A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom
AA non-degenerate scattering theory for the wave equationon extremal Reissner–Nordstr¨om
Y. Angelopoulos, S. Aretakis, and D. GajicOctober 18, 2019
Abstract
It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degeneratescattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts asan unstable blueshift mechanism in the backwards direction. In the extremal case, however, the redshifteffect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction.In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equationon extremal Reissner–Nordstr¨om backgrounds. We make use of physical-space energy norms which arenon-degenerate both at the event horizon and at null infinity. As an application of our theory we presenta construction of a large class of smooth exponentially decaying modes. We also derive scattering resultsin the black hole interior region.
Contents r -weighted estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Forwards r -weighted estimates revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Higher-order energies and time integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Estimates near spacelike infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.5 Scattering and regularity in black hole interiors . . . . . . . . . . . . . . . . . . . . . . . . . . 241 a r X i v : . [ m a t h . A P ] O c t The forwards evolution map 24
Scattering theories for the wave equation (cid:3) g ψ = 0 . (1.1)on black hole backgrounds provide useful insights in studying the evolution of perturbations “at infinity”. Inthis article we construct a new scattering theory for scalar perturbations on extremal Reissner–Nordstr¨om.Our theory makes crucial use of the vanishing of the surface gravity on the event horizon and our methodsextend those of the horizon instability of extremal black holes in the forward-in-time evolution. In theremainder of this section we will briefly recall scattering theories for sub-extremal backgrounds and in thenext section we will provide a rough version of the main theorems.We will first review the scattering theories of the wave equation (1.1) on Schwarzschild backgrounds.Let T denote the standard stationary Killing vector field on Schwarzschild spacetime. Since T is globallycausal in the domain of outer communications, the energy flux associated to T is non-negative definite.This property played a crucial role in the work of Dimock and Kay [25, 26] where a T -scattering theory onSchwarzschild, in the sense of Lax–Phillips [42], was developed. Subsequently, the T -scattering theory wasunderstood by Nicolas [50], following the notion of scattering states by Friedlander [29].The T -energy scattering theory on Schwarzschild applies also when the standard Schwarzschild timefunction t is replaced by a time function corresponding to a foliation by hypersurfaces intersecting the futureevent horizon and terminating at future null infinity. This is convenient since it allows one to bound energiesas measured by local observers. Recall that T is timelike in the black hole exterior and null on the eventhorizon. For this reason, the T energy flux across an achronal hypersurface intersecting the event horizon ispositive-definite away from the horizon and degenerate at the horizon. Hence, the associated norm for the T -energy scattering theory is degenerate at the event horizon. On the other hand, it has been shown [22, 23]that Schwarzschild does not admit a non-degenerate scattering theory where the norm on the achronalhypersurface is defined in terms of the energy flux associated to a globally timelike vector field N . This is2 a) The Dimock–Kay T -scattering mapin the sense of Lax–Phillips. (b) The Nicolas T -scattering mapin the sense of Friedlander. Figure 1: The T -scattering maps on Schwarzschild spacetime.due to the celebrated redshift effect which turns into a blueshift instability mechanism when seen from thebackwards scattering point of view. (a) The T -scattering map. (b) The N -scattering mapfails to be surjective. Figure 2: The T and N scattering maps on Schwarzschild.It is important to note that one can indeed define a backwards scattering map for non-degenerate high-regularity norms on an achronal hypersurface if the data on H + and I + are sufficiently regular and decayexponentially fast with sufficiently large rate.Figure 3: Higher-order non-degenerate backwards scattering on Schwarzschild.A fully nonlinear version of the above backwards scattering construction was presented in [18]. As far3s the Kerr family is concerned, Dafermos, Rodnianski and Shlapentokh-Rothman [22] derived a degeneratescattering theory in terms of the energy flux associated to a globally causal vector field V which is null onthe event horizon and timelike in the exterior region. Similarly to the Schwarzschild case, the sub-extremalKerr backgrounds do not admit a non-degenerate scattering theory in the exterior region.Finally we present results regarding the black hole interior region. Fournodavlos and Sbierski [27] derivedthe asymptotics of the wave equation at the singular boundary { r = 0 } . On the other hand, Luk–Oh [45]showed that the forward evolution of smooth compactly supported initial data on sub-extremal RN is H singular at the Cauchy horizon.Figure 4: Blow-up of W , norm in any neighborhood of the Cauchy horizon.Similar instability results for the wave equation on Kerr interiors were presented by Dafermos–Shlapentokh-Rothman [23] and independently by Luk–Sbierski [46] (see also [39, 38, 28]). Specifically, the authors assumedtrivial data on the past event horizon and arbitrary polynomially decaying data on past null infinity andshowed that local (non-degenerate) energies blow up in a neighborhood of any point at the Cauchy horizon.Figure 5: Blow-up of W , norm from scattering data on H − and I − . In this section we present a rough version of our main theorems. Precise statements of the theorems canbe found in Section 4. First note that the standard stationary Killing vector field T is causal everywherein the domain of outer communications of ERN. Hence, one trivially extends the T -scattering theory to theextremal case. Here Σ denotes a spacelike-null hypersurfaces intersecting H + and terminating at I + .4igure 6: The T -scattering theory for ERN.The following theorem provides a bijective scattering theory for weighted and non-degenerate norms onERN. A rough schematic definition of these norms is the following (cid:107) rψ (cid:107) E H + ∩J +(Σ0) = (cid:90) H + ∩J + (Σ ) v · J T [ ψ ] + ..., (cid:107) rψ (cid:107) E I + ∩J +(Σ0) = (cid:90) I + ∩J + (Σ ) u · J T [ ψ ] + ..., (cid:107) rψ (cid:107) E Σ0 = (cid:90) Σ J N [ ψ ] + ( ∂ ρ ( rψ )) + ..., (1.2)on H + and I + , and (cid:107) rψ (cid:107) E Σ0 = (cid:90) Σ J N [ ψ ] + ( ∂ ρ ( rψ )) + ..., (1.3)on Σ . Note that the E Σ − norm is non-degenerate both at the event horizon and at null infinity (the laterunderstood in an appropriate conformal sense; see Section 2.4). The omitted terms involve smaller weights.Here J T and J N denote the energy fluxes associated to the vector fields T and N and ∂ ρ is a tangential toΣ derivative such that ∂ ρ r = 1. Let E H + ∩J + (Σ ) , E I + ∩J + (Σ ) , E Σ denote the closure of smooth compactlysupported data under the corresponding norms schematically defined above. Theorem 1. (Rough version of Theorem 4.1) The scattering map defined between the energy spaces E Σ and E H + ∩J + (Σ ) ⊕ E H + ∩J + (Σ ) in the black hole exterior of ERN is bounded and bijective. Figure 7: A non-degenerate scattering theory on ERN.The above theorem is in stark contrast to the sub-extremal case where the backwards evolution is singularat the event horizon (contrast Figure 7 with Figure 2). In the extremal case it immediately follows from theabove theorem that for all scattering data with finite T − flux which however does not lie in the weightedenergy classes E H + ∩J + (Σ ) and E I + ∩J + (Σ ) along H + and I + the non-degenerate norm E Σ must blow up.5sing techniques of [23] one can construct special such scattering data for which the N -flux, specifically,across Σ blows up.The following theorem concerns the scattering of initial data with higher regularity. Theorem 2. (Rough version of Theorem 4.2) The scattering map defined between appropriate higher-order,degenerate energy spaces in the black hole exterior of ERN is bounded and bijective.
Figure 8: Higher-order degenerate scattering theory on ERN.The above theorem is of particular importance in constructing special solutions with high regularity. Wenext present a result for the black hole interior of ERN.
Theorem 3. (Rough version of Theorem 4.3) The scattering map in the black hole interior of ERN definedbetween weighted energy spaces is bounded and bijective.
Figure 9: Scattering theory in the black hole interior of ERN.We will next provide a few applications of the above theorems. The first application has to do with therelation of decay along H + and I + and regularity of the data on the hypersurface Σ . Theorem 4. (Rough version of Theorem 4.4) Solutions to the wave equation (1.1) on ERN with sufficientlyfast polynomially decay rates along H + and I + have finite W k, norm in the domain of dependence of Σ . Theorem 5. (Rough version of Theorem 4.5) Consider smooth scattering data which are exponential intime functions with identical decay rates on H + and on I + . There exists a unique exponentially decayingsmooth solution to the equation (1.1) which admits these data. We refer to such solutions as mode solutions. See also Remarks 4.3 and 4.4 for a discussion about therelation of our modes solutions and the quasinormal modes.Finally, we have the following application for the black hole interior of ERN.
Theorem 6. (Rough version of Theorem 4.6) Solutions to the wave equation (1.1) on ERN with finite E Σ energy norm on the hypersurface Σ have finite W , norm in the black hole interior region up to andincluding the Cauchy horizon. Figure 11: Finiteness of W , norm in the black hole interior of ERN.Contrast Figure 11 with Figure 4 in the sub-extremal case. See also Remark 4.5. A closely related topic to the scattering theories on black holes is the black hole stability problem for theforward-in-time evolution. Intense research has been done for both sub-extremal and extremal black holesin this direction. Decay results for the wave equation on the full sub-extremal Kerr family were derived in[21]. Definitive stability results of the linearized gravity system for Schwarzschild and Reissner–Nordstr¨om7ere presented in [19] and [34, 35], respectively. The non-linear stability of Schwarzschild in a symmetryrestricted context was presented in [41]. The rigorous study of linear waves on extremal black holes wasinitiated by the second author in [7, 8, 9, 11, 10] where it was shown that scalar perturbations are unstablealong the event horizon in the sense that higher-order transversal derivatives asymptotically blow up towardsthe future. Precise late-time asymptotics were derived in [4]. These asymptotics led to a novel observationalsignature of ERN [3] where it was shown that the horizon instability of ERN is in fact “observable” byobservers at null infinity. For a detailed study of this signature we refer to the recent [14]. For works onextremal Kerr spacetimes we refer to the works [44, 15, 37]. Extentions of the horizon instability have beenpresented in various settings [2, 51, 53, 49, 36, 13, 17]. For a detailed review of scalar perturbations onextremal backgrounds we refer to [12].
The methods developed in this article have applications beyond extremal black holes. Indeed, they maybe also applied in the construction of non-degenerate scattering theories with weighted energy norms inmore general asymptotically flat spacetimes without a local redshift effect at the horizon (which acts as ablueshift effect in backwards evolution). One such example would be the Minkowski spacetime; see Section 5.Since our methods involve weighted and non-degenerate energies, we expect them to be particularly usefulfor developing a scattering theory for nonlinear wave equations satisfying the classical null condition, asweighted energies need to be controlled in order to obtain global well-posedness for the (forwards) initialvalue problem [40]. It would be moreover interesting to explore the generalization of our methods to thesetting of perturbations of Minkowski in the context of a scattering problem for the Einstein equations. Seealso [43] for work in this direction.Another interesting direction to explore is the construction of dynamically extremal black holes settlingdown to extremal Reissner–Nordstr¨om with inverse polynomial rates from initial data along the future eventhorizon and future null infinity, which would involve a generalization of the backwards evolution estimatesin this article to the setting of the Einstein equations. Note that the construction of dynamically extremalblack holes settling down exponentially follows from an application of the methods of [18]. However, whereasit is conjectured in [18] that a scattering construction of dynamically sub-extremal black holes settling downinverse polynomially will generically result in spacetimes with a weak null singularity at the event horizon,our methods suggest that the event horizon of dynamically extremal black holes will generically be moreregular (with the regularity depending on the assumed polynomial decay rate).
We provide in this section an overview of the remainder of the paper. • In Section 2, we introduce the extremal Reissner–Nordstr¨om geometry and spacetime foliations. Wealso introduce the main notation used throughout the rest of the paper. • We introduce in Section 3 the main Hilbert spaces which appear as domains for our scattering maps. • Having introduced the main notation and Hilbert spaces, we subsequently give precise statement ofthe main theorems of the paper in Section 4. • In Section 5, we outline the main new ideas introduced in the present paper and we provide a sketchof the key proofs. • We construct in Section 6 the forwards scattering map F , mapping initial data on a mixed spacelike-null hypersurface to the traces of the radiation field at the future event horizon and future null infinity.We moreover construct restrictions to this map which involve additionally higher-order, degeneratenorms. • In Section 7, we construct the backwards evolution map B , which send initial data for the radiation fieldat the future event horizon and future null infinity to the trace of the solution at a mixed spacelike-nullhypersurface and is the inverse of F . Similarly, we construct restrictions of B involving higher-order,degenerate norms. 8 We prove in Section 8 additional energy estimates (in forwards and backwards time direction) that allowus to construct invertible maps F ± that send initial data along the asymptotically flat hypersurface { t = 0 } to the future event horizon/null infinity and past event horizon/null infinity, respectively. Thecomposition S = F + ◦ F − − defines the scattering map, which may be thought of as the key object inour non-degenerate scattering theory. • In Section 9 we construct a scattering map S int in a subset of the black hole interior of extremalReissner–Nordstr¨om. • In the rest of the paper, we provide several applications of the scattering theory developed in theaforementioned sections. In Section 10, we apply the backwards estimates of Section 7 to constructarbitrarily regular solutions to (1.1) from data along future null infinity and the future event horizon.As a corollary, we construct in Section 11 smooth mode solutions from data at infinity and the eventhorizon.
The second author (S.A.) acknowledges support through NSERC grant 502581, an Alfred P. Sloan Fellowshipin Mathematics and the Connaught Fellowship 503071.
Consider the 1-parameter family of extremal Reissner-Nordstr¨om spacetimes ( M ext , g M ), where M ext = R × [ M, ∞ ) × S is a manifold-with-boundary. In ( v, r, θ, ϕ ) coordinates, g can be expressed as follows: g M = − Ddv + 2 dvdr + r /g S , (2.1)where D ( r ) = (1 − M r − ) , with M > θ, ϕ ) are spherical coordinates on S . Wedenote the boundary as follows H + := ∂ M ext = { r = M } . We refer to H + as the future event horizon . Thecoordinate vector field T := ∂ v is a Killing vector field that generates the time-translation symmetry of thespacetime.Consider u = v − r ∗ ( r ), with r ∗ ( r ) = r − M − M ( r − M ) − + 2 M log (cid:18) r − MM (cid:19) . We moreover denote t = ( v + u ) and we will also employ the notation u + := u , u − := v , v + := v and v − := u .We can change to the coordinate chart ( u, r, θ, ϕ ) on the manifold ˚ M ext = M ext \ H + , in which g can beexpressed as follows: g M = − Ddu − dudr + r /g S , (2.2)and ˚ M ext = R u × ( M, ∞ ) r × S . By employing the coordinate chart ( u, r, θ, ϕ ), we can moreover smoothlyembed ˚ M ext into a different manifold-with-boundary M (cid:48) ext = R × [ M, ∞ ) × S , where we denote H − := ∂ M (cid:48) ext = { r = M } . We refer to H − as the past event horizon . In these coordinates T = ∂ u .Finally, it will also be convenient to employ the Eddington–Finkelstein double null coordinate chart( u, v, θ, ϕ ) in ˚ M ext , in which g takes the following form: g M = − Ddudv + r /g S . (2.3)Here, ( u, v ) ∈ R × R .In these coordinates T = ∂ u + ∂ v . We moreover introduce the following vector field notation in ( u, v, θ, ϕ )coordinates: L := ∂ v , := ∂ u . We have that L ( r ) = D and L ( r ) = − D . Note that in ( v, r ) coordinates, we can express: ∂ r = 2 D − L. Let / ∇ denote the induced covariant derivative on the spheres of constant ( u, v ). Then we denote thefollowing rescaled covariant derivative: / ∇ S = r / ∇ . The rescaled covariant derivative / ∇ S is the standard covariant derivative on the unit round sphere.Consider the following rescaled radial coordinate on ˚ M ext : x := r . The metric g M takes the followingform in ( u, x, θ, ϕ ) coordinates: g M = − Ddu + 2 r dudx + r /g S . We can then express ˚ M ext = R u × (0 , M ] x × S . We can embed M ext into the manifold-with-boundary (cid:99) M ext = (cid:18) R u × (cid:20) , M (cid:21) x × S (cid:19) ∪ H + . We denote I + := R u × { } x × S and refer to this hypersurface as future null infinity . By considering aconformally rescaled metric ˆ g M = r − g M = − Dx du + 2 dudx + /g S in ( u, x, θ, ϕ ) coordinates, we can extend ˆ g M smoothly to (cid:99) M ext so that I + embeds as a genuine null boundarywith respect to ˆ g M . This interpretation, however, will not be necessary for our purposes.Similarly, we can embed M (cid:48) ext = R v × (0 , M ] x × S into the manifold-with-boundary (cid:100) M (cid:48) ext = R v × (cid:20) , M (cid:21) x × S and define past null infinity as the hypersurface I − := R v × { } x × S , which can be interpreted as a nullboundary with respect to a smooth extension of ˆ g . By employing ( v, r, θ, ϕ ) coordinates it follows immediately that we can smoothly embed ( M ext , g M ) intothe manifold M = R v × (0 , ∞ ) r × S , where the metric g takes on the form (2.1). We will refer to the subset M int = R v × (0 , M ] r × S as the black hole interior . By defining u = v − r ∗ ( r ) in ˚ M int = R v × (0 , M ) r × S ,with r ∗ ( r ) = r − M + M ( M − r ) − + 2 M log (cid:18) M − rM (cid:19) we can also introduce ( u, r, θ, ϕ ) coordinates on ˚ M int , in which the metric takes the expression (2.2). In thesecoordinates, it immediately follows that we can embed ˚ M int into a larger manifold (cid:102) M = R u × (0 , ∞ ) r × S .Let us denote the manifold-with-boundary (cid:102) M int = R u × (0 , M ] r × S and the boundary CH + := ∂ (cid:102) M int = { r = M } ⊂ (cid:102) M , which we refer to as the inner horizon or the Cauchy horizon (the latter terminology follows from the globallyhyperbolic spacetime regions considered in Section 2.3).Finally, it is also useful to work in Eddington–Finkelstein double-null coordinates ( u, v, θ, ϕ ) in ˚ M int , inwhich the metric g takes the form (2.3), with ( u, v ) ∈ { ( u (cid:48) , v (cid:48) ) ∈ R | r ( u (cid:48) , v (cid:48) ) > } . Furthermore, as in ˚ M ext ,we have that L ( r ) = D ( r ) and L ( r ) = − D ( r ). The positivity of L ( r ) in ˚ M int illustrates the following characteristic property of extremal Reissner–Nordstr¨om black holes:the spheres foliating the black hole interior are not trapped. .3 Foliations We introduce the function v Σ ( r ) : [ M, ∞ ) → R , defined as follows: v Σ ( M ) = v > dv Σ dr ( r ) = h ( r ),where h : [ M, ∞ ) → R is a piecewise smooth non-negative function satisfying h ( r ) = 0, when r ∈ [ M, r H ],with M < r H < M and h = 2 D − ( r ) for r ∈ [ r I , ∞ ), with 2 M < r I < ∞ . Furthermore, in r H < r < r H , h is smooth and satisfies 0 < h < D − .Consider the corresponding hypersurface Σ = { ( v, r, θ, ϕ ) ∈ M ext | v = v Σ ( r ) } . Then N v := Σ | { r ∈ ( M,r H ) } is an ingoing null hypersurface intersecting H + , tangential to L and N u := Σ | { r ∈ [ r I , ∞ ) } is an outgoing nullhypersurface, tangential to L . Furthermore, Σ | { r ∈ ( r H ,r I ) } is spacelike. We denote u Σ ( r ) := v Σ ( r ) − r ∗ andobserve that u := lim r →∞ u Σ ( r ) = u r H < ∞ . Without loss of generality, we can assume that u > v appropriately large for fixed r H and r I ). We will consider the coordinate chart ( ρ := r | Σ , θ, ϕ ) on Σ.We denote with D ± ( S ) the future and past domain of dependence, respectively, of a spacelike or mixedspacelike-null hypersurface S . Let R := D + (Σ). We can foliate R as follows: R = (cid:91) τ ∈ [0 , ∞ ) Σ τ , where Σ τ denote the hypersurfaces induced by the flowing Σ along T , with Σ = Σ.Denote with N τ = { v = τ + v , M ≤ r ≤ r H } the ingoing null part of Σ τ and with N τ = { u = τ + u , r ≥ r I } the outgoing part.We can extend R (with respect to the ( u, x, θ, ϕ ) coordinate chart) into the extended manifold-with-boundary (cid:99) M ext by attaching the boundary I + ≥ u := I + ∩ { u ≥ u } : (cid:98) R := R ∪ I + ≥ u . Note that we can similarly consider D − (Σ (cid:48) ) where Σ (cid:48) is the time-reversed analogue of Σ (the roles of u and v reversed) that intersects H − and define, with respect to ( v, x, θ, ϕ ) coordinates and v (cid:48) ∈ R theanalogue of u and also define I −≤ v (cid:48) := I − ∩ { v ≤ v (cid:48) } .The hypersurface Σ naturally extends to a hypersurface (cid:98) Σ in (cid:98) R , with endpoints on H + and I + , and canbe equipped with the coordinate chart ( χ = x | (cid:98) Σ , θ, ϕ ).We moreover define H + ≥ v = H + ∩ { v ≥ v } .Let u int <
0. We will denote with N int v the hypersurface { r ( u int , v ) < r < M | v = v } ⊂ M int .Furthermore, we let Σ int0 := Σ ∪ N int v . We denote furthermore (cid:101)
Σ := { t = 0 } ,D u (cid:48) := D + (cid:16)(cid:101) Σ ∩ { u ≤ u (cid:48) } (cid:17) ,D v (cid:48) := D + (cid:16)(cid:101) Σ ∩ { v ≤ v (cid:48) } (cid:17) . We foliate the regions D − u , with u >
0, by outgoing null hypersurfaces that we also denote N u (cid:48) . Inthis setting N u (cid:48) = { u (cid:48) = u | v ≥ | u |} . It is also useful to consider a foliation by ingoing null hypersurfaces I v (cid:48) = { v = v (cid:48) | − v ≤ u ≤ − u } .Similarly, we foliate D − v by ingoing hypersurfaces N v (cid:48) = { v (cid:48) = v | u ≥ | v |} and outgoing hypersurfaces H u (cid:48) = { u = u (cid:48) | − u ≤ v ≤ − v } .We moreover consider the following null hypersurfaces in D + (Σ int0 ) ∩ ˚ M int : N int v (cid:48) = { v = v (cid:48) | | u | ≤ | u int |} and H int u (cid:48) = { u = u (cid:48) v ≥ v } . We refer to Figure 12 for an illustration of the above foliations and hypersurfaces.11igure 12: A Penrose diagrammatic representation of the four main spacetime regions (shaded) of theextremal Reissner–Nordstr¨om manifold M where we derive energy estimates, together with their respectivefoliations.We use the following notation for the standard volume form on the unit round sphere: dω = sin θdθdϕ .Let n τ and n (cid:101) Σ be the normal vector fields to Σ τ and (cid:101) Σ, respectively. We denote with dµ τ , dµ (cid:101) Σ the inducedvolume forms on Σ τ and (cid:101) Σ respectively. On the null segments N τ and N τ , n τ and dµ Σ τ are not uniquelydefined, so we take the following conventions: n τ | N τ = L,dµ τ | N τ = r dωdv, n τ | N τ = L,dµ τ | N τ = r dωdu. We moreover use the notation dµ g M for the natural volume form on M ext or (cid:102) M int . Note that in ( u, v, θ, ϕ )coordinates on either ˚ M ext or ˚ M int , we can express: dµ g M = Dr dωdudv. We use the notation dµ ˆ g M for the natural volume form on (cid:99) M ext (corresponding to the metric ˆ g M ). In( u, x, θ, ϕ ) coordinates on (cid:99) M ext \ H + , we can express: dµ ˆ g M = dωdudx. Let n ∈ N . Suppose K ⊂ (cid:98) R is compact. Then the Sobolev spaces W n, ( K ) are defined in a coordinate-independent way with respect to the following norm: || f || W n, ( K ) := (cid:88) ≤ k + k + k ≤ n (cid:90) K | / ∇ k S ( r L ) k ( D − L ) k f | dµ ˆ g M . Recall that we can write in ( v, r, θ, ϕ ) coordinates: 2 D − L = ∂ r , which is a regular vector field in (cid:98) R .Furthermore, we can express in ( u, x, θ, ϕ ) coordinates: r L = 12 D∂ x , which implies that r L is also regular in (cid:98) R . Hence, W n, ( K ) is a natural choice of Sobolev space withrespect to the conformal metric ˆ g M . 12f K int ⊂ M int is compact, we instead define W n, ( K int ) in a coordinate-independent way with respectto the following norm: || f || W n, ( K int ) := (cid:88) ≤ k + k + k ≤ n (cid:90) K int | / ∇ k ( D − L ) k L k f | dµ g M In ( u, r, θ, ϕ ) coordinates, we can express 2 D − L = ∂ r , which is a regular vector field in (cid:102) M int . We can alsoexpress L = ∂ u − D∂ r , in ( u, r, θ, ϕ ) coordinates, which clearly is also regular (cid:102) M int . We have that W n, ( K int ) is therefore a naturalchoice of Sobolev space with respect to g M .We define the Sobolev spaces W , ( N int v ) with respect to the following norm: || f || W , ( N int v ) := (cid:88) ≤ k + k ≤ (cid:90) Mr ( u ,v ) (cid:90) S | / ∇ k ( D − L ) k f | (cid:12)(cid:12) N int v r dωdr. Let f, g be positive real-valued functions. We will make use of the notation f (cid:46) g when there exists aconstant C > f ≤ C · g . We will denote f ∼ g when f (cid:46) g and g (cid:46) f . We will also employ thealternate notation f ∼ c,C g , with f, g for 0 < c ≤ C positive constants, to indicate: c · g ≤ f ≤ C · g. We use the “big O” notation O (( r − M ) p ) and O ( r − p ), p ∈ R to group functions f of r satisfying | f | (cid:46) ( r − M ) p , | f | (cid:46) r − p , respectively. In this section, we will introduce the Hilbert spaces on which we will define scattering maps. Before we cando so, we will need existence and uniqueness (in the smooth category) for the Cauchy problem for (1.1) onextremal Reissner–Nordstr¨om.
