Linear Waves in the Interior of Extremal Black Holes II

We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr–Newman black holes. These results are the analogues of results obtained in Gajic (Commun Math Phys 353(2), 717–770, 2017) for extremal Reissner–Nordström and stand in stark contrast to previously established results for the subextremal case, where the local energy was shown to generically blow up at the Cauchy horizon.

For result (A), we considered Cauchy initial data for φ on an asymptotically flat spacelike hypersurface intersecting the event horizon, which decay suitably fast towards spacelike infinity. For results (B), (C) and (D) we imposed stronger decay estimates in affine time on φ and its tangential derivatives along the event horizon than those that had previously been established in [6,7] for φ arising from Cauchy data. The required decay estimates have been obtained in [3] for suitable Cauchy data. For result (E) we assumed more precise asymptotics of φ along the event horizon, which are motivated by the numerical results in [27] and have not yet been shown to hold for φ arising from generic, suitably decaying Cauchy data in a mathematically rigorously setting.
In this paper, we shall prove the analogues of (A), (B) and (C) for axisymmetric solutions φ to (1.1) in the black hole interior of extremal Kerr-Newman spacetimes; see Theorem 1-4 below. The Kerr-Newman spacetimes are a three-parameter family, characterised by a mass M , a rotation parameter a and a charge e [31]. Extremal Kerr-Newman spacetimes constitute a twoparameter subfamily of spacetimes, satisfying the constraint M 2 = a 2 + e 2 ; they can be viewed as a continuous family that connects the extremal Reissner-Nordström solutions (a 2 = 0) to the extremal Kerr solutions (a 2 = M 2 ). For an overview of the geometry of Kerr-Newman spacetimes, see [10].
In [8], polynomial decay in affine time of axisymmetric φ and its tangential derivatives was shown to hold along the event horizon of extremal Kerr (a 2 = M 2 ) for suitably decaying Cauchy initial data. To obtain the analogue of (A) for axisymmetric φ in the extremal Kerr interior, we will assume the decay rates that follow from [8]. For the analogue of (A) for axisymmetric φ in extremal Kerr-Newman spacetimes with a 2 < M 2 and, moreover, for the analogues of (B) and (C) for axisymmetric φ in any extremal Kerr-Newman spacetime, Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4007 we assume polynomial decay in time of φ along the event horizon that is conjectured, but has not yet been proved, to hold. Note that the methods involved in proving results (D) and (E) rely fundamentally on the spherical symmetry of φ and the background spacetime. For this reason, they do not carry over to extremal Kerr-Newman. In addition, we will show that we can drop the axisymmetry assumption on φ and prove the analogues of (A), (B) and (C) in extremal Kerr-Newman spacetimes that are sufficiently close to extremal Reissner-Nordström, i.e. with a sufficiently small rotation parameter a; see Theorems 5-7 below. We refer to this subfamily of extremal Kerr-Newman as slowly rotating extremal Kerr-Newman. We assume, again, the decay for φ along the event horizon that is expected to hold for suitably decaying Cauchy initial data in this setting. This assumption is now also necessary for the analogue of (A), as the required polynomial decay has not yet been proved to hold for φ (without axisymmetry) along the event horizon of slowly rotating extremal Kerr-Newman.
The analogue of (A) has recently been obtained for the wave equation in subextremal Reissner-Nordström (e 2 < M 2 ) [21] and subextremal Kerr (a 2 < M 2 ) [20] by Franzen (see also the results of Hintz [25] in the very slowly rotating setting, where a 2 M 2 ), whereas the analogue of (B) has been shown to fail in subextremal Reissner-Nordström for generic Cauchy data [29] by Luk-Oh. See also related results concerning instabilities in subextremal Kerr [19,30].
The results of this paper are related to Christodoulou's formulation of the strong cosmic censorship conjecture [14]. Indeed, the analogue of the H 1 loc extendibility result (B) in extremal Kerr, if also applicable in the context of the vacuum Einstein equations, would provide a construction of dynamical black hole interiors arising from perturbations of extremal Kerr spacetimes which are extendible beyond their Cauchy horizons with Christoffel symbols that are locally L 2 with respect to spacetime integration, which is precisely the regularity class considered by Christodoulou. As such, the corresponding initial data would not lie in the class of initial data to which the strong cosmic censorship applies (and would therefore certainly be expected to be non-generic). See also related conjectures in the introduction of [23].

Linear Waves in the Exterior Region of Extremal Kerr
We will review in this section several results for the wave Eq. (1.1) in the exterior region of extremal Kerr.
Aretakis considered in [8] axisymmetric solutions φ to (1.1) in the exterior region of extremal Kerr, arising from Cauchy data on a spacelike hypersurface Σ intersecting the event horizon H + ; see Fig. 1. He established polynomial decay in time for φ everywhere in the exterior, including along H + .
In [4], he, moreover, proved the existence of conserved quantities, the Aretakis constants, along H + for solutions φ (that need not be axisymmetric). If non-vanishing, these constants constitute an obstruction to the decay of either φ itself or its transversal derivative. Since axisymmetric solutions φ have been shown to decay along H + , this means that, generically, their transversal derivatives cannot decay. Furthermore, higher-order transversal derivatives will generically blow up in infinite time along H + . These non-decay and blow-up results have been dubbed "the Aretakis instability" in the literature [27].
Lucietti-Reall generalised the Aretakis constants to higher-spin equations in extremal Kerr in [28]. In particular, they showed that conserved quantities also form an obstruction to the decay of solutions to the Teukolsky equation, which governs the evolution of perturbations of certain components of the curvature tensor in the context of the linearised Einstein equations.
In [9] Aretakis extended the results of [8] to show non-decay and blow-up of higher-order derivatives of φ even in the case of data with vanishing Aretakis constants. There is still no proof of pointwise and energy boundedness or decay for non-axisymmetric φ in the exterior region of extremal Kerr (cf. a complete picture of the boundedness and decay properties of the linear wave equation in the full subextremal range of Kerr-Newman) spacetimes has recently been obtained in [15,17]).
The main difficulty when studying non-axisymmetric φ in the exterior of extremal Kerr is that the geometric phenomena of superradiance, the trapping of null geodesics and the degeneration of the local red-shift effect at H + are strongly coupled (in contrast with the subextremal case); see the discussion in the introduction of [8] for more details. Based on numerical studies of quasinormal modes on extremal Kerr [2,12], the expectation is that φ with a fixed azimuthal number m = 0 arising from initial data supported away from H + will decay slower than axisymmetric φ (for which m = 0) both away from H + and along H + .

