A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström

It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect 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 equation on extremal Reissner–Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.

on black hole backgrounds provide useful insights in studying the evolution of perturbations "at infinity". In this article we construct a new scattering theory for scalar perturbations on extremal Reissner-Nordström. Our theory makes crucial use of the vanishing of the surface gravity on the event horizon and our methods extend those of the horizon instability of extremal black holes in the forward-in-time evolution. In the remainder of this section we will briefly recall scattering theories for sub-extremal backgrounds and in the next 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 spacetime backgrounds. Let T denote the standard stationary Killing vector field on a Schwarzschild spacetime. Since T is globally causal 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 [26,27] where a T -scattering theory on Schwarzschild, in the sense of Lax-Phillips [43], was developed (Fig. 1a). Subsequently, the T -scattering theory was understood by Nicolas [51], following the notion of scattering states by Friedlander [30] (Fig. 1b).
The T -energy scattering theory on Schwarzschild applies also when the standard Schwarzschild time function t is replaced by a time function corresponding to a foliation by hypersurfaces intersecting the future event horizon and terminating at future null infinity (Fig. 2a). This is convenient since it allows one to bound energies as measured by local observers. Recall that T is timelike in the black hole exterior and null on the event horizon. For this reason, the T -energy flux across an achronal hypersurface intersecting the event horizon is positive-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 [23,24] that Schwarzschild does not admit a non-degenerate scattering theory where the norm on the achronal hypersurface is defined in terms of the energy flux associated to a globally timelike vector field N (Fig. 2b) and the norms on the event horizon and null infinity are also defined in terms of energy flux associated with N , but with additional, arbitrarily fast polynomially decaying weights in time. This is due to the celebrated redshift effect which turns into a blueshift instability mechanism when seen from the backwards scattering point of view.
It is important to note that one can counter the blue-shift mechanism and 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 decay exponentially fast with sufficiently large rate (Fig. 3). A fully nonlinear version of this statement, in the context of the vacuum Einstein equations, was presented in [19].
As far as the Kerr family is concerned, Dafermos, Rodnianski and Shlapentokh-Rothman [23] derived a degenerate scattering theory in terms of the energy flux associated to a globally causal vector field V which is null on the event horizon and timelike in the exterior region. Similarly to the Schwarzschild case, the sub-extremal Kerr backgrounds do not admit a non-degenerate scattering theory in the exterior region. Let us also note that a T -energy scattering theory on Oppenheimer-Snyder spacetimes, describ- The T -scattering map.
The N -scattering map fails to be surjective.  ing Schwarzschild-like black holes arising from gravitational collapse, was developed in [1].
Finally we present some results regarding the black hole interior region. Luk-Oh [46] showed that the forward evolution of smooth compactly supported initial data on sub-extremal Reissner-Nordström (RN) is W 1,2 -singular at the Cauchy horizon (Fig. 4).
Similar instability results for the wave equation on Kerr interiors were presented by Luk-Sbierski [47] and independently by Dafermos-Shlapentokh-Rothman [24] (see also [29,39,40]). Specifically, in [24] the authors assumed trivial data on the past event horizon and arbitrary, non-trivial polynomially decaying data on past null infinity and showed that local (non-degenerate) energies blow up in a neighborhood of any point at the Cauchy horizon (Fig. 5). The interior of Schwarzschild was considered by Fournodavlos and Sbierski [28], who derived asymptotics for the wave equation at the singular boundary {r = 0}.

Overview of the main theorems.
In this section we present a rough version of our main theorems. Theorems A and B are straightforward extensions of known results, so we will only sketch their proofs, whereas Theorems 1-6 are entirely novel results that require new techniques and whose precise statements of the theorems can be found in Sect. 4.
First of all, note that the standard stationary Killing vector field T is causal everywhere in the domain of outer communications of ERN. From this, it follows that the T -energy scattering theory in Schwarzschild can easily be extended to ERN (see Fig. 6): The T -scattering theory in Schwarzschild extends to extremal Reissner-Nordström.
Proof. Follows by applying the methods in Section 9.6 of [23] together with the decay estimates derived in [8].
In the following theorem, we show that in ERN we can in fact go beyond T -energy scattering by providing a bijective scattering theory for weighted and non-degenerate norms on ERN; see Fig. 7 for an illustration. Here, 0 will denote a spacelike-null hypersurface intersecting H + and terminating at I + . Note that the E 0 −norm is non-degenerate both at the event horizon and at null infinity (the latter understood in an appropriate conformal sense; see Sect. 2.4). The omitted terms involve either smaller weights or extra degenerate factors and additional angular or time derivatives. Here J T and J N denote the energy fluxes associated to the vector fields T and N and ∂ ρ is a tangential to 0 derivative such that ∂ ρ r = 1. Let E H + ∩J + ( 0 ) , E I + ∩J + ( 0 ) , E 0 denote the closure of smooth compactly supported data under the corresponding norms schematically defined above. The above theorem is in stark contrast to the sub-extremal case where the backwards evolution is singular at the event horizon (contrast Fig. 7 with Fig. 2).
By the bijective properties of Theorem 1, we can moreover conclude immediately that all scattering data along H + and I + with finite T -energy but with infinite weighted norm (as in (1.2)) will have an infinite weighted non-degenerate energy on 0 . The above theorem however does not specify which of the horizon-localized N -energy or the weighted energy for {r > R 0 }, for some large R 0 > 0, is infinite. The following theorem shows that there are characteristic data for which the solutions specifically have infinite horizon-localized N -energy. This immediately implies that the unweighted nondegenerate N -energy forward scattering map fails to be invertible, in other words we can find data with finite characteristic N -energies but with infinite standard (unweighted) N -energy at 0 . Theorem B. There exists solutions ψ to (1.1) on ERN that are smooth away from the event horizon H + with finite T -energy flux along H + and future null infinity I + , such that either: (i) ψ| H + vanishes, but r ψ| I + satisfies (1 + u) p (∂ u (r ψ)) 2 sin θ dθ dϕdu = ∞ if and only if p ≥ 2 and ψ has infinite unweighted N -energy flux along 0 ∩ {r ≤ r 0 }, with r 0 > r + arbitrarily close to the horizon radius r + , or (1 + v) p (∂ v (r ψ)) 2 sin θ dθ dϕdv = ∞ if and only if p ≥ 2 and ψ has infinite weighted N -energy flux along ∩{r ≥ R 0 } with R 0 > 0 arbitrarily large.
The following theorem concerns the scattering of initial data with higher regularity; see Fig. 8 for an illustration. • between a weighted higher-order energy space on (H + ∩ J + ( 0 ), I + ∩ J + ( 0 )) and a degenerate higher-order energy space on 0 are bounded and bijective.
The above theorem is of particular importance in constructing special solutions with high regularity. We next present a scattering result for the black hole interior of ERN ( Fig. 9) that extends the results derived in [31].

Theorem 3. (Rough version of Theorem 4.3) The scattering map in the black hole interior of ERN defined between weighted energy spaces is bounded and bijective.
We will now provide a few applications of the above theorems. The first application has to do with the relation of decay along H + and I + and regularity of the data on the hypersurface 0 (see Fig. 10).    Contrast Fig. 11 with Fig. 4 in the sub-extremal case. See also Remark 4.5.