Theorem 3.1.
1. Consider (Ψ , Ψ (cid:48) ) ∈ C ∞ (Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) . Then there exists a uniquesolution ψ ∈ C ∞ ( D + (Σ )) to (1.1) such that ψ Σ = Ψ and n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } = Ψ (cid:48) .2. Consider (Ψ , Ψ (cid:48) ) ∈ C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) . Then there exists a unique solution ψ ∈ C ∞ ( D + ( (cid:101) Σ) ∪ H + ) to (1.1) such that ψ (cid:101) Σ = Ψ and n (cid:101) Σ ψ | (cid:101) Σ = Ψ (cid:48) . We denote with C ∞ ( (cid:98) Σ ) the space of smooth functions on the hypersurface (cid:98) Σ , with respect to thecoordinate chart ( χ, θ, ϕ ) introduced in Section 2.3. We denote with C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) the space ofsmooth function on the restriction Σ ∩ { r H ≤ r ≤ r I } , with respect to the coordinate chart ( ρ, θ, ϕ ).Let us introduce the stress-energy tensor T [ ψ ] of (1.1), defined as follows with respect to a coordinatebasis: T αβ [ ψ ] := ∂ α ψ∂ β ψ − g αβ ( g − ) κλ ∂ κ ψ∂ λ ψ. Given a vector field X on M , we define the corresponding X - energy current J X as follows:( J X [ ψ ]) α = T αβ [ ψ ] X β . We will denote the radiation field of ψ as follows: φ := rψ. We define the following energy space 13 efinition 3.1.
Define the norm || · || E T Σ0 as follows: let ( r Ψ , Ψ (cid:48) ) ∈ C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) ,then || (Ψ , Ψ (cid:48) ) || E T Σ0 := (cid:90) Σ J T [ ψ ] · n dµ . where ψ denotes the smooth extension of Ψ to R that satisfies ψ | Σ = Ψ and n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } = Ψ (cid:48) and solves (1.1) (see Theorem 3.1), so that all derivatives of ψ above can be expressed solely in terms ofderivatives of Ψ and Ψ (cid:48) .We also define the norm || · || E Σ0 on C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) as follows: || (Ψ , Ψ (cid:48) ) || E Σ0 := (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n dµ , We denote with E T Σ and E Σ the completions of C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) with respect to thenorms || · || E T Σ0 and || · || E Σ0 , respectively. Note that, by construction, E Σ ⊂ E T Σ . Definition 3.2.
Define the norm || · || E T (cid:101) Σ as follows: let (Ψ , Ψ (cid:48) ) ∈ C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) , then || (Ψ , Ψ (cid:48) ) || E T (cid:101) Σ := (cid:90) (cid:101) Σ J T [ ψ ] · n (cid:101) Σ dµ (cid:101) Σ , where ψ denotes the smooth extension of Ψ to D + ( (cid:101) Σ) that satisfies ψ | (cid:101) Σ = Ψ and n (cid:101) Σ ψ | (cid:101) Σ = Ψ (cid:48) and solves (1.1) (see Theorem 3.1), so that all derivatives of ψ above can be expressed solely in terms of derivatives of Ψ and Ψ (cid:48) .We also define the norm || · || E (cid:101) Σ on C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) as follows: || (Ψ , Ψ (cid:48) ) || E (cid:101) Σ := (cid:88) j =0 (cid:90) (cid:101) Σ ∩{ v ≥ v r I } r − j ( LT j φ ) + r − j ( LT j φ ) + r − j | / ∇ S T j φ | dv + (cid:90) (cid:101) Σ ∩{ u ≥ u r H } ( r − M ) − j ( LT j φ ) + ( r − M ) − j ( LT j φ ) + ( r − M ) j | / ∇ S T j φ | dv + (cid:88) m +2 | α |≤ (cid:90) (cid:101) Σ J T [ T m Ω α ψ ] · n (cid:101) Σ dµ (cid:101) Σ . We denote with E T (cid:101) Σ and E (cid:101) Σ the completions of C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) with respect to the norms || · || E T (cid:101) Σ and || · || E (cid:101) Σ , respectively. Note that, by construction, E (cid:101) Σ ⊂ E T (cid:101) Σ . We denote with C ∞ c ( H + ≥ v ) and C ∞ c ( I + ≥ u ) the spaces of smooth, compactly supported functions on H + ≥ v and I + ≥ u , respectively. Definition 3.3.
Let u , v > . Define the norms || · || E T H + ≥ v and || · || E T I + ≥ u as follows: let (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) , then || Φ || E T H + ≥ v := (cid:90) H + ≥ v ( ∂ v Φ) dωdv, || Φ || E T I + ≥ u := (cid:90) I + ≥ u ( ∂ u Φ) dωdu. e also define the norms || · || E H + ≥ v and || · || E I + ≥ u as follows: let (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) , then || Φ || E H + ≥ v := (cid:88) j =0 (cid:90) H + ≥ v v − j ( ∂ j +1 v Φ) + | / ∇ S Φ | dωdv, || Φ || E I + ≥ u := (cid:88) j =0 (cid:90) I + ≥ u u − j ( ∂ j +1 u Φ) + | / ∇ S Φ | dωdu. Then we denote with E T H + ≥ v ⊕ E T I + ≥ u and E H + ≥ v ⊕ E I + ≥ u the completions of C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) withrespect to the product norms associated to || · || E T H + ≥ v and || · || E T I + ≥ u and || · || E H + ≥ v and || · || E I + ≥ u , respectively.Note that E H + ≥ v ⊕ E I + ≥ u ⊂ E T H + ≥ v ⊕ E T I + ≥ u . Definition 3.4.
Define the norms || · || E T H± and || · || E T I± on respectively H ± and I ± as follows: let (Φ , Φ) ∈ C ∞ c ( H ± ) ⊕ C ∞ c ( I ± ) , then || Φ || E T H± := (cid:90) H ± ( ∂ v ± Φ) dωdv ± , || Φ || E T I± := (cid:90) I ± ( ∂ u ± Φ) dωdu ± , with respect to the coordinate charts ( u ± , v ± , θ, ϕ ) .We also define the norms || · || E H± and || · || E I± on respectively H ± and I ± as follows: let (Φ , Φ) ∈ C ∞ c ( H ± ) ⊕ C ∞ c ( I ± ) , then || Φ || E H± := (cid:88) j =0 (cid:90) H ± (1 + | v ± | ) − j ( ∂ j +1 v ± Φ) + | / ∇ S Φ | + | / ∇ S ∂ v ± Φ | dωdv ± , || Φ || E I± := (cid:88) j =0 (cid:90) I ± (1 + | u ± | ) − j ( ∂ j +1 u ± Φ) + | / ∇ S Φ | + | / ∇ S ∂ u ± Φ | dωdu ± . Then we denote with E T H ± ⊕ E T I ± and E H ± ⊕ E I ± the completions of C ∞ c ( H ± ) ⊕ C ∞ c ( H ± ) with respect tothe product norms associated to || · || E T H± and || · || E T H± , and || · || E H± and || · || E H± ,respectively.Note that E H ± ⊕ E I ± ⊂ E T H ± ⊕ E T I ± . In this section, we will introduce analogues of the Hilbert spaces introduced in Section 3.1, but with normsdepending on degenerate higher-order derivatives.
Definition 3.5.
Define the norm || · || E n ;Σ0 as follows: let ( r Ψ , Ψ (cid:48) ) ∈ C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) ,then || (Ψ , Ψ (cid:48) ) || E n ;Σ0 := (cid:88) j =0 (cid:88) m +2 | α | +2 k ≤ n (cid:90) N u r k − j ( L k T m + j Ω α φ ) dωdv + (cid:90) N v ( r − M ) − − k + j ( L k T m + j Ω α φ ) dωdu + (cid:88) m +2 | α |≤ n +2 | α |≤ n (cid:90) Σ J T [ T m Ω α ψ ] · n dµ . e denote with E n ;Σ the completion of C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) with respect to the norm || · || E n ;Σ0 . Definition 3.6.
Define the norm || · || E n ; (cid:101) Σ as follows: let (Ψ , Ψ (cid:48) ) ∈ C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) , then || (Ψ , Ψ (cid:48) ) || E n ; (cid:101) Σ := (cid:88) j =0 (cid:88) m +2 | α | +2 k ≤ n (cid:90) (cid:101) Σ ∩{ v ≥ v r I } r k − j ( L k +1 Ω α T j + m φ ) + r k − j | / ∇ S L k Ω α T j + m φ | + r k +2 − j ( L k +1 Ω α T j + m φ ) + r k − j | / ∇ S L k Ω α T j + m φ | dωdr + (cid:88) m +2 | α | +2 k ≤ n (cid:90) (cid:101) Σ ∩{ v ≥ v r I } r k | / ∇ S L k +1 Ω α T m φ | + r k − | / ∇ S L k Ω α T m φ | + r k | / ∇ S L k +1 Ω α T m φ | + r k − | / ∇ S L k Ω α T m φ | dωdr + (cid:88) j =0 (cid:88) m +2 | α | +2 k ≤ n (cid:90) (cid:101) Σ ∩{ u ≥ u r H } ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k + j | / ∇ S L k Ω α T j + m φ | + ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k + j | / ∇ S L k Ω α T j + m φ | dωdr ∗ + (cid:88) m +2 | α | +2 k ≤ n (cid:90) (cid:101) Σ ∩{ u ≥ u r H } ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | + ( r − M ) − k +2 | / ∇ S L k Ω α T m φ | + ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | + ( r − M ) − k +2 | / ∇ S L k Ω α T m φ | dωdr ∗ + (cid:88) m +2 | α |≤ n +2 (cid:90) (cid:101) Σ J T [ T m Ω α ψ ] · n (cid:101) Σ dµ (cid:101) Σ , where ψ denotes the smooth extension of Ψ to R that satisfies ψ | (cid:101) Σ = Ψ and n (cid:101) Σ ψ | (cid:101) Σ = Ψ (cid:48) and solves (1.1) (see Theorem 3.1), so that all derivatives of ψ above can be expressed solely in terms of derivatives of Ψ and Ψ (cid:48) . We denote with E n ; (cid:101) Σ the completion of C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) with respect to the norm || · || E (cid:101) Σ . Definition 3.7.
Let n ∈ N and u , v > . Define the higher-order norms || · || E n ; H + ≥ v and || · || E n ; I + ≥ u asfollows: let (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) , then || Φ || E n ; H + ≥ v := (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) H + ≥ v v k +2 − j ( L k + m + j Ω α Φ) + | / ∇ S L m Ω α Φ | dωdv, || Φ || E n ; I + ≥ u := (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) I + ≥ u u k +2 − j ( L k + m + j Ω α Φ) + | / ∇ S L m Ω α Φ | dωdu Then we denote with E n ; H + ≥ v ⊕ E n ; I + ≥ u the completion of C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) with respect to the norms || · || E n ( H + ≥ v ) and || · || E n ; I + ≥ u . Note that for all n ∈ N , E n ; H + ≥ v ⊕ E n ; I + ≥ u ⊂ E H + ≥ v ⊕ E I + ≥ u ⊂ E T H + ≥ v ⊕ E T I + ≥ u . Definition 3.8.
Let n ∈ N . Define the higher-order norms || · || E n ; H± and || · || E n ; I± , as follows: let (Φ , Φ) ∈ C ∞ c ( H ± ) ⊕ C ∞ c ( I ± ) , then || Φ || E n ; H± := (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) H ± (1 + | v ± | ) k − j ( ∂ k + m + jv ± Ω α Φ) + (1 + | v ± | ) k | / ∇ S ∂ k + mv ± Ω α Φ | + (1 + | v ± | ) k | / ∇ S ∂ k +1+ mv ± Ω α Φ | dωdv ± , | Φ || E n ; I± := (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) I ± (1 + | u ± | ) k +2 j ( ∂ k + m + ju ± Ω α T − n Φ) + (1 + | u ± | ) k | / ∇ S ∂ k + mu ± Ω α Φ | + (1 + | u ± | ) k | / ∇ S ∂ k +1+ mu ± Ω α Φ | dωdu ± , with respect to the coordinate charts ( u ± , v ± , θ, ϕ ) .Then we denote with E n ; H ± ⊕ E n ; I ± the completion of C ∞ c ( H ± ) ⊕ C ∞ c ( I ± ) with respect to the norms || · || E n ; H± and || · || E n ; I± . In this section, we introduce additional energy spaces that play a role in a non-degenerate scattering theoryfor the extremal Reissner–Nordstr¨om black hole interior.
Definition 3.9.
Let v > and u int < . Define the norms || · || E H + ≥ v and || · || E CH + ≤ u int as follows: let Φ ∈ C ∞ c ( H + ≥ v ) and Φ ∈ C ∞ c ( CH + ≤ u int ) , then || Φ || E H + ≥ v := (cid:90) H + ≥ v v ( ∂ v Φ) + | / ∇ S Φ | dωdv, || Φ || E CH + ≤ u int := (cid:90) CH + ≥ u int u ( ∂ u Φ) + | / ∇ S Φ | dωdu. Then we denote with E int H + ≥ v and E int CH + ≤ u int the completions of C ∞ c ( H + ≥ v ) and C ∞ c ( CH + ≤ u int ) with respectto the norms || · || E H + ≥ v and || · || E CH + ≤ u int , respectively. Definition 3.10.
Let v > and u int < . Define the norms || · || E N int v and || · || E H int u int as follows: let Φ ∈ C ∞ ( N int v ) and Φ ∈ C ∞ ( H int u int ) , then || Φ || E N int v := (cid:90) N int v u ( ∂ u Φ) + D | / ∇ S Φ | dωdu, || Φ || E H int u int := (cid:90) H int u int v ( ∂ v Φ) + D | / ∇ S Φ | dωdv. Then we denote with E N int v and E H int u int the completions of C ∞ ( N int v ) and C ∞ ( H int u int ) with respect to thenorms || · || E N int v and || · || E H int u int , respectively. In this section, we give precise statements of the results proved in this paper. We refer to Sections 2 and 3 foran introduction to the notation and definitions of the objects appearing in the statements of the theorems.
We first state the main theorems that establish a non-degenerate scattering theory in extremal Reissner–Nordstr¨om.
Theorem 4.1.
The following linear maps F : C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) → E H + ≥ v ⊕ E I + ≥ u , (cid:102) F ± : C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) → E H ± ⊕ E I ± , ith F (Ψ , Ψ (cid:48) ) = ( rψ | H + ≥ v , rψ | I + ≥ u ) , (cid:102) F ± (Ψ , Ψ (cid:48) ) = ( rψ | H ± , rψ | I ± ) are well-defined.Furthermore, their unique extensions F : E Σ → E H + ≥ v ⊕ E I + ≥ u , (cid:102) F ± : E (cid:101) Σ → E H ± ⊕ E I ± are bijective and bounded linear operators, and S := F + ◦ F − − : E H − ⊕ E I − → E H + ⊕ E I + is also a bijective bounded linear operator. We refer to the maps F and F ± as a forwards evolution maps , F − and (cid:102) F − ± and backwardsevolution maps and S as the scattering map . Remark 4.1.
An analogous result holds with respect to the degenerate energy spaces E T Σ , E T (cid:101) Σ , E T H + ≥ v , E T I + ≥ u , E T H ± and E T I ± . This follows easily from an analogue of Proposition 9.6.1 in [22] applied to the setting ofextremal Reissner–Nordstr¨om; see also Sections 6.5, 7.4 and 8.3. They advantage of Theorem 4.1 is the useof non-degenerate and weighted energy norms that also appear when proving global uniform boundedness anddecay estimates for solutions to (1.1) . The following theorem extends Theorem 4.1 by considering degenerate and weighted higher-order energyspaces.
Theorem 4.2.
Let n ∈ N . We can restrict the codomains of the linear maps F and (cid:102) F ± defined in Theorem4.1, to arrive at F n : C ∞ ( (cid:98) Σ ) × C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) → E n ; H + ≥ v ⊕ E n ; I + ≥ u , (cid:102) F n ; ± : C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ) → E n ; H ± ⊕ E n ; I ± , which are well-defined.Furthermore, the unique extensions F : E n ;Σ → E n ; H + ≥ v ⊕ E n ; I + ≥ u , (cid:102) F n ; ± : E n ; (cid:101) Σ → E n ; H ± ⊕ E n ; I ± are bijective and bounded linear operators and S n := F n ;+ ◦ F − n ; − : E n ; H − ⊕ E n ; I − → E n ; H + ⊕ E n ; I + is also a bijective bounded linear operator. Both Theorem 4.1 and Theorem 4.2 follow by combining Propositions 6.16 and 7.11, Corollary 7.12 andPropositions 8.11 and 8.14.We additionally construct a scattering map restricted to the black hole interior.
Theorem 4.3.
Let u int < with | u int | suitably large. The following linear map: S int : C c ( H + v ≥ v ) × C ∞ ( N int v ) → E CH + ≤ u int ⊕ E H int u int , with S int ( rψ | H + ≥ v , rψ | N int v ) = ( rψ | CH + ≤ u int , rψ H int u int ) is well-defined and extends uniquely as a bijective, bounded linear operator: S int : E H + v ≥ v ⊕ E N int v → E CH + ≤ u int ⊕ E H int u int . Theorem 4.3 is a reformulation of Proposition 9.2.18 .2 Applications
In this section, we state some applications of the non-degenerate scattering theory of Section 4.1.In Theorem 4.4 below, we show that we can obtain unique solutions to (1.1) with arbitrary high
Sobolevregularity (with respect to the differentiable structure on (cid:98) R ) from suitably regular and polynomially decaying scattering data on H + and I + in an L -integrated sense. Theorem 4.4.
Let n ∈ N and let (Φ , Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) . Assume that lim v →∞ v n +1 | Φ | ( v, θ, ϕ ) < ∞ and lim u →∞ u n +1 | Φ | ( u, θ, ϕ ) < ∞ . Define the integral functions ( T − n Φ , T − n Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) as follows: T − n Φ( v, θ, ϕ ) := (cid:90) ∞ v (cid:90) ∞ v . . . (cid:90) ∞ v n − Φ( v n , θ, ϕ ) dv n . . . dv ,T − n Φ( u, θ, ϕ ) = (cid:90) ∞ u (cid:90) ∞ u . . . (cid:90) ∞ u n − Φ( u n , θ, ϕ ) du n . . . du . and assume moreover that (cid:12)(cid:12)(cid:12)(cid:12) T − n Φ (cid:12)(cid:12)(cid:12)(cid:12) E n ; I + ≥ u + (cid:12)(cid:12)(cid:12)(cid:12) T − n Φ (cid:12)(cid:12)(cid:12)(cid:12) E n ; H + ≥ v < ∞ . (4.1) Then there exists a unique corresponding solution ψ to (1.1) that satisfies rψ | H + ≥ v = Φ , rψ | I + ≥ u = Φ and rψ ∈ W n +1 , ( (cid:98) R ) . Theorem 4.4 follows from Proposition 10.5.
Remark 4.2.
Theorem 4.4 illustrates a stark difference in the setting of extremal Reissner–Nordstr¨om withthe sub-extremal setting, where generic polynomially decaying data along the future event horizon and futurenull infinity (with an arbitrarily fast decay rate) lead to blow-up of the non-degenerate energy along Σ ; see[22, 23]. As a corollary of Theorem 4.4, we can moreover construct smooth solutions and in particular smoothsolutions with an exact exponential time dependence.
Theorem 4.5.
Let (Φ , Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) and assume that (Φ , Φ) and all derivatives up to anyorder decay superpolynomially in v and u , respectively. (i) Then there exists a corresponding smooth solution ψ to (1.1) on R such that r ˆ ψ can moreover besmoothly extended to ˆ R with respect to the differentiable structure on ˆ R . (ii) Assume additionally that Φ( v, θ, ϕ ) = f H ( θ, ϕ ) e − iωv and Φ( u, θ, ϕ ) = f I ( θ, ϕ ) e − iωu for f H , f I ∈ C ∞ ( S ) , with ω ∈ C such that Im ω < . Then we can express r · ψ ( τ, ρ, θ, ϕ ) = f ( ρ, θ, ϕ ) e − iω · τ , with f ∈ C ∞ ( ˆΣ) and lim ρ ↓ M f ( ρ, θ, ϕ ) = f H ( θ, ϕ ) , lim ρ →∞ f ( ρ, θ, ϕ ) = f I ( θ, ϕ ) . We refer to ψ as mode solutions . Theorem 4.5 (i) follows from Corollary 11.1 and Theorem 4.5 (ii) follows from Proposition 11.2.
Remark 4.3.
Note that in order for an analogous result to Theorem 4.5 (i) to hold in sub-extremal Reissner–Nordstr¨om, one needs to consider scattering data (Φ , Φ) that are superexponentially decaying, and hence itcannot be used to prove the analogue of Theorem 4.5 (ii) . Nevertheless, the existence of a more restricted classof smooth solutions that behave exponentially in time with arbitrary ω such that Im ω < in sub-extremalReissner–Nordstr¨om can be established by restricting to fixed spherical harmonics and applying standardasymptotic ODE analysis. emark 4.4. One can apply the results of [4] to show that the time translations acting on L -based Sobolevspaces S ( τ (cid:48) ) : W k +1 , ( (cid:98) Σ) × W k, ( (cid:98) Σ ∩ { r H < r ≤ r I } ) → W k +1 , ( (cid:98) Σ) × W k, ( (cid:98) Σ ∩ { r H < r ≤ r I } ) , (Ψ , Ψ (cid:48) ) (cid:55)→ ( ψ | τ = τ (cid:48) , T ψ | τ = τ (cid:48) ) , with ψ the solution to (1.1) associated to (Ψ , Ψ (cid:48) ) , form a continuous semi-group, such that S ( τ (cid:48) ) = e τ (cid:48) A ,with A the corresponding densely defined infinitesimal generator A that formally agrees with T : A ( ψ | τ = τ (cid:48) , T ψ | τ = τ (cid:48) ) = ( T ψ | τ =0 , T ψ | τ =0 ) . The results of [54] imply that, in the setting of asymptotically de Sitter or anti de Sitter spacetimes,quasi-normal modes or resonances are smooth mode solutions that can be interpreted as eigenfunctions of A and the corresponding frequencies ω form a discrete set in the complex plane (cf. the normal modes andfrequencies of an idealised vibrating string or membrane).The smooth mode solutions of Theorem 4.5 (ii) (and those obtained in the sub-extremal setting by ODEarguments as sketched in Remark 4.3) form an obstruction to extending this interpretation to the asymp-totically flat setting. Indeed, all the mode solutions of Theorem 4.5 (ii) are eigenfunctions of A but thecorresponding set of frequencies ω , which is the entire open lower-half complex plane, is certainly not dis-crete. In order to maintain the viewpoint of [54], one has to consider smaller function spaces that exclude the smooth mode solutions of Theorem 4.5 (ii); see [33]. Theorem 4.6.
