C E aux,I (cid:96) =0 ,n u − p − δ , (6.8) for p ∈ (0 , , any δ ∈ (0 , small enough, C . = C ( D, R, p, δ, (cid:96) ) , and E aux,I (cid:96) =0 ,m . = (cid:90) N u r − δ ( L ( P (cid:96) Φ ( m ) )) dωdv + m − (cid:88) s =0 (cid:90) N u r ( L ( P (cid:96) Φ ( s ) )) dωdv + (cid:88) l ≤ m (cid:90) Σ u J T [ T l P (cid:96) ψ ] · n u dµ Σ u . The proof of the above Lemma is quite standard and can be done in the same way asthe proof of Lemma 6.1 using the results of Propositions 5.2 and 5.3. Lemma 6.4. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) . Thenwe have that for k ∈ { , . . . , (cid:96) − } and for every u ≥ u that: (cid:90) N u r p ( L ( P (cid:96) Φ ( k ) )) dωdv ≤ C E aux,I (cid:96) (cid:54) =0 ,n u (cid:96) − k ) − p − δ , (6.9) for p ∈ (0 , , any δ ∈ (0 , small enough, and C . = C ( D, R, p, δ, (cid:96) ) .If additionally we assume that I (cid:96) [ ψ ] = 0 , we have for k ∈ { , . . . , (cid:96) − } and for every u ≥ u that: (cid:90) N u r p ( L ( P (cid:96) Φ ( k ) )) dωdv ≤ C E aux,I (cid:96) =0 ,(cid:96) u (cid:96) − k ) − p − δ , (6.10) for p ∈ (0 , , any δ ∈ (0 , small enough, and C . = C ( D, R, p, δ, (cid:96) ) . The proof of the above Lemma uses the previous one as well as its proof, along withHardy’s inequality (A.1). For details see Section 7 of [AAG18d].By making use of the results of Lemmas 6.3 and 6.4 (for P (cid:96) Φ ( (cid:96) ) and P (cid:96) +1 Φ ( (cid:96) ) respectively,for the latter assuming that I (cid:96) +1 [ ψ ] (cid:54) = 0) and by making use of the r p -weighted estimatesof Proposition 5.6 (along with the same tools as in the proof of Lemma 6.1) we have thefollowing result: 30 emma 6.5. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) . Thenfor every u ≥ u we have that: (cid:90) N u r p ( L ( P ≥ (cid:96) +2 Φ ( (cid:96) ) )) dωdv ≤ C E aux,(cid:96) +2 u − p − δ , for p ∈ (0 , , for some constant C , for any δ > and for an initial norm E aux,(cid:96) +2 . = (cid:90) N u r ( L ( P ≥ (cid:96) +2 Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P ≥ (cid:96) +2 Φ ( (cid:96) +1) )) dωdv + (cid:90) N u r ( L ( P ≥ (cid:96) +2 Φ ( (cid:96) +2) )) dωdv, Using the last two results we get the following: Corollary 6.6. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) . Thenwe have that (cid:90) N u r p ( L ( P ≥ (cid:96) Φ ( (cid:96) ) )) dωdv ≤ C E aux − decomp − n u − p − δ , (6.11) for p ∈ (0 , , δ > , C = C ( D, R, p, δ, (cid:96) ) , and E aux − decomp − n . = E aux,I n (cid:54) =0 ,n + E aux,(cid:96) +2 + (cid:88) l ≤ n +2 (cid:90) Σ u J N [ T l P ≥ (cid:96) ψ ] · n u dµ Σ u . If additionally we assume that I (cid:96) [ ψ ] = 0 , and we have that (cid:90) N u r p ( L ( P ≥ (cid:96) Φ ( (cid:96) ) )) dωdv ≤ C E aux − decomp − n u − p − δ , (6.12) for p ∈ (0 , , δ > , C = C ( D, R, p, δ, (cid:96) ) , and E aux − decomp − . = E aux,I (cid:96) =0 ,(cid:96) + E aux,(cid:96) +2 + (cid:88) l ≤ (cid:96) +2 (cid:90) Σ u J N [ T l P ≥ (cid:96) ψ ] · n u dµ Σ u . Using the previous two results and the red-shift estimate (3.5), we can obtain thefollowing energy estimates (which can also be viewed as r p -weighted estimates for p = 0): Proposition 6.7. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then we have that for every τ ≥ τ that: (cid:90) Σ τ J N [ P (cid:96) ψ ] · n τ dµ Σ τ ≤ C E aux,I (cid:96) (cid:54) =0 ,(cid:96) τ (cid:96) +3 − δ , (6.13) for any δ ∈ (0 , small enough, C . = C ( D, R, δ, (cid:96) ) , and E aux,I (cid:96) (cid:54) =0 ,(cid:96) as defined in Lemma6.4.If additionally we assume that I (cid:96) [ ψ ] = 0 , we have for k ∈ { , . . . , (cid:96) − } and for every u ≥ u that: (cid:90) Σ u J N [ P (cid:96) ψ ] · n u dµ Σ u ≤ C E aux,I (cid:96) =0 ,(cid:96) u (cid:96) +5 − δ , (6.14) for any δ ∈ (0 , small enough, C . = C ( D, R, δ, (cid:96) ) , and E aux,I (cid:96) =0 ,(cid:96) as defined in Lemma6.4. Corollary 6.8. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) . Thenwe have that: (cid:90) Σ u J N [ P ≥ (cid:96) ψ ] · n u dµ Σ u ≤ C E aux − decomp − n u (cid:96) +3 − δ , (6.15) for all u ≥ u , for any δ > , for some C = C ( D, R ) and where E aux − decomp − n wasdetermined in Remark 6.6.If we additionally assume that I (cid:96) [ ψ ] = 0 , we have that (cid:90) Σ u J N [ P ≥ (cid:96) ψ ] · n u dµ Σ u ≤ C E aux − decomp − u (cid:96) +5 − δ , (6.16) for all u ≥ u , for any δ > , for some C = C ( D, R ) and where E aux − decomp − wasdetermined in Remark 6.6. Using Propositions 5.6, 5.7, 5.8, and 5.9 we get the following decay estimates for the r p -weighted quantities and for the energies after commuting with T (for details see theanalogous estimates from section 7 of [AAG18d]): Proposition 6.9. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then we have for every u ≥ u that: (cid:90) N u r p ( L ( P (cid:96) T m Φ ( (cid:96) ) )) dωdv ≤ C E aux − T,I (cid:96) (cid:54) =0 ,(cid:96),m u m − p − δ , (6.17) for p ∈ (0 , , any δ ∈ (0 , small enough, C . = C ( D, R, m, p, δ, (cid:96) ) , and E aux − T,I (cid:96) (cid:54) =0 ,q,m . = m (cid:88) j =0 (cid:88) k ≤ (cid:96) (cid:90) N u r m +3 − j − δ ( L m +1 − j ( P (cid:96) T k Φ ( q ) )) dωdv + q − (cid:88) s =0 (cid:88) k ≤ (cid:96) (cid:90) N u r m +2 − j ( L m +1 − j ( P (cid:96) T k Φ ( s ) )) dωdv + (cid:88) l ≤ q + m + (cid:96) +1 (cid:90) Σ u J N [ T l P (cid:96) ψ ] · n u dµ Σ u . For k ∈ { , . . . , (cid:96) } , m ∈ N , and for every u ≥ u we have that: (cid:90) N u r p ( L ( P (cid:96) T m Φ ( k ) )) dωdv ≤ C E aux − T,I (cid:96) (cid:54) =0 ,(cid:96),m u (cid:96) − k )+2 m − p − δ , (6.18) for p ∈ (0 , , any δ ∈ (0 , small enough, and C . = C ( D, R, m, p, δ, (cid:96) ) .We also have that for every u ≥ u : (cid:90) Σ u J N [ P (cid:96) T m ψ ] · n u dµ Σ u ≤ C E aux − T,I (cid:96) (cid:54) =0 ,(cid:96) u (cid:96) +2 m +3 − δ , (6.19) for any δ ∈ (0 , small enough, C . = C ( D, R, m, δ, (cid:96) ) . n the case that we additionally assume I (cid:96) [ ψ ] = 0 , for every u ≥ u we have that: (cid:90) N u r p ( L ( P (cid:96) T m Φ ( (cid:96) ) )) dωdv ≤ C E aux − T,I (cid:96) =0 ,(cid:96),m u m − p − δ , (6.20) for p ∈ (0 , , any δ ∈ (0 , small enough, C . = C ( D, R, m, δ, (cid:96) ) , and E aux − T,I (cid:96) =0 ,q,m . = m (cid:88) j =0 (cid:88) k ≤ (cid:96) (cid:90) N u r m +5 − j − δ ( L m +1 − j ( P (cid:96) T k Φ ( q ) )) dωdv + q − (cid:88) s =0 (cid:88) k ≤ (cid:96) (cid:90) N u r m +2 − j ( L m +1 − j ( P (cid:96) T k Φ ( s ) )) dωdv + (cid:88) l ≤ q + m + (cid:96) +1 (cid:90) Σ u J N [ T l P (cid:96) ψ ] · n u dµ Σ u . for m ∈ N , and for every u ≥ u we have that: (cid:90) N u r p ( L ( P (cid:96) T m Φ ( k ) )) dωdv ≤ C E aux − T,I (cid:96) =0 ,(cid:96),k,m u (cid:96) − k )+2 m − p − δ , (6.21) for p ∈ (0 , , any δ ∈ (0 , small enough, C . = C ( D, R ) , and we also have that for every u ≥ u : (cid:90) Σ u J N [ P (cid:96) T m ψ ] · n u dµ Σ u ≤ C E aux − T,I (cid:96) =0 ,(cid:96) u (cid:96) +2 m +5 − δ , (6.22) for any δ ∈ (0 , small enough, C . = C ( D, R, m, δ, (cid:96) ) . By writing P ≥ (cid:96) T m Φ ( (cid:96) ) as a sum of P (cid:96) T m Φ ( (cid:96) ) , P (cid:96) +1 T m Φ ( (cid:96) ) and P ≥ (cid:96) +2 T m Φ ( (cid:96) ) , and ap-plying the previous results to the first two terms, and working similarly for the last termusing the results of the previous section, we can obtain almost sharp energy decay estimatesfor P ≥ (cid:96) T m Φ ( (cid:96) ) . Corollary 6.10. Fix (cid:96) ∈ N and consider a smooth solution ψ to (1.1) satisfying (3.2) .Then we have that (cid:90) N u r p ( L ( P ≥ (cid:96) T m Φ ( (cid:96) ) )) dωdv ≤ C E aux − decomp − n − m u m − p − δ , (6.23) for all u ≥ u , for any δ > , where C = C ( D, R ) , and where E aux − decomp − n − m . = (cid:88) k ≤ (cid:96) (cid:16) (cid:90) N u r − δ ( L ( P (cid:96) T m + k Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P (cid:96) +1 T m + k Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P ≥ (cid:96) +2 T m + k Φ ( (cid:96) ) )) dωdv + (cid:88) | α |≤ m,j ≤ m + (cid:96) (cid:90) N u r − δ | / ∇ α S ( L ( P ≥ (cid:96) +2 T j Φ ( (cid:96) ) )) | dωdv (cid:17) (cid:88) l ≤ m + (cid:96) +1 (cid:90) Σ u J N [ P ≥ (cid:96) T l ψ ] · n u dµ Σ u . If we additionally assume that I (cid:96) [ ψ ] = 0 , then we have that (cid:90) N u r p ( L ( P ≥ (cid:96) T m Φ ( (cid:96) ) )) dωdv ≤ C E aux − decomp − − m u m − p − δ , (6.24) for all u ≥ u , for any δ > , where C = C ( D, R ) , and where E aux − decomp − − m . = (cid:88) k ≤ (cid:96) (cid:16) (cid:90) N u r − δ ( L ( P (cid:96) T m Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P (cid:96) +1 T m Φ ( (cid:96) ) )) dωdv + (cid:90) N u r ( L ( P ≥ (cid:96) +2 T m Φ ( (cid:96) ) )) dωdv + (cid:88) | α |≤ m,j ≤ m + (cid:96) (cid:90) N u r − δ | / ∇ α S ( L ( P ≥ (cid:96) +2 T j Φ ( (cid:96) ) )) | dωdv (cid:17) + (cid:88) l ≤ m + (cid:96) +1 (cid:90) Σ u J N [ P ≥ (cid:96) T l ψ ] · n u dµ Σ u . We note that the angular derivatives present in the norms of the aforementioned Corol-lary appear due to the use of equation (4.5) in Proposition 5.9. In this section we will present bound ∂ r derivatives of a linear wave supported on angularfrequencies ≥ (cid:96) for some (cid:96) ≥ 1, by T derivatives, via r − k -weighted energy estimates, wherethe range of k depends on (cid:96) . These estimates will allow us to obtain almost sharp upperbounds on fixed r hypersurfaces (something that will be done in the following sections).Our estimates have the form of a weighted