Related works.
A closely related topic to the scattering theories on black holes is the black hole stability problem for the forward-in-time evolution. Intense research has been done for both sub-extremal and extremal black holes in this direction. Decay results for the wave equation on the full sub-extremal Kerr family were derived in [22]. Definitive stability results of the linearized gravity system for Schwarzschild and Reissner-Nordström were presented in [20] and [35,36], respectively. The non-linear stability of Schwarzschild in a symmetry restricted context was presented in [42]. The rigorous study of linear waves on extremal black holes was initiated by the second author in [8][9][10][11][12] where it was shown that scalar perturbations are unstable along the event horizon in the sense that higher-order transversal derivatives asymptotically blow up towards the future. The stronger regularity properties of scalar perturbations in the interior of extremal black hole spacetimes compared to sub-extremal black holes was derived by the third author in [31,32]. Precise late-time asymptotics were derived in [5]. These asymptotics led to a novel observational signature of ERN [4] where it was shown that the horizon instability of ERN is in fact "observable" by observers at null infinity. For a detailed study of this signature we refer to the recent [15]. For works on extremal Kerr spacetimes we refer to the works [16,38,45]. Extentions of the horizon instability have been presented in various settings [3,14,18,37,50,52,54]. For a detailed review of scalar perturbations on extremal backgrounds we refer to [13].

1.4.
Discussion on nonlinear problems. The methods developed in this article have applications beyond extremal black holes. Indeed, they may be also applied in the construction of non-degenerate scattering theories with weighted energy norms in more general asymptotically flat spacetimes without a local redshift effect at the horizon (which acts as a blueshift effect in backwards evolution). One such example would be the Minkowski spacetime; see Sect. 5. Since our methods involve weighted and nondegenerate energies, we expect them to be particularly useful for developing a scattering theory for nonlinear wave equations satisfying the classical null condition, as weighted energies need to be controlled in order to obtain global well-posedness for the (forwards) initial value problem [41]. It would be moreover interesting to explore the generalization of our methods to the setting of perturbations of Minkowski in the context of a scattering problem for the Einstein equations. See also [44] for work in this direction. Another interesting direction to explore is the construction of dynamically extremal black holes settling down to extremal Reissner-Nordström with inverse polynomial rates from initial data along the future event horizon and future null infinity, which would involve a generalization of the backwards evolution estimates in this article to the setting of the Einstein equations. Note that the construction of dynamically extremal black holes settling down exponentially follows from an application of the methods of [19]. However, whereas it is conjectured in [19] that a scattering construction of dynamically sub-extremal black holes settling down inverse 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 may generically be more regular (with the regularity depending on the assumed polynomial decay rate).
1.5. Overview of paper. We provide in this section an overview of the remainder of the paper.
• In Sect. 2, we introduce the extremal Reissner-Nordström geometry and spacetime foliations. We also introduce the main notation used throughout the rest of the paper. • We introduce in Sect. 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 of the main theorems of the paper in Sect. 4.
• In Sect. 5, we outline the main new ideas introduced in the present paper and we provide a sketch of the key proofs. • We construct in Sect. 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, degenerate norms. • In Sect. 7, we construct the backwards evolution map B, which send initial data for the radiation field at the future event horizon and future null infinity to the trace of the solution at a mixed spacelike-null hypersurface and is the inverse of F . Similarly, we construct restrictions of B involving higher-order, degenerate norms. • We prove in Sect. 8 additional energy estimates (in forwards and backwards time direction) that allow us 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. The composition S = F + • F −1 − defines the scattering map, which may be thought of as the key object in our nondegenerate scattering theory. • In Sect. 9 we construct a scattering map S int in a subset of the black hole interior of extremal Reissner-Nordström. • In the rest of the paper, we provide several applications of the scattering theory developed in the aforementioned sections. In Sect. 10, we apply the backwards estimates of Sect. 7 to construct arbitrarily regular solutions to (1.1) from data along future null infinity and the future event horizon. As a corollary, we construct in Sect. 11 smooth mode solutions from data at infinity and the event horizon.

Geometry and Notation
where D(r ) = (1 − Mr −1 ) 2 , with M > 0 the mass parameter, and (θ, ϕ) are spherical coordinates on S 2 . We denote the boundary as follows H + := ∂M ext = {r = M}. We refer to H + as the future event horizon. The coordinate vector field T := ∂ v is a Killing vector field that generates the time-translation symmetry of the spacetime.
We can change to the coordinate chart (u, r, θ, ϕ) on the manifoldM ext = M ext \H + , in which g can be expressed as follows: Finally, it will also be convenient to employ the Eddington-Finkelstein double null coordinate chart (u, v, θ, ϕ) inM ext , in which g takes the following form: In these coordinates T = ∂ u + ∂ v . We moreover introduce the following vector field notation in (u, v, θ, ϕ) coordinates: We have that L(r ) = 1 2 D and L(r ) = − 1 2 D. Note that in (v, r ) coordinates, we can express: Let / ∇ denote the induced covariant derivative on the spheres of constant (u, v). Then we denote the following rescaled covariant derivative: The rescaled covariant derivative / ∇ S 2 is the standard covariant derivative on the unit round sphere.
Consider the following rescaled radial coordinate onM ext : x := 1 r . The metric g M takes the following form in (u, x, θ, ϕ) coordinates: We can then expressM ext = R u × (0, 1 M ] x × S. We can embed M ext into the manifold-with-boundary We denote I + := R u × {0} x × S 2 and refer to this hypersurface as future null infinity. By considering a conformally rescaled metriĉ we can also introduce (u, r, θ, ϕ) coordinates onM int , in which the metric takes the expression (2.2). In these coordinates, it immediately follows that we can embedM int into a larger manifold M = R u × (0, ∞) r × S 2 . Let us denote the manifold-withboundary M int = R u × (0, M] r × S 2 and the boundary which we refer to as the inner horizon or the Cauchy horizon (the latter terminology follows from the globally hyperbolic spacetime regions considered in Sect. 2.3). Finally, it is also useful to work in Eddington-Finkelstein double-null coordinates (u, v, θ, ϕ) inM int , in which the metric g takes the form ( Consider the corresponding hypersurface Then N v 0 := | {r ∈(M,r H )} is an ingoing null hypersurface intersecting H + , tangential to L and N u 0 := | {r ∈[r I ,∞)} is an outgoing null hypersurface, tangential to L. Furthermore, | {r ∈(r H ,r I )} is spacelike. We denote u (r ) := v (r ) − 2r * and observe that Without loss of generality, we can assume that u 0 > 0 (by taking v 0 appropriately large for fixed r H and r I ). We will consider the coordinate chart (ρ := r | , θ, ϕ) on .
We denote with D ± (S) the future and past domain of dependence, respectively, of a spacelike or mixed spacelike-null hypersurface S. Let R := D + ( ). We can foliate R as follows: where τ denote the hypersurfaces induced by flowing along T , with 0 = . We can extend R (with respect to the (u, x, θ, ϕ) coordinate chart) into the extended manifold-with-boundary M ext by attaching the boundary I + ≥u 0 := I + ∩ {u ≥ u 0 }: Note that we can similarly consider D − ( ) where is the time-reversed analogue of (the roles of u and v reversed) that intersects H − and define, with respect to (v, x, θ, ϕ) coordinates and v 0 ∈ R the analogue of u 0 and also define I − ≤v 0 The hypersurface naturally extends to a hypersurface in R, with endpoints on H + and I + , and can be equipped with the coordinate chart (χ = x| , θ, ϕ).
We moreover define H We denote furthermore We foliate the regions D −u 0 , with u 0 > 0, by outgoing null hypersurfaces that we also denote N u . In this setting N u = {u = u | v ≥ |u|}. It is also useful to consider a foliation by ingoing null hypersurfaces We moreover consider the following null hypersurfaces in We refer to Fig. 12 for an illustration of the above foliations and hypersurfaces. We use the following notation for the standard volume form on the unit round sphere: dω = sin θ dθ dϕ. Let n τ and n be the normal vector fields to τ and , respectively. We denote with dμ τ , dμ the induced volume forms on τ and respectively. On the null segments N τ and N τ , n τ and dμ τ are not uniquely defined, so we take the following conventions: We moreover use the notation dμ g M for the natural volume form on M ext or M int . Note that in (u, v, θ, ϕ) coordinates on eitherM ext orM int , we can express: We use the notation dμĝ M for the natural volume form on M ext (corresponding to the metricĝ M ). In (u, x, θ, ϕ) coordinates on M ext \H + , we can express: 2.4. Additional notation. Let n ∈ N 0 . Suppose K ⊂ R is compact. Then the Sobolev spaces W n,2 (K ) are defined in a coordinate-independent way with respect to the following norm: Recall that we can write in (v, r, θ, ϕ) coordinates: 2D −1 L = ∂ r , which is a regular vector field in R. Furthermore, we can express in (u, x, θ, ϕ) coordinates: which implies that r 2 L is also regular in R. Hence, W n,2 (K ) is a natural choice of Sobolev space with respect to the conformal metricĝ M . If K int ⊂ M int is compact, we instead define W n,2 (K int ) in a coordinate-independent way with respect to the following norm: In (u, r, θ, ϕ) coordinates, we can express 2D −1 L = ∂ r , which is a regular vector field in M int . We can also express in (u, r, θ, ϕ) coordinates, which clearly is also regular M int . We have that W n,2 (K int ) is therefore a natural choice of Sobolev space with respect to g M . We define the Sobolev spaces W 1,2 (N int v 0 ) with respect to the following norm: Let f, g be positive real-valued functions. We will make use of the notation f g when there exists a constant C > 0 such that f ≤ C · g. We will denote f ∼ g when f g and g f . We will also employ the alternate notation f ∼ c,C g, with f, g for 0 < c ≤ C positive constants, to indicate: We use the "big O" notation O((r − M) p ) and O(r − p ), p ∈ R to group functions f of r satisfying respectively.