Let u be suitably large. Then there exist a constant C = C ( M, u , v ) > such that we canestimate in the black hole interior: || ψ || W , ( D + (Σ int ,u ) ∩ (cid:102) M int ) ≤ C (cid:88) j ≤ (cid:16) || ( T j ψ | Σ , n Σ T j ψ | Σ ) || E Σ0 + || T j ψ || W , ( N int v ) (cid:17) . Theorem 4.6 follows from Corollary 9.3.
Remark 4.5.
Theorem 4.6 addresses the question of whether ψ ∈ W , in the black hole interior of extremalReissner–Nordstr¨om for localized, low regularity initial data, which was raised as an open problem in [24].For smooth and localized data, this statement follows from [30, 4]. Indeed, Theorem 4.6 demonstrates thatboundedness of a non-degenerate energy with weights that grow in r (together with boundedness of energiesinvolving additional derivatives that are tangential to the event horizon) is sufficient to establish ψ ∈ W , .Theorem 4.6 can straightforwardly be extended to the Λ > setting of extremal Reissner–Nordstr¨om–de Sitter black holes, where there is no need to include r -weights in the non-degenerate energy norm that issufficient to establish ψ ∈ W , . See also [1] for the results in the interior of extremal Reissner–Nordstr¨om–deSitter. In this section, we provide an overview of the main techniques that are used in the proofs of the theoremsstated in Section 4. We will highlight the key new ideas and estimates that are introduced in this paper.The proof of the main theorems Theorem 4.1 and Theorem 4.2 can roughly be split into four parts:
Showing that the linear maps F , F − and F n , F − n that appear in Theorem 4.1 and Theorem 4.2 arewell-defined when considering as a domain spaces of either smooth or smooth and compactly supportedfunctions. Proving uniform boundedness properties of these linear maps with respect to weighted Sobolev norms.This allows one to immediately extended the linear maps to the completions of the spaces of smooth(and compactly) supported functions with respect to appropriately weighted Sobolev norms.
Constructing the linear maps S and S n . Constructing S int (independently from above). 20 he heart of this paper consists of establishing 2.) and 3.) by proving uniform estimates forsmooth (and compactly supported) data along Σ , (cid:101) Σ and H ± ∪I ± . An overview of the correspondingestimates and techniques leading to is given in Section 5.1 – 5.3. Part follows by complementingthese estimates with additional estimates in D ± ( (cid:101) Σ) near the past limit points of I + and H + , which is brieflydiscussed in Section 5.4. We briefly discuss the black hole interior estimates involved in in Section 5.5.Part follows from local estimates combined with soft global statements that have already beenestablished in the literature. We give an overview of the logic of the arguments in this section.The forwards map F : C ∞ ( (cid:98) Σ ) ⊕ C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) → E H + ≥ v ⊕ E I + ≥ u is well-defined by global existence and uniqueness for (1.1) combined with the finiteness (and decay) of theradiation field rψ , see for example the results in [7, 8, 4].In order to show that the backwards map F − : C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) → E Σ is well-defined, we first need to make sense of the notion of prescribing initial data “at infinity”; that is to say,we need to show as a preliminary step that we can associate to each pair (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u )a unique solution ψ to (1.1), such that rψ | H + ≥ v = Φ and rψ | I + ≥ u = Φ. This may be viewed as a semi-globalproblem. We construct ψ as the limit of a sequence of solutions ψ i arising from a sequence of local initial valueproblems with fixed initial data (Φ , Φ) imposed on the null hypersurfaces H +0 ≤ τ ≤ τ ∗ ∪{ v = v i , u ≤ u ≤ u ( τ ∗ ) } and trivial data on Σ τ ∗ ∩ { v ≤ v i } , such that v i → ∞ as i → ∞ . A very similar procedure was carried out inthe physical space construction of scattering maps on Schwarzschild in Proposition 9.6.1 in [22]. One couldalternatively interpret I + as a genuine null hypersurface with respect to the conformally rescaled metric ˆ g M ,which turns the semi-global problem into a local problem. r -weighted estimates We introduce time-reversed analogues of the r p -weighted estimates of Dafermos–Rodnianski [20] and the( r − M ) − p -weighted estimates of [4]. We first illustrate key aspects of these estimates in the setting of thestandard wave equation on Minkowski. We can foliate the causal future of a null cone C in Minkowski byoutgoing spherical null cones C u = { t − r = u } , with t, r the standard spherical Minkowski coordinates and u ≥
0. Let us denote ∂ v = ( ∂ t + ∂ r ) and ∂ u = ( ∂ t − ∂ r ). We consider smooth, compactly supported initialdata on I + ∩ { ≤ u ≤ u } , with u > ψ vanishes along C u .The r p -weighted estimates applied backwards in time with p = 1 and p = 2 give (cid:90) C u r · ( ∂ v ( rψ )) dωdv (cid:46) (cid:90) C u r · ( ∂ v ( rψ )) dωdv + (cid:90) u u (cid:90) C u T ( ∂ t , ∂ v ) r dωdvdu, (cid:90) C u r · ( ∂ v ( rψ )) dωdv (cid:46) (cid:90) C u r · ( ∂ v ( rψ )) dωdv + (cid:90) u u (cid:90) C u r · ( ∂ v ( rψ )) dωdvdu, for u > u > r p -weighted estimates, the spacetime integrals on the right-hand sidesabove have a bad sign. Hence, in order to obtain control of r -weighted energies along C u , we need to startby controlling (cid:90) u u (cid:90) C u T ( ∂ t , ∂ v ) r dωdr. Note that standard ∂ t -energy conservation implies that for any 0 < u (cid:48) < u : (cid:90) C u (cid:48) T ( ∂ t , ∂ v ) r dωdr = (cid:90) C u T ( ∂ t , ∂ v ) r dωdr + (cid:90) I + ∩{ u (cid:48) ≤ u ≤ u } ( ∂ u ( rψ )) dωdu. (5.1) We use the notation F − to indicated the backwards map, but we still have to show that this map is indeed a two-sidedinverse of F . In contrast with [dafrodshl], we take the limit with respect to L ∞ norms of the weighted quantities ( r ∂ v ) k ( rψ ) with k ∈ N . ψ is vanishing along C u , we can integrate the above equation in u (cid:48) to obtain (cid:90) u u (cid:20)(cid:90) C u T ( ∂ t , ∂ v ) r dωdr (cid:21) du = (cid:90) u u (cid:34)(cid:90) I + ∩{ u (cid:48) ≤ u ≤ u } ( ∂ u ( rψ )) dωdu (cid:35) du (cid:48) . We can integrate by parts to convert one u -integration into an additional u weight: (cid:90) ∞ u (cid:34)(cid:90) I + ∩{ u ≥ u (cid:48) } ( ∂ u ( rψ )) dωdu (cid:35) du (cid:48) = (cid:90) I + ∩{ u ≥ u } ( u − u ) · ( ∂ u ( rψ )) dωdu. (5.2)By applying both the p = 1 and p = 2 estimates above, and integrating by parts once more along I + as in(5.2), we obtain: (cid:90) C r · ( ∂ v ( rψ )) dωdv + (cid:90) C T ( ∂ t , ∂ v ) r dωdr (cid:46) (cid:90) I + ∩{ u ≥ } ( u + 1) · ( ∂ u ( rψ )) dωdu (5.3)Comparing (5.3) with (5.1) with u (cid:48) = 0, we see that we can obtain stronger, weighted uniform controlalong C , provided we control an appropriately weighted energy along I + . One may compare this to the(modified) energy estimate obtained by using the Morawetz conformal vector field K = u ∂ u + v ∂ v , whichis the generator of the inverted time translation conformal symmetries, as a vector field multiplier insteadof ∂ t [47]; see also Section 5.4.The main difference in the setting of extremal Reissner–Nordstr¨om is that the r p -estimates above onlyapply in the spacetime region where r ≥ r I , with r I suitably large, and they have to be complemented byan analogous hierarchy of ( r − M ) − p weighted estimates in a region { r ≤ r H } near H + , i.e. with r H − M sufficiently small. Roughly speaking, the analogue of the p = 2 weighted energy near H + corresponds to therestriction of the following non-degenerate energy (in ( v, r ) coordinates): (cid:90) Σ ∩{ r ≤ r H } T ( N, ∂ r ) r dωdr, where N is a timelike vector field in { M ≤ r ≤ r H } .It is in controlling the non-degenerate energy in the backwards direction that we make essential use ofthe extremality of extremal Reissner–Nordstr¨om or the degeneracy of the event horizon. Indeed, if we wereto consider instead sub -extremal Reissner–Nordstr¨om, we would fail to obtain control of a non-degenerateenergy near H + with polynomially decaying data along H + ∪I + due to the blueshift effect (the time reversedredshift effect); see [22, 23]. In order to control the boundary terms arising from restricting the r -weighted estimates near I + and H + ,we apply the Morawetz estimate derived in [7] in the backwards direction. Note that the presence of trappednull geodesics along the photon sphere at r = 2 M does not lead to a loss of derivatives in the analogue of(5.3). This is because the backwards estimates, in contrast with the forwards estimates (see Section 5.2), donot require an application of a Morawetz estimate with non-degenerate control at the photon sphere. r -weighted estimates revisited We consider again the setting of Minkowski to illustrate the main ideas. In order to construct a bijectionfrom an r -weighted energy space on C to a u -weighted energy space on I + , we need to complement thebackwards estimate (5.3) with the following forwards estimate: (cid:90) C r · ( ∂ v ( rψ )) dωdv + (cid:90) C T ( ∂ t , ∂ v ) r dωdr (cid:38) (cid:90) I + ∩{ u ≥ } ( u + 1) · ( ∂ u ( rψ )) dωdu (5.4)Note that a standard application of the r p -weighted estimates (combined with energy conservation (5.1) anda Morawetz estimate), see [20], is the following energy decay statement: (cid:90) C u T ( ∂ t , ∂ v ) r dωdr (cid:46) (1 + u ) − (cid:20)(cid:90) C r · ( ∂ v ( rψ )) dωdv + (cid:90) C T ( ∂ t , ∂ v ) r dωdr (cid:21) . In this case, one may however assume sufficiently fast exponentially decay along H + ∪ I + to beat the blueshift effect andobtain boundedness of the non-degenerate energy, see [18]. (cid:90) C r · ( ∂ v ( rψ )) dωdv + (cid:90) C T ( ∂ t , ∂ v ) r dωdr (cid:38) (cid:90) I + ∩{ u ≥ } ( u + 1) − (cid:15) · ( ∂ u ( rψ )) dωdu with (cid:15) >
0. In order to take (cid:15) = 0, we instead revisit the r p -estimates and, rather than deriving energy decayalong C u , we observe that the r p -estimates (together with (5.1) and a Morawetz estimate) provide directly control over (cid:90) ∞ (cid:90) ∞ u (cid:34)(cid:90) I + ∩{ u ≥ u (cid:48) } ( ∂ u ( rψ )) dωdu (cid:48)(cid:48) (cid:35) du (cid:48) du. After integrating by parts twice in u as in (5.2), we arrive at (5.4).We arrive at an analogous estimate to (5.4) in the extremal Reissner–Nordstr¨om setting by followingthe same ideas, both near I + and near H + . The main difference is that whenever we apply a Morawetzestimate, we lose a derivative because of the trapping of null geodesics, which we have to take into accountwhen defining the appropriate energy spaces. Given suitably regular and suitably decaying scattering data H + and I + , we can apply Theorem 4.1 toconstruct a corresponding solution ψ ∈ C ∩ W , (with respect the differentiable structure on (cid:98) R ) to (1.1)such that rψ approaches the scattering data as r → M or r → ∞ .In the setting of (1.1) on Minkowski with coordinates ( u, x, θ, ϕ ), where x := r +1 (so that x ↓ r → ∞ and x ↑ r ↓ rψ ∈ W , ([ u , u ] u × (0 , x × S ) for any 0 ≤ u < u < ∞ . Inorder to show that moreover rψ ∈ W , ([ u , u ] u × (0 , x × S ), we first consider T ψ . By rearranging andrescaling (1.1) in Minkowski, we have that in ( u, x ) coordinates:2 ∂ x T ( rψ ) = ∂ x ( x ∂ x ( rψ )) + / ∆ S ( rψ )with x ∂ x ( rψ ) = 2 L ( rψ ). So, we obtain that T ( rψ ) ∈ W , ([ u , u ] u × (0 , x × S )if we can show that L ( rψ ) ∈ W , ([ u , u ] u × (0 , x × S ) and / ∆ S ( rψ ) ∈ W , ([ u , u ] u × (0 , x × S ) . Since / ∆ S commutes with the operator (cid:3) g , both in Minkowski and in extremal Reissner–Nordstr¨om, we canimmediately obtain / ∆ S ( rψ ) ∈ W , from Theorem 4.1 (or its Minkowski analogue). Moreover, L ( rψ ) ∈ W , follows from bounding uniformly in u the integral: (cid:90) C u r ( L ( rψ ) + r ( L ( rψ ))) dωdv. Hence, we have to establish control over improved r -weighted energies where rψ is replaced by L ( rψ ) and L ( rψ ). Analogous improved r -weighted energies have appeared previously in the setting of forwards es-timates in [52, 6, 4], see also the related energies in [48]. The backwards analogues of the correspondingimproved r -weighted estimates form the core of the proof of Theorem 4.2.To pass from T ( rψ ) ∈ W , to rψ ∈ W , , we apply the above estimates to solutions ψ (1) to (1.1), suchthat T ψ (1) = ψ . Such solutions ψ (1) can easily be constructed by considering initial scattering data that are time integrals of the scattering data H + in v and I + in u , assuming moreover that rψ (1) | H + and rψ (1) | I + vanish as v → ∞ and u → ∞ , respectively.In fact, we can show by an extension of the arguments above that T n ( rψ ) ∈ W n, for all n ≥
2, assumingsuitably regular and decaying data along H + and I + , so we can conclude that ψ ∈ W n +1 , , provided thescattering data decays suitably fast in time. In order to obtain more regularity, we need faster polynomial23ecay along H + ∪ I + . This is the content of Theorem 4.4. By considering smooth and superpolynomially decaying data along H + ∪ I + and applying standard Sobolev inequalities, we can in fact take n arbitrarilyhigh and show that ψ ∈ C ∞ ( (cid:98) R ); see Theorem 4.5.Note that time integrals ψ (1) also play an important role in [5, 4] for spherical symmetric solutions. In thatsetting, one needs to solve an elliptic PDE (which reduces to an ODE in spherical symmetry) to construct ψ (1) , which is contrast with the backwards problem, where the construction is much simpler because we canintegrate the scattering data in time to obtain data leading to ψ (1) . The backwards and forwards estimates sketched in Section 5.1 and Section 5.2 allow us to construct abijection between weighted energy spaces on Σ and H + ≥ v ∪ I + ≥ u . In order to construct the bijection S between energy spaces on H − ∪ I − and H + ∪ I + we need to additionally construct a bijection betweenappropriate energy spaces on Σ ∪ H + v ≤ v ∪ I + u ≤ u and (cid:101) Σ = { t = 0 } . Without loss of generality, we can pickΣ so that Σ ∩ { r H ≤ r ≤ r I } = (cid:101) Σ ∩ { r H ≤ r ≤ r I } , and we are left with only proving energy estimates in the regions D − u and D − v , see Section 2.3 and Figure12. While r -weighted estimates are still suitable in the forwards direction in D − u and D − v , they are not suitable in the backwards direction. We therefore consider energy estimates for the radiation field rψ withthe vector field multiplier K = u ∂ u + v ∂ v , both in D − u and D − v in order to arrive at the analogue of the p = 2 estimate. In Minkowski space, K corresponds to the generator of a conformal symmetry, the invertedtime translations. It is a Killing vector field of the rescaled metric r − m , where m is the Minkowski metric.Hence, K may be thought of as the analogue of ∂ t when considering rψ instead of ψ and r − m instead of m .In particular, when considering K as a vector field multiplier in a spacetime region of Minkowski, one canobtain a weighted energy conservation law for rψ . Since r is large in D − u in extremal Reissner–Nordstr¨om, K may be thought of as an “approximate Killing vector field” of the rescaled metric r − g .Another useful property of K is that it is invariant under the Couch–Torrence conformal symmetry [16]that maps D − u to D − v . It therefore plays the same role when used as a vector field multiplier for theradiation field in D − v as it does in D − u .In order to obtain the analogue of the r p -weighted estimate with p = 1 for T ψ , we apply instead thevector field multiplier Y = v∂ v − u∂ u in D − u and Y = u∂ u − v∂ v in D − v .We construct S : C ∞ c ( H − ) ⊕ C ∞ c ( I − ) → E H + ⊕ E I + by first observing that the spacetime is invariant under the map t (cid:55)→ − t , so the above discussion on F − can be applied to associate to each pair (Φ , Φ) ∈ C ∞ c ( H − ) ⊕ C ∞ c ( I − ) a solution ψ ∈ D − ( (cid:101) Σ) such that( ψ | (cid:101) Σ , n (cid:101) Σ ψ | (cid:101) Σ ∩{ r H ≤ r ≤ r I } ) lie in a suitable energy space. We show that in fact ( ψ | Σ , n ψ | Σ ∩{ r H ≤ r ≤ r I } ) ∈ E Σ ,so we can apply (the extension of) F to obtain a pair of radiation fields (Φ (cid:48) , Φ) ∈ E H + ⊕ E I + . We derive estimates for the radiation field in M int using once again the vector field K = u ∂ u + v ∂ v . Recallfrom Section 5.4 that the favourable properties of K as a vector field multiplier are related to its role asan approximate conformal symmetry generator near infinity and its invariance under the Couch–Torrenceconformal symmetry. The equation for the radiation field takes the same form in M int and M ext near H + ifone considers the standard Eddington–Finkelstein double-null coordinates in ˚ M int and in ˚ M ext . Therefore, K (now defined with respect to ( u, v ) coordinates in ˚ M int ) remains useful in the black hole interior. Theusefulness of K in the interior of extremal black holes was already observed in [30, 31, 32]. In this section, we present the energy estimates in the forwards time direction that are relevant for definingthe forwards evolution map F (see Section 6.5). 24 .1 Preliminary estimates We make use of the following Hardy inequalities:
Lemma 6.1 (Hardy inequalities) . Let p ∈ R \ {− } and let f : [ a, b ] → R be a C function with a, b ≥ .Then (cid:90) ba x p f ( x ) dx ≤ p + 1) − (cid:90) ba x q +2 (cid:12)(cid:12)(cid:12)(cid:12) dfdx (cid:12)(cid:12)(cid:12)(cid:12) dx + 2 b p +1 f ( b ) , for p > − , (6.1) (cid:90) ba x p f ( x ) dx ≤ p + 1) − (cid:90) ba x q +2 (cid:12)(cid:12)(cid:12)(cid:12) dfdx (cid:12)(cid:12)(cid:12)(cid:12) dx + 2 a p +1 f ( a ) , for p < − . (6.2) Proof.
See the proof of Lemma 2.2 in [6].We define the angular momentum operators Ω i , with i = 1 , ,
3, as follows:Ω = sin ϕ∂ θ + cot θ cos ϕ∂ ϕ , Ω = − cos ϕ∂ θ + cot θ sin ϕ∂ ϕ , Ω = ∂ ϕ . We denote for α = ( α , α , α ) ∈ N Ω α = Ω α Ω α Ω α . We now state the following standard inequalities on S : Lemma 6.2 (Angular momentum operator inequalities) . Let f : S → R be a C function. Then we canestimate (cid:90) S | / ∇ S f | dω = (cid:88) | α | =1 (cid:90) S (Ω α f ) dω, (6.3) (cid:90) S | / ∇ S f | + | / ∇ S f | dω ∼ (cid:88) | α | =1 (cid:90) S | / ∇ S Ω α f | dω ∼ (cid:88) ≤| α |≤ (cid:90) S (Ω α f ) dω. (6.4) Lemma 6.3 (Degenerate energy conservation) . Let ψ be a smooth solution to (1.1) . Then (cid:90) Σ τ J T [ ψ ] · n τ dµ τ ∼ (cid:90) N τ [( Lψ ) + | / ∇ ψ | ] r dωdu + (cid:90) Σ τ ∩{ r H ≤ r ≤ r I } ( Lψ ) + ( Lψ ) + | / ∇ ψ | dµ τ + (cid:90) N τ [( Lψ ) + | / ∇ ψ | ] r dωdv, (cid:90) H + J T [ ψ ] · L r dωdv = (cid:90) H + ( Lφ ) dωdv, (cid:90) I + J T [ ψ ] · L r dωdu = (cid:90) I + ( Lφ ) dωdu and div J T [ ψ ] ≡ . Proof.
See for example [7, 8].
We now recall some regularity properties of the radiation field at null infinity, which do not immediatelyfollow from Theorem 3.1, and are derived in [4]. 25 emma 6.4.
Let ψ be a smooth solution to (1.1) . Then for all n ∈ N , we have that LL ((2 D − r L ) n φ ) = [ − nr − + O ( r − )] L ((2 D − r L ) n φ )+ Dr − / ∆ S ((2 D − r L ) n φ )+ n − (cid:88) k =0 O ( r − )(2 D − r L ) k φ. (6.5) Proof.
By (1.1) we obtain the following equation for φ : LLφ = − DD (cid:48) r φ + D r / ∆ S φ, (6.6)which implies (6.5) with n = 0. We obtain n ≥ Proposition 6.5.
Let (Ψ , Ψ (cid:48) ) ∈ C ∞ (cid:16)(cid:98) Σ (cid:17) ⊕ C ∞ c (Σ ∩ { r H ≤ r ≤ r I } ) . Then for all k, l ∈ N and α ∈ N , lim v →∞ ( r L ) k T l Ω α φ ( u, v, θ, ϕ ) < ∞ . In particular, the limit r · ψ | I + ( u, θ, ϕ ) := lim v →∞ rψ ( u, v, θ, ϕ ) exists for all u ≥ and defines a smooth function on I + ≥ u .Proof. The k ≤ k ≥ k = 1 using instead the commutedequation (6.5). See also Proposition 6.2 of [4]. The two main ingredients for establishing energy decay estimates forwards in time are
Morawetz estimates away from H + and I + (Theorem 6.6 below) and hierarchies of r p - and ( r − M ) − p -weighted estimates in a neighbourhood of the event horizon and future null infinity (Theorem 6.7 below). Theorem 6.6 (Morawetz/integrated local energy decay estimate, [7]) . Let ≤ τ < τ < ∞ and M < r < r < M < r < r < ∞ , then for all k, l ∈ N and α ∈ N there exists a constant C = C ( r i , M, Σ , k, l, α ) > , such that (cid:90) τ τ (cid:34)(cid:90) Σ τ ∩ ( { r ≤ r ≤ r }∪{ r ≤ r ≤ r } ) ( ∂ v T k ∂ lr Ω α ψ ) + | / ∇ T k ∂ lr Ω α ψ | + ( T k ∂ lr Ω α ψ ) + ( ∂ r T k ∂ lr Ω α ψ ) dµ Σ τ (cid:35) dτ ≤ C k + l + | α | (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n Σ τ dµ Σ τ . (6.7) Furthermore, we have that for any
M < r < r < ∞ : (cid:90) τ τ (cid:34)(cid:90) Σ τ ∩{ r ≤ r ≤ r } J T [ ψ ] · n Σ τ dµ Σ τ (cid:35) dτ ≤ C ( r , r , Σ ) (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n Σ τ dµ Σ τ . (6.8) Theorem 6.7.
Let ψ be a solution to (1.1) arising from initial data (Ψ , Ψ (cid:48) ) ∈ C ∞ (cid:16)(cid:98) Σ (cid:17) ⊕ C ∞ c (Σ ∩ { r H ≤ r ≤ r I } ) . Let k ∈ N and k ≤ p ≤ k , then we can estimate for ll ≤ τ ≤ τ : (cid:90) N τ ( r − M ) − p ( L k +1 φ ) dωdu + (cid:90) N τ r p ( L k +1 φ ) dωdv + (cid:90) H + ∩{ τ ≤ τ ≤ τ } ( r − M ) − p | / ∇ S L k φ | dωdv + (cid:90) I + ∩{ τ ≤ τ ≤ τ } r p − | / ∇ S L k φ | dωdu + (cid:90) A τ τ ( r − M ) − p ( L k +1 φ ) + (2 − p )( r − M ) − p | / ∇ S L k φ | dωdudv + (cid:90) A τ τ r p − ( L k +1 φ ) + (2 − p ) r p − | / ∇ S L k φ | dωdvdτ ≤ C k (cid:88) j =0 (cid:90) N τ ( r − M ) − p +2 j ( L k − j φ ) dωdu + (cid:90) N τ r p − j ( L k − j φ ) dωdv + C k (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n Σ τ dµ Σ τ . (6.9) Proof.