Linear Waves in the Interior Region of Extremal Kerr-Newman
In this section, we will give an overview of the main theorems proved in this paper; we will state more detailed versions of the theorems in Sect. 3. In Sect. 2 we will give the precise definitions of the spacetime regions of interest in extremal Kerr-Newman that are mentioned in the paragraphs below and we will present the construction of double-null coordinates that cover these regions.
In this paper, we will restrict to a spacetime rectangle D u0,v0 , which is a subset of M ∪ CH + , where M denotes the extremal Kerr-Newman manifold and CH + is the inner horizon of extremal Kerr-Newman. We take D u0,v0 to be the intersection of the causal future of the event horizon segment H + ∩{v ≥ v 0 } and the causal past of the inner horizon segment CH + ∩{u ≤ u 0 }, with respect to the manifold-with-boundary M∪CH + , where v 0 and u 0 are chosen suitably, such that restriction of D u0,v0 to the interior region is entirely contained within the domain of the (u, v) Eddington-Finkelstein-type double-null coordinates; see Fig. 1. Note that we have defined D u0,v0 to include a segment of CH + .
We can employ an ingoing null coordinate U (u) in M∩D u0,v0 , which can be extended across H + , and an outgoing null coordinate V (v), which can be extended beyond CH + , to express D u0,v0 as the following set: where U = 0 at H + and V = 0 at CH + .
We equip H + and H v0 , the ingoing null hypersurface in D u0,v0 which is a subset of {v = v 0 }, with characteristic initial data for the wave Eq. (1.1).
We can also consider solutions φ arising from Cauchy initial data for (1.1) on an asymptotically flat spacelike hypersurface Σ in extremal Kerr-Newman. We will choose a hypersurface Σ that has a non-trivial intersection with the black hole interior; see the discussion in [5] for why this is a natural choice. As a consequence of the geometry of the interior of extremal Kerr-Newman, Σ must be incomplete; see Fig. 1. We restrict to the future domain of dependence of Σ, which we denote by D + (Σ). The inner horizon CH + contains part of the boundary of D + (Σ), so we will sometimes refer to CH + as the Cauchy horizon. Theorem 1 (L ∞ boundedness and C 0 extendibility for axisymmetric solutions). Let φ be an axisymmetric solution to (1.1) in extremal Kerr-Newman arising from suitably regular characteristic initial data on H v0 ∪ H + , such that for some > 0, where / ∇ denotes derivatives tangential to 2-spheres S 2 −∞,v that foliate H + . Then, there exists a constant C = C(M, a, ) > 0 and a natural norm D 0 > 0 on initial data for φ, such that Theorems 3.5 and 3.6 together form a more precise version of Theorem 1. In view of the decay results along H + in [8] for φ in extremal Kerr arising from Cauchy initial data on a spacelike hypersurface Σ, we can reformulate Theorem 1 if we restrict to the subfamily of extremal Kerr spacetimes, where we consider suitably regular Cauchy data along Σ in accordance with the results of [8]: Theorem 2 (L ∞ boundedness and C 0 extendibility for axisymmetric solutions in extremal Kerr). Let φ be an axisymmetric solution to (1.1) in extremal Kerr arising from suitably regular and decaying data on Σ. Then, there exists a constant C = C(M, Σ) > 0 and a natural norm D 0 > 0 on initial data for φ, such that |φ| ≤ CD 0 , everywhere in D + (Σ). Moreover, φ admits a C 0 extension beyond CH + .
Then, φ admits an extension beyond CH + that is H 1 loc with respect to spacetime integration.
In [23] we reformulated Theorem 2 of [23] by imposing Cauchy data on a spacelike hypersurface instead of characteristic data on the event horizon to obtain Theorem 3 of [23]. We made use of the improved decay results along the event horizon of extremal Reissner-Nordström that are proved in [3]. However, as the decay estimates for φ along H + that are necessary for (1.2) to hold have not yet been obtained for suitable data on Σ in any extremal Kerr-Newman spacetime with a = 0, we cannot yet reformulate Theorem 3 above by imposing Cauchy data on Σ. 1 Theorem 3 follows from Theorem 3.2 after applying the estimates (1.3) and (2.19).
We can further conclude that φ can be extended beyond CH + in the Hölder space C 0,α with α < 1. This result is the analogue of Theorem 5 of [23].
The precise necessary initial decay requirements along H + appear in Theorem 3.7.

Slowly Rotating Extremal Kerr-Newman.
We now restrict to the slowly rotating subfamily of extremal Kerr-Newman spacetimes, satisfying 0 ≤ |a| < a c , where a c is the parameter described above. In particular, this subfamily excludes extremal Kerr. We will state analogues of the results from Sect. 1.2.1 in slowly rotating extremal Kerr-Newman without the restriction to axisymmetric solutions of (1.1).
In slowly rotating extremal Kerr-Newman we can obtain L ∞ boundedness and C 0 extendibility without an axisymmetry assumption on φ.
1 For Kerr-Newman spacetimes with a 2 < a 2 c , the Hawking vector field, which is a Killing vector field that is null along H + and is precisely defined in Sect. 2.4, will also be timelike in the exterior region in a neighbourhood of H + . We expect this geometric property would significantly simplify the difficulties in the analysis of non-axisymmetric solutions in the exterior region of slowly rotating Kerr-Newman spacetimes compared to the extremal Kerr case and would lead to better decay estimates for φ along the event horizon than expected in extremal Kerr; see also the discussion in Sect. 1.1.

Part 0: Constructing a Double-Null Foliation.
Before carrying out any estimates involving the wave equation, we first construct a suitable doublenull foliation of the interior region of extremal Kerr-Newman. As Kerr-Newman spacetimes with a = 0 are not spherically symmetric, in contrast with Reissner-Nordström spacetimes, the existence of global double-null coordinates in the interior region is not immediate. In [22], a suitable global doublenull foliation of extremal Kerr-Newman is constructed, which covers both the exterior and interior regions, following ideas of [32]. We will use the results of [22] here.

Part 1: Vector Field Multipliers and Energy Estimates (Theorem 3).
We obtain uniform bounds on weighted L 2 norms of φ along null hypersurfaces by means of energy estimates. Energy estimates are derived by using the vector field method; see for example [26] for a general overview and the discussion in Sect. 2.5 for further particulars. Energy estimates for axisymmetric φ are obtained very similarly to the energy estimates in extremal Reissner-Nordström in [23]; we use the following vector field multiplier: with p = q = 2, where u and v are double-null coordinates obtained in Part 0 that are akin to the Eddington-Finkelstein double-null coordinates in extremal Reissner-Nordström. See Sect. 2.2 for an overview of the construction and main properties of the Eddington-Finkelstein-type double-null coordinates u and v in extremal Kerr-Newman, and see Sect. 2.5 for more details regarding N p,q .
As in extremal Reissner-Nordström, the energy estimates rely crucially on the following polynomial decay rate of the g uv component of the metric in Eddington-Finkelstein-type double-null coordinates: see Sect. 2.3 for the corresponding estimates. The above bounds play an important role in the proof of Theorem 3; see Sect. 4.
If we drop the axisymmetry assumption on φ, we have to take into account additional error terms in the energy estimates; most notably, extra error terms arise that involve the non-vanishing torsion of the double-null foliation, denoted by ζ. The torsion can be expressed as a commutator, where L and L are vector fields that are tangent to null generators of the outgoing and ingoing null hypersurfaces, respectively. See Sect. 2.3.
In the a = 0 case, L and L are coordinate vector fields, so they commute, and ζ vanishes everywhere. If a = 0, ζ does not vanish. It turns out, however, that axisymmetric φ still satisfy ζ(φ) = 0 if a = 0, so the error terms involving ζ do not form an obstruction for axisymmetric φ.
In the case of non-axisymmetric φ, we can estimate the error terms involving ζ by invoking the Hawking