Main energy spaces.
In this section, we will introduce the Hilbert spaces on which we will define scattering maps. Before we can do so, we will need existence and uniqueness (in the smooth category) for the Cauchy problem for (1.1) on extremal Reissner-Nordström.

Consider characteristic initial data
We denote with C ∞ ( 0 ) the space of smooth functions on the hypersurface 0 , with respect to the coordinate chart (χ, θ, ϕ) introduced in Sect. 2.3. We denote with C ∞ ( 0 ∩ {r H ≤ r ≤ r I }) the space of smooth function on the restriction 0 ∩ {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 coordinate basis: Given a vector field X on M, we define the corresponding X -energy current J X as follows: We will denote the radiation field of ψ as follows: We define the following energy space where ψ denotes the (unique) smooth local extension of in R that satisfies ψ| 0 = and n 0 ψ| 0 ∩{r H ≤r ≤r I } = and solves (1.1) (see Theorem 3.1), so that all derivatives of ψ above can be expressed solely in terms of derivatives of and .
We also define the norm ||·|| E 0 on C ∞ ( 0 )×C ∞ ( 0 ∩{r H ≤ r ≤ r I }) as follows: We denote with E T 0 and E 0 the completions of C ∞ ( 0 )×C ∞ ( 0 ∩{r H ≤ r ≤ r I }) with respect to the norms || · || E T 0 and || · || E 0 , respectively. Note that, by construction, where ψ denotes the (unique) smooth local extension of to D + ( ) that satisfies ψ| = and n ψ| = and solves (1.1) (see Theorem 3.1), so that all derivatives of ψ above can be expressed solely in terms of derivatives of and .
We also define the norm || · || E on C ∞ c ( ) × C ∞ c ( ) as follows: We denote with E T and E the completions of C ∞ c ( ) × C ∞ c ( ) with respect to the norms || · || E T and || · || E , respectively. Note that, by construction, We denote with C ∞ c (H + ≥v 0 ) and C ∞ c (I + ≥u 0 ) the spaces of smooth, compactly supported functions on H + ≥v 0 and I + ≥u 0 , respectively.
Note that 3.2. Degenerate higher-order energy spaces. In this section, we will introduce analogues of the Hilbert spaces introduced in Sect. 3.1, but with norms depending on degenerate higher-order derivatives.
Definition 3.5. Define the norm ||·|| E n; 0 as follows: We denote with E n; 0 the completion of C ∞ ( 0 ) × C ∞ ( 0 ∩ {r H ≤ r ≤ r I }) with respect to the norm || · || E n; 0 . Definition 3.6. Define the norm || · || E n; as follows: let where ψ denotes the smooth extension of to R that satisfies ψ| = and n ψ| = and solves (1.1) (see Theorem 3.1), so that all derivatives of ψ above can be expressed solely in terms of derivatives of and .
We denote with E n; the completion of Definition 3.7. Let n ∈ N 0 and u 0 , v 0 > 0. Define the higher-order norms || · || E n;H + ≥v 0 and || · || E n;I + ≥u 0 as follows: Then we denote with E n;H + ≥v 0 Note that for all n ∈ N 0 , Definition 3.8. Let n ∈ N 0 . Define the higher-order norms || · || E n;H ± and || · || E n;I ± , as follows: with respect to the coordinate charts (u ± , v ± , θ, ϕ). Then we denote with E n;H ± ⊕ E n; with respect to the norms || · || E n;H ± and || · || E n;I ± .

Black hole interior energy spaces.
In this section, we introduce additional energy spaces that play a role in a non-degenerate scattering theory for the extremal Reissner-Nordström black hole interior.

Main Theorems
In this section, we give precise statements of the results proved in this paper. We refer to Sects. 2 and 3 for an introduction to the notation and definitions of the objects appearing in the statements of the theorems.

Non-degenerate scattering theory results.
We first state the main theorems that establish a non-degenerate scattering theory in extremal Reissner-Nordström.

Theorem 4.1. The following linear maps
Here, ψ denotes the unique solution to (1.1) with initial data ( , ) in accordance with statements 2. and 3. of Theorem 3.1.

Furthermore, their unique extensions
are bijective and bounded linear operators, and is also a bijective bounded linear operator.
We refer to the maps F and F ± as a forwards evolution maps, F −1 and F −1 ± and backwards evolution maps and S as the scattering map.

Remark 4.1. An analogous result holds with respect to the degenerate energy spaces
This follows easily from an analogue of Proposition 9.6.1 in [23] applied to the setting of extremal Reissner-Nordström; see also Sects. 6.5, 7.4 and 8.3. They advantage of Theorem 4.1 is the use of non-degenerate and weighted energy norms that also appear when proving global uniform boundedness and decay estimates for solutions to (1.1).
The following theorem extends Theorem 4.1 by considering degenerate and weighted higher-order energy spaces.