hierarchy, where the weights depend on thelowest angular frequencies on which our linear wave is supported. Note that here we makethe choice D − h = cr η + O ( r − ) for some c (cid:54) = 0 and η > S hypersurfaces as noted before), for ∂ ρ = ∂ r + hT . Theorem 7.1. Assume that (cid:88) k ≤ lim ρ →∞ rT k ψ < ∞ , lim ρ →∞ r ∂ r ψ =0 . Let ψ be a solution to (1.1) and let (cid:96) ≥ . Then we can estimate (cid:90) ∞ r + (cid:90) S (cid:104) r − − k ( ∂ ρ ( Dr ∂ ρ ψ ≥ (cid:96) )) + Dr − k | / ∇ S ∂ ρ ψ ≥ (cid:96) | + r − − k ( / ∆ S ψ ≥ (cid:96) ) (cid:105) dωdρ ≤ C ( D, (cid:96), k ) (cid:90) ∞ r + (cid:90) S (cid:104) r − k ( ∂ ρ T ψ ≥ (cid:96) ) + r − k − η ( T ψ ≥ (cid:96) ) + r − k ( T ψ ≥ (cid:96) ) (cid:105) dωdρ, (7.1)34 or all − < k < (cid:96) − , and any η > .Proof. Recall that in ( v, r, θ, ϕ ) ingoing Eddington–Finkelstein coordinates, and with ∂ ρ = ∂ r + hT , we have that the wave equation (1.1) becomes: ∂ ρ ( Dr ∂ ρ ψ ) + / ∆ S ψ = r F T , (7.2)where F T := (2 + O ( r − − η )) ∂ ρ T ψ + O ( r − − η ) T ψ + (2 r − + O ( r − − η )) T ψ. Therefore, by squaring and integrating both sides of (7.2) and multiplying by r − − k , wearrive at (cid:90) ∞ r + (cid:90) S r − − k ( ∂ ρ ( Dr ∂ ρ ψ )) r + r − − k ( / ∆ S ψ ) + 2 r − − k ∂ ρ ( Dr ∂ ρ ψ ) / ∆ S ψ dωdρ ≤ C (cid:90) ∞ r + (cid:90) S r − k F T dωdρ. (7.3)We first consider the mixed derivative term on the left-hand side of (7.3). We integrateover S and integrate by parts in ρ and the angular variables. (cid:90) ∞ r + (cid:90) S r − − k ∂ ρ ( Dr ∂ ρ ψ ) / ∆ S ψ dωdρ = (cid:90) ∞ r + (cid:90) S k ) r − − k D∂ ρ ψ / ∆ S ψ − Dr − k ∂ ρ ψ / ∆ S ∂ ρ ψ dωdρ = (cid:90) ∞ r + (cid:90) S k ) r − − k D∂ ρ ψ / ∆ S ψ + 2 Dr − k | / ∇ S ∂ ρ ψ | dωdρ, (7.4)where we used that all resulting boundary terms vanish by D ( r + ) = 0 and the asymptoticson ψ as ρ → ∞ in the assumptions.We apply (7.2) again to estimate2(2 + k ) r − − k D∂ ρ ψ / ∆ S ψ = − k ) r − − k D∂ ρ ψ∂ ρ ( Dr ∂ ρ ψ )+ 2(2 + k ) r − − k D∂ ρ ψ · r F T = − (2 + k ) r − − k ∂ ρ (( Dr ∂ ρ ψ ) ) + 2(2 + k ) r − − k D∂ ρ ψ · r F T ≥ − (2 + k ) r − − k ∂ ρ (( Dr ∂ ρ ψ ) ) − (cid:15) (2 + k ) r − k D ( ∂ ρ ψ ) − (2 + k ) (cid:15) − r − − k r F T , where we will take (cid:15) > − (cid:90) ∞ r + (2 + k ) r − − k ∂ ρ (( Dr ∂ ρ ψ ) ) dρ = − (cid:90) ∞ r + (2 + k )(3 + k ) r − − k ( Dr ∂ ρ ψ ) dρ, where we used that the boundary term vanishes at r = r + and moreover lim ρ →∞ r − k D ( ∂ ρ ψ ) =0 if k > − 3. We furthermore apply (7.2) once more estimate to express the ( / ∆ S ψ ) termon the left-hand side of (7.3) in terms of ( ∂ ρ (( Dr ∂ ρ ψ )) : r − − k ( / ∆ S ψ ) ≥ (1 − (cid:15) ) r − − k ( ∂ ρ (( Dr ∂ ρ ψ )) + (cid:18) − (cid:15) (cid:19) r − k r F T , (cid:15) > (cid:90) ∞ r + (cid:90) S (2 − (cid:15) ) r − − k ( ∂ ρ ( Dr ∂ r ψ )) + 2 Dr − k | / ∇ S ∂ r ψ | − (2 + k )(3 + k + (cid:15) ) r − − k ( Dr ∂ ρ ψ ) dωdρ ≤ C (cid:90) ∞ r + (cid:90) S r − k F T dωdρ (7.5)We apply a Hardy inequality to absorb part of the third term on the left-hand side intothe first term (cid:90) ∞ r + 12 (1 − (cid:15) )(3 + k + (cid:15) ) r − − k ( Dr ∂ ρ ψ ) dρ ≤ − (cid:15) ) (cid:18) k + (cid:15) k (cid:19) (cid:90) ∞ r + r − − k ( ∂ ρ ( Dr ∂ ρ ψ )) dρ, where we used that lim ρ →∞ r − − k ( Dr ∂ ρ ψ ) = 0 if k > − (cid:15) i > (cid:20)(cid:18) k − 12 (1 − (cid:15) )(3 + k + (cid:15) ) (cid:19) (3 + k + (cid:15) ) (cid:21) r − − k ( Dr ∂ ρ ψ ) into the left-hand side of (7.5). At this point we localize in angular frequencies ≥ (cid:96) and wetherefore need (cid:90) ∞ r + (cid:90) S (cid:20)(cid:18) k − 12 (1 − (cid:15) )(3 + k + (cid:15) ) (cid:19) (3 + k + (cid:15) ) (cid:21) r − k D ( ∂ ρ ψ ≥ (cid:96) ) dωdρ ≤ (cid:96) ( (cid:96) + 1) (cid:90) ∞ r + (cid:90) S r − k D ( ∂ ρ ψ ≥ (cid:96) ) dωdρ ≤ (cid:90) ∞ r + (cid:90) S r − k D | / ∇ S ∂ ρ ψ ≥ (cid:96) | dωdρ, where we applied a Poincar´e inequality to ψ ≥ (cid:96) in the final inequality.We can absorb the very right-hand side of the above equation into the | / ∇ S ∂ r ψ ≥ (cid:96) | termon the left-hand side of (7.5), provided used that | D | ≤ k satisfies the inequality:(4 + 2 k − (1 − (cid:15) )(3 + k + (cid:15) ))(3 + k + (cid:15) ) ≤ (cid:96) ( (cid:96) + 1) , which we can rewrite as(1 + k + (cid:15) k + 3 (cid:15) − (cid:15) + (cid:15) (cid:15) )(3 + k + (cid:15) ) ≤ (cid:96) ( (cid:96) + 1) . (7.6)Equivalently, whenever (1 + k )(3 + k ) < (cid:96) ( (cid:96) + 1) . (7.7)36e can find (cid:15) , (cid:15) suitably small such that (7.6) is satisfied. The inequality (7.7) holds when − − (cid:96) < k < (cid:96) − 1. Then we use the above estimates to obtain: (cid:90) ∞ r min (cid:90) S r − − k ( ∂ ρ ( Dr ∂ ρ ψ )) + Dr − k | / ∇ S ∂ ρ ψ | dρ ≤ C ( (cid:96), k ) (cid:90) ∞ r min r − k F T dωdρ. (7.8)Hence, by applying Young’s inequality to the terms in F T , we are left with: (cid:90) ∞ r min (cid:90) S r − − k ( ∂ ρ ( Dr ∂ ρ ψ ≥ (cid:96) )) + Dr − k | / ∇ S ∂ ρ ψ ≥ (cid:96) | + r − − k ( / ∆ S ψ ≥ (cid:96) ) dωdρ ≤ C k,(cid:96) (cid:90) ∞ r min (cid:90) S r − k ( ∂ ρ T ψ ≥ (cid:96) ) + r − k ( T ψ ≥ (cid:96) ) + r − k − η ( T ψ ≥ (cid:96) ) dωdρ. We demonstrate now a localized version of the previous result, which can be seen as a“spacelike” redshift estimate. Proposition 7.2. Assume that (cid:88) k ≤ lim ρ →∞ rT k ψ < ∞ , lim ρ →∞ r ∂ r ψ =0 . Let ψ be a solution to (1.1) and let (cid:96) ≥ . Then we have that (cid:90) Rr + (cid:90) S ( ∂ ρ ψ ≥ (cid:96) ) dωdρ + (cid:90) S | / ∇ S ψ ≥ (cid:96) | dω (cid:12)(cid:12)(cid:12)(cid:12) ρ = r + ≤ C (cid:90) ∞ r + (cid:90) S (cid:104) r − k ( ∂ ρ T ψ ≥ (cid:96) ) + r − k − η ( T ψ ≥ (cid:96) ) + r − k ( T ψ ≥ (cid:96) ) (cid:105) dωdρ, for any − < k < (cid:96) − , any η > , and where C = C ( D, (cid:96), k ) .Proof. We multiply equation (7.2) with ∂ ρ ψ and after integrating it in { r + ≤ r ≤ R } weget that: (cid:90) Rr + (cid:90) S (cid:2) ( ∂ ρ ψ ≥ (cid:96) ) (cid:0) ∂ ρ ( Dr ∂ ρ ψ ≥ (cid:96) ) (cid:1) + ( ∂ ρ ψ ≥ (cid:96) )( / ∆ S ψ ≥ (cid:96) ) (cid:3) dωdρ = (cid:90) Rr + (cid:90) S ( Dr ) (cid:48) ( ∂ ρ ψ ≥ (cid:96) ) dωdρ + 12 (cid:90) S ( D ( R ) R )( ∂ ρ ψ ≥ (cid:96) ) dω (cid:12)(cid:12)(cid:12)(cid:12) ρ = R + 12 (cid:90) S | / ∇ S ψ ≥ (cid:96) | dω (cid:12)(cid:12)(cid:12)(cid:12) ρ = r + − (cid:90) S | / ∇ S ψ ≥ (cid:96) | dω (cid:12)(cid:12)(cid:12)(cid:12) ρ = R , which implies that (cid:90) Rr + (cid:90) S ( Dr ) (cid:48) ( ∂ ρ ψ ≥ (cid:96) ) dωdρ + 12 (cid:90) S ( D ( R ) R )( ∂ ρ ψ ≥ (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = R 37 12 (cid:90) S | / ∇ S ψ ≥ (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12) ρ = r + (cid:46) (cid:15) (cid:90) Rr + (cid:90) S | F T | + (cid:15) (cid:90) Rr + (cid:90) S ( ∂ ρ ψ ≥ (cid:96) ) + 12 (cid:90) S | / ∇ S ψ ≥ (cid:96) | (cid:12)(cid:12)(cid:12)(cid:12) ρ = R , and by absorbing the second term of the right hand side in the left hand side, and by usingestimate (7.1) of Theorem 7.1 for the last term (which we can use for any − < k < (cid:96) − Corollary 7.3. Assume that (cid:88) k ≤ lim ρ →∞ rT k ψ < ∞ , lim ρ →∞ r ∂ r ψ =0 . Let ψ be a solution to (1.1) and let (cid:96) ≥ . Then we have that (cid:90) Rr + (cid:90) S (cid:104) ( ∂ ρ ψ ≥ (cid:96) ) + | / ∇ S ( ∂ ρ ψ ≥ (cid:96) ) | (cid:105) dωdρ ≤ C (cid:90) Rr + (cid:90) S | ∂ ρ F T | dωdρ + C (cid:90) ∞ r + (cid:90) S (cid:104) r − k ( ∂ ρ T ψ ≥ (cid:96) ) + r − k − η ( T ψ ≥ (cid:96) ) + r − k ( T ψ ≥ (cid:96) ) (cid:105) dωdρ, for any − < k < (cid:96) − , any η > , and where C = C ( D, (cid:96), k ) .Proof. We consider the equation ∂ ρ (( Dr ) ∂ ρ ψ ≥ (cid:96) ) + ( Dr ) (cid:48) ∂ ρ ψ ≥ (cid:96) + 2 ∂ ρ ψ ≥ (cid:96) = ∂ ρ ¯ F T , (7.9)where¯ F T = (2 hDr − r ) ∂ ρ T ψ ≥ (cid:96) +(2 hr − h Dr +2 h Dr − hr ) T ψ ≥ (cid:96) +[( hDr ) (cid:48) − hDr +2 hDr − r ] T ψ ≥ (cid:96) . We multiply equation (7.9) by r∂ ρ ψ ≥ (cid:96) and we integrate it over { r + ≤ r ≤ R } and we havethat (cid:90) Rr + (cid:90) S (cid:104) r ( Dr ) (cid:48) ( ∂ ρ ψ ≥ (cid:96) ) − ( Dr )( ∂ ρ ψ ≥ (cid:96) ) (cid:105) dωdρ + (cid:90) Rr + (cid:90) S (cid:104) | / ∇ S ( ∂ ρ ψ ≥ (cid:96) ) | − ( ∂ ρ ψ ≥ (cid:96) ) (cid:105) dωdρ ≤ (cid:90) Rr + (cid:90) S | r ( ∂ ρ ψ ≥ (cid:96) )( ∂ ρ ¯ F T ) | dωdρ, and we get the required estimate by applying Cauchy-Schwarz on the term in the right-hand side, absorbing the term involving ∂ ρ ψ ≥ (cid:96) from the left-hand side (which has a termwith ∂ ρ ψ ≥ (cid:96) and no degeneracy in D by the choice of R as the second term of the left-handside above can be absorbed by the first), and by applying Theorem 7.1 to the term withthe minus sign of the left-hand side. 38ote that by the Corollary above and Theorem 7.1 we also get that (cid:90) Rr + (cid:90) S (cid:104) r − k ( ∂ ρ ψ ≥ (cid:96) ) + r − k | / ∇ S ( ∂ ρ ψ ≥ (cid:96) ) | (cid:105) dωdρ ≤ C (cid:90) Rr + (cid:90) S | ∂ ρ F T | dωdρ + C (cid:90) ∞ r + (cid:90) S (cid:104) r − k ( ∂ ρ T ψ ≥ (cid:96) ) + r − k − η ( T ψ ≥ (cid:96) ) + r − k ( T ψ ≥ (cid:96) ) (cid:105) dωdρ, for − < k < (cid:96) − 1, any η > 0, and where C = C ( D, (cid:96), k ) is a constant.Now we generalize Theorem 7.1 for higher ∂ ρ derivatives. Theorem 7.4. Fix some (cid:96) ≥ . Assume that (cid:88) l ≤ m +1 lim ρ →∞ rT l ψ < ∞ , lim ρ →∞ r l ∂ l +1 r ψ =0 for l ≤ m. Then for all − − m < k < (cid:96) − m − and ≤ m ≤ (cid:96), we have that (cid:90) ∞ r + (cid:90) S (cid:104) r − − k (cid:0) ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ ≥ (cid:96) (cid:1)(cid:1) + r − k ( ∂ m +1 ρ ψ ≥ (cid:96) ) (cid:105) dωdρ + (cid:90) ∞ r + (cid:90) S (cid:104) r − k D | / ∇ S ( ∂ m +1 ρ ψ ≥ (cid:96) ) | + r − − k ( / ∆ S ( ∂ mρ ψ ≥ (cid:96) )) (cid:105) dωdρ ≤ C m +1 (cid:88) s =1 (cid:90) ∞ r + (cid:90) S (cid:104) r − k − m − s +1) ( ∂ sρ T ψ ≥ (cid:96) ) + r − k − η − m − s +1) ( ∂ s − ρ T ψ ≥ (cid:96) ) (cid:105) dωdρ + (cid:90) ∞ r + r − k − m − s +1) ( ∂ s − ρ T ψ ≥ (cid:96) ) dωdρ, for some C = C ( D, m, k, (cid:96) ) and for any η > .Proof. We will argue by induction. The base case is m = 0 which was proven through acombination of Theorem 7.1 and Corollary 7.2. We assume that the desired estimate istrue for n = m − 1, and we also make the following induction hypothesis which due toCorollary 7.3 is true in the base case: we assume that for some small enough r c < ∞ that (cid:90) ∞ r + (cid:90) S (cid:104) r − k (cid:48) ( ∂ m +1 ρ ψ ≥ (cid:96) ) + r − k (cid:48) | / ∇ S ( ∂ mρ ψ ≥ (cid:96) ) | dωdρ (7.10) (cid:46) (cid:90) r c r + (cid:90) S | ∂ m +1 ρ F T | dωdρ (7.11)+ m (cid:88) s =1 (cid:90) ∞ r + (cid:90) S (cid:104) r − k (cid:48) − s − ( ∂ sρ T ψ ≥ (cid:96) ) + r − k (cid:48) − η − s − ( ∂ s − ρ T ψ ≥ (cid:96) ) + r − k (cid:48) − s − ( ∂ s − ρ T ψ ≥ (cid:96) ) (cid:105) dωdρ, (7.12)39or − − m − < k (cid:48) < (cid:96) − m − − 1. Now we examine the case n = m . We havethe equation ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ ≥ (cid:96) (cid:1) + m ( Dr ) (cid:48) ∂ m +1 ρ ψ ≥ (cid:96) + m ( m + 1) ∂ mρ ψ ≥ (cid:96) + / ∆ S ( ∂ mρ ψ ≥ (cid:96) ) = ∂ mρ ¯ F T , (7.13)with F T as given after equation (7.9). We rearrange it as ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ ≥ (cid:96) (cid:1) + / ∆ S ( ∂ mρ ψ ≥ (cid:96) ) = ∂ mρ ¯ F T − m ( Dr ) (cid:48) ∂ m +1 ρ ψ ≥ (cid:96) − m ( m + 1) ∂ mρ ψ ≥ (cid:96) , and after multiplying it by r − − k and integrating in ρ we have that: (cid:90) ∞ r + (cid:90) S r − − k (cid:104)(cid:0) ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ ≥ (cid:96) (cid:1)(cid:1) + ( / ∆ S ( ∂ mρ ψ ≥ (cid:96) )) (cid:105) dωdρ + 2 (cid:90) ∞ r + (cid:90) S r − − k ( ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ ≥ (cid:96) (cid:1) ) · ( / ∆ S ( ∂ mρ ψ ≥ (cid:96) )) dωdρ ≤ (cid:90) ∞ r + (cid:90) S r − − k ( ∂ mρ ¯ F T ) dωdρ + 2 (cid:90) ∞ r + (cid:90) S r − − k (cid:104) m [( Dr ) (cid:48) ] ( ∂ m +1 ρ ψ ≥ (cid:96) ) + m ( m + 1) ( ∂ mρ ψ ≥ (cid:96) ) (cid:105) dωdρ. For the cross term we have that2 (cid:90) ∞ r + (cid:90) S r − − k ( ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ (cid:1) ) · ( / ∆ S ( ∂ mρ ψ )) (cid:105) dωdρ = (cid:90) ∞ r + (cid:90) S r − − k Dr ) | / ∇ S ( ∂ m +1 ρ ψ ) | dωdρ + (cid:90) ∞ r + (cid:90) S r − − k (cid:104) k ) D (cid:48) r − k )(1 + k ) D (cid:105) | / ∇ S ( ∂ mρ ψ ) | dωdρ. Using the last two computations we now have that (cid:90) ∞ r + (cid:90) S r − − k (cid:104)(cid:0) ∂ ρ (cid:0) ( Dr ) ∂ m +1 ρ ψ ≥ (cid:96) (cid:1)(cid:1) + Dr − k | / ∇ S ( ∂ m +1 ρ ψ ≥ (cid:96) ) | + ( / ∆ S ( ∂ mρ ψ ≥ (cid:96) )) (cid:105) dωdρ (cid:46) (cid:90) ∞ r + (cid:90) S r − − k ( ∂ mρ ¯ F T ) dωdρ + 2 (cid:90) ∞ r + (cid:90) S r − − k (cid:104) m [( Dr ) (cid:48) ] ( ∂ m +1 ρ ψ ≥ (cid:96) ) + m ( m + 1) ( ∂ mρ ψ ≥ (cid:96) ) (cid:105) dωdρ + (cid:90) ∞ r + (cid:90) S r − − k | / ∇ S ( ∂ mρ ψ ≥ (cid:96) ) | dωdρ. For the last three terms of the right-hand side we now use the induction hypothesis, andspecifically estimate (7.10). Note that we can do this because now as − − m < k < (cid:96) − m − − − m − < k + 2 < (cid:96) − m − − 1. This finishes theproof of the desired estimate. Additionally to the elliptic estimates we will also need some higher order red-shift estimates.Recall that N := T − ∂ r for r ∈ [ r + , r ], N := T for r ≥ r ,for r + ≤ r < r , in ingoing ( v, r, θ, ϕ ) coordinates.40 emma 8.1. Fix some (cid:96) ≥ , and assume that ψ solves (1.1) . Then for some r , r suchthat r + < r < r and r − r being small enough, ≤ m ≤ (cid:96) and any τ ≤ τ ≤ τ , wehave that: (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) S τ J N [ N m +1 T ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r ≤ r } J T [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ + C (cid:88) k + j = m +1 ,k ≤ m (cid:90) τ τ (cid:90) S τ J N [ N k T j ψ ≥ (cid:96) ] · n τ dµ S τ dτ + C (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r } J T [ N m ψ ≥ (cid:96) ] · n τ dµ S τ dτ, (8.1) for some C = C ( D, h ) > .Proof. From the proof of the above Lemma in the case m = 0 which was established in[DR13] we have the following estimate: (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) S τ J N [ N m +1 T ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r ≤ r } J T [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ + C (cid:90) τ τ (cid:90) S τ J T [ N m T ψ ≥ (cid:96) ] · n τ dµ S τ dτ + C (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r } J T [ N m ψ ≥ (cid:96) ] · n τ dµ S τ dτ + C (cid:90) τ τ (cid:90) S τ | G m,(cid:96) | dµ S τ dτ, (8.2)for some constant C , where G m,(cid:96) is defined as (cid:3) g ( N m ψ (cid:96) ) = κ m ∂ m +1 r ψ (cid:96) + G m,(cid:96) , where G m,(cid:96) = m − (cid:88) k =0 O ( r − − k )( ∂ m − − kr T ψ ≥ (cid:96) ) + m − (cid:88) k =0 O ( r − − k )( ∂ m − − kr N ψ ≥ (cid:96) ) . Note that the previous estimate holds true as κ m > G m,(cid:96) we get that: (cid:90) τ τ (cid:90) S τ | G m,(cid:96) | dµ S τ dτ ≤ C (cid:88) k + j = m +1 ,k ≤ m (cid:90) τ τ (cid:90) S τ J N [ N k T j ψ ≥ (cid:96) ] · n τ dµ S τ dτ, which finishes the proof of the desired estimate.Using the previous Lemma we can now show decay for the following energies:41 roposition 8.2. Fix some (cid:96) ≥ and assume that ψ solves (1.1) . Assume that I (cid:96) [ ψ ] (cid:54) = 0 , and that (cid:88) j + j ≤ m,j ≤ K +1 (cid:90) Σ τ J N [ N j +1 T j + j ψ ≥ (cid:96) ] · n τ dµ Σ τ < ∞ , (8.3) for some m such that ≤ m ≤ (cid:96) , and for some K ∈ N .Then for any u ≥ u we have that (cid:90) S τ J N [ N m +1 T K ψ ≥ (cid:96) ] · n u dµ S τ ≤ C E aux − decomp − n − K τ m +1)+2 K − δ , (8.4) for any δ > , some C = C ( D, R ) , and for E aux − decomp − n − K as in Corollary 6.10.If instead we assume that I (cid:96) [ ψ ] = 0 , and maintain the assumption (8.3) , we have for any u ≥ u that (cid:90) S u J N [ N m +1 T K ψ ≥ (cid:96) ] · n u dµ Σ τ ≤ C E aux − decomp − − K u m +1)+2 K − δ , (8.5) for any δ > , some C > , and for E aux − decomp − − K as in Corollary 6.10.Proof. First we assume that I (cid:96) (cid:54) = 0. We use estimate (8.1) and we examine one by one theterms in the right-hand side. For the first one we have for any τ < τ and r , r as inLemma 8.1 that: (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r ≤ r } J T [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) τ τ (cid:90) S τ J N [ N m T ψ ≥ (cid:96) ] · n τ dµ S τ dτ + (cid:90) τ τ (cid:90) S τ ( ∂ mρ T ψ ≥ (cid:96) ) dµ S τ dτ, using Theorem 7.4. We can also improve the above estimate by applying Hardy’s inequality(A.2) to the second term of the last estimate hence arriving at: (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r ≤ r } J T [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) τ τ (cid:90) S τ J N [ N m T ψ ≥ (cid:96) ] · n τ dµ S τ dτ. Moreover using Hardy’s inequality (A.2) and Theorem 7.4 we get that (cid:90) τ τ (cid:90) S τ ∩{ r ≤ r } J T [ N m ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) τ τ (cid:90) S τ J N [ N m T ψ ≥ (cid:96) ] · n τ dµ S τ dτ. We have the following estimate in the end: (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N m +1 ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) S τ J N [ N m +1 T ψ ≥ (cid:96) ] · n τ dµ S τ C (cid:88) k + j = m +1 ,k ≤ m (cid:90) τ τ (cid:90) S τ J N [ N k T j ψ ≥ (cid:96) ] · n τ dµ S τ dτ. We will use the last inequality to obtain the desired estimate in an inductive manner. Westart with the case m = 0 where we note that the previous estimate takes the followingform: (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) S τ J N [ N T ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) τ τ (cid:90) S τ J N [ T ψ ≥ (cid:96) ] · n τ dµ S τ dτ, and using the decay estimates from Proposition 6.9 and Corollary 6.6 we get that: (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) S τ J N [ N T ψ ≥ (cid:96) ] · n τ dµ S τ + C E aux − T,I (cid:96) (cid:54) =0 , u − (cid:15) , for any (cid:15) > 0. The desired estimate now follows from the Gr¨onwall-type estimate (A.6).For m = 1 we follow the same process and we have that: (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) S τ J N [ N T ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:88) k + j =2 ,k ≤ (cid:90) τ τ (cid:90) S τ J N [ N k T j ψ ≥ (cid:96) ] · n τ dµ S τ dτ, and by the decay estimates from Proposition 6.9 and Corollary 6.10 and the estimate wejust proved for m = 0 we have that: (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + (cid:90) τ τ (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ dτ ≤ C (cid:90) S τ J N [ N ψ ≥ (cid:96) ] · n τ dµ S τ + C (cid:90) S τ J N [ N T ψ ≥ (cid:96) ] · n τ dµ S τ + C E aux − decomp − n − + E aux − decomp − n − u − (cid:15) , for any (cid:15) > 0. The desired estimate now follows again from the Gr¨onwall-type estimate(A.6). We work in the same way for all other m ≤ (cid:96) .For I (cid:96) = 0 the proof is identical, we just use the relevant decay energy estimates fromProposition 6.9, Corollaries 6.6 and 6.10. 