# Integrated Local Energy Decay for the Damped Wave Equation on Stationary
Space-Times

Collin Kofroth  Department of Mathematics, University of North Carolina,
Chapel Hill<EMAIL_ADDRESS>
###### Abstract.

We prove integrated local energy decay for the damped wave equation on
stationary, asymptotically flat space-times in $(1+3)$ dimensions. Local
energy decay constitutes a powerful tool in the study of dispersive partial
differential equations on such geometric backgrounds. By utilizing the
geometric control condition to handle trapped trajectories, we are able to
recover high frequency estimates without any loss. We may then apply known
estimates from the work of Metcalfe, Sterbenz, and Tataru in the medium and
low frequency regimes in order to establish local energy decay. This
generalizes the integrated version of results established by Bouclet and Royer
from the setting of asymptotically Euclidean manifolds to the full Lorentzian
case.

Funding. The author gratefully acknowledges the partial support afforded by
the F. Ivy Carroll Summer Research Fellowship
awarded by the UNC-CH Graduate School and partial funding from NSF grant
DMS-2054910 (PI: Jason Metcalfe).

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Problem Setup and Main Results
    3. 1.3 Cutoff Notation
    4. 1.4 Acknowledgements
  2. 2 High Frequency Analysis
    1. 2.1 Introduction
    2. 2.2 Dynamical Framework
    3. 2.3 Results on the Flow
    4. 2.4 Escape Function Construction
    5. 2.5 Starting Energy Estimates and Case Reductions
    6. 2.6 Proof of the High Frequency Estimate
  3. 3 Local Energy Decay
    1. 3.1 Medium Frequencies
    2. 3.2 Low Frequencies
    3. 3.3 Establishing Local Energy Decay

##  1\. Introduction

###  1.1. Background

The goal of this paper is to establish local energy decay for the damped wave
equation on asymptotically flat space-times with time-independent metrics (in
the sense that $\partial_{t}$ is a Killing field) subject to the geometric
control condition. The primary advance in this work is recovering the high
frequency local energy estimate present in [mst20] for waves on non-trapping
space-times. Since the aforementioned work only utilized the non-trapping
assumption at high frequencies, this establishes the key step in extending
time-integrated versions of previously-known results for damped waves on
product manifolds (see [BR14]) to the full Lorentzian case. From the proven
high frequency estimate, we may apply known results in [mst20] to conclude
local energy decay to complete this extension.

Local energy estimates are a collection of rich and well-studied quantities
within the field of dispersive partial differential equations, originally
introduced on Minkowski space in classical works such as [Mora66, Mora68,
Mora75], [MRS77]. A particularly important class of local energy estimates are
the integrated local energy estimates; if $u$ solves the homogeneous flat wave
equation

$(\partial_{t}^{2}-\Delta)u=0,\qquad\Delta=\sum\limits_{j=1}^{n}\partial_{x_{j}}^{2}$

in spatial dimension $n\geq 3$, then the integrated local energy estimate
which we are interested in takes the form

(1.1) $\displaystyle\sup_{j\geq 0}\left(\left\lVert\langle
x\rangle^{-1/2}\partial
u\right\rVert_{L^{2}_{t}L^{2}_{x}\big{(}\mathbb{R}_{+}\times\\{\langle
x\rangle\approx 2^{j}\\}\big{)}}+\left\lVert\langle
x\rangle^{-3/2}u\right\rVert_{L^{2}_{t}L^{2}_{x}\big{(}\mathbb{R}_{+}\times\\{\langle
x\rangle\approx 2^{j}\\}\big{)}}\right)\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}},$