Theorem 4.2.
Let n ∈ N 0 . We can restrict the codomains of the linear maps F and F ± defined in Theorem 4.1, to arrive at which are well-defined. Furthermore, the unique extensions

are bijective and bounded linear operators and
is also a bijective bounded linear operator.
Both Theorems 4.1 and 4.2 follow by combining Propositions 6.16 and 7.11, Corollary 7.12 and Propositions 8.11 and 8.14.
We additionally construct a scattering map restricted to the black hole interior.

Theorem 4.3.
Let u int < 0 with |u int | suitably large. The following linear map: Furthermore, uniquely as a bijective, bounded linear operator: Theorem 4.3 is a reformulation of Proposition 9.2.

Applications.
In this section, we state some applications of the non-degenerate scattering theory of Sect. 4.1.
In Theorem 4.4 below, we show that we can obtain unique solutions to (1.1) with arbitrary high Sobolev regularity (with respect to the differentiable structure on R) from suitably regular and polynomially decaying scattering data on H + and I + in an L 2 -integrated sense.
and assume moreover that  Reissner-Nordström with the sub-extremal setting, where generic polynomially decaying data along the future event horizon and future null infinity (with an arbitrarily fast decay rate) lead to blow-up of the non-degenerate energy along 0 ; see [23,24].
As a corollary of Theorem 4.4, we can moreover construct smooth solutions and in particular smooth solutions with an exact exponential time dependence.
and assume that ( , ) and all derivatives up to any order decay superpolynomially in v and u, respectively.
(i) Then there exists a corresponding smooth solution ψ to (1.1) on R such that rψ can moreover be smoothly extended toR with respect to the differentiable structure on R.
with ω ∈ C such that Im ω < 0. Then we can express and We refer to ψ as mode solutions.

Remark 4.3.
Note that in order for an analogous result to Theorem 4.5 (i) to hold in sub-extremal Reissner-Nordström, one needs to consider scattering data ( , ) that are superexponentially decaying, and hence it cannot be used to prove the analogue of Theorem 4.5 (ii). Nevertheless, the existence of a more restricted class of smooth solutions that behave exponentially in time with arbitrary ω such that Im ω < 0 in sub-extremal Reissner-Nordström can be established by restricting to fixed spherical harmonics and applying standard asymptotic ODE analysis.
Remark 4.4. One can apply the results of [5] to show that the time translations acting on L 2 -based Sobolev spaces with ψ the solution to (1.1) associated to ( , ), form a continuous semi-group, such that S(τ ) = e τ A , with A the corresponding densely defined infinitesimal generator A that formally agrees with T : The results of [55] 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 and frequencies of an idealised vibrating string or membrane).
The smooth mode solutions of Theorem 4.5 (ii) (and those obtained in the subextremal setting by ODE arguments as sketched in Remark 4.3) form an obstruction to extending this interpretation to the asymptotically flat setting. Indeed, all the mode solutions of Theorem 4.5 (ii) are eigenfunctions of A but the corresponding set of frequencies ω, which is the entire open lower-half complex plane, is certainly not discrete. In order to maintain the viewpoint of [55], one has to consider smaller function spaces that exclude the smooth mode solutions of Theorem 4.5 (ii); see [34]. Theorem 4.6. Let u 0 be suitably large. Then there exists a constant C = C(M, u 0 , v 0 ) > 0 such that we can estimate in the black hole interior: Theorem 4.6 follows from Corollary 9.3.
Remark 4.5. Theorem 4.6 addresses the question of whether ψ ∈ W 1,2 loc in the black hole interior of extremal Reissner-Nordström for localized, low regularity initial data, which was raised as an open problem in [25]. For smooth and localized data, this statement follows from [5,31]. Indeed, Theorem 4.6 demonstrates that boundedness of a nondegenerate energy with weights that grow in r (together with boundedness of energies involving additional derivatives that are tangential to the event horizon) is sufficient to establish ψ ∈ W 1,2 loc . Theorem 4.6 can straightforwardly be extended to the > 0 setting of extremal Reissner-Nordström-de Sitter black holes, where there is no need to include r -weights in the non-degenerate energy norm that is sufficient to establish ψ ∈ W 1,2 loc . See also [2] for the results in the interior of extremal Reissner-Nordström-de Sitter.

Overview of Techniques and Key Ideas
In this section, we provide an overview of the main techniques that are used in the proofs of the theorems stated in Sect. 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: 1.) Showing that the linear maps F , F −1 and F n , F −1 n that appear in Theorem 4.1 and Theorem 4.2 are well-defined when considering as a domain spaces of either smooth or smooth and compactly supported functions.

2.)
Proving uniform boundedness properties of these linear maps with respect to weighted Sobolev norms. This allows one to immediately extend the linear maps to the completions of the spaces of smooth (and compactly) supported functions with respect to appropriately weighted Sobolev norms. 3.) Constructing the linear maps S and S n . 4.) Constructing S int (independently from above).
The heart of this paper consists of establishing 2.) and 3.) by proving uniform estimates for smooth (and compactly supported) data along 0 , and H ± ∪ I ± . An overview of the corresponding estimates and techniques leading to 2.) is given in Sects. 5.1-5.3. Part 3.) follows by complementing these estimates with additional estimates in D ± ( ) near the past limit points of I + and H + , which is briefly discussed in Sect. 5.4. We briefly discuss the black hole interior estimates involved in 4.) in Sect. 5.5. Part 1.) follows from local estimates combined with soft global statements that have already been established in the literature. We give an overview of the logic of the arguments in this section.

The forwards map
is well-defined by global existence and uniqueness for (1.1) combined with the finiteness (and decay) of the radiation field r ψ, see for example the results in [5,8,9].
In order to show that the backwards map 2 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 = . This may be viewed as a semi-global problem. We construct ψ as the limit of a sequence of solutions ψ i arising from a sequence of local initial value problems with fixed initial data ( , ) imposed on the null hypersurfaces A very similar procedure was carried out in the physical space construction of scattering maps on Schwarzschild in Proposition 9.6.1 in [23]. 3 One could alternatively interpret I + as a genuine null hypersurface with respect to the conformally rescaled metricĝ M , which turns the semi-global problem into a local problem.