Part 2: Commutation Vector Fields and Pointwise Estimates (Theorems 1 and 5).
We subsequently use the uniformly bounded weighted L 2 norms from Part 1 to obtain a uniform bound for the L ∞ norm of φ everywhere in the interior and to prove continuous extendibility across CH + . For this purpose, we apply standard Sobolev inequalities on the spheres S 2 u,v corresponding to the double-null foliation, i.e. we can estimate where / ∇ denotes the covariant derivative restricted to S 2 u,v . Moreover, we apply the fundamental theorem of calculus along the null generators of ingoing null hypersurfaces, together with a (weighted) Cauchy-Schwarz inequality, to arrive at the following estimate: 3) with p > 1; see also the proof of Proposition 7.1. The second term on the right-hand side of the inequality can be controlled by a weighted energy along an ingoing null hypersurface.
In order to estimate || / ∇ k φ|| L 2 (S 2 u,v ) with k ≥ 1, we also need to consider appropriately weighted energies for angular derivatives of φ. Replacing φ by 2 An interesting question that remains open is whether a loss of derivatives in the initial energies is necessary to prevent the -loss in the a = 0 case. That is to say, whether it is possible to construct a sequence of suitably regular and decaying initial data along H + and H v 0 for which the uniform constant appearing in energy estimates without a loss of derivatives blows up as we move along the sequence. Note that this construction is not possible if we restrict to axisymmetric solutions.
Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4015 in the estimates from Part 1 results in error terms that cannot be controlled using the methods mentioned in Part 1.
Obtaining estimates for angular derivatives of φ in L 2 (S 2 u,v ) turns out not to be a problem in extremal Reissner-Nordström, as the spacetime is spherically symmetric, which means that the angular momentum operators O i with i = 1, 2, 3, which are Killing vector fields generating the isometries of spherical symmetry, control all derivatives tangential to the round spheres of the double-null foliation; see for example Sect. 2.1 of [23] for explicit expressions of O i with respect to spherical polar coordinates. Since the vector fields O i are Killing, they commute with the operator g , so the functions O i (φ) are also solutions to (1.1). Any energy estimate for φ therefore automatically holds for In extremal Kerr-Newman with a = 0, however, the only angular momentum operator that remains a Killing vector field is Φ, the generator of rotations about the axis of symmetry. Fortuitously, there exists a second-order operator Q, the Carter operator, which also commutes with g . This operator is closely related to the conserved Carter constant; see [11]. See also Andersson-Blue [1], for example, for more details on the Carter constant and operator, and for applications of the commutation property of Q.
The operator Q, together with the vector fields Φ and T , the Killing vector field corresponding to time-translation symmetry, controls the derivatives of φ that are tangent to the spheres of the Boyer-Lindquist foliation of Kerr-Newman. To obtain control over derivatives tangent to the spheres S 2 u,v , (which do not coincide with the Boyer-Lindquist spheres if a = 0), we need to additionally commute g with the vector fields L and L.
In contrast with the error terms arising from commuting g with / ∇, or ∂ ϑ A , the error terms corresponding to a commutation with L and L can be controlled via the methods of Part 1 by using profusely the Killing property of Q, Φ and T ; see the estimates in Sect. 6.2. As a result, we are able to prove Theorems 1 and 5; see Sects. 7.1 and 7.2.

Part 3: Decay Estimates (Theorems 4, 6 and 7).
In the final step, we consider the difference function ψ = φ − φ| H + , such that ψ vanishes along H + . The function ψ has the advantage that it can be shown to decay uniformly in u. By treating the wave equation as a transport equation for Lφ along ingoing null generators, we can use the u-decay of ψ to obtain v-decay of ||Lφ|| L 2 (S 2 with the rate v −2+ , for any > 0. If φ is axisymmetric, we can in fact improve this decay rate to v −2 log(v). By commuting further with L and L and applying standard Sobolev inequalities on S 2 u,v , this allows one to obtain pointwise decay for |Lφ| with the rates v −2+ and v −2 log(v), respectively.
The outgoing derivative corresponding to double-null coordinates that cover the region beyond CH + in the maximal analytic extension of extremal Kerr-Newman, denoted by ∂ V , is related to L as follows: Ann. Henri Poincaré Since we cannot remove the in the decay rate v −2+ of |Lφ|, we are unable to infer boundedness of ∂ V φ at CH + or C 1 extendibility of φ at CH + . We can, nevertheless, infer that φ is extendible as a C 0,α function beyond CH + , for any α < 1, if the initial data along H v0 ∪ H + are suitably regular and decaying, thereby proving Theorems 4 and 7; see Sect. 7.3. Moreover, we can integrate the v-decaying L 2 (S 2 u,v ) norm of Lφ in the v-direction, in slowly rotating extremal Kerr-Newman, to obtain boundedness of Hu v 2 (Lφ) 2 and also H v v 2 Ω 2 | / ∇φ| 2 . In this way we get rid of some of the -loss in the weights that was present in the energy estimates of Part 1 and arose from the obstruction of ζ to the energy estimates for φ without the axisymmetry assumption. This improvement comes at the expense of requiring decay of higher-order derivatives in the initial data, compared to the estimates in Part 1. See Sect. 7.3 for more details. In particular, we can infer Theorem 6.

Outline
In Sect. 2 we introduce some notation and state estimates relating to the double-null foliation of the interior of extremal Kerr-Newman (Part 0 of Sect. 1.3) that are relevant in the rest of the paper. We state the theorems that are proved in the paper in Sect. 3. We prove energy estimates for axisymmetric solutions φ to (1.1) in extremal Kerr-Newman in Sect. 4. Subsequently, we prove energy estimates in slowly rotating extremal Kerr-Newman in Sect. 5, completing Part 1 of Sect. 1.3. In Sect. 6, we commute with L and L to arrive at energy estimates for higher-order derivatives of φ. Finally, we use the higherorder energy estimates to prove pointwise estimates of φ (Part 2 of Sect. 1.3). Moreover, we obtain pointwise decay in v of Lφ in Sect. 7 by making use of higher-order energy estimates, completing Part 3 of Sect. 1.3.

The Geometry of Extremal Kerr-Newman
We will first introduce the extremal Kerr-Newman spacetimes in Boyer-Lindquist and Kerr-star coordinates and subsequently present more convenient double-null coordinates, by foliating the spacetime with suitable ingoing and outgoing null hypersurfaces, covering both the exterior and interior regions of extremal Kerr-Newman. Sections 2.2 and 2.3 are based on a more elaborate discussion on double-null foliations of Kerr-Newman that can be found in [22].

Boyer-Lindquist and Kerr-star Coordinates
Fix the mass parameter M > 0 and the rotation parameter a ∈ R, such that |a| ≤ M , and let, moreover, the charge parameter e satisfy e 2 = M 2 − a 2 .

