Uniform boundedness and continuity at the Cauchy horizon for linear waves on Reissner-Nordstr\"om-AdS black holes

Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying $\Box_g \psi - \mu \psi =0$ on the interior of Anti-de Sitter (AdS) black holes. We prescribe initial data on a spacelike hypersurface of a Reissner--Nordstr\"om--AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously that such waves only decay at a sharp logarithmic rate (in contrast to a polynomial rate as in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, the question of uniform boundedness in the black hole interior and continuity at the Cauchy horizon has remained up to now open. We answer this question in the affirmative.


Introduction
We initiate the study of (massive) linear waves satisfying g ψ − µψ = 0 (1.1) on the interior of asymptotically Anti-de Sitter (AdS) black holes (M, g). In the context of asymptotically AdS spacetimes it is natural to consider (possibly negative) mass parameters µ satisfying the Breitenlohner-Freedman [6] bound µ > 3 4 Λ, where Λ < 0 is the cosmological constant of the underlying spacetime. In particular, this covers the conformally invariant operator with µ = 3 2 Λ. We will consider Reissner-Nordström-AdS (RN-AdS) black holes [7] which can be viewed as the simplest model in the context of the question of stability of the Cauchy horizon. These spacetimes are spherically symmetric solutions of the Einstein equations Ric µν − 1 2 Rg µν + Λg µν = 8πT µν (EE) coupled to the Maxwell equations via the energy momentum tensor T µν . Our main result Theorem 1 (see Theorem 3.1 in Section 3 for its precise formulation) is the statement of uniform boundedness in the black hole interior and continuity at the Cauchy horizon of solutions to (1.1) arising from initial data on a spacelike hypersurface on RN-AdS. We moreover assume Dirichlet (reflecting) boundary conditions at infinity. Our result is surprising because in contrast to black hole backgrounds with non-negative cosmological constants (Λ ≥ 0), the decay of ψ in the exterior region for asymptotically AdS black holes (Λ < 0) is only logarithmic as shown by Holzegel-Smulevici [38] (cf. polynomial [58,18,1] (Λ = 0) and exponential [5,24] (Λ > 0)). Indeed, the logarithmic decay is too slow to adapt the mechanism exploited in previous studies of black hole interiors [13,25,16]. The proof of our main theorem will now follow a new approach, combining physical space estimates with Fourier based estimates exploited in the scattering theory developed in [42].
In the rest of the introduction we will give some background on the problem and formulate our main result Theorem 1.
The Cauchy horizon and the Strong Cosmic Censorship Conjecture. The main motivation for studying linear waves on black hole interiors is to shed light on one of the most fundamental puzzles in general relativity: The Kerr(-de Sitter or -Anti-de Sitter) and Reissner-Nordström (-de Sitter or -Anti-de Sitter) black holes share the property that in addition to the event horizon H, they hide another horizon, the so-called Cauchy horizon CH, in their interiors. This Cauchy horizon defines the boundary beyond which initial data on a spacelike hypersurface (together with boundary conditions at infinity in the asymptotically AdS case) no longer uniquely determine the spacetime as a solution of (EE). In particular, these spacetimes admit infinitely many smooth extensions beyond their Cauchy horizons solving (EE). This severe violation of determinism is conjectured to be an artifact of the high degree of symmetry in those explicit spacetimes and generically, due to blue-shift instabilities, it is expected that a singularity ought to form at or before the Cauchy horizon. This is known as the Strong Cosmic Censorship Conjecture (SCC) [56,9]. A full resolution of the SCC conjecture would also include a precise description of the breakdown of regularity at or before the Cauchy horizon.
We first present the C 0 formulation of SCC (see [9,16]), which can be seen as the strongest inextendibility statement in this context. of the Cauchy horizon without symmetry assumptions and the validity of Conjecture 2 (even restricted to a neighborhood of Kerr) have yet to be understood.
Linear waves and SCC for asymptotically AdS black holes. The situation is changed radically if one considers asymptotically Anti-de Sitter (Λ < 0) spacetimes. Due to the timelike nature of null infinity I = I A ∪ I B , see for example Fig. 1, these spacetimes are not globally hyperbolic. For well-posedness of (EE) and (1.1) it is required to impose also boundary conditions at infinity. The most natural conditions are Dirichlet (reflecting) boundary conditions, see [28]. Before we address the question of stability of the Cauchy horizon, it is essential to understand the behavior in the exterior region of Kerr-AdS or Reissner-Nordström-AdS.
Logarithmic decay for linear waves on the exterior of Kerr-AdS and Reissner-Nordström-AdS. For the massive linear wave equation (1.1) on Kerr-AdS and Reissner-Nordström-AdS, Holzegel-Smulevici showed in [38] stability in the exterior region. Indeed, they proved that solutions decay at least at logarithmic rate towards i + (cf. polynomial (Λ = 0) and exponential (Λ > 0)) assuming the Hawking-Reall [32] bound 1 r + > |a|l and the Breitenlohner-Freedman [6] bound µ > 3 4 Λ. Moreover, they showed that solutions of (1.1) with fixed angular momentum actually decay exponentially on the exterior of Reissner-Nordström-AdS. (This is in contrast to the asymptotically flat case, in which fixed angular momentum solutions of (1.1) decay polynomially on the exterior of Reissner-Nordström.) However, their main insight was that a suitable infinite sum of such rapidly decaying fixed angular momentum solutions, possessing finite energy in some weighted norm, indeed achieves the logarithmic decay rate [40]. This is due to the presence of stable trapping. Note that this sharpness can also be concluded from later work showing the existence of quasinormal modes converging to the real axis at an exponential rate as the real part of the frequency and angular momentum tend to infinity [63,31]. (For some asymptotically flat five dimensional black holes a similar inverse logarithmic lower bound was shown in [2].) Strong Cosmic Censorship for AdS black holes. With the logarithmic decay on the exterior in hand, we turn to the question of the stability of the Cauchy horizon. Indeed, the logarithmic decay rate on the exterior is too slow to follow the methods involving the red-shift vector field and the vector field S as in (1.2) (see discussion before) to prove uniform boundedness and C 0 (continuous) extendibility at the Cauchy horizon of solutions to (1.1). More specifically, after propagating the logarithmic decay through the red-shift region, the energy flux associated to S is infinite on a {r = const.} hypersurface in the black hole interior due to the slow logarithmic decay towards i + . Thus, the question of whether to expect the validity of Conjecture 1 for asymptotically AdS black holes appears to be completely open.
The present paper is an attempt to shed some first light on SCC in the asymptotically AdS case: We will show that, despite the slow decay on the exterior, boundedness in the interior and continuous extendibility to the Cauchy horizon still holds for solutions of (1.1) on Reissner-Nordström-AdS black holes (cf. Theorem 1). The additional phenomenon which we exploit to prove boundedness is that the trapped frequencies responsible for slow decay have high energy with respect to the T vector field and can be bounded using the scattering theory developed in [42]. Thus, for Reissner-Nordström-AdS, the analog of Conjecture 1 is false on the linear level, just as in the Λ ≥ 0 cases. See however our remarks on Kerr-AdS later in the introduction.
The massive linear wave equation on Reissner-Nordström-AdS. As mentioned above, we will consider the massive linear wave equation for AdS radius l 2 := − 3 Λ on a fixed subextremal Reissner-Nordström-AdS black hole. Moreover, we assume the so-called Breitenlohner-Freedman bound [6] for the Klein-Gordon mass parameter α < 9 4 , which includes the conformally invariant case α = 2. This bound is required to obtain well-posedness [37,62,61] of (1.3).
Recall from the discussion above that solutions with fixed angular momentum actually decay exponentially in the exterior region. For such solutions with fixed , uniform boundedness with upper bound C = C in the interior and continuity at the Cauchy horizon can be shown using the methods involving the vector field S as in (1.2). Note however that this does not imply that a general solution remains bounded in the interior as the constant C is not summable: L =0 C ∼ e L → +∞ as L → ∞. Note in particular that, as a result of this, one cannot study the new non-trivial aspect of this problem restricted to spherical symmetry. (Nevertheless, see [3] for a discussion of the Ori model for RN-AdS black holes.) Main theorem: Uniform boundedness and continuity at the Cauchy horizon. We now state a rough version of our main result. See Theorem 3.1 for the precise statement.
Theorem 1 (Rough version of Theorem 3.1). Let ψ be a solution to (1.3) arising from smooth and compactly supported initial data (ψ 0 , ψ 1 ) posed on a spacelike hypersurface Σ 0 as depicted in Fig. 1. Then, ψ remains uniformly bounded in the black hole interior where C is constant depending on the parameters M, Q, l, α, the choice of Σ 0 and on some higher order Sobolev norm of the initial data (ψ 0 , ψ 1 ). Moreover, ψ can be extended continuously across the Cauchy horizon.
As we have explained above, the main difficulty compared to the asymptotically flat case, where the analysis was carried out entirely in physical space and requires inverse polynomial decay in the exterior [25], is the slow decay of ψ along the event horizon. Our strategy is to decompose the solution ψ in a low and high frequency part ψ = ψ + ψ with respect to the Killing field T = ∂ ∂t and treat each term separately. For the low frequency part ψ , we will show a superpolynomial decay rate in the exterior, see already Proposition 4.8. This decay is sufficient so as to follow the method of [25] with vector fields of the form (1.2) to show boundedness and continuity at the Cauchy horizon, up to the additional difficulty caused by the fact that we allow a possibly negative Klein-Gordon mass parameter. The violation of the dominant energy condition due to the presence of a negative mass term can be overcome with twisted derivatives [62,41], which provide a useful framework to replace Hardy inequalities for the lower order terms in this context.
For the high frequency part ψ , which is exposed to stable trapping and does in general only decay at a sharp logarithmic rate in the exterior, the key ingredient is the scattering theory developed in [42] (see discussion above). More specifically, the uniform bounds for the transmission and reflections coefficients T and R for |ω| ≥ ω 0 proved in [42] turn out to be useful for the high frequency part ψ . These bounds allow us to control |ψ | at the Cauchy horizon by the T -energy norm on the event horizon commuted with angular derivatives. The T -energy flux on the event horizon is in turn bounded from initial data by a simple application of the T -energy identity in the exterior. In particular, no quantitative decay along the event horizon is used for the high frequency part ψ . This is what allows us to overcome the problem of slow logarithmic decay.
Outlook on Kerr-AdS. We strongly believe that our arguments also apply to axially symmetric solutions ψ of (1.3) on a Kerr-AdS black hole. For general non-axisymmetric solutions, however, the question of uniform boundedness and continuity at the Cauchy horizon is less clear. Indeed, specific high frequency solutions which decay at a logarithmic decay rate can be considered as "low frequency" solutions when frequency is measured with respect to the Killing generator of the Cauchy horizon. In fact, it might well be the case that for solutions of (1.3) on Kerr-AdS there is C 0 blow-up at the Cauchy horizon, supporting the validity of Conjecture 1 after all in this context! Instability of asymptotically AdS spacetimes? Turning to the fully nonlinear dynamics, there is another scenario which could happen. Recall that Minkowski space (Λ = 0) and de Sitter space (Λ > 0) are proved to be nonlinearly stable [27,10]. Anti-de Sitter space (Λ < 0), however, is expected to be nonlinearly unstable with Dirichlet conditions imposed at infinity. This was recently proved in [50,49,52,51] for appropriate matter models. See also the original conjecture in [15] and the numerical results in [4]. Similarly, for Kerr-AdS (or Reissner-Nordström-AdS), the slow logarithmic decay on the linear level proved in [40] could in fact give rise to nonlinear instabilities in the exterior. 2 If indeed the exterior of Kerr-AdS was nonlinearly unstable, linear analysis like that in the present paper would be manifestly inadequate and the question of the validity of Strong Cosmic Censorship would be thrown even more open! Refer to the introduction of [16] for a more elaborate discussion.
Outline. This paper is organized as follows. In Section 2 we set up the spacetime and summarize relevant previous work. In Section 3 we state and prove our main result Theorem 3.1. Parts of the proof require a separate analysis which are treated in Section 4 and Section 5.