See Proposition 7.6 of [4].By combining Theorem 6.6 and Theorem 6.7 with Lemma 6.3 and applying the mean-value theoremalong a dyadic sequence of times (“the pigeonhole principle”), one can obtain energy decay in time alongthe foliation Σ τ ; see for example [7, 8] and [4] for an application of this procedure in extremal Reissner–Nordstr¨om.In the present article, however, we will not apply the mean-value theorem, bur rather derive uniformboundedness estimates for various time-integrated energies on the left-hand side (see Proposition 6.8). Wewill then use these time-integrated energy estimates to obtain estimates for energy fluxes along H + and I + with growing time weights inside the integrals (Corollary 6.10). Proposition 6.8.
There exists a constant C = C ( M, Σ , r H , r I ) > such that (cid:90) ∞ (cid:90) ∞ τ (cid:90) Σ τ (cid:48) J T [ ψ ] · n τ (cid:48) dµ τ (cid:48) dτ (cid:48) dτ ≤ C (cid:34) (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n Σ dµ Σ (cid:35) . (6.10) and (cid:90) ∞ (cid:90) ∞ τ (cid:34)(cid:90) I + ≥ τ (cid:48) J T [ ψ ] · L r dωdu (cid:35) dτ (cid:48) dτ + (cid:90) ∞ (cid:90) ∞ τ (cid:34)(cid:90) H + ≥ τ (cid:48) J T [ ψ ] · L dωdv (cid:35) dτ (cid:48) dτ + (cid:90) H + ≥ v | / ∇ S φ | dωdv + (cid:90) I + ≥ u | / ∇ S φ | dωdu ≤ C (cid:34) (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n dµ (cid:35) . (6.11) Proof.
Note first of all that for all τ ≥ (cid:90) τ τ (cid:90) N τ J T [ ψ ] · L r dωdvdτ ∼ (cid:90) τ τ (cid:90) N τ r ( L ( r − φ )) + r − | / ∇ S φ | dωdvdτ (cid:46) (cid:90) τ τ (cid:90) N τ r − φ + ( Lφ ) + r − | / ∇ S φ | dωdvdτ (cid:46) (cid:90) τ τ (cid:90) N τ ( Lφ ) + r − | / ∇ S φ | dωdτ + (cid:90) Σ τ J T [ ψ ] · n τ dµ τ , (6.12)27here in the final inequality we applied Lemma 6.1 and (6.7), using that φ attains a finite limit at I + , byProposition 6.5.Similarly, we have that (cid:90) τ τ (cid:90) N τ J T [ ψ ] · L r dωdudτ (cid:46) (cid:90) τ τ (cid:90) N τ ( Lφ ) + ( r − M ) | / ∇ S φ | dωdudτ + (cid:90) Σ τ J T [ ψ ] · n τ dµ τ . (6.13)We combine (6.12) and (6.13) together with (6.8) to obtain the estimate: (cid:90) ∞ τ (cid:90) Σ τ (cid:48) J T [ ψ ] · n τ (cid:48) dµ τ (cid:48) (cid:46) (cid:90) ∞ τ (cid:90) N τ (cid:48) ( Lφ ) + r − | / ∇ S φ | dωdvdτ (cid:48) + (cid:90) ∞ τ (cid:90) N τ (cid:48) ( Lφ ) + ( r − M ) | / ∇ S φ | dωdudτ (cid:48) + (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n τ dµ τ . We now apply (6.9) with k = 0 and p = 1 to obtain: (cid:90) ∞ τ (cid:90) Σ τ (cid:48) J T [ ψ ] · n τ (cid:48) dµ τ (cid:48) (cid:46) (cid:90) N τ r ( Lφ ) dωdv + (cid:90) N τ ( r − M ) − ( Lφ ) dωdu + (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n τ dµ τ . (6.14)By Lemma 6.3 and (6.14), we immediately obtain also (cid:90) ∞ τ (cid:34)(cid:90) I + ≥ τ (cid:48) J T [ ψ ] · L r dω (cid:35) du + (cid:90) ∞ τ (cid:34)(cid:90) H + ≥ τ (cid:48) J T [ ψ ] · L r dω (cid:35) dv (cid:46) (cid:90) N τ r ( Lφ ) dωdv + (cid:90) N τ ( r − M ) − ( Lφ ) dωdu + (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n τ dµ τ . (6.15)We integrate once more in τ and apply (6.9) with k = 0 and p = 2 to obtain (6.10). Equation (6.11)follows from (6.10) by applying Lemma 6.3 applied in the region D + (Σ τ (cid:48) ), together with (6.9) with p = 2and k = 0.The following simple lemma is crucial in order to bound energy norms along H + and I + with time-weightsinside the integrals. Lemma 6.9.
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 . (6.16) Proof.
We integrate the left-hand side of (6.16) by parts to obtain (cid:90) ∞ x ( x − x ) n f ( x ) dx = − (cid:90) ∞ x ( x − x ) n ddx (cid:20)(cid:90) ∞ x f ( x ) dx (cid:21) dx = n (cid:90) ∞ x ( x − x ) n − (cid:90) ∞ x f ( x ) dx dx + ( x − x ) n (cid:90) ∞ x f ( x ) dx (cid:12)(cid:12)(cid:12) x = ∞ x = x = n (cid:90) ∞ x ( x − x ) n − (cid:90) ∞ x f ( x ) dx dx + lim x →∞ x n (cid:90) ∞ x f ( x (cid:48) ) dx (cid:48) . Note that for n ≥ x →∞ x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ x f ( x (cid:48) ) dx (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) = lim x →∞ sup x (cid:48) ≥ x x (cid:48) n +1 | f ( x (cid:48) ) | x n (cid:90) ∞ x x − n − dx (cid:48) = 028nd hence, (cid:90) ∞ x ( x − x ) n f ( x ) dx = n (cid:90) ∞ x ( x − x ) n − (cid:20)(cid:90) ∞ x f ( x ) dx (cid:21) dx . We then keep integrating by parts to arrive (6.16), using thatlim x →∞ x n − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ x (cid:90) ∞ x . . . (cid:90) ∞ x k f ( x (cid:48) ) dx (cid:48) dx k . . . dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x (cid:48) ≥ x x n +1 | f ( x (cid:48) ) | lim x →∞ · x − = 0 . Corollary 6.10.
There exists a constant C = C ( M, Σ , r H , r I ) > such that (cid:88) j =0 (cid:90) H + ≥ v v − j ( L j +1 φ ) + | / ∇ S φ | dωdv + (cid:90) I + ≥ u u − j ( L j +1 φ ) + | / ∇ S φ | dωdv ≤ C (cid:34) (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n Σ dµ Σ (cid:35) . (6.17) Proof.
First of all, by Theorem 5.1 from [4] it follows that for 0 ≤ j ≤ lim sup v →∞ v − j (cid:90) S ( LT j φ | H + ) dω < ∞ , lim sup u →∞ u − j (cid:90) S ( LT j φ | I + ) dω < ∞ . We can therefore apply Proposition 6.8 together with Lemma 6.9 with n = 2 to obtain the desiredestimate for the j = 0 term. The j = 1 estimate follows by replacing φ with T φ and applying (6.15) andLemma 6.9 with n = 1. Finally, we obtain the j = 0 estimate by replacing ψ with T ψ and applying Lemma6.3.We will complement (6.17) in Corollary 6.10 with an estimate involving additional angular derivatives.The motivation for this comes from the energy estimates in Section 8.1. Corollary 6.11.
There exists a constant C = C ( M, Σ , r H , r I ) > such that (cid:88) j =0 (cid:90) H + ≥ v v − j ( L j +1 φ ) + | / ∇ S φ | + | / ∇ S Lφ | dωdv + (cid:90) I + ≥ u u − j ( L j +1 φ ) + | / ∇ S φ | + | / ∇ S Lφ | dωdu ≤ C (cid:34) (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n Σ dµ Σ + (cid:88) | α | =1 (cid:90) Σ J T [Ω α ψ ] · n Σ dµ Σ (cid:35) . Proof.
We apply (6.17) and add the Lemma 6.3 estimate applied to Ω α ψ , where | α | = 1. In fact, Theorem 5.1 from [4] provides much stronger, quantitative L ( S ) time-decay estimates, but we do not requirethose here. .4 Higher-order estimates In this section we will derive the analogue of Corollary 6.10 for T n φ with n ≥
1, but with stronger growingweights in u and v on the left-hand side (depending on n ). Proposition 6.12.
Let n ∈ N . Then, there exists a constant C = C ( M, Σ , r H , r I , n ) > , such that (cid:90) ∞ τ (cid:90) ∞ τ n +1 (cid:90) ∞ τ n . . . (cid:90) ∞ τ (cid:34)(cid:90) Σ τ (cid:48) J T [ T n ψ ] · n τ (cid:48) dµ τ (cid:48) (cid:35) dτ (cid:48) dτ . . . dτ n +1 + (cid:90) ∞ τ (cid:90) ∞ τ n . . . (cid:90) ∞ τ (cid:34)(cid:90) H + ≥ τ | / ∇ S T n φ | dωdv (cid:35) dτ . . . dτ n + (cid:90) ∞ τ (cid:90) ∞ τ n . . . (cid:90) ∞ τ (cid:34)(cid:90) I + ≥ τ | / ∇ S T n φ | dωdu (cid:35) dτ . . . dτ n ≤ C (cid:34) (cid:88) j =0 (cid:88) m + | α | + k ≤ n (cid:90) N τ r k − j ( L k T m + j Ω α φ ) dωdv + (cid:90) N τ ( r − M ) − − k + j ( L k T m + j Ω α φ ) dωdu + (cid:88) k ≤ n +2 (cid:90) Σ τ J T [ T k ψ ] · n τ dµ τ (cid:35) . (6.18) Proof.
We will derive (6.18) by induction. Observe that the n = 0 case follows immediately from (6.8). Now,suppose (6.18) holds for all n = N . Then, by replacing T N ψ with T N +1 ψ (using that T commutes with thewave operator (cid:3) g ) and setting τ = τ N +2 , we have that (cid:90) ∞ τ N +2 (cid:90) ∞ τ N +1 (cid:90) ∞ τ N . . . (cid:90) ∞ τ (cid:34)(cid:90) Σ τ (cid:48) J T [ T N +1 ψ ] · n τ (cid:48) dµ τ (cid:48) (cid:35) dτ . . . dτ N +1 dτ (cid:48) ≤ C (cid:34) (cid:88) j =0 (cid:88) m + | α | + k ≤ N (cid:90) N τ N +2 r k − j ( L k T m + j +1 Ω α φ ) dωdv + (cid:90) N τ N +2 ( r − M ) − − k + j ( L k T m + j +1 Ω α φ ) dωdu + (cid:88) k ≤ N +2 (cid:90) Σ τ N +2 J T [ T k +1 ψ ] · n τ dµ τ (cid:35) . Now, we apply the following identities LT m +1 φ = L T m φ + LLT m φ = L T m φ + 14 Dr − / ∆ S T m φ + O ( r − ) T m φ, (6.19) LT m +1 φ = L T m φ + LLT m φ = L T m φ + 14 Dr − / ∆ S T N φ + O (( r − M ) ) T m φ, (6.20)and we integrate once more in τ to obtain: (cid:90) ∞ τ N +3 (cid:90) ∞ τ N +2 (cid:90) ∞ τ N +1 (cid:90) ∞ τ N . . . (cid:90) ∞ τ (cid:34)(cid:90) Σ τ (cid:48) J T [ T N +1 ψ ] · n τ (cid:48) dµ τ (cid:48) (cid:35) dτ . . . dτ N +2 dτ (cid:48) ≤ C (cid:34) (cid:88) j =0 (cid:88) m + | α | + k ≤ N (cid:90) ∞ τ N +3 (cid:90) N τ N +2 r k − j ( L k T m + j Ω α φ ) + r k − j − | / ∆ S L k T m + j Ω α φ | dωdvdτ (cid:48) + (cid:90) ∞ τ N +3 (cid:90) N τ N +2 ( r − M ) − − k + j ( L k T m + j Ω α φ ) + ( r − M ) − − k + j +4 ( / ∆ S L k T m + j Ω α φ ) dωdudτ (cid:48) + (cid:88) k ≤ N +3 (cid:90) ∞ τ N +3 (cid:90) Σ τ N +2 J T [ T k ψ ] · n τ dµ τ dτ (cid:48) (cid:35) , k ≤ N + 1 and p = 2 k + 1 when j = 0 and k ≤ N and p = 2 k when j = 1, togetherwith Lemma 6.2, to obtain (cid:90) ∞ τ N +1)+1 (cid:90) ∞ τ N +2 (cid:90) ∞ τ N +1 (cid:90) ∞ τ N . . . (cid:90) ∞ τ (cid:34)(cid:90) Σ τ (cid:48) J T [ T N +1 ψ ] · n τ (cid:48) dµ τ (cid:48) (cid:35) dτ . . . dτ N +2 dτ (cid:48) ≤ C (cid:34) (cid:88) j =0 (cid:88) m + | α | + k ≤ N +1 (cid:90) N τ N +3 r k − j +1 ( L k T m + j Ω α φ ) dωdv + (cid:90) N τ N +3 ( r − M ) − − k + j − ( L k T m + j Ω α φ ) dωdu + (cid:88) k ≤ N +3 (cid:90) ∞ τ N +3 (cid:34)(cid:90) Σ τ N +2 J T [ T k ψ ] · n τ N +2 dµ τ N +2 (cid:35) dτ τ N +2 (cid:35) . Subsequently, apply (6.9) again, with k ≤ N + 1 and p = 2 k + 2 when j = 0 and k ≤ N and p = 2 k + 1 when j = 1.Finally, since we are integrating two more times in τ compared to the n = N estimate, we can alsoinclude on the left-hand side of the above estimate the terms (cid:90) ∞ τ (cid:90) ∞ τ N +2 . . . (cid:90) ∞ τ (cid:34)(cid:90) H + ≥ τ | / ∇ S T N +1 φ | dωdv (cid:35) dτ . . . dτ N +2 + (cid:90) ∞ τ (cid:90) ∞ τ N +2 . . . (cid:90) ∞ τ (cid:34)(cid:90) I + ≥ τ | / ∇ S T N +1 φ | dωdu (cid:35) dτ . . . dτ N +2 to obtain (6.18) with n = N + 1. Corollary 6.13.
Let n ∈ N . Then, there exists a constant C = C ( M, Σ , r H , r I , n ) > , such that (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) H + ≥ v v k +2 − j ( L k + m + j Ω α φ ) + v k | / ∇ S L k + m Ω α φ | dωdv + (cid:90) I + ≥ u u k +2 − j ( L k + m + j Ω α φ ) + u k | / ∇ S L k + m Ω α φ | dωdu ≤ C (cid:34) (cid:88) j =0 (cid:88) m +2 | α | +2 k ≤ n (cid:90) N v r k − j ( L k T m + j Ω α φ ) dωdv + (cid:90) N u ( r − M ) − − k + j ( L k T m + j Ω α φ ) dωdu (cid:35) + (cid:88) m +2 | α |≤ n +2 | α |≤ n (cid:90) Σ J T [ T m Ω α ψ ] · n dµ . (6.21) Proof.
We apply (6.18), with n replaced by k ≤ n and φ replaced by T m Ω α φ with | α | ≤ n − k and m ≤ n − k − | α | suitably chosen, and combine it with Lemma 6.3, Lemma 6.9 to derive (6.21). The decayof L k + m + j Ω α φ | H + and L k + m + j Ω α φ | I + that is required in order to be able to apply Lemma 6.9 followsfrom Theorem 5.1 of [4].We will complement (6.21) in Corollary 6.13 with an estimate involving additional angular derivatives.The motivation for this comes from the energy estimates in Section 8.2.31 orollary 6.14. Let n ∈ N . Then, there exists a constant C = C ( M, Σ , r H , r I , n ) > , such that (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) H + ≥ v v k +2 − j ( L k + m + j Ω α φ ) + v k | / ∇ S L k + m Ω α φ | + v k | / ∇ S L k +1+ m Ω α φ | dωdv + (cid:90) I + ≥ u u k +2 − j ( L k + m + j Ω α φ ) + u k | / ∇ S L k + m Ω α φ | + u k | / ∇ S L k +1+ m Ω α φ | dωdu ≤ C (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:34) (cid:90) N u r k +2 − j ( L k +1 Ω α T j + m φ ) + r k | / ∇ S L k +1 Ω α T m φ | dωdv + (cid:90) N v ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | dωdu (cid:35) + C (cid:88) | α | + m ≤ n +2 (cid:90) Σ J T [Ω α T m ψ ] · n dµ . (6.22) In this section, we will use the uniform estimates derived in Section 6.3 and 6.4 in order to construct theforward evolution map between suitable weighted energy spaces.
Proposition 6.15.
Let (Ψ , Ψ (cid:48) ) ∈ C ∞ ( (cid:98) Σ ) ⊕ C ∞ (Σ ∩ { r H ≤ r ≤ r I } ) . Then the corresponding solution ψ to (1.1) satisfies ( r · ψ | H + ≥ v , r · ψ | I + ≥ u ) ∈ E T H + ≥ v ⊕ E T I + ≥ u . and furthermore, || r · ψ | H + ≥ v || E T H + ≥ v + || r · ψ | I + ≥ u || E T I + ≥ u = || (Ψ , Ψ (cid:48) ) || E T Σ0 . Proof.
Follows from Lemma 6.3, [4] and Lemma A.1.
Definition 6.1.
Define the forwards evolution map F : C ∞ ( (cid:98) Σ ) ⊕ C ∞ (Σ ∩{ r H ≤ r ≤ r I } ) → E T H + ≥ v ⊕E T I + ≥ u as the following linear operator: F (Ψ , Ψ (cid:48) ) = ( r · ψ | H + ≥ v , r · ψ | I + ≥ u ) , where ψ is the unique solution to (1.1) with ( ψ | Σ , n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } ) = (Ψ , Ψ (cid:48) ) . Then F extends uniquelyto a linear bounded operator: F : E T Σ → E T H + ≥ v ⊕ E T I + ≥ u . Proposition 6.16.
Let n ∈ N . Then F is a bounded linear operator from C ∞ ( (cid:98) Σ ) to E n ; H + ≥ v ⊕ E n ; I + ≥ u ,which can uniquely be extended as as a bounded linear operator F n : E n ;Σ → E n ; H + ≥ v ⊕ E n ; I + ≥ u . We moreover have that F n = F | E n ;Σ0 .Proof. First of all, we assume that (Ψ , Ψ (cid:48) ) ∈ C ∞ c ( (cid:98) Σ ) ⊕ C ∞ (Σ ∩ { r H ≤ r ≤ r I } ). We apply Proposition6.12 to obtain estimates for the corresponding solution ψ : D + (Σ ) → R . By [4], it follows in particular thatlim v →∞ φ | H + = 0 and lim u →∞ φ | H + = 0 for all 0 ≤ k ≤ n . Furthermore, by Corollary 6.13, we have thatthere exists a constant C > || φ | H + || E n ; H + ≥ v + || φ | I + || E n ; I + ≥ u ≤ C · || (Ψ , Ψ (cid:48) ) || E n ;Σ0 . F n (Ψ , Ψ (cid:48) ) = ( φ | H + , φ | I + ) ∈ E n ; H + ≥ v ⊕ E n ; I + ≥ u , so || F n || ≤ √ C . Thenby a standard functional analytic argument, F n extends uniquely to the completion E n ; H + ≥ v ⊕ E n ; I + ≥ u andthe extension F n also satisfies || F n || ≤ √ C . In this section we will construct a map from suitably weighted energy spaces on H + and I + to suitablyweighted energy spaces on Σ . The construction will proceed in two steps. As a first step, we constructin Section 7.1 a map with the domain C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ). In other words, we establish semi-global existence and uniqueness for the backwards scattering initial value problem.In the second step, this will be promoted to global existence and uniqueness in Section 7.4 by using theglobal, uniform weighted energy estimates of Section 7.2 that are valid on the completion of C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) with respect to the associated energy norms. In this section we will associate to a pair C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) a unique solution to (1.1) in D + (Σ ) suchthat r · ψ | H + = Φ and r · ψ | I + = Φ. This association is central to the definition of the backwards evolutionmap (see Definition 7.1).Figure 13: A Penrose diagrammatic representation of the spacetime regions in consideration in Proposition7.1. Proposition 7.1.