where $\partial=(\partial_{t},\nabla)$ denotes the space-time gradient, and
$\left\langle x\right\rangle=(1+|x|^{2})^{1/2}$ denotes the Japanese bracket
of $x$. This estimate is known to hold in the flat setting through a positive
commutator argument using the multiplier introduced in the appendix of [SR05].
In such a case, we will say that (integrated) local energy decay holds. This
is a quantitative statement on dispersion, and it heuristically expresses that
the energy of the wave must decay quickly enough within compact spatial sets
to be integrable in time. Estimates of this form have significant utility, as
they have been used to prove other important measures of dispersion such as
Strichartz estimates (see [BT07, BT08], [JSS90, JSS91], [MMT08], [MMTT10],
[MT09, MT12], [Tat08], [Toh12], and the references therein) and pointwise
decay estimates (see [Hi22], [Looi21], [MTT12], [Morg20], [MW21], [Tat13], and
references in these works). Additionally, local energy estimates have
applications to nonlinear wave equations where one can develop estimates on an
appropriate linearization of the problem, viewing the nonlinearity as a
perturbation. These techniques have been applied in many works; see e.g.
[BH10], [KSS02, KSS04], [MS06, MS07], [SW10], and the citations contained in
them. We will be focused on establishing local energy decay rather than
demonstrating its utility via applications.

In [mst20], the authors proved that local energy decay holds for a broad class
of stationary wave operators if and only if

  1. (1)

The space-time is non-trapping: There are no null bicharacteristic rays which
stay within a compact set for all time.

  2. (2)

The operator satisfies certain spectral assumptions: Upon replacing time
derivatives in the wave operator with a complex parameter, one requires that
this family of operators have no eigenvalues in the lower half-plane nor real
resonances/embedded eigenvalues (see [mst20] for more precise definitions);
equivalently, one requires analytic continuation of the inverse (resolvent) of
this family of operators to the entire lower half plane and continuous
extension to the real line.

They also established results for almost stationary operators, though that is
not the context of the work presented here. While the authors employed a non-
trapping hypothesis, their work did not require product structure on their
space-times, which makes their work highly influential in our own.

Although the absence of trapping is known to be necessary for waves to
experience local energy decay (see [Ral69], [Sb15]), one can recover weak
local energy decay estimates with a prescribed loss at high frequencies for
certain types of trapping (see [Burq98], [Chr08], [Ika82, Ika88], [MMTT10],
[NZ09], [Toh12], [WZ11], and the contained references). When the trapping is
sufficiently weak/unstable, then this loss is nominal (in fact, logarithmic);
this is the case for both the Schwarzschild ([MMTT10]) and Kerr ([Toh12])
space-times. Both space-times possess non-trivial trapped sets, which
constitute regions where light remains for all time. Although one can extract
weak local energy decay estimates, the trapping still generates an immutable
barrier to full local energy decay. We will not be working in a scenario that
generates loss, although we would be remiss if we did not briefly mention weak
local energy decay and essential space-times that enjoy it.

The study of damped waves also possesses a deep history, especially on compact
manifolds. The seminal work [RT74] introduced the geometric control condition,
which required that all null bicharacteristic rays intersect the damping
region, and they used it to show that the energy of solutions to damped
hyperbolic equations on compact product manifolds enjoys exponential decay in
time. The uniform exponential bound is equivalent to so-called strong
stabilization, whereby one can bound the energy at an arbitrary time by the
initial energy multiplied by a monotone-decreasing, non-negative function
tending to zero as $t\rightarrow\infty$. This established the sufficiency of
geometric control for strong stabilization in such settings, while [Ral69]
demonstrated necessity (also, see [Leb96]). The work [BLR92] showed
sufficiency for observability and control on compact manifolds with boundary
where the observability/control region is contained within the boundary. While
there is notably less literature in the non-compact setting, it was proven in
[BR14] that local energy decay holds for the damped wave equation on
asymptotically Euclidean space-times with time-independent metrics under the
assumption of geometric control on trapped geodesics. The authors proved
dissipative Mourre estimates to obtain uniform resolvent bounds in different
frequency regimes in order to apply a limiting absorption argument. This
approach is highly dependent on the metric coefficients being independent of
time and the product structure (asymptotically Euclidean metrics contain no
metric cross terms). This result was improved in [R2018] to estimates in the
(weighted) energy space.