Backwards r -weighted estimates.
We introduce time-reversed analogues of the r p -weighted estimates of Dafermos-Rodnianski [21] and the (r − M) − p -weighted estimates of [5]. We first illustrate key aspects of these estimates in the setting of the standard wave equation on Minkowski. We can foliate the causal future of a null cone C 0 in Minkowski by outgoing spherical null cones C u = {t − r = u}, with t, r the standard spherical Minkowski coordinates and u ≥ 0. Let us denote . We consider smooth, compactly supported initial data on The r p -weighted estimates applied backwards in time with p = 1 and p = 2 give In contrast with the usual forwards r p -weighted estimates, the spacetime integrals on the right-hand sides above have a bad sign. Hence, in order to obtain control of r -weighted energies along C u 1 , we need to start by controlling Note that standard ∂ t -energy conservation implies that for any 0 < u < u 2 : Hence, using that ψ is vanishing along C u 2 , we can integrate the above equation in u to obtain We can integrate by parts to convert one u-integration into an additional u weight: 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: Comparing (5.3) with (5.1) with u = 0, we see that we can obtain stronger, weighted uniform control along C 0 , 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 2 ∂ u +v 2 ∂ v , which is the generator of the inverted time translation conformal symmetries, as a vector field multiplier instead of ∂ t [48]; see also Sect. 5.4. The main difference in the setting of extremal Reissner-Nordström is that the r pestimates above only apply in the spacetime region where r ≥ r I , with r I suitably large, and they have to be complemented by an 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 the restriction of the following non-degenerate energy (in (v, r ) coordinates): It is in controlling the non-degenerate energy in the backwards direction that we make essential use of the extremality of extremal Reissner-Nordström or the degeneracy of the event horizon. Indeed, if we were to consider instead sub-extremal Reissner-Nordström, we would fail to obtain control of a non-degenerate energy near H + with polynomially decaying data along H + ∪I + due to the blueshift effect (the time reversed redshift effect); see [23,24]. 4 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 [8] in the backwards direction. Note that the presence of trapped null geodesics along the photon sphere at r = 2M 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 Sect. 5.2), do not require an application of a Morawetz estimate with non-degenerate control at the photon sphere.

Forwards r -weighted estimates revisited.
We consider again the setting of Minkowski to illustrate the main ideas. In order to construct a bijection from an rweighted energy space on C 0 to a u-weighted energy space on I + , we need to complement the backwards estimate (5.3) with the following forwards estimate: Note that a standard application of the r p -weighted estimates (combined with energy conservation (5.1) and a Morawetz estimate), see [21], is the following energy decay statement: One can apply this estimate along a suitable dyadic sequence and combine it with energy conservation (5.1) to arrive at the estimate with > 0. In order to take = 0, we instead revisit the r p -estimates and, rather than deriving energy decay along C u , we observe that the r p -estimates (together with (5.1) and a Morawetz estimate) provide directly control over 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-Nordström setting by following the same ideas, both near I + and near H + . The main difference is that whenever we apply a Morawetz estimate, we lose a derivative because of the trapping of null geodesics, which we have to take into account when defining the appropriate energy spaces.

Higher-order energies and time integrals.
Given suitably regular and suitably decaying scattering data on H + and I + , we can apply Theorem 4.1 to construct a corresponding solution ψ ∈ C 0 ∩ W 1,2 loc (with respect the differentiable structure on R) to (1.1) such that r ψ approaches the scattering data as r → M or r → ∞.
In the setting of (1.1) on Minkowski with coordinates (u, , we first consider T ψ. By rearranging and rescaling (1.1) in Minkowski, we have that in (u, x) coordinates: Since / S 2 commutes with the operator g , both in Minkowski and in extremal Reissner-Nordström, we can immediately obtain / S 2 (r ψ) ∈ W 1,2 from Theorem 4.1 (or its Minkowski analogue). Moreover, L(r ψ) ∈ W 2,2 follows from bounding uniformly in u the integral: Hence, we have to establish control over improved r -weighted energies where r ψ is replaced by L(r ψ) and L 2 (r ψ). Analogous improved r -weighted energies have appeared previously in the setting of forwards estimates in [5,7,53], see also the related energies in [49]. The backwards analogues of the corresponding improved r -weighted estimates form the core of the proof of Theorem 4.2.
To pass from T (r ψ) ∈ W 2,2 to r ψ ∈ W 2,2 , we apply the above estimates to solutions ψ (1) to (1.1), such that 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 r ψ (1) | H + and r ψ (1) | I + vanish as v → ∞ and u → ∞, respectively.
In fact, we can show by an extension of the arguments above that T n (r ψ) ∈ W 1+n,2 loc for all n ≥ 2, assuming suitably regular and decaying data along H + and I + , so we can conclude that ψ ∈ W n+1,2 loc , provided the scattering data decays suitably fast in time. In order to obtain more regularity, we need faster polynomial decay 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 arbitrarily high and show that ψ ∈ C ∞ ( R); see Theorem 4.5.
Note that time integrals ψ (1) also play an important role in [5,6] for spherical symmetric solutions. In that setting, 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 can integrate the scattering data in time to obtain data leading to ψ (1) . While r -weighted estimates are still suitable in the forwards direction in D −u 0 and D −v 0 , they are not suitable in the backwards direction. We therefore consider energy estimates for the radiation field r ψ with the vector field multiplier K = u 2 ∂ u + v 2 ∂ v , both in D −u 0 and D −v 0 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 inverted time translations. It is a Killing vector field of the rescaled metric r −2 m, where m is the Minkowski metric. Hence, K may be thought of as the analogue of ∂ t when considering r ψ instead of ψ and r −2 m instead of m. In particular, when considering K as a vector field multiplier in a spacetime region of Minkowski, one can obtain a weighted energy conservation law for r ψ. Since r is large in D −u 0 in extremal Reissner-Nordström, K may be thought of as an "approximate Killing vector field" of the rescaled metric r −2 g.

Estimates
Another useful property of K is that it is invariant under the Couch-Torrence conformal symmetry [17] that maps D −u 0 to D −v 0 . It therefore plays the same role when used as a vector field multiplier for the radiation field in D −v 0 as it does in D −u 0 .
In order to obtain the analogue of the r p -weighted estimate with p = 1 for T ψ, we apply instead the vector field multiplier first observing that the spacetime is invariant under the map t → −t, so the above discussion on F −1 can be applied to associate to each pair ( , ) ∈ C ∞ c (H − ) ⊕ C ∞ c (I − ) a solution ψ ∈ D − ( ) such that (ψ| , n ψ| ∩{r H ≤r ≤r I } ) lie in a suitable energy space. We show that in fact (ψ| 0 , n 0 ψ| 0 ∩{r H ≤r ≤r I } ) ∈ E 0 , so we can apply (the extension of) F to obtain a pair of radiation fields ( , ) ∈ E H + ⊕ E I + .

Scattering and regularity in black hole interiors.
We derive estimates for the radiation field in M int using once again the vector field K = u 2 ∂ u +v 2 ∂ v . Recall from Sect. 5.4 that the favourable properties of K as a vector field multiplier are related to its role as an approximate conformal symmetry generator near infinity and its invariance under the Couch-Torrence conformal symmetry. The equation for the radiation field takes the same form in M int and M ext near H + if one considers the standard Eddington-Finkelstein double-null coordinates inM int and inM ext . Therefore, K (now defined with respect to (u, v) coordinates inM int ) remains useful in the black hole interior. The usefulness of K in the interior of extremal black holes was already observed in [31][32][33].

The Forwards Evolution Map
In this section, we present the energy estimates in the forwards time direction that are relevant for defining the forwards evolution map F (see Sect. 6.5).

Preliminary estimates.
We make use of the following Hardy inequalities: Proof. See the proof of Lemma 2.2 in [7].
We denote for α = (α 1 , . We now state the following standard inequalities on S 2 : Lemma 6.2 (Angular momentum operator inequalities). Let f : S 2 → R be a C 2 function. Then we can estimate Proof. See for example [8,9].