D. Gajic
Ann. Henri Poincaré We can similarly introduce outgoing Kerr-star coordinates In these coordinates it is easy to see that M int can be smoothly embedded into a bigger spacetime M , by patching M int to a spacetime M ext that is isometric to M ext . The manifold M in is embedded in the patched spacetime as the region {0 < r < M} and M ext is embedded as the region {r > M}. The corresponding boundary {r = M } of M int and M ext in the patched spacetime lies in the causal future of M int and is denoted by CH + . We refer to this boundary as the inner horizon. We can write M = M int ∪ M ext ∪ CH + , or As M ext is isometric to M ext , we can repeat the above procedure ad infinitum to extend the manifold M∪M further and form an infinite sequence of patched manifolds containing regions isometric to either M ext or M int , glued across horizons. The resulting spacetime M is called maximal analytically extended extremal Kerr-Newman, and it is depicted in Fig. 2. For the remainder of this paper we will, however, mainly direct our attention to the subset M ∪ CH + .

Double-Null Coordinates
In the sections below, we will consider energy fluxes along ingoing and outgoing null hypersurfaces in M. It is therefore more natural to work in double-null coordinates in M rather than Kerr-star coordinates.
We first consider M int , covered by Kerr-star coordinates. If we can construct a tortoise function r * to be of the form r * (r, θ), such that the functions then the level sets {u = constant} and {v = constant} are null hypersurfaces. We will follow a construction of r * that was introduced by Pretorius-Israel in [32] and allows for suitable, double-null coordinates. We will assume that a = 0. In the a = 0 case we consider Eddington-Finkelstein double-null coordinates; see Sect. 2 of [23].
In [22] the construction of r * from [32] is used to extend the local doublenull coordinates in M int to obtain a smooth, global Eddington-Finkelstein- are diffeomorphic to 2-spheres and we, moreover, obtain quantitative bounds on the metric components in double-null coordinates (see Sect. 2.3).
The metric g on M int ∩ {r > e 2 2M } can then be written as follows: and ϕ * ∈ (0, 2π). The metric components in (2.5) are given by they are of the same form as the metric components with respect to the doublenull coordinates considered in [18]. A precise definition of the functions f i is given in [22], but for the purposes of this paper we only need the estimates on the metric components that are stated in Sect. 2.3 and are derived in [22].
As we approach H + along constant v hypersurface, the coordinate u goes to −∞. We can, however, introduce a rescaled ingoing null coordinate in order to further extend the double-null coordinates and additionally cover the region Fix v 0 ∈ R and define the function U : . We can interpret U as a smooth, negative function U : In [22] it is shown function U : M int ∩ {r > e 2 2M } → R extends smoothly with respect to Kerr-star coordinates to the bigger manifold M ∩ {r > e 2 2M }, such that U = 0 along H + and U < 0 in M ext and moreover, the metric is well defined and non-degenerate with respect to the chart (U, v, θ * , ϕ * ) on M ∩ {r > e 2 2M }: By introducing another function f 5 (see [22]) we can shift the angular coordinate ϕ * to a new coordinate ϕ * ∈ (0, 2π) and the metric can be written in (ũ,ṽ, θ * , ϕ * ) coordinates, To distinguish these coordinates from the previous double-null coordinates, we have denoted them with tildes (ũ,ṽ, ϑ), whereṽ = v,ũ = u and θ * = θ * . Now, fix u 0 ∈ R and define the function V : . In [22] it is shown that we can extend V as a smooth function to the bigger manifold M ∩ {r > e 2 2M }, such that V = 0 along CH + and V > 0 in M ext and moreover, the metric is well defined and non-degenerate with respect to the chart (ũ, V , θ * , ϕ * ) on M ∩ {r > e 2 2M }: We will use the notation for points on H + and CH + , respectively, for the sake of convenience. These points lie in the domain of either the (U, v) or (ũ, V ) double-null coordinates.
We will consider the following null hypersurfaces: and we refer to the hypersurfaces H v and H u as ingoing and outgoing null hypersurfaces, respectively.
We will fix |u 0 | and v 0 to be suitable large such that Consider the null vector fields L and L, which are tangent to the generators of the outgoing and ingoing null hypersurfaces, respectively, and satisfy Lv = 1 and Lu = 1.
The vector field L can be naturally expressed in the chart (u, v, θ * , ϕ * ). Indeed, Note that we can alternatively express L in (U, v, θ * , ϕ * ) coordinates: From the above expression it is clear that L can be extended smoothly as a vector field across H + (where it vanishes). The vector field L can similarly be expressed in the chart (ũ, v, θ * , ϕ * ): Note that we can also express L in (u, V , θ * , ϕ * ) coordinates: From the above expression it is clear that, analogously to L, L can be extended smoothly as a vector field across CH + (where it vanishes).

Estimates for Metric Components and Connection Coefficients in Double-Null Coordinates
In this section, we will present an overview of relevant estimates for the metric components g αβ in Eddington-Finkelstein-type double-null coordinates, their derivatives and components (and derivatives) of the Jacobian matrix relating Eddington-Finkelstein-type double-null coordinates to Boyer-Lindquist coordinates. All these estimates are obtained in [22]. We first define the following notation to separate out leading-order terms in v + |u|.
We obtain in [22] the following estimates for the metric components g αβ in M int ∩ {r > r 0 > e 2 2M } in the Eddington-Finkelstein-type coordinates (u, v, θ * , ϕ * ) introduced above: Theorem 2.1 (Estimates for metric components in double-null coordinates, [22]). , and estimate for n ≤ N , with N ∈ N, We define the connection coefficients , where A = 1, 2 and e 3 = Ω −1 L and e 4 = Ω −1 L are renormalised null vector fields, such that g(e 3 , e 4 ) = −2. We have the following relations between connection coefficients and metric derivatives: where we also have that See "Appendix A" for the derivations of the above identities and for further properties the connection coefficients and their expressions in terms of derivatives of g αβ . Theorem 2.2 (Estimates for connection coefficients in double-null coordinates, [22]).
where we made use the following notation (ii) Moreover, we can expand

Killing Vector Fields
The vector field T = it corresponds to time-translation symmetry in extremal Kerr-Newman. Note that T is not causal everywhere in M. The subset of M ext in which T is not causal is called the ergoregion. Similarly, there is a subset of M int in which T fails to be causal everywhere (cf. T is timelike everywhere away from the horizons in extremal Reissner-Nordström, where a = 0). We denote the Killing vector field corresponding to axial symmetry in extremal Kerr-Newman by Φ. In Kerr-star coordinates, we can write Φ = ∂ ∂(ϕKS) * . However, we can also write Φ = ∂ ϕ * in Eddington-Finkelstein-type double-null coordinates, or Φ = ∂ ϕ in Boyer-Lindquist coordinates.
The Carter operator is a second-order differential operator that can be expressed as follows: where Δ S 2 is the Laplacian with respect to the metric on the round sphere (of area radius 1). Since T and Φ are Killing vector fields, we have that It turns out that the Carter operator also commutes with the wave operator: See [1] for a derivation of the above commutator identity.
We can define the Hawking vector field H in D u0,v0 by We can also express H by as a linear combination of the Killing vector fields T and Φ, where In the literature, the constant ω H + is commonly referred to as the angular velocity of the Kerr-Newman black hole.
In order for the energy fluxes with respect to H along null hypersurfaces to be non-negative definite, we need H to be causal. We have that The maximum value of R 2 sin 2 θ is obtained at θ = π 2 , Consequently, by applying the estimates in Theorem 2.1, we obtain 1], to obtain an equivalent inequality: One can solve the above cubic equation to obtain 0 < a c (M ) < M, such that g(H, H) < 0 for all 0 ≤ |a| < a c and v + |u| suitably large.
We define slowly rotating extremal Kerr-Newman spacetimes to be the subfamily of extremal Kerr-Newman spacetimes satisfying 0 ≤ |a| < a c . Note that extremal Kerr (|a| = M ) is not a slowly rotating extremal Kerr-Newman spacetime.