Let τ ∞ > and −∞ < u −∞ , v −∞ ≤ u , v and define u ∞ := u + τ ∞ and v ∞ := v + τ ∞ .Let (Φ , Φ) ∈ C ∞ c ( H + ) ⊕ C ∞ c ( I + ) such that supp Φ ⊂ H + v −∞
A variant of Proposition 7.1 was established in Proposition 9.1.4 of [22] in the setting ofsub-extremal Kerr. Note however that Proposition 7.1 establishes in addition qualitative bounds on theradiation field rψ and weighted higher-order derivatives thereof in the form of the inequality (7.1) , which willbe necessary in the backwards-in-time estimates of Section 7.2.Proof of Proposition 7.1. Observe first of all that ψ i is well-defined by local existence and uniqueness withsmooth initial data on Σ τ ∞ ∪ { v = V i } .Apply the divergence theorem with J T in the region { r ≥ r I } bounded to the past by I v = { v (cid:48) = v } ∩ { u ≤ u ≤ u ∞ } and Σ and to the future by I V i := { v (cid:48) = V i } ∩ { u ≤ u ≤ u ∞ } and Σ τ ∞ to obtain: (cid:90) I v J T [ ψ i ] · L r dωdu ≤ (cid:90) I Vi J T [ ψ i ] · L r dωdu, which is equivalent to (cid:90) I v r ( Lψ i ) + 14 Dr − | / ∇ S ψ i | dωdu ≤ (cid:90) I Vi r ( L ( r − Φ)) + 14 Dr − | / ∇ S Φ | dωdu. By applying the fundamental theorem of calculus in u , integrating from u (cid:48) = τ ∞ to u (cid:48) = u , together withCauchy–Schwarz, we therefore obtain (cid:20)(cid:90) S ψ i dω (cid:21) ( u, v ) (cid:46) (cid:90) τ ∞ u r − ( u (cid:48) , v ) du (cid:48) · (cid:90) I Vi r ( Lψ i ) dωdu (cid:48) , where we used that ψ i | Σ τ ∞ = 0, from which it follows that (cid:20)(cid:90) S φ i dω (cid:21) ( u, v ) (cid:46) u ∞ (cid:90) I v ( Lφ i ) dωdu (cid:46) u ∞ (cid:90) I v r ( Lψ i ) + ψ i dωdu (cid:46) u ∞ (cid:90) I Vi r ( L ( r − Φ)) + 14 Dr − | / ∇ S Φ | dωdu (cid:48) . (7.2)Now, we can use (7.2) and (6.5) with n = 0 together with the fundamental theorem of calculus in the u -direction to obtain (cid:20)(cid:90) S r ( Lφ i ) dω (cid:21) ( u, v ) (cid:46) C ( τ ∞ , u ) · (cid:88) | α |≤ (cid:90) I Vi r ( L ( r − Ω α Φ)) + 14 Dr − | / ∇ S Ω α Φ | dωdu (cid:48) . n ∈ N wehave in { r ≥ r I } : (cid:20)(cid:90) S (( r L ) n φ i ) dω (cid:21) ( u, v ) (cid:46) C ( τ ∞ , u ) · (cid:88) | α |≤ n (cid:90) I Vi r ( L ( r − Ω α Φ)) + 14 Dr − | / ∇ S Ω α Φ | dωdu (cid:48) . We can immediately apply the above argument to Ω α φ and T k for any α ∈ N , k ∈ N , together with astandard Sobolev inequality on S to obtain the following i - independent estimate: for all k ∈ N and α ∈ N ,there exists a constant C ( τ ∞ , u ) >
0, such that | ( r L ) n T k Ω α φ i | ( u, v, θ, ϕ ) ≤ C ( τ ∞ , u −∞ ) · (cid:88) | α (cid:48) |≤ n +2 (cid:90) I Vi ( LT k Ω α + α (cid:48) Φ) + r − ( T k Ω α + α (cid:48) Φ) + 14 Dr − | / ∇ S T k Ω α + α (cid:48) Φ | dωdu. (7.3)We obtain a similar estimate in the region { r ≤ r H } by reversing the roles of u and v (integrating in the v -direction) and replacing r by ( r − M ) − : | (( r − M ) − L ) n T k Ω α φ i | ( u, v, θ, ϕ ) ≤ C ( τ ∞ , v −∞ ) · (cid:88) | α (cid:48) |≤ n +2 (cid:90) H + v −∞≤ v ≤ v ∞ ( LT k Ω α + α (cid:48) Φ) dωdu. (7.4)Given (cid:101) V > n ≥ N , we have by (7.3) and (7.4) that for I ≥ V I > (cid:101) V , φ i is uniformly bounded in i for all i ≥ I with respect to the C k norm on J + ( (cid:98) Σ ) ∩ J − ( (cid:98) Σ τ ∞ ) ∩ { v ≤ (cid:101) V } ⊂ (cid:98) R with respect to the differentiable structure on (cid:98) R and therefore, by Arzel`a–Ascoli, there exists a subsequence φ i k that converges in C k ( J + ( (cid:98) Σ ) ∩ J − ( (cid:98) Σ τ ∞ ) ∩ { v ≤ (cid:101) V } ), for any k ∈ N , to a smooth function φ on J + ( (cid:98) Σ ) ∩ J − ( (cid:98) Σ τ ∞ ) ∩ { v ≤ (cid:101) V } .We can extend the domain of φ to J + ( (cid:98) Σ ) ∩ J − ( (cid:98) Σ τ ∞ ) as follows: we replace (cid:101) V above with (cid:101) V (cid:48) > (cid:101) V ,applying Arzel`a-Ascoli to the subsequence φ i k (starting from k suitably large) in the corresponding largerspacetime region and passing to a further subsequence. By uniqueness of limits, the resulting limit, whichwe note by φ (cid:48) has to agree with φ when v ≤ (cid:101) V .The above C k convergence moreover implies that (cid:3) g ψ = 0, with ψ = r − φ , φ | H + = Φ and( ψ | Σ τ ∞ , n Σ τ ∞ ( ψ ) | Σ τ ∞ ∩{ r H ≤ r ≤ r I } ) = (0 , . We also have by (7.3) that for any (cid:15) >
0, there exist a
V >
K >
0, such that for all v ≥ V and k > K in the region { r ≥ r I } : | rψ ( u, v, θ, ϕ ) − Φ( u, θ, ϕ ) | ≤ | rψ ( u, v, θ, ϕ ) − rψ k ( u, v, θ, ϕ ) | + | rψ i k ( u, v, θ, ϕ ) − Φ( u, θ, ϕ ) |≤ | rψ ( u, v, θ, ϕ ) − rψ i k ( u, v, θ, ϕ ) | + r − ( u, v ) (cid:90) V i k v r | Lφ i k | dv (cid:48) ≤ (cid:15). for all u ∈ ( − u −∞ , u ∞ ] and ( θ, ϕ ) ∈ S . Hence,lim v →∞ rψ ( u, v, θ, ϕ ) = Φ( u, θ, ϕ ) . We can analogously use (7.3) to obtain for all j, k, l ∈ N :lim v →∞ / ∇ j S ( r ∂ v ) k ∂ lu ( r · ψ )( u, v, θ, ϕ ) < ∞ . Furthermore, by replacing ψ by T l Ω α ψ we can conclude that with respect to the differentiable structure inˆ R , the restriction rψ | I + is a smooth function on I + , satisfying rψ | I + = Φ. We can therefore conclude 1 . ) ofthe proposition. 35ow suppose (cid:101) ψ is another smooth solution to (cid:3) g (cid:101) ψ = 0, such that M (cid:101) ψ | H + v −∞≤ v ≤ v ∞ = Φ ,r · (cid:101) ψ | I + u −∞≤ u ≤ u ∞ = Φ , ( (cid:101) ψ | Σ τ ∞ , n Σ τ ∞ ( (cid:101) ψ ) | Σ τ ∞ ∩{ r H ≤ r ≤ r I } ) = (0 , T -energy estimate, we have that (cid:90) (cid:101) Σ J T [ ˜ ψ − ψ ] · n (cid:101) Σ dµ (cid:101) Σ = 0 , so (cid:101) ψ = ψ , which concludes 2 . ) of the proposition. In this section, we will derive estimates for the solutions ψ to (1.1) constructed in Proposition 7.1 thatare uniform in τ ∞ . This is crucial for constructing solutions with scattering data that is not compactlysuppported.The main tool we will develop is this section is a hierarchy of r -weighted estimates in the backwards timedirection. However, we will first state a backwards Morawetz estimate that follows immediately from theresults in [7], i.e. an analogue of Theorem 6.6 in the backwards time direction.In this section, we will always assume that ψ is a solution to (1.1) arising from smooth and compactlysupported scattering data (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ), as in Proposition (7.1), i.e. let τ ∞ > ⊂ H + v ≤ v<τ ∞ + v and supp Φ ⊂ I + u ≤ u<τ ∞ + u . Proposition 7.2 (Backwards Morawetz/integrated local energy decay estimate,[7]) . Let ≤ τ < τ < ∞ and M < r < r < M < r < r < ∞ , then for all k, l ∈ N and α ∈ N there exists a constant C = C ( r i , M, Σ , k, l, α ) > , such that (cid:90) τ τ (cid:34)(cid:90) Σ τ ∩ ( { r ≤ r ≤ r }∪{ r ≤ r ≤ r } ) ( ∂ v T k ∂ lr Ω α ψ ) + | / ∇ T k ∂ lr Ω α ψ | + ( T k ∂ lr Ω α ψ ) + ( ∂ r T k ∂ lr Ω α ψ ) dµ Σ τ (cid:35) dτ ≤ C k + l + | α | (cid:88) j =0 (cid:90) H + ∩{ τ ≤ v ≤ τ } J T [ T j ψ ] · L dωdv + (cid:90) I + ∩{ τ ≤ u ≤ τ } r J T [ T j ψ ] · Ldωdu + C (cid:90) Σ τ J T [ T j ψ ] · n Σ τ dµ Σ τ . (7.5) Proof.
The proof of (7.5) follows directly from the Morawetz estimates established in [7].In the propositions below, we derive the “backwards analogues” of the hierarchies from Proposition 6.7.
Proposition 7.3.
Let ≤ p ≤ , then there exists a constant C ( M, Σ , r I , r H ) > , such that for all ≤ τ ≤ τ ≤ τ ∞ : (cid:90) N τ ( r − M ) − p ( Lφ ) dωdu + (cid:90) N τ r p ( Lφ ) dωdv ≤ C (cid:90) A τ τ ( r − M ) − p +1 ( Lφ ) + (2 − p )( r − M ) − p | / ∇ S φ | dωdudτ + C (cid:90) A τ τ r p − ( Lφ ) + (2 − p ) r p − | / ∇ S φ | dωdvdτ + C (cid:90) N τ ( r − M ) − p ( Lφ ) dωdu + C (cid:90) N τ r p ( Lφ ) dωdv + C (cid:90) H + ∩{ τ ≤ τ ≤ τ } ( r − M ) − p | / ∇ S φ | dωdv + C (cid:90) I + ∩{ τ ≤ u ≤ τ } r p − | / ∇ S φ | dωdu + C (cid:90) H + ∩{ τ ≤ τ ≤ τ } J T [ ψ ] · L dωdv + C (cid:90) I + ∩{ τ ≤ u ≤ τ } r J T [ ψ ] · L dωdu + C (cid:90) Σ τ J T [ ψ ] · n Σ τ dµ Σ τ , (7.6) Proof.
Recall that φ satisfies the equation: LLφ = 14 DD (cid:48) r − φ + Dr − / ∆ S φ. (7.7)Therefore, L ( r p ( Lφ ) ) = − p r p − ( Lφ ) + 2 r p Lφ · LLφ = − p r p − ( Lφ ) + O ( r p − ) φ · Lφ + ( r p − + O ( r p − )) Lφ · / ∆ S φ = − p r p − ( Lφ ) + O ( r p − ) φ · Lφ + / ∇ S ( r p − Lφ · / ∇ S φ ) − L (cid:18) r p − | / ∇ S φ | (cid:19) −
12 (2 − p ) r p − | / ∇ S φ | + O ( r p − ) / ∆ S φ · Lφ = − p r p − ( Lφ ) + O ( r p − ) φ · Lφ + / ∇ S ( r p − Lφ · / ∇ S φ ) − L (cid:18) r p − | / ∇ S φ | (cid:19) −
12 (2 − p ) r p − | / ∇ S φ | + O ( r p − ) LLφ · Lφ By reordering the terms, we therefore obtain: L (( r p + O ( r p − ))( Lφ ) ) + L (cid:18) r p − | / ∇ S φ | (cid:19) − / ∇ S ( r p − Lφ · / ∇ S φ )= (cid:16) − p r p − + O ( r p − ) (cid:17) ( Lφ ) −
12 (2 − p ) r p − | / ∇ S φ | + O ( r p − ) φ · Lφ. (7.8)Let χ denote a cut-off function and consider χφ . 37e integrate both sides of (7.8) in spacetime to obtain: − (cid:90) N τ ( r p + O ( r p − ))( L ( χφ )) dωdr + (cid:90) N τ ( r p + O ( r p − ))( L ( χφ )) dωdr − (cid:90) I + ∩{ τ ≤ u ≤ τ } r p − | / ∇ S χφ | dωdτ = (cid:90) τ τ (cid:90) N τ (cid:16) p r p − + O ( r p − ) (cid:17) ( L ( χφ )) + 12 (2 − p ) r p − | / ∇ S χφ | + O ( r p − ) χφ · L ( χφ ) dωdrdτ + (cid:88) | α | + | α |≤ (cid:90) τ τ (cid:90) N τ R χ [ ∂ α φ · ∂ α φ ] dωdrdτ ≤ (cid:90) τ τ (cid:90) N τ Cr p − ( L ( χφ )) + 12 (2 − p ) r p − | / ∇ S χφ | dωdrdτ + C ( r , r ) (cid:90) H + ∩{ τ ≤ v ≤ τ } J T [ ψ ] · ∂ v dωdv + C ( r , r ) (cid:90) I + ∩{ τ ≤ u ≤ τ } r J T [ ψ ] · ∂ u dωdu + C ( r , r ) (cid:90) Σ τ J T [ ψ ] · n Σ τ dµ Σ τ , (7.9)where we applied Lemma 6.1 and (7.5) to arrive at the inequality above. See also the derivations in theproof of Lemma 6.3 in [4] in the special case n = 0.We can repeat the above steps in the region where r ≤ r H by reversing the roles of L and L and replacing r p with ( r − M ) − p ; see the proof of Lemma 6.3 in [4] for more details.We subsequently apply Proposition 7.3 to arrive at uniform weighted energy estimates along Σ . Proposition 7.4.
Then there exists a constant C ( M, Σ , r I , r H ) > , such that (cid:90) N v ( r − M ) − ( Lφ ) dωdu + (cid:90) N u r ( Lφ ) dωdv + (cid:90) Σ J T [ ψ ] · n dµ ≤ C (cid:90) H + ≥ v v ( Lφ ) dωdv + C (cid:90) I + ≥ u u ( Lφ ) dωdu. (7.10) We moreover have that (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n dµ ≤ C (cid:88) j =0 (cid:90) H + ≥ v v − j ( LT j φ ) + | / ∇ S φ | dωdv + C (cid:90) I + ≥ u u − j ( LT j φ ) + | / ∇ S φ | dωdu. (7.11) Proof.
By applying Lemma 6.1 and Lemma 6.3, it follows that (cid:90) ∞ τ (cid:90) N τ ( Lφ ) + r − | / ∇ S φ | dωdvdτ (cid:48) + (cid:90) ∞ τ (cid:90) N τ ( Lφ ) + r − | / ∇ S φ | dωdudτ (cid:48) ≤ C (cid:90) ∞ τ (cid:90) N τ J T [ ψ ] · L r dωdvdτ (cid:48) + C (cid:90) ∞ τ (cid:90) N τ J T [ ψ ] · L r dωdudτ (cid:48) ≤ C (cid:90) ∞ τ (cid:90) H + ≥ τ (cid:48) J T [ ψ ] · L r dωdvdτ (cid:48) + C (cid:90) ∞ τ (cid:90) I + ≥ τ J T [ ψ ] · L r dωdudτ (cid:48) ≤ C (cid:90) ∞ τ (cid:90) H + ≥ τ (cid:48) ( Lφ ) dωdvdτ (cid:48) + C (cid:90) ∞ τ (cid:90) I + ≥ τ ( Lφ ) dωdudτ (cid:48) . (7.12)38e now apply (7.6) with p = 1, together with (7.12) to conclude that (cid:90) N τ ( r − M ) − ( Lφ ) dωdu (cid:90) N τ r ( Lφ ) dωdv ≤ C (cid:90) ∞ τ (cid:90) H + ≥ τ ( Lφ ) dωdvdτ + C (cid:90) ∞ τ (cid:90) I + ≥ τ ( Lφ ) dωdudτ + C (cid:90) H + ≥ τ ( Lφ ) dωdv + C (cid:90) I + ≥ τ ( Lφ ) dωdu. Next, apply (7.6) with p = 2 to obtain (cid:90) N τ ( r − M ) − ( Lφ ) dωdu + (cid:90) N τ r ( Lφ ) dωdv ≤ C (cid:90) ∞ τ (cid:90) ∞ τ (cid:90) H + ≥ τ ( Lφ ) dωdvdτ dτ + C (cid:90) ∞ τ (cid:90) ∞ τ (cid:90) I + ≥ τ ( Lφ ) dωdudτ dτ + C (cid:90) ∞ τ (cid:90) H + ≥ τ ( Lφ ) dωdvdτ + C (cid:90) ∞ τ (cid:90) I + ≥ τ ( Lφ ) dωdudτ + C (cid:90) H + ≥ τ ( Lφ ) dωdv + C (cid:90) I + ≥ τ ( Lφ ) dωdu. We apply Lemma 6.9 to rewrite the right-hand side above to arrive at: (cid:90) N τ ( r − M ) − ( Lφ ) dωdu + (cid:90) N r ( Lφ ) dωdv ≤ C (cid:90) H + ≥ v H +( τ ) v ( Lφ ) dωdv + C (cid:90) I + ≥ u I +( τ ) u ( Lφ ) dωdu, (7.13)which leads to (7.10) when we take τ = 0.By applying the above estimates to T ψ and T ψ we moreover obtain: (cid:88) j =0 (cid:90) N ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N r − j ( LT j φ ) dωdv ≤ C (cid:88) j =0 (cid:90) H + ≥ v v − j ( ∂ v T j φ ) dωdv + C (cid:90) I + ≥ u u − j ( ∂ u T j φ ) dωdu. We conclude the proof by combining the above proposition with [Tenergy conservation] to obtain (cid:88) j =0 (cid:90) N ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ ψ ] · n dµ ≤ C (cid:88) j =0 (cid:90) H + ≥ v v − j ( LT j φ ) dωdv + C (cid:90) I + ≥ u u − j ( LT j φ ) dωdu. Remark 7.2.
Note that in contrast with the estimates in Proposition 6.8, there is no loss of derivatives(caused by the application of (6.8) ) on the right-hand side of (7.10) . We will complement (7.14) in Proposition 7.4 with an estimate involving additional angular derivatives.The motivation for this comes from the energy estimates in Section 8.1.
Corollary 7.5.
Then there exists a constant C ( M, Σ , r I , r H ) > , such that (cid:88) j =0 (cid:90) N v ( r − M ) − j ( LT j φ ) dωdu + (cid:90) N u r − j ( LT j φ ) dωdv + (cid:88) j =0 (cid:90) Σ J T [ T j ψ ] · n dµ + (cid:88) | α | =1 (cid:90) Σ J T [Ω α ψ ] · n dµ ≤ C (cid:88) j =0 (cid:90) H + ≥ v v − j ( LT j φ ) + | / ∇ S φ | + | / ∇ S Lφ | dωdv + C (cid:90) I + ≥ u u − j ( LT j φ ) + | / ∇ S φ | + | / ∇ S Lφ | dωdu. (7.14) Proof.
We apply (7.14) together with Lemma 6.3 applied to Ω α ψ , with | α | = 1.39 .3 Higher-order estimates By commuting (7.7) with L k , we arrive at LL ( L k φ ) = ( r − + O ( r − )) / ∆ S L k φ + k (cid:88) j =0 O ( r − − j ) L k − j φ + k (cid:88) j =1 O ( r − − j ) / ∆ S L k − j φ. (7.15)Similarly, we can commute (7.7) with L k to obtain: LL ( L k φ ) = [ M − ( r − M ) + O (( r − M ) ] / ∆ S L k φ + k (cid:88) j =0 O (( r − M ) j ) L k − j φ + k (cid:88) j =1 O (( r − M ) j ) / ∆ S L k − j φ. (7.16) Proposition 7.6.
Fix k ∈ N . Let k ≤ p ≤ k , then we can estimate for all ≤ τ ≤ τ ≤ τ ∞ : (cid:90) N τ ( r − M ) − p ( L k +1 φ ) dωdr + (cid:90) N τ r p ( L k +1 φ ) dωdv ≤ C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) A τ τ ( r − M ) − p +2 j ( L k +1 − j Ω α φ ) + (2 − p )( r − M ) − p +2 j | / ∇ S L k − j Ω α φ | dωdudτ + C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) A τ τ r p − − j ( L k +1 − j Ω α φ ) + (2 − p ) r p − − j | / ∇ S L k − j Ω α φ | dωdvdτ + C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) N τ ( r − M ) − p +2 j ( L k +1 − j Ω α φ ) dωdu + C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) N τ r p − j ( L k +1 − j Ω α φ ) dωdv + C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) H + ∩{ τ ≤ τ ≤ τ } ( r − M ) − p − j | / ∇ S L k − j Ω α φ | dωdv + C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) I + ∩{ τ ≤ τ ≤ τ } r p − − j | / ∇ S L k − j Ω α φ | dωdu + C k (cid:88) j =0 (cid:88) | α |≤ j (cid:90) H + ∩{ τ ≤ τ ≤ τ } J T [ T k − j Ω α ψ ] · L dωdv + C (cid:90) I + ∩{ τ ≤ τ ≤ τ } r J T [ T k − j Ω α ψ ] · L dωdu + C k (cid:88) j =0 (cid:90) Σ τ J T [ T j ψ ] · n Σ τ dµ Σ τ . (7.17) Proof.
The proof is a straightforward generalisation of the proof of Proposition 7.3: we repeat the steps inthe proof of Proposition 7.3, but we replace φ with either L k φ (when { r ≥ r I } ) or L k φ (when { r ≤ r H } ),and we use (7.15) and (7.16). Proposition 7.7.
Let n ∈ N and let ψ be a solution to (1.1) such that ψ | Σ τ ∞ = 0 and n τ ∞ ψ | Σ τ ∞ = 0 forsome τ ∞ < ∞ . Then there exists a constant C ( M, Σ , r I , r H , n ) > such that (cid:88) j =0 (cid:88) m + k + | α |≤ n (cid:90) N τ r k ( L k T m + j Ω α φ ) dωdv + (cid:90) N τ ( r − M ) − − k ( L k T m + j Ω α φ ) dωdu ≤ (cid:88) j =0 (cid:88) | α | + m ≤ n (cid:90) I + ≥ u I +( τ ) u − j +2 m ( ∂ u T m + j Ω α φ ) + | / ∇ S Ω α T m φ | dωdu + (cid:88) j =0 (cid:88) | α | + m ≤ n (cid:90) H + ≥ v H +( τ ) v − j +2 m ( ∂ v T m + j Ω α φ ) + + | / ∇ S Ω α T m φ | dωdv. (7.18)40 roof. We first consider the n = 1 case. Note that by (7.6) with k = 1 and p = 3: (cid:90) N τ r ( L φ ) dωdv + (cid:90) N τ ( r − M ) − ( L φ ) dωdu ≤ C (cid:88) | α |≤ (cid:90) ∞ τ (cid:90) N τ (cid:48) r ( L φ ) + ( L Ω α φ ) + r − | / ∇ S Ω α φ | dωdvdu + C (cid:88) | α |≤ (cid:90) ∞ τ (cid:90) N τ (cid:48) ( r − M ) − ( L φ ) + ( L Ω α φ ) + ( r − M ) | / ∇ S Ω α φ | dωdudv + (cid:88) | α | + m ≤ (cid:90) I + ≥ u I +( τ ) ( ∂ u T m Ω α φ ) du + (cid:88) | α | + m ≤ (cid:90) H + ≥ v H +( τ ) ( ∂ v T m Ω α φ ) dωdv (6.19) and (6.20) ≤ C (cid:88) | α |≤ (cid:90) ∞ τ (cid:90) N τ (cid:48) r ( LT φ ) + ( L Ω α φ ) + r − | / ∇ S Ω α φ | + r − φ dωdvdu + C (cid:88) | α |≤ (cid:90) ∞ τ (cid:90) N τ (cid:48) ( r − M ) − ( LT φ ) + ( L Ω α φ ) + ( r − M ) | / ∇ S Ω α φ | + ( r − M ) φ dωdudv + (cid:88) | α | + m ≤ (cid:90) I + ≥ u I +( τ ) ( ∂ u T m Ω α φ ) du + (cid:88) | α | + m ≤ (cid:90) H + ≥ v H +( τ ) ( ∂ v T m Ω α φ ) dωdv (7.13), Lemma 6.9 and Lemma 6.3 ≤ (cid:88) | α | + m ≤ (cid:90) I + ≥ u I +( τ ) u m ( ∂ u T m Ω α φ ) du + (cid:88) | α | + m ≤ (cid:90) H + ≥ v H +( τ ) v m ( ∂ v T m Ω α φ ) dωdv. (7.19)Now, we apply (7.6) with k = 1 and p = 4: (cid:90) N τ r ( L φ ) dωdv + (cid:90) N τ ( r − M ) − ( L φ ) dωdu ≤ C (cid:88) | α |≤ (cid:90) ∞ τ (cid:90) N τ (cid:48) r ( L φ ) + r ( L Ω α φ ) dωdvdu + C (cid:88) | α |≤ (cid:90) ∞ τ (cid:90) N τ (cid:48) ( r − M ) − ( L φ ) + ( r − M ) − ( L Ω α φ ) dωdudv + (cid:88) | α | + m ≤ (cid:90) I + ≥ u I +( τ ) ( ∂ u T m Ω α φ ) du + (cid:88) | α | + m ≤ (cid:90) H + ≥ v H +( τ ) ( ∂ v T m Ω α φ ) dωdv (7.19) ≤ (cid:88) | α | + m ≤ (cid:90) I + ≥ u I +( τ ) u m ( ∂ u T m Ω α φ ) du + (cid:88) | α | + m ≤ (cid:90) H + ≥ v H +( τ ) v m ( ∂ v T m Ω α φ ) dωdv. By replacing φ on the left-hand side of (7.19) with T j φ and applying Proposition 7.4 to T m Ω α φ , we thereforeobtain: (cid:88) j =0 (cid:88) m + k + | α |≤ (cid:90) N τ r k − j ( L k +1 T m + j Ω α φ ) dωdv + (cid:90) N τ ( r − M ) − − k + j ( L k +1 T m + j Ω α φ ) dωdu ≤ (cid:88) j =0 (cid:88) | α | + m ≤ (cid:90) I + ≥ u I +( τ ) u − j +2 m ( ∂ u T m + j Ω α φ ) + | / ∇ S Ω α φ | dωdu + (cid:88) j =0 (cid:88) | α | + m ≤ (cid:90) H + ≥ v H +( τ ) v − j +2 m ( ∂ v T m + j Ω α φ ) + | / ∇ S Ω α φ | dωdv, where we applied Proposition 7.4 and Lemma 6.9 to arrive at the final inequality.The general n case now follows easily via an inductive argument, where we apply (7.6) with k = n and p = 2 n + 1 and p = 2 n + 2.Proposition 7.7 combined with Lemma 6.3 immediately implies the following:41 orollary 7.8. Let n ∈ N . Then there exists a constant C ( M, r I , r H , n ) > such that (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) N u r k − j ( L k +1 T m + j Ω α φ ) dωdv + (cid:90) N v ( r − M ) − − k + j ( L k +1 T m + j Ω α φ ) dωdu + (cid:88) m +2 | α |≤ n +2 | α |≤ n (cid:90) Σ J T [ T m Ω α ψ ] · n dµ ≤ C (cid:34) (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:90) I + ≥ u u k − j ( LT m + j Ω α φ ) + | / ∇ S Ω α T m φ | du + (cid:90) H + ≥ v v k − j ( LT m + j Ω α φ ) + | / ∇ S Ω α T m φ | dv (cid:35) . (7.20)We will complement (7.20) in Corollary 7.8 with an estimate involving additional angular derivatives.The motivation for this comes from the energy estimates in Section 8.2. Corollary 7.9.