The Reissner-Nordström-AdS black hole
We are ultimately interested in the behavior of solutions to (1.3) to the future of a spacelike hypersurface Σ 0 as depicted in Fig. 1. For technical reasons (Fourier space decompositions are non-local operations) we will however construct also parts to the past of Σ 0 . In the following will define the spacetime pictured in Fig. 2. Note that P defines black hole parameters in the subextremal range. From now on, we will consider fixed parameters M, Q, l, α, where (M, Q, l) ∈ P and α < 9 4 . Note that M is the mass parameter, Q the charge parameter of the black hole and l = − 3 Λ is the Antide Sitter radius. For this specific choice of parameters we will also write ∆(r) := ∆ M,Q,l (r) and denote by 0 < r − < r + the positive roots of ∆. Now, let the two exterior regions R A , R B and the black hole region B be smooth four dimensional manifolds diffeomorphic to R 2 × S 2 . On R A , R B and B we introduce global 3 coordinate charts: If it is clear from the context which coordinates are being used, we will omit their subscripts throughout the paper. Again, on the manifolds R A , R B and B we define-using the coordinates (t, r, θ, ϕ) on each of the patches-the Reissner-Nordström-Anti-de Sitter metric On each of R A , R B and B, we define time orientations using the vector field We will also define the tortoise coordinate r * by in R A , R B and B independently. This defines r * up to an unimportant constant. Then, in each of the regions R A , R B and B, we define null coordinates by v = r * + t and u = r * − t, where for example for the v coordinate on R A , we will use the notation v R A and analogously for the other regions. Note that throughout the paper we will use the notation for derivatives ∂ ∂r * .
Patching the regions R A , R B and B together. Now, we patch the regions R A , R B and B together. We begin by attaching the future (resp. past) event horizon H + A (resp. H − A ) to R A by formally 4 setting Finally, we attach the past (resp. future) bifurcation sphere B − (resp. B + ) to B as We shall also set CH : The resulting manifold will be called M RNAdS . Note that, g extends to a smooth Lorentzian metric on M RNAdS which we will call g RNAdS and in particular, (M RNAdS , g RNAdS ) is a time oriented smooth Lorentzian manifold with corners. We illustrate the constructed spacetime as a Penrose diagram in Fig. 2. Note that the vector field ∂ t defined on R A , R B and B, respectively, extends to a smooth Killing field on M RNAdS , which we will from now on call T . Moreover, the standard angular momentum operators W i for i = 1, 2, 3, the generators of so(3) defined as We will impose initial data on a spacelike hypersurface Σ 0 to be made precise in the following. Note that we can choose for convenience that the spacelike hypersurface Σ 0 lies to the future of the past bifurcation sphere B − . Indeed, by general theory (an energy estimate in a compact region) this can be assumed without loss of generality [17]. More precisely, let Σ 0 be a 3 dimensional connected, complete and spherically symmetric spacelike hypersurface extending to the conformal infinity I = I A ∪ I B . Moreover, assume that A possible choice of Σ 0 is denoted in Fig. 3. We are ultimately interested in the shaded region to the future of Σ 0 . For the rest of the paper, we will consider such a Σ 0 to be fixed.