The Divergence Theorem and Integration Norms
In this section, we will introduce some basic notation regarding integration in M∩D u0,v0 . We will state the divergence theorem, which is the main ingredient of the vector field method; see also the discussion in Sect. 1.3 of [23].
Let V be a vector field in a Lorentzian manifold (N , g). We consider the stress-energy tensor T[φ] corresponding to (1.1), with components denote the energy current corresponding to V , which is obtained by applying V as a vector field multiplier, i.e. in components An energy flux is an integral of J V [φ] contracted with the normal to a hypersurface with the natural volume form corresponding to the metric induced on the hypersurface. We apply the divergence theorem to relate the energy flux along the boundary of a spacetime region to the spacetime integral of the divergence of the energy current J V . If the boundary has a null segment, there is no natural volume form or normal; these are assumed compatible with the divergence theorem. That is to say, if we take 2M } gives the following identity: Here, we introduced the following notation: for vector fields V and W . Moreover, in the notation on the left-hand side of (2.14), we integrate over spacetime with respect to the standard volume form, i.e. let f : M ∩ D u0,v0 → R be a suitably regular function and U an open subset of M, then where det / g is expressed in (2.6). When integrating over H u and H v we used the following convention in the notation on the right-hand side of (2.14): Note that by a change of variables we can alternatively express the above integrals in terms of (U, θ * , ϕ * ) or ( V , θ * , ϕ * ) coordinates, respectively: In the notation of [13] we decompose the divergence term appearing in (2.14) in the following way: We can estimate in M∩D u0,v0 , with |u 0 |, v 0 ≥ 1 without loss of generality, We rewrite the estimates above by using the following notation: Let use introduce the following natural L 2 norms: Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4029 Now consider a compact subset K ⊂ M ∩ {r > e 2 2M }, such that, moreover, K ⊂ D u0,v0 . Then, we define the following spacetime L 2 norms: where / ∇ denotes the induced covariant derivative on S 2 u,v . We can, in particular, estimate We define the weighted null-directed vector field N p,q in M int ∩ D u0,v0 as follows: with 0 ≤ p, q ≤ 2. In particular, in (U, v, ϑ) coordinates, we can express If p ≤ 2, N p,q can be extended as a smooth vector field across H + into M ext . In (ũ, V , ϑ) coordinates, we have that If q ≤ 2, N p,q can be extended as a smooth vector field beyond CH + in M ext . The energy currents with respect to the constant u and constant v null hypersurfaces are given by where we inserted the expressions for T αβ from "Appendix A". In "Appendix A" we show that the current K Np,q , compatible to J Np,q , is given by In extremal Kerr-Newman spacetimes with |a| < a c , we consider, moreover, the vector field Y p in the region if |u 1 | and v 1 are chosen suitably large. We use, moreover, that H is a Killing vector field to easily obtain an expression for K Yp , where non-negativity, in the case that φ is not axisymmetric, follows from the timelike character of H.

Precise Statements of the Main Theorems
In this section we present more precise versions of the main results proved in this paper, which are stated in Sect. is axisymmetric, the extension φ to M int ∪ H + ∩ D u0,v0 must also be axisymmetric.
The above proposition can be proved by reducing the characteristic initial value problem to a Cauchy problem with initial data on a spacelike hypersurface, as done in [33], and then appealing to a global existence and uniqueness result for the standard Cauchy problem for linear wave equations; see for example [16]. Observe that Proposition 3.1 does not provide any information about the asymptotic behaviour of φ towards CH + . We will state in the subsections below further quantitative and qualitative properties of φ, relating to boundedness and extendibility of φ and its derivatives beyond CH + , under the assumption of suitable additional decay requirements along H + .

Energy Estimates Along Null Hypersurfaces
Consider solutions φ to (1.1) that arise from the characteristic initial data in Proposition 3.1. We will first show that we can prove boundedness of weighted L 2 norms for φ along null hypersurfaces, under additional assumptions on suitable initial L 2 norms along H + . We will treat separately the case of axisymmetric solutions φ on extremal Kerr-Newman with 0 ≤ |a| ≤ M , and the case of general solutions φ on slowly rotating extremal Kerr-Newman, with 0 ≤ |a| < a c . In the next section, we will give an overview of the theorems regarding L ∞ estimates for φ. Unless specified differently, we consider (1.1) on an extremal Kerr-Newman background with 0 ≤ |a| ≤ M . Theorem 3.2. Take 0 < q ≤ 2. Let φ be a solution to (1.1) corresponding to axisymmetric initial data from Proposition 3.1 satisfying Then, there exists a constant C = C(a, M, u 0 , v 0 , q) > 0 such that for all Theorem 3.2 is proved in Proposition 4.2. Theorem 3 follows immediately by using, moreover, Theorem 3.5 below and the estimate (2.19). Note, moreover, that by using (2.15) and (2.17) one can easily see that the assumption of φ along H v0 is certainly satisfied if φ is smooth along H + with respect to (U, θ * , ϕ * ), which, in particular, is the case if one considers φ arising from smooth initial data along a hypersurface Σ intersecting H + . Theorem 3.3. Let φ be a solution to (1.1), with |a| < a c , corresponding to initial data from Proposition 3.1 satisfying Let 0 ≤ p < 2 and let > 0 be arbitrarily small. Then, there exists a constant C = C(a, M, u 0 , v 0 , p, q, ) > 0, such that for all H u and H v , for η > 0 arbitrarily small. Then, there exists a constant C = C(a, M, v 0 , u 0 , η) > 0 such that, Theorem 3.4 is proved in Corollary 7.7. Theorem 6 now follows from Theorem 3.4, combined with Theorem 3.5 below and the estimate (2.19).

Pointwise Estimates and Continuous Extendibility Beyond CH +
We can use the energy estimates in the subsection above to obtain L ∞ estimates in M ∩ D u0,v0 , and we can, moreover, show that φ is continuously extendible beyond CH + . Here, we treat the restriction to axisymmetric φ and the restriction to slowly rotating extremal Kerr-Newman simultaneously.
Theorem 3.5. Either take 0 < p < 2 and let φ be a solution to (1.1), with |a| < a c , corresponding to initial data from Proposition 3.1 without any symmetry assumptions, or take 0 ≤ p ≤ 2 and let φ be a solution to (1.1), with 0 ≤ |a| ≤ M , corresponding to axisymmetric initial data from Proposition 3.1.
Assume that, for > 0 arbitrarily small, Then, there exists a constant C = C(a, M, v 0 , u 0 , ) > 0 such that Assume furthermore that for some q > 1.
Then, φ can be extended as a C 0 function beyond CH + .
Theorem 3.6 is proved in Proposition 7.2. We can infer Theorems 1 and 5 from Theorems 3.5 and 3.6.
As Theorem 2 is formulated in terms of Cauchy initial data for φ on an asymptotically flat hypersurface Σ in extremal Kerr, we also need to appeal to the decay estimates in the exterior of extremal Kerr. In particular, boundedness of a non-degenerate energy and τ −1− -decay of the (degenerate) T -energy for axisymmetric solutions, with respect to a suitable spacelike foliation Σ τ of the extremal Kerr exterior, which are proved in Theorems 2 and 3 of [8], are sufficient to show that (3.1) holds for suitable Cauchy data for φ, so that Theorem 2 can be viewed as a corollary of Theorem 1.
Finally, we obtain v-decay estimates for S 2 u,v (Lφ) 2 dμ / g , which are needed to show that φ can be extended as a C 0,α (with α < 1) across CH + : Theorem 3.7. Let φ be a solution to (1.1) corresponding to initial data from Proposition 3.1 without any symmetry assumptions. Let k ∈ N 0 and denote D 2k := j 1 +j 2 +2j 3 +j 4 ≤2k