In this article, we combine the approaches of [BR14] and [mst20] to establish
high frequency local energy estimates for damped waves on stationary,
asymptotically flat space-times and explain how such a result can be readily
combined with the existing work in [mst20] to prove local energy decay. We
underscore that we are not requiring the product structure evident in [BR14]
nor [R2018] but, instead, allow for the full Lorentzian formulation. Non-
product metrics possess non-trivial cross terms and are called non-static, of
which the Kerr metric constitutes an important example. We most closely keep
to the framework present in [mst20], which does not assume product structure
and has results for even more general asymptotically flat non-trapping space-
times (such as non-stationary ones). We again stress their use of a non-
trapping hypothesis, which we replace by imposing geometric control. Trapping
is an intrinsically high-frequency phenomenon, so only their high frequency
work is affected by the trapping. Hence, this is the portion of the argument
that needs modification to ensure local energy decay, and this is where the
influence of [BR14] comes into play. Since the medium and low frequency
analyses (as well as the procedure of combining the different frequency regime
estimates into the full local energy decay estimate) do not depend on the non-
trapping hypothesis nor use the damping themselves, the corresponding results
in [mst20] readily apply (i.e. our problem essentially becomes a special case
here). We omit the details of such results in this work, but we will explain
why they apply in our context.

###  1.2. Problem Setup and Main Results

Let $(\mathbb{R}^{4},g)$ be a Lorentzian manifold with coordinates
$(t,x)\in\mathbb{R}\times\mathbb{R}^{3},$ where $g$ has signature $(-+++).$ We
will consider damped wave operators of the form

$P=\Box_{g}+iaD_{t},\qquad\Box_{g}=D_{\alpha}g^{\alpha\beta}D_{\beta},$

where $a\in C_{c}^{\infty}(\mathbb{R}^{3})$ is non-negative and positive on an
open set, and $D_{\alpha}=\frac{1}{i}\partial_{\alpha},\ \alpha=0,1,2,3$.
Greek indices will generally range over such values, whereas Latin indices
will run over the integers $1,2,$ and $3$. Notice that we are using the
standard Einstein summation convention, which we will do throughout this work.
We will also subject $g$ to an asymptotic flatness condition. More precisely,
we first define the norm

$\left\lVert h\right\rVert_{AF}=\sum\limits_{|\alpha|\leq
2}\left\lVert\left\langle
x\right\rangle^{|\alpha|}\partial^{\alpha}h\right\rVert_{\ell^{1}_{j}L^{\infty}([0,T]\times
A_{j})},$

where $A_{j}=\\{\left\langle x\right\rangle\approx 2^{j}\\}$ for $j\geq 0$
denote inhomogeneous dyadic regions, and $\ell^{1}_{j}$ denotes the $\ell^{1}$
norm over the $j$ index. The notation $A\lesssim B$ means that $A\leq CB$ for
some $C>0,$ and the notation $A\approx B$ means that $B\lesssim A\lesssim B$.
In the definition of the $A_{j}$’s, we require that these implicit constants
are compatible to cover $\mathbb{R}^{3}.$ This allows us to define the $AF$
topology.

######  Definition 1.1.

We say that $P$ is asymptotically flat if $\left\lVert
g-m\right\rVert_{AF}<\infty$, where $m$ denotes the Minkowski metric, and

$\left\lVert\left\langle
x\right\rangle^{|\alpha|}\partial^{\alpha}g\right\rVert_{\ell_{j}^{1}L^{\infty}([0,T]\times
A_{j})}\lesssim_{\alpha}1$

for all $\alpha\in\mathbb{N}^{3}$ with $|\alpha|\geq 3.$

The latter condition will be necessary for certain functions appearing in this
work to be symbolic in the Kohn-Nirenberg sense. We remark that the dyadic
summability assumptions on our metric are weaker than the long-range
perturbation condition present in [BR14] (which provides a symbolic-type decay
estimate for derivatives of the metric in $x$ in terms of $\left\langle
x\right\rangle^{-\rho},$ with $\rho>0$ fixed). In [BR14], the damping is not
assumed to be compactly-supported, but rather non-negative everywhere and
subject to a similar symbolic estimate to the metric (with an additional power
of decay). Since the damping is a helpful term which will only be necessary
within a compact spatial set (this is made explicit with the introduction of
the parameter $R_{0}$ on the next page), it is unnecessarily beneficial for us
to assume that is it non-negative everywhere.