Conventions
With a b for a ∈ R and b ≥ 0 we mean that there exists a constant C(M, Q, l, α, Σ 0 ) with a ≤ Cb. If C(M, Q, l, α, Σ 0 ) depends on an additional parameter, say , we will write a b. We also use a ∼ b for some a, b ≥ 0 if there exist constants C 1 (M, Q, l, α, Σ 0 ), C 2 (M, Q, l, α, Σ 0 ) > 0 with C 1 a ≤ b ≤ C 2 a. We shall also make use of the standard Landau notation O and o [54]. To be more precise, let X be a point set (e.g. |g(x)| ≤ C(M, Q, l, α). We write O (g(x)) if the constant C depends on an additional parameter . For the standard volume form in spherical coordinates (ϕ, θ) on the sphere S 2 we will use the notation dσ S 2 := sin θdϕdθ. Finally, let the Japanese symbol be defined as x :

Wave equation and mixed boundary value Cauchy problem
We are interested in solutions to the massive wave equation (1.3) associated to the metric g RNAdS on a subextremal Reissner-Nordström AdS black hole with black hole parameters M, Q, l as in (2.3). In view of the timelike boundaries I A and I B , we need to specify boundary conditions on I A and I B in addition to prescribing data on the spacelike hypersurface Σ 0 , cf. Fig. 3. We will use Dirichlet (reflecting) boundary conditions which can be viewed as the most natural conditions in the context of stability of the Cauchy horizon. In principle, however, in view of [62], we could also use more general boundary conditions like Neumann or Robin conditions. We will now introduce an appropriate foliation and norms in order to state the well-posedness statement.
We will foliate R A ∪ R B ∪ H + A ∪ H + B ∪ B with spacelike hypersurfaces. To do so, we let T be a smooth future-directed causal vector field on R A ∪ R B ∪ H + A ∪ H + B ∪ B with the properties that and that T is a future-directed timelike vector field on B. Now, define the leaves where Φ T is the flow generated by T and t * ∈ R is its affine parameter. We have illustrated some leaves in Fig. 4.

Further coordinates in the exterior region
In the region R A ∪ H + A , we moreover define a global (up to the well-known degeneracy on S 2 ) coordinate system (t * , r, ϕ, θ), where t * is the affine parameter of the flow generated by T . Note that on Similarly, we can define such a coordinate system on R B . Figure 4: Illustration of the foliation with leaves Σ τ defined in (2.11).

Norms on hypersurfaces Σ t *
By construction Σ t * intersects R A , R B and B. We will now define norms on Σ t * which are adaptations of the norms introduced in [37]. We define where each of the terms appearing in (2.12) will be defined in the following.
Norms in the interior region. We begin by defining the first term in (2.12). We define · 2 as the standard Sobolev norm of order k on the Riemannian manifold (Σ t * ∩ B, g RNAdS Σ t * ∩B ).
Norms in the exterior region. Due to the symmetry of the regions R A and R B , we will only define the norms on R A in the following. The norms on R B are be constructed analogously. We use the coordinates (t * , r, θ, ϕ) in R A to define the norms and similarly for higher order norms. Here and in the following we denote with / ∇ and / g the induced covariant derivative and the induced metric, respectively, on spheres of constant (t * , r). We will also use the notation | / ∇ψ| 2 := / g( / ∇ψ, / ∇ψ). Now having defined (2.12), we will define energies in the following.

Energies on hypersurfaces Σ t *
We set for i = 1, 2, where all terms in (2.13) will be defined in the following.
Energies in the interior region. In the interior region we are not concerned with r-weights and define the energies as Energies in the exterior region. To define the energies in the exterior region, it is convenient to start with defining the following energy densities and their integrals as Note that we will write E B i for the analogous energy restricted to R B . Also remark the following relation between the norms and energies defined above Energy-momentum tensor. Our energies are based on the energy momentum tensor associated to (1.3) which is defined as For a smooth vector field X we also define The term K X is often referred to as the "bulk term" and satisfies if φ is a solution to (1.3). Note that if X is Killing, then K X vanishes. More generally, integrating (2.19) one obtains an energy identity relating boundary and bulk terms. For more details about the energy-momentum tensor and its usage for standard energy estimates we refer to [17].

Well-posedness
Having set up the spacetime and the norms, we will restate the well-posedness result for (1.3) as a mixed boundary value-Cauchy problem. For asymptotically AdS spacetimes, well-posedness was first proved in [37]. 37]). Let the Reissner-Nordström-AdS parameters (M, Q, l) and the Klein-Gordon mass α < 9 4 be as in be prescribed on the spacelike hypersurface Σ 0 and impose Dirichlet (reflecting) boundary conditions on I = I A ∪ I B .
Then, there exists a smooth solution ψ ∈ C ∞ (M RNAdS \ CH) of (1.3) such that ψ Σ0 = ψ 0 , T ψ Σ0 = ψ 1 . The solution ψ is also unique in the class C(R t * , H 1,0 RNAdS (Σ t * )) and satisfies the energy identity The analogous energy identity holds in R B . In particular, (2.20) shows that the T -energy flux through I = I A ∪ I B vanishes.
Moreover, the T -energy flux through the event horizon is bounded by initial data Remark 2.2. The well-posedness statement in Theorem 2.1 holds true for a more general class of initial data, called a H 2 AdS initial data triplet, see [37].

Boundedness and decay in the exterior region
In the exterior regions R A and R B we have energy decay and boundedness results which have been proved in [37,36,38,40] 6 . We summarize these results in

23)
and similarly for higher order norms. Moreover, we have the energy decay statements for t * ≥ 0 and the pointwise decay for t * ≥ 0 in the exterior region R A and similarly in R B . Moreover, just like for Schwarzschild-AdS (cf. [38]), fixed angular frequencies decay exponentially. More precisely, let Y m denote the spherical harmonics and let ψ be a solution to (1.3) arising from smooth and compactly supported data on Σ 0 . If there exists an L ∈ N with ψ, Y m L 2 (S 2 ) = 0 for ≥ L, then for t * ≥ 0 and a constant C(M, Q, l, α) > 0 only depending on the parameters M, Q, l, α.
Remark 2.4. Note that (2.26) also implies pointwise exponential decay for ψ (assuming ψ, Y m L 2 (S 2 ) = 0 for ≥ L) and all higher derivatives of ψ using standard techniques like commuting with T and W i , elliptic estimates as well as applying a Sobolev embedding.
Remark 2.5. The previous decay estimates have only been stated to the future of Σ 0 in the region R A , nevertheless, they also hold in R B . Moreover, they also hold true to the past of Σ 0 for an appropriate foliation for which the leaves intersect H − A and H − B , and are transported along the flow of −T for R A ∪ H − A and along the flow of T for R B ∪ H − B . Notation. In the main part of the paper we will makes use of the Fourier transform and convolution associated to the coordinate t in (t, r, θ, ϕ) coordinates as in (2.4). We denote F T as the Fourier transform (and F −1 T as its inverse) defined as in the coordinates (t, r, ϕ, θ) of R A , R B and B, respectively. Here, we assume that t → f (t, r, θ, ϕ) is (at least) a tempered distribution and (2.27), in general, is to be understood in the distributional sense. Moreover, the convolution * associated to the coordinate t is defined as where we again assume that t → f (t, r, θ, ϕ) is a tempered distribution and t → g(t, r, θ, ϕ) is a Schwartz function. Here, (2.28), in general, is to be understood in the distributional sense.