43 Pointwise decay estimates Φ ( (cid:96) ) In this Section we demonstrate how to obtain almost sharp decay estimates for the radiationfields P ≥ (cid:96) Φ ( k ) , k ∈ { , . . . , (cid:96) } .The method of proof is quite standard (and has been used in the Lemma 6.1) so we onlygive an outline. For a function f and a cut-off χ as in 5.1 ( χ ( r ) = 0 for r ≤ R , χ ( r ) = 1for r ≥ R + 1, and χ is smooth) we have that: (cid:90) S ( χf ) ( u, v ) dω =2 (cid:90) vv R ( χf ) · ( L ( χf )) dωdv (cid:48) ≤ (cid:18)(cid:90) N u r ( χf ) dωdv (cid:19) / (cid:18)(cid:90) N u r ( L ( χf )) dωdv (cid:19) / ≤ (cid:18)(cid:90) N u ( L ( χf )) dωdv (cid:19) / (cid:18)(cid:90) N u r ( L ( χf )) dωdv (cid:19) / . We will fix an (cid:96) ≥ 1, and we will apply the above computation for f = P ≥ (cid:96) Φ ( k ) with k ∈ { , . . . , (cid:96) } , and we will obtain decay for the radiation fields by using the energy decayestimates of Section 6 (specifically the decay of the energy flux, and the decay for therelevant r p -weighted energy for p = 2). Theorem 9.1. Let ψ be a smooth solution of (7.3) satisfying (3.2) and fix some (cid:96) ≥ .Then for k ∈ { , . . . , (cid:96) } we have in the region { r ≥ R } for all u ≥ u that: (cid:90) S ( P ≥ (cid:96) Φ ( k ) ) dω ≤ C E aux − decomp − n − u (cid:96) − k ) − δ , (9.1) for any δ > , for C = C ( D, R ) and where the quantity E aux − decomp − n − is given inCorollary 6.10.If instead we assume that I (cid:96) [ ψ ] = 0 , then for k ∈ { , . . . , (cid:96) } we have in the region { r ≥ R } for all u ≥ u that: (cid:90) S ( P ≥ (cid:96) Φ ( k ) ) dω ≤ C E aux − decomp − − u (cid:96) − k ) − δ , (9.2) for any δ > , for C = C ( D, R ) and where the quantity E aux − decomp − − is given inCorollary 6.10. We can also apply the same method to P ≥ (cid:96) T m Φ ( k ) for any m ∈ N and k ∈ { , . . . , (cid:96) } and in this case we get the following result. Theorem 9.2. Let ψ be a smooth solution of (7.3) satisfying (3.2) , fix some (cid:96) ≥ , andfix some m ∈ N . Then for k ∈ { , . . . , (cid:96) } we have in the region { r ≥ R } for all u ≥ u that: (cid:90) S ( P ≥ (cid:96) T m Φ ( k ) ) dω ≤ C E aux − decomp − n − m u m +2( (cid:96) − k ) − δ , (9.3) for any δ > , for C = C ( D, R ) and where the quantity E aux − decomp − n − m is given inCorollary 6.10. f instead we assume that I (cid:96) [ ψ ] = 0 , then for k ∈ { , . . . , (cid:96) } we have in the region { r ≥ R } for all u ≥ u that: (cid:90) S ( P ≥ (cid:96) T m Φ ( k ) ) dω ≤ C E aux − decomp − − m u m +2( (cid:96) − k ) − δ , (9.4) for any δ > , for C = C ( D, R ) and where the quantity E aux − decomp − − m is given inCorollary 6.10. P ≥ (cid:96) ψ and its derivatives In this section we will use the elliptic estimates and the energy decay estimates from Section6 to derive almost sharp decay estimates for a frequency localized linear wave P ≥ (cid:96) ψ andits ∂ r derivatives. Theorem 9.3. Fix some (cid:96) ≥ and let ψ be a smooth solution of (1.1) satisfying (3.2) ,and assume that I (cid:96) [ ψ ] (cid:54) = 0 . Then for ≤ k ≤ (cid:96) we have that (cid:90) S r − (cid:96) + η +2 k ( ∂ kr P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − n − k + (cid:96) +1 u (cid:96) +4 − η − δ , (9.5) and (cid:90) S r − (cid:96) + η +2 k +1 ( ∂ kr P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − n − k + (cid:96) +1 u (cid:96) +3 − η − δ , (9.6) where C = C ( D, R, k, δ, (cid:96) ) , and where E aux − decomp − n − k + (cid:96) +1 is given in Corollary 6.10.If instead we assume that I (cid:96) [ ψ ] = 0 , then for ≤ k ≤ (cid:96) we have that (cid:90) S r − (cid:96) + η +2 k ( ∂ kr P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − − k u (cid:96) +6 − η − δ , (9.7) and (cid:90) S r − (cid:96) + η +2 k +1 ( ∂ kr P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − − k u (cid:96) +5 − η − δ , (9.8) where C = C ( D, R, k, δ, (cid:96) ) , and where E aux − decomp − − k + (cid:96) +1 is given in Corollary 6.10.Proof. We will demonstrate the cases k = 0 and k = (cid:96) , the proof of the remaining cases isdone in the same way. The case k = 0 : By the fundamental theorem of calculus we have for any r ≥ r + andsome η > (cid:90) S r − (cid:96) + η ( P ≥ (cid:96) ψ ) dω = − (cid:90) ∞ r (cid:90) S ∂ ρ (cid:16) r − (cid:96) + η ( P ≥ (cid:96) ψ ) (cid:17) dωdρ =(2 − η ) (cid:90) ∞ r (cid:90) S r − (cid:96) − η ( P ≥ (cid:96) ψ ) dωdρ − (cid:90) ∞ r (cid:90) S r − (cid:96) + η P ≥ (cid:96) ψ · ( ∂ ρ P ≥ (cid:96) ψ ) dωdρ (2 − η ) (cid:90) ∞ r (cid:90) S r − (cid:96) − η ( P ≥ (cid:96) ψ ) dωdρ + (cid:18)(cid:90) ∞ r (cid:90) S r − (cid:96) − η ( P ≥ (cid:96) ψ ) dωdρ (cid:19) / (cid:18)(cid:90) ∞ r (cid:90) S r − (cid:96) +1+ η ( ∂ ρ P ≥ (cid:96) ψ ) dωdρ (cid:19) / . = I + II. We are going to use now the elliptic estimates (7.1) of Theorem 7.1. For the term I weapply estimate (7.1) for k = 2 (cid:96) − − η using the lowest order term and we have that I =(2 − η ) (cid:90) ∞ r (cid:90) S r − (cid:96) − η ( P ≥ (cid:96) ψ ) dωdρ ≤ C (cid:90) ∞ r + (cid:90) S (cid:104) r − (cid:96) +3+ η ( ∂ ρ T P ≥ (cid:96) ψ ) + r − (cid:96) +1+ η − η (cid:48) ( T P ≥ (cid:96) ψ ) + r − (cid:96) +1+ η ( T P ≥ (cid:96) ψ ) (cid:105) dωdρ. We repeat the same process (cid:96) times by using again the elliptic estimates of Lemma 7.1always using the two lowest order terms without the D weight, but for T k P ≥ (cid:96) ψ , k ∈{ , . . . , (cid:96) } in the place of P ≥ (cid:96) ψ (in the last estimate we use the estimate for the lowestorder term of Theorem 7.1 for T ψ ≥ (cid:96) and T ψ ≥ (cid:96) in the place of ψ ≥ (cid:96) ). In the end we get forsome C = C ( D, R, (cid:96) ) that I ≤ C (cid:90) ∞ r (cid:90) S r η ( ∂ ρ T (cid:96) +1 P ≥ (cid:96) ψ ) dωdρ. The same process gives us for some C = C ( D, R, (cid:96) ) that II ≤ C (cid:90) ∞ r (cid:90) S r η ( ∂ ρ T (cid:96) +1 P ≥ (cid:96) ψ ) dωdρ. The two previous estimates give us now that: (cid:90) S r − (cid:96) + η ( P ≥ (cid:96) ψ ) dω ≤ C (cid:90) ∞ r (cid:90) S r η ( ∂ ρ T (cid:96) +1 P ≥ (cid:96) ψ ) dωdρ ≤ (cid:90) S τ J N [ T (cid:96) +1 P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:90) N τ r η ( L ( P ≥ (cid:96) T (cid:96) +1 φ )) dωdv, and the decay follows from the results of Corollary 6.10. The case k = (cid:96) : By the fundamental theorem of calculus we have for any r ≥ r + that: (cid:90) S r η ( ∂ (cid:96)ρ P ≥ (cid:96) ψ ) dω = − (cid:90) ∞ r (cid:90) S ∂ ρ (cid:16) r η ( ∂ (cid:96)ρ P ≥ (cid:96) ψ ) (cid:17) dωdρ = − η (cid:90) ∞ r (cid:90) S r η − ( ∂ (cid:96)ρ P ≥ (cid:96) ψ ) dωdρ − (cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96)ρ P ≥ (cid:96) ψ ) · ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdρ ≤ − (cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96)ρ P ≥ (cid:96) ψ ) · ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdρ ≤ (cid:18)(cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdρ (cid:19) / · (cid:18)(cid:90) ∞ r (cid:90) S r − η ( ∂ (cid:96)ρ P ≥ (cid:96) ψ ) dωdρ (cid:19) / ≤ C (cid:18)(cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdρ (cid:19) / (cid:32) (cid:96) +1 (cid:88) s =1 (cid:104) (cid:90) ∞ r (cid:90) S r η − s − (cid:96) +1) ( ∂ sρ T P ≥ (cid:96) ψ ) + r η − η (cid:48) − (cid:96) − s +1) ( ∂ s − ρ T P ≥ (cid:96) ψ ) dωdρ + (cid:90) ∞ r (cid:90) S r η − (cid:96) − s +1) ( ∂ s − ρ T P ≥ (cid:96) ψ ) dωdρ (cid:105)(cid:33) / , where in the last inequality we used the higher order elliptic estimates of Theorem 7.4 with m = (cid:96) and k = − − η . Let us keep only the highest order term (in ∂ ρ derivatives) fromthe very last term as the rest can be turned into it by repeating the same process. We areleft with the following: (cid:32) (cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdr (cid:33) / · (cid:32) (cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96) +1 ρ T P ≥ (cid:96) ψ ) dωdr (cid:33) / ≤ (cid:32) (cid:90) S τ J N [ N (cid:96) P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:96) (cid:88) s =1 (cid:104) (cid:90) ∞ r (cid:90) S r η − (cid:96) − s ) ( ∂ sρ T P ≥ (cid:96) ψ ) dωdρ + (cid:90) ∞ r (cid:90) S ( r − η − η (cid:48) − (cid:96) − s ) ( ∂ s − ρ T P ≥ (cid:96) ψ ) + r − η − (cid:96) − s ) ( ∂ s − ρ T P ≥ (cid:96) ψ ) ) dωdρ (cid:105)(cid:33) / × (cid:32) (cid:90) S τ J N [ N (cid:96) T P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:96) (cid:88) s =1 (cid:104) (cid:90) ∞ r r η − (cid:96) − s ) ( ∂ sρ T P ≥ (cid:96) ψ ) dωdρ + (cid:90) ∞ r (cid:90) S ( r η − η (cid:48) − (cid:96) − s ) ( ∂ s − ρ T P ≥ (cid:96) ψ ) + r η − (cid:96) − s ) ( ∂ s − ρ T P ≥ (cid:96) ψ ) ) dωdρ (cid:105)(cid:33) / , where we broke the integrals into parts close and away from the horizon, and close to thehorizon we just bound them by the aforementioned N energies, and away we use again theresults of Theorem 7.4 for m = (cid:96) − k = 1 − η and k = − − η . Repeating the sameprocess (and applying Hardy’s inequality (A.1) once) we are left with the following: (cid:32) (cid:88) m + m = (cid:96) (cid:90) S τ J N [ N m T m P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:90) ∞ r (cid:90) S r η ( ∂ ρ T (cid:96) P ≥ (cid:96) ψ ) dωdρ (cid:33) / × (cid:32) (cid:88) m + m = (cid:96) (cid:90) S τ J N [ N m T m +1 P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:90) ∞ r (cid:90) S r η ( ∂ ρ T (cid:96) +1 P ≥ (cid:96) ψ ) dωdρ (cid:33) / ≤ (cid:32) (cid:88) m + m = (cid:96) (cid:90) S τ J N [ N m T m P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:90) N