We will primarily be interested in when $\partial_{t}$ is a Killing field for
$g$, in which case we say that $P$ is stationary.

######  Definition 1.2.

We say that $P$ is stationary if $g$ is independent of $t$.

Next, we introduce

  * •

the parameters $R_{0}$ and c, which are such that

$\displaystyle\left\lVert g-m\right\rVert_{AF_{>R_{0}}}$
$\displaystyle\leq\textbf{c}\ll 1,$

where the subscript denotes the restriction of the norm to $\\{|x|>R_{0}\\}.$
The parameter c should be viewed as being fixed first, after which we find an
$R_{0}$ for which the above holds. Without loss of generality, we will assume
that $\operatorname{supp}a\subset\\{|x|\leq R_{0}\\}$ (as it is unnecessarily
beneficial outside of this set).

  * •

the sequence $(c_{j})_{j\geq\log_{2}R_{0}}$ satisfying

$\displaystyle\left\lVert g-m\right\rVert_{AF(A_{j})}\lesssim
c_{j},\qquad\sum\limits_{j}c_{j}\lesssim\textbf{c},$

where $\left\lVert\cdot\right\rVert_{AF(A_{j})}$ denotes the restriction of
the $AF$ norm to the dyadic region $A_{j}.$ We may assume, without any loss of
generality, that the sequence is slowly-varying, i.e.

$c_{j}/c_{k}\leq 2^{\delta|k-j|},\qquad\delta\ll 1.$

This sequence will be utilized when working in spatial weights within dyadic
regions.

These parameters tell us that, outside of a large enough spatial ball, the
operator $P$ is a uniformly small perturbation of the flat wave operator
$\Box_{m}=\partial_{t}^{2}-\Delta$ (which we simply denote as $\Box$). The
sequence $(c_{j})$ provides a quantitative measure on the size of the $AF$
norm throughout each spatial dyadic region outside of this ball.

We will also assume throughout that the vector field $\partial_{t}$ is
uniformly time-like, which essentially constitutes a choice of coordinates.
This condition, coupled with the signature of the metric, ensures that
$D_{i}g^{ij}D_{j}$ is uniformly elliptic, i.e.

(1.2) $g^{ij}\xi_{i}\xi_{j}\approx|\xi|^{2},\qquad\xi\neq 0,$

where $|\cdot|$ denotes the standard Euclidean norm. This follows from the
positive-definiteness of the momentum-energy tensor when applied to time-like
vector fields.

Next, we define the local energy norms

$\displaystyle\left\lVert u\right\rVert_{LE}$ $\displaystyle=\sup_{j\geq
0}\left\lVert\langle
x\rangle^{-1/2}u\right\rVert_{L^{2}_{t}L^{2}_{x}\big{(}\mathbb{R}_{+}\times
A_{j}\big{)}},$ $\displaystyle\left\lVert u\right\rVert_{LE^{1}}$
$\displaystyle=\left\lVert\partial u\right\rVert_{LE}+\left\lVert\langle
x\rangle^{-1}u\right\rVert_{LE}.$

A predual-type norm to the $LE$ norm is the $LE^{*}$ norm, which is defined as

$\left\lVert
f\right\rVert_{LE^{*}}=\sum\limits_{j=0}^{\infty}\left\lVert\langle
x\rangle^{1/2}f\right\rVert_{L^{2}_{t}L^{2}_{x}\big{(}\mathbb{R}_{+}\times
A_{j}\big{)}}.$

Here, $L^{p}_{t}L^{q}_{x}$ denotes the Bochner space
$L^{p}(\mathbb{R}_{+},L^{q}(\mathbb{R}^{3})).$ In the particular case of
$p,q=2,$ then this is a Hilbert space; we will use
$\left\langle\cdot,\cdot\right\rangle$ to denote the inner product on
$L^{2}_{t}L^{2}_{x}$. Lastly, we define the sum-space norm