Assume that
(i) Let |a| < a c and assume also that Then, we can estimate for > 0 arbitrarily small. Then, we can estimate (iii) Either restrict to axisymmetric data in from Proposition 3.1, or let |a| < a c . Assume that Then, φ can be extended in C 0,α , for all α < 1.

Energy Estimates for Axisymmetric Solutions
We will first restrict to axisymmetric solutions to (1.1) on extremal Kerr-Newman spacetimes with 0 ≤ |a| ≤ M . In this section we will always use φ to denote a solution to (1.1), with 0 ≤ |a| ≤ M , corresponding to axisymmetric initial data from Proposition 3.1.
We will frequently make use of a Grönwall-type lemma.
By applying a (weighted) Cauchy-Schwarz inequality, we can further estimate for η > 0, Similarly, by reversing the roles of u and v, we obtain (4.5) Recall from (ii) of Theorem 2.2 that we can expand Consequently, we can rewrite (4.5) to obtain K Np,q (4.6) First, let 0 ≤ p < 2. Then, the term between square brackets in front of | / ∇φ| 2 will become positive in the region |u| > v, as we approach H + , which means that K Np,q angular will be negative, and we are not able to control it. We therefore restrict to p = 2.
If p = 2 and q < 2, the term inside the square brackets is negative for suitably large v, so we can estimate If p = 2 and q = 2, a cancellation occurs in the leading-order terms between square brackets, so we can estimate K Np,q If we fix p = 2, we can therefore estimate for all 0 ≤ q ≤ 2, with > 0 arbitrarily small and C = C (M, u 0 , v 0 , ) > 0. We will fix 0 < < 1.
We combine the estimates above for K N2,q null and K N2,q angular to obtain, for 0 ≤ q ≤ 2, Finally, we can apply Lemma 4.1 with the choices where we use that h and k are integrable for 0 < η < min{q, 1} and 0 < < 1, to arrive at the estimate in the proposition. We therefore need the restriction q > 0 if p = 2.
We have now proved Theorem 3.2.

Energy Estimates in Slowly Rotating Extremal Kerr-Newman
We now drop the axisymmetry assumptions on solutions to (1.1) on extremal Kerr-Newman. We do, however, restrict to the subfamily of slowly rotating extremal Kerr-Newman spacetimes, with 0 ≤ |a| < a c ; see Sect. 2.5. In this section we will always use φ to denote a solution to (1.1), with 0 ≤ |a| < a c , corresponding to initial data from Proposition 3.1 without symmetry assumptions.
Even without an axisymmetry assumption on φ, we can still obtain energy estimates with respect to vector fields N p,q if we restrict to subsets of D u0,v0 with a finite spacetime volume. We introduce the hypersurfaces γ α and γ β , with α ≥ 1 and β ≥ 1, such that We define f α (u, v) as follows: We define f β (u, v) as follows: where v 1 is taken suitably large, such that −g(df β , df β ) ≥ C, for v > v 1 , with C > 0 a constant. Moreover, we can choose h β such that f β is a smooth function on (−∞, u 0 ] × [v 0 , ∞) and for all (u, v) such that h β (u, v) = 0, we can uniformly bound −g(dh β , dh β )(u, v) ≥ C.  Consequently, γ α and γ β are spacelike hypersurfaces. Denote See Fig. 3. It is easy to verify that the spacetime volumes of A and A are finite, if we take α > 1 and β > 1.

Energy Estimates in AA
We first consider energy estimates with respect to N p,q . Since φ is no longer assumed to be axisymmetric, K mixed [φ] does not necessarily vanish. To deal with a non-vanishing K mixed [φ], we first consider the regions A and A of finite spacetime volume.
By applying Cauchy-Schwarz, we can further estimate, for η > 0, where we used that v ≤ C|u| 1 α in J − (γ α ), for some constant C > 0. Similarly, we apply Cauchy-Schwarz to estimate Combined with the estimates from Proposition 4.2 for K Np,q angular and K Np,q null , we can apply Lemma 4.1 with where we use that h and k are integrable for 0 < η < min{q, 1, 2 − q α , α − 1} and 0 < < 1. Since we assumed that α > 1, we obtain the estimate in (i).
Consider now the region A.
Furthermore, we can actually improve the estimate for K Np,q angular from Proposition 4.2 when restricted to A, by including the cases 0 < p ≤ 2, with 0 ≤ q < 2. This improvement will in fact be necessary to prove Proposition 5.3.
Indeed, we can estimate where in the second inequality we used that (q − 2)v + q|u| < 0 in A if v + |u| is suitably large and q < 2, which follows from the inequality |u| < 2−q q v, which holds in A if v + |u| is suitably large and β > 1.

Energy Estimates in B
We are left with proving a suitable energy estimate in the region B. In Kerr-Newman spacetimes with 0 ≤ |a| < a c , we can obtain an energy estimate away from H + with respect to the vector field Y p , defined by if we restrict to a region {v ≥ v 1 }, where v 1 is taken suitably large, such that M − r is sufficiently small, so as to ensure that Y p is a causal vector field everywhere in B ∩ {v ≥ v 1 }; see the discussion in Sect. 2.4.  C(a, M, u 0 , v 0 , v 1 , p) > 0 such that Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4041 where non-negativity in the case that φ is not axisymmetric requires that H is causal. Moreover, there exists a constant C > 0 such that we can estimate If we apply the divergence theorem in the region B ∩ {v ≥ v 1 }, the bulk term is therefore of a good sign.
We can now obtain energy estimates in the entire region D u0,v0 by combining the results from Propositions 5.1 and 5.2.
where we need β > 1, 0 ≤ p ≤ 2 and 0 < q ≤ 2, or 0 < p ≤ 2 and 0 ≤ q ≤ 2. Moreover, we need Similarly, we can estimate where p ≥ p and p ≥ qβ. Now we apply Proposition 5.1 in the region A to estimate where we need α > 1, and we require Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4043 If we combine the restrictions on p, q, p , q and q , we obtain We now consider the region {v 0 ≤ v ≤ v 1 }. Since the region B ∩ {v ≤ v 1 } is compact, we do not need to appeal to the estimates with respect to the vector fields Y p from Proposition 5.2. Instead, we use the vector fields N p,q , as in Proposition 5.1, making use of the compactness of B ∩ {v ≤ v 1 } to, in particular, estimate the previously problematic K Np,q mixed [φ] error term. We arrive at the estimate: The estimate in the proposition now follows by adding the estimates in {v ≥ v 1 } and {v ≤ v 1 } together.
Remark 5.1. For > 0 arbitrarily small we can always choose α and β in Proposition 5.3 suitably close to 1, so that we can take p = 2− and q = 2− .
We have now proved Theorem 3.3.

