Morawetz estimates without relative degeneration and exponential decay on Schwarzschild-de Sitter spacetimes

We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild-de Sitter spacetimes with parameters $(M,\Lambda)$. These are integrated decay statements whose bulk energy density, though degenerate at highest order, is everywhere comparable to the energy density of the boundary fluxes. As a corollary, we prove that solutions of the wave equation decay exponentially on the exterior region. The main ingredients are a previous Morawetz estimate of Dafermos-Rodnianski and an additional argument based on commutation with a vector field which can be expressed in the form $r\sqrt{1-\frac{2M}{r}-\frac{\Lambda}{3}r^2}\frac{\partial}{\partial r}$, where $\partial_r$ here denotes the coordinate vector field corresponding to a well chosen system of hyperboloidal coordinates. Our argument gives exponential decay also for small first order perturbations of the wave operator. In the limit $\Lambda=0$, our commutation corresponds to the one introduced by Holzegel-Kauffman.


Introduction and motivation
Einstein's equation (1.1) Ric[g] − Λg = 0 in the absence of matter with non-negative cosmological constant Λ ≥ 0 has been extensively studied by both the mathematics and physics communities over the past century. Black hole solutions of (1.1) are of particular interest. We will specifically here consider the Schwarzschild-de Sitter spacetime (M ext , g M,Λ ) with where dσ S 2 is the standard metric of the unit sphere. This represents a black hole in an expanding cosmological universe, see [22,35,31]. Making contact with a recent result of Holzegel-Kauffman [20], we shall also consider the Λ = 0 case, i.e. the Schwarzschild spacetime (M S , g M ) with g M = g M, Λ=0 . The aim of this paper will be to revisit the study of the scalar wave equation (1.3) g ψ = 0, on the background (1.2), studied in [3,10,26,33,34,13].
To give some motivation, we recall that over the years a number of methods have been developed to attack problems governed by linear and non-linear equations of hyperbolic type, including (1.1). E-mail address: gm615@cam.ac.uk. Date: November 19, 2021. A fundamental insight is the central role of generalizations of the energy concept as a tool for the global analysis of (1.1). To a large extent, this concept can be understood directly in physical space, i.e. in the 'time domain'. Physical space energy based methods have the advantage of displaying incredible resiliance when passing from statements concerning linear equations, e.g. the scalar wave equation (1.3), to understanding non-linear problems, e.g. Einstein's equation (1.1). A spectacular example of the success of this approach is the proof of the stability of Minkowski spacetime by Christodoulou and Klainerman, see the monograph [7], and the more recent alternative proof [23]. Therefore, it is desirable, when possible, to have a physical space, purely energy based, understanding of boundedness and decay properties of (1.3). This is the goal of the present work.
In the present paper we are specifically interested in the the exterior region of subextremal Schwarzschild-de Sitter bounded between the event H + and cosmologicalH + horizons, see already the dark shaded region of Figure 1. For an analysis of linear waves on the cosmological region see [30], and analysis of linear waves on the black hole interior see [16,18,15]. The extremal case has not been studied systematically for Λ > 0. It is subject to the Aretakis instability [1]. The wave equation (1.3) on the subextremal Schwarzschild-de Sitter exterior with Λ > 0 has been studied in the past by two different, though related, approaches.
One approach was initiated in [10] by Dafermos-Rodnianski, where the wave equation (1.3) was studied using only energy estimates. Their energy estimates, which can in fact be expressed exclusively in physical space, prove faster than any polynomial decay in the shaded region of Figure 1, along a suitable foliation. Specifically, by assigning data on a spacelike hyperboloidal hypersurface (1.4) Σ, that connects the event H + and cosmological horizonH + , they prove faster than any polynomial decay, in τ , of the energy flux through the hypersurface which is the push forward of Σ by the Killing vector field ∂ t . This decay result, in turn, follows from a Morawetz estimate, which is of the form (1.6) τ2 τ1 dτ φτ (Σ) where (r,t, θ, φ) are suitably defined hyperboloidal coordinates. Here, ψ ∞ is a constant which can also be bounded from initial data. Note that the estimate (1.6) manifestly already excludes finite frequency growing modes and, finally, is non-degenerate at the horizons H + ,H + , exploiting thus the red-shift. The estimate, however, degenerates at r = 3M due to the presence of trapped null geodesics, as necessitated by [32,29].
Another approach was initiated in [3] by Bony-Häfner, where they proved exponential decay for (1.3), restricted, however, away from the horizons H + ,H + , based on results concerning the asymptotic distribution of quasinormal modes shown previously by Sá Barreto and Zworski in [28]. Following these proofs, a number of authors worked on the problem, see [26,33], and finally exponential decay was proved for the slowly rotating Kerr-de Sitter black hole by Dyatlov in [13,12] without restriction away from the horizons. The papers of Dyatlov appeal to resolvent estimates in the complex plane and further machinery developed in microlocal analysis. Remarkably, building on all these results, Hintz and Vasy [19] proved non-linear stability for the slowly rotating Kerr-de Sitter spacetime.