$\left\lVert
f\right\rVert_{LE^{*}+L^{1}_{t}L^{2}_{x}}=\inf_{f=f_{1}+f_{2}}\left(\left\lVert
f_{1}\right\rVert_{LE^{*}}+\left\lVert
f_{2}\right\rVert_{L^{1}_{t}L^{2}_{x}}\right).$

If we wish for the time interval to be e.g. $[0,T]$ in the above norms, then
we will use the notation $\left\lVert u\right\rVert_{LE[0,T]},\left\lVert
u\right\rVert_{LE^{1}[0,T]},\left\lVert
u\right\rVert_{LE^{*}[0,T]},\left\lVert
u\right\rVert_{LE^{*}+L^{1}_{t}L^{2}_{x}[0,T]}$ (although we will write
$LE^{*}[0,T]+L^{1}_{t}L^{2}_{x}[0,T]$ when referring to this space outside of
norm subscripts), etc. A subscript of $c$ on any of these spaces denotes
compact spatial support.

There are two additional function spaces that will be utilized extensively in
this work. The first is the class of Schwartz functions
$\mathcal{S}(\mathbb{R}^{4})$, which will be useful for approximation
arguments. The second is a particular collection of functions which is often
the natural class to study wave equations.

######  Definition 1.3.

Let $T>0$. We define the class $\mathcal{W}_{T}$ to be the space of all
functions $u\in C^{2}([0,T]\times\mathbb{R}^{3})$ for which there exists $R>0$
so that $u(t,x)=0$ for all $t\in[0,T]$ and $|x|>R$. That is,

$\mathcal{W}_{T}=\\{u\in C^{2}([0,T]\times\mathbb{R}^{3}):(\exists
R>0)(\forall|x|>R)(\forall t\in[0,T])\ \ u(t,x)=0\\}.$

We are interested in Cauchy problems of the form

$\left\\{\begin{aligned} Pu&=f\in LE^{*}[0,T]+L^{1}_{t}L^{2}_{x}[0,T],\\\
u[0]&=(u(0),\partial_{t}u(0))\in\dot{H}^{1}\oplus L^{2}.\end{aligned}\right.$

######  Remark 1.4.

The decay conditions on $u\in\mathcal{W}_{T}$ are not as restrictive as they
might initially appear. If the Cauchy data is compactly-supported, then the
condition is free by finite speed of propagation. If it is not, then one can
approximate the data (which generically lives in the energy space) by
compactly-supported data. The regularity conditions on $u$ are also not
restrictive, as one can perform density arguments to reduce to the case of
increased regularity. ∎

Now, we state the pertinent local energy estimates for such problems.

######  Definition 1.5.

We say that (integrated) local energy decay holds for an asymptotically flat
wave operator if the following estimate holds for all $T>0$:

(1.3) $\left\lVert u\right\rVert_{LE^{1}[0,T]}+\left\lVert\partial
u\right\rVert_{L^{\infty}_{t}L^{2}_{x}[0,T]}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\left\lVert
Pu\right\rVert_{LE^{*}+L^{1}_{t}L^{2}_{x}[0,T]}$

for all $u\in\mathcal{W}_{T}$ such that $u[0]\in\dot{H}^{1}\oplus L^{2}$, with
the implicit constant being independent of $T$.

The notion of an asymptotically flat wave operator is more broad than an
asymptotically flat damped wave operator. They need not feature a damping
term, and they are allowed to possess general lower-order terms which are
asymptotically flat in an appropriate sense (see Definition 1.1 in [mst20] for
a precise definition).

