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

Georgios Mavrogiannis  University of Cambridge, Department of Pure Mathematics
and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

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 [20].

##  1\. Introduction and motivation

Einstein’s equation

(1.1) $Ric[g]-\Lambda g=0$

in the absence of matter with non-negative cosmological constant $\Lambda\geq
0$ 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 $(\mathcal{M}_{\textit{ext}},g_{M,\Lambda})$ with

(1.2) $g_{M,\Lambda}=-\left(1-\frac{2M}{r}-\frac{1}{3}\Lambda
r^{2}\right)dt^{2}+\left(1-\frac{2M}{r}-\frac{1}{3}\Lambda
r^{2}\right)^{-1}dr^{2}+r^{2}d\sigma_{\mathbb{S}^{2}},$

where $d\sigma_{\mathbb{S}^{2}}$ 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 $\Lambda=0$ case, i.e. the Schwarzschild spacetime
$(\mathcal{M}_{\textit{S}},g_{M})$ with $g_{M}=g_{M,\Lambda=0}$. The aim of
this paper will be to revisit the study of the scalar wave equation

(1.3) $\Box_{g}\psi=0,$

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). 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
$\mathcal{H}^{+}$ and cosmological $\bar{\mathcal{H}}^{+}$ 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 $\Lambda>0$. It is subject to the Aretakis instability [1].

Figure 1. The Schwarzschild–de Sitter spacetime

The wave equation (1.3) on the subextremal Schwarzschild–de Sitter exterior
with $\Lambda>0$ 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) $\Sigma,$

that connects the event $\mathcal{H}^{+}$ and cosmological horizon
$\bar{\mathcal{H}}^{+}$, they prove faster than any polynomial decay, in
$\tau$, of the energy flux through the hypersurface

(1.5) $\phi_{\tau}(\Sigma),$

which is the push forward of $\Sigma$ by the Killing vector field
$\partial_{t}$. This decay result, in turn, follows from a Morawetz estimate,
which is of the form

(1.6)
$\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\phi_{\tau}(\Sigma)}(\psi-\psi_{\infty})^{2}+(\partial_{r}\psi)^{2}+\left(1-\frac{3M}{r}\right)^{2}\left(|\not{\nabla}\psi|^{2}+(\partial_{\bar{t}}\psi)^{2}\right)\lesssim\int_{\phi_{\tau_{1}}(\Sigma)}(\partial_{r}\psi)^{2}+(\partial_{\bar{t}}\psi)^{2}+|\not{\nabla}\psi|^{2},$

where $(r,\bar{t},\theta,\phi)$ are suitably defined hyperboloidal
coordinates. Here, $\psi_{\infty}$ 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
$\mathcal{H}^{+},\bar{\mathcal{H}}^{+}$, 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
$\mathcal{H}^{+},\bar{\mathcal{H}}^{+}$, 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 $\Lambda=0$ 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
$(r,\bar{t},\theta,\phi)$, see already Section 2, in which the metric takes
the form

(1.7)
$g_{M,\Lambda}=-\left(1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\right)(d\bar{t})^{2}-2\frac{1-\frac{3M}{r}}{\sqrt{1-9M^{2}\Lambda}}\sqrt{1+\frac{6M}{r}}d\bar{t}dr+\frac{27M^{2}}{1-9M^{2}\Lambda}\frac{1}{r^{2}}(dr)^{2}+r^{2}d\sigma_{\mathbb{S}^{2}}.$

The leaf

(1.8) $\Sigma=\\{\bar{t}=0\\}$

connects the event horizon $\mathcal{H}^{+}$ with the cosmological horizon
$\bar{\mathcal{H}}^{+}$, as depicted in Figure 2.

We introduce the commutation vector field

(1.9)
$\displaystyle\mathcal{G}=r\sqrt{1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}}\frac{\partial}{\partial
r},$

where $\partial_{r}$ is the coordinate vector field associated to
$(r,\bar{t},\theta,\phi)$. (The simple form of this vector field is intimately
related to the precise choice of coordinate $\bar{t}$. Note that $\mathcal{G}$
is orthogonal to the Killing vector field $\partial_{\bar{t}}$ at $r=3M$.)

Figure 2. The vector field $\mathcal{G}$

We define the energy density

(1.10)
$\displaystyle\mathcal{E}(\mathcal{G}\psi,\psi)\>\dot{=}\>\mathbb{T}(\partial_{\bar{t}},n)[\mathcal{G}\psi]+\mathbb{T}(n,n)[\psi],$

where $\mathbb{T}$ is the energy momentum tensor of the wave equation, and $n$
is the normal to the foliation $\phi_{\tau}(\Sigma)$, see already Section 2.

The energy density (1.10) is a non-negative definite quantity which contains
up to second derivatives of $\psi$. We have the property

(1.11)
$\mathcal{E}(\mathcal{G}\psi,\psi)\sim(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+\left(1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\right)(\partial_{r}\mathcal{G}\psi)^{2}+|\not{\nabla}\mathcal{G}\psi|^{2}+(\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}.$

In particular, we note that $\mathcal{E}(\mathcal{G}\psi,\psi)$ controls the
$H^{1}$ energy density. Note, however, that
$\mathcal{E}(\mathcal{G}\psi,\psi)$ does not control the full $H^{2}$ energy
density.

###  1.2. The main result

Let

(1.12) $\phi_{\tau}(\Sigma)=\\{\bar{t}=\tau\\},$

for $\tau\geq 0$. 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 $(\Lambda>0)$ black hole background
$\left(\mathcal{M}_{\textit{ext}},g_{M,\Lambda}\right)$, dark shaded region of
Figure 1, satisfy

(1.13)
$\int_{\phi_{\tau_{2}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\lesssim\int_{\phi_{\tau_{1}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right),$

for $0\leq\tau_{1}<\tau_{2}$.

######  Remark 1.1.

Theorem 1 remains true with $\Sigma$ 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
$\left(1-\frac{3M}{r}\right)$ 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

$\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\lesssim
e^{-c\tau}\int_{\Sigma}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)$

and

$\displaystyle\sup_{\phi_{\tau}(\Sigma)}|\psi-\psi_{\infty}|\lesssim\sqrt{E}e^{-c\tau},$

for $0\leq\tau$. Here $E$ is an appropriate higher order energy of the initial
data of $\psi$, and $\psi_{\infty}$ 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 $\psi$ be a solution of

(1.14) $\Box_{g_{M,\Lambda}}\psi=\epsilon a^{j}\partial_{j}\psi,$

where the vector field $a=a^{j}\partial_{j}$ is suitably bounded to first
order, and $\epsilon$ sufficiently small. Then, the following estimate holds

$\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\lesssim
e^{-c\tau}\int_{\Sigma}\mathcal{E}\left(\mathcal{G}\psi,\psi\right).$

Also, if $a=a^{j}\partial_{j}$ is suitably bounded up to second order, we
obtain

$\displaystyle\sup_{\phi_{\tau}(\Sigma)}|\psi-\psi_{\infty}|\lesssim\sqrt{E}e^{-c\tau},$

for $0\leq\tau$, where $E$ is an appropriate integral quantity defined on
$\Sigma$, and $\psi_{\infty}$ 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 $(a,M,\Lambda)$ 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 ($|a|\ll M,\Lambda$), or alternatively, in the full
subextremal case of parameters but where the solution is assumed axisymmetric.

###  1.3. The $\Lambda=0$ Schwarzschild limit

In the Schwarzschild limit $\Lambda=0$ the commutation vector field
$\mathcal{G}$ (2.18) reduces to the vector field introduced in [20], see
already Section 7. Note, however, that in [20] the commutation vector field
$\mathcal{G}$ was expressed in coordinates that were there denoted as
$(t,R^{\star},\theta,\phi)$. Those coordinates, although regular at
$\mathcal{H}^{+}$, do not coincide with our regular hyperboloidal coordinates.
Thus, in those coordinates, $\mathcal{G}$ did not have the simple form (1.9)
for $\Lambda=0$.

###  1.4. Acknowledgments

The author thanks his supervisor M. Dafermos for numerous useful comments and
helpful discussions. Moreover the author acknowledges the assistance of G.
Moschidis, Y. Shlapentokh Rothman, D. Gajic and C. Kehle for various
insightful conversations and important remarks.

##  2\. Preliminaries

###  2.1. The subextremality conditions

We will use the following notation extensively.

######  Definition 2.1.

Let $M>0$ and $\Lambda\geq 0$. Then, we define the function $\mu$ by

(2.1) $1-\mu_{M,\Lambda}=1-\frac{2M}{r}-\frac{1}{3}\Lambda r^{2}.$

We will often simply denote it as $\mu$.

We define the following.

######  Definition 2.2.

Let $\Lambda\geq 0$. Then, the set of sub-extremal black hole parameters is

$\displaystyle\mathcal{B}_{\Lambda}$ $\displaystyle=\\{M>0:\>1-\mu=0\textit{
admits only distinct real roots}\\}=\\{M:0<M<\frac{1}{3\sqrt{\Lambda}}\\}.$

For $M\in\mathcal{B}_{\Lambda}$, we denote the two positive real roots of
$1-\mu$ as

(2.2) $r_{+}(M,\Lambda)<\bar{r}_{+}(M,\Lambda),$

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 $r_{+},\bar{r}_{+}$.

###  2.2. The spacetimes in regular hyperboloidal coordinates

We now define the metric of Schwarzschild–de Sitter.

######  Definition 2.3.

Let $\Lambda>0$ and $M\in\mathcal{B}_{\Lambda}$. We define the following
manifold with boundary

(2.3)
$\displaystyle\mathcal{M}_{\textit{ext}}=[r_{+},\bar{r}_{+}]_{r}\times\mathbb{R}_{\bar{t}}\times\mathbb{S}^{2}_{(\theta,\phi)},\>\>\mathcal{M}_{\textit{ext},\delta}=[r_{+}-\delta,\bar{r}_{+}+\delta]_{r}\times\mathbb{R}_{\bar{t}}\times\mathbb{S}^{2}_{(\theta,\phi)}$

for a $\delta>0$ sufficiently small, and the metric

(2.4)
$g_{M,\Lambda}=-(1-\mu)(d\bar{t})^{2}-2\xi(r)d\bar{t}dr+\frac{1-\xi^{2}(r)}{1-\mu}(dr)^{2}+r^{2}d\sigma_{\mathbb{S}^{2}}$

where $d\sigma_{\mathbb{S}^{2}}$ is the standard metric of the unit sphere and

(2.5)
$\xi(r)=\frac{1-\frac{3M}{r}}{\sqrt{1-9M^{2}\Lambda}}\sqrt{1+\frac{6M}{r}},\quad\frac{1-\xi^{2}(r)}{1-\mu}=\frac{27M^{2}}{1-9M^{2}\Lambda}\frac{1}{r^{2}}.$

We refer to the tuple $(r,\bar{t},\theta,\phi)$ as regular hyperboloidal
coordinates. Note the inverse metric components

(2.6) $\displaystyle g^{rr}=(1-\mu),\quad
g^{\bar{t}\bar{t}}=-\frac{1-\xi^{2}(r)}{1-\mu},\quad g^{\bar{t}r}=-\xi(r).$

See also Appendix A for the Christoffel symbols of the metric (2.4).

###  2.3. The time orientation

We take the vector field

(2.7) $\frac{\partial}{\partial\bar{t}}.$

to be future oriented. This defines a time orientation for
$\mathcal{M}_{\textit{ext}}$.

###  2.4. The event $\mathcal{H}^{+}$ and cosmological $\bar{\mathcal{H}}^{+}$
horizons

Now we can define the following boundaries.

######  Definition 2.4.

We define the following boundaries to the manifolds
$\mathcal{M}_{\textit{ext}},\mathcal{M}_{\textit{ext},\delta}$

(2.8)
$\displaystyle\mathcal{H}^{+}=\\{(r,\bar{t},\theta,\phi):\bar{t}>-\infty,r=r_{+}\\},\>\bar{\mathcal{H}}^{+}=\\{(r,\bar{t},\theta,\phi):\bar{t}>-\infty,r=\bar{r}_{+}\\},$
$\displaystyle\mathcal{H}^{+}_{\delta}=\\{(r,\bar{t},\theta,\phi):\bar{t}>-\infty,r=r_{+}-\delta\\},\>\bar{\mathcal{H}}^{+}_{\delta}=\\{(r,\bar{t},\theta,\phi):\bar{t}>-\infty,r=\bar{r}_{+}+\delta\\}$

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 $\mathcal{H}^{+}$, cross
$\bar{\mathcal{H}}^{+}$, 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
Schwarzschild–de Sitter coordinates,
$(r,t,\theta,\phi)\in(r_{+},\bar{r}_{+})\times\mathbb{R}\times\mathbb{S}^{2}$,
by the following transformation

(2.9) $t=\bar{t}+H(r),\quad
H(r)=\int_{3M}^{r}\frac{\xi(\tilde{r})}{1-\mu}d\tilde{r},$

where $\xi(r)$ is given by (2.5). These coordinates cover the region
$\mathcal{M}_{\textit{ext}}^{o}$.