For nonlinear applications, arbitrarily fast polynomial decay is in fact more than sufficient, in principle, to obtain stability. Nonetheless, it is curious that the physical space argument of [10] only seemed to give this type of decay and not the full exponential decay. Indeed, this is connected precisely with the degeneration at r = 3M in the Morawetz estimate (1.6), referred to above.
The purpose of our paper is to overcome this difficulty and to show exponential decay for (1.3) on Schwarzschild-de Sitter, by an elementary additional physical space argument. We do so by proving a different type of local energy estimate, which, though still degenerate at r = 3M , is relatively non-degenerate, i.e. its bulk term is not degenerate with respect to its boundary term, see already (1.13). One ingredient of our proof is the Morawetz estimate (1.6). However, on top of the Morawetz estimate (1.6), we will require an additional commutation by a vector field which can again be thought to capture some of the properties of trapping, see already (1.9). In the Λ = 0 case, this will recover a recent construction of Holzegel-Kauffman [20]. Our physical space commutation, in the high frequency limit, connects with the work of previous authors on 'lossless estimates' and 'non-trapping estimates', e.g. see [2,4,5,21,27,17,14,26].
In our companion [24], we use the results of the present paper to prove global well posedness and exponential decay for the solutions of quasilinear wave equation and semilinear wave equations on Schwarzschild-de Sitter.
Before stating our main results, we introduce the commutation and the energies which are of fundamental importance to this paper.
1.1. The commutation vector fields and the energy. We use a system of regular hyperboloidal coordinates (r,t, θ, φ), see already Section 2, in which the metric takes the form The leaf connects the event horizon H + with the cosmological horizonH + , as depicted in Figure 2. We introduce the commutation vector field where ∂ r is the coordinate vector field associated to (r,t, θ, φ). (The simple form of this vector field is intimately related to the precise choice of coordinatet. Note that G is orthogonal to the Killing vector field ∂t at r = 3M .) We define the energy density (1.10) E(Gψ, ψ)= T(∂t, n)[Gψ] + T(n, n)[ψ], where T is the energy momentum tensor of the wave equation, and n is the normal to the foliation φ τ (Σ), see already Section 2. Figure 2. The vector field G The energy density (1.10) is a non-negative definite quantity which contains up to second derivatives of ψ. We have the property In particular, we note that E(Gψ, ψ) controls the H 1 energy density. Note, however, that E(Gψ, ψ) does not control the full H 2 energy density.
for τ ≥ 0. We shall refer below to the energy density (1.10) and the commutation vector field (1.9). Our main theorem is an energy estimate without relative degeneration.
Theorem 1 (rough version). Solutions of the wave equation (1.3) on the exterior of the subextremal Schwarzschild-de Sitter (Λ > 0) black hole background (M ext , g M,Λ ), dark shaded region of Figure 1, satisfy Remark 1.1. Theorem 1 remains true with Σ replaced by a general spacelike hypersurface connecting the event with the cosmological horizons.
Remark 1.2. The estimate (1.13) differs from the Morawetz estimate (1.6) in that, in the former, the same energy density appears in both the bulk term on the left hand side and the initial hypersurface flux term on the right hand side while in the latter, certain derivatives in the bulk term have 1 − 3M r weights relative to their flux terms. In this sense, estimate (1.13) is relatively non-degenerate. Note that this is still compatible with the obstructions of [32,29] due to trapping at r = 3M . It is this relative non-degeneracy that will allow us to immediately obtain exponential decay.
As a corollary of Theorem 1 we have exponential decay: Corollary 1 (rough version). With the assumptions of Theorem 1, we have the exponential decay estimates and sup for 0 ≤ τ . Here E is an appropriate higher order energy of the initial data of ψ, and ψ ∞ is a constant that can be controlled by initial data.
We can apply the arguments of Theorem 1 to obtain a second corollary, concerning small first order perturbations of the wave operator.
Corollary 2 (rough version). Let ψ be a solution of where the vector field a = a j ∂ j is suitably bounded to first order, and sufficiently small. Then, the following estimate holds Also, if a = a j ∂ j is suitably bounded up to second order, we obtain for 0 ≤ τ , where E is an appropriate integral quantity defined on Σ, and ψ ∞ is a constant that can be controlled by initial data.
In our forthcoming [25], we prove a Morawetz estimate on Kerr-de Sitter spacetimes with parameters (a, M, Λ) for the wave equation (1.3), and more generally for the Klein-Gordon equation, and use it in conjunction with a generalization of the methods introduced here, to again prove an analogue of Theorem 1 and exponential decay. We specifically establish exponential decay for slow rotation ( a ≪ M, Λ), or alternatively, in the full subextremal case of parameters but where the solution is assumed axisymmetric.
1.3. The Λ = 0 Schwarzschild limit. In the Schwarzschild limit Λ = 0 the commutation vector field G (2.18) reduces to the vector field introduced in [20], see already Section 7. Note, however, that in [20] the commutation vector field G was expressed in coordinates that were there denoted as (t, R ⋆ , θ, φ). Those coordinates, although regular at H + , do not coincide with our regular hyperboloidal coordinates. Thus, in those coordinates, G did not have the simple form (1.9) for Λ = 0.