τ r η ( L ( T (cid:96) +1 P ≥ (cid:96) φ )) dωdv (cid:33) / × (cid:32) (cid:88) m + m = (cid:96) (cid:90) S τ J N [ N m T m +1 P ≥ (cid:96) ψ ] · n τ dµ S τ + (cid:90) N τ r η ( L ( T (cid:96) +1 P ≥ (cid:96) φ )) dωdv (cid:33) / , after applying again Theorem 7.1, and now the result follows from Corollary 6.10 andProposition 8.2.The same proof as the one of the aforementioned theorem, gives us the following aswell: 47 heorem 9.4. Fix some (cid:96) ≥ and let ψ be a smooth solution of (1.1) satisfying (3.2) ,and assume that I (cid:96) [ ψ ] (cid:54) = 0 . Then for ≤ k ≤ (cid:96) we have that (cid:90) S r − (cid:96) + η +2 k ( ∂ kr T P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − n − k + (cid:96) +2 u (cid:96) +6 − η − δ , (9.9) and (cid:90) S r − (cid:96) + η +2 k +1 ( ∂ kr T P ≥ (cid:96) ψ (cid:96) ) dω ≤ C E aux − decomp − n − k + (cid:96) +2 u (cid:96) +5 − η − δ , (9.10) where C = C ( D, R, k, δ, (cid:96) ) , and E aux − decomp − n − k + (cid:96) +2 is given in Corollary 6.10. Finally we record one more estimate (that is not almost sharp but one power off) thatwill be used later. Theorem 9.5. Fix some (cid:96) ≥ and let ψ be a smooth solution of (1.1) satisfying (3.2) ,and assume that I (cid:96) [ ψ ] (cid:54) = 0 . Then we have that (cid:90) S r η ( ∂ (cid:96) +1 r P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − n − (cid:96) u (cid:96) +4 − η − δ , (9.11) and (cid:90) S r η ( ∂ (cid:96) +1 r T P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − n − (cid:96) u (cid:96) +6 − η − δ , (9.12) while if we assume that I (cid:96) [ ψ ] = 0 we have that (cid:90) S r η ( ∂ (cid:96) +1 r P ≥ (cid:96) ψ ) dω ≤ C E aux − decomp − − (cid:96) u (cid:96) +6 − η − δ , (9.13) where E aux − decomp − n − (cid:96) and E aux − decomp − − (cid:96) are given in Corollary 6.10, and where C = C ( D, R, δ, (cid:96) ) .Proof. We will only show the case of P ≥ (cid:96) ψ as the case of T P ≥ (cid:96) ψ is similar.Using once again the fundamental theorem of calculus we have that: (cid:90) S r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dω = − (cid:90) ∞ r (cid:90) S ∂ ρ (cid:16) r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) (cid:17) dωdρ = − η (cid:90) ∞ r (cid:90) S r η − ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdρ − (cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) · ( ∂ (cid:96) +2 ρ P ≥ (cid:96) ψ ) dωdρ ≤ − (cid:90) ∞ r (cid:90) S r η ( ∂ (cid:96)ρ P ≥ (cid:96) +1 ψ ) · ( ∂ (cid:96) +2 ρ P ≥ (cid:96) ψ ) dωdρ ≤ (cid:18)(cid:90) ∞ r (cid:90) S r − η (( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dωdρ (cid:19) / · (cid:18)(cid:90) ∞ r (cid:90) S r η (( ∂ (cid:96) +2 ρ P ≥ (cid:96) ψ ) dωdρ (cid:19) / , m = (cid:96) − m = (cid:96) using the highest order term (after separating the integrals in regionsclose and away from the horizon), and in the end we get that (cid:90) S r η ( ∂ (cid:96) +1 ρ P ≥ (cid:96) ψ ) dω ≤ C (cid:88) m + m = (cid:96) (cid:90) S τ J N [ N m T m P ≥ (cid:96) ψ ] · n τ dµ S τ / × (cid:88) m + m = (cid:96) +1 (cid:90) S τ J N [ N m T m P ≥ (cid:96) ψ ] · n τ dµ S τ / , and the result follows from the decay estimates of Proposition 8.2. 10 Precise late-time asymptotics when I (cid:96) (cid:54) = 0 In this section we will derive the precise late-time asymptotics of a frequency localizedsolution P (cid:96) ψ of (1.1), for some (cid:96) ≥ 1. In order to achieve this we will make use of the (cid:96) -th Newman–Penrose charge and the almost sharp upper bounds obtained in the previoussections.Note that in this section we will always work with a solution ψ of (1.1) that is localizedat angular frequency (cid:96) ≥ 1, so by ψ we always mean P (cid:96) ψ . Recall the definition of Φ ( k ) given in Proposition 4.1. The (cid:96) -th Newman–Penrose chargefor (cid:96) ≥ 1, as we have seen before, is given by I (cid:96) [ ψ ]( θ, ϕ ) = lim r →∞ Φ ( (cid:96) +1) ( u, r, θ, ϕ ) , for ψ = P (cid:96) ψ a frequency localized solution of (1.1). Note that in this Section we will alwaysassume that our solution of (1.1) is localized at frequency (cid:96) for some (cid:96) ≥ (cid:96) -th Newman-Penrose charge and the decay estimatesfrom the previous sections to derive first asymptotics for L Φ ( (cid:96) ) , then for Φ ( k ) , k ∈ { , . . . , (cid:96) − } , and finally for φ (where φ = rψ ) in the following region close to I + : B α (cid:96) = { r ≥ R } ∩ { ( u, v ) : u ≤ u ≤ v − v α (cid:96) } , for some large R , and for (cid:96) +22 (cid:96) +3 < α (cid:96) < 1. Note that the boundary of B α (cid:96) is the curve γ α (cid:96) = { ( u, v ) : v − u = v α (cid:96) } on which we have in particular that v ∼ u . Recall also that v − u ∼ r for r ≥ R .We need some auxiliary lemmas. The first one is a direct consequence of Corollary 4.2. Lemma 10.1. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) . Wehave that: L (cid:18) r (cid:96) L Φ ( (cid:96) ) (cid:19) = O ( r − − (cid:96) )( L Φ ( (cid:96) ) ) + (cid:96) (cid:88) k =0 O ( r − − (cid:96) )Φ ( k ) . (10.1)49he second one has to do with a preliminary upper bound on the (cid:96) -th Newman–Penrosecharge in the region B α (cid:96) . Lemma 10.2. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) . Assumethat: r ∂ r Φ ( (cid:96) ) (cid:12)(cid:12) u = u = I (cid:96) [ ψ ]( θ, ϕ ) + O ( v − (cid:15) ) , (10.2) for some (cid:15) > , and I (cid:96) [ ψ ] (cid:54) = 0 . Then for all ( u, v ) ∈ B α (cid:96) we have that: | v L Φ ( (cid:96) ) ( u, v ) | ≤ CE aux,I (cid:96) (cid:16) O ( v − β ) (cid:17) , (10.3) for some β > and for a weighted initial energy E aux,I (cid:96) that depends on the sum of initialenergies given in Lemma 6.3 and I (cid:96) [ ψ ] .Proof. We integrate in u the quantity L ( v L Φ ( (cid:96) ) ) for ( u, v ) ∈ B α (cid:96) and we have that: v L Φ ( (cid:96) ) ( u, v ) = e − (cid:82) uu ( − (cid:96)r − + O ( r − )) du (cid:48) v L Φ ( (cid:96) ) ( u , v )+ e − (cid:82) uu ( − (cid:96)r − + O ( r − )) du (cid:48) (cid:96) (cid:88) k =0 (cid:90) uu e (cid:82) u (cid:48) u ( − (cid:96)r − + O ( r − )) du (cid:48)(cid:48) v O ( r − )Φ ( k ) du (cid:48) . Note that e − (cid:82) uu ( − (cid:96)r − + O ( r − )) du (cid:48) = e O ( r − ( u,v ))+ O ( r − ( u ,v )) r (cid:96) ( u, v ) r (cid:96) ( u , v ) , and for ( u, v ) ∈ B α (cid:96) : v L Φ ( (cid:96) ) ( u , v ) = v r ( u, v ) ( r ∂ r Φ ( (cid:96) ) )( u , v ) − M v r ( u, v ) ( r ∂ r Φ ( (cid:96) ) )( u , v ) + e v r ( u, v ) ( r ∂ r Φ ( (cid:96) ) )( u , v )=4 I (cid:96) [ ψ ] + O ( v − β ) , for some β > 0, as L = D∂ r and v = u + 2 r . Then we have that: | v L Φ ( (cid:96) ) | ( u, v ) ≤ I (cid:96) [ ψ ] + O ( v − β )+ C (cid:96) (cid:88) k =0 (cid:90) uu v r | Φ ( k ) | du (cid:48) ≤ I (cid:96) [ ψ ] + O ( v − β )+ Cv − β (cid:96) (cid:88) k =0 (cid:90) uu v β r E aux,I (cid:96) ( u (cid:48) ) (cid:96) − k ) − δ du (cid:48) ≤ I (cid:96) [ ψ ] + O ( v − β )+ Cv − β (cid:96) (cid:88) k =0 (cid:90) uu E aux,I (cid:96) r − βα(cid:96) ( u (cid:48) ) (cid:96) − k ) − δ du (cid:48) ≤ I (cid:96) [ ψ ] + O ( v − β ) + CE aux,I (cid:96) v − β , for C a constant depending on u and (cid:96) . Note that we used that v α (cid:96) (cid:46) r and u α (cid:96) (cid:46) r in B α (cid:96) and estimate (9.1). 50sing the previous two lemmas we get the following result: Proposition 10.3. Fix some (cid:96) ≥ , consider a solution ψ of the wave equation (1.1) thatis localized at frequency (cid:96) , and also assume that (10.2) holds true for some (cid:15) > and I (cid:96) [ ψ ] (cid:54) = 0 . Then there exists η > small enough such that for all ( u, v ) ∈ B α (cid:96) we have that: L Φ ( (cid:96) ) ( u, v, θ, ϕ ) = 2 (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ ) ( v − u ) (cid:96) v (cid:96) +2 + O ( v − η ) ( v − u ) (cid:96) ( v − u ) (cid:96) +2 . (10.4) Proof. For ( u, v ) ∈ B α (cid:96) by integrating in u equation (10.1) we get that: v (cid:96) +2 r − (cid:96) L Φ ( (cid:96) ) ( u, v ) = v (cid:96) +2 r − (cid:96) ( u , v ) L Φ ( (cid:96) ) ( u , v ) + (cid:90) uu v (cid:96) O ( r − (cid:96) − )( v L Φ ( (cid:96) ) )( u (cid:48) , v ) du (cid:48) + (cid:96) (cid:88) k =0 (cid:90) uu v (cid:96) +2 O ( r − (cid:96) − )Φ ( k ) ( u (cid:48) , v ) du (cid:48) . By 10.2 we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) uu v (cid:96) O ( r − (cid:96) − )( v L Φ ( (cid:96) ) )( u (cid:48) , v ) du (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ≤ CE aux,I (cid:96) v − η (cid:90) uu r − (cid:96) − (cid:96) + ηα(cid:96) du (cid:48) ≤ CE aux,I (cid:96) v − η r γ α ( v ) − (cid:96) − (cid:96) + ηα(cid:96) = O ( v − η ) , if α (cid:96) > (cid:96) + η (cid:96) +1 for some η > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:96) (cid:88) k =0 (cid:90) uu v (cid:96) +2 O ( r − (cid:96) − ) (cid:101) Φ ( k ) ( u (cid:48) , v ) du (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CE aux,I (cid:96) v − η (cid:96) (cid:88) k =0 (cid:90) uu r − (cid:96) − η/α (cid:96) ( u (cid:48) ) (cid:96) − k ) − δ du (cid:48) ≤ Cv − η (cid:90) uu r − (cid:96) − (cid:96) +2+ η +2 (cid:15)α(cid:96) ( u (cid:48) ) − − (cid:15) du (cid:48) ≤ Cv − η r γ α ( v ) − (cid:96) − (cid:96) +2+ η +2 (cid:15)α(cid:96) = O ( v − η ) , if α (cid:96) > (cid:96) +2+ η +2 (cid:15) (cid:96) +3 , for some (cid:15) > η > 0. Note that (cid:96) +2+ η +2 (cid:15) (cid:96) +3 > (cid:96) + η (cid:96) +1 and that we usedthe decay estimates (9.1).Using that L = D∂ r and v = u + 2 r , we moreover have that v (cid:96) +2 r − (cid:96) ( u , v ) L Φ ( (cid:96) ) ( u , v ) = 12 v (cid:96) +2 r − (cid:96) ( u , v ) ∂ r Φ ( (cid:96) ) ( u , v )= 12 v (cid:96) +2 r − (cid:96) ( u , v )( r ∂ r Φ ( (cid:96) ) )( u , v ) − M v (cid:96) +2 r − (cid:96) − ( u , v )( r ∂ r Φ ( (cid:96) ) )( u , v )+ e v (cid:96) +2 r − (cid:96) − ( u , v )( r ∂ r Φ ( (cid:96) ) )( u , v )=2 (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ ) + O ( v − η ) . where we used the properties of v and r in B α (cid:96) and the assumption (10.2).Using the last three results and equation (A.15) from Lemma A.7 we finally get thefollowing Proposition which provides us with precise asymptotics for all the radiation fieldsin P (cid:96) Φ ( k ) , k ∈ { , . . . , (cid:96) } in B α (cid:96) . 