We rewrite the metric (2.4) by using the transformation (2.9) to obtain

(2.10)
$g_{M,\Lambda}=-\left(1-\mu\right)dt^{2}+\left(1-\mu\right)^{-1}dr^{2}+r^{2}d\sigma_{\mathbb{S}^{2}}.$

We distinguish between the coordinate vector field

(2.11) $\boldsymbol{\frac{\partial}{\partial r}}$

with respect to the Schwarzschild–de Sitter coordinate system
$(r,t,\theta,\phi)$, and the coordinate vector field

(2.12) $\frac{\partial}{\partial r}$

with respect to the regular hyperboloidal coordinates
$(r,\bar{t},\theta,\phi)$.

The following equality holds

(2.13) $\frac{\partial}{\partial\bar{t}}=\frac{\partial}{\partial t},$

in the region $\mathcal{M}^{o}_{\textit{ext}}$ that the Schwarzschild–de
Sitter coordinates are defined.

###  2.7. Tortoise coordinate

We define the tortoise coordinate

(2.14) $r^{\star}\>\dot{=}\>\int_{3M}^{r}\frac{1}{1-\mu}dr.$

The expression

(2.15) $\frac{\partial}{\partial r^{\star}}$

will denote the vector field taken in $(r^{\star},t,\theta,\phi)$ coordinates.
Note that $\frac{\partial}{\partial
r^{\star}}=(1-\mu)\boldsymbol{\frac{\partial}{\partial r}}$. The vector field
(2.15) extends smoothly to a tangential vector field along $\mathcal{H}^{+}$
and $\bar{\mathcal{H}}^{+}$.

###  2.8. Chain rule between coordinate vector fields

By a simple chain rule, we have

(2.16) $\boldsymbol{\frac{\partial}{\partial r}}=\frac{\partial}{\partial
r}+\frac{d\bar{t}}{dr}\frac{\partial}{\partial\bar{t}}.$

We have

$\frac{d\bar{t}}{dr}=-\frac{dH}{dr}(r),$

where, for $H(r)$, see Section 2.6. Note that at $r=3M$ the vector field
$\partial_{r}$ is in the direction of $\partial_{r^{\star}}$, since

(2.17) $\frac{\partial}{\partial
r}=\sqrt{\frac{27M^{2}}{1-9M^{2}\Lambda}}\frac{\partial}{\partial r^{\star}}.$

###  2.9. The vector field $\mathcal{G}$ in Schwarzschild–de Sitter
coordinates

We define the vector field

(2.18) $\mathcal{G}=r\sqrt{1-\mu}\frac{\partial}{\partial r}.$

Note that $\mathcal{G}$ is $C^{0}$ on the horizons
$\mathcal{H}^{+},\bar{\mathcal{H}}^{+}$, but not $C^{1}$. We extend the vector
field $\mathcal{G}$, of (2.18), beyond the horizons
$\mathcal{H}^{+},\bar{\mathcal{H}}^{+}$ such that

(2.19) $\mathcal{G}\equiv 0,\quad\text{in }\\{r\leq
r_{+}\\}\cup\\{r\geq\bar{r}_{+}\\}.$

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 $\mathcal{G}$ is
in the direction of $\partial_{r^{\star}}$ and is thus orthogonal to
$\partial_{\bar{t}}$. 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.

######  Lemma 2.1.

In $\mathcal{M}^{o}_{\textit{ext}}$ the vector field (2.18) can be written as

(2.20) $\mathcal{G}=G_{1}(r)\frac{\partial}{\partial
r^{\star}}+G_{2}(r)\frac{\partial}{\partial t},$

in Schwarzschild–de Sitter coordinates. The functions $G_{1},G_{2}$ are
defined as follows

$G_{1}(r)=\sqrt{\frac{r^{2}}{1-\mu}},\>\>G_{2}(r)=\frac{r}{\sqrt{1-\mu}}\left(1-\frac{3M}{r}\right)\frac{1}{\sqrt{1-9M^{2}\Lambda}}\sqrt{1+\frac{6M}{r}}.$

###### Proof.

We note

(2.21) $\displaystyle\left(G_{1}\frac{\partial}{\partial
r^{\star}}+G_{2}\frac{\partial}{\partial t}\right)$
$\displaystyle=r\sqrt{1-\mu}\frac{\partial}{\partial
r}+\frac{r}{\sqrt{1-\mu}}\left(-(1-\mu)\frac{dH}{dr}+\frac{(1-\frac{3M}{r})}{\sqrt{1-9M^{2}\Lambda}}\sqrt{1+\frac{6M}{r}}\right)\frac{\partial}{\partial
t}=r\sqrt{1-\mu}\frac{\partial}{\partial r},$

since
$(1-\mu)\frac{dH}{dr}=\frac{1-\frac{3M}{r}}{\sqrt{1-9M^{2}\Lambda}}\sqrt{1+\frac{6M}{r}}$.
∎

###  2.10. Wave operator

We denote by $\not{\nabla}$ the covariant derivative with respect to
$r^{2}d\sigma_{\mathbb{S}^{2}}$.

The wave operator is

(2.22)
$\displaystyle\Box_{g_{M,\Lambda}}\psi=-\frac{1-\xi^{2}(r)}{1-\mu}\partial_{t}^{2}\psi+(1-\mu)\partial_{r}^{2}\psi$
$\displaystyle+2\xi(r)\partial_{r}\partial_{t}\psi+\not{\nabla}^{A}\not{\nabla}_{A}\psi$
$\displaystyle+\left(-\partial_{r}\xi+2r^{-1}\xi(r)\right)\partial_{\bar{t}}\psi+\left(2r^{-1}(1-\mu)+\partial_{r}(1-\mu)\right)\partial_{r}\psi$

where $\not{\Delta}=\not{\nabla}^{A}\not{\nabla}_{A}$. Note, moreover, that we
define the following expression

(2.23) $|\not{\nabla}\psi|^{2}=\not{\nabla}^{A}\psi\not{\nabla}_{A}\psi.$

Furthermore, we denote as $\Omega_{\alpha}$, for $\alpha=1,2,3$ the standard
vector fields

(2.24)
$\Omega_{1}=\partial_{\phi},\quad\Omega_{2}=\cos{\phi}\>\partial_{\theta}-\sin{\phi}\cot{\theta}\>\partial_{\phi},\quad\Omega_{3}=-\left(\sin{\phi}\>\partial_{\theta}+\cos{\phi}\cot{\theta}\>\partial_{\phi}\right),$

that generate the lie algebra $so(3)$. Note
$[\Box_{g_{M,\Lambda}},\Omega_{\alpha}]=0,[\Box_{g_{M,\Lambda}},\partial_{\bar{t}}]=0$.

###  2.11. Spacelike hypersurfaces

In Schwarzschild–de Sitter a prototype hypersurface that is spacelike and
connects the event horizon $\mathcal{H}^{+}$ with the cosmological horizon
$\bar{\mathcal{H}}^{+}$ would be

(2.25) $\Sigma=\\{\bar{t}=0\\}.$

Our results also hold for general spacelike hypersurfaces connecting
$\mathcal{H}^{+}$ and $\bar{\mathcal{H}}^{+}$. (For convenience, however, we
always work with $\Sigma$ fixed as above.)

###  2.12. Spacelike foliations and causal domains

We push forward the hypersurface $\Sigma$, see Section 2.11, under the flow
$\phi_{\tau}$ of the vector field $\partial_{\bar{t}}$ to obtain the family of
hypersurfaces

(2.26) $\phi_{\tau}(\Sigma)=\\{\bar{t}=\tau\\}.$

We define the following spacetime domains.

######  Definition 2.6.

For $\tau_{1}<\tau_{2}$ define the spacetime domain

(2.27) $D(\tau_{1},\tau_{2})\>\dot{=}\>J^{+}(\phi_{\tau_{1}}(\Sigma))\cap
J^{-}(\phi_{\tau_{2}}(\Sigma))=\\{(r,\bar{t},\theta,\phi):\tau_{1}\leq\bar{t}\leq\tau_{2}\\},$

and

(2.28)
$D(\tau,\infty)\>\dot{=}\>J^{+}(\phi_{\tau}(\Sigma))=\\{(r,\bar{t},\theta,\phi):\tau\leq\bar{t}<\infty\\}.$

The domains of Definition 2.6 are both globally hyperbolic, with
$\phi_{\tau_{1}}(\Sigma)$ and $\phi_{\tau}(\Sigma)$, respectively, as Cauchy
hypersurfaces.

###  2.13. Normals of spacelike hypersurfaces

The unit normal vector fields of the foliation of Section 2.11 can be computed
from the gradient $\nabla\bar{t}$ which in regular hyperboloidal coordinates
is

(2.29)
$\nabla\bar{t}=-\frac{1-\xi^{2}(r)}{1-\mu}\frac{\partial}{\partial\bar{t}}-\xi(r)\frac{\partial}{\partial
r}=-\frac{27M^{2}}{1-9M^{2}\Lambda}\frac{1}{r^{2}}\frac{\partial}{\partial\bar{t}}-\frac{1-\frac{3M}{r}}{\sqrt{1-9M^{2}\Lambda}}\sqrt{1+\frac{6M}{r}}\frac{\partial}{\partial
r}.$

Note moreover that $g(\nabla\bar{t},\nabla\bar{t})\leq b(M,\Lambda)<0$ on
$[r_{+},\bar{r}_{+}]$. The normal of the relevant foliation will be

(2.30)
$n_{\phi_{\tau}(\Sigma)}=-\frac{\nabla\bar{t}}{\sqrt{-g(\nabla\bar{t},\nabla\bar{t})}}=\sqrt{\frac{1-\xi^{2}}{1-\mu}}\frac{\partial}{\partial\bar{t}}+\frac{\xi(r)}{\sqrt{\frac{1-\xi^{2}}{1-\mu}}}\frac{\partial}{\partial
r},$

often denoted simply as $n$.

###  2.14. Volume forms

The volume form of a spacetime domain is

(2.31) $dg=r^{2}\sin\theta d\bar{t}drd\theta d\phi$

with respect to the $\left(r,\bar{t},\theta,\phi\right)$ coordinates.

By pulling back the spacetime volume form (2.31) into hypersurfaces of
constant $\bar{t}$, we obtain that the $\\{\bar{t}=\tau\\}$ hypersurfaces
admit the volume form

(2.32) $dg_{\\{\bar{t}=c\\}}=r\sqrt{\frac{27M^{2}}{1-9M^{2}\Lambda}}\sin\theta
drd\theta d\phi.$

We define the normals of the event and cosmological horizons respectively as

(2.33)
$n_{\mathcal{H}^{+}}=\partial_{\bar{t}},\>n_{\bar{\mathcal{H}}^{+}}=\partial_{\bar{t}}.$

With the above choice of normals, the corresponding volume forms of the
respective null hypersurfaces take the form

$dg_{\mathcal{H}^{+}}=r^{2}\sin\theta
d\bar{t}d\sigma_{\mathbb{S}^{2}},\>\>dg_{\bar{\mathcal{H}}^{+}}=r^{2}\sin\theta
d\bar{t}d\sigma_{\mathbb{S}^{2}}.$

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

(2.34)
$\int_{0}^{\tau}d\tau\int_{\phi_{\tau}(\Sigma)}fdg_{\\{\bar{t}=\tau\\}}\sim\int\int_{D(0,\tau)}\frac{1}{r}fdg,$

where the constants in the above similarity depend only on the black hole mass
$M$ and do not degenerate in the limit $\Lambda\rightarrow 0$. For fixed
$\Lambda>0$ 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.

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 $\psi$ a solution of

(2.35) $\Box_{g}\psi=F,$

we define the energy momentum tensor

(2.36)
$\mathbb{T}_{\mu\nu}[\psi]\>\dot{=}\>\partial_{\mu}\psi\partial_{\nu}\psi-\frac{1}{2}g_{\mu\nu}\Big{(}g^{\gamma\epsilon}\partial_{\gamma}\psi\partial_{\epsilon}\psi\Big{)}.$

The energy current with respect to a vector field $X$ is

(2.37) $J^{X}_{\mu}[\psi]\>\dot{=}\>\mathbb{T}_{\mu\nu}[\psi]X^{\nu},$

with divergence

(2.38)
$\nabla^{\mu}J^{X}_{\mu}=X(\psi)F+\frac{1}{2}\mathbb{T}_{\mu\nu}\>^{(X)}\pi^{\mu\nu},$

where

(2.39)
${}^{(X)}\pi^{\mu\nu}\dot{=}\>\frac{1}{2}\big{(}\nabla^{\mu}X^{\nu}+\nabla^{\nu}X^{\mu}\big{)}$

is the deformation tensor.

Lastly, for $X,n$ future causal vector fields, we have that

(2.40) $J^{X}_{\mu}[\psi]n^{\mu}=\mathbb{T}(X,n)[\psi]\geq 0$

We apply the divergence theorem in the region $D(\tau_{1},\tau_{2})$ to obtain
the following Proposition.

######  Proposition 2.1.

Let $\psi$ satisfy the equation (2.35) on $D(\tau_{1},\tau_{2})$. Then, with
the notation above, the following holds

(2.41)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}J^{X}_{\mu}[\psi]n^{\mu}_{\phi_{\tau_{2}}(\Sigma)}+\int_{\mathcal{H}^{+}\cap
D(\tau_{1},\tau_{2})}J^{X}_{\mu}[\psi]n^{\mu}_{\mathcal{H}^{+}}+\int_{\bar{\mathcal{H}}^{+}\cap
D(\tau_{1},\tau_{2})}J^{X}_{\mu}[\psi]n^{\mu}_{\bar{\mathcal{H}}^{+}}$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\nabla^{\mu}J^{X}_{\mu}[\psi]$
$\displaystyle=\int_{\phi_{\tau_{1}}(\Sigma)}J^{X}_{\mu}[\psi]n^{\mu}.$

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].