Note that, due to global energy conservation for the flat wave problem, the
general definition of local energy decay that we have given here is consistent
with the integrated local energy estimate for the flat wave equation in (1.1)
(in the inhomogeneous case, one applies Hölder’s inequality to the forcing).
This estimate is known to hold whenever $P$ is a small asymptotically flat
perturbation of $\Box$ (see [Ali06], [MS06, MS07], [MT12]). In [mst20], the
authors considered large $AF$ perturbations and proved that, for stationary
problems, the local energy decay estimate (1.3) is equivalent to assuming that
the wave operator $P$ is non-trapping and has no negative eigenfunctions
($L^{2}$ eigenfuctions with corresponding eigenvalues in the lower half-plane)
nor real resonant states (outgoing non-$L^{2}$ eigenfunctions with real
“eigenvalues,” which are called resonances); see Definitions 2.2, 2.4, and 2.8
in [mst20] for more precise statements. The non-trapping hypothesis only arose
during their proof of a high frequency estimate (Theorem 2.11 in [mst20]),
which took the form

(1.4) $\displaystyle\left\lVert
u\right\rVert_{LE^{1}[0,T]}+\left\lVert\partial
u\right\rVert_{L^{\infty}_{t}L^{2}_{x}[0,T]}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{LE[0,T]}+\left\lVert
Pu\right\rVert_{LE^{*}+L^{1}_{t}L^{2}_{x}[0,T]}.$

The implicit constant in the above estimate is crucially independent of $T$.
This estimate does not require $u$ to be truncated to large time frequencies,
but this is the context in which it is used in proving local energy decay.

The added spatial weight in the error term does not play a particular role in
making this high frequency. Rather, it is the weight that naturally arises
when performing a bootstrapping argument in the proof of the estimate; it is
largely unimportant since this estimate can be reduced to studying solutions
with compact spatial support (see Section 2.5).

######  Remark 1.6.

To see this as an estimate on the high frequencies, let
$u\in\mathcal{S}(\mathbb{R}^{4})$ be frequency-supported in time for $\tau$ in
the range $1\ll\tau_{1}\leq|\tau|<\infty$ ((1.4) is only applied for such $u$
in the proof of local energy decay). Then, we can use Plancherel’s theorem in
$t$ to obtain that

(1.5) $\displaystyle\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{LE_{t,x}}$
$\displaystyle\approx\left\lVert\left\langle
x\right\rangle^{-2}\hat{u}(\tau,x)\right\rVert_{LE_{\tau,x}}\lesssim\frac{1}{\tau_{1}}\left\lVert\left\langle
x\right\rangle^{-2}\tau\hat{u}(\tau,x)\right\rVert_{LE_{\tau,x}}\lesssim\frac{1}{\tau_{1}}\left\lVert
u\right\rVert_{LE^{1}_{t,x}}.$

For large enough $\tau_{1},$ this term can be absorbed into the left-hand side
of (1.4), providing local energy decay for solutions restricted to high
frequencies.

In fact, we may apply the high frequency estimate (1.4) to $u(t-T/2)$ to get
(after dropping the uniform energy piece) that

$\left\lVert u\right\rVert_{LE^{1}[-T/2,T/2]}\lesssim\left\lVert\partial
u(-T/2)\right\rVert_{L^{2}}+\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{LE[-T/2,T/2]}+\left\lVert
Pu\right\rVert_{LE^{*}[-T/2,T/2]}.$

Since the implicit constant is independent of $T$, we may take the limit as
$T\rightarrow\infty$ and apply the prior work in (1.5) to obtain that

$\left\lVert u\right\rVert_{LE^{1}}\lesssim\left\lVert
Pu\right\rVert_{LE^{*}}.$

This is the context that we will apply the estimate to establish local energy
decay. ∎

Our first main theorem is the following, which states that we recover the high
frequency estimate (1.4) of [mst20] when working with damped waves and
replacing the non-trapping hypothesis with the geometric control condition.

######  Theorem 1.7.

Let $P$ be a stationary, asymptotically flat damped wave operator satisfying
the geometric control condition, and suppose that $\partial_{t}$ is uniformly
time-like. Then, the high frequency local energy estimate (1.4) holds for all
for all $u\in\mathcal{W}_{T}$ such that $u[0]\in\dot{H}^{1}\oplus L^{2}$. The
implicit constant is independent of $T$.

The geometric control condition, initially introduced in [RT74] for
dissipative hyperbolic equations on compact product manifolds, requires that
every trapped null bicharacteristic ray intersects the damping region. We will
make this more precise in Section 2.2.