Preliminaries
2.1. The subextremality conditions. We will use the following notation extensively.
We will often simply denote it as µ.
We define the following.
Definition 2.2. Let Λ ≥ 0. Then, the set of sub-extremal black hole parameters is For M ∈ B Λ , we denote the two positive real roots of 1 − µ as which will correspond to the area radius of the event and cosmological horizons respectively, see Definition 2.4. We shall often denote these simply as r + ,r + .

2.2.
The spacetimes in regular hyperboloidal coordinates. We now define the metric of Schwarzschild-de Sitter.
Definition 2.3. Let Λ > 0 and M ∈ B Λ . We define the following manifold with boundary for a δ > 0 sufficiently small, and the metric where dσ S 2 is the standard metric of the unit sphere and We refer to the tuple (r,t, θ, φ) as regular hyperboloidal coordinates. Note the inverse metric components See also Appendix A for the Christoffel symbols of the metric (2.4).

The time orientation.
We take the vector field to be future oriented. This defines a time orientation for M ext .

2.4.
The event H + and cosmologicalH + horizons. Now we can define the following boundaries.
Definition 2.4. We define the following boundaries to the manifolds M ext , M ext,δ which we call future event horizon and future cosmological horizon respectively.
2.5. The photon sphere. The hypersurface {r = 3M } is called the 'photon sphere'. All future directed null geodesics either cross H + , crossH + , or asymptote to r = 3M . We will refer to the ones asymptoting to r = 3M as 'future trapped null geodesics'.
2.6. The Schwarzschild-de Sitter coordinates. We define the following coordinates.
Definition 2.5. From regular hyperboloidal coordinates, Definition 2.3, we define the Schwarzschildde Sitter coordinates, (r, t, θ, φ) ∈ (r + ,r + ) × R × S 2 , by the following transformation where ξ(r) is given by (2.5). These coordinates cover the region M o ext . We rewrite the metric (2.4) by using the transformation (2.9) to obtain We distinguish between the coordinate vector field (2.11) ∂ ∂r with respect to the Schwarzschild-de Sitter coordinate system (r, t, θ, φ), and the coordinate vector field (2.12) ∂ ∂r with respect to the regular hyperboloidal coordinates (r,t, θ, φ).
The following equality holds in the region M o ext that the Schwarzschild-de Sitter coordinates are defined.
2.8. Chain rule between coordinate vector fields. By a simple chain rule, we have We have where, for H(r), see Section 2.6. Note that at r = 3M the vector field ∂ r is in the direction of ∂ r ⋆ , since 2.9. The vector field G in Schwarzschild-de Sitter coordinates. We define the vector field Note that G is C 0 on the horizons H + ,H + , but not C 1 . We extend the vector field G, of (2.18), beyond the horizons H + ,H + such that This vector field is suggested by the good commutation property of Proposition 4.1. Note that in view of (2.17), at r = 3M the vector field G is in the direction of ∂ r ⋆ and is thus orthogonal to ∂t. This is significant because this is precisely the derivative that does not degenerate in estimate (1.6). We express the vector field (2.18) in Schwarzschild-de Sitter coordinates.
in Schwarzschild-de Sitter coordinates. The functions G 1 , G 2 are defined as follows Proof. We note (2.21) 2.10. Wave operator. We denote by ∇ the covariant derivative with respect to r 2 dσ S 2 . The wave operator is Note, moreover, that we define the following expression Furthermore, we denote as Ω α , for α = 1, 2, 3 the standard vector fields 2.11. Spacelike hypersurfaces. In Schwarzschild-de Sitter a prototype hypersurface that is spacelike and connects the event horizon H + with the cosmological horizonH + would be Our results also hold for general spacelike hypersurfaces connecting H + andH + . (For convenience, however, we always work with Σ fixed as above.) 2.12. Spacelike foliations and causal domains. We push forward the hypersurface Σ, see Section 2.11, under the flow φ τ of the vector field ∂t to obtain the family of hypersurfaces We define the following spacetime domains.
2.13. Normals of spacelike hypersurfaces. The unit normal vector fields of the foliation of Section 2.11 can be computed from the gradient ∇t which in regular hyperboloidal coordinates is The normal of the relevant foliation will be often denoted simply as n.
2.14. Volume forms. The volume form of a spacetime domain is with respect to the (r,t, θ, φ) coordinates. By pulling back the spacetime volume form (2.31) into hypersurfaces of constantt, we obtain that the {t = τ } hypersurfaces admit the volume form We define the normals of the event and cosmological horizons respectively as With the above choice of normals, the corresponding volume forms of the respective null hypersurfaces take the form In all integrals without explicit volume form, it is to be understood that the volume forms are taken to be the ones defined in this Section.

2.15.
Coarea formula. Let f be a continuous non-negative function. Then, note the coarea formula where the constants in the above similarity depend only on the black hole mass M and do not degenerate in the limit Λ → 0. For fixed Λ > 0 the r factor of (2.34) is of course inessential.
2.16. Penrose diagrams. The reader familiar with the Penrose diagrammatic representation may wish to refer to Figure 3.