##  3\. The Main theorems

###  3.1. The homogeneous wave equation

The following is the central theorem of this paper.

######  Theorem 1 (detailed version).

We fix the parameters $M,\Lambda>0$. Then, there exists a constant
$C=C(M,\Lambda)>0$, such that for $\psi$ a sufficiently regular solution of
the wave equation (1.3) on $D(\tau_{1},\tau_{2})$, we have

(3.1)
$\int_{\phi_{\tau_{2}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)+\int\int_{D(\tau_{1},\tau_{2})}\left(\frac{1}{(1-\mu)}(\partial_{t}\mathcal{G}\psi)^{2}+\mathcal{E}(\mathcal{G}\psi,\psi)\right)\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right),$

for all $0\leq\tau_{1}<\tau_{2}$, where
$\mathcal{E}\left(\mathcal{G}\psi,\psi\right)$ is defined in (1.10).

Since
$\int\int_{D(\tau_{1},\tau_{2})}\mathcal{E}(\mathcal{G}\psi,\psi)\sim\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)$
we in particular have

(3.2)
$\int_{\phi_{\tau_{2}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)+\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right).$

Note that this Theorem also holds for the domain
$D_{\delta}(\tau_{1},\tau_{2})$ in the place of $D(\tau_{1},\tau_{2})$, for a
sufficiently small $\delta>0$, where the constant $C$ is independent of
$\delta$.

As an immediate corollary we have exponential decay.

######  Corollary 1 (detailed version).

There exist positive constants $c(M,\Lambda)>0,C(M,\Lambda)>0$, such that for
$\psi$ a sufficiently regular solution of the wave equation (1.3) on
$D(0,\tau)$, we have

$\displaystyle\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\leq
Ce^{-c\tau}\int_{\Sigma}\mathcal{E}\left(\mathcal{G}\psi,\psi\right),$

and also the pointwise decay

$\displaystyle\sup_{\phi_{\tau}(\Sigma)}|\psi-\psi_{\infty}|\leq
C\sqrt{E}e^{-c\tau},$

where $\psi_{\infty}$ is a constant satisfying $|\psi_{\infty}|\leq
C\left(\sup_{\Sigma}|\psi|+\sqrt{\int_{\Sigma}\mathbb{T}(n,n)[\psi]}\right)$,
and

(3.3) $E=\sum_{0\leq i\leq 1}\sum_{1\leq\alpha\leq
3}\int_{\Sigma}\mathcal{E}(\mathcal{G}\Omega_{\alpha}^{i}\psi,\Omega_{\alpha}^{i}\psi),$

where $\Omega_{\alpha}$ are defined in (2.24).

###  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.

######  Theorem 2 (detailed version).

Let F be a sufficiently regular function on $D(\tau_{1},\tau_{2})$. We have
that, for sufficiently regular solutions of

(3.4) $\Box_{g_{M,\Lambda}}\psi=F$

on $D(\tau_{1},\tau_{2})$, the following holds

(3.5)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\left(\frac{1}{(1-\mu)}(\partial_{t}\mathcal{G}\psi)^{2}+\mathcal{E}(\mathcal{G}\psi,\psi)\right)$
$\displaystyle\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)\left|\mathcal{G}F\right|^{2}+|F|^{2},$

for all $0\leq\tau_{1}<\tau_{2}$, and for some constant $C(M,\Lambda)$.

Note that this Theorem also holds for the domain
$D_{\delta}(\tau_{1},\tau_{2})$ in the place of $D(\tau_{1},\tau_{2})$ for a
sufficiently small $\delta>0$, where the constant $C$ is independent of
$\delta$.

We have the following Corollary.

######  Corollary 2 (detailed version).

Let $a=a^{j}\partial_{j}$ be a vector field, where

(3.6)
$a^{\bar{t}},a^{r},g^{\theta\theta}(a^{\theta})^{2}+g^{\phi\phi}(a^{\phi})^{2},[\mathcal{G},a]^{\bar{t}},[\mathcal{G},a]^{r},g^{\theta\theta}([\mathcal{G},a]^{\theta})^{2}+g^{\phi\phi}([\mathcal{G},a]^{\phi})^{2}$

are smooth and bounded functions on $D(0,\infty)$. For $\epsilon>0$
sufficiently small, there exists a constant $C(M,\Lambda)>0$ such that
sufficiently regular solutions of the equation

(3.7) $\Box_{g_{M,\Lambda}}\psi=\epsilon a^{j}\partial_{j}\psi$

satisfy the following estimate

(3.8)
$\int_{\phi_{\tau_{2}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)+\int\int_{D(\tau_{1},\tau_{2})}\left(\frac{1}{(1-\mu)}(\partial_{t}\mathcal{G}\psi)^{2}+\mathcal{E}(\mathcal{G}\psi,\psi)\right)\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right).$

Also, there exist constants $C(M,\Lambda)>0$, $c(M,\Lambda)>0$ depending only
on the black hole parameters such that

(3.9) $\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\leq
Ce^{-c\tau}\int_{\Sigma}\mathcal{E}(\mathcal{G}\psi,\psi).$

Moreover, let
$[\Omega_{\alpha}\mathcal{G},a]^{\bar{t}},[\Omega_{\alpha}\mathcal{G},a]^{r},g^{\theta\theta}([\Omega_{\alpha}\mathcal{G},a]^{\theta})^{2}+g^{\phi\phi}([\Omega_{\alpha}\mathcal{G},a]^{\phi})^{2}$
be bounded for all $\alpha$, where $\Omega_{\alpha}$ are defined in equation
(2.24). Then, we obtain

(3.10) $\sup_{\phi_{\tau}(\Sigma)}|\psi-\psi_{\infty}|\leq
C\sqrt{E}e^{-c\tau},$

where $\psi_{\infty}$ is a constant satisfying $|\psi_{\infty}|\leq
C\left(\sup_{\Sigma}|\psi|+\sqrt{\int_{\Sigma}\mathbb{T}(n,n)[\psi]+\mathbb{T}(n,n)[\mathcal{G}\psi]}\right)$,
and

(3.11) $E=\sum_{0\leq i\leq 1}\sum_{1\leq\alpha\leq
3}\int_{\Sigma}\mathcal{E}(\mathcal{G}\Omega_{\alpha}^{i}\psi,\Omega_{\alpha}^{i}\psi).$

###  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 $D(\tau_{1},\tau_{2})$. We have
that, for a sufficiently regular solution of

(3.12) $\Box_{g_{M,\Lambda}}\psi=F$

on $D(\tau_{1},\tau_{2})$, and for any $j\geq 3$, there exists a constant
$C=C(j,M,\Lambda)$ such that the following higher order estimate holds

(3.13) $\displaystyle E_{\mathcal{G},j}[\psi](\tau_{2})$
$\displaystyle\quad\quad+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{1-\mu}\sum_{0\leq
i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{1+i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right)^{2}+\int_{\tau_{1}}^{\tau_{2}}d\tau
E_{\mathcal{G},j}[\psi](\tau)$ $\displaystyle\quad\leq
CE_{\mathcal{G},j}[\psi](\tau_{1})+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)\sum_{0\leq
i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}F\right)^{2}+\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+C\int_{\\{\bar{t}=\tau_{2}\\}}\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}(1-\mu)^{2i_{3}+1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}F\right)^{2}+\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$

for all $0\leq\tau_{1}\leq\tau_{2}$, where

(3.14) $\displaystyle E_{\mathcal{G},j}[\psi](\tau)$
$\displaystyle=\sum_{\alpha}\int_{\\{\bar{t}=\tau\\}}\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq j-1,i_{3}\geq
1}(1-\mu)^{2i_{3}-1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}\psi\right)^{2}+\sum_{1\leq
i_{1}+i_{2}\leq
j-1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}.$

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
$D_{\delta}(\tau_{1},\tau_{2})$ in the place of $D(\tau_{1},\tau_{2})$, for a
sufficiently small $\delta>0$, where the constant $C$ is independent of
$\delta$.

###### Proof.

See Section 6. ∎

##  4\. 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].

######  Theorem 4.1 (Theorem 1.1 in [10]).

There exists a constant $B=B(\Sigma,M,\Lambda)>0$ such that, for $\psi$
satisfying the wave equation (1.3) in $D(\tau_{1},\tau_{2})$

(4.1)
$\int\int_{D(\tau_{1},\tau_{2})}(\psi-\psi_{\infty})^{2}+(\partial_{r}\psi)^{2}+\left(1-\frac{3M}{r}\right)^{2}\left((\partial_{\bar{t}}\psi)^{2}+\left|\not{\nabla}\psi\right|^{2}\right)\leq
B\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{n}[\psi]n^{\mu},$

and

(4.2) $\left(\int_{\phi_{\tau_{2}}(\Sigma)}+\int_{\mathcal{H}^{+}\cap
D(\tau_{1},\tau_{2})}+\int_{\bar{\mathcal{H}}^{+}\cap
D(\tau_{1},\tau_{2})}\right)J^{n}_{\mu}[\psi]n^{\mu}\leq
B\int_{\phi_{\tau_{1}}(\Sigma)}J^{n}_{\mu}[\psi]n^{\mu}$

where the constant $\psi_{\infty}$ satisfies $|\psi_{\infty}|\leq
C\left(\sup_{\phi_{\tau_{1}}(\Sigma)}|\psi|+\sqrt{\int_{\phi_{\tau_{1}}(\Sigma)}J^{n}_{\mu}[\psi]n^{\mu}}\right)$.

######  Remark 4.1.

At $r=3M$, corresponding to trapping, note that the vector field
$\partial_{r}$ does not degenerate and is, in fact, in the direction of
$\partial_{r^{\star}}$ by (2.17). We have used this relation to translate
Theorem 4.1 from the coordinates of [10].

######  Remark 4.2.

The proof of Theorem 4.1 of Dafermos–Rodnianski actually made use of a
spherical harmonics decomposition. It would be interesting to give a
completely physical space proof, as has been done for Schwarzschild, see [9].

######  Remark 4.3.

For a solution of the inhomogeneous equation

(4.3) $\Box_{g_{M,\Lambda}}\psi=F,$

one sees that the same estimate applies, with an additional term

(4.4) $\int\int_{D(\tau_{1},\tau_{2})}|\partial_{\bar{t}}\psi\cdot
F|+|F|^{2},$

on the right hand side of both equations of Theorem 4.1. Moreover, in this
case, $\psi_{\infty}$ is bounded by

(4.5) $|\psi_{\infty}|\leq
C\left(\sup_{\phi_{\tau_{1}}(\Sigma)}|\psi|+\sqrt{\int_{\phi_{\tau_{1}(\Sigma)}}J^{n}_{\mu}[\psi]n^{\mu}+\int\int_{D(0,\infty)}|\partial_{\bar{t}}\psi\cdot
F|+|F|^{2}}\right).$

###  4.2. The equation for $\mathcal{G}\psi$ and the $\partial_{\bar{t}}$
energy identity

Before we begin the proof, we derive the equation satisfied by
$\mathcal{G}\psi$.

######  Proposition 4.1.

Let $\psi$ satisfy the wave equation (1.3). Then, the following holds

(4.6)
$\Box_{g_{M,\Lambda}}\mathcal{G}\psi=\big{[}\Box_{g_{M,\Lambda}},\mathcal{G}\big{]}\psi=\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\partial_{\bar{t}}\mathcal{G}\psi+E_{1}(r)\partial_{\bar{t}}\psi+E_{2}(r)\partial_{r}\psi$

where

(4.7) $\displaystyle E_{1}(r)$ $\displaystyle=\frac{2\sqrt{1-\mu}(-1+2\Lambda
r^{2})}{r},$ $\displaystyle E_{2}(r)$ $\displaystyle=\frac{9M^{2}-24M\Lambda
r^{3}+r^{4}\Lambda(9-2r^{2}\Lambda)}{9r^{3}\sqrt{1-\mu}}.$

###### Proof.

For convenience let

(4.8) $f(r)=r\sqrt{1-\mu}.$

Let $\varphi$ be a smooth function on $\mathcal{M}_{\textit{ext}}$, and note
the commutation

(4.9) $\displaystyle\big{[}\Box_{g_{M,\Lambda}},f(r)\frac{\partial}{\partial
r}\big{]}\varphi$
$\displaystyle=\Box(f(r)\partial_{r}\varphi)-f(r)\partial_{r}\Box_{g_{M,\Lambda}}\varphi=\Box_{g_{M,\Lambda}}f\partial_{r}\varphi+2\nabla^{c}f(r)\nabla_{c}\partial_{r}\varphi+f(r)\left(\Box_{g_{M,\Lambda}}\partial_{r}\varphi-\partial_{r}\Box_{g_{M,\Lambda}}\varphi\right)$
$\displaystyle=\Box_{g_{M,\Lambda}}f(r)\partial_{r}\varphi+2g^{rr}\partial_{r}f(r)\partial_{r}^{2}\varphi+2g^{r\bar{t}}\partial_{r}f(r)\partial_{\bar{t}}\partial_{r}\varphi+f(r)\left(\Box_{g_{M,\Lambda}}\partial_{r}\varphi-\partial_{r}\Box_{g_{M,\Lambda}}\varphi\right),$

where

(4.10)
$\Box_{g_{M,\Lambda}}\partial_{r}\varphi-\partial_{r}\Box_{g_{M,\Lambda}}\varphi=-\left(\partial_{r}g^{rr}\partial_{r}^{2}\varphi+2\partial_{r}g^{r\bar{t}}\partial_{\bar{t}}\partial_{r}\varphi+\partial_{r}g^{\bar{t}\bar{t}}\partial_{\bar{t}}^{2}\varphi+\partial_{r}\left(\frac{1}{r^{2}}\right)r^{2}\not{\Delta}\varphi+\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\varphi\right).$