######  Remark 1.8.

The implicit constant in the bound depends on $R_{0}$. In fact, much of our
work will implicitly depend on $R_{0}$ due to our applications of asymptotic
flatness. It is essential to note that this parameter is fixed second (with c
being fixed first), after which our other parameters (such as the scaling
parameter $\gamma$ and the high-frequency parameter $\lambda$ which will both
be introduced in Chapter 3) will be chosen (and hence depend on it). We will
not track the dependence on $R_{0}$ within our implicit constants any longer.
Our constants throughout will not depend on $T$, however. ∎

Our second main theorem is local energy decay.

######  Theorem 1.9.

Let $P$ be a stationary, asymptotically flat damped wave operator satisfying
the geometric control condition, and suppose that $\partial_{t}$ is uniformly
time-like. Then, local energy decay holds, with the implicit constant in (1.3)
independent of $T$.

######  Remark 1.10.

Unlike [mst20], we do not require any spectral hypotheses. The damping
eliminates the possibility of non-zero real resonances, and a zero resonance
cannot occur since $P|_{D_{t}=0}$ is elliptic, which is independent of the
damping. The latter is discussed further in Section 3.2. ∎

This follows rather directly from our high frequency estimate and the existing
work in [mst20]. We will cite the necessary result from [mst20] and explain
why they apply here.

The structure of the paper is as follows. In Section 2.2, we will introduce
the Hamiltonian formalism required to define trapping and geometric control,
then we will state a key lemma (Lemma 2.4) for the proof of Theorem 1.7;
namely, we construct an appropriate escape function and lower-order correction
to allow for a positive commutator argument proof of the theorem. In Section
2.3, we establish various results on the bicharacteristic flow that are vital
for proving Lemma 2.4, which we establish in Section 2.4. In Section 2.5, we
will establish and cite supplemental energy estimates and provide multiple
case reductions to simplify the proof of Theorem 1.7. In Section 2.6, we will
prove Theorem 1.7. Finally, Section 3 will present the applicable theorems
from [mst20] which are required to establish local energy decay for our
problem and discuss why they apply in our setting. Sections 3.1 and 3.2
introduce the relevant medium and low frequency estimates, respectively, and
Section 3.3 provides a discussion on the proof of Theorem 1.9.

###  1.3. Cutoff Notation

For the remainder of paper, we will fix the cutoffs

$\chi\in C_{c}^{\infty}\text{ non-increasing},\ \chi\equiv 1\text{ for
}|x|\leq 1,\ \chi\equiv 0\text{ for }|x|>2,$
$\chi_{<R}(|x|)=\chi\left(\frac{|x|}{R}\right),\qquad\chi_{>R}=1-\chi_{<R},$

and

$\chi_{R}\in C_{c}^{\infty},\ 0\leq\chi_{R}\leq 1,\
\operatorname{supp}\chi_{R}\subset\\{|x|\approx R\\}.$

We will choose $\chi$ such that its square root is smooth (otherwise, we
replace $\chi$ with $\chi^{2}$; we only use $\chi$ for notational
convenience). When working in frequency variables, we will often add the
variable into the subscript to make the dependence clear (e.g.
$\chi_{|\xi|>\lambda}$). We will also occasionally write $r=|x|.$

###  1.4. Acknowledgements

This work was performed as a portion of the Ph.D. dissertation of the author
at the University of North Carolina at Chapel Hill (see [Kof22]). The author
would like to thank his advisor, Jason Metcalfe, for suggesting this problem,
engaging in countless helpful conversations, and carefully reading multiple
drafts of this work. He would also like to thank Yaiza Canzani, Hans
Christianson, Jeremy Marzuola, and both referees of this article for providing
many beneficial comments and references.