Exterior region
H +H + Figure 3. The foliation of the Schwarzschild-de Sitter exterior 2.17. Currents and the divergence theorem. We will employ the energy momentum tensor and the relevant current it produces.
Definition 2.7. Let g be a smooth Lorentzian metric. For ψ a solution of (2.35) g ψ = F, we define the energy momentum tensor The energy current with respect to a vector field X is Lastly, for X, n future causal vector fields, we have that We apply the divergence theorem in the region D(τ 1 , τ 2 ) to obtain the following Proposition.
Proposition 2.1. Let ψ satisfy the equation (2.35) on D(τ 1 , τ 2 ). Then, with the notation above, the following holds (2.41) In the above, in accordance with our conventions from Section 2.14, all integrals are taken with respect to the volume form of the respective hypersurfaces or spacetime domain.
For a further study on currents related to partial differential equations, see the monograph of Christodoulou [6].
Note that this Theorem also holds for the domain D δ (τ 1 , τ 2 ) in the place of D(τ 1 , τ 2 ), for a sufficiently small δ > 0, where the constant C is independent of δ.
As an immediate corollary we have exponential decay.
3.2. The inhomogeneous wave equation and absorption of small error terms. The following theorem concerns the inhomogeneous wave equation on the Schwarzschild-de Sitter background.
Note that this Theorem also holds for the domain D δ (τ 1 , τ 2 ) in the place of D(τ 1 , τ 2 ) for a sufficiently small δ > 0, where the constant C is independent of δ.
We have the following Corollary.
Corollary 2 (detailed version). Let a = a j ∂ j be a vector field, where (3.6) at, a r , g θθ (a θ ) 2 g M,Λ ψ = a j ∂ j ψ satisfy the following estimate Also, there exist constants C(M, Λ) > 0, c(M, Λ) > 0 depending only on the black hole parameters such that where Ω α are defined in equation (2.24). Then, we obtain 3.3. The higher order statement. The following Theorem is the higher order statement of Theorem 2. We will use the following result in our companion paper [24] to prove stability of solutions of the quasilinear wave equation.
Theorem 3. Let F be a sufficiently regular function on D(τ 1 , τ 2 ). We have that, for a sufficiently regular solution of (3.12) g M,Λ ψ = F on D(τ 1 , τ 2 ), and for any j ≥ 3, there exists a constant C = C(j, M, Λ) such that the following higher order estimate holds The j = 2 case is the same without the hypersurface error terms in the right hand side of (3.13).
Note that this Theorem also holds for the domain D δ (τ 1 , τ 2 ) in the place of D(τ 1 , τ 2 ), for a sufficiently small δ > 0, where the constant C is independent of δ.

Proof of Theorem 1
4.1. The Morawetz estimate of [10]. As mentioned earlier, this proof utilizes a Morawetz estimate for the wave equation on Schwarzschild-de Sitter, which we find in [10].
and (4.2)    ∂tψ ⋅ F + F 2 , on the right hand side of both equations of Theorem 4.1. Moreover, in this case, ψ ∞ is bounded by

The equation for
Gψ and the ∂t energy identity. Before we begin the proof, we derive the equation satisfied by Gψ.