Radiation field at null infinity.
We now recall some regularity properties of the radiation field at null infinity, which do not immediately follow from Theorem 3.1, and are derived in [5]. Proof. By (1.1) we obtain the following equation for φ: which implies (6.5) with n = 0. We obtain n ≥ 0 by induction.
. Then for all k, l ∈ N 0 and α ∈ N 3 0 , In particular, the limit exists for all u ≥ 0 and defines a smooth function on I + ≥u 0 . Proof. The k ≤ 1 case follows from Section 3 of [7] by using (6.6). We obtain the k ≥ 2 case via an induction argument, where in the induction step we simply repeat the argument for k = 1 using instead the commuted equation (6.5). See also Proposition 6.2 of [5].
By combining Theorems 6.6 and 6.7 with Lemma 6.3 and applying the mean-value theorem along a dyadic sequence of times ("the pigeonhole principle"), one can obtain energy decay in time along the foliation τ ; see for example [8,9] and [5] for an application of this procedure in extremal Reissner-Nordström.
In the present article, however, we will not apply the mean-value theorem, bur rather derive uniform boundedness estimates for various time-integrated energies on the lefthand side (see Proposition 6.8). We will 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).
Proof. Note first of all that for all τ ≥ 0 τ2 τ1 Nτ where in the final inequality we applied Lemma 6.1 and (6.7), using that φ attains a finite limit at I + , by Proposition 6.5. Similarly, we have that We combine (6.12) and (6.13) together with (6.8) to obtain the estimate: We now apply (6.9) with k = 0 and p = 1 to obtain: (6.14) By Lemma 6.3 and (6.14), we immediately obtain also 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 + ( τ ), together with (6.9) with p = 2 and k = 0.
The following simple lemma is crucial in order to bound energy norms along H + and I + with time-weights inside the integrals.
Proof. We integrate the left-hand side of (6.16) by parts to obtain Note that for n ≥ 1: We then keep integrating by parts to arrive (6.16), using that Proof. First of all, by Theorem 5.1 from [5] it follows that for 0 ≤ j ≤ 2 the following qualitative statements hold: 5 We can therefore apply Proposition 6.8 together with Lemma 6.9 with n = 2 to obtain the desired estimate for the j = 0 term. The j = 1 estimate follows by replacing φ with T φ and applying (6.15) and Lemma 6.9 with n = 1. Finally, we obtain the j = 0 estimate by replacing ψ with T 2 ψ and applying Lemma 6.3.
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 Sect. 8.1.

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 growing weights in u and v on the left-hand side (depending on n).
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 the wave operator g ) and setting τ = τ 2N +2 , we have that Now, we apply the following identities and we integrate once more in τ to obtain: where we moreover applied Lemma 6.1 (together with a standard averaging argument near the boundaries) and Theorem 6.6 to control the lowest order derivative terms on the right-hand sides of (6.19) and (6.20). Now, apply (6.9) with k ≤ N + 1 and p = 2k + 1 when j = 0 and k ≤ N and p = 2k when j = 1, together with Lemma 6.2, to obtain Subsequently, apply (6.9) again, with k ≤ N + 1 and p = 2k + 2 when j = 0 and k ≤ N and p = 2k + 1 when j = 1.
Finally, since we are integrating two more times in τ compared to the n = N estimate, we can also include on the left-hand side of the above estimate the terms to obtain (6.18) with n = N + 1.
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 Sect. 8.2. Corollary 6.14. Let n ∈ N 0 . Then, there exists a constant C = C(M, , r H , r I , n) > 0, such that (6.22)

Construction of the forwards evolution map.
In this section, we will use the uniform estimates derived in Sects. 6.3 and 6.4 in order to construct the forward evolution map between suitable weighted energy spaces.

Definition 6.1. Define the forwards evolution map
as the following linear operator: where ψ is the unique solution to (1.1) with (ψ| 0 , n 0 ψ| 0 ∩{r H ≤r ≤r I } ) = ( , ). Then F extends uniquely to a linear bounded operator: .
We moreover have that F n = F | E n; 0 .

The Backwards Evolution Map
In this section we will construct a map from suitably weighted energy spaces on H + and I + to suitably weighted energy spaces on 0 . The construction will proceed in two steps. As a first step, we construct in Sect. 7.1 a map with the domain C ∞ c (H + ≥v 0 ) ⊕ C ∞ c (I + ≥u 0 ). 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 Sect. 7.4 by using the global, uniform weighted energy estimates of Sect. 7.2 that are valid on the completion of C ∞ c (H + ≥v 0 ) ⊕ C ∞ c (I + ≥u 0 ) with respect to the associated energy norms.
7.1. Initial value problem with compactly supported scattering data. In this section we will associate to a pair ( , ) ∈ C ∞ c (H + ≥v 0 ) ⊕ C ∞ c (I + ≥u 0 ) a unique solution to (1.1) in D + ( 0 ) such that r · ψ| H + = and r · ψ| I + = . This association is central to the definition of the backwards evolution map (see Definition 7.1).
(7.1)  [23] in the setting of sub-extremal Kerr. Note however that Proposition 7.1 establishes in addition qualitative bounds on the radiation field r ψ and weighted higher-order derivatives thereof in the form of the inequality (7.1), which will be necessary in the backwards-in-time estimates of Sect. 7.2.

2.) (Uniqueness) If
Proof of Proposition 7.1. Observe first of all that ψ i is well-defined by local existence and uniqueness with smooth initial data on 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 = v} ∩ {u 0 ≤ u ≤ u ∞ } and 0 and to the future by I V i := {v = V i } ∩ {u 0 ≤ u ≤ u ∞ } and τ ∞ to obtain: which is equivalent to By applying the fundamental theorem of calculus in u, integrating from u = τ ∞ to u = u, together with Cauchy-Schwarz, we therefore obtain where we used that ψ i | τ∞ = 0, from which it follows that 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 Similarly, we can use (6.5) and Lemma 6.2 in a simple induction argument to conclude that for all n ∈ N we have in {r ≥ r I }: We can immediately apply the above argument to α φ and T k for any α ∈ N 3 0 , k ∈ N 0 , together with a standard Sobolev inequality on S 2 to obtain the following i-independent estimate: for all k ∈ N 0 and α ∈ N 3 0 , there exists a constant C(τ ∞ , u 0 ) > 0, such that |(r 2 L) n T k α φ i | 2 (u, v, θ, ϕ) 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) −1 : Given V > 0 arbitrarily large and n ≥ N , we have by (7.3) and (7.4) that for I ≥ 1 such that V I > V , φ i is uniformly bounded in i for all i ≥ I with respect to the C k norm on  6 We can extend the domain of φ to J + ( 0 )∩ J − ( τ ∞ ) as follows: we replace V above with V > V , applying Arzelà-Ascoli to the subsequence φ i k (starting from k suitably large) in the corresponding larger spacetime region and passing to a further subsequence. By uniqueness of limits, the resulting limit, which we note by φ has to agree with φ when v ≤ V .
We also have by (7.3) that for any > 0, there exist a V > 0 and K > 0, such that for all v ≥ V and k > K in the region {r ≥ r I }: We can analogously use (7.3) to obtain for all j, k, l ∈ N 0 : Furthermore, by replacing ψ by T l α ψ we can conclude that with respect to the differentiable structure inR, the restriction r ψ| I + is a smooth function on I + , satisfying r ψ| I + = . We can therefore conclude 1.) of the proposition. Now suppose ψ is another smooth solution to g ψ = 0, such that By a global T -energy estimate, we have that so ψ = ψ, which concludes 2.) of the proposition.