Main theorem and frequency decomposition
Now, we are in the position to state our main result Moreover, ψ extends continuously to the Cauchy horizon, i.e. ψ ∈ C 0 (M RNAdS ).
Remark 3.2. The data term D[ψ] in (3.2) can be controlled by the initial data (ψ 0 , ψ 1 ) such that (3.1) can be written in terms of initial data as for a constant C(M, Q, l, α, Σ 0 ) only depending on the parameters M, Q, l, α and the choice of initial hypersurface Σ 0 .
Remark 3.3. Theorem 3.1 can be extended to a more general class of initial data using standard density arguments. In the context of uniform boundedness and continuity at the Cauchy horizon, it is enough to consider smooth and localized initial data. Nevertheless, note that for more general initial data in appropriate Sobolev spaces, already well-posedness becomes more delicate [37].
Proof of Theorem 3.1. We split up the proof in four steps, where Step 3 and Step 4 are the main parts relying on Section 4 and Section 5.
Step 1: Decomposition into low and high frequencies. Let be as in the assumption of Theorem 3.1. Now, in R A , R B and in B, define the low frequency part ψ and the high frequency part ψ as From Proposition A.4 in the appendix we know that the low and high frequency parts ψ and ψ in (3.5) are well-defined and ψ and ψ extend to smooth solutions of (1.3) on M RNAdS \ CH. The cut-off frequency ω 0 = ω 0 (M, Q, l, α) > 0 will be chosen in the proof of Proposition 4.5 only depending on M, Q, l, α. For convenience we can also assume that χ ω0 is a symmetric function which implies that ψ and ψ will be realvalued as long as ψ was real valued. This concludes Step 1.
Having decomposed the solution in low and high frequency parts ψ and ψ , we shall now see how the initial data D[ψ ] and D[ψ ], respectively, can be bounded by the initial data D[ψ] of ψ.
Step 2: Estimating the initial data of the decomposed solution. This step is the content of the following proposition.
Proposition 3.4. Let ψ be as in (3.4) and ψ , ψ be as in (3.5) and recall the definition of D[·] from (3.2). Then, (3.7) Proof. Since ψ = ψ + ψ , it suffices to obtain a bound of the type . Because of the Dirichlet conditions imposed at infinity, the energy fluxes through I A and I B vanish (see (2.20)), and we estimate is a higher order energy on the hypersurfacẽ to be made precise in the following. Note also that the normal vector field on R A ∩Σ 0 is nΣ 0 = r √ ∆ ∂ t . More precisely, due to the support properties of the initial data, there exists a spherically symmetric compact set K : Estimate (3.8) follows from general theory [17], that is a (higher order) energy estimate followed by an application of Grönwall's lemma. In order to estimate the energy on the compact hypersurface K we decompose K in K ∩ R A and K ∩ R B and estimate the energy on each of those slices independently. Again, in view of the fact that R A and R B can be treated analogously, we only show the estimate in R A . Note that all the terms of where we have used boundedness of higher order energies in the exterior which are proved in [36,38] and restated in Theorem 2.3. Also note that we can interchange the derivatives with the convolution since T is a Killing vector field. Thus, we conclude thatD[ψ ] D [ψ] and again by Cauchy stability and the vanishing of the energy flux at I (see (2.20 The previous analysis in Step 1 and Step 2 allows us to treat the low and high frequency parts ψ and ψ completely independently.
Step 3: Uniform boundedness for ψ and ψ . This step is at the heart of the paper and will be proved in Section 4 and Section 5. According to Proposition 4.22 and Proposition 5.3, Thus, in view of Step 2, we conclude which shows (3.1).
Step 4: Continuous extendibility beyond the Cauchy horizon. Again, this is proved Section 4 and Section 5. In particular, in Proposition 4.23 and Proposition 5.4 it is proved that ψ and ψ , respectively, are continuously extendible beyond the Cauchy horizon. Thus, ψ = ψ + ψ can be continuously extended beyond the Cauchy horizon which concludes the proof.

Low frequency part ψ
We will begin this section by showing that ψ decays superpolynomially in the exterior regions R A and R B (Section 4.1). This strong decay in the exterior regions then leads to uniform boundedness of ψ in the interior B and continuous extendibility of ψ beyond the Cauchy horizon. This will be shown in Section 4.2. In the following, it suffices to only consider R A because the region R B can be treated completely analogously.