Let n ∈ N . Then there exists a constant C ( M, r I , r H , n ) > such that (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:34) (cid:90) N u r k +2 − j ( L k +1 Ω α T j + m φ ) + r k | / ∇ S L k +1 Ω α T m φ | dωdv + (cid:90) N v ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | dωdu (cid:35) + (cid:88) | α | + m ≤ n +2 (cid:90) Σ J T [Ω α T m ψ ] · n dµ ≤ C (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) H + ≥ v v k +2 − j ( L k + m + j Ω α φ ) + v k | / ∇ S L k + m Ω α φ | + v k | / ∇ S L k +1+ m Ω α φ | dωdv + (cid:90) I + ≥ u u k +2 − j ( L k + m + j Ω α φ ) + u k | / ∇ S L k + m Ω α φ | + u k | / ∇ S L k +1+ m Ω α φ | dωdu. (7.21) In this section, we apply the uniform estimates derived in Sections 7.2 and 7.3 to construct the backwardsevolution maps B on appropriate energy spaces. Proposition 7.10.
Let (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) , then the corresponding solution ψ to (1.1) satisfies ( ψ | Σ , n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } ) ∈ E T Σ and || ( ψ | Σ , n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } ) || E T Σ0 = || Φ || E T H + ≥ v + || Φ || E T I + ≥ u . Proof.
From Proposition 7.1 it follows that ψ | Σ ∈ C ∞ (Σ) and n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } ∈ C ∞ (Σ ∩ { r H ≤ r ≤ r I } ). The remaining statment follows from Lemma 6.3.Using Proposition 7.10, together with the standard general construction of the unique extensions ofbounded linear operators to the completion of their domains, we can define the backwards evolution map asfollows: 42 efinition 7.1. The backwards evolution map is the map B : C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) → E T (Σ ) , such that B (Φ , Φ) = ( ψ | Σ , n Σ ψ | Σ ∩{ r ≤ r ≤ r } ) , where ψ is the unique solution to (cid:3) g ψ = 0 with ( M ψ | H + ≥ v , rψ | I + ≥ u ) = (Φ , Φ) . The map B uniquely extendsto a unitary linear operator, which we will also denote with B : B : E T H + ⊕ E T I + → E T Σ . In the proposition below, we show that we can consider restriction of B to suitably weighted energyspaces. Proposition 7.11.
Let n ∈ N . The backwards evolution map B is a bounded linear operator from C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ) to E n ;Σ , which can uniquely be extended as as the bounded linear operator B n : E n ; H + ≥ v ⊕ E n ; I + ≥ u → E n ;Σ . We moreover have that B n = B | E n ; H + ≥ v ⊕E n ; I + ≥ u .Proof. By Proposition 7.1 it follows that the solution ψ corresponding to (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u )satisfies φ | Σ ∈ C ∞ ( (cid:98) Σ ) and n Σ ψ | Σ ∈ C ∞ (Σ ∩ { r H ≤ r ≤ r I } ). By Corollary 7.8 it follows moreover that || ( ψ | Σ , n Σ ψ | Σ ) || E n ;Σ0 ≤ C || Φ || E n ; H + ≥ v + C || Φ || E n ; I + ≥ v , so || B || ≤ C . We can infer that, in particular, ( ψ | Σ , n Σ ψ | Σ ) ∈ E n ;Σ . The map B extends uniquely to thecompletion E H + ≥ v ⊕ E I + ≥ u and satisfies || B || ≤ √ C . Corollary 7.12.
The map F : E T Σ → E T H + ≥ v ⊕ E T I + ≥ u is a bijection with inverse B = F − and for each n ∈ N , the restrictions F n : E n ;Σ → E n ; H + ≥ v ⊕ E n ; I + ≥ u are also bijections with inverses B n = F − n .Proof. Let (Φ , Φ) ∈ C ∞ c ( H + ≥ v ) ⊕ C ∞ c ( I + ≥ u ), then the corresponding solution ψ to (1.1) satisfies φ | Σ ∈ C ∞ ( (cid:98) Σ ) and n Σ ψ | Σ ∈ C ∞ (Σ ∩ { r H ≤ r ≤ r I } ), and hence F ( φ | Σ , n Σ ψ | Σ ) = ( φ | H + , φ | I + ) is well-defined and ( φ | H + , φ | I + ) = (Φ , Φ). We conclude that F ◦ B = id on a dense subset. By boundedness of F ◦ B we can conclude that F ◦ B = id on the full domain. Hence, F must be surjective and in factbijective (we have already established injectivity). It immediately follows then that B ◦ F = id. The aboveargument can also be applied to F n and B n . The aim of this section is to extend the estimates of Section 6 and 7 from the hypersurface Σ to thehypersurface (cid:101) Σ. This will allow us to construct the scattering map S , a bijective map between (time-weighted) energy spaces on H − ∪ I − and H + ∪ I + . The estimates in this section will therefore concernthe “triangular” regions bounded to the future by the null hypersurfaces N and N and to the past by (cid:101) Σ = { t = 0 } . In the proposition below we derive energy estimates with respect to the vector field multiplier K = v L + u L ,which is commonly referred to as the Morawetz conformal vector field . The main purpose of K is to derivebackwards energy estimates along (cid:101) Σ with r -weighted initial data along N − u and N − v which are analogousto the r -weighted boundary terms in the estimates in Proposition 7.3 with p = 2. The geometric significance of K is that it generates the inverted translation conformal symmetry on the Minkowski space-time. roposition 8.1. Let u −∞ , v −∞ < , with | u −∞ | , | v −∞ | arbitrarily large. There exist constants C, c = C, c ( M, r I , r H , u , v ) > , such that (cid:90) (cid:101) Σ ∩{ u ≤ v ≤− u −∞ } r ( Lφ ) + r ( Lφ ) + | / ∇ S φ | dv (8.1) ∼ C,c (cid:90) N − u r ( Lφ ) + r − | / ∇ S φ | dωdv + (cid:90) I + ≤− u u ( Lφ ) + | / ∇ S φ | dωdu, (cid:90) (cid:101) Σ ∩{ u ≤ u ≤− v −∞ } ( r − M ) − ( Lφ ) + ( r − M ) − ( Lφ ) + | / ∇ S φ | du (8.2) ∼ C,c (cid:90) N − v D − ( Lφ ) + D | / ∇ S φ | dωdu + (cid:90) H + ≤− v v ( Lφ ) + | / ∇ S φ | dωdv. Proof.
By (6.6) it follows that L ( u ( Lφ ) ) + L ( v ( Lφ ) ) = 2 u LLφ · Lφ + 2 v LLφ · Lφ = 12 u r D / ∆ S φ · Lφ + 12 v r D / ∆ S φ · Lφ + ( u Lφ + v Lφ ) O ( r − ) φ. After integrating by parts on S , we therefore obtain: (cid:90) S L ( u ( Lφ ) ) + L ( v ( Lφ ) ) + 14 L ( v r − D | / ∇ S φ | ) + 14 L ( u r − D | / ∇ S φ | ) dω = (cid:90) S (cid:18) L (cid:18) u Dr (cid:19) + L (cid:18) v Dr (cid:19)(cid:19) | / ∇ S φ | dω + (cid:90) S ( u Lφ + v Lφ ) O ( r − ) φ dω. (8.3)We first consider estimates in the backwards time direction. We integrate (8.3) in spacetime and we use thefollowing identity: L (cid:18) u Dr (cid:19) + L (cid:18) v Dr (cid:19) = O ( r − ) log r (8.4)to estimatesup u (cid:90) N u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + sup v (cid:90) I v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu ≤ (cid:90) N − u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + (cid:90) I + u −∞≤ u ≤− u u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + C (cid:90) ∞ u (cid:90) − u − min { v, | u −∞ |} r − | φ | · ( u | Lφ | + v | Lφ | ) dωdudv + C (cid:90) ∞ u (cid:90) − u − min { v, | u −∞ |} r − log r | / ∇ S φ | dωdudv. (8.5)Using that r ∼ v + | u | (cid:46) v in the integration region, we can further estimate: (cid:90) ∞ u (cid:90) − u − min { v, | u −∞ |} r − log r | / ∇ S φ | dωdudv ≤ C (cid:90) ∞ u v − log v dv · sup v (cid:90) I v v r − D | / ∇ S φ | dωdv ≤ C(cid:15) sup v (cid:90) I v v r − D | / ∇ S φ | dωdv for (cid:15) > r I > v − (cid:46) r − in the integration region). Note thatwe can absorb the very right-hand side above into the left-hand side of (8.5) when (cid:15) > r − | φ | ( u | Lφ | + v | Lφ | ) ≤ r − − η ( u ( Lφ ) + v ( Lφ ) ) + r − η ( u + v ) φ .
44e absorb the spacetime integral of ( Lφ ) and ( Lφ ) to the left-hand side of (8.5), using that r is suitablylarge and ( v + | u | ) (cid:46) r in the integration region. In order to absorb the φ term, we first observe that byassumption, we are considering φ such that φ | I + is well-defined and is compactly supported in u > u −∞ , solim v →∞ φ ( u −∞ , v ) = 0 . Therefore, by Cauchy–Schwarz, we can estimate (cid:20)(cid:90) S ( φ − φ | I + ) dω (cid:21) ( u, v ) ≤ (cid:90) S (cid:18)(cid:90) ∞ v ( Lφ ) dv (cid:48) (cid:19) dω ≤ v − (cid:90) ∞ v (cid:90) S v (cid:48) ( Lφ ) dωdv (cid:48) ≤ v − sup u (cid:90) N u v (cid:48) ( Lφ ) dωdv (cid:48) . Furthermore, similarly we have that (cid:20)(cid:90) S φ | I + dω (cid:21) ( u ) ≤ u − (cid:90) uu −∞ (cid:90) S u (cid:48) ( Lφ ) dωdu (cid:48) ≤ u − sup v (cid:90) I v u (cid:48) ( Lφ ) dωdu (cid:48) . Hence, (cid:20)(cid:90) S φ dω (cid:21) ≤ ( u − + v − ) (cid:20) sup u (cid:90) N u v (cid:48) ( Lφ ) dωdv (cid:48) + sup v (cid:90) I v u (cid:48) ( Lφ ) dωdu (cid:48) (cid:21) , so we can estimate: (cid:90) ∞ u (cid:90) − u − min { v, | u −∞ |} (cid:90) S ( u + v ) r − η φ dωdudv (cid:46) (cid:15) (cid:20) sup u (cid:90) N u v (cid:48) ( Lφ ) dωdv (cid:48) + sup v (cid:90) I v u (cid:48) ( Lφ ) dωdu (cid:48) (cid:21) , with (cid:15) > r I suitably large. As a result, we obtainsup u (cid:90) N u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + sup v (cid:90) I v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu ≤ C (cid:90) N − u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + C (cid:90) I + ≤− u u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu. (8.6)We integrate (8.3) and apply (8.6) to obtain: (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( Lφ ) + r ( Lφ ) + | / ∇ S φ | dv ≤ C (cid:90) N − u v ( Lφ ) + u r − D | / ∇ S φ | dωdv + (cid:90) I + ≤− u u ( Lφ ) + v r − D | / ∇ S φ | dωdu + C (cid:90) ∞ v r I (cid:90) − u − min { v, | u −∞ |} r − | φ | · ( u | Lφ | + v | Lφ | ) dωdudv + C (cid:90) ∞ v r I (cid:90) − u − min { v, | u −∞ |} r − log r | / ∇ S φ | dωdudv ≤ C (cid:90) N − u v ( Lφ ) + u r − D | / ∇ S φ | dωdv + (cid:90) I + ≤− u u ( Lφ ) + v r − D | / ∇ S φ | dωdu. Analogously, we have that (cid:90) S L ( u ( Lφ ) ) + L ( v ( Lφ ) ) + 14 L ( v r − D | / ∇ S φ | ) + 14 L ( u r − D | / ∇ S φ | ) dω = (cid:90) S (cid:18) L (cid:18) u Dr (cid:19) + L (cid:18) v Dr (cid:19)(cid:19) | / ∇ S φ | dω + (cid:90) S ( u Lφ + v Lφ ) O (( r − M ) ) φ dω. (8.7)45nd L (cid:18) u Dr (cid:19) + L (cid:18) v Dr (cid:19) = O (( r − M ) ) | log( r − M ) | , (8.8)so that we can estimatesup v (cid:90) N v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + sup u (cid:90) H u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv ≤ (cid:90) N v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + (cid:90) H + ≤− v v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + C (cid:90) ∞ u r H (cid:90) − v − min { u, | v −∞ |} ( r − M ) | φ | · ( u | Lφ | + v | Lφ | ) dωdudv + C (cid:90) ∞ u r H (cid:90) − v − min { u, | v −∞ |} log(( r − M ) − ) D | / ∇ S φ | dωdudv. (8.9)Using that ( r − M ) − ∼ u + | v | (cid:46) u , we estimate further: (cid:90) ∞ u r H (cid:90) − v − min { u, | v −∞ |} log(( r − M ) − ) D | / ∇ S φ | dωdudv ≤ C (cid:90) ∞ u r H u − log u du · sup u (cid:90) H u u r − D | / ∇ S φ | dωdv ≤ (cid:15) sup u (cid:90) H u u r − D | / ∇ S φ | dωdv for (cid:15) > r H − M > (cid:15) > r − M ) | φ | ( u | Lφ | + v | Lφ | ) ≤ ( r − M ) η ( u ( Lφ ) + v ( Lφ ) ) + ( r − M ) − η ( u + v ) φ and absorb the corresponding spacetime integral to the left-hand side of (8.9), using that (cid:20)(cid:90) S φ dω (cid:21) ≤ ( u − + v − ) (cid:34) sup u (cid:90) N u u (cid:48) ( Lφ ) dωdu (cid:48) + sup v (cid:90) H u v (cid:48) ( Lφ ) dωdv (cid:48) (cid:35) , which follows from Cauchy–Schwarz combined with the assumption that φ | H + ( v ) = 0 for v ≤ v −∞ . We areleft withsup v (cid:90) N v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + sup u (cid:90) H u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv ≤ C (cid:90) N v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + C (cid:90) H + ≤− v v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv (8.10)and hence, (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − ( Lφ ) + ( r − M ) − ( Lφ ) + | / ∇ S φ | du ≤ C (cid:90) H + ≤− v v ( Lφ ) + | / ∇ S φ | dωdv + C (cid:90) N − v D − ( Lφ ) + D | / ∇ S φ | dωdu. We now consider the forwards time direction. First of all, we are assuming compact support on (cid:101) Σ ∩{ v r I ≤ v ≤ − u −∞ } , so for | u −∞ | , | v −∞ | suitably large, we have that φ vanishes along N − u −∞ , N − v −∞ , I + ∩ { u ≤ u −∞ } and H + ∩ { v ≤ v −∞ } , by the domain of dependence property of the wave equation.We then apply the estimates (8.6) and (8.10) to obtain: (cid:90) N − u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + (cid:90) I + ≤− u u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu c (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( Lφ ) + r ( Lφ ) + | / ∇ S φ | dv, (cid:90) N v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + (cid:90) H + ≤− v ∞ v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv ≤ c (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − ( Lφ ) + ( r − M ) − ( Lφ ) + | / ∇ S φ | dωdu, for a suitably small positive constant c > r , u and v , applied to T φ rather than φ . The r -weighted energies along N − u and N − v appearing in the proposition below appear asenergy flux terms in Proposition 7.3 with p = 1. Proposition 8.2.
Let u −∞ , v −∞ < , with | u −∞ | , | v −∞ | arbitrarily large. There exists constants C, c = C, c ( M, r I , r H , u , v ) > , such that (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( LT φ ) + r ( LT φ ) + r − | / ∇ S T φ | dωdr (8.11)+ (cid:88) | α |≤ (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } ( L Ω α φ ) + ( L Ω α φ ) + r − | / ∇ S Ω α φ | dωdr ∼ c,C (cid:88) | α |≤ (cid:90) N − u v ( LT φ ) + r − | / ∇ S T φ | + ( L Ω α φ ) + r − | / ∇ S Ω α φ | dωdv + (cid:88) j =0 (cid:90) I + ≤− u | u | j ( LT j φ ) + | / ∇ S Lφ | dωdu, (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − ( LT φ ) + ( r − M ) − ( LT φ ) + ( r − M ) | / ∇ S T φ | du (8.12)+ (cid:88) | α |≤ (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( L Ω α φ ) + ( L Ω α φ ) + ( r − M ) | / ∇ S Ω α φ | du ∼ c,C (cid:88) | α |≤ (cid:90) N − v u ( LT φ ) + ( r − M ) | / ∇ S T φ | + ( L Ω α φ ) + ( r − M ) | / ∇ S Ω α φ | dωdu + (cid:88) j =0 (cid:90) H + ≤− v | v | j ( LT j φ ) + | / ∇ S Lφ | dωdv. Remark 8.1.
The energy estimates (8.11) and (8.12) are associated to the vector field multiplier Y = v∂ v − u∂ u near infinity and Y = − v∂ v + u∂ u near the horizon. In contrast with the vector field K that playsa role in Proposition 8.1, Y does not correspond to a (conformal) symmetry generator in Minkowski.Proof of Proposition 8.2. First of all, we have immediately that by Lemma 6.3 and [Hardy] (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } ( Lφ ) + ( Lφ ) + r − | / ∇ S φ | dωdv (8.13) ∼ c,C (cid:90) N u ( Lφ ) + r − | / ∇ S φ | dωdv + (cid:90) I + ≤− u ( Lφ ) dωdu, (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( Lφ ) + ( Lφ ) + ( r − M ) | / ∇ S φ | dωdu (8.14) ∼ c,C (cid:90) N − v ( Lφ ) + ( r − M ) | / ∇ S φ | dωdu + (cid:90) H + ≤− v ( Lφ ) dωdv.
47e can moreover replace φ with Ω α φ in the above estimates, with | α | ≤
1, due to the commutation propertiesof Ω i and (cid:3) g .By (6.6) it follows that L ( | u | ( LT φ ) ) + L ( v ( LT φ ) ) = 2 | u | LLT φ · LT φ + 2 vLLT φ · LT φ = 12 | u | r D / ∆ S T φ · LT φ + 12 vr D / ∆ S T φ · LT φ + ( | u | LT φ + vLT φ ) O ( r − ) T φ.
After integrating by parts on S , we therefore obtain: (cid:90) S L ( | u | ( LT φ ) ) + L ( v ( LT φ ) ) + 14 L ( vr − D | / ∇ S T φ | ) + 14 L ( | u | r − D | / ∇ S T φ | ) dω = (cid:90) S (cid:18) L (cid:18) vD r (cid:19) − L (cid:18) uD r (cid:19)(cid:19) | / ∇ S T φ | dω + (cid:90) S ( | u | LT φ + vLT φ ) O ( r − ) T φ dω. (8.15)Note that L (cid:18) vD r (cid:19) − L (cid:18) uD r (cid:19) = 18 ( v + u ) D ddr ( Dr − ) = − t r − + O ( r − ))Hence, after integrating (8.15) in spacetime, the | / ∇ S T φ | term on the right-hand side will have a good signif we consider forwards-in-time estimates and a bad sign if we consider backwards-in-time estimates.In the backwards-in-time case, we use that T = ∂ u + ∂ v and t = ( v − | u | ) and | u | + v (cid:46) r in theintegration region, together with Lemma 6.2 to estimate: (cid:90) − u ∞ v r I (cid:90) − v − u t · r − | / ∇ S T φ | dωdudv (cid:46) (cid:88) | α | =1 (cid:90) − u −∞ v r I (cid:90) − v − u r − (cid:2) r ( L Ω α ψ ) + r ( L Ω α ψ ) (cid:3) dωdudv (cid:46) (cid:88) | α | =1 sup v (cid:90) I v r ( L Ω α ψ ) dωdu + sup u (cid:90) N u r ( L Ω α ψ ) dωdv (cid:46) (cid:88) | α |≤ sup v (cid:90) I v J T [Ω α ψ ] · L dωdu + sup u (cid:90) N u J T [Ω α ψ ] · L dωdv (cid:46) (cid:88) | α |≤ (cid:90) N − u J T [Ω α ψ ] · L dωdv + (cid:90) I + ≤− u J T [Ω α ψ ] · L dωdu, where we arrived at the last inequality by applying Lemma 6.3.
Note that in this step we needed touse that our solution to (1.1) a time derivative, i.e. it is of the form
T ψ ! We moreover apply Young’s inequality to estimate r − | T φ | ( | u || LT φ | + v | LT φ | ) ≤ r − − η ( | u | ( LT φ ) + v ( LT φ ) ) + r − η ( | u | + v )( T φ ) ≤ r − − η ( | u | ( LT φ ) + v ( LT φ ) ) + r − η ( | u | + v )(( Lφ ) + ( Lφ ) ) . We can absorb the spacetime integrals of the terms on the very right-hand side into the following flux terms:sup u (cid:90) N u v ( LT φ ) + 14 | u | r − D | / ∇ S T φ | dωdv + sup v (cid:90) I v | u | ( LT φ ) + 14 vr − D | / ∇ S T φ | dωdu and sup u (cid:90) N u J T [ ψ ] · L dωdv + sup v (cid:90) I v J T [ ψ ] · L dωdu.
Integrating the identity (8.15) in u and v and applying the above estimates therefore gives the followinginequality:sup u (cid:90) N u v ( LT φ ) + 14 | u | r − D | / ∇ S T φ | dωdv + sup v (cid:90) I v | u | ( LT φ ) + 14 vr − D | / ∇ S T φ | dωdu ≤ C (cid:90) N − u v ( LT φ ) + 14 | u | r − D | / ∇ S T φ | dωdv + C (cid:90) I + ≤− u | u | ( LT φ ) + 14 vr − D | / ∇ S T φ | dωdu + C (cid:88) | α |≤ (cid:90) N − u J T [Ω α ψ ] · Ldωdv + (cid:90) I + ≤− u J T [Ω α ψ ] · Ldωdu (8.16)48nd hence, using (8.15) and the above estimate once more, now in combination with (8.16), we arrive at (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( LT φ ) + r ( LT φ ) + r − | / ∇ S T φ | dωdv ≤ C (cid:90) N − u v ( LT φ ) + 14 | u | r − D | / ∇ S T φ | dωdv + C (cid:90) I + ≤− u | u | ( LT φ ) + 14 vr − D | / ∇ S T φ | dωdu + C (cid:88) | α |≤ (cid:90) N − u J T [Ω α ψ ] · Ldωdv + (cid:90) I + ≤− u ( L Ω α φ ) dωdu. We repeat the above arguments near H + by considering L ( | v | ( LT φ ) ) + L ( u ( LT φ ) )and reversing the roles of u and v and L and L , in order to obtain the near-horizon estimate in the backwardstime direction. We omit further details of this step.Now, we consider the forwards time direction. By repeating the arguments above in the forwards timedirection, using that the ψ and n (cid:101) Σ ψ are initially compactly supported and taking | u −∞ | and | v −∞ | appro-priately large, we obtain moreover thatsup u (cid:90) N u v ( LT φ ) + 14 | u | r − D | / ∇ S T φ | dωdv + sup v (cid:90) I v | u | ( LT φ ) + 14 vr − D | / ∇ S T φ | dωdv ≤ C (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( LT φ ) + r ( LT φ ) + r − | / ∇ S T φ | dωdv + C (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } J T [ ψ ] · n (cid:101) Σ dµ (cid:101) Σ . Note that, in contrast with the backwards-in-time estimates, there is no need for an additional angularderivative in the T -energy term on the right hand side. The analogous estimate near H + proceeds byrepeating the above arguments, interchanging the roles of u and v and replacing r by ( r − M ) − . Corollary 8.3.