Now, by computing the right hand side of equation (4.10) for $\psi$, a
solution of the wave equation (1.3), we obtain

(4.11)
$\displaystyle-\left(\partial_{r}g^{rr}\partial_{r}^{2}\psi+2\partial_{r}g^{r\bar{t}}\partial_{\bar{t}}\partial_{r}\psi+\partial_{r}g^{\bar{t}\bar{t}}\partial_{\bar{t}}^{2}\psi+\partial_{r}\left(\frac{1}{r^{2}}\right)r^{2}\not{\Delta}\psi+\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\psi\right)$
$\displaystyle\quad=-\Bigg{(}\partial_{r}g^{rr}\partial_{r}^{2}\psi+2\partial_{r}g^{r\bar{t}}\partial_{\bar{t}}\partial_{r}\psi+\partial_{r}g^{\bar{t}\bar{t}}\partial_{\bar{t}}^{2}\psi$
$\displaystyle\quad\quad\quad-\frac{2}{r}\left(-g^{rr}\partial_{r}^{2}\psi-2g^{r\bar{t}}\partial_{\bar{t}}\partial_{r}\psi-g^{\bar{t}\bar{t}}\partial_{\bar{t}}^{2}\psi+g^{ab}\Gamma^{c}\>_{ab}\partial_{c}\psi\right)+\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\psi\Bigg{)}$
$\displaystyle\quad=-\Bigg{(}\partial_{r}^{2}\psi\left(\partial_{r}g^{rr}+\frac{2}{r}g^{rr}\right)+\partial_{\bar{t}}^{2}\psi\left(\partial_{r}g^{\bar{t}\bar{t}}+\frac{2}{r}g^{\bar{t}\bar{t}}\right)+\partial_{\bar{t}}\partial_{r}\psi\left(2\partial_{r}g^{r\bar{t}}+\frac{4}{r}g^{r\bar{t}}\right)$
$\displaystyle\quad\quad\quad+\frac{2}{r}g^{ab}\Gamma^{c}\>_{ab}\partial_{c}\psi+\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\psi\Bigg{)}.$

Finally, by using equation (4.9) for $\psi$, a solution of the wave equation
(1.3), together with (4.11), we obtain

(4.12) $\displaystyle\big{[}\Box_{g_{M,\Lambda}},f(r)\frac{\partial}{\partial
r}\big{]}\psi$
$\displaystyle=\left(2\partial_{r}f(r)-f(r)\left(\partial_{r}g^{rr}+\frac{2}{r}g^{rr}\right)\right)\partial_{r}^{2}\psi+\left(2g^{r\bar{t}}\partial_{r}f(r)-2f(r)\left(\partial_{r}g^{r\bar{t}}+\frac{2}{r}g^{r\bar{t}}\right)\right)\partial_{{\bar{t}}}\partial_{r}\psi$
$\displaystyle\quad-f(r)\left(\partial_{r}g^{\bar{t}\bar{t}}+\frac{2}{r}g^{\bar{t}\bar{t}}\right)\partial_{{\bar{t}}}^{2}\psi+\Box_{g_{M,\Lambda}}f(r)\partial_{r}\psi-\frac{2}{r}f(r)g^{ab}\Gamma^{c}\>_{ab}\partial_{c}\psi-f(r)\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\psi.$

By revisiting the metric in regular hyperboloidal coordinates, see (2.4), we
note

(4.13)
$\displaystyle\partial_{r}g^{\bar{t}\bar{t}}+\frac{2}{r}g^{\bar{t}\bar{t}}=0,$

also

(4.14) $\displaystyle
2\partial_{r}f(r)g^{rr}-f(r)\left(\partial_{r}g^{rr}+\frac{2}{r}g^{rr}\right)=0,$

and finally

(4.15)
$2g^{r\bar{t}}\partial_{r}f(r)-2f(r)\left(\partial_{r}g^{r\bar{t}}+\frac{2}{r}g^{r\bar{t}}\right)=\frac{2\sqrt{1-9M^{2}\Lambda}}{\sqrt{1-\mu}\sqrt{1+\frac{6M}{r}}}.$

Therefore, we conclude

(4.16) $\displaystyle\big{[}\Box_{g_{M,\Lambda}},f(r)\frac{\partial}{\partial
r}\big{]}\psi$
$\displaystyle=\frac{2\sqrt{1-9M^{2}\Lambda}}{\sqrt{1-\mu}\sqrt{1+\frac{6M}{r}}}\partial_{r}\partial_{{\bar{t}}}\psi+\Box_{g_{M,\Lambda}}f(r)\partial_{r}\psi-\frac{2}{r}f(r)g^{ab}\Gamma^{c}\>_{ab}\partial_{c}\psi-f(r)\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\psi$
$\displaystyle=\frac{2\sqrt{1-9M^{2}\Lambda}}{r(1-\mu)\sqrt{1+\frac{6M}{r}}}\partial_{{\bar{t}}}\mathcal{G}\psi+\Box_{g_{M,\Lambda}}f(r)\partial_{r}\psi-\frac{2}{r}f(r)g^{ab}\Gamma^{c}\>_{ab}\partial_{c}\psi-f(r)\partial_{r}\left(g^{ab}\Gamma^{c}\>_{ab}\right)\partial_{c}\psi.$

Now, by revisiting the wave operator (2.22), we compute

(4.17)
$g^{ab}\Gamma^{c}\>_{ab}\partial_{c}\psi=\left(-\partial_{r}\xi+2r^{-1}\xi(r)\right)\partial_{t}\psi+\left(2r^{-1}(1-\mu)+\partial_{r}(1-\mu)\right)\partial_{r}\psi,$

and

(4.18)
$\partial_{r}(g^{ab}\Gamma^{c}\>_{ab})\partial_{c}\psi=\partial_{r}\left(-\partial_{r}\xi+2r^{-1}\xi(r)\right)\partial_{\bar{t}}\psi+\partial_{r}\left(2r^{-1}(1-\mu)+\partial_{r}(1-\mu)\right)\partial_{r}\psi.$

By examining equations (4.16) and (4.17), (4.18) we conclude the result. ∎

######  Remark 4.4.

From relations (4.13), (4.14) we can see how we arrived at our choice of
coordinates, Definition 2.3, and our vector field (2.18). Specifically, in the
class of coordinate tranformations $\bar{t}=t-H(r)$ we select $H(r)$ such that
equation (4.13) holds. Moreover, we select the function $f$, such that (4.14)
holds. By this choice of coordinates $(r,\bar{t},\theta,\phi)$ and vector
field $\mathcal{G}$, the expression of equation (4.15) is positive. This
reflects the good unstable structure of trapping.

We apply a $\partial_{\bar{t}}$-multiplier estimate to the equation (4.6).

######  Proposition 4.2.

Let $\psi$ satisfy the wave equation (1.3). Then, we have the following

(4.19) $\displaystyle\left(\int_{\mathcal{H}^{+}\cap
D(\tau_{1},\tau_{2})}+\int_{\bar{\mathcal{H}}^{+}\cap
D(\tau_{1},\tau_{2})}+\int_{\phi_{\tau_{2}}(\Sigma)}\right)J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+\int\int_{D(\tau_{1},\tau_{2})}\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\left(\partial_{{\bar{t}}}\mathcal{G}\psi\right)^{2}$
$\displaystyle\quad=\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}-\int\int_{D(\tau_{1},\tau_{2})}\left(E_{1}(r)\partial_{r}\psi+E_{2}(r)\partial_{{\bar{t}}}\psi\right)\partial_{{\bar{t}}}\mathcal{G}\psi.$

###### Proof.

We have

(4.20)
$\nabla^{\mu}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]=\partial_{{\bar{t}}}\mathcal{G}\psi\Box_{g_{M,\Lambda}}\mathcal{G}\psi,$

and we have already computed

(4.21)
$\Box_{g_{M,\Lambda}}\mathcal{G}\psi=\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\partial_{{\bar{t}}}\mathcal{G}\psi+E_{1}(r)\partial_{r}\psi+E_{2}(r)\partial_{{\bar{t}}}\psi$

in Proposition 4.1. We apply the divergence theorem, see equation (2.41), to
$\mathcal{G}\psi$ and conclude the result. ∎

###  4.3. 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
$\Lambda\rightarrow 0$. This shall be useful in the next Section 7, where we
treat the $\Lambda=0$ Schwarzschild case.

We will also need the following elementary pointwise lemma later.

######  Lemma 4.1.

Let $E_{1}(r),E_{2}(r)$ be defined as in Proposition 4.1. Then, we have the
following

(4.22) $\displaystyle\Big{|}\int\int_{D(\tau_{1},\tau_{2})}$
$\displaystyle\left(E_{1}(r)\partial_{{\bar{t}}}\psi+E_{2}(r)\partial_{r}\psi\right)\partial_{{\bar{t}}}\mathcal{G}\psi\Big{|}$
$\displaystyle\leq\int\int_{D(\tau_{1},\tau_{2})}\left(\frac{r}{2\epsilon}(1-\mu)\left(E_{2}(r)\partial_{r}\psi\right)^{2}+\frac{\epsilon}{2}\frac{1}{(1-\mu)r}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}\right)$
$\displaystyle\quad+\int\int_{D(\tau_{1},\tau_{2})}|\partial_{r}(1-\mu)|(\partial_{\bar{t}}\psi)^{2}.$

We want to generate all the derivatives of $\mathcal{G}\psi$ from terms
appearing in equation (4.19).

######  Lemma 4.2.

There exists a constant $C(M,\Lambda)>0$, which does not degenerate as
$\Lambda\rightarrow 0$, such that, if $\psi$ solves the wave equation (1.3),
then we obtain

(4.23) $\displaystyle\int\int_{D(\tau_{1},\tau_{2})}$
$\displaystyle\frac{1-\mu}{r}\left(\partial_{r}\mathcal{G}\psi\right)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}$
$\displaystyle\leq
C\Bigg{(}\int\int_{D(\tau_{1},\tau_{2})}r^{-1}(1-\mu)^{-1}\left(\partial_{\bar{t}}\mathcal{G}\psi\right)^{2}+r^{-2}(\partial_{r}\psi)^{2}+r^{-2}(\partial_{t}\psi)^{2}$
$\displaystyle\quad\quad\quad+\int_{\phi_{\tau_{2}}(\Sigma)}r^{-1}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+(1-\mu)r(\partial_{r}\mathcal{G}\psi)^{2}+r(\partial_{r}\psi)^{2}$
$\displaystyle\quad\quad\quad+\int_{\phi_{\tau_{1}}(\Sigma)}r^{-1}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+(1-\mu)r(\partial_{r}\mathcal{G}\psi)^{2}+r(\partial_{r}\psi)^{2}\Bigg{)}.$

###### Proof.

We begin from the equation for $\mathcal{G}\psi$, see the commutation (4.6)

(4.24)
$\displaystyle-\frac{1-\xi^{2}(r)}{1-\mu}\partial_{\bar{t}}^{2}\mathcal{G}\psi+(1-\mu)\partial_{r}^{2}\mathcal{G}\psi+2\xi(r)\partial_{r}\partial_{\bar{t}}\mathcal{G}\psi+\not{\nabla}^{A}\not{\nabla}_{A}\mathcal{G}\psi-\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\partial_{\bar{t}}\mathcal{G}\psi$
$\displaystyle=\left(E_{1}(r)-\left(\partial_{r}\xi-2r^{-1}\xi(r)\right)\right)\partial_{\bar{t}}\psi+\left(E_{2}(r)-\left(2r^{-1}(1-\mu)+\partial_{r}(1-\mu)\right)\right)\partial_{r}\psi$

and multiply it with $\frac{1}{r}\mathcal{G}\psi$

(4.25)
$\displaystyle-\frac{1-\xi^{2}(r)}{1-\mu}\frac{1}{r}\mathcal{G}\psi\partial_{t}^{2}\mathcal{G}\psi$
$\displaystyle+(1-\mu)\frac{1}{r}\mathcal{G}\psi\partial_{r}^{2}\mathcal{G}\psi+2\xi(r)\frac{1}{r}\mathcal{G}\psi\partial_{r}\partial_{\bar{t}}\mathcal{G}\psi+\frac{1}{r}\mathcal{G}\psi\not{\nabla}^{A}\not{\nabla}_{A}\mathcal{G}\psi$
$\displaystyle-\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\frac{1}{r}\mathcal{G}\psi\partial_{\bar{t}}\mathcal{G}\psi=\frac{1}{r}\mathcal{G}\psi\textit{RHS}(\psi)$

where RHS is the right hand side of equation (4.24).

We integrate (4.25) over $D(\tau_{1},\tau_{2})$. Then, we perform the
necessary integration by parts and Young’s inequalities to get the result. ∎

We want to generate a non-degenerate $(\partial_{t}\psi)^{2}$ from terms
appearing in equation (4.19).