51 roposition 10.4. Fix some (cid:96) ≥ . For a solution ψ of (1.1) that is localized at angularfrequency (cid:96) , let k ∈ { , . . . , (cid:96) } , and also assume that (10.2) holds true for some (cid:15) > and I (cid:96) [ ψ ] (cid:54) = 0 . Then there exist < η k < suitably small such that for all ( u, v ) ∈ B α (cid:96) : Φ ( (cid:96) ) ( u, v, θ, ϕ ) = 2 (cid:96) I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) (cid:18) v − uv (cid:19) (cid:96) +1 u − + O ( u − − η ) , (cid:101) Φ ( (cid:96) − k ) ( u, v, θ, ϕ ) = 2 (cid:96) +2 k I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · (2 (cid:96) ) · . . . · (2 (cid:96) + 1 − k ) (cid:18) v − uv (cid:19) (cid:96) +1 − k u − − k + O ( u − − k − η k ) . In particular, φ ( u, v, θ, ϕ ) = 2 (cid:96) I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · . . . · ( (cid:96) + 1) (cid:18) v − uv (cid:19) (cid:96) +1 u − − (cid:96) + O ( u − − (cid:96) − η (cid:96) +1 )= 2 (cid:96) I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · . . . · ( (cid:96) + 1) (cid:18) u − v (cid:19) (cid:96) +1 + O ( u − − (cid:96) − η (cid:96) +1 ) . Proof. For all k ∈ { , . . . , (cid:96) } recall thatΦ ( k ) = k (cid:88) m =0 ( d k,m + O ( r − )) r k +1+ m ∂ kr ψ, for some constants d k,m (that depends on the constants α n,k given in Proposition 4.1). Bythe estimates of Theorem 9.3 we can therefore estimate: | Φ ( k ) ( u, v γ α ( u )) | ≤ C (cid:15) E aux − decomp − n − k u − (cid:96) − / (cid:15) r (cid:96) + k +1 / ≤ C (cid:15) E aux − decomp − n − k u − − ( (cid:96) − k )+ (cid:15) − ( k + (cid:96) +1 / − α (cid:96) ) , where E aux − decomp − n − k is as in the estimates of Theorem 9.3. Note that for k = (cid:96) inparticular we have that: | Φ ( (cid:96) ) ( u, v γ α ( u )) | ≤ C (cid:15) E aux − decomp − n − (cid:96) u − (cid:96) − / (cid:15) r (cid:96) +1 / ≤ C (cid:15) E aux,I (cid:96) (cid:54) =0 ,k,(cid:96) u − (cid:15) − (2 (cid:96) +1 / − α (cid:96) ) . Choose (cid:15) > η such that 0 < η < (2 (cid:96) + 1 / − α (cid:96) ) − (cid:15) . Using (10.4)together with (A.15) we find thatΦ ( (cid:96) ) ( u, v ) = Φ ( (cid:96) ) ( u, v γ α(cid:96) ( u )) + (cid:90) vv γα(cid:96) ( u ) (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ ) ( v (cid:48) − u ) (cid:96) v (cid:48) (cid:96) +2 (1 + O ( v (cid:48)− η )) dv (cid:48) = 2 (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ )2 (cid:96) + 1 (cid:18) v − uv (cid:19) (cid:96) +1 u − (1 + O ( u − η (cid:48) ) + O ( v − η (cid:48) )) + O ( u − − η ) . Hence, using the definition of Φ ( (cid:96) ) from Proposition 4.1, we get asymptotics as above for (cid:101) Φ (cid:96) that imply ∂ r (cid:101) Φ ( (cid:96) − ( u, v ) = 2 (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ )2 (cid:96) + 1 1 r (cid:18) v − uv (cid:19) (cid:96) +1 u − (1 + O ( u − η (cid:48) ) + O ( v − η (cid:48) ))+ 1 r O ( u − − η ) , L (cid:101) Φ ( (cid:96) − ( u, v ) = 2 (cid:96) I (cid:96) [ ψ ]( θ, ϕ )2 (cid:96) + 1 2 v − u (cid:18) v − uv (cid:19) (cid:96) +1 u − (1 + O ( u − η (cid:48) ) + O ( v − η (cid:48) ))+ 1( v − u ) O ( u − − η )+ O (( v − u ) − ) (cid:104) (cid:96) I (cid:96) [ ψ ]( θ, ϕ )2 (cid:96) + 1 2 v − u (cid:18) v − uv (cid:19) (cid:96) +1 u − (1 + O ( u − η (cid:48) ) + O ( v − η (cid:48) ))+ 2 v − u O ( u − − η ) (cid:105) . This then implies that L (cid:101) Φ ( (cid:96) − ( u, v ) = 2 (cid:96) +2 I (cid:96) [ ψ ]( θ, ϕ )2 (cid:96) + 1 ( v − u ) (cid:96) − v (cid:96) +1 u − (1 + O ( u − η (cid:48) ) + O ( v − η (cid:48) )) + O ( u − − η (cid:48) ) , for some η (cid:48) > α (cid:96) . Integrating the above equation in v from the curve γ α (cid:96) we get that: (cid:101) Φ ( (cid:96) − ( u, v ) =Φ ( (cid:96) − ( u, v α (cid:96) )+ 2 (cid:96) +2 I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1)(2 (cid:96) ) (cid:18) v − uv (cid:19) (cid:96) u − + O ( u − − η (cid:48)(cid:48) ) , for some η (cid:48)(cid:48) > 0. Using the estimates (9.3) for k = (cid:96) − | (cid:101) Φ ( (cid:96) − ( u, v γ α ( u )) | ≤ C (cid:15) E aux − dr,(cid:96) u − (cid:96) − / (cid:15) r (cid:96) − / ≤ C (cid:15) E aux − dr,(cid:96) u − (cid:15) − (2 (cid:96) − / − α ) which then implies that: (cid:101) Φ ( (cid:96) − ( u, v ) = 2 (cid:96) +2 I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1)(2 (cid:96) ) (cid:18) v − uv (cid:19) (cid:96) u − + O ( u − − η ) , for some η > 0. We get asymptotics for all the remaining (cid:101) Φ ( (cid:96) − k ) ’s by repeating the processall the way to k = (cid:96) .An immediate consequence of the previous Proposition is to obtain asymptotics for ψ in B α (cid:96) , now starting from asymptotics for ψ (cid:96) /r (cid:96) and using this to build the asymptoticsof all higher derivatives (hence following the reverse path compared to Proposition 10.4).In particular for any m ∈ N as r ∼ v − u in B α (cid:96) , there exist 0 < ¯ η k < k ∈ { , . . . , (cid:96) } suitably small such that for all ( u, v ) ∈ B α (cid:96) : ψr (cid:96) ( u, v, θ, ϕ ) = 2 (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · . . . · ( (cid:96) + 1) u − − (cid:96) v − − (cid:96) + O ( u − − (cid:96) − ¯ η (cid:96) ) , (10.5)and ∂ (cid:96) − kr ψr k ( u, v, θ, ϕ ) = (cid:101) A (cid:96),k I (cid:96) [ ψ ] u − − k v − − (cid:96) + k + O ( u − − (cid:96) − ¯ η (cid:96) − k ) , (10.6)for constants (cid:101) A (cid:96),k depending on (cid:96) and k . 53 P (cid:96) ψ The goal of this section is to propagate the asymptotics of a linear wave P (cid:96) ψ obtained in B α (cid:96) to the rest of the spacetime. Lemma 10.5. Fix (cid:96) ≥ . Let ψ (cid:96) . = P (cid:96) ψ be a frequency localized linear wave by assumingthat ψ solves (1.1) , then for all k ≥ : ∂ ρ (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) =( Dr ) k [ − / ∆ S − k ( k + 1)] ∂ kr ψ (cid:96) + ( Dr ) k r · k (cid:88) j =0 O ( r − k + j ) ∂ jr T ψ (cid:96) + ( Dr ) k O ( r − η ) ∂ k +1 r T ψ (cid:96) . (10.7) Proof. We will prove (10.7) by induction. By (7.2), we have that (10.7) holds for k = 0.Now suppose (10.7) holds for k ∈ N , then we will show that (10.7) also holds for k replacedby k + 1.We have by applying the Leibniz rule multiple times that( Dr ) − ( k +1) ∂ ρ (( Dr ) k +2 ∂ k +2 r ψ (cid:96) ) = ( Dr ) − k ∂ ρ (( Dr ) k +1 ∂ k +2 r ψ (cid:96) ) + ( Dr ) (cid:48) ∂ k +2 r ψ (cid:96) = ( Dr ) − k ∂ ρ ∂ r (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) − ( k + 1)( Dr ) − k ∂ ρ (( Dr ) k ( Dr ) (cid:48) ∂ k +1 r ψ (cid:96) ) + ( Dr ) (cid:48) ∂ k +2 r ψ (cid:96) = ( Dr ) − k ∂ r ∂ ρ (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) − ( Dr ) h (cid:48) ∂ k +1 r T ψ − ( k + 1)( Dr ) (cid:48) ( Dr ) − k ∂ ρ (( Dr ) k ∂ k +1 r ψ (cid:96) ) − ( k + 1)( Dr ) (cid:48)(cid:48) ∂ k +1 r ψ (cid:96) + ( Dr ) (cid:48) ∂ k +2 r ψ (cid:96) , where in the last inequality we used that ∂ ρ ∂ r = ∂ r ∂ ρ − h (cid:48) T . We can further expand theright-hand side above to obtain( Dr ) − ( k +1) ∂ ρ (( Dr ) k +2 ∂ k +2 r ψ (cid:96) ) = ∂ r (cid:104) ( Dr ) − k ∂ ρ (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) (cid:105) + k ( Dr ) (cid:48) ( Dr ) − ( k +1) ∂ ρ (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) − ( Dr ) h (cid:48) ∂ k +1 r T ψ − ( k + 1)( Dr ) (cid:48) ( Dr ) − k ∂ ρ (( Dr ) k ∂ k +1 r ψ (cid:96) ) − ( k + 1)( Dr ) (cid:48)(cid:48) ∂ k +1 r ψ + ( Dr ) (cid:48) ∂ k +2 r ψ (cid:96) = ∂ r (cid:104) ( Dr ) − k ∂ ρ (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) (cid:105) + k ( k + 1)(( Dr ) (cid:48) ) ( Dr ) − ∂ k +1 r ψ (cid:96) + k ( Dr ) (cid:48) ∂ ρ ∂ k +1 r ψ − ( Dr ) h (cid:48) ∂ k +1 r T ψ (cid:96) − k ( k + 1)(( Dr ) (cid:48) ) ( Dr ) − ∂ k +1 r ψ (cid:96) − ( k + 1)( Dr ) (cid:48) ∂ ρ ∂ k +1 r ψ (cid:96) − ( k + 1)( Dr ) (cid:48)(cid:48) ∂ k +1 r ψ + ( Dr ) (cid:48) ∂ k +2 r ψ (cid:96) = ∂ r (cid:104) ( Dr ) − k ∂ ρ (( Dr ) k +1 ∂ k +1 r ψ (cid:96) ) (cid:105) − ( k + 1)( Dr ) (cid:48)(cid:48) ∂ k +1 r ψ (cid:96) − [( Dr ) h (cid:48) + h ( Dr ) (cid:48) ] ∂ k +1 r T ψ (cid:96) . By applying ∂ r to both sides of (10.7) and using that ( Dr ) (cid:48)(cid:48) = 2 and D − h = c ( r η ) +54 ( r − ), for c (cid:54) = 0 and some η > 0, we can rewrite the above equation as follows:( Dr ) − ( k +1) ∂ ρ (( Dr ) k +2 ∂ k +2 r ψ (cid:96) ) = (cid:2) − / ∆ S − k ( k + 1) − k + 1) (cid:3) ∂ k +1 r ψ (cid:96) + r k +1 (cid:88) j =0 O ( r − k − j ) ∂ jr T ψ (cid:96) + O ( r − η ) ∂ k +2 r T ψ (cid:96) = (cid:2) − / ∆ S − ( k + 1)( k + 2) (cid:3) ∂ k +1 r ψ (cid:96) + r k +1 (cid:88) j =0 O ( r − ( k +1)+ j ) ∂ jr T ψ (cid:96) + O ( r − η ) ∂ k +2 r T ψ (cid:96) . We use the above equation to obtain almost sharp decay for ∂ (cid:96) +1 r ψ (cid:96) . Proposition 