Let u −∞ , v −∞ < , with | u −∞ | , | v −∞ | arbitrarily large. There exists constants C, c = C, c ( M, r I , r H , u , v ) > , such that (cid:88) j =0 (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r − j ( LT j φ ) + r − j ( LT j φ ) + r − j | / ∇ S T j φ | dωdr (8.17)+ (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } | / ∇ S Lφ | + | / ∇ S Lφ | + r − | / ∇ S φ | dωdr ∼ c,C (cid:88) j =0 (cid:90) N − u r − j ( LT j φ ) + r − | / ∇ S T j φ | + | / ∇ S Lφ | + r − | / ∇ S φ | dωdv + (cid:88) j =0 (cid:90) I + ≤− u (1 + | u | ) − j ( LT j φ ) + | / ∇ S φ | + | / ∇ S Lφ | dωdu, (cid:88) j =0 (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − j ( LT j φ ) + ( r − M ) − j ( LT j φ ) + ( r − M ) j | / ∇ S T j φ | du (8.18)+ (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } | / ∇ S Lφ | + | / ∇ S Lφ | + ( r − M ) | / ∇ S φ | du ∼ c,C (cid:88) j =0 (cid:90) N − v ( r − M ) − j ( LT j φ ) + ( r − M ) | / ∇ S T j φ | + | / ∇ S Lφ | + ( r − M ) | / ∇ S φ | dωdu + (cid:88) j =0 (cid:90) H + ≤− v (1 + | v | ) − j ( LT j φ ) + | / ∇ S φ | + | / ∇ S Lφ | dωdv. roof. We combine (8.14), (8.14) (and apply it to T φ ), Proposition 8.1 and Proposition 8.2. We moreoverapply Lemma 6.2. The aim of this section is to derive analogues of the estimates in Proposition 8.1 for higher-order derivativesof ψ (with additional growing weights). The key vector field that plays a role in this step is S = uL + vL .This vector field is also called the scaling vector field because it generates the scaling conformal symmetryin Minkowski. Even though the exact symmetry property is lost in extremal Reissner–Nordstr¨om, we willsee below that the vector field still has favourable commutation properties with the operator LL . Lemma 8.4.
Let n ∈ N and S = uL + vL . Then LL ( S n φ ) = 14 Dr − / ∆ S ( S n φ ) − DD (cid:48) r S n φ + n max { n − , } (cid:88) k =0 O ( r − ) S k φ + O ( r − ) log r / ∆ S S k φ, (8.19) LL ( S n φ ) = 14 Dr − / ∆ S ( S n φ ) − DD (cid:48) r S n φ + n max { n − , } (cid:88) k =0 O (( r − M ) ) S k φ + O (( r − M ) ) log(( r − M ) − ) / ∆ S S k φ. (8.20) Proof.
We will derive (8.19) and (8.20) inductively. Note that (8.19) and (8.20) hold for n = 0 by (6.6).Now assume (8.19) and (8.20) hold for n = N with N ≥ C function f : LL ( Sf ) = LL ( uLf + vLf )= ( uL + vL + 2)( LLf ) . For any p ≥ S ( O ( r − p )) = O ( r − p ) ,S ( O (( r − M ) p )) = O (( r − M ) p ) . Furthermore, we can expand Dr − = 4( v − u ) + O ( r − ) log r,Dr − = 4( v − u ) + O (( r − M ) ) log(( r − M ) − ) . Hence, S ( Dr − ) = − v − u ) + O ( r − ) log r = − Dr − + O ( r − ) log r,S ( Dr − ) = − v − u ) + O (( r − M ) ) log(( r − M ) − ) = − Dr − + O (( r − M ) ) log(( r − M ) − ) , and we obtain, using the above observations and applying (8.19) with n = N : LL ( S N +1 φ ) = 14 Dr − / ∆ S ( S N +1 φ ) − DD (cid:48) r S N +1 φ + N max { N − , } (cid:88) k =0 O ( r − ) S k +1 φ + O ( r − ) log r / ∆ S S k +1 φ + 14 ( S ( Dr − ) + 2 Dr − ) / ∆ S S N φ + N (cid:88) k =0 O ( r − ) S k φ + O ( r − ) log r / ∆ S S k φ = 14 Dr − / ∆ S ( S N +1 φ ) − DD (cid:48) r S N +1 φ + ( N + 1) N (cid:88) k =0 O ( r − ) S k +1 φ + O ( r − ) log r / ∆ S S k +1 φ. Hence, we can conclude that (8.19) must hold for all n ∈ N . It follows analogously that (8.20) must holdfor all n ∈ N . 50ince the vector field S does not commute with (cid:3) g , we do not immediately obtain Lemma 6.3 for S n ψ replacing ψ , with n ∈ N . However, we show in Proposition 8.5 that, when considering φ instead of ψ , anequivalent energy boundedness statement holds. Proposition 8.5.
Let n ∈ N . There exists constants c, C = c, C ( M, (cid:101) Σ , r I , r H , , u , v , n ) > , such that (cid:88) ≤ k ≤ n (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } ( LS k φ ) + ( LS k φ ) + r − | / ∇ S S k φ | dωdv (8.21) ∼ c,C (cid:88) ≤ k ≤ n (cid:34)(cid:90) N − u ( LS k φ ) + r − | / ∇ S S k φ | dωdv + (cid:90) I + ≤− u ( LS k φ ) dωdu (cid:35) , (cid:88) ≤ k ≤ n (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( LS k φ ) + ( LS k φ ) + D | / ∇ S S k φ | du (8.22) ∼ c,C (cid:88) ≤ k ≤ n (cid:34)(cid:90) N − v ( LS k φ ) + D | / ∇ S S k φ | dωdu + (cid:90) H + ≤− v ( LS k φ ) dωdv (cid:35) . Proof.
We establish the estimate (8.21) inductively. We prove the n = 0 case first and then assume that(8.21) holds for 0 ≤ k ≤ n − k = n case. We will in fact do both of these steps at thesame time in the argument below. By Lemma 8.4, we have that L (( LS n φ ) ) + L ( v ( LS n φ ) ) = 2 LLS n φ · LS n φ + 2 LLS n φ · LS n φ = 12 1 r D / ∆ S S n φ · LS n φ + 12 1 r D / ∆ S S n φ · LS n φ + ( LS n φ + LS n φ ) (cid:32) n (cid:88) k =0 O ( r − ) S k φ + n − (cid:88) k =0 log rO ( r − ) / ∆ S S k φ (cid:33) . (8.23)Furthermore,12 1 r D / ∆ S S n φ · LS n φ + 12 1 r D / ∆ S S n φ · LS n φ = − L (cid:18) r D | / ∇ S S n φ | (cid:19) − L (cid:18) r D | / ∇ S S n φ | (cid:19) . We subsequently integrate both sides of (8.23) in u , v and S and we apply Young’s inequality to absorb allthe spacetime integrals either into the corresponding boundary integrals as in the proof of Proposition 8.1,or (if n ≥
1) also into the left-hand sides of the estimates contained in (8.21) with 0 ≤ k ≤ n − Proposition 8.6.
Let n ∈ N . There exists constants c, C = c, C ( M, r I , r H , n, u , v ) > , such that (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r k ( L k +1 Ω α φ ) + r k ( L k +1 Ω α φ ) (8.24)+ r k ( | / ∇ S L k Ω α φ | + | / ∇ S L k Ω α φ | ) dωdv ∼ c,C (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k (cid:90) N − u r k ( L k +1 Ω α φ ) + r − k | / ∇ S L k Ω α φ | dωdv + (cid:90) I + ≤− u u k ( L k Ω α φ ) + u k | / ∇ S L k Ω α φ | dωdu, (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − − k ( L k +1 Ω α φ ) + ( r − M ) − − k ( L k +1 Ω α φ ) (8.25)+ ( r − M ) − k ( | / ∇ S L k Ω α φ | + | / ∇ S L k Ω α φ | ) dωdu ∼ c,C (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k (cid:90) N − v ( r − M ) − − k ( L k +1 Ω α φ ) + ( r − M ) − k | / ∇ S L k Ω α φ | dωdu + (cid:90) H + ≤− v v k ( L k Ω α φ ) + v k | / ∇ S L k Ω α φ | dωdv. roof. We can apply the same arguments as in Proposition 8.1, replacing φ by S k φ , with 0 ≤ k ≤ n andapplying the more general equations (8.19) and (8.20) instead of (6.6) to obtain: (cid:88) ≤ k ≤ n (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( LS k φ ) + r ( LS k φ ) + | / ∇ S S k φ | dωdv ∼ c,C (cid:88) ≤ k ≤ n (cid:34)(cid:90) N − u r ( LS k φ ) + r − | / ∇ S S k φ | dωdv + (cid:90) I + ≤− u u ( LS k φ ) + | / ∇ S S k φ | dωdu (cid:35) , (cid:88) ≤ k ≤ n (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − ( LS k φ ) + ( r − M ) − ( LS k φ ) + | / ∇ S S k φ | du ∼ c,C (cid:88) ≤ k ≤ n (cid:34) (cid:90) N − v ( r − M ) − ( LS k φ ) + ( r − M ) | / ∇ S S k φ | dωdu + (cid:90) H + ≤− v v ( LS k φ ) + | / ∇ S S k φ | dωdv (cid:35) . We conclude the proof by rewriting S k φ in terms of u and v derivatives and we moreover apply Lemma8.4 to rewrite all mixed u and v derivatives. Furthermore, we apply Lemma 6.2 to replace the angularderivatives by derivatives of the form Ω α . Proposition 8.7.
Let n ∈ N . Then there exists constants c, C = c, C ( M, r I , r H , u , v , n ) > , such that (cid:88) j =0 (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k +1 − j (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r k + j ( L k +1 Ω α T j φ ) + r k − j | / ∇ S L k Ω α T j φ | (8.26)+ r k + j ( L k +1 Ω α T j φ ) + r k − j | / ∇ S L k Ω α T j φ | dωdr + (cid:88) | α |≤ n +1 (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } J T [Ω α ψ ] · n (cid:101) Σ dµ (cid:101) Σ ∼ c,C (cid:88) j =0 (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k +1 − j (cid:90) N − u r k + j ( L k +1 Ω α T j φ ) + r k − j | / ∇ S L k Ω α T j φ | dωdv + (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k (cid:90) I + ≤− u | u | k +1 ( L k +1 Ω α T φ ) + | u | k ( L k +1 Ω α φ ) dωdu + (cid:88) | α |≤ n +1 (cid:90) N − u J T [Ω α ψ ] · L dωdv + (cid:88) | α |≤ n +1 (cid:90) I + ≤− u ( L Ω α φ ) dωdu, (cid:88) j =0 (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k +1 − j (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − k − j ( L k +1 Ω α T j φ ) + ( r − M ) − k +2 − j | / ∇ S L k Ω α T j φ | (8.27)+ ( r − M ) − k − j ( L k +1 Ω α T j φ ) + ( r − M ) − k +2 − j | / ∇ S L k Ω α T j φ | dωdr ∗ + (cid:88) | α |≤ n +1 (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } J T [Ω α ψ ] · n (cid:101) Σ dµ (cid:101) Σ ∼ c,C (cid:88) j =0 (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k +1 − j (cid:90) N − v ( r − M ) − k − j ( L k +1 Ω α T j φ ) + ( r − M ) − k +2 | / ∇ S L k Ω α T j φ | dωdu + (cid:88) ≤ k ≤ n (cid:88) | α |≤ n − k (cid:90) H + ≤− v | v | k +1 ( L k +1 Ω α T φ ) + | v | k ( L k +1 Ω α φ ) + | v | k | / ∇ S L k Ω α φ | dωdv + (cid:88) | α |≤ n +1 (cid:90) N − v J T [Ω α ψ ] · L dωdu + (cid:88) | α |≤ n +1 (cid:90) H + ≤− v ( L Ω α φ ) dωdv. roof. We repeat the arguments in the proof of Proposition 8.2, applying the equations in Lemma 8.4 thatintroduce additional terms, which can be absorbed straightforwardly. Furthermore, rather than using Lemma6.3, we apply Proposition 8.5 where necessary. We then obtain: (cid:88) ≤ k ≤ n (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r ( LT S k φ ) + r ( LT S k φ ) + r − | / ∇ S T S k φ | dωdr + (cid:88) | α |≤ (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } ( L Ω α S k φ ) + ( L Ω α φ ) + r − | / ∇ S Ω α S k φ | dωdr ∼ c,C (cid:88) ≤ k ≤ n (cid:88) | α |≤ (cid:90) N − u r ( LT S k φ ) + r − | / ∇ S T S k φ | + ( L Ω α S k φ ) + r − | / ∇ S Ω α S k φ | dωdv + (cid:88) j =0 (cid:90) I + ≤− u | u | j ( LT j S k φ ) + | / ∇ S LS k φ | dωdu, (cid:88) ≤ k ≤ n (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − ( LT S k φ ) + ( r − M ) − ( LT S k φ ) + ( r − M ) | / ∇ S T S k φ | du + (cid:88) | α |≤ (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( L Ω α S k φ ) + ( L Ω α S k φ ) + ( r − M ) | / ∇ S Ω α S k φ | du ∼ c,C (cid:88) ≤ k ≤ n (cid:88) | α |≤ (cid:90) N − v ( r − M ) − ( LT S k φ ) + ( r − M ) | / ∇ S T S k φ | + ( L Ω α S k φ ) + ( r − M ) | / ∇ S Ω α S k φ | dωdu + (cid:88) j =0 (cid:90) H + ≤− v | v | j ( LT j S k φ ) + | / ∇ S LS k φ | dωdv. We conclude the proof by replacing the S k derivatives by u and v derivatives with weights in | u | and | v | , andmoreover applying Lemma 8.4 to rewrite all mixed u and v derivatives in terms of pure u or v derivatives,angular derivatives and lower-order derivatives. Corollary 8.8.
Let n ∈ N . Then there exists constants c, C = c, C ( M, r I , r H , u , v , n ) > , such that (cid:88) j =0 (cid:88) | α | + k ≤ n (cid:34) (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r k − j ( L k +1 Ω α T j φ ) + r k − j | / ∇ S L k Ω α T j φ | + r k | / ∇ S L k +1 Ω α φ | + r k − | / ∇ S L k Ω α φ | + r k +2 − j ( L k +1 Ω α T j φ ) + r k − j | / ∇ S L k Ω α T j φ | + r k | / ∇ S L k +1 Ω α φ | + r k − | / ∇ S L k Ω α φ | dωdr (cid:35) ∼ c,C (cid:88) j =0 (cid:88) | α | + k ≤ n (cid:34) (cid:90) N − u r k +2 − j ( L k +1 Ω α T j φ ) + r k | / ∇ S L k +1 Ω α φ | dωdv + (cid:90) I + ≤− u | u | k +2 − j ( L k +1 Ω α T j φ ) + | u | k | / ∇ S L k +1 Ω α φ | + | u | k | / ∇ S L k Ω α φ | dωdu (cid:35) + (cid:88) | α |≤ n +1 (cid:90) N − u J T [Ω α ψ ] · L dωdv + (cid:88) | α |≤ n +1 (cid:90) I + ≤− u ( L Ω α φ ) dωdu, (8.28)53 nd (cid:88) j =0 (cid:88) | α | + k ≤ n (cid:34) (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − k − j ( L k +1 Ω α T j φ ) + ( r − M ) − k + j | / ∇ S L k Ω α T j φ | + ( r − M ) − k | / ∇ S L k +1 Ω α φ | + ( r − M ) − k +2 | / ∇ S L k Ω α φ | + ( r − M ) − k − j ( L k +1 Ω α T j φ ) + ( r − M ) − k + j | / ∇ S L k Ω α T j φ | + ( r − M ) − k | / ∇ S L k +1 Ω α φ | + ( r − M ) − k +2 | / ∇ S L k Ω α φ | dωdr ∗ (cid:35) ∼ c,C (cid:88) j =0 (cid:88) | α | + k ≤ n (cid:34) (cid:90) N − v ( r − M ) − k − j ( L k +1 Ω α T j φ ) + ( r − M ) − k | / ∇ S L k +1 Ω α φ | dωdu + (cid:90) H + ≤− v | v | k +2 − j ( L k +1 Ω α T j φ ) + | v | k | / ∇ S L k +1 Ω α φ | + | v | k | / ∇ S L k Ω α φ | dωdv (cid:35) + (cid:88) | α |≤ n +1 (cid:90) N − v J T [Ω α ψ ] · L dωdu + (cid:88) | α |≤ n +1 (cid:90) H + ≤− v ( L Ω α φ ) dωdv. (8.29) Proof.
Follows immediately after combining the results of Proposition 8.6 and Proposition 8.7.By commuting (cid:3) g additionally with T and applying Lemma 6.3, we arrive at energy estimates along N u and N v (rather than N − u and N − v ) with the same weights and number of derivatives as the energy fluxesthat appear in Corollary 6.13 and Corollary 7.8. Corollary 8.9.
Let n ∈ N . Then there exists constants c, C = c, C ( M, r I , r H , u , v , n ) > , such that (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:34) (cid:90) (cid:101) Σ ∩{ v r I ≤ v ≤− u −∞ } r k − j ( L k +1 Ω α T j + m φ ) + r k − j | / ∇ S L k Ω α T j + m φ | + r k | / ∇ S L k +1 Ω α T m φ | + r k − | / ∇ S L k Ω α T m φ | + r k +2 − j ( L k +1 Ω α T j + m φ ) + r k − j | / ∇ S L k Ω α T j + m φ | + r k | / ∇ S L k +1 Ω α T m φ | + r k − | / ∇ S L k Ω α T m φ | dωdr (cid:35) ∼ c,C (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:34) (cid:90) N − u r k +2 − j ( L k +1 Ω α T j + m φ ) + r k | / ∇ S L k +1 Ω α T m φ | dωdv + (cid:90) I + ≤− u (1 + | u | ) k +2 − j ( L k +1 Ω α T j + m φ ) + (1 + | u | ) k | / ∇ S L k +1 Ω α T m φ | + (1 + | u | ) k | / ∇ S L k Ω α φ | dωdu (cid:35) + (cid:88) | α | + m ≤ n +2 (cid:90) N − u J T [Ω α T m ψ ] · L dωdv + (cid:88) | α | + m ≤ n +2 (cid:90) I + ≤− u ( L Ω α T m φ ) dωdu, (8.30)54 nd (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:34) (cid:90) (cid:101) Σ ∩{ u r H ≤ u ≤− v −∞ } ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k + j | / ∇ S L k Ω α T j + m φ | + ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | + ( r − M ) − k +2 | / ∇ S L k Ω α T m φ | + ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k + j | / ∇ S L k Ω α T j + m φ | + ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | + ( r − M ) − k +2 | / ∇ S L k Ω α T m φ | dωdr ∗ (cid:35) ∼ c,C (cid:88) j =0 (cid:88) | α | +2 k + m ≤ n (cid:34) (cid:90) N − v ( r − M ) − k − j ( L k +1 Ω α T j + m φ ) + ( r − M ) − k | / ∇ S L k +1 Ω α T m φ | dωdu + (cid:90) H + ≤− v (1 + | v | ) k +2 − j ( L k +1 Ω α T j + m φ ) + (1 + | v | ) k | / ∇ S L k +1 Ω α T m φ | + (1 + | v | ) k | / ∇ S L k Ω α φ | dωdv (cid:35) + (cid:88) | α | + m ≤ n +2 (cid:90) N − v J T [Ω α T m ψ ] · L dωdu + (cid:88) | α | + m ≤ n +2 (cid:90) H + ≤− v ( L Ω α T m φ ) dωdv. (8.31) In this section we will construct the scattering map, which is a map from energy spaces on I − and H − toenergy spaces on I + and H + . First, we need to define what we mean by the solution to (1.1) in J + ( (cid:101) Σ)arising from scattering data along H + ∪ I + .We introduce the following hypersurface: let s <
0, then (cid:101) Σ s := (cid:101) Σ ∩ { s < r ∗ < | s |} ∪ N s ∪ N s . Definition 8.1.
Let s < s < and define the solutions ψ s i : D + ( (cid:101) Σ s i ) → R as the unique smooth solutionsto (1.1) corresponding to scattering data (Φ , Φ) ∈ C ∞ c ( H + ) ⊕ C ∞ c ( I + ) in accordance with Proposition 7.1.Then, by uniqueness, ψ s | D + ( (cid:101) Σ s ) = ψ s , so we can define the function ψ : D + ( (cid:101) Σ) → R as follows: let p ∈ D + ( (cid:101) Σ) , then there exists an s ∗ > suchthat p ∈ D + ( (cid:101) Σ s ∗ ) . Let ψ ( p ) = ψ s ∗ ( p ) . It follows immediately that ψ is a uniquely determined smooth solution to (1.1) , such that lim v →∞ rψ ( u, v, θ, ϕ ) =Φ( u, θ, ϕ ) and M ψ | H + = Φ . Proposition 8.10.
Let (Ψ , Ψ (cid:48) ) ∈ ( C ∞ c ( (cid:101) Σ)) . Then the corresponding solution ψ to (1.1) satisfies ( r · ψ | H ± , r · ψ | I ± ) ∈ E T H ± ⊕ E T I ± . and furthermore, the following identity holds || r · ψ | H ± || E T H± + || r · ψ | I ± || E T I± = || (Ψ , Ψ (cid:48) ) || E T (cid:101) Σ . Proof.
Follows from Lemma 6.3 and Proposition 6.15 (combined with an analogue of Proposition 6.15 in thepast-direction, making use of the time-symmetry of the spacetime).
Definition 8.2.
Define the evolution maps (cid:102) F ± : ( C ∞ c ( (cid:101) Σ)) → E T H ± ⊕ E T I ± as the following linear operator: (cid:102) F ± (Ψ , Ψ (cid:48) ) = ( r · ψ | H ± , r · ψ | I ± ) , here ψ is the unique solution to (1.1) with ( ψ | (cid:101) Σ , n (cid:101) Σ ψ | (cid:101) Σ ) = (Ψ , Ψ (cid:48) ) . Then (cid:102) F ± extends uniquely to a linearbounded operator, also denoted (cid:102) F ± : (cid:102) F ± : E T (cid:101) Σ → E T H ± ⊕ E T I ± . Proposition 8.11.
Let n ∈ N . Then for all n ∈ N (cid:102) F ± ( C ∞ c ( (cid:101) Σ)) ) ⊆ E n ; H ± ⊕ E n ; I ± , (8.32) and (cid:102) F ± can uniquely be extended as as the following bounded linear operator (cid:102) F n ; ± : E n ; (cid:101) Σ → E n ; H ± ⊕ E n ; I ± . We moreover have that (cid:102) F n ; ± = (cid:102) F ± | E n ; (cid:101) Σ .Proof. Without loss of generality, we restrict our considerations to (cid:102) F + . We choose Σ so thatΣ ∩ { r H ≤ r ≤ r I } = (cid:101) Σ ∩ { r H ≤ r ≤ r I } . Let ψ denote the solution to (1.1) corresponding to initial data (Ψ , Ψ (cid:48) ) ∈ C ∞ c ( (cid:101) Σ)) . We apply Corollary 8.9to conclude that || ( ψ | Σ , n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } ) || E n ;Σ0 ≤ C || (Ψ , Ψ (cid:48) ) || E n ; (cid:101) Σ . We then apply the bounded operator F n from Corollary 6.16 to arrive at (8.32). The extension propertyfollows immediately from the uniform boundedness of (cid:102) F + with respect to the desired norms. Proposition 8.12.
Let (Φ , Φ) ∈ C ∞ c ( H ± ) ⊕ C ∞ c ( I ± ) . Then the corresponding solution ψ according toDefinition 8.1 satisfies ψ | (cid:101) Σ ( r, θ, ϕ ) → as r → ∞ and r ↓ M and || ( ψ | (cid:101) Σ , n (cid:101) Σ ψ | (cid:101) Σ ) || E T (cid:101) Σ = || Φ || E T H± + || Φ || E T I± . Proof.
By applying the fundamental theorem of calculus, we have that for suitably large r ∗ > ψ (0 , r ∗ , θ, ϕ ) ≤ r (cid:90) N − r ∗ T ( ∂ t , L ) r dωdv ≤ (cid:90) H + ( Lφ ) dωdv + (cid:90) I + ( Lφ ) dωdu. so ψ | (cid:101) Σ ( r, θ, ϕ ) → r → ∞ . By considering r ∗ < | r ∗ | suitably large, we can conclude analogouslythat ψ | (cid:101) Σ ( r, θ, ϕ ) → r → ∞ and r ↓ M .The energy conservation statement simply follows from applying Lemma 6.3. Definition 8.3.