r , Proof. For convenience let Let ϕ be a smooth function on M ext , and note the commutation (4.9) Now, by computing the right hand side of equation (4.10) for ψ, a solution of the wave equation (1.3), we obtain Finally, by using equation (4.9) for ψ, a solution of the wave equation (1.3), together with (4.11), we obtain (4.12) By revisiting the metric in regular hyperboloidal coordinates, see (2.4), we note (4.13) ∂ r gtt + 2 r gtt = 0, also (4.14) 2∂ r f (r)g rr − f (r) ∂ r g rr + 2 r g rr = 0, and finally Therefore, we conclude (4.16) Now, by revisiting the wave operator (2.22), we compute Proof. We have and we have already computed in Proposition 4.1. We apply the divergence theorem, see equation (2.41), to Gψ and conclude the result.

Auxilliary estimates.
Remark 4.5. In the proof of this Section we shall keep track of the powers of r in our estimates, such that the constants in our estimates do not degenerate as Λ → 0. This shall be useful in the next Section 7, where we treat the Λ = 0 Schwarzschild case.
We will also need the following elementary pointwise lemma later.
We want to generate all the derivatives of Gψ from terms appearing in equation (4.19).
Proof. We begin from the equation for Gψ, see the commutation (4.6) and multiply it with 1 r Gψ where RHS is the right hand side of equation (4.24). We integrate (4.25) over D(τ 1 , τ 2 ). Then, we perform the necessary integration by parts and Young's inequalities to get the result.
We want to generate a non-degenerate (∂ t ψ) 2 from terms appearing in equation (4.19).
Proof. We begin by We perform an integration by parts, and write the right hand side of the above as We conclude the proof.
Note that the horizon hypersurface terms on H + ,H + , on the left hand side of equation (4.26), are non-negative, see (2.40), so we may drop them from our estimates. We multiply the estimate of Lemma 4.2 with a smallness parameter and add it to (4.26) to obtain (4.27) Finally, we add in the Morawetz estimate (4.1) of Theorem 4.1, multiplied with a large parameter and use the boundedness estimate (4.2), to conclude (4.28) We recall the definition of E(Gψ, ψ) in equation (1.10) and note equation (1.11). We conclude (3.1). Moreover, we conclude (3.2), in view of the relation of the coaread formula of Section 2.15, namely We need the following lemma.
for all τ 2 > τ 1 ≥ 0 and some k > 0. Then, there exists constants c, C > 0 depending only on k such that: Proof. This is elementary. See for example [10].
Now we can infer exponential decay.
Proof of Corollary 1. The estimate is an immediate consequence of equation ( We note that inequality (4.30) holds for ∂ ī t Ω j α ψ, for all indices i, j, in the place of ψ, where Ω α are defined in equation (2.24), since the following hold: Therefore, one can prove, by commuting with Ω α and then a Sobolev estimate, the pointwise estimate where ψ ∞ is from Theorem 4.1, and satisfies ψ ∞ ≤ C sup Σ ψ + ∫ Σ J n µ [ψ]n µ .
Remark 4.6. Note that from Theorem 3 and the use of Sobolev estimates we can obtain pointwise estimates for arbitrary regular higher order derivatives where for E G,k+3 see (3.14).