######  Lemma 4.3.

There exists a constant $C(M,\Lambda)>0$, that does not degenerate as
$\Lambda\rightarrow 0$, such that for all $\epsilon>0$, if $\psi$ satisfies
the wave equation (1.3), then we have

$\displaystyle\frac{C}{\epsilon}\int\int_{D(\tau_{1},\tau_{2})}\frac{1-\mu}{r^{3}}(\partial_{{\bar{t}}}\psi)^{2}\leq\frac{1}{2}$
$\displaystyle\int\int_{D(\tau_{1},\tau_{2})}\frac{2\sqrt{1-9M^{2}\Lambda}}{r^{4}(1-\mu)\sqrt{1+\frac{6M}{r}}}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}$
$\displaystyle+\frac{C}{\epsilon^{2}}\left(\int\int_{D(\tau_{1},\tau_{2})}\frac{(r-3M)^{2}}{r^{4}\sqrt{1-9M^{2}\Lambda}}(\partial_{{\bar{t}}}\psi)^{2}\right).$

###### Proof.

We begin by

$\int\int_{D(\tau_{1},\tau_{2})}\frac{1-\mu}{r^{2}}(\partial_{\bar{t}}\psi)^{2}\frac{1}{r}dg=\int_{\mathbb{S}^{2}}\int_{0}^{\tau}\int_{r_{+}}^{\bar{r}_{+}}\frac{1-\mu}{r^{2}}\frac{\partial}{\partial
r}(r-3M)(\partial_{t}\psi)^{2}\sin{\theta}drd\bar{t}d\sigma.$

We perform an integration by parts, and write the right hand side of the above
as

$\displaystyle\int_{\mathbb{S}^{2}}\int_{0}^{\tau}\big{[}\frac{1-\mu}{r^{2}}(r-3M)(\partial_{\bar{t}}\psi)^{2}\big{]}_{r_{+}}^{\bar{r}_{+}}$
$\displaystyle\quad-\int\int_{D(\tau_{1},\tau_{2})}\left(\frac{\partial}{\partial
r}\left(\frac{1-\mu}{r^{2}}\right)(r-3M)(\partial_{\bar{t}}\psi)^{2}+(r-3M)\frac{1-\mu}{r^{2}}2\partial_{\bar{t}}\psi\frac{\partial}{\partial
r}\partial_{\bar{t}}\psi\right)r\sin{\theta}drd\bar{t}d\sigma$
$\displaystyle=-\int\int_{D(\tau_{1},\tau_{2})}\left(\frac{\partial}{\partial
r}\left(\frac{1-\mu}{r^{2}}\right)(r-3M)(\partial_{\bar{t}}\psi)^{2}+(r-3M)\frac{1-\mu}{r^{2}}2\partial_{\bar{t}}\psi\frac{\partial}{\partial
r}\partial_{\bar{t}}\psi\right)r\sin{\theta}drd\bar{t}d\sigma$
$\displaystyle\quad\leq
C\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r^{6}}(r-3M)^{2}(\partial_{\bar{t}}\psi)^{2}+\frac{1}{\epsilon
r^{8}\sqrt{1-9M^{2}\Lambda}}(r-3M)^{2}(\partial_{\bar{t}}\psi)^{2}+\epsilon\frac{\sqrt{1-9M^{2}\Lambda}}{r^{4}(1-\mu)\sqrt{1+\frac{6M}{r}}}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}.$

We conclude the proof. ∎

###  4.4. Proofs of Theorem 1 and Corollary 1

We are now ready to conclude Theorem 1 and the exponential decay Corollary 1.

###### Proof of Theorem 1.

We first use Lemma 4.1 on equation (4.19), in conjunction with Lemmata 4.1,
4.3 to obtain

(4.26) $\displaystyle\Big{(}\int_{\mathcal{H}^{+}\cap D(\tau_{1},\tau_{2})}$
$\displaystyle+\int_{\bar{\mathcal{H}}^{+}\cap
D(\tau_{1},\tau_{2})}+\int_{\phi_{\tau_{2}}(\Sigma)}\Big{)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\left(\partial_{{\bar{t}}}\mathcal{G}\psi\right)^{2}+\frac{1-\mu}{r^{4}}(\partial_{\bar{t}}\psi)^{2}$
$\displaystyle\quad\lesssim\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+\int\int_{D(\tau_{1},\tau_{2})}r^{-2}(\partial_{\bar{t}}\psi)^{2}+r^{-2}(\partial_{r}\psi)^{2}$

Note that the horizon hypersurface terms on
$\mathcal{H}^{+},\bar{\mathcal{H}}^{+}$, 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)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\left(J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}\right)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1-\mu}{r^{4}}(\partial_{\bar{t}}\psi)^{2}$
$\displaystyle\quad\lesssim\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+\int\int_{D(\tau_{1},\tau_{2})}r^{-2}(\partial_{\bar{t}}\psi)^{2}+r^{-2}(\partial_{r}\psi)^{2}$
$\displaystyle\quad\quad+\int_{\phi_{\tau_{2}}(\Sigma)}(\partial_{r}\psi)^{2}+\int_{\phi_{\tau_{1}}(\Sigma)}(\partial_{r}\psi)^{2}.$

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)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\left(J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}\right)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+(1-\mu)(\partial_{r}\mathcal{G}\psi)^{2}+|\not{\nabla}\mathcal{G}\psi|^{2}+(\partial_{r}\psi)^{2}+(\partial_{{\bar{t}}}\psi)^{2}+|\not{\nabla}\psi|^{2}$
$\displaystyle\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}.$

We recall the definition of $\mathcal{E}(\mathcal{G}\psi,\psi)$ 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

(4.29)
$\int\int_{D(\tau_{1},\tau_{2})}\mathcal{E}(\mathcal{G}\psi,\psi)\sim\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi).$

∎

We need the following lemma.

######  Lemma 4.4.

Let $f:[0,\infty)\rightarrow\mathbb{R}$ a non-negative continuous function
satisfying

$f(\tau_{2})+\int_{\tau_{1}}^{\tau_{2}}f(\tau)d\tau\leq kf(\tau_{1}),$

for all $\tau_{2}>\tau_{1}\geq 0$ and some $k>0$. Then, there exists constants
$c,C>0$ depending only on $k$ such that:

$f(\tau)\leq Ce^{-c\tau}f(0),$

for all $\tau\geq 0$.

###### Proof.

This is elementary. See for example [10]. ∎

Now we can infer exponential decay.

###### Proof of Corollary 1.

The estimate

(4.30) $\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)\lesssim
e^{-c\tau}\int_{\Sigma}\mathcal{E}(\mathcal{G}\psi,\psi)$

is an immediate consequence of equation (3.2) of Theorem 1 and Lemma 4.4,
where $f(\tau)=\int_{\phi_{\tau}(\Sigma)}\mathcal{E}(\mathcal{G}\psi,\psi)$.

We note that inequality (4.30) holds for
$\partial_{\bar{t}}^{i}\Omega_{\alpha}^{j}\psi$, for all indices $i,j$, in the
place of $\psi$, where $\Omega_{\alpha}$ are defined in equation (2.24), since
the following hold:
$[\Box_{g_{M,\Lambda}},\partial_{\bar{t}}]=0,[\Box_{g_{M,\Lambda}},\Omega_{\alpha}]=0$.

Therefore, one can prove, by commuting with $\Omega_{\alpha}$ and then a
Sobolev estimate, the pointwise estimate

(4.31)
$\sup_{\phi_{\tau}(\Sigma)}|\psi-\psi_{\infty}|^{2}\lesssim\int_{\phi_{\tau}(\Sigma)}\sum_{0\leq
i\leq 1}\sum_{0\leq\alpha\leq
3}\mathcal{E}(\mathcal{G}\Omega_{\alpha}^{i}\psi,\Omega_{\alpha}\psi)\lesssim
e^{-c{\tau}}\int_{\Sigma}\sum_{0\leq i\leq 1}\sum_{0\leq\alpha\leq
3}\mathcal{E}(\mathcal{G}\Omega_{\alpha}^{i}\psi,\Omega_{\alpha}\psi),$

where $\psi_{\infty}$ is from Theorem 4.1, and satisfies $|\psi_{\infty}|\leq
C\left(\sup_{\Sigma}|\psi|+\sqrt{\int_{\Sigma}J^{n}_{\mu}[\psi]n^{\mu}}\right)$.
∎

######  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

(4.32) $\sup_{\phi_{\tau}(\Sigma)}\sum_{1\leq i+j+l\leq
k}|\partial_{r}^{i}\partial_{\bar{t}}^{j}\Omega_{\alpha}^{l}\psi|^{2}\lesssim
E_{\mathcal{G},k+3}e^{-c\tau},$

where for $E_{\mathcal{G},k+3}$ see (3.14).

##  5\. Proof of Theorem 2

We recall

(5.1)
$\big{[}\Box_{g_{M,\Lambda}},\mathcal{G}\big{]}\psi=\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\partial_{{\bar{t}}}\mathcal{G}\psi+E_{1}(r)\partial_{\bar{t}}\psi+E_{2}(r)\partial_{r}\psi$

from which we easily deduce

(5.2)
$\Box_{g_{M,\Lambda}}\mathcal{G}\psi=\frac{2\sqrt{1-9M^{2}\Lambda}}{(1-\mu)r\sqrt{1+\frac{6M}{r}}}\partial_{{\bar{t}}}\mathcal{G}\psi+E_{1}(r)\partial_{\bar{t}}\psi+E_{2}(r)\partial_{r}\psi+\mathcal{G}F.$

Now, we follow the arguments of the Section 4. Specifically, we conclude that
there exists a constant $C=C(M,\Lambda)>0$, such that for $\psi$ a
sufficiently regular solution of the inhomogeneous wave equation (3.4) on
$D(\tau_{1},\tau_{2})$, we obtain

(5.3)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}$
$\displaystyle\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+C\int\int_{D(\tau_{1},\tau_{2})}\left|\partial_{\bar{t}}\mathcal{G}\psi\mathcal{G}F\right|+C\int\int_{D(\tau_{1},\tau_{2})}\left|\partial_{\bar{t}}\mathcal{G}\psi\left(E_{1}(r)\partial_{\bar{t}}\psi+E(r)\partial_{r}\psi\right)\right|.$

Then, we use appropriate Young’s inequalities on the right hand side of (5.3)
to conclude

(5.4)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}$
$\displaystyle\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)\left|\mathcal{G}F\right|^{2}+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)\left|E_{1}(r)\partial_{\bar{t}}\psi+E_{2}(r)\partial_{r}\psi\right|^{2}.$

Now, we sum in (5.4) the Morawetz estimate of Theorem 4.1, (in the
inhomogeneous form given by Remark 4.3), to conclude

(5.5)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\left(J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}\right)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+(\partial_{r}\psi)^{2}+(\partial_{{\bar{t}}}\psi)^{2}+|\not{\nabla}\psi|^{2}$
$\displaystyle\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)\left|\mathcal{G}F\right|^{2}+|\partial_{\bar{t}}\psi\cdot
F|,$

where we generated all the derivatives of $\mathcal{G}\psi$ and $\psi$ 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)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\left(J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}\right)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+(\partial_{r}\psi)^{2}+(\partial_{{\bar{t}}}\psi)^{2}+|\not{\nabla}\psi|^{2}$
$\displaystyle\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)\left|\mathcal{G}F\right|^{2}+|F|^{2}.$

Now, Theorem 2 is a trivial consequence of equation (5.6), since we have the
property (1.11).