Higher-Order Energy Estimates
In order to obtain L ∞ bounds from the L 2 bounds derived in Sects. 4 and 5, we need to derive similar L 2 bounds for higher-order derivatives of φ. In this section we will use φ to denote a solution to (1.1) corresponding to initial data from Proposition 3.1. We will always specify whether we are assuming φ arises from axisymmetric data in Proposition 3.1, or the rotation parameter a is restricted to the range 0 ≤ |a| < a c .

Elliptic Estimates on S 2 u,v
In this section we will show that the angular derivatives on the Eddington-Finkelstein-type spheres S 2 u,v can be controlled by derivatives with respect to the Killing vector field Φ, the null-directed vector fields L and L and the Carter operator Q; see Sect. 2.4.
Note that norms of the angular derivatives of functions on Boyer-Lindquist spheres of constant t and r with respect to the corresponding induced spherical metric can easily been seen to be comparable to analogous norms with the induced metric on the Boyer-Lindquist spheres replaced by the metric on the unit round sphere. This follows from the fact that r is bounded 4044 D. Gajic Ann. Henri Poincaré away from zero and infinity in the region of interest in the black hole interior. Similarly, the induced volume form on the Boyer-Lindquist spheres is comparable to the natural volume form on the unit round sphere. As a preliminary step to considering norms on Eddington-Finkelsteintype spheres S 2 u,v , we will need that L 2 norms of angular derivatives of any function f restricted to the Boyer-Lindquist spheres with respect to the unit round sphere can be controlled solely by L 2 norms of T (f ), Φ(f ) and Q(f ). Therefore, the L 2 norms with respect to the actual induced metric on the Boyer-Lindquist spheres can also be controlled similarly. The lemma below can be found in [8].
where ∇ S 2 denotes the covariant derivative on S 2 and dμ S 2 = sin θdθdϕ.
Proof. By decomposing f into spherical harmonics f on S 2 , one can show that where Δ S 2 denotes the Laplacian on S 2 and O k denotes the operators of the form O j1 with O i angular momentum operators; see for example Sect. 2.1 of [23] for explicit expressions of O i .
The estimate (6.1) follows by using the definition of Q to rewrite the left-hand side above and applying Cauchy-Schwarz.
We would similarly like to control the angular derivatives in the coordinates (θ * , ϕ * ) by using the operators Q, T and Φ that commute with g . However, since the tangent spaces to the Boyer-Lindquist spheres and the spheres S 2 u,v are not spanned by the same tangent vectors, we need to include L and L derivatives in our estimate.
For the sake of convenience, we change from the chart (θ * , ϕ * ) on the 2-spheres S 2 u,v to the chart (θ * , ϕ), because the induced metric on S 2 u,v then becomes diagonal:  18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4045 for all |u| ≥ |u 0 | and v ≥ v 0 .
Let v 1 > v 0 and |u 1 | > u 0 be suitably large. Then, we can estimate Proof. By Theorem 2.1 there exist uniform constants C, c > 0 such that where det / g is the determinant with respect to the coordinate basis corresponding to the chart (θ * , ϕ * ), which is equal to the determinant of the matrix of / g with respect to the coordinate basis corresponding to the chart (θ * , ϕ).
Consider the first-order angular derivatives. We can write By the chain rule, we have that By applying (2.10) and (2.12), we find that We can now conclude the following: Now consider the second-order angular derivatives. We can estimate By applying the chain rule we find that Consequently, by applying the estimates from Theorem 2.1, we obtain where we used, moreover, that the vector field Φ commutes with all the vector fields L, L, ∂ θ * , ∂ θ and with b ϕ * . We can further estimate We now turn to ( / ∇ A ∂ C ) E . The only non-vanishing components are given by: where we used the estimates from Theorem 2.1 to arrive at the inequalities on the right-hand sides.
We can now estimate, by applying (6.4), (6.9) By combining (6.7), the first inequality in (6.8), and (6.9) and making use of the smallness of |b ϕ * | for suitably large v 1 and |u 1 |, we obtain (6.3). We Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4047 can also estimate We have now obtained the estimate (6.2). We can easily commute L and L with Γ above.
Remark 6.1. Observe that there is a loss of derivatives on the right-hand side of (6.2), but no loss of derivatives on the right-hand side of (6.3). Since (6.2) will be used to obtain L ∞ estimates from energy estimates, this (additional) loss of derivatives will be present in the pointwise estimates of Sect. 7. See also Remark 6.2 about the loss of derivatives in the energy estimates themselves.

Commutator Estimates
We can use the elliptic estimates in Proposition 6.2 to control angular error terms that arise from commuting g with L and L. We first derive a general expression for the commutator [ g , W m V n ], where V and W are vector fields. Lemma 6.3. Let V and W be vector fields and n ≥ 1, then Proof. Use the expression for the wave Eq. (B.2) in "Appendix B", together with Ann. Henri Poincaré Proposition 6.4. Either restrict to |a| < a c and let 0 ≤ p < 2 and 0 ≤ q < 2, or restrict to axisymmetric φ, with p = 2 and 0 < q ≤ 2. Then, there exist α = α(p, q) > 1 and β = β(p, q) > 1, such that for u 1 suitably large and |u| ≥ |u 1 |, where 0 < qβα ≤ 2, > 0 can be taken arbitrarily small and For axisymmetric φ, we can replace N 2,qβα on the right-hand side by N 2,q .
where 0 < qβα ≤ 2, > 0 can be taken arbitrarily small and For axisymmetric φ, we can replace N 2,qβα on the right-hand side by N 2,q .
Proof. For any vector field V , we have that Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4049 We can commute g with L and L and apply Lemma 6.3 to obtain Moreover, we have that By applying the estimates from Sect. 2.3, we obtain Consequently, we can apply Cauchy-Schwarz to obtain We obtain similar estimates for 2Ω  We also commute with higher-order derivatives along null vector fields in the region {v ≥ v 1 } ∪ {|u| ≥ |u 1 |}. In this case, we do need to keep track of the behaviour in v + |u| of the error terms arising from commuting with L and L. Lemma 6.6. Let n ∈ N 0 . Then, there exists a constant C = C(a, M, v 0 , u 0 , n) > 0, such that Proof. By Lemma 6.3, we have that We have that It follows that By making use of the estimates for ∂ n r * b ϕ * from Theorem 2.1, it follows that Furthermore, we have that so we obtain: We once again make use of the estimates for ∂ n r * ∂ θ * b ϕ * from Theorem 2.1 to estimate Recall from Proposition 6.4 that We can therefore estimate Hence, we obtain by using the estimates for ∂ k r * / g AB from Theorem 2.1, We can easily estimate the remaining terms in 2Ω 2 g (L j1 L j2 φ), applying the estimates from Sect. 2.3, to conclude that (6.11) must hold.
where 0 < qβα ≤ 2, > 0 can be taken arbitrarily small and For axisymmetric φ, we can replace N 2,qβα on the right-hand side by N 2,q .
Proof. We have that From (6.11) it follows that We can apply (6.3) to obtain We can further estimate We have that Furthermore, We conclude that, for either v ≥ v 1 or |u| ≥ |u 1 |, with v 1 and |u 1 | suitably large We can therefore use the above estimate to obtain The estimates in the region {|u| ≥ |u 1 |} proceed similarly.
Let us define the following higher-order energy norms for k ∈ N 0 : We combine the results of Propositions 6.7 and 6.5 to obtain an energy estimate in the entire region D u0,v0 . Corollary 6.8. Let k ∈ N 0 . Restrict to |a| < a c and let 0 ≤ p < 2 and 0 ≤ q < 2, or restrict to axisymmetric φ, with p = 2 and 0 < q ≤ 2.