Proof of Theorem 2
We recall from which we easily deduce Now, we follow the arguments of the Section 4. Specifically, we conclude that there exists a constant C = C(M, Λ) > 0, such that for ψ a sufficiently regular solution of the inhomogeneous wave equation (3.4) on D(τ 1 , τ 2 ), we obtain (5.3) ∂tGψ (E 1 (r)∂tψ + E(r)∂ r ψ) .
Then, we use appropriate Young's inequalities on the right hand side of (5.3) to conclude (5.4) where we generated all the derivatives of Gψ and ψ on the left hand side of our estimate (5.5), by Lemmata 4.2, 4.3. We apply a Young's inequality on the last term on the right hand side of (5.5) and conclude (5.6) Now, Theorem 2 is a trivial consequence of equation (5.6), since we have the property (1.11).
Proof of Corollary 2. Suppose that F = a j ∂ j ψ, where a j are smooth and bounded with Ga j bounded, for all j. Now, we use equation (5.5) to obtain (5.7) If is sufficiently small the terms on the right hand side can be absorbed, after also using a Hardy inequality. Finally, for the pointwise result, we commute the inhomogeneous equation (3.4) with the vector fields Ω α , see equation (2.24), to conclude that (5.7) holds for Ω α ψ in the place of ψ. Moreover, we know that [Ω α G, a]t, [Ω α G, a] r , g θθ ([Ω α G, a] θ ) 2 + g φφ ([Ω α G, a] φ ) 2 are bounded. We use a Sobolev inequality and conclude that (5.8) sup where, in view of equation (4.5), the following holds and