Exterior estimates
We will now consider ψ in the exterior region R A and show an integrated energy decay estimate which will eventually lead to the superpolynomial decay for ψ . First, however, we review the separation of variables for solutions to (1.3).
for r ∈ (r − , r + ) and r ∈ (r + , ∞). In the regions B and R A , respectively, set where (Y m ) |m|≤ are the standard spherical harmonics.
Next, we prove that the potential V has a local maximum for large enough angular parameter 0 . Proposition 4.4. There exists an˜ 0 (M, Q, l, α) ∈ N such that for all ≥˜ 0 , the potential V has a local maximum r ,max > r + and V ≥ 0 for r + ≤ r ≤ r ,max . Moreover, r ,max → r max : Proof. Note that for large enough, V is non-negative in a neighborhood of r + with r ≥ r + . Also, V vanishes at r = r + . Hence, it suffices to show that dV dr is negative somewhere for r ≥ r + . But note that for some function F (r) which is independent of . Now, first choose r > r + large enough only depending on M, Q such that the last term is negative. Then, choose large enough such that it dominates the first term which proves that a r ,max as in the statement exists. The limiting behavior r ,max → 3 2 M + 9 4 M 2 − 2Q 2 as → ∞ also follows from (4.7). This concludes the proof. Now, we are in the position to prove a frequency localized integrated decay estimate in the exterior region for the bounded frequencies |ω| ≤ 2ω 0 . (4.9) Proof. In view of [38, Proposition 7.4 and Section 9.3] it suffices to prove (4.8) for ≥ 0 (M, Q, l, α) for some fixed 0 (M, Q, l, α) ∈ N 0 . Let r 0 , r 1 depending only on M, Q, l, α be such that r + < r 0 < r 1 < r max − δ, where r max is defined in Proposition 4.4. Here, δ = δ( 0 ) > 0 is such that V ≥ 0 for all r + ≤ r ≤ r max − δ, cf. Proposition 4.4. We can make δ( 0 ) as small as we want by choosing 0 sufficiently large. Now, we choose ω 0 (M, Q, l, α) > 0 small enough and 0 large enough such that V − ω 2 + ∆ 4l 2 r 2 ( + 1) + ∆ r 2 for r ≥ r 0 , (4.10) and for all |ω| ≤ 2ω 0 , ≥ 0 . For smooth f (r * ) andh(r * ), we define the currents Thus, We choose a smooth f ≤ 0 such that • f is monotonically increasing, • f ≤ −c 1 for r + ≤ r ≤ r 1 and some c 1 (M, Q, l) > 0, • |f | ∆, and a smoothh ≥ 0 such that • |h | 1 for r 0 < r 1 , •h = 1 for r ≥ r 1 .
Then, we have for r + ≤ r ≤ r max − δ and to control the negative signed term in (4.18), yields Note that we use lim r→∞ |r 1 2 u| = 0 and lim r→∞ |r − 1 2 u | = 0 to apply the Hardy inequality. To obtain control of |u | 2 in the region r ≥ r max − δ in (4.20) we just add a small portion of the integral over (4.18). This proves With the frequency localized integrated energy decay estimate of Proposition 4.5 we will now prove a local integrated energy decay estimate in physical space. Indeed, a naive application of Plancherel's theorem to (4.8) gives a global integrated energy estimate. However, localizing this energy decay requires some sort of cut-off which does not respect the compact frequency support. Nevertheless, by carefully choosing a localization, we can show that the error term decays superpolynomially in time. At this point we shall remark that we do expect ψ to decay exponentially. However, for our problem, superpolynomial decay in the exterior is (more than) sufficient. Proposition 4.6. Let ψ be as in (3.5). Then, for any q > 1 and τ 2 ≥ 2τ 1 ≥ 0, we have the integrated energy decay estimate where C(q) > 0 is a constant only depending on q. Moreover, this directly implies for the T -energy.
Proof. In order to show (4.22) we will first construct an auxiliary solution Ψ of (1.3). We set initial data for Ψ on Σ τ1 as (Ψ 0 , Ψ 1 ) := (ψ , T ψ ) Στ 1 ∩R A . Then, we will define data Ψ 2 on H + A ∩ {t * ≤ τ 1 } such that the data can be extended to a C k function in a neighborhood of H + A ∩ {t * = τ 1 } for some finite regularity k. Choosing the regularity k large enough will guarantee well-posedness. More precisely, in local coordinates (t * , r, θ, ϕ) and for r = r + , we define for t * ≤ τ 1 and some uniquely determined (λ j ) 1≤j≤k such that R × S 2 (t * , ϕ, θ) → Ψ 2 (t * , r + , ϕ, θ) for t * ≤ τ 1 ψ (t * , r + , ϕ, θ) for t * > τ 1 (4.25) is C k . Indeed, the function is smooth everywhere except at t * = τ 1 . Now, we consider the mixed boundary Figure 5: In the darker shaded region J + (Σ τ1 ) ∩ R A we have that Ψ = ψ and in the lighter shaded region we can estimate the energy of Ψ in terms of ψ . This holds true as Ψ 2 is the C k reflection of ψ from H + value-Cauchy-characteristic problem, where we impose data as follows. On the null hypersurface H + A ∩{t * ≤ τ 1 } we impose Ψ 2 . This null cone intersects the spacelike hypersurface Σ τ1 on which we have prescribed (Ψ 0 , Ψ 1 ) as data. As before, we assume Dirichlet condition on I A . For fixed k > 0 large enough, this is a wellposed problem and can be solved backwards and forwards in R A [53, Theorem 2]. We will call the arising solution Ψ and by uniqueness note that Ψ = ψ on (R A ∪ H + A ) ∩ J + (Σ τ1 ). Indeed, analogously to ψ , we have Ψ ∈ C(R t * ; H 1,0 AdS (Σ t * ∩R A ))∩C 1 (R t * ; H 0,−2 AdS (Σ t * ∩R A )). Moreover, Ψ decays logarithmically and Ψ, Y m Y m decays exponentially towards i + and i − on a {r = const.} hypersurface. 7 Refer to Fig. 5 for a visualization of the Cauchy-characteristic problem with Dirichlet boundary conditions.
Analogously to ψ = ψ + ψ , we decompose the new solution Ψ in low and high frequencies Ψ = Ψ + Ψ : We define where χ 2ω0 is a smooth cutoff function such that χ 2ω0 = 1 for |ω| ≤ ω 0 and χ 2ω0 = 0 for |ω| ≥ 2ω 0 . Now, note that from the T -energy identity in (2.21) we have as the flux through I A vanishes in view of the Dirichlet boundary condition at I A . Here, we use the notation We have used the estimate which follows from our construction of the initial data. Thus, Now, note that u[Ψ ] defined as satisfies the assumptions of Proposition 4.6 such that (4.8) holds true for u[Ψ ]. We now integrate the frequency localized energy estimate (4.8) associated to u[Ψ ] in ω and sum over all spherical harmonics. There are two main terms appearing and we will estimate them in the following. This step is similar to [38, Sections 9.1 and 9.3] so we will be rather brief. An application of Plancherel's theorem for the integrated left hand side of (4.8) yields To estimate the boundary term on the right hand side of (4.8), we first decompose u[Ψ ] as u[Ψ ] = a(ω, m, )u 1 + b(ω, m, )u 2 , where u 1 , u 2 are defined as the unique solutions to the radial o.d.e. (4.3) in the exterior satisfying u 1 = e iωr * + O (r − r + ) and u 2 = e −iωr * + O (r − r + ) as r → r + (r * → −∞). Here, a = a(ω, , m) and b = b(ω, , m) are the unique coefficients of the decomposition. Then, in view of (4.9) and as r → r + . Now, using that ωa(ω), ωb(ω) are in L 1 ω (R) and in L 2 ω (R) (note that they have compact support), an application of the Riemann-Lebesgue Lemma, the Fourier inversion theorem and Plancherel's theorem shows that m R |ω| 2 (|a(ω, , m)| 2 + |b(ω, , m)| 2 )dω Thus, we conclude the global integrated energy decay statement and in particular, Here, we have also used (4.34) and the fact that H + dvol Στ 1 follows from (4.29).
Finally, we have also used that the Schwartz functionχ 2ω0 decays superpolynomially at any power q > 1. This concludes the proof in view of (4.35).
In order to remove the degeneracy of the T -energy at the event horizon, we will use the by now standard red-shift vector field [17]. As usual, the red-shift vector field N is a future-directed T invariant timelike vector field which has a positive bulk term K N ≥ 0 near the event horizon. In a compact r region bounded away from the event horizon H + A , the bulk term K N of N is sign-indefinite but this will be absorbed in the spacetime integral of the T current in Proposition 4.6. Also, note that N = T for large enough r. In the negative mass AdS setting, we refer to [36,Section 4.2] for an explicit construction of the red-shift vector field N . Note that the red-shift vector field N has the property that for ψ as in (3.5).
Proposition 4.7. Let ψ be as in (3.5). Then for any τ 2 ≥ 2τ 1 ≥ 0, we have and in particular, Proof. We apply the energy identity (the spacetime integral of (2.18)) with the red-shift vector field N for ψ in the region After taking care of the negative lower order term via a Hardy inequality and absorbing the sign-indefinite bulk of N away from the horizon in the spacetime integral of J T on the right hand side (see [36,Section 4] for further details), we arrive at Note that the integral along the horizon H + A ∩{2τ1≤t * ≤τ2} J N µ [ψ ]n µ H dvol H is sign-indefinite due to the (possible) negative mass. However, this can be absorbed in the bulk term using an of the integrated bulk term of the red-shift vector field N and some of the bulk term from the T vector field, cf. [36,Equation (70)]. Moreover, using the integrated energy decay estimate of the T vector field from Proposition 4.6, we conclude Now we obtain Proposition 4.8. Let ψ be defined as in (3.5). Then, for any q > 1 and τ ≥ 0 we have and Proof. In view of Proposition 4.7 it suffices to prove (4.44). Upon setting we have from Proposition 4.7 that for any t 2 ≥ 2t 1 ≥ 0. The claim follows now from Lemma 4.9 below.
for any q > 1, 0 ≤ 2t 1 ≤ t 2 and some α(q) > 0 only depending on q. Then, for all q > 1, there exists a constant C(α(q), q) > 0 only depending on α and q such that for all t ≥ 0.