10.6. Fix (cid:96) ≥ . Let ψ (cid:96) . = P (cid:96) ψ be a frequency localized linear wave byassuming that ψ solves (1.1) . Assume also that I (cid:96) [ ψ ] (cid:54) = 0 . Then we have that | ∂ (cid:96) +1 r ψ (cid:96) | ≤ CE aux − decomp − n − (cid:96) +1 (1 + τ ) − (cid:96) − (cid:15) , (10.8) where C = C ( D, R, (cid:96) ) and E aux − decomp − n − (cid:96) +1 is as in (9.9) .Proof. If we apply (10.7) to ψ (cid:96) , with k = (cid:96) , we obtain ∂ ρ (( Dr ) (cid:96) +1 ∂ (cid:96) +1 r ψ (cid:96) ) = ( Dr ) (cid:96) r · (cid:96) (cid:88) j =0 O ( r − (cid:96) + j ) ∂ jr T ψ (cid:96) + ( Dr ) (cid:96) O ( r − η ) ∂ (cid:96) +1 r T ψ. (10.9)Now, we integrate (10.9) in ρ from ρ = r + and use Theorem 9.3 to obtain( Dr ) (cid:96) +1 | ∂ (cid:96) +1 r ψ (cid:96) | (cid:46) (cid:96) (cid:88) j =0 (cid:90) ρr + ( Dr ) (cid:96) r − (cid:96) + j +1 | ∂ jr T ψ (cid:96) | dρ (cid:48) + (cid:90) ρr + ( Dr ) (cid:96) r − η | ∂ (cid:96) +1 r T ψ (cid:96) | dρ (cid:48) (cid:46) (cid:90) ρr + r · ( Dr ) (cid:96) dρ (cid:48) · || ∂ (cid:96) +1 r T ψ (cid:96) || L ∞ + (cid:96) (cid:88) j =0 || r − (cid:96) + j ∂ kr T ψ (cid:96) || L ∞ ≤ CE aux − decomp − n − (cid:96) +1 ( Dr ) (cid:96) +1 · (1 + τ ) − (cid:96) − (cid:15) , where we used estimates (9.9). We can therefore conclude that | ∂ (cid:96) +1 r ψ (cid:96) | ≤ CE aux − decomp − n − (cid:96) +1 (1 + τ ) − (cid:96) − (cid:15) . The above proposition is the key to propagating the asymptotics of ψ (cid:96) from the curve ρ = ρ γ α ( τ ) to the region r + ≤ ρ ≤ ρ γ α(cid:96) ( τ ), where the curve γ α (cid:96) is defined as in the previoussubsection. 55 roposition 10.7. Fix (cid:96) ≥ . Let ψ (cid:96) . = P (cid:96) ψ be a frequency localized linear wave byassuming that ψ solves (1.1) . Assume also that I (cid:96) [ ψ ] (cid:54) = 0 . For all ≤ k ≤ (cid:96) and for all ρ ≤ ρ γ α(cid:96) ( τ ) , where γ α (cid:96) and α (cid:96) are defined in subsection 10.1,we have that r k − (cid:96) ∂ kr ψ (cid:96) ( τ, ρ ) = A (cid:96),k I (cid:96) (1 + τ ) − (cid:96) − + O ((1 + τ ) − (cid:96) − − (cid:15) ) , for some (cid:15) > , where the quantities A (cid:96),k are given by Proposition 10.5.Proof. By (10.6), we have that there exists a constant A (cid:96),(cid:96) (depending on (cid:101) A (cid:96),(cid:96) ), such that ∂ (cid:96)r ψ (cid:96) ( τ, ρ γ α ( τ )) = A (cid:96),(cid:96) I (cid:96) [ ψ (cid:96) ](1 + τ ) − (cid:96) − + O ((1 + τ ) − (cid:96) − − (cid:15) ) , where we note that we suppress the dependence on ( θ, ϕ ) for I (cid:96) . We can then integrate in ρ and use (10.8) together with estimates (9.9) to obtain for r + ≤ ρ ≤ ρ γ α(cid:96) ( τ ) and (cid:15) > (cid:12)(cid:12)(cid:12) ∂ (cid:96)r ψ (cid:96) ( τ, ρ ) − A (cid:96) I (cid:96) (1 + τ ) − (cid:96) − (cid:15) (cid:12)(cid:12)(cid:12) ≤ (cid:90) ρ γα(cid:96) ( τ ) ρ | ∂ ρ ∂ (cid:96)r ψ (cid:96) | dρ (cid:48) (cid:46) τ − (cid:96) − (cid:15) · ρ γ α ( τ ) (cid:46) τ − (cid:96) − α (cid:96) + (cid:15) = τ − (cid:96) − − (cid:15) (cid:48) , where (cid:15) (cid:48) = 1 − (cid:15) − α (cid:96) > ∂ (cid:96) − r ψ (cid:96) we integrate in r from ρ to ρ γ α(cid:96) and we have that: ∂ (cid:96) − r ψ (cid:96) ( τ, ρ ) = ∂ (cid:96) − r ψ (cid:96) ( τ, ρ γ α(cid:96) ) − (cid:90) ρ γα(cid:96) ρ ∂ ρ (cid:48) ( ∂ (cid:96) − r ψ (cid:96) ) dρ (cid:48) = ∂ (cid:96) − r ψ (cid:96) ( τ, ρ γ α(cid:96) ) − (cid:90) ρ γα(cid:96) ρ ∂ (cid:96)r ψ (cid:96) dr = ρ γ α(cid:96) ∂ (cid:96) − r ψ (cid:96) ( τ, ρ γ α(cid:96) ) ρ γ α(cid:96) − ρ γ α(cid:96) [ A (cid:96),(cid:96) I (cid:96) [ ψ (cid:96) ](1 + τ ) − (cid:96) − + O ((1 + τ ) − (cid:96) − − (cid:15) )]+ ρ [ A (cid:96),(cid:96) I (cid:96) [ ψ (cid:96) ](1 + τ ) − (cid:96) − + O ((1 + τ ) − (cid:96) − − (cid:15) )] , and notice that there is an exact cancellation in the term ρ γ α(cid:96) ∂ (cid:96) − r ψ (cid:96) ( ρ γ α(cid:96) ) ρ γ α(cid:96) − ρ γ α(cid:96) [ A (cid:96),(cid:96) I (cid:96) [ ψ (cid:96) ](1 + τ ) − (cid:96) − + O ((1 + τ ) − (cid:96) − − (cid:15) )] = 0 . For the remaining derivatives we work in the same way and we notice that there is acancellation of the form: A (cid:96),k +1 (cid:96) − k = A (cid:96),k for 0 ≤ k ≤ (cid:96) − (cid:101) A (cid:96),k from (10.6).56 For ψ a solution of (1.1), let again ψ (cid:96) . = P (cid:96) ψ, for a fixed (cid:96) ≥ 1. In this section we will invert the time translation operator T . Theorem 11.1. Fix some (cid:96) ≥ and consider a frequency localized solution ψ (cid:96) . = P (cid:96) ψ of (1.1) , where ψ solves (1.1) . Assume that ( ψ (cid:96) , n Σ ψ (cid:96) ) | Σ ∈ ( C ∞ c (Σ )) .Then there exists a unique frequency localized (at frequency (cid:96) ) smooth solution of (1.1) (cid:101) ψ (cid:96) such that T (cid:101) ψ (cid:96) = ψ (cid:96) , satisfying the boundary conditions lim r →∞ r∂ (cid:96) +1 ρ (cid:101) ψ (cid:96) = ( Dr ) (cid:96) +1 ∂ (cid:96) +1 ρ (cid:101) ψ (cid:96) | H + = 0 .Moreover we have that I (1) (cid:96) [ ψ ]( θ, ϕ ) . = I (cid:96) [ (cid:101) ψ ]( θ, ϕ ) = lim r →∞ (cid:2) − r ∂ r L ( (cid:96) ) ( ψ (cid:96) ) (cid:3) , for L ( (cid:96) ) ( ψ (cid:96) ) the solution of the system L (0) ( ψ (cid:96) ) := r · (cid:90) ∞ r ( r (cid:48) − r ) (cid:96) D ( r (cid:48) ) r (cid:48) ) (cid:96) +1 F [ ψ (cid:96) ]( r (cid:48) ) dr (cid:48) , L ( n ) ( ψ (cid:96) ) := (cid:101) L ( n ) ( ψ (cid:96) ) + n (cid:88) k =1 α n,k L ( n − ( ψ (cid:96) ) , where the α n,k ’s are as in Proposition 4.1, where F [ ψ (cid:96) ]( r ) . = (cid:90) rr + ( D ( r (cid:48) ) r (cid:48) ) (cid:96) f ( r (cid:48) ) ∂ (cid:96) +1 r (cid:48) ψ (cid:96) dr (cid:48) + (cid:96) (cid:88) j =0 (cid:90) rr + ( D ( r (cid:48) ) r (cid:48) ) (cid:96) r (cid:48) f j ( r (cid:48) ) ∂ jr (cid:48) ψ (cid:96) dr (cid:48) , for f and f j are given by the equation: ∂ ρ (( Dr ) (cid:96) +1 ∂ (cid:96) +1 ρ (cid:101) ψ (cid:96) ) = ( Dr ) (cid:96) f ( r ) ∂ (cid:96) +1 ρ T (cid:101) ψ (cid:96) + (cid:96) (cid:88) j =0 f j ( r )( Dr ) (cid:96) r∂ jρ T (cid:101) ψ (cid:96) , (11.1) and where (cid:101) L ( k ) ( ψ (cid:96) ) . = ( − k ( r ∂ r ) k (cid:18) (cid:96) ! r · (cid:90) ∞ r ( r (cid:48) − r ) (cid:96) D ( r (cid:48) ) r (cid:48) ) (cid:96) +1 F [ ψ (cid:96) ]( r (cid:48) ) dr (cid:48) (cid:19) . Remark 11.1. Note that the constants in the iterative system for (cid:101) L k are the same as inthe system for Φ ( k ) in Proposition 4.1.Proof. The existence and uniqueness of the time-inverse ψ (cid:96) follows by solving equation(11.1) with appropriate conditions at infinity and at the horizon (taking lim r →∞ r∂ (cid:96) +1 ρ (cid:101) ψ (cid:96) =( Dr ) (cid:96) +1 ∂ (cid:96) +1 ρ (cid:101) ψ (cid:96) | H + = 0 is enough) on the initial hypersurface Σ u . With the initial dataobtained from the solutions of the previous ODE problem, we can now have a globalsolution ψ (cid:96) in all of the domain of outer communications up to and including the eventhorizon. 57e integrate equation (11.1) (which in schematic form is (10.9)) in ρ and we have thatfor any r ≥ r + : ∂ (cid:96) +1 ρ (cid:101) ψ (cid:96) ( r ) = 1( D ( r ) r ) (cid:96) +1 (cid:90) rr + ( D ( r (cid:48) ) r (cid:48) ) (cid:96) f ( r (cid:48) ) ∂ (cid:96) +1 r (cid:48) T (cid:101) ψ (cid:96) dρ + 1( D ( r ) r ) (cid:96) +1 (cid:96) (cid:88) j =0 (cid:90) rr + f j ( r (cid:48) )( D ( r (cid:48) ) r (cid:48) ) (cid:96) r (cid:48) ∂ jr (cid:48) T (cid:101) ψ (cid:96) dρ. For F as in the statement of the theorem, we have that f ( r ) ∼ O ( r − η ) , f j ( r ) ∼ O ( r (cid:48)− (cid:96) + j ) , hence F [ ψ (cid:96) ]( r ) ∼ (cid:90) rr + ( D ( r (cid:48) ) r (cid:48) ) (cid:96) O ( r (cid:48) − η ) ∂ (cid:96) +1 r (cid:48) T (cid:101) ψ (cid:96) dρ + (cid:96) (cid:88) j =0 (cid:90) rr + O ( r (cid:48)− (cid:96) + j )( D ( r (cid:48) ) r (cid:48) ) (cid:96) r (cid:48) ∂ jr (cid:48) T (cid:101) ψ (cid:96) dρ, For ∂ (cid:96) +1 ρ we now have that ∂ (cid:96) +1 ρ ψ (cid:96) ( r ) = 1( D ( r ) r ) (cid:96) +1 F [ (cid:101) ψ (cid:96) ]( r ) . (11.2)Note that generically we have that F [ ψ (cid:96) ]( r ) ∼ r (cid:96) +1 , we have that 1( D ( r ) r ) (cid:96) +1 F [ (cid:101) ψ (cid:96) ]( r ) ∼ r − (cid:96) − , so after integrating equation (11.2) (cid:96) times and applying Lemma A.5 we get that: ∂ ρ (cid:101) ψ (cid:96) ( r (cid:96) − ) = ( − (cid:96) ( (cid:96) − (cid:90) ∞ r (cid:96) − ( r − r (cid:96) − ) (cid:96) − D ( r ) r ) (cid:96) +1 F [ ψ (cid:96) ]( r ) dρ, for any r (cid:96) − ≥ r + . We use the assumption of compact at this point to make sure the largeintegral is finite. Then integrating one more time we get that: (cid:101) ψ (cid:96) ( r (cid:96) ) =( − (cid:96) +1 (cid:96) ! (cid:90) ∞ r (cid:96) ( r − r (cid:96) ) (cid:96) D ( r ) r ) (cid:96) +1 F [ (cid:101) ψ (cid:96) ]( r ) dρ (11.3)+ ( − (cid:96) ( (cid:96) − 1) lim r →∞ r (cid:96) +1 (cid:18) D ( r ) r ) (cid:96) +1 F [ ψ (cid:96) ]( r ) (cid:19) . (11.4)Note also that F depends on T (cid:101) ψ (cid:96) . At this point we use the assumption that T (cid:101) ψ (cid:96) = ψ (cid:96) , and the decay assumption on ψ (cid:96) . The rest follows by an application of Proposition 4.1 on ψ (cid:96) defined as above.Note also that from the formulas above we have that¯Φ ( k ) = O ( r − (cid:96) + k + η (cid:48) ) for any η (cid:48) > k ∈ { , . . . , (cid:96) − } , ( k ) but with (cid:101) ψ in the place of ψ . This implies thatlim r →∞ ¯Φ ( k ) = 0 for k ∈ { , . . . , (cid:96) − } . Remark 11.2. Note that for F [ ψ (cid:96) ]( r ) = (cid:96) (cid:88) j =0 (cid:90) rr + ( D ( r (cid:48) ) r (cid:48) ) (cid:96) r (cid:48) O ( r − (cid:96) + j ) ∂ jr (cid:48) ψ (cid:96) dρ + (cid:90) rr + ( D ( r (cid:48) ) r (cid:48) ) (cid:96) O ( r (cid:48) − η ) ∂ (cid:96) +1 r (cid:48) ψ (cid:96) dρ. = F [ ψ (cid:96) ]( r ) + F [ ψ (cid:96) ]( r ) , we have that F [ ψ (cid:96) ]( r ) = O (log r ) , F [ ψ (cid:96) ]( r ) = O ( r − η ) . We note that in the case of non-compactly data for ψ (cid:96) with vanishing (cid:96) -th Newman–Penrosecharget I (cid:96) [ (cid:101) ψ ] = 0 satisfying lim r →∞ r ∂ r [( r ∂ r ) (cid:96) ( rψ (cid:96) )] < ∞ , we need to use the precise formof F in order to express I (cid:96) [ (cid:101) ψ ] in terms of ψ (cid:96) . Remark 11.3. Let us note that due to the last observation in the proof above, once wetake the limits in the iterative expression for the L n ’s to compute the constants at infinity,only two terms will be non-zero (the ones at the top two highest derivative levels). One canshow that the α (cid:96),(cid:96) − ’s depends only on M , we note that at least in the case of compactlysupported data, the time-inverted Newman–Penrose charge I (1) (cid:96) depend only on the mass M and do not depend on the charge e of the Reissner–Nordstr¨om spacetime that we areworking in.Moreover, starting from the formula (11.3) and computing lim r →∞ L ( (cid:96) ) ( (cid:101) ψ (cid:96) ) = lim r →∞ [ T − ( L ( (cid:96) ) ( ψ (cid:96) )] , we note that I (1) (cid:96) [ (cid:101) ψ (cid:96) ] = c ( (cid:96), M ) lim r →∞ [ T − ( L ( (cid:96) ) ( ψ (cid:96) ))] , for a function c that depends only on the mass M and the frequency (cid:96) , and we can thenexpress the time-inverted Newman–Penrose charge as an integral over future null infinity: I (1) (cid:96) [ (cid:101) ψ (cid:96) ]( θ, ϕ ) = c ( (cid:96), M ) (cid:90) ∞ u L ( (cid:96) ) ( ψ (cid:96) )( τ, θ, ϕ ) (cid:12)(cid:12) I + dτ. 12 Precise late-time asymptotics when I (cid:96) = 0 Using the time-inversion construction of the previous Section we can now prove asymptoticsfor a frequency localized linear wave. First we derive asymptotics for T derivatives. Westart with a basic Lemma that is a generalization of Lemma 10.2.Note that again in this section we will always work with a solution ψ of (1.1) that islocalized at angular frequency (cid:96) ≥ 1, so by ψ we always mean P (cid:96) ψ .59 emma 12.1. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) ≥ .Under the assumption (10.2) (which we make for some (cid:15) > ) and I (cid:96) [ ψ ] (cid:54) = 0 for all ( u, v ) ∈ B α (cid:96) we have that: m (cid:88) k =0 | v k L k +1 Φ ( (cid:96) ) ( u, v ) | ≤ CE aux − decomp − n − k (cid:16) O ( v − β ) (cid:17) , (12.1) for some β > , for C = C ( D, R ) , and for E aux − decomp − n − k as in Theorem 9.3 . The proof is similar to that of Lemma 10.2 where we use equation (5.6). Next we provea generalization of Proposition 10.3. Proposition 12.2. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) ≥ .Assume that (10.2) holds true for some (cid:15) > and I (cid:96) [ ψ ] (cid:54) = 0 . Then for m ∈ N there exists η > small enough such that for all ( u, v ) ∈ B α (cid:96) we have that: L m +1 Φ ( (cid:96) ) ( u, v, θ, ϕ ) = 2 (cid:96) +1 I (cid:96) [ ψ ]( θ, ϕ )( v − u ) (cid:96) L m (cid:18) v (cid:96) +2 (cid:19) + O ( v − η )( v − u ) (cid:96) L m (cid:18) v − u ) (cid:96) +2 (cid:19) . (12.2)The proof follows the one of Proposition 10.3 using the equation L (cid:18) r (cid:96) L m +1 Φ ( (cid:96) ) (cid:19) = O ( r − − (cid:96) )( L m +1 Φ ( (cid:96) ) )+ (cid:88) m + m = m,m Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) ≥ .Assume that (10.2) holds true for some (cid:15) > and I (cid:96) [ ψ ] (cid:54) = 0 . Then for any m ∈ N thereexist < η k < suitably small such that for all ( u, v ) ∈ B α (cid:96) : T m φ ( u, v, θ, ϕ ) = 2 (cid:96) I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · . . . · ( (cid:96) + 1) ( v − u ) (cid:96) +1 T m ( u − − (cid:96) v − − (cid:96) ) + O ( u − − (cid:96) − m − η (cid:96) +1 ) . As in the case of I (cid:96) (cid:54) = 0, an immediate consequence of the previous Proposition is toobtain asymptotics for ψ in B α (cid:96) . Proposition 12.4. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) ≥ .Assume that (10.2) holds true for some (cid:15) > and I (cid:96) [ ψ ] (cid:54) = 0 . Then for any m ∈ N thereexist < η k < suitably small such that for all ( u, v ) ∈ B α (cid:96) : T m (cid:18) ∂ (cid:96) − kr ψ ( τ, r, θ, ϕ ) r k (cid:19) = 2 (cid:96) I (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · . . . · ( (cid:96) + 1) T m ( u − − k v − − (cid:96) + k ) + O ( u − − (cid:96) − m − η (cid:96) +1 ) . T m ψ (cid:96) everywhere in R . Proposition 12.5. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) ≥ .Assume also that I (cid:96) [ ψ (cid:96) ] (cid:54) = 0 . Then we have for all τ ≥ u and any r + ≤ r < ∞ that T m (cid:18) ∂ kr ψ (cid:96) ( τ, r ) r (cid:96) − k (cid:19) = A (cid:96),k,m τ − (cid:96) − − m + O ( τ − (cid:96) − − m − (cid:15) ) , (12.3) for some (cid:15) > . Using the construction of the time-inverted Newman–Penrose charges of Section 11 andthe previous Proposition of the current section for m = 1, we can obtain precise asymptoticsfor frequency localized waves P (cid:96) ψ with vanishing (cid:96) -th Newman–Penrose constant. We havethe following result: Proposition 12.6. Let ψ be a solution of (1.1) that is localized at angular frequency (cid:96) ≥ .Assume that (10.2) holds true for some (cid:15) > and I (cid:96) [ ψ ] = 0 , I (1) (cid:96) [ ψ ] (cid:54) = 0 , for I (1) (cid:96) [ ψ ] the time inverted Newman–Penrose charge from Section 11. Then we have that: φ ( u, v, θ, ϕ ) = − (cid:96) I (1) (cid:96) [ ψ ]( θ, ϕ )(2 (cid:96) + 1) · · · · · ( (cid:96) + 2) ( v − u ) (cid:96) +1 u − − (cid:96) v − − (cid:96) + O ( u − − (cid:96) − η v − − (cid:96) ) for ( u, v ) ∈ B α (cid:96) and for some η > , (12.4) and ψ ( τ, r, θ, ϕ ) r (cid:96) = A (1) (cid:96), τ − (cid:96) − + O ( τ − (cid:96) − − η ) for any τ ≥ u , r + ≤ r < ∞ , and some η > , (12.5) where A (1) (cid:96), is the quantity A (cid:96), , given in (12.3) but depending on I (1) (cid:96) instead of I (cid:96) . A Basic inequalities We gather here several auxiliary results that we use throughout the paper. Lemma A.1. Let q ∈ R \ {− } and ζ : [ r , ∞ ) → R a C function. Then, (cid:90) ∞ r r q ζ ( r ) dr ≤ q + 1) (cid:90) ∞ r r q +2 ( ∂ r ζ ) ( r ) dr (A.1)if ζ ( r ) = 0 and lim r →∞ r q +1 ζ ( r ) = 0 , (cid:90) ∞ r ζ ( r ) dr ≤ (cid:90) ∞ r ( r − r ) ( ∂ r ζ ) ( r ) dr if lim r →∞ rζ ( r ) = 0 . (A.2) Lemma A.2 (Poincar´e inequality on S ) . Fix (cid:96) ≥ . Then (cid:90) S ζ ≥ (cid:96) dω ≤ (cid:96) ( (cid:96) + 1) r − (cid:90) S | / ∇ ζ ≥ (cid:96) | dω. (A.3)61 n the case ζ is supported on a single angular mode the inequality becomes an equality: (cid:90) S ψ (cid:96) dω = 1 (cid:96) ( (cid:96) + 1) r − (cid:90) S | / ∇ ψ (cid:96) | dω. (A.4) Furthermore, (cid:90) S r | / ∇ ζ | dω ≤ (cid:90) S ( / ∆ S ζ ) dω. (A.5) Lemma A.3. There exists a numerical constant C > such that sup ω ∈ S ζ ( ω ) ≤ C (cid:88) | k |≤ (cid:90) S (Ω k ζ ) ( ω ) dω. (A.6) Lemma A.4. Let ζ : R + × [ R, ∞ ) → R be a function such that the following inequalitieshold: (cid:90) ∞ R r p − (cid:15) ζ ( τ, r ) dr ≤ C (1 + τ ) − q , (A.7) (cid:90) ∞ R r p +1 − (cid:15) ζ ( τ, r ) dr ≤ C (1 + τ ) − q +1 , (A.8) for some τ -independent constants C , C > , q ∈ R and (cid:15) ∈ (0 , .Then (cid:90) ∞ R r p f ( τ, r ) dr (cid:46) max { C , C } (1 + τ ) − q + (cid:15) . (A.9)For the proofs of all the above inequalities see Section 2.5 of [AAG18d]. Lemma A.5. Let f ∈ C ([ x , ∞ )) . Let n ∈ N such that lim x →∞ x n +1 | f ( x ) | = 0 . Then (cid:90) ∞ x ( x − x ) n f ( x ) dx = n ! (cid:90) ∞ x (cid:90) ∞ x . . . (cid:90) ∞ x n f ( x n +1 ) dx n +1 dx n . . . dx . (A.10)For a proof of the aforementioned lemma see Lemma 6.9 of [AAG20a]. Lemma A.6. Let ζ : [0 , ∞ ) → R be a continuous, positive function. Assume that for all ≤ τ ≤ τ < ∞ , ζ ( τ ) + β (cid:90) τ τ ζ ( s ) ds ≤ ζ ( τ ) + E ( τ − τ )( τ + 1) − p , (A.11) with E , β, p > being constants and moreover, for all ≤ τ ≤ τ ≤ τ ζ ( τ ) + β (cid:90) τ τ ζ ( s ) ds ≤ ζ ( τ ) + C ( τ − τ ) ζ ( τ ) . (A.12) Then we have that ζ ( τ ) ≤ (cid:0) C β − (cid:1) ζ ( τ ) (A.13) for all τ ≥ τ and there exists a constant C = C ( C , E , β, p ) > , such that ζ ( τ ) ≤ C ( ζ ( τ ) + E )(1 + τ ) − p , (A.14) for all τ ≥ τ . Lemma A.7. For any (cid:96) ≥ there exists η (cid:48) = η (cid:48) ( η, α ) > such that for any v ≥ v γ α(cid:96) (cid:90) vv γα(cid:96) ( u ) ( v (cid:48) − u ) k ( v (cid:48) ) k +2 (1 + O (( v (cid:48) ) − η )) dv (cid:48) = (cid:20) ( v − u ) k +1 ( k + 1) uv k +1 (cid:21) (1 + O ( v − η (cid:48) ) + O ( u − η (cid:48) )) , (A.15) where γ α (cid:96) = { ( u, v ) : v = u = v α (cid:96) } for (cid:96) +22 (cid:96) +3 < α (cid:96) < . Moreover for any m ∈ N we havethat: (cid:90) vv γα(cid:96) ( u ) ( v (cid:48) − u ) k L m (cid:18) v (cid:48) ) k +2 (cid:19) (1 + O (( v (cid:48) ) − η )) du (cid:48) = (cid:20) ( v − u ) k +1 T m (cid:18) k + 1) uv k +1 (cid:19)(cid:21) (1 + O ( v − η (cid:48) ) + O ( u − η (cid:48) )) . (A.16) Proof. We first note that 1 k + 1 L (cid:34) u (cid:18) v − uv (cid:19) k (cid:35) = ( v − u ) k v k +2 , and estimate (A.15) follows from the above computation. 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