###### Proof of Corollary 2.

Suppose that

$F=\epsilon a^{j}\partial_{j}\psi,$

where $a^{j}$ are smooth and bounded with $\mathcal{G}a^{j}$ bounded, for all
$j$. Now, we use equation (5.5) to obtain

(5.7)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\left(J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}\right)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{{\bar{t}}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+(\partial_{r}\psi)^{2}+(\partial_{{\bar{t}}}\psi)^{2}+|\not{\nabla}\psi|^{2}$
$\displaystyle\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}J_{\mu}^{\partial_{\bar{t}}}[\mathcal{G}\psi]n^{\mu}+J^{n}_{\mu}[\psi]n^{\mu}$
$\displaystyle\quad\quad+C\int\int_{D(\tau_{1},\tau_{2})}\epsilon^{2}\left(\partial_{j}\psi\mathcal{G}a^{j}\right)^{2}+\epsilon^{2}(\mathcal{G}\psi)^{2}+\epsilon^{2}(a^{j}\mathcal{G}\partial_{j}\psi)^{2}$

since

$\displaystyle\mathcal{G}F$
$\displaystyle=\epsilon\left(\partial_{j}\psi\mathcal{G}a^{j}\right)+\epsilon\left(a^{j}\mathcal{G}\partial_{j}\psi\right).$

If $\epsilon$ 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 $\Omega_{\alpha}$, see equation (2.24), to conclude
that (5.7) holds for $\Omega_{\alpha}\psi$ in the place of $\psi$. Moreover,
we know that
$[\Omega_{\alpha}\mathcal{G},a]^{\bar{t}},[\Omega_{\alpha}\mathcal{G},a]^{r},g^{\theta\theta}([\Omega_{\alpha}\mathcal{G},a]^{\theta})^{2}+g^{\phi\phi}([\Omega_{\alpha}\mathcal{G},a]^{\phi})^{2}$
are bounded. We use a Sobolev inequality and conclude that

(5.8) $\sup_{\phi_{\tau}(\Sigma)}|\psi-\psi_{\infty}|\leq C\sqrt{E}e^{-c\tau}$

where, in view of equation (4.5), the following holds

(5.9) $|\psi_{\infty}|\leq
C\left(\sup_{\Sigma}|\psi|+\sqrt{\int_{\Sigma}\mathbb{T}(n,n)[\psi]+\mathbb{T}(\partial_{\bar{t}},n)[\mathcal{G}\psi]}\right)$

and

(5.10) $E=\sum_{0\leq i\leq 1}\sum_{1\leq\alpha\leq
3}\int_{\Sigma}\mathcal{E}(\mathcal{G}\Omega_{\alpha}^{i}\psi,\Omega_{\alpha}^{i}\psi).$

∎

##  6\. Proof of Theorem 3

We define the auxiliary energy

(6.1) $E_{j}[\psi](\tau)=\sum_{1\leq i_{1}+i_{2}+i_{3}\leq
j}\sum_{\alpha=1,2,3}\int_{\phi_{\tau}(\Sigma)}\left(\partial_{\bar{t}}^{i_{1}}\partial_{r}^{i_{2}}\Omega_{\alpha}^{i_{3}}\psi\right)^{2}.$

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 $\psi$ satisfy the inhomogeneous wave equation (3.12). Then, we obtain the
following higher order Morawetz estimate

(6.2) $\displaystyle
E_{j}[\psi](\tau_{2})+\int\int_{D(\tau_{1},\tau_{2})}\left(1-\frac{3M}{r}\right)^{2}\sum_{i_{1}+i_{2}=j}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\psi\right)^{2}+\sum_{i_{1}+i_{2}+i_{3}=j,i_{3}\geq
1}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}$
$\displaystyle\qquad\quad\quad\quad\quad\quad\quad+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-1}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}$
$\displaystyle\leq
CE_{j}[\psi](\tau_{1})+C\int\int_{D(\tau_{1},\tau_{2})}\left|\partial_{\bar{t}}^{j}\psi\cdot\partial_{\bar{t}}^{j-1}F\right|+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$
$\displaystyle\qquad\qquad\qquad+C\int_{\phi_{\tau_{2}}(\Sigma)}\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$

for a constant $C(j,M,\Lambda)>0$.

###### 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)
$\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)+\int\int_{D(\tau_{1},\tau_{2})}\frac{1}{r(1-\mu)}(\partial_{t}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}\left|\not{\nabla}\mathcal{G}\psi\right|^{2}+(\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}$
$\displaystyle\quad\quad\quad\quad\quad\leq
C\int_{\phi_{\tau_{1}}(\Sigma)}\mathcal{E}\left(\mathcal{G}\psi,\psi\right)+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)r\left|\mathcal{G}F\right|^{2}+|F|^{2},$

where we have kept certain $r$ factor explicitly for integrands related to
$\mathcal{G}$, for comparison with the case $\Lambda=0$. (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 $j\geq 3$, by commuting the inhomogeneous wave equation (3.12)
with

(6.4) $\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l},\qquad k+l\leq j-2$

we obtain

(6.5) $\displaystyle\int_{\phi_{\tau_{2}}(\Sigma)}\sum_{0\leq
k+l=j-2}\sum_{\alpha}\mathcal{E}\left(\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi,\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\psi\right)$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\sum_{0\leq
k+l=j-2}\sum_{\alpha}\left(\frac{1}{r(1-\mu)}(\partial_{t}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi)^{2}+\frac{1}{r}\left|\not{\nabla}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi\right|^{2}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad+\sum_{0\leq
k+l=j-2}\sum_{\alpha}\left((\partial_{\bar{t}}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\psi)^{2}+(\partial_{r}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\psi)^{2}+|\not{\nabla}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\psi|^{2}\right)$
$\displaystyle\leq C\int_{\phi_{\tau_{1}}(\Sigma)}\sum_{0\leq
k+l=j-2}\sum_{\alpha}\mathcal{E}\left(\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi,\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\psi\right)$
$\displaystyle\qquad+C\int\int_{D(\tau_{1},\tau_{2})}\sum_{0\leq
k+l=j-2}\sum_{\alpha}\left((1-\mu)r\left(\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}F\right)^{2}+|\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}F|^{2}\right).$

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) $\displaystyle E^{\prime}_{\mathcal{G},j}[\psi](\tau_{2})$
$\displaystyle+\int\int_{D(\tau_{1},\tau_{2})}\sum_{0\leq
k+l=j-2}\sum_{\alpha}\left(\frac{1}{r(1-\mu)}(\partial_{t}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi)^{2}+\frac{1}{r}\left|\not{\nabla}\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}\psi\right|^{2}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-1}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}$
$\displaystyle\leq
CE^{\prime}_{\mathcal{G},j}[\psi](\tau_{1})+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)r\sum_{0\leq
k+l\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{k}\Omega_{\alpha}^{l}\mathcal{G}F\right)^{2}+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$
$\displaystyle\qquad\qquad\qquad+C\int_{\phi_{\tau_{2}}(\Sigma)}\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$

where we used the auxiliary energy

(6.7)
$E^{\prime}_{\mathcal{G},j}[\psi](\tau)=\int_{\\{\bar{t}=\tau\\}}\sum_{0\leq
i_{1}+i_{2}=j-2}\sum_{\alpha}J_{\mu}^{\partial_{\bar{t}}}[\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi]n^{\mu}+E_{j-1}[\psi](\tau).$

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 $\mathcal{G}$, on the left hand side of (6.6), except for

(6.8) $\partial_{r}^{i_{3}}\mathcal{G}\psi,\qquad i_{3}=j-1$

with the appropriate degenerative weight. For that purpose, we return to the
equation satisfied by $\mathcal{G}\psi$, see (4.6), which reads

(6.9) $\displaystyle
g^{rr}\partial_{r}^{2}\mathcal{G}\psi=-\left(g^{\bar{t}\bar{t}}\partial_{t}^{2}\mathcal{G}\psi+2g^{r\bar{t}}\partial_{t}\partial_{r}\mathcal{G}\psi+\not{\Delta}\mathcal{G}\psi\right)+\frac{2\sqrt{1-9M^{2}\Lambda}}{\sqrt{1-\mu}\sqrt{1+\frac{6M}{r}}}\partial_{\bar{t}}\partial_{r}\psi+E_{1}(r)\partial_{\bar{t}}\psi+E_{2}(r)\partial_{r}\psi+\mathcal{G}F,$

where recall $g^{rr}=1-\mu$. We differentiate (6.9) by

(6.10) $\partial_{r}^{i_{3}-2},$

and then we square the result and note that there exists a constant
$C(j,M,\Lambda)>0$ such that

(6.11) $\displaystyle(1-\mu)^{2}(\partial_{r}^{i_{3}}\mathcal{G}\psi)^{2}$
$\displaystyle\leq
C(j,M,\Lambda)\left(\left(\partial_{r}^{i_{3}-2}\left(\frac{1}{\sqrt{1-\mu}}\right)\frac{2\sqrt{1-9M^{2}\Lambda}}{\sqrt{1+\frac{6M}{r}}}(\partial_{r}\partial_{\bar{t}}\psi)\right)^{2}+\dots+\left(\partial_{r}^{i_{3}-2}\mathcal{G}F\right)^{2}\right)$

where, in inequality (6.11) the three terms displayed are from left to right
the highest $\partial_{r}$ derivative term, the fastest degenerating $(1-\mu)$
term and the contribution of the inhomogeneity $F$. We rewrite (6.11) as

(6.12) $\displaystyle(1-\mu)^{2}(\partial_{r}^{i_{3}}\mathcal{G}\psi)^{2}$
$\displaystyle\leq
C(j,M,\Lambda)\left(\frac{1}{(1-\mu)^{2i_{3}-3}}\left(\partial_{r}\partial_{\bar{t}}\psi\right)^{2}+\dots+\left(\partial_{r}^{i_{3}-2}\mathcal{G}F\right)^{2}\right)$

and therefore, by multiplying (6.12) with $\frac{(1-\mu)^{2i_{3}-3}}{r}$, so
that no terms blow up at the roots of $1-\mu=0$, we obtain

(6.13)
$\displaystyle\frac{(1-\mu)^{2i_{3}-1}}{r}(\partial_{r}^{i_{3}}\mathcal{G}\psi)^{2}$
$\displaystyle\leq
C(j,M,\Lambda)\left(\frac{1}{r^{3}(1-\mu)}\left(\partial_{\bar{t}}\mathcal{G}\psi\right)^{2}+\dots+\frac{(1-\mu)^{2i_{3}-3}}{r}\left(\partial_{r}^{i_{3}-2}\mathcal{G}F\right)^{2}\right).$

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) $\displaystyle E^{\prime}_{\mathcal{G},j}[\psi](\tau_{2})$
$\displaystyle\quad\quad+\int\int_{D(\tau_{1},\tau_{2})}\sum_{0\leq
i_{1}+i_{2}=j-2}\sum_{\alpha}\frac{1}{r(1-\mu)}\left(\partial_{\bar{t}}^{1+i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right)^{2}+\sum_{1\leq
i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\frac{1}{r}\left|\not{\nabla}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right|^{2}$
$\displaystyle\quad\quad+\int\int_{D(\tau_{1},\tau_{2})}\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq j-1,i_{3}\geq
1}\sum_{\alpha}\frac{(1-\mu)^{2i_{3}-1}}{r}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}\psi\right)^{2}+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-1}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{2}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}$
$\displaystyle\quad\leq
CE^{\prime}_{\mathcal{G},j}[\psi](\tau_{1})+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)r\sum_{0\leq
i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}F\right)^{2}+\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}.$
$\displaystyle\qquad\qquad\qquad\qquad+C\int_{\phi_{\tau_{2}}(\Sigma)}\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$

for all $\tau_{1}\leq\tau_{2}$ and for all $j\geq 2$.

Now, we want to estimate from below
$E^{\prime}_{\mathcal{G},j}[\psi](\tau_{2})$ by
$E_{\mathcal{G},j}[\psi](\tau_{2})$, in the energy estimate (6.14), for all
orders $j\geq 3$ at the expense of producing hypersurface error terms. We note
that there exists a constant $c(j,M,\Lambda)$ such that

(6.15) $\displaystyle
c(j,M,\Lambda)\int_{\\{\bar{t}=\tau\\}}\sum_{i_{1}+i_{2}=j-2}\sum_{\alpha}\left(\frac{1}{r}(\partial_{\bar{t}}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi)^{2}+(1-\mu)r(\partial_{r}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi)^{2}+r\left|\not{\nabla}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right|^{2}\right)$
$\displaystyle\qquad+c(j,M,\Lambda)E_{j-1}[\psi](\tau)$ $\displaystyle\leq
E^{\prime}_{\mathcal{G},j}[\psi](\tau).$

Then, we want to obtain the top order derivatives

(6.16) $\partial_{r}^{i_{3}}\mathcal{G}\psi,\qquad i_{3}=j-1,$

on the left hand side, with the appropriate degenerative weights of $(1-\mu)$.
Therefore, we multiply the pointwise estimate (6.12) with $(1-\mu)^{2i_{3}-3}$
and sum it to (6.15) to obtain that there exist constants

(6.17) $c(j,M,\Lambda),C(j,M,\Lambda)>0$ (6.18) $\displaystyle
c(j,M,\Lambda)\int_{\\{\bar{t}=\tau\\}}\sum_{0\leq i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\mathcal{E}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi,\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\psi\right)+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq j-1,i_{3}\geq
1}\sum_{\alpha}r(1-\mu)^{2i_{3}-1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}\psi\right)^{2}$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{1\leq i_{1}+i_{2}+i_{3}\leq
j-1,i_{3}\geq
1}\sum_{\alpha}r\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}$
$\displaystyle\quad\leq E^{\prime}_{\mathcal{G},j}[\psi](\tau)$
$\displaystyle\qquad+C(j,M,\Lambda)\int_{\\{\bar{t}=\tau\\}}\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}(1-\mu)^{2i_{3}+1}r\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}F\right)^{2}+\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}r\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}.$

Note that the left hand side of (6.18) is similar to
$E_{\mathcal{G},j}[\psi](\tau)$.