Define the backwards evolution maps (cid:101) B ± : C ∞ c ( H ± ) ⊕ C ∞ c ( I ± ) → E T (cid:101) Σ as the followinglinear operator: (cid:101) B ± (Φ , Φ) = ( ψ | (cid:101) Σ , n (cid:101) Σ ψ | (cid:101) Σ ) , where ψ is the corresponding unique solution to (1.1) as defined in Definition 8.1. Then (cid:101) B ± extends uniquelyto a linear bounded operator, also denoted (cid:101) B ± : (cid:101) B ± : E T H ± ⊕ E T I ± → E T (cid:101) Σ . Proposition 8.13.
The linear operator (cid:102) F ± : E T (cid:101) Σ → E T H ± ⊕ E T I ± is bijective with (cid:101) B ± = (cid:102) F − ± .Proof. Follows by the same arguments as in the proof of Proposition 7.12.
Proposition 8.14.
Let n ∈ N . Then for all n ∈ N (cid:101) B ± ( C ∞ c ( H ± ) ⊕ C ∞ c ( I ± )) ⊆ E n ; (cid:101) Σ , (8.33) and (cid:101) B ± can uniquely be extended as as the following bounded linear operator (cid:101) B n ; ± : E n ; H ± ⊕ E n ; I ± → E n ; (cid:101) Σ . We moreover have that (cid:101) B n ; ± = (cid:101) B ± | E n ; (cid:101) Σ and (cid:101) B n ; ± = (cid:102) F − n ; ± . roof. Without loss of generality, we consider (cid:101) B + . We choose Σ so thatΣ ∩ { r H ≤ r ≤ r I } = (cid:101) Σ ∩ { r H ≤ r ≤ r I } . We apply B n from Proposition 7.11 to conclude that the ψ corresponding to initial data (Φ , Φ) ∈ C ∞ c ( H + ) ⊕ C ∞ c ( I + ) satisfies || ( ψ | Σ , n Σ ψ | Σ ∩{ r H ≤ r ≤ r I } ) || E n ;Σ0 ≤ C ( || Φ || E n ; H + + || Φ || E n ; I + ) . Hence, we can apply Corollary 8.9 to obtain (8.33). The extension property then follows from the uniformityof all estimates involved. The inversion follows by repeating the arguments in the proof of Proposition7.12.
Definition 8.4.
We define the scattering matrix S : E T H − ⊕ E T I − → E T H + ⊕ E T I + as the following boundedlinear operator: S := (cid:102) F + ◦ (cid:101) B − . Let n ∈ N . Then we define the restricted scattering matrix S n : E n ; H − ⊕ E n ; I − → E n ; H + ⊕ E n ; I + as thefollowing bounded linear operator: S n := (cid:102) F n ;+ ◦ (cid:101) B n ; − . In this section, we obtain some additional estimates in the black hole interior, which allow use to constructa non-degenerate interior scattering map.
Proposition 9.1.
Let u int < with | u int | suitably large. Then there exist constants c, C = c, C ( M, u , v ) > such that (cid:90) H int u int v ( Lφ ) + D | / ∇ S φ | dv + (cid:90) CH + ≤ u int u ( Lφ ) + | / ∇ S φ | dωdu ∼ c,C (cid:90) N int v u ( Lφ ) + D | / ∇ S φ | dωdu + (cid:90) H + ≥ v v ( Lφ ) + | / ∇ S φ | dv. (9.1) Proof.
Observe first that (8.7) and (8.9) hold also in M int , with respect to the Eddington–Finkelstein double-null coordinates ( u, v ). Hence, we can estimate, for ψ arising from data along H + ≥ v and N int v :sup v (cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + sup u (cid:90) H int u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv ≤ (cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + (cid:90) H + ≥ v v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv + Cv − (cid:15) (cid:90) u −∞ (cid:90) ∞ v ( r − M ) | φ | · ( u | Lφ | + v | Lφ | ) dωdudv + Cv − (cid:15) (cid:90) u −∞ (cid:90) ∞ v log(( r − M ) − ) D | / ∇ S φ | dωdudv. (9.2)Using that ( r − M ) − ∼ v + | u | in M int ∩ D + (Σ ∪ N int v ), we can absorb the last two integrals on theright-hand side into the left-hand side for | u | suitably large, in order to obtainsup v (cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + sup u (cid:90) H int u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv ≤ C (cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + C (cid:90) H + ≥ v v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv. { v k } we can bound for any n > m ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) N int vn u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu − (cid:90) N int vm u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( v m ) − (cid:15) (cid:34)(cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + (cid:90) H + ≥ v v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv (cid:35) + (cid:90) H + ≥ vm v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv. So we can conclude that (cid:90) N int vk u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu is a Cauchy sequence, so it must converge as k → ∞ . Furthermore, the limit is independent of the choice ofsequence. Hence, (cid:90) CH + ≤ u int u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu =: lim v →∞ (cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu is well-defined.Similarly, if we take ψ to arise from data along CH + ≤ u int and H int u int , we can apply (8.7) and (8.9) to showthat sup v (cid:90) N int v u ( Lφ ) + 14 v r − D | / ∇ S φ | dωdu + sup u (cid:90) H int u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv ≤ c (cid:90) H int u int v ( Lφ ) + D | / ∇ S φ | dv + 1 c (cid:90) CH + ≤ u int u ( Lφ ) + | / ∇ S φ | dωdu. and it follows analogously that (cid:90) H + ≥ v v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdu =: lim u →−∞ (cid:90) N int u v ( Lφ ) + 14 u r − D | / ∇ S φ | dωdv is well-defined.The estimate (9.1) then follows by combining the above estimates. Proposition 9.2.
Let u int < with | u int | suitably large. Let S int : C c ( H + v ≥ v ) × C ∞ ( N int v ) → E CH + ≤ u int ⊕E H int u int be defined as follows: S int ( rψ | H + ≥ v , rψ | N int v ) = ( rψ | CH + ≤ u int , rψ H int u int ) . Then S int extends uniquely as a bijective, bounded linear operator: S int : E H + v ≥ v ⊕ E N int v → E CH + ≤ u int ⊕ E H int u int . Proof.
The construction of S int and its inverse, on a domain of smooth, compactly supported functions,follow immediately from the estimates in the proof of Proposition 9.1, where rψ | CH + ≤ u int (in the forwardsdirection) and rψ | H + ≥ v (in the backwards direction) can understood in a limiting sense, as in Proposition9.1, and it follows that rψ | CH + ≤ u int ∈ E CH + ≤ u int and rψ | H + ≥ v ∈ E H + v ≥ v by the fundamental theorem of calculus: (cid:18)(cid:90) S φ dω (cid:19) ( u, v ) ≤ | u | − (cid:90) N int v u (cid:48) ( Lφ ) dωdu (cid:48) in the forwards direction and58 (cid:90) S φ dω (cid:19) ( u, v ) ≤ | v | − (cid:90) H int u v (cid:48) ( Lφ ) dωdv (cid:48) in the backwards direction , and (a straightforward variation of) 2.) of Lemma A.1. The extendibility follows moreover from the unifor-mity of the estimates in Proposition 9.1. Corollary 9.3.
Let u int < with | u int | suitably large. Then there exist a constant C = C ( M, u , v ) > such that we can estimate with respect to ( u, r ) coordinates: (cid:90) H int u int ( ∂ r φ ) + ( ∂ u φ ) + | / ∇ S φ | dωdr ≤ C (cid:34) (cid:88) j ≤ (cid:90) N int v u ( ∂ u T j φ ) + D | / ∇ S T j φ | dωdu + (cid:90) H + ≥ v v ( ∂ v T j φ ) + | / ∇ S T j φ | dωdv (cid:35) (9.3) Furthermore, (cid:90) H int u int ( ∂ r φ ) + ( ∂ u φ ) + | / ∇ S φ | dωdr ≤ C (cid:34) (cid:88) j ≤ (cid:90) N int v u ( ∂ u T j φ ) + D | / ∇ S T j φ | dωdu + (cid:88) j ≤ || ( T j ψ | Σ , n T j ψ | Σ ) || E Σ0 (cid:35) (9.4) Proof.
We use that ∂ v r | H int u int ∼ v − , together with (cid:18)(cid:90) S φ dω (cid:19) ( u, v ) ≤ | u | − (cid:90) N int v u (cid:48) ( Lφ ) dωdu (cid:48) and we apply the estimates of Proposition 9.1, replacing ψ with T j ψ , j = 0 ,
1, to arrive at (9.3). We obtain(9.4) by appealing additionally to
Remark 9.1.
One can easily extend the estimate in Corollary 9.3 to smaller values of | u int | (provided r > r min > ), by applying a standard Gr¨onwall inequality.
10 Application 1 : Regularity at the event horizon and null infin-ity
As an application of the maps B n constructed in Proposition 7.11, we can show that we can associatearbitrarily regular solutions to suitably polynomially decaying scattering data along H + and I + . First ofall, we will show that by considering T k ψ , rather than ψ , we obtain higher-regularity near H + and I + .Before we address these regularity properties, we will relate the differential operators ( r L ) k and (( r − M ) − L ) k to r k L k and ( r − M ) − k L k . Lemma 10.1.
Let ψ be a solution to (1.1) . Then we can express for all k ∈ N : r LL (( r L ) k φ ) = O ( r ) L (( r L ) k φ ) + k (cid:88) j =0 O (1)( r L ) j φ + O (1)( r L ) j / ∆ S φ. and ( r − M ) − LL ((( r − M ) − L ) k φ ) = O (( r − M ) − ) L (( r L ) k φ ) + k (cid:88) j =0 O (1)( r L ) j φ + O (1)( r L ) j / ∆ S φ. roof. The identities can be obtained inductively by applying (7.7) and commuting LL with r L and r L .See Lemma 6.1 in [4] for more details. Proposition 10.2.
Let ψ be a solution to (1.1) . For all k ≥ we have that: ( r LT ) k ( φ ) = (cid:88) m + i +2 s ≤ k ≤ s ≤ k,m ≥ ≤ i ≤ k − m (cid:88) j =0 O ( r m − j ) L m − j / ∆ s S T i φ. (10.1) and (( r − M ) − LT ) k ( φ ) = (cid:88) m + i +2 s ≤ k ≤ s ≤ k,m ≥ ≤ i ≤ k − m (cid:88) j =0 O (( r − M ) − m + j ) L m − j / ∆ s S T i φ. (10.2) Proof.
We will do a proof by induction. We have that (10.1) and (10.2) hold for k = 1. Suppose (10.1) and(10.2) hold for 1 ≤ k ≤ n . We will show below that (10.1) and (10.2) also hold for k = n + 1.Writing T = L + L , we can express( r LT ) n +1 φ = r L ( r LT ) n φ + r LL (( r LT ) n φ )and apply Lemma 10.1 to obtain( r LT ) n +1 φ = r L ( r LT ) n φ + O ( r ) L (( r LT ) n φ ) + n (cid:88) k =0 O (1)( r LT ) k T n − k φ + O (1)( r LT ) k / ∆ S T n − k φ. Now, we take apply (10.1) for 0 ≤ k ≤ n (taking appropriate derivatives on both sides of the equation) toobtain: ( r LT ) n +1 φ = (cid:88) m + i +2 s ≤ n +20 ≤ s ≤ n,m ≥ ≤ i ≤ n − m (cid:88) j =0 O ( r m − j ) L m − j / ∆ s S T i φ + n (cid:88) k =1 (cid:88) m + i +2 s ≤ k +20 ≤ s ≤ k +1 ,m ≥ n − k ≤ i ≤ k − n − k ) m (cid:88) j =0 O ( r m − j ) L m − j / ∆ s S T i φ + O (1) T n φ + O (1) / ∆ S T n φ = (cid:88) m + i +2 s ≤ n +1)0 ≤ s ≤ n +1 ,m ≥ ≤ i ≤ n m (cid:88) j =0 O ( r m − j ) L m − j / ∆ s S T i φ. We apply an analogous argument, using that L ( O (( r − M ) p ) = O (( r − M ) p +1 ), to also conclude that (10.2)holds for k = n + 1. Proposition 10.3.
Let n ∈ N . Suppose that (Φ , Φ) ∈ E n ; I + ≥ u ⊕ E n ; H + ≥ v . Then we have that the corre-sponding solution ψ to (1.1) satisfies T n ( rψ ) ∈ W n +1 , ( (cid:98) R ) . Proof.
By Proposition 7.11, we have that B n (Φ , Φ) ∈ E n ;Σ . Hence, (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) N u r k − j ( L k +1 T m + j Ω α φ ) dωdv + (cid:90) N v ( r − M ) − − k + j ( L k +1 T m + j Ω α φ ) dωdu + (cid:88) m +2 | α |≤ n +2 | α |≤ n (cid:90) Σ J T [ T m Ω α ψ ] · n dµ < ∞ .
60e subsequently apply Proposition 10.2 to obtain in ( v, r ) coordinates: (cid:88) k + m + α ≤ n (cid:90) N v r ( L ( r L ) k T m Ω α ( T n φ )) dωdv + (cid:90) N u ( ∂ kr T m Ω α ( T n φ )) dωdr + (cid:88) m +2 | α |≤ n +2 | α |≤ n (cid:90) Σ J T [ T m Ω α ( T n ψ )] · n dµ < ∞ . We conclude the proof by integrating the above norm locally in τ . Definition 10.1.
Consider (Φ , Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) such that (cid:90) ∞ v | Φ | dv < ∞ , (cid:90) ∞ u | Φ | du < ∞ . Then we define the time-integrals T − Φ and T − Φ of Φ and Φ as follows: T − Φ( v, θ, ϕ ) = − (cid:90) ∞ v Φ( v (cid:48) , θ, ϕ ) dv (cid:48) ,T − Φ( u, θ, ϕ ) = − (cid:90) ∞ u Φ( u (cid:48) , θ, ϕ ) du (cid:48) Let n ≥ and δ > and suppose that lim v →∞ v n + δ | Φ | ( v, θ, ϕ ) < ∞ and lim u →∞ u n + δ | Φ | ( u, θ, ϕ ) < ∞ . Thenwe define the n -th order time-integrals T − n Φ and T − n Φ of Φ and Φ inductively as follows: T − n Φ( v, θ, ϕ ) = − (cid:90) ∞ v T − ( n − Φ( v (cid:48) , θ, ϕ ) dv (cid:48) ,T − n Φ( u, θ, ϕ ) = − (cid:90) ∞ u T − ( n − Φ( u (cid:48) , θ, ϕ ) du (cid:48) , with T Φ := Φ and T Φ := Φ . Lemma 10.4.
Let n ∈ N and let (Φ , Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) . Assume that lim v →∞ v n + δ | Φ | ( v, θ, ϕ ) < ∞ and lim u →∞ u n + δ | Φ | ( u, θ, ϕ ) < ∞ for some δ > and assume moreover that || T − n Φ || E n ; I + ≥ u + || T − n Φ || E n ; H + ≥ v < ∞ . (10.3) Then T n ( T − n ψ ) = ψ, with ψ the solution associated to (Φ , Φ) and T − n ψ the solution associated to ( T − n Φ , T − n Φ) .Proof. By (10.3), we can conclude thatlim u →∞ T n ( rT − n ψ )( u, v, θ, ϕ ) = L n ( T − n Φ) = Φ , lim v →∞ T n ( rT − n ψ )( u, v, θ, ϕ ) = L n ( T − n Φ) = ΦHence, by uniqueness of ψ given (Φ , Φ), we conclude that T n ( T − n ψ ) = ψ. roposition 10.5. Let n ∈ N and let (Φ , Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) . Assume that lim v →∞ v n + δ | Φ | ( v, θ, ϕ ) < ∞ and lim u →∞ u n + δ | Φ | ( u, θ, ϕ ) < ∞ for some δ > and assume moreover that || T − n Φ || E n ; I + ≥ u + || T − n Φ || E n ; H + ≥ v < ∞ . (10.4) Then rψ ∈ W n +1 , ( (cid:98) R ) . Proof.
By the assumptions on the limiting behaviour of Φ and Φ, together with (10.4), we can apply Propo-sition A.1 to conclude that ( T − n Φ , T − n Φ) ∈ E n ; H + ≥ v ⊕ E n ; I + ≥ u . Then we can apply Proposition 10.3together with Lemma 10.4 to conclude the proof.
11 Application 2: A scattering construction of smooth solutions
We make use of the results in Section 10 to construct smooth solutions from scattering data.
Corollary 11.1.
Let (Φ , Φ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) such that lim v →∞ v p | L k Ω α Φ | ( v, θ, ϕ ) = 0 , lim u →∞ u p | L k Ω α Φ | ( u, θ, ϕ ) = 0 , for all p ∈ R , k ∈ N and α ∈ N . Then rψ ∈ C ∞ ( (cid:98) R ) . Proof.
By the initial data assumptions, we have that T − n Φ and T − n Φ are well-defined and satisfy (10.4) forall n ∈ N . Hence we arrive at the desired statement by applying Proposition 10.5 together with standardSobolev embeddings.Corollary 11.1 allows us to construct smooth “mode solutions” with an arbitrary frequency ω withpostitive imaginary part: Proposition 11.2.
Let ω ∈ C with Im ω < . Let Φ( v, θ, ϕ ) = f H ( θ, ϕ ) e − iωv and Φ( u, θ, ϕ ) = f I ( θ, ϕ ) e − iωu for f H , f I ∈ C ∞ ( S ) . Then there exists a unique smooth solution ψ to (1.1) on ˆ R , such that r · ψ ( τ, ρ, θ, ϕ ) = f ( ρ, θ, ϕ ) e − iω · τ , with f ∈ C ∞ ( ˆΣ) and lim ρ ↓ M f ( ρ, θ, ϕ ) = f H ( θ, ϕ ) , lim ρ →∞ f ( ρ, θ, ϕ ) = f I ( θ, ϕ ) . Proof.
The initial data satisfy the assumptions of Corollary 11.1, so we have that rψ ∈ C ∞ ( (cid:98) R ) andlim u →∞ rT ψ ( u, v, θ, ϕ ) = T Φ( v, θ, ϕ ) , lim v →∞ rT ψ ( u, v, θ, ϕ ) = T Φ( u, θ, ϕ ) . Furthermore, the specific choice of (Φ , Φ) ensures that T Φ + iω Φ = 0 ,T Φ + iω Φ = 0 . Hence, by uniqueness of the associated solution to (1.1), linearity and Lemma 10.4, we have that
T ψ + iωψ = 0so ψ ( τ, ρ, θ, ϕ ) = f ( ρ, θ, ϕ ) e − iω · τ for some f ∈ C ∞ ( ˆΣ).62 Basic estimates
Lemma A.1.
Let ( f, g ) ∈ C ∞ ( H + ≥ v ) ⊕ C ∞ ( I + ≥ u ) .1.) We have that ( f, g ) ∈ E T H + ≥ v ⊕ E T I + ≥ u if || f || E T H + ≥ v + || g || E T I + ≥ u < ∞ . and lim v →∞ f < ∞ , lim u →∞ g < ∞ .2.) Let n ∈ N . Then ( f, g ) ∈ E n ; H + ≥ v ⊕ E n ; I + ≥ u if || f || E n ; H + ≥ v + || g || E n ; I + ≥ u < ∞ and lim v →∞ f = 0 , lim u →∞ g = 0 .Proof. We will prove 2 . ). The proof of 1 . ) proceeds very similarly. Without loss of generality, we can restrictto f . The estimates for g proceed entirely analogously. We introduce a smooth cut-off χ : [ v , ∞ ) → R such that χ ( v ) = 1 for all v ≤ v and χ = 0 for all v ≥ v . Then | χ ( k ) | ≤ C k . Rescale χ by defining χ i : [ v , ∞ ) → R , with i ∈ N , as follows: χ i ( v ) := χ ( vi ). Then | χ ( k ) i | ≤ i − k C k for all 0 ≤ k ≤ n and hence | χ ( k ) i | ≤ C k v − k .Now, define f i = χ i · f , then f i ∈ C ∞ c ( H + ≥ v ). Furthermore, by applying the Leibniz rule successively andusing that | χ ( k ) i | ≤ C k v − k for k ≥
1, we obtain || f − f i || E n ; H + ≥ v ≤ C (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) H + ≥ v i v k +2 − j ( ∂ k + m + jv Ω α f ) + v k | / ∇ S ∂ k + mv Ω α f | dωdv + C (cid:88) j =0 (cid:88) m +2 k ≤ n (cid:90) H +2 v i ≤ v ≤ v i v k − j | χ (1+ k + m + j ) | f dωdv. and (cid:88) j =0 (cid:88) m +2 k ≤ n (cid:90) H +2 v i ≤ v ≤ v i v k − j | χ (1+ k + m + j ) | f dωdv ≤ C (cid:90) H +2 v i ≤ v ≤ v i f dωdv. Note that since f ( v, θ, ϕ ) → v → ∞ , we can estimate f ( v, θ, ϕ ) ≤ v − (cid:90) ∞ v v (cid:48) ( ∂ v f ) ( v (cid:48) , θ, ϕ ) dv, so (cid:90) H +2 v i ≤ v ≤ v i f dωdv ≤ (cid:90) H + v ≥ v i v ( ∂ v f ) dωdv. Hence, || f − f i || E n ; H + ≥ v → i → ∞ and we can conclude that f lies in the completion of C ∞ c ( H + ≥ v ) withrespect to the norm || · || E n ; H + ≥ v . Lemma A.2.
Let ( f, g ) ∈ ( C ∞ ( (cid:101) Σ)) .1.) Then ( f, g ) ∈ E n ; (cid:101) Σ if || ( f, g ) || E T (cid:101) Σ < ∞ and lim r →∞ f = 0 . .) Let n ∈ N . Then ( f, g ) ∈ E n ; (cid:101) Σ if || ( f, g ) || E n ; (cid:101) Σ < ∞ and lim r →∞ rf = 0 .Proof. We will prove 2 . ). The proof of 1 . ) proceeds very similarly. The proof proceeds analogously to theproof of Lemma A.1. We first introduce a cut-off χ : ( −∞ , ∞ ) → R such that χ ( r ∗ ) = 1 for all | r ∗ | ≤ r ,with r >
0, and χ ( r ∗ ) = 0 for all | r ∗ | ≥ r . Then we define χ i : ( −∞ , ∞ ) → R as follows: χ i ( r ∗ ) = χ ( r ∗ i ).Observe that | χ ( k ) i | ≤ C k (1 + | r ∗ | ) − k . Define f i := χ i · f and g i = χ i · f , then ( f i , g i ) ∈ C ∞ c ( (cid:101) Σ) × C ∞ c ( (cid:101) Σ).Furthermore, if ψ denotes the solution corresponding to the initial data ( f, g ), we can estimate || ( f − f i , g − g i ) || E n ; (cid:101) Σ ≤ C (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) (cid:101) Σ ∩{ r ∗ ≥ r i } r k − j ( ∂ k +1 v Ω α T j + m φ ) + r k − j | / ∇ S ∂ kv Ω α T j + m φ | + r k +2 − j ( ∂ k +1 u Ω α T j + m φ ) + r k − j | / ∇ S ∂ ku Ω α T j + m φ | dωdr ∗ + C (cid:88) j =0 (cid:88) m +2 k +2 | α |≤ n (cid:90) (cid:101) Σ ∩{ r ∗ ≤− r i } ( r − M ) − k − j ( ∂ k +1 v Ω α T j + m φ ) + ( r − M ) − k + j | / ∇ S ∂ kv Ω α T j + m φ | + ( r − M ) − k − j ( ∂ k +1 u Ω α T j + m φ ) + ( r − M ) − k + j | / ∇ S ∂ ku Ω α T j + m φ | dωdr ∗ + C (cid:88) m ≤ n +2 (cid:90) (cid:101) Σ ∩{| r ∗ |≥ r i } J T [ T m ψ ] · n (cid:101) Σ dµ (cid:101) Σ + C (cid:90) (cid:101) Σ ∩{ r i ≤| r ∗ |≤ r i } φ dωdr ∗ . Furthermore, using that lim | r ∗ |→∞ φ = 0, (cid:90) (cid:101) Σ ∩{ r i ≤| r ∗ |≤ r i } φ dωdr ∗ ≤ C (cid:90) (cid:101) Σ ∩{| r ∗ |≥ r i } r ( Lφ ) dωdr ∗ . Hence, || ( f − f i , g − g i ) || E n ; (cid:101) Σ → i → ∞ , so ( f, g ) are in the completion of ( C ∞ c ( (cid:101) Σ)) with respect to thenorm || · || E n ; (cid:101) Σ . References [1] A. Almakroudi. Boundedness of linear waves on the Interior of Extremal Reissner–Nordstr¨om–de SitterBlack Holes.
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