Interior estimates
Having obtained the superpolynomial decay for ψ in the exterior and in particular on the event horizon, we will now use this to show uniform boundedness in the black hole interior. The approach we take is similar to [25], however, due to the negative mass and hence the violation of the dominant energy condition, we will use twisted derivatives introduced in [62, 6].

Metric in the interior
In the interior we will also use null coordinates (u B , v B ) introduced in Section 2.1. Throughout this section we shall drop the index B. Then, setting where r = r(u, v), we write the metric in the interior as Note that in the interior we have r − < r(u, v) < r + and dr * = r 2 ∆ dr.

Twisted derivatives and twisted energy momentum tensor
Definition 4.10 (Twisted derivative). For a smooth and nowhere vanishing function f we define the twisted derivative∇ and its formal adjoint∇ * We shall refer to f as the twisting function. where the potential V is given by Now, we also associate a twisted energy-momentum tensor to the twisted derivatives. 3) and f as where V is as in (4.55).
We will now compute the divergence of the twisted energy-momentum tensor.
Now, assume that φ moreover satisfies (1.3) and X is a smooth vector field. Set Then, Finally, note that if the twisting function f associated to∇ is chosen such that V ≥ 0, thenT µν satisfies the dominant energy condition, i.e. if X is a future pointing causal vector field, then so is −J X .
In Proposition A.1 in the appendix we have written down the components of the twisted energy-momentum tensor, the twisted 1-jetsJ X and the twisted bulk termK X in null components. We will use the notation C u1 := {u = u 1 }, C v1 = {v = v 1 } for null cones and Σ r1 = {r = r 1 } for spacelike hypersurfaces in the interior. Furthermore, we set and analogously for Σ and C. Recall that u + v = 2r * . Will make also use of the following notation. For anỹ r ∈ (r − , r + ) we set vr(u) := 2r * (r) − u, ur(v) := 2r * (r) − v and for hypersurfaces with constant u, v, r we denote n u , n v , n r as their normals. 8 Now, we are in the position to propagate the superpolynomial decay on the horizon established in Proposition 4.8 further into the interior. To do so we will make use of the twisted red-shift.

Twisted red-shift in the black hole interior
Proposition 4.14. There exist a r red ∈ (r − , r + ), a constant b(M, Q, l, α) > 0, a nowhere vanishing smooth function f associated to the twisted energy momentum tensor and a future directed timelike vector field N such that for R red := {r red ≤ r ≤ r + } ∩ {v ≥ 1} and for ψ as in (3.5).
Proof. We choose the ansatz N = N u ∂ u +N v ∂ v for our red-shift vector field. We will first estimate the twisted 1-jetJ and then the twisted bulk termK.
whereḟ := df dr . Thus, choosing f = e −r gives Note that for r red < r + close enough to r + , we have for all r red ≤ r ≤ r + and some constant c red > 0 only depending on the black hole parameters. The constant c red > 0 does not decrease, when we choose r red even closer r + . By choosing r red close enough to r + , we ensure that V 1 in r red ≤ r ≤ r + . This finally shows that if we take N as a future directed vector field, the 1-jet J N µ n µ v is positive definite. We will construct the explicit form of N in the bulk term estimate.
Bulk termK N . Now, we will estimate the bulk term. We will choose the components of the timelike vector field N = N u ∂ u + N v ∂ v as Note that N is smooth in R red . Moreover, for fixed δ 1 , δ 2 > 0 (only depending on the black hole parameters), we can choose r red close enough to r + such that N is future directed in R red . Then, note that In the following we will show that We will start with the sign-indefinite term appearing in (4.69). We estimate it as follows where we have applied an -weighted Young's inequality. We have also used that-by choosing r red closer to r + -we can make Ω 2 uniformly smaller than any constant, in particular smaller than δ 1 and δ 2 once those are fixed. Choosing small enough, we absorb the term Ω 4 |∇ u ψ | 2 of (4.75) in the first term of (4.69). Then, choosing δ 2 (δ 1 , ) small enough, we can also absorb the term 1 |∇ v ψ | 2 in the first term of (4.69). Completely analogously and by potentially choosing δ 2 and δ 1 even smaller, we estimate the terms of the form 1 Ω 2 Re(∇ u ψ ∇ v ψ ) arising from (4.72) and (4.73). Next, note that, in view of V 1 and −∂v(f 2 V) 2f 2 Ω 2 , we choose δ 1 small enough such that we absorb error terms coming from (4.72) and (4.73) in the term with the good sign in (4.70). By doing so we also have to make δ 2 ( , δ 1 ) > 0 small enough. Finally, once δ 1 and δ 2 are fixed, note that we can make terms involving higher orders of Ω 2 arbitrarily small by choosing r red close to r + . This finally shows (4.74) and concludes the proof.
With the help of the constructed twisted red-shift current, we obtain Proposition 4.15. Let r 0 ∈ [r red , r + ). Let ψ defined as in (3.5) and recall that from Proposition 4.8 we have

(4.79)
for any 1 ≤ v 1 ≤ v 2 . Here, we use the notation Proof. The strategy is very similar to [25]. For any 1 ≤ v 1 ≤ v 2 we start by applying the energy identity (spacetime integral of (4.60)) in the region R(v 1 , v 2 ) to obtain (4.80) From Proposition 4.8 we have that Upon definingẼ we obtainẼ for any v ≥ 1. This follows from an argument very similar to Lemma 4.9. Note that we have by general theory [17] for v ≥ 1 which proves (4.77). The estimates (4.78) and (4.79) now follow from applying the energy identity again in the region R(v 1 , v 2 ).