Proof of Theorem 3
We define the auxiliary energy We begin by noting the following Proposition, which is a higher order analogue of the inhomogeneous version of Theorem 4.1 Proposition 6.1. Let ψ satisfy the inhomogeneous wave equation (3.12). Then, we obtain the following higher order Morawetz estimate Proof. The proof of this Proposition follows from Theorem 4.1 and additional redshift commutations of the Lecture notes [11] and elliptic estimates. Now we prove Theorem 3.
Proof of Theorem 3. We start by recalling the result of Theorem 2 namely (6.3) where we have kept certain r factor explicitly for integrands related to G, for comparison with the case Λ = 0. (We note however that we drop the r factor in the lower order terms.) The inequality (6.3) already gives the result of the Theorem for j = 2. Now, for any j ≥ 3, by commuting the inhomogeneous wave equation (3.12) with we obtain (6.5) Then, to obtain all the lower order terms on the bulk of the left hand side of (6.5) we sum in the Morawetz estimate (6.2) of Proposition 6.1, at order j − 1, and after appropriate Young's inequalities on the contribution of the F error term we obtain (6.6) where we used the auxiliary energy Note that in the energy estimate (6.6) there is no degeneration, at the low order, on the photon sphere r = 3M , because we repeated a Poincare type argument at top order, see Lemma 4.3.
Note that we control all the desired higher order derivatives (at order j) related to G, on the left hand side of (6.6), except for with the appropriate degenerative weight. For that purpose, we return to the equation satisfied by Gψ, see (4.6), which reads (6.9) where recall g rr = 1 − µ. We differentiate (6.9) by (6.10) ∂ i3−2 r , and then we square the result and note that there exists a constant C(j, M, Λ) > 0 such that (6.11) where, in inequality (6.11) the three terms displayed are from left to right the highest ∂ r derivative term, the fastest degenerating (1 − µ) term and the contribution of the inhomogeneity F . We rewrite (6.11) as and therefore, by multiplying (6.12) with (1−µ) 2i 3 −3 r , so that no terms blow up at the roots of 1−µ = 0, we obtain For the j = 3 case of inequality (6.13) with all its terms displayed see already Remark 6.1. Therefore, by using the integrated inequality (6.6) and the pointwise estimate (6.13) we obtain (6.14) for all τ 1 ≤ τ 2 and for all j ≥ 2. Now, we want to estimate from below E ′ G,j [ψ](τ 2 ) by E G,j [ψ](τ 2 ), in the energy estimate (6.14), for all orders j ≥ 3 at the expense of producing hypersurface error terms. We note that there exists a constant c(j, M, Λ) such that Then, we want to obtain the top order derivatives on the left hand side, with the appropriate degenerative weights of (1−µ). Therefore, we multiply the pointwise estimate (6.12) with (1 − µ) 2i3−3 and sum it to (6.15) to obtain that there exist constants Note that the volume form of (7.7) is (7.8) dg = r 2 sin θdtdθdφ and the volume form of (7.6) is and the normal of the {t = c} hypersurface is where ξ(r) = 1 − 3M r 1 + 6M r . The vector field G here takes the form is now defined with G as in (7.11), where N is a time translation invariant strictly timelike vector field, see the Lecture notes [11] that away from the horizon H + is equal to ∂t. We also obtain (7.13)Ẽ (Gψ, ψ) ∼ M 1 r (∂tGψ) 2 + r 1 − 2M r (∂ r Gψ) 2 + r ∇Gψ 2 + 1 r (∂tψ) 2 + r(∂ r ψ) 2 + r ∇ψ 2 ∼ M r 1 r 2 (∂tGψ) 2 + 1 − 2M r (∂ r Gψ) 2 + ∇Gψ 2 + 1 r 2 (∂tψ) 2 + (∂ r ψ) 2 + ∇ψ 2 .
Remark 7.1. Note that the vector field (7.11) coincides with the vector field already described by Holzegel-Kauffman [20].
Note that the divergence theorem of equation (2.41), holds with I + in the place ofH + . The energy flux at null infinity I + , and the event horizon H + , are nonnegative and we will may drop them from our estimates, see equation (7.4).
The Morawetz estimate we will need was proved in [8] by Dafermos-Rodnianski.
Recalling Remark 4.5, we repeat similar arguments to the ones used in the previous Section 4, up to estimate (4.27). Note, however, that we use the commutation vector field (7.11) and the energy density (1.10) with Λ = 0. Now, adding the Morawetz estimate of Theorem 7.1, noting the weights and absorbing relevant terms, we conclude the following.
for any η ∈ (0, 1]. Proof. By repeating the arguments of Section 4, and by keeping track of the weights in r, we conclude the following estimate ∂tGψ ⋅ GF + ∂tGψ (E 2 (r)∂ t ψ + E 1 (r)∂ r ψ) , where for E 1 (r), E 2 (r) see Proposition 4.6. Then, to obtain all of the derivatives of Gψ on the left hand side of (7.17) we use Lemma 4.2, by keeping the weights in r and by an additional F error term to obtain (7.18) φτ (Σ) To control all the first order terms on the right hand side of (7.18) we use the boundedness estimate and the Morawetz estimate of Theorem 7.1 and obtain (7.19) φτ (Σ)Ẽ (Gψ, ψ) for any η ∈ (0, 1], where we used Young's inequalities in the lower order terms on the right hand side of (7.17). Note that no degeneration is present on the photon sphere r = 3M since we repeated the Poincare type inequality of Lemma 4.3. Finally, by using the appropriate Young's inequalities on the right hand side of (7.19), we conclude the result.
Moreover, we have the Corollary.