Backwards energy estimates.
In this section, we will derive estimates for the solutions ψ to (1.1) constructed in Proposition 7.1 that are uniform in τ ∞ . This is crucial for constructing solutions with scattering data that is not compactly suppported. The main tool we will develop is this section is a hierarchy of r -weighted estimates in the backwards time direction. However, we will first state a backwards Morawetz estimate that follows immediately from the results in [8], i.e. an analogue of Theorem 6.6 in the backwards time direction.
In the propositions below, we derive the "backwards analogues" of the hierarchies from Proposition 6.7. Proposition 7.3. Let 0 ≤ p ≤ 2, then there exists a constant C(M, , r I , r H ) > 0, such that for all 0 ≤ τ 1 ≤ τ 2 ≤ τ ∞ : Proof. Recall that φ satisfies the equation: Therefore, By reordering the terms, we therefore obtain: (7.8) Let χ denote a cut-off function and consider χφ.
We integrate both sides of (7.8) in spacetime to obtain: where we applied Lemma 6.1 and (7.5) to arrive at the inequality above. See also the derivations in the proof of Lemma 6.3 in [5] 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 [5] for more details.
We subsequently apply Proposition 7.3 to arrive at uniform weighted energy estimates along 0 . (7.10) We moreover have that Proof. By applying Lemmas 6.1 and 6.3, it follows that We now apply (7.6) with p = 1, together with (7.12) to conclude that Next, apply (7.6) with p = 2 to obtain We apply Lemma 6.9 to rewrite the right-hand side above to arrive at: (7.13) which leads to (7.10) when we take τ = 0. By applying the above estimates to T ψ and T 2 ψ we moreover obtain: We conclude the proof by combining the above proposition with Lemma 6.3 to obtain 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 Sect. 8.1.

Higher-order estimates.
By commuting (7.7) with L k , we arrive at Similarly, we can commute (7.7) with L k to obtain: Proof. The proof is a straightforward generalisation of the proof of Proposition 7.3: we repeat the steps in the 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 0 and let ψ be a solution to (1.1) such that ψ| τ∞ = 0 and n τ ∞ ψ| τ∞ = 0 for some τ ∞ < ∞. Then there exists a constant C(M, , r I , r H , n) > 0 such that Proof. We first consider the n = 1 case. Note that by (7.6) with k = 1 and p = 3: (6.19) and (6.20) (7.13),Lemma 6.9 and Lemma 6. (7.20) Now, we apply (7.6) with k = 1 and p = 4: By replacing φ on the left-hand side of (7.20) with T j φ and applying Proposition 7.4 to T m α φ, we therefore obtain: 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 = 2n + 1 and p = 2n + 2. Proposition 7.7 combined with Lemma 6.3 immediately implies the following: We will complement (7.21) in Corollary 7.8 with an estimate involving additional angular derivatives. The motivation for this comes from the energy estimates in Sect. 8.2. Nu 0 Proof. From Proposition 7.1 it follows that ψ| 0 ∈ C ∞ ( ) and n 0 ψ| 0 ∩{r H ≤r ≤r I } ∈ C ∞ ( 0 ∩ {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 of bounded linear operators to the completion of their domains, we can define the backwards evolution map as follows: Definition 7.1. The backwards evolution map is the map B : where ψ is the unique solution to g ψ = 0 with (Mψ| H + ≥v 0 , r ψ| I + ≥u 0 ) = ( , ). The map B uniquely extends to a unitary linear operator, which we will also denote with B: In the proposition below, we show that we can consider restriction of B to suitably weighted energy spaces. Proposition 7.11. Let n ∈ N 0 . The backwards evolution map B is a bounded linear operator from C ∞ c (H + ≥v 0 ) ⊕ C ∞ c (I + ≥u 0 ) to E n; 0 , which can uniquely be extended as as the bounded linear operator We moreover have that B n = B| E n;H + ≥v 0 ⊕E n;I + ≥u 0 . Proof. By Proposition 7.1 it follows that the solution ψ corresponding to . By Corollary 7.8 it follows moreover that so ||B|| ≤ C. We can infer that, in particular, (ψ| 0 , n 0 ψ| 0 ) ∈ E n; 0 . The map B extends uniquely to the completion E H + , then the corresponding solution ψ to (1.1) satisfies φ| 0 ∈ C ∞ ( 0 ) and n 0 ψ| 0 ∈ C ∞ ( 0 ∩ {r H ≤ r ≤ r I }), and hence F (φ| 0 , n 0 ψ| 0 ) = (φ| H + , φ| I + ) is well-defined and (φ| H + , φ| I + ) = ( , ). 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 fact bijective (we have already established injectivity). It immediately follows then that B • F = id. The above argument can also be applied to F n and B n .

The Scattering Map
The aim of this section is to extend the estimates of Sects. 6 and 7 from the hypersurface 0 to the hypersurface . 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 concern the "triangular" regions bounded to the future by the null hypersurfaces N 0 and N 0 and to the past by = {t = 0}.

Weighted energy estimates near spacelike infinity.
In the proposition below we derive energy estimates with respect to the vector field multiplier K = v 2 L + u 2 L, which is commonly referred to as the Morawetz conformal vector field. 7 The main purpose of K is to derive backwards energy estimates along with r -weighted initial data along N −u 0 and N −v 0 which are analogous to the r -weighted boundary terms in the estimates in Proposition 7.3 with p = 2.
Proof. By (6.6) it follows that After integrating by parts on S 2 , we therefore obtain: We first consider estimates in the backwards time direction. We integrate (8.3) in spacetime and we use the following identity: Using that r ∼ v + |u| v in the integration region, we can further estimate: for > 0 arbitrarily small given r I > 0 suitably large (and v −1 r −1 in the integration region). Note that we can absorb the very right-hand side above into the left-hand side of (8.5) when > 0 is suitably small. We apply Young's inequality to estimate We absorb the spacetime integral of (Lφ) 2 and (Lφ) 2 to the left-hand side of (8.5), using that r is suitably large and (v + |u|) r in the integration region. In order to absorb the φ 2 term, we first observe that by assumption, we are considering φ such that φ| I + is well-defined and is compactly supported in u > u −∞ , so Therefore, by Cauchy-Schwarz, we can estimate Furthermore, similarly we have that Hence, so we can estimate: with > 0 suitably small given r I suitably large. As a result, we obtain We integrate (8.3) and apply (8.6) to obtain: Analogously, we have that and so that we can estimate Using that (r − M) −1 ∼ u + |v| u, we estimate further: for > 0 arbitrarily small given r H − M > 0 suitably small. Note that we can absorb the very right-hand side above into the left-hand side of (8.9) when > 0 is suitably small. We apply Young's inequality to estimate and absorb the corresponding spacetime integral to the left-hand side of (8.9), using that which follows from Cauchy-Schwarz combined with the assumption that φ| H and hence, We now consider the forwards time direction. First of all, we are assuming compact support on 0 ∩ {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: for a suitably small positive constant c > 0.
We complement Proposition 8.1 with estimates involving lower weights in r , u and v, applied to T φ rather than φ. The r -weighted energies along N −u 0 and N −v 0 appearing in the proposition below appear as energy flux terms in Proposition 7.3 with p = 1.
We can moreover replace φ with α φ in the above estimates, with |α| ≤ 1, due to the commutation properties of i and g . By (6.6) it follows that After integrating by parts on S 2 , we therefore obtain: Hence, after integrating (8.15) in spacetime, the | / ∇ S 2 T φ| 2 term on the right-hand side will have a good sign if 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 = 1 2 (v − |u|) and |u| + v r in the integration region, together with Lemma 6.2 to estimate: where we arrived at the last inequality by applying Lemma 6.3. Note that in this step we needed to use 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 We can absorb the spacetime integrals of the terms on the very right-hand side into the following flux terms: Integrating the identity (8.15) in u and v and applying the above estimates therefore gives the following inequality: (8.16) and hence, using (8.15) and the above estimate once more, now in combination with (8.16), we arrive at We repeat the above arguments near H + by considering and reversing the roles of u and v and L and L, in order to obtain the near-horizon estimate in the backwards time direction. We omit further details of this step. Now, we consider the forwards time direction. By repeating the arguments above in the forwards time direction, using that the ψ and n ψ are initially compactly supported and taking |u −∞ | and |v −∞ | appropriately large, we obtain moreover that Note that, in contrast with the backwards-in-time estimates, there is no need for an additional angular derivative in the T -energy term on the right hand side. The analogous estimate near H + proceeds by repeating the above arguments, interchanging the roles of u and v and replacing r by (r − M) −1 .