No-shift region
In this region we propagate the decay towards i + from the red-shift region to the blue-shift region using a T = ∂ t invariant vector field X and a t-independent twisting function f . Take r red fixed from Proposition 4.14 and let r blue (M, Q, l) > r − be close to r − which we will fix later in the proof of Proposition 4.21. As our vector field we will choose for some β ns > 0 large enough such that uniformly in [r blue , r red ]. In particular, since r ∈ [r blue , r red ] is bounded away from r + , r − , we havẽ Proposition 4.16. Let ψ defined as in (3.5). For any r 0 ∈ [r blue , r red ], q > 1 and any v * ≥ 1 we have (4.90) Moreover, for any 1 < p < q we also have Proof. We apply the energy identity (spacetime integral of (4.60)) with X = ∂ u + ∂ v (cf. (4.86)) and f ns as in (4.87) in the region The choice of f ns guarantees the twisted dominated energy condition for the twisted energy-momentum tensor. Together with the coarea formula as well as the facts that [r * (r 0 ), r * (r red )] is compact and X is T invariant, we conclude forr ∈ [r 0 , r red ], we also have E(vr(u r0 (v * )),r) ≤B 1 r≤r≤r red E(vr(u r0 (v * )),r)dr + E(v r red (u r0 (v * )), r red ) (4.94) for a constantB 1 =B 1 (M, Q, l, α, Σ 0 ). An application of Grönwall's inequality yields Note that v * − v r red (u r0 (v * ))) = const. and hence by Proposition 4.15 we have Then, applying (4.95) and (4.96) forr = r 0 proves (4.90). Finally, (4.91) is a consequence of the fact that v p ∼ u p (using r blue ≤ r ≤ r red ) and the following well-known lemma.  18. From now on we will consider p and q as fixed and constants appearing in , and ∼ can additionally depend on 1 < p < q.
By doing the analogous analysis in the neighborhood of the left component of i + we obtain Proposition 4.19. Let ψ defined as in (3.5). Then, for any r 0 ∈ [r blue , r + ) we have Commuting with angular momentum operators (W i ) 1≤i≤3 , an application of the Sobolev embedding H 2 (S 2 ) → L ∞ (S 2 ) and using the fact that p > 1, we also conclude Proposition 4.20. Let ψ defined as in (3.5). Then, (4.98) Finally, we will use the decay towards i + to show uniform boundedness in the interior and continuity all the way up to and including the Cauchy horizon for ψ .

Blue-shift region
In the blue-shift region we shall use a different twisting function. Recall that we would like to have V 1, where Now, we set f := e r and obtain Note that for r blue > r − close enough to r − , we have for all r blue ≥ r ≥ r − and some constant c blue > 0 only depending on the black hole parameters. Thus, we obtain V 1 uniformly in the blue-shift region r blue ≥ r ≥ r − by choosing r blue close enough to r − . In the blue-shift region we define the vector field for some potentially large N > 0 and p > 1 as in Remark 4.18. We will show in the following that sup θ,ϕ |ψ (u 0 , v 0 , θ, ϕ)| is uniformly bounded from initial data D[ψ ] independently of (u 0 , v 0 ) ∈ J + (Σ r blue )∩B.
To do so, we will apply the energy identity (spacetime integral of (4.60)) in the region which we depict in Fig. 6.

This leads to
where ψ is defined in (3.5). In the following we will show, that after choosing N > 0 large enough and an appropriate integration by parts to control error terms, we can control the flux terms by initial data. This gives Proposition 4.21. Let ψ defined as in (3.5). Then, and for any (u 0 , v 0 ) ∈ J + (Σ r blue ). Commuting with the angular momentum operators (W i ) 1≤i≤3 also gives Proof. The general strategy of the proof is to apply (4.104) and to show that R fK S N dvol ≥ 0 + boundary terms, (4.108) where the boundary terms are small (lower orders in Ω) and by choosing r blue closer to r − , can be absorbed in the positive flux terms on the left hand side of (4.104). In the first part, we compute the flux terms for our vector field S N defined in (4.102). Then, in the second part, we will estimate the bulk term and indeed show (4.108). From this we will then deduce (4.105).
Part I: Flux terms of S N . We obtain three flux terms from (4.104). The future flux terms read (cf. Proposition A.1) and The past flux term on the spacelike hypersurface Σ r blue is uniformly bounded by initial data from Proposition 4.19: Part II: Bulk term of S N . We will now estimate the bulk term To estimate all terms, we will also integrate by parts and substitute terms of the form ∂ u ∂ v ψ using the equation g ψ = 0. The boundary terms arising from the integration by parts will then be absorbed in the future flux terms appearing in Part I: Flux terms of S N . In the following we shall treat each terms ofK X as in (A.4) with X = S N individually.
First term of (A.4). The first term of (A.4) is non-negative: This means that-by choosing N > 0 large enough-we will be able to absorb sign-indefinite terms of the form r N −1 v p |∇ v ψ | 2 and r N −1 u p |∇ u ψ | 2 . This will be used in the following. Before we treat the second term appearing in (A.4), which is sign-indefinite, we look at the angular and potential term in the second line of (A.4).
Angular and potential term: Second line of (A.4). Now, we look at the term involving angular derivatives.
In the region R f we have Also recall that we have chosen the twisting function such that V 1.
Second, sign-indefinite term of (A.4). Now, note that the second term in the first line of (A.4) is sign-indefinite, however, we can absorb it in other positive terms after integrating by parts in the region R f as we will see in the following. In order to integrate by parts, it is useful to express the twisted derivatives with ordinary derivatives. The integration by parts will generate boundary terms. As mentioned above, we estimate these boundary terms with the fluxes in the energy identity. This will be done later in (4.119) and we will not write the boundary terms explicitly in the following. We will also have to control (sign-indefinite) ordinary derivatives by positive terms in (4.112) and (4.113). Note that this is possible since where the right hand side of (4.115) is controlled by , (4.112), (4.113) and potentially choosing r blue closer to r − . The analogous statement holds true for u p |∂ u ψ | 2 .
The integrated term we have to estimate reads We first look at Using the explicit form of f and noting that we have control over ( v p + u p )Ω 4 |ψ | 2 from (4.113), it suffices to estimate Now, note that the second term from (4.116) is controlled by (4.112) and (4.113) using Cauchy's inequality. Now, in both terms, the first and third term of (4.116), we integrate by parts in u. We also use Re ψ ∂ u ψ = 1 2 ∂ u (|ψ | 2 ). Then, it follows that-up to boundary contributions which will be dealt with below in (4.119)-we have to control the terms (4.117) The first and third term of (4.117) are controlled by (4.112) and (4.113) and by potentially choosing r blue even closer to r − . For the second term of (4.117) we will use (1.3) which reads Replacing ∂ u ∂ v ψ and integrating by parts on the sphere, we estimate all but one term of (4.117) using (4.113) and (4.112). The term which we cannot estimate with (4.113) and (4.112) is of the form This is of a similar form as the third term in (4.116), which we control-as before-via an integration by parts in u. Finally we have controlled all terms except for boundary terms arising from the integration by parts. The first boundary terms arose from integrating by parts the first term in (4.116). It is of the form Cu 0 ∩{vr blue (u0)≤v≤v0} which we control as where we have used that r p * (Ω 2 ) 1 4 1 for r * ≥ r * (r blue ). Using (4.121) we absorb (4.120) in the flux term (4.109) by potentially choosing r blue closer to r − such that Ω 2 is uniformly small in the blue-shift region. Completely analogously, we control the other boundary terms which arose from integrating by parts. Now, we are left with the terms of the last two lines in (A.4).
Terms from last two lines of (A.4). We will only look at the terms with v weights as the terms involving u weights are estimated completely analogously. It suffices to estimate the terms Since ∂v(f 2 V) 2f 2 Ω 2 , we control the first and second term of (4.122) using (4.113) by potentially choosing r blue closer to r − . The last term of (4.122) is be estimated as Finally, we have estimated and absorbed all sign-indefinite terms in the energy identity to obtain (4.108). Thus, we have proved (4.105), which concludes the first part of the proof.
Part III: Proof of (4.106) and (4.107). Now, observe that the estimate (4.106) follows from (4.105) and (4.115). More precisely, the error arising from interchanging the twisted derivatives with partial derivatives on C u are estimated as Finally, note that the error term on the right hand side is controlled as in (4.119). This works for C v completely analogously which concludes the proof.