Uniform Boundedness of φ
We can use the higher-order energy estimates in the previous section to obtain a uniform pointwise bound on φ. As in the previous section, φ always denotes a solution to (1.1) arising from initial data prescribed in Proposition 3.1. We will always indicate whether we are assuming axisymmetry of φ or the restriction 0 ≤ |a| < a c for the rotation parameter a. Proposition 7.1. Let n ∈ N 0 . Restrict to 0 ≤ |a| < a c and take 0 ≤ p < 2, or restrict to axisymmetric φ and take 0 ≤ p ≤ 2. Let > 0 arbitrarily small and take 0 < q < 2. There exists a constant C = C(a, M, v 0 , u 0 , q, ) > 0 such that, Proof. By the fundamental theorem of calculus applied to integrating along ingoing null geodesics, together with Cauchy-Schwarz, we can estimate Hv∩{|u |≥|u|} for q > 0. To arrive at a pointwise estimate, we apply the standard Sobolev inequality on the spheres S 2 u,v , together with Proposition 6.2: We now combine the results of Propositions 6.2, 6.4, 6.5 and Corollary 6.8; in particular, we commute Γ in the terms above to act directly on φ, in order to arrive at the estimate in the proposition.
We have now proved Theorem 3.5.

Extendibility of φ in C 0
We can use Proposition 7.1 to show that φ can be extended as a continuous function beyond the Cauchy horizon CH + . As this extension is independent of the characteristic data, it is non-unique.
is well defined, so φ can be extended as a C 0 function to the region beyond CH + .
Proof. Consider a sequence of points x k in D u0,v0 \H + , such that lim k→∞ x k = x CH + . The sequence {x k } is, in particular, a Cauchy sequence. We will show that the sequence of points (φ)(x k ) must also be a Cauchy sequence, from which it follows immediately that the sequence converges to a finite number as k → ∞.
By the fundamental theorem of calculus, a Sobolev inequality on S 2 and Cauchy-Schwarz, we can estimate for q > 0 Similarly, we find that for q > 1: Finally, we can estimate by Cauchy-Schwarz on S 2 where we need q > 0. By the above estimates it follows that φ(x k ) must also be a Cauchy sequence if the energies on the right-hand sides are finite.
Finally, as in Proposition 7.1, we can estimate the energies on the righthand sides of the above estimates by the initial energy E Γ;q [φ].
Vol. 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4063 Remark 7.1. By employing the higher-order energy estimates from Sect. 6.2 the above proposition can in fact easily be repeated to obtain also C 0 extendibility of | / ∇ j1 L j2 φ| 2 across CH + for any j 1 , j 2 ∈ N 0 , provided the initial data are taken to be suitably regular (and decaying).
We have now proved Theorem 3.6.

Decay of Lφ
Consider the function φ H + : M ∩ D u0,v0 → R defined by In particular, Lφ H + = 0. We consider ψ := φ − φ H + . We can improve the pointwise decay in ψ with respect to |u| and use the wave Eq. (1.1) to obtain decay of |Lφ| in v.
Moreover, we will show in this section that we can obtain boundedness of which is an improvement over Corollary 5.3. Note that the analogous statement for axisymmetric solutions already follows from Proposition 4.2.

Proposition 7.3. Denote
Let 0 < p < 2, 0 < q < 2 and 0 ≤ s ≤ 1 if |a| < a c . Let p = 2, 0 < q ≤ 2 and 0 ≤ s ≤ 1 if φ is axisymmetric. Then, for every > 0, there exists a constant C = C(M, u 0 , v 0 , p, q, s, ) > 0, such that for all H u and H v in D u0,v0 , Proof. We have that Consequently, we can estimate, By applying Stokes' theorem in D u0,v0 we obtain the following error term: By Cauchy-Schwarz, we can estimate for η > 0 We further estimate for 0 ≤ s ≤ 1, Hence, where we used that The remaining terms in E Np,q [ψ] and the terms in K Np,q can be estimated as in Proposition 4.2 and the propositions in Sect. 5.
Let 0 < p < 2, 0 < q ≤ 2 and 0 ≤ s ≤ 1 if |a| < a c . Let p = 2, 0 < q ≤ 2 and 0 ≤ s ≤ 1 if φ is axisymmetric. Let n ∈ N 0 . Then, for every > 0, there exists a constant C = C(M, a, n, u 0 , v 0 , p, q, s, ) > 0, such that j1+j2=n Hu Proof. We have that We can repeat the proof of Proposition 6.7, but we have to additionally estimate the contribution of the final term in the above expression for 2Ω 2 g (L j1 L j2 ψ). We can estimate We can therefore deal with the corresponding term in E Np,q [L j1 L j2 ψ] in the same way as in the proof of Proposition 7.3.
Proposition 7.5. Let s ≤ 1 and 0 ≤ p < 2 for 0 ≤ |a| < a c . For axisymmetric φ we let 0 ≤ |a| ≤ M and we can also take p = 2. Then, there exists a constant C = C(M, v 0 , u 0 , p, s) > 0 such that, Proof. We obtain estimates for angular derivatives along S 2 u,v from the higherorder energy estimates in Proposition 7.4 in the same way as in Proposition 7.1.
We can split Using Lemma 6.6, we therefore obtain j1+j2=n |L((det / g) We can therefore repeat the proof of Proposition 7.6, using appropriate higherorder energy estimates, to obtain with q > 0 arbitrarily small. We now apply (6.2) together with a standard Sobolev inequality on S 2 u,v to obtain the following L ∞ estimate:

Appendix A. Energy Currents in Kerr-Newman
We consider a spacetimes (N , g) equipped with a double-null foliation (u, v, ϑ 1 , ϑ 2 ), such that the metric is given by Here, u, v solve the Eikonal equation and the (topological) spheres (S 2 u,v , / g) are covered by coordinates ϑ A , with A = 1, 2, foliate the null hypersurfaces {u = const.} and {v = const.}. Let In the (L, L, ∂ ϑA ) basis, the metric components are given by We can define a renormalised ingoing null vector e 3 and outgoing null vector e 4 , satisfying g(e 3 , e 4 ) = −2 by The inverse metric in the basis (e 3 , e 4 , ∂ A ) can therefore be expressed in the double-null coordinate basis as With respect to the basis (L, L, ∂ ϑA ) the inverse metric is given by Define the second fundamental forms χ AB and χ by Lemma A.2. We can express,