8.2.
Higher-order estimates. The aim of this section is to derive analogues of the estimates in Proposition 8.1 for higher-order derivatives of 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 symmetry in Minkowski. Even though the exact symmetry property is lost in extremal Reissner-Nordström, we will see below that the vector field still has favourable commutation properties with the operator L L.
Lemma 8.4. Let n ∈ N 0 and S = uL + vL. Then 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 ≥ 0. Note first of all that for an arbitrary C 2 function f : For any p ≥ 0 we have that: Furthermore, we can expand

S(Dr
and we obtain, using the above observations and applying (8.19) with n = N : Hence, we can conclude that (8.19) must hold for all n ∈ N 0 . It follows analogously that (8.20) must hold for all n ∈ N 0 .
Since the vector field S does not commute with 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, an equivalent energy boundedness statement holds.
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 − 1 in order to prove the k = n case. We will in fact do both of these steps at the same time in the argument below. By Lemma 8.4, we have that We subsequently integrate both sides of (8.23) in u, v and S 2 and we apply Young's inequality to absorb all the 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 − 1.
Proposition 8.6. Let n ∈ N 0 . There exists constants c, Proof. We can apply the same arguments as in Proposition 8.1, replacing φ by S k φ, with 0 ≤ k ≤ n and applying the more general equations (8.19) and (8.20) instead of (6.6) to obtain: We conclude the proof by rewriting S k φ in terms of u and v derivatives and we moreover apply Lemma 8.4 to rewrite all mixed u and v derivatives. Furthermore, we apply Lemma 6.2 to replace the angular derivatives by derivatives of the form α .
Proof. We repeat the arguments in the proof of Proposition 8.2, applying the equations in Lemma 8.4 that introduce additional terms, which can be absorbed straightforwardly. Furthermore, rather than using Lemma 6.3, we apply Proposition 8.5 where necessary. We then obtain: We conclude the proof by replacing the S k derivatives by u and v derivatives with weights in |u| and |v|, and moreover 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.
Proof. Follows immediately after combining the results of Propositions 8.6 and 8.7.
By commuting g additionally with T and applying Lemma 6.3, we arrive at energy estimates along N u 0 and N v 0 (rather than N −u 0 and N −v 0 ) with the same weights and number of derivatives as the energy fluxes that appear in Corollaries 6.13 and 7.8.   It follows immediately that ψ is a uniquely determined smooth solution to (1.1), such that lim v→∞ r ψ(u, v, θ, ϕ) = (u, θ, ϕ) and Mψ| H + = .
Proposition 8.10. Let ( , ) ∈ (C ∞ c ( )) 2 . Then the corresponding solution ψ to (1.1) satisfies and furthermore, the following identity holds Proof. Follows from Lemma 6.3 and Proposition 6.15 (combined with an analogue of Proposition 6.15 in the past-direction, making use of the time-symmetry of the spacetime).

Definition 8.2.
Define the evolution maps F ± : (C ∞ c ( )) 2 → E T H ± ⊕ E T I ± as the following linear operator: where ψ is the unique solution to (1.1) with (ψ| , n ψ| ) = ( , ). Then F ± extends uniquely to a linear bounded operator, also denoted F ± : Proposition 8.11. Let n ∈ N 0 . Then for all n ∈ N 0 F ± (C ∞ c ( )) 2 ) ⊆ E n;H ± ⊕ E n;I ± , (8.32) and F ± can uniquely be extended as as the following bounded linear operator F n;± : E n; → E n;H ± ⊕ E n;I ± .
We moreover have that F n;± = F ± | E n; .
Proof. Without loss of generality, we restrict our considerations to F + . We choose 0 so that 0 ∩ {r H ≤ r ≤ r I } = ∩ {r H ≤ r ≤ r I }.
We then apply the bounded operator F n from Corollary 6.16 to arrive at (8.32). The extension property follows immediately from the uniform boundedness of F + with respect to the desired norms. Proof. By applying the fundamental theorem of calculus, we have that for suitably large r * > 0 so ψ| (r, θ, ϕ) → 0 as r → ∞. By considering r * < 0 with |r * | suitably large, we can conclude analogously that ψ| (r, θ, ϕ) → 0 as r → ∞ and r ↓ M. The energy conservation statement simply follows from applying Lemma 6.3. where ψ is the corresponding unique solution to (1.1) as defined in Definition 8.1. Then B ± extends uniquely to a linear bounded operator, also denoted B ± : Proposition 8.13. The linear operator F ± : E T → E T H ± ⊕ E T I ± is bijective with B ± = F −1 ± . Proof. Follows by the same arguments as in the proof of Proposition 7.12. Proposition 8.14. Let n ∈ N 0 . Then for all n ∈ N 0 B ± (C ∞ c (H ± ) ⊕ C ∞ c (I ± )) ⊆ E n; , (8.33) and B ± can uniquely be extended as as the following bounded linear operator B n;± : E n;H ± ⊕ E n;I ± → E n; .
We moreover have that B n;± = B ± | E n; and B n;± = F −1 n;± .
Using that (r − M) −1 ∼ v + |u| in M int ∩ D + ( 0 ∪ N int and it follows analogously that is well-defined. The estimate (9.1) then follows by combining the above estimates.
Proposition 9.2. Let u int < 0 with |u int | suitably large. Let S int : be defined as follows: ).
Then S int extends uniquely as a bijective, bounded linear operator: 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 r ψ| CH + 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 Corollary 6.10.
Remark 9.1. One can easily extend the estimate in Corollary 9.3 to smaller values of |u int | (provided r > r min > 0 in the spacetime region under consideration), by applying a standard Grönwall inequality.

Application 1: Regularity at the Event Horizon and Null Infinity
As an application of the maps B n constructed in Proposition 7.11, we can show that we can associate arbitrarily regular solutions to suitably polynomially decaying scattering data along H + and I + . First of all, 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 Proof. The identities can be obtained inductively by applying (7.7) and commuting L L with r 2 L and r 2 L. See Lemma 6.1 in [5] for more details. . Then we have that the corresponding solution ψ to (1.1) satisfies T n (r ψ) ∈ W n+1,2 loc ( R).