Uniform boundedness and continuity at the Cauchy horizon for bounded frequencies
Now, Proposition 4.21 allows us to prove the uniform boundedness.
Proposition 4.23. Let ψ be as defined in (3.5). Then, ψ is continuously extendible beyond the Cauchy horizon CH.
Proof. Similarly to (4.125) we have uniformly in u 0 , ϕ, θ. The same estimate holds after interchanging the roles of u and v. After commuting the equation with W 3 , we have from (4.124) for some constantC < ∞ depending on the initial data. (Recall that we assumed our initial data to be smooth and compactly supported.) Thus, for ϕ 1 ≤ ϕ 2 , we have uniformly in u 0 , v 0 , θ 0 . A similar estimate holds true for θ. Applications of the fundamental theorem of calculus and a triangle inequality finally yield the continuity result for ψ .

High frequency part ψ
In the previous section we have shown the uniform boundedness for the low frequency part ψ . Now, we turn to ψ , the high frequency part. The key ingredient for the proof of the uniform boundedness for |ψ | in the interior is (a) the uniform boundedness of transmission and reflection coefficients associated to the radial o.d.e. (4.3) which is proved in [42] for Λ = 0, together with (b) the finiteness of the (commuted) T -energy flux on the event horizon given by (2.21). Now, recall the radial o.d.e. (4.3) which reads −u +V u = ω 2 u in the interior, where V decays exponentially as r * → +∞(r → r − ) and r * → −∞(r → r + ). For ω = 0, so in particular for |ω| > ω0 2 , the radial o.d.e. admits the following pairs of mode solutions (u 1 , u 2 ) and (v 1 , v 2 ), where u 1 and u 2 are solutions to (4.3) satisfying u 1 = e iωr * + O (r − r + ) and u 2 = e −iωr * + O (r − r + ) as r * → −∞. Similarly, v 1 and v 2 satisfy v 1 = e iωr * + O (r − r − ) and v 2 = e −iωr * + O (r − r − ) as r * → +∞. Now, for ω = 0, the transmission and reflection coefficients T(ω, ) and R(ω, ) are defined as the unique coefficients satisfying See [42] for more details. In the following we will state the uniform boundedness of T(ω, ) and R(ω, ) for |ω| ≥ ω0 2 . In [42,Proposition 4.7,Proposition 4.8] this has been proven for Λ = 0. However, the proof of Proposition 4.7 and Proposition 4.8 in [42] also applies if we include a non-vanishing cosmological constant. 9 Another result which we will use from [42] is the representation formula for ψ in the separated picture. It is essential that |ω| ≥ ω0 2 to apply the same steps as in [ where and Proof of Lemma 5.2. This proof is very similar to [42, Proof of Propostion 5.1] so we will be rather brief. Let ψ as in (3.5). Since the expansion in spherical harmonics converges pointwise, it suffices to prove (5.5) for ψ m := ψ , Y m S 2 Y m for fixed m, . Now, define u[ψ m ] as in (4.2) such that in view of |ω| ≥ ω0 2 , we conclude that ω → a(ω, m, ) is in L 1 (R) for fixed , m. Recall that the Wronskian W(f, g) := f g − f g is independent of r * for two solutions of the radial o.d.e. (4.3). We have also used that u 2 L ∞ 1 and u 2 L ∞ 1 + |ω| for |ω| ≥ ω0 2 (cf. [42,Propostion 4.7 and Proposition 4.8]). Similarly, we have that ω → b(ω, m, l) is in L 1 (R). Using We will now prove the uniform boundedness for ψ . Proposition 5.3. Let ψ be as defined in (3.5). Then, (5.12) Proof. We start with the representation of ψ as in (5.5 Finally, on the right hand side of (5.13) we only see the commuted T -energy flux. An application of the T -energy identity in the exterior and an energy estimate in a compact spacetime region shows that the commuted T -energy flux on the event horizon is controlled from the initial data (cf. (2.21) in Theorem 2.1). Thus, in view of (5.13) we conclude |ψ (r, t, ϕ, θ)| 2 E 1 [ψ ](0) + Proposition 5.4. Let ψ be as defined in (3.5). Then, ψ is continuously extendible across the Cauchy horizon CH.
for 0 ≤ v 1 ≤ v 2 , where D[ψ] is as in (3.2). In the analogous step from (4.42) to (4.43), we estimate the spacetime integral as from which we can deduce that t → ψ(t, r, ϕ, θ) is slowly growing. Similarly, as t → −∞, we obtain the same conclusion. Now, commuting with ∂ t , the angular momentum operators W i and using elliptic estimates it follows that higher order derivatives are also slowly growing which concludes the proof. Proof. From Proposition A.2 we know that t → ψ(t, r, ϕ, θ) is a tempered distribution in the interior. Thus, ψ defined (3.5) is well defined as F −1 T [χ ω0 ] is a Schwartz function. Moreover, ψ is smooth because ψ is smooth itself and by Proposition A.2 we have that higher derivatives t → ∂ k ψ(t, r, ϕ, θ) are tempered distributions, too. Now, this also implies that ψ ∈ C ∞ (B) solves (1.3) which concludes the proof in view of ψ = ψ + ψ .
Proposition A.4. Let ψ ∈ C ∞ (M RNAdS \ CH) be defined as in (3.4). Then, there exist ψ ∈ C ∞ (M RNAdS \ CH) and ψ ∈ C ∞ (M RNAdS \ CH), two solutions of (1.3) with where χ ω0 is defined in (3.6) and ψ (t, r, ϕ, θ) = in all coordinate patches (t R A , r R A , θ R A , ϕ R A ), (t R B , r R B , θ R B , ϕ R B ) and (t B , r B , θ B , ϕ B ) in the regions R A , R B and B, respectively.
Proof. First, from Theorem 2.3 we know that ψ and all higher derivatives decay logarithmically on the exterior regions R A and R B . 12 Hence, ψ and all higher derivatives are smooth tempered distributions in the exterior regions R A and R B as functions of t R A and t R B , respectively. Thus, the Fourier projections ψ (A.9) is well-defined in R A and R B and it follows by Lebesgue's dominated convergence that ψ is a smooth solution of (1.3). Moreover, from Corollary A.3 we deduce that ψ is also a well-defined smooth solution of (1.3) in the interior B. Finally, ψ , defined a priori only in R A , R B and B, extends to a smooth solution of (1.3) on M RNAdS \ CH. This follows from using regular coordinates near the respective event horizons (outgoing Eddington-Finkelstein coordinates (v, r, θ, ϕ), where v(t, r) = t + r * , r(t, r) = r, θ = θ, ϕ = ϕ near H A and ingoing Eddington-Finkelstein coordinates near H B ) and writing ψ again as a convolution in this coordinate system ψ = 1 √ 2π F −1 T [χ ω0 ] * ψ. Note that T = ∂ v in this coordinate system. This concludes the proof in view of ψ = ψ + ψ .
Proof. The proof follows the same lines as the proof Proposition A.2 with the difference that we have exponential decay on the event horizon where D[ψ] is as in (3.2). Note that (A.10) follows from [38,Section 12]. Analogously to the proof of Proposition A.2 we can propagate this decay to any {r = const.} hypersurface in the interior. As before, by commuting with ∂ t and W i as well as using elliptic estimates, we see that on {r = const.}, ψ and higher derivatives ∂ k ψ decay exponentially towards both components of i + . This concludes the proof.