Finally, by using the energy estimate (6.14) in conjunction with the pointwise
estimate (6.18) we conclude the result of the Corollary. ∎

######  Remark 6.1.

For an example of the computation (6.13) note that at order $j=3$ we obtain

(6.19)
$\displaystyle\frac{(1-\mu)^{3}}{r}\left(\partial_{r}^{2}\mathcal{G}\psi\right)^{2}$
$\displaystyle\leq\frac{C(M,\Lambda)}{r}\Bigg{(}(1-\mu)\left(g^{\bar{t}\bar{t}}\partial_{t}^{2}\mathcal{G}\psi+2g^{r\bar{t}}\partial_{t}\partial_{r}\mathcal{G}\psi+\not{\Delta}\mathcal{G}\psi\right)^{2}+(1-\mu)\frac{1}{(1-\mu)}(\partial_{\bar{t}}\partial_{r}\psi)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad+(1-\mu)E_{1}^{2}(\partial_{\bar{t}}\psi)^{2}+(1-\mu)E_{2}^{2}(\partial_{r}\psi)^{2}+(1-\mu)(\mathcal{G}F)^{2}\Bigg{)}$
$\displaystyle\leq\frac{C(M,\Lambda)}{r}\Bigg{(}(1-\mu)\left(g^{\bar{t}\bar{t}}\partial_{t}^{2}\mathcal{G}\psi+2g^{r\bar{t}}\partial_{t}\partial_{r}\mathcal{G}\psi+\not{\Delta}\mathcal{G}\psi\right)^{2}+(\partial_{\bar{t}}\partial_{r}\psi)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad+(1-\mu)E_{1}^{2}(r)(\partial_{\bar{t}}\psi)^{2}+(1-\mu)E_{2}^{2}(r)(\partial_{r}\psi)^{2}+(1-\mu)(\mathcal{G}F)^{2}\Bigg{)}$
$\displaystyle\leq
C(M,\Lambda)\Bigg{(}\frac{1-\mu}{r^{5}}\left(\partial_{t}^{2}\mathcal{G}\psi\right)^{2}+\frac{1-\mu}{r}\left(\partial_{t}\partial_{r}\mathcal{G}\psi\right)^{2}+\frac{1-\mu}{r}\left(\not{\Delta}\mathcal{G}\psi\right)^{2}+\frac{1}{r^{3}(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\frac{1-\mu}{r^{3}}(\partial_{\bar{t}}\psi)^{2}+\frac{1}{r^{7}}(\partial_{r}\psi)^{2}+\frac{1-\mu}{r}\left(\mathcal{G}F\right)^{2}\Bigg{)},$

where the constant $C(M,\Lambda)$ does not degenerate in the limit
$\Lambda\rightarrow 0$.

##  7\. The Schwarzschild case $\Lambda=0$

We can also prove the equivalent of Theorem 2 on the asymptotically flat
Schwarzschild exterior. This result had been obtained previously by [20]. For
completeness, we give a treatment here in our set up.

We study the inhomogeneous wave equation

(7.1) $\Box_{g_{M}}\psi=F.$

Before stating the theorem let us introduce some preliminary notions,
specifically for the Schwarzschild case.

First, we define regular hyperboloidal coordinates $(\bar{t},r,\theta,\phi)$
on which the metric takes the form

(7.2)
$g_{M}=-\left(1-\frac{2M}{r}\right)(d\bar{t})^{2}-2\left(1-\frac{3M}{r}\right)\sqrt{1+\frac{6M}{r}}d\bar{t}dr+\frac{27M^{2}}{r^{2}}(dr)^{2}+r^{2}d\sigma_{\mathbb{S}^{2}}.$

These correspond to the regular hyperboloidal coordinates of Section 2.2, when
we take $\Lambda=0$. Moreover, we can define the Schwarzschild coordinates
$(t,r,\theta,\phi)$ similarly to the Schwarzschild–de Sitter coordinates, see
Section 2.

We may attach null infinity

(7.3)
$\mathcal{I}^{+}=\\{(\bar{t},r=+\infty,\theta,\phi)\in\mathbb{R}\times\\{+\infty\\}\times\mathbb{S}^{2}\\},$

as a boundary in the obvious way. Formally, we consider its normal to be
$n_{\mathcal{I}^{+}}=\frac{\partial}{\partial\bar{t}}$ and note

(7.4)
$J^{T}_{\mu}[\psi]n_{\mathcal{I}^{+}}^{\mu}=\left(\partial_{\bar{t}}\psi\right)^{2}.$

Let

(7.5) $\tilde{\Sigma}=\\{\bar{t}=0\\}.$

Then the leaves of the foliation

(7.6) $\phi_{\tau}(\tilde{\Sigma})=\\{\bar{t}=\tau\\}$

connect the event horizon $\mathcal{H}^{+}$ with future null infinity
$\mathcal{I}^{+}$. Also, we define the spacetime domain

(7.7) $\tilde{D}(\tau_{1},\tau_{2})=\\{\tau_{1}\leq\bar{t}\leq\tau_{2}\\}.$

The reader familiar with the Penrose diagram may want to consult Figure 4.

Figure 4. The foliation of the Schwarzschild exterior

Note that the volume form of (7.7) is

(7.8) $dg=r^{2}\sin{\theta}d\bar{t}d\theta d\phi$

and the volume form of (7.6) is

(7.9) $dg_{\\{\bar{t}=c\\}}=3\sqrt{3}Mr\sin{\theta}drd\theta d\phi.$

and the normal of the $\\{\bar{t}=c\\}$ hypersurface is

(7.10)
$n_{\phi_{\tau}(\Sigma)}=\frac{\sqrt{27}M}{r}\frac{\partial}{\partial\bar{t}}+r\frac{\xi(r)}{\sqrt{27}M}\frac{\partial}{\partial
r},$

where $\xi(r)=\left(1-\frac{3M}{r}\right)\sqrt{1+\frac{6M}{r}}$.

The vector field $\mathcal{G}$ here takes the form

(7.11) $\mathcal{G}=r\sqrt{1-\frac{2M}{r}}\frac{\partial}{\partial
r}=\frac{r}{\sqrt{1-\mu}}\frac{\partial}{\partial
r^{\star}}+\frac{r}{\sqrt{1-\mu}}\left(1-\frac{3M}{r}\right)\sqrt{1+\frac{6M}{r}}\frac{\partial}{\partial
t}$

and the energy density,

(7.12)
$\displaystyle\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)\>\dot{=}\>\mathbb{T}(\partial_{\bar{t}},n)[\mathcal{G}\psi]+\mathbb{T}(N,n)[\psi],$

is now defined with $\mathcal{G}$ 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 $\mathcal{H}^{+}$ is equal to
$\partial_{\bar{t}}$. We also obtain

(7.13) $\displaystyle\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)$
$\displaystyle\sim_{M}\frac{1}{r}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+r\left(1-\frac{2M}{r}\right)(\partial_{r}\mathcal{G}\psi)^{2}+r|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1}{r}(\partial_{\bar{t}}\psi)^{2}+r(\partial_{r}\psi)^{2}+r|\not{\nabla}\psi|^{2}$
$\displaystyle\sim_{M}r\left(\frac{1}{r^{2}}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+\left(1-\frac{2M}{r}\right)(\partial_{r}\mathcal{G}\psi)^{2}+|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1}{r^{2}}(\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}\right).$

######  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
$\mathcal{I}^{+}$ in the place of $\bar{\mathcal{\mathcal{H}}}^{+}$. The
energy flux at null infinity $\mathcal{I}^{+}$, and the event horizon
$\mathcal{H}^{+}$, 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.

######  Theorem 7.1 (Theorem 1.1 in [8]).

Let $\psi$ satisfy equation (7.1) on a fixed Schwarzschild background. Then,
for initial data vanishing at infinity, for all $0<\eta\leq 1$ there exists a
constant $C=C(M,\eta)$ such that

(7.14)
$\displaystyle\int\int_{\tilde{D}(0,\tau)}r^{-1}\left(1-\frac{3M}{r}\right)^{2}\left(r^{-\eta}(\partial_{\bar{t}}\psi)^{2}+\left|\not{\nabla}\psi\right|^{2}\right)$
$\displaystyle+r^{-1-\eta}\left(\partial_{r}\psi\right)^{2}+r^{-3-\eta}\psi^{2}$
$\displaystyle\leq
C\int_{\tilde{\Sigma}}J^{N}_{\mu}[\psi]n^{\mu}+C\int\int_{\tilde{D}(0,\tau)}|\partial_{\bar{t}}\psi\cdot
F|+r^{1+\eta}|F|^{2}$

and

(7.15)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}J^{N}_{\mu}[\psi]n^{\mu}\leq
C\int_{\tilde{\Sigma}}J^{N}_{\mu}[\psi]n^{\mu}+C\int\int_{\tilde{D}(0,\tau)}|\partial_{\bar{t}}\psi\cdot
F|+r^{1+\eta}|F|^{2}$

where $N$ is the redshift vector field of Dafermos–Rodnianski, see the Lecture
notes [11].

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
$\Lambda=0$. Now, adding the Morawetz estimate of Theorem 7.1, noting the
weights and absorbing relevant terms, we conclude the following.

######  Theorem 7.2.

There exists a constant $C(M,\eta)>0$, such that if $\psi$ satisfies

$\Box_{g_{M}}\psi=F$

on the spacetime domain $\tilde{D}(0,\tau)$, the following holds

(7.16)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)+\int\int_{\tilde{D}(0,\tau)}$
$\displaystyle\frac{1}{r(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1}{r^{1+\eta}}\left((\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}\right)$
$\displaystyle\quad\leq
C\int_{\tilde{\Sigma}}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)+C\int\int_{\tilde{D}(0,\tau)}r|1-\mu|\left|\mathcal{G}F\right|^{2}+r^{1+\eta}|F|^{2}$

for any $\eta\in(0,1]$.

###### Proof.

By repeating the arguments of Section 4, and by keeping track of the weights
in $r$, we conclude the following estimate

(7.17)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}J^{\partial_{\bar{t}}}_{\mu}[\mathcal{G}\psi]n^{\mu}+\int\int_{\tilde{D}(0,\tau)}\frac{1}{r(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}$
$\displaystyle\quad\lesssim\int_{\tilde{\Sigma}}J^{\partial_{\bar{t}}}_{\mu}[\mathcal{G}\psi]n^{\mu}+\int\int_{\tilde{D}(0,\tau)}\partial_{\bar{t}}\mathcal{G}\psi\cdot\mathcal{G}F+\partial_{\bar{t}}\mathcal{G}\psi\left(E_{2}(r)\partial_{t}\psi+E_{1}(r)\partial_{r}\psi\right),$

where for $E_{1}(r),E_{2}(r)$ see Proposition 4.6. Then, to obtain all of the
derivatives of $\mathcal{G}\psi$ 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)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}J^{\partial_{\bar{t}}}_{\mu}[\mathcal{G}\psi]n^{\mu}+\int\int_{\tilde{D}(0,\tau)}\frac{1}{r(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}\left(\partial_{r}\mathcal{G}\psi\right)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}$
$\displaystyle\quad\lesssim\int_{\tilde{\Sigma}}J^{\partial_{\bar{t}}}_{\mu}[\mathcal{G}\psi]n^{\mu}$
$\displaystyle\quad\quad+\int\int_{\tilde{D}(\tau_{1},\tau_{2})}r^{-2}(\partial_{r}\psi)^{2}+r^{-2}(\partial_{t}\psi)^{2}+\int_{\phi_{\tau_{2}}(\tilde{\Sigma})}r(\partial_{r}\psi)^{2}+\int_{\phi_{\tau_{1}}(\tilde{\Sigma})}r(\partial_{r}\psi)^{2}$
$\displaystyle\quad\quad+\int\int_{\tilde{D}(0,\tau)}\partial_{\bar{t}}\mathcal{G}\psi\cdot\mathcal{G}F+\partial_{\bar{t}}\mathcal{G}\psi\left(E_{2}(r)\partial_{t}\psi+E_{1}(r)\partial_{r}\psi\right)+\left|\frac{1}{r}\mathcal{G}\psi\cdot
F\right|.$

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)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)$
$\displaystyle+\int\int_{\tilde{D}(0,\tau)}\frac{1}{r(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1}{r^{1+\eta}}\left((\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}\right)$
$\displaystyle\quad\lesssim\int_{\tilde{\Sigma}}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)+\int\int_{\tilde{D}(0,\tau)}|\partial_{\bar{t}}\mathcal{G}\psi\cdot\mathcal{G}F|+|\partial_{\bar{t}}\psi\cdot
F|+r^{1+\eta}|F|^{2},$

for any $\eta\in(0,1]$, 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.

######  Corollary 7.1.

Under the assumptions of the above theorem, and the additional assumptions
that, for the vector field $a=a^{j}\partial_{j}$ and the scalar function $b$
the following

(7.20)
$a^{\bar{t}},a^{r},g^{\theta\theta}(a^{\theta})^{2}+g^{\phi\phi}(a^{\phi})^{2},b(r)$

are smooth functions on $\tilde{D}(0,\infty)$, with

(7.21)
$a^{\bar{t}}(r,\cdot),a^{r}(r,\cdot),\left(g^{\theta\theta}(a^{\theta})^{2}+g^{\phi\phi}(a^{\phi})^{2}\right)(r,\cdot),|[\mathcal{G},a]^{\bar{t}}(r,\cdot)|\leq\frac{C}{r^{2}}$

and

(7.22) $b(r,\cdot),|\mathcal{G}b|(r,\cdot)\leq\frac{C}{r^{3}},$

then we have the following.

There exists a constant $C=C(M,\eta)$ such that for solutions of

(7.23) $\Box_{g_{M}}\psi=\epsilon a^{j}\partial_{j}\psi+\epsilon b(r)\psi,$

and for $\epsilon$ sufficiently small, we have

(7.24)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)$
$\displaystyle\quad+\int\int_{\tilde{D}(0,\tau)}\frac{1}{r(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+\frac{1-\mu}{r}(\partial_{r}\mathcal{G}\psi)^{2}+\frac{1}{r}|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1}{r^{1+\eta}}\left((\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}\right)$
$\displaystyle\leq
C\int_{\tilde{\Sigma}}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi),$

and

(7.25)
$\displaystyle\int_{\phi_{\tau}(\tilde{\Sigma})}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi)$
$\displaystyle\quad+\int_{0}^{\tau}d\tau\int_{\phi_{\tau}(\tilde{\Sigma})}\frac{1}{(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}+(1-\mu)(\partial_{r}\mathcal{G}\psi)^{2}+|\not{\nabla}\mathcal{G}\psi|^{2}+\frac{1}{r^{\eta}}\left((\partial_{\bar{t}}\psi)^{2}+(\partial_{r}\psi)^{2}+|\not{\nabla}\psi|^{2}\right)$
$\displaystyle\leq
C\int_{\tilde{\Sigma}}\tilde{\mathcal{E}}(\mathcal{G}\psi,\psi),$

since by the relation of the volume forms, see Section 2.14, we have

(7.26)
$\int_{0}^{\tau}d\tau\int_{\phi_{\tau}(\Sigma)}\left(\cdot\right)\sim\int\int_{D(0,\tau)}\frac{1}{r}\left(\cdot\right),$

where the constants in the above similarity depend only on the black hole mass
$M$.

######  Remark 7.2.

The bulk term, on the left hand side of inequality (7.25), and the
hypersurface term, on the right hand side of inequality (7.25), have different
weights in $r$. Specifically, some of the terms of the right hand side, have
larger weights in $r$. It is because of this reason that one does not obtain
exponential decay for $\psi$, on a Schwarzschild exterior.

######  Remark 7.3.

The reason we can include the zero’th order term, with component $b(r)$, in
equation (7.23), as opposed to equation (3.7) on the Schwarzschild–de Sitter
case, is the lack of the $\psi_{\infty}$ term in (7.14) for initial data
vanishing at infinity.

######  Remark 7.4.

Our Corollary 7.1, without the
$\frac{1}{r(1-\mu)}(\partial_{\bar{t}}\mathcal{G}\psi)^{2}$ term and slightly
different weights of $r$ on the left hand side, coincides with Theorem 4.1 of
[20].

Note that we have the following higher order Theorem

######  Theorem 7.3.

Under the assumption of Theorem 7.2, for all $j\geq 3$ there exists a positive
constant $C=C(j,M,\eta)>0$ such that

(7.27) $\displaystyle\tilde{E}_{\mathcal{G},j}[\psi](\tau_{2})$
$\displaystyle\quad+\int\int_{\tilde{D}(\tau_{1},\tau_{2})}\sum_{0\leq
i_{1}+i_{2}=j-2}\sum_{\alpha}\frac{1}{r(1-\mu)}\left(\partial_{\bar{t}}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right)^{2}+\frac{1}{r}\left|\not{\nabla}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi\right|^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\sum_{1\leq
i_{1}+i_{2}+i_{3}=j-1}\sum_{\alpha}\frac{(1-\mu)^{2i_{3}-1}}{r}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}\psi\right)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\sum_{1\leq i_{1}+i_{2}+i_{3}\leq
j-1,i_{2}<j-1}\sum_{\alpha}\frac{1}{r^{1+\eta}}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}+\frac{1}{r}\left|\not{\nabla}\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right|^{2}$
$\displaystyle\quad\leq
C\tilde{E}_{\mathcal{G},j}[\psi](\tau_{1})+C\int\int_{D(\tau_{1},\tau_{2})}(1-\mu)r\sum_{0\leq
i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}F\right)^{2}+\sum_{0\leq
i_{1}+i_{2}+i_{3}\leq
j-2}\sum_{\alpha}r^{1+\eta}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$
$\displaystyle\quad\quad\quad+C\int_{\\{\bar{t}=\tau_{2}\\}}\sum_{0\leq
i_{1}+i_{2}\leq
j-3}\sum_{\alpha}\tilde{\mathcal{E}}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}F,\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}F\right)+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}r(1-\mu)^{2i_{3}+1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}F\right)^{2}$
$\displaystyle\qquad\qquad\qquad+\sum_{0\leq i_{1}+i_{2}+i_{3}\leq
j-3}\sum_{\alpha}r\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}F\right)^{2}$

for all $0\leq\tau_{1}\leq\tau_{2}$, where

(7.28) $\displaystyle\tilde{E}_{\mathcal{G},j}[\psi](\tau)$
$\displaystyle=\int_{\\{\bar{t}=\tau\\}}\sum_{0\leq i_{1}+i_{2}\leq
j-2}\sum_{\alpha}\tilde{\mathcal{E}}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\mathcal{G}\psi,\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\psi\right)+\sum_{1\leq
i_{1}+i_{2}+i_{3}\leq j-1,i_{3}\geq
1}\sum_{\alpha}r(1-\mu)^{2i_{3}-1}\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\mathcal{G}\psi\right)^{2}$
$\displaystyle\qquad\qquad+\sum_{1\leq i_{1}+i_{2}+i_{3}\leq j-1,i_{3}\geq
1}\sum_{\alpha}r\left(\partial_{\bar{t}}^{i_{1}}\Omega_{\alpha}^{i_{2}}\partial_{r}^{i_{3}}\psi\right)^{2}$

where $N$ is the redshift vector field of Dafermos–Rodnianski, see the Lecture
notes [11]. The $j=2$ case is the same without the hypersurface error terms on
the right hand side of (7.27). For the volume forms of the spacetime domains
and of the hypersurface terms see Section 2.14.

###### Proof.

The proof of this Theorem is essentially the same as Theorem 3 of the
Schwarzschild–de Sitter case. Note that in the proof of Theorem 3 we kept some
‘key’ weights in $r$ so that the proof can be read for the asymptotically flat
case. ∎

######  Remark 7.5.

In order to write inequality (7.27) in its most compact form we had to
slightly differ its presentation from the relevant inequality of Theorem 3,
since the weights in $r$ are now significant.

##  Appendix A The Christoffel symbols of the metric (2.4)

The non-zero Christoffel symbols, of the metric (2.4), are

(A.1)
$\displaystyle\Gamma^{\bar{t}}\>_{\bar{t}\bar{t}}=\frac{\sqrt{1+\frac{6M}{r}}(r-3M)(-3M+r^{2}\Lambda)}{3r^{3}\sqrt{1-9M^{2}\Lambda}},\quad\Gamma^{\bar{t}}\>_{\bar{t}r}=\frac{9M^{2}(3M-r^{3}\Lambda)}{r^{4}(1-9M^{2}\Lambda)}$
$\displaystyle\Gamma^{\bar{t}}\>_{rr}=\frac{27M^{2}(9M^{2}+3Mr+r^{2})}{\sqrt{1+\frac{6M}{r}}r^{5}\left(1-9M^{2}\Lambda\right)^{3/2}},\quad\Gamma^{\bar{t}}\>_{\theta\theta}=\frac{\sqrt{1+\frac{6M}{r}}(r-3M)}{\sqrt{1-9M^{2}\Lambda}},$
$\displaystyle\Gamma^{\bar{t}}\>_{\phi\phi}=\frac{\sqrt{1+\frac{6M}{r}}(r-3M)\sin{\theta}}{\sqrt{1-9M^{2}\Lambda}},\quad\Gamma^{r}\>_{\bar{t}\bar{t}}=\frac{(-3M+r^{3}\Lambda)(6M-3r+r^{3}\Lambda)}{9r^{3}},$
$\displaystyle\Gamma^{r}\>_{\bar{t}r}=-\frac{\sqrt{1+\frac{6M}{r}}(r-3M)(-3M+r^{3}\Lambda)}{3r^{3}\sqrt{1-9M^{2}\Lambda}},\quad\Gamma^{r}_{rr}=-\frac{9M^{2}(3M-r^{3}\Lambda)}{r^{4}(1-9M^{2}\Lambda)}$
$\displaystyle\Gamma^{r}\>_{\theta\theta}=-r\left(1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\right),\quad\Gamma^{r}\>_{\phi\phi}=-r\left(1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}\right)\sin{\theta}$
$\displaystyle\Gamma^{\theta}\>_{r\theta}=\frac{1}{r},\quad\Gamma^{\theta}\>_{\phi\phi}=-\frac{\cos{\theta}}{2},\quad\Gamma^{\phi}\>_{r\phi}=\frac{1}{r},\quad\Gamma^{\phi}\>_{\theta\phi}=\frac{\cot{\theta}}{2}.$

## References

  * [1] Stefanos Aretakis.  Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I.  Comm. Math. Phys., 307(1):17–63, 2011.
  * [2] Antônio Sá Barreto and Maciej Zworski.  Distribution of resonances for spherical black holes.  Math. Res. Lett., 4(1):103–121, 1997.
  * [3] Jean-François Bony and Dietrich Häfner.  Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric.  Comm. Math. Phys., 282(3):697–719, 2008.
  * [4] N. Burq.  Smoothing effect for Schrödinger boundary value problems.  Duke Math. J., 123(2):403–427, 2004.
  * [5] Nicolas Burq, Colin Guillarmou, and Andrew Hassell.  Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics.  Geom. Funct. Anal., 20(3):627–656, 2010.
  * [6] Demetrios Christodoulou.  The action principle and partial differential equations, volume 146 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2000.
  * [7] Demetrios Christodoulou and Sergiu Klainerman.  The global nonlinear stability of the Minkowski space, volume 41 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1993.
  * [8] Mihalis Dafermos, Gustav Holzegel, and Igor Rodnianski.  Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: The case $|a|\ll M$.  Ann. PDE, 5(1):Paper No. 2, 118, 2019.
  * [9] Mihalis Dafermos and Igor Rodnianski.  A note on energy currents and decay for the wave equation on a Schwarzschild background.  September 2007.  arXiv:0710.0171.
  * [10] Mihalis Dafermos and Igor Rodnianski.  The wave equation on Schwarzschild-de Sitter spacetimes, September 2007\.  arXiv:0709.2766.
  * [11] Mihalis Dafermos and Igor Rodnianski.  Lectures on black holes and linear waves.  In Evolution equations, volume 17 of Clay Math. Proc., pages 97–205. Amer. Math. Soc., Providence, RI, 2013.
  * [12] Semyon Dyatlov.  Exponential energy decay for Kerr–de Sitter black holes beyond event horizons.  Math. Res. Lett., 18(5):1023–1035, 2011.
  * [13] Semyon Dyatlov.  Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole.  Comm. Math. Phys., 306(1):119–163, 2011.
  * [14] Semyon Dyatlov.  Spectral gaps for normally hyperbolic trapping.  Ann. Inst. Fourier (Grenoble), 66(1):55–82, 2016.
  * [15] Grigorios Fournodavlos and Jan Sbierski.  Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole.  Arch. Ration. Mech. Anal., 235(2):927–971, 2020.
  * [16] Anne T. Franzen.  Boundedness of massless scalar waves on Kerr interior backgrounds.  Ann. Henri Poincaré, 21(4):1045–1111, 2020.
  * [17] Peter Hintz and Andras Vasy.  Non-trapping estimates near normally hyperbolic trapping.  Math. Res. Lett., 21(6):1277–1304, 2014.
  * [18] Peter Hintz and András Vasy.  Analysis of linear waves near the Cauchy horizon of cosmological black holes.  J. Math. Phys., 58(8):081509, 45, 2017.
  * [19] Peter Hintz and András Vasy.  The global non-linear stability of the Kerr–de Sitter family of black holes.  Acta Math., 220(1):1–206, 2018.
  * [20] Gustav Holzegel and Christopher Kauffman.  A note on the wave equation on black hole spacetimes with small non-decaying first order terms, May 2020.  arXiv:2005.13644.
  * [21] Mitsuru Ikawa.  Decay of solutions of the wave equation in the exterior of several convex bodies.  Ann. Inst. Fourier (Grenoble), 38(2):113–146, 1988.
  * [22] Friedrich Kottler.  Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie.  Annalen der Physik, 361(14):401–462, January 1918.
  * [23] Hans Lindblad and Igor Rodnianski.  The global stability of Minkowski space-time in harmonic gauge.  Ann. of Math. (2), 171(3):1401–1477, 2010.
  * [24] Georgios Mavrogiannis.  Quasilinear wave equations on Schwarzschild de Sitter.  preprint.
  * [25] Georgios Mavrogiannis.  Relatively non degenerate estimates on Kerr de Sitter and applications.  in preparation.
  * [26] Richard Melrose, Antônio Sá Barreto, and András Vasy.  Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space.  Comm. Partial Differential Equations, 39(3):512–529, 2014.
  * [27] Stéphane Nonnenmacher and Maciej Zworski.  Quantum decay rates in chaotic scattering.  Acta Math., 203(2):149–233, 2009.
  * [28] Antônio Sá Barreto and Maciej Zworski.  Distribution of resonances for spherical black holes.  Math. Res. Lett., 4(1):103–121, 1997.
  * [29] Jan Sbierski.  Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes.  Anal. PDE, 8(6):1379–1420, 2015.
  * [30] Volker Schlue.  Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies.  Comm. Math. Phys., 334(2):977–1023, 2015.
  * [31] Karl Schwarzschild.  Über das Gravitationsfeld eines Massenpunktes nach der EinsteinschenTheorie.  Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften1, page 189–196, 1916.
  * [32] J. M. Stewart.  Solutions of the wave equation on a Schwarzschild space-time with localized energy.  Proc. Roy. Soc. London Ser. A, 424(1866):239–244, 1989.
  * [33] András Vasy.  The wave equation on asymptotically de Sitter-like spaces.  Adv. Math., 223(1):49–97, 2010.
  * [34] András Vasy.  Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov).  Invent. Math., 194(2):381–513, 2013.
  * [35] Hermann Weyl.  Uber die statischen kugelsymmetrischen Losungen von Einsteins kosmologischen Gravitationsgleichungen.  Phys. Z. 20, pages 31–34, 1919.