##  2\. High Frequency Analysis

###  2.1. Introduction

In this section, we will establish Theorem 1.7. The notions of trapping and
geometric control are intrinsically dynamical, so we will provide a thorough
discussion of the relevant theory. Namely, we must introduce the
bicharacteristic flow generated by the principal symbol of the damped wave
operator and the properties that it satisfies. From here, we will construct an
escape function and correction term in order to apply a utilize commutator
argument and prove the theorem. The constructed symbols will satisfy an
appropriate positivity bound, which will allow us to apply the sharp Gårding
inequality upon swapping to the framework of pseudodifferential operators. The
symbols will also be supported in an unbounded range of frequencies
$[\lambda,\infty$) with $\lambda\gg 1$. This will be fundamentally important
for bootstrapping error terms resulting from employment of pseudodifferential
calculus.

###  2.2. Dynamical Framework

In order to state the geometric control condition more precisely, we must
first outline our dynamical framework, which is rooted in the Hamiltonian
dynamics pertaining to the principal symbol of the operator $P$. Since we
assumed that $\partial_{t}$ is uniformly time-like, the signature of the
metric and the cofactor expansion for the inverse metric tell us that
$g^{00}\lesssim-1$, as well. This allows us to divide through by $-g^{00}$ and
preserve the assumptions on the operator coefficients (see [MT12]). Hence, we
may assume (without loss of generality) that $g^{00}=-1$.

After these modifications, the principal symbol of $P$ is

$p(\tau,x,\xi)=-(\tau^{2}-2\tau g^{0j}(x)\xi_{j}-g^{ij}(x)\xi_{i}\xi_{j}).$

This is considered as a smooth function on $T^{*}\mathbb{R}^{4}\setminus o$,
where $o$ denotes the zero section. This symbol generates a
bicharacteristic/Hamiltonian flow on $\mathbb{R}\times T^{*}\mathbb{R}^{4}$
given by
$\varphi_{s}(w)=\left(t_{s}(w),\tau_{s}(w),x_{s}(w),\xi_{s}(w)\right)$ which
solves

$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}&=\partial_{\tau}p(\varphi_{s}(w)),&\qquad\dot{\tau}_{s}&=-\partial_{t}p(\varphi_{s}(w)),\\\
\dot{x}_{s}&=\nabla_{\xi}p(\varphi_{s}(w)),&\qquad\dot{\xi}_{s}&=-\nabla_{x}p(\varphi_{s}(w)),\end{array}\right.$

with initial data $w\in T^{*}\mathbb{R}^{4}.$ Explicitly, one can write the
system as

$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}&=-2\tau_{s}+2g^{0j}(x_{s})[\xi_{s}]_{j},&\dot{\tau}_{s}&=0,\\\
(\dot{x}_{s})_{k}&=2\tau_{s}g^{0k}(x_{s})+2g^{kj}(x_{s})[\xi_{s}]_{j},&\qquad(\dot{\xi}_{s})_{k}&=-2\tau_{s}\partial_{x_{k}}g^{0j}(x_{s})[\xi_{s}]_{j}-\partial_{x_{k}}g^{ij}(x_{s})[\xi_{s}]_{i}[\xi_{s}]_{j}.\end{array}\right.$

Since $g$ is smooth and asymptotically flat, and $\partial_{t}$ is uniformly
time-like, we have a unique, smooth, globally-defined flow with smooth
dependence on the data. We will have particular interest in null
bicharacteristics, i.e. those with initial data lying in the zero set of $p$
(also called the characteristic set of $P$ and denoted $\text{Char}(P)$).
Using the flow $\varphi_{s}$, we define the forward and backward trapped and
non-trapped sets with respect to $\varphi_{s}$, respectively, as

$\displaystyle\Gamma_{tr}$ $\displaystyle=\left\\{w\in
T^{*}\mathbb{R}^{4}\setminus o:\sup_{s\geq
0}|x_{s}(w)|<\infty\right\\}\cap\text{Char}(P),$ $\displaystyle\Lambda_{tr}$
$\displaystyle=\left\\{w\in T^{*}\mathbb{R}^{4}\setminus o:\sup_{s\geq
0}|x_{-s}(w)|<\infty\right\\}\cap\text{Char}(P),$
$\displaystyle\Gamma_{\infty}$ $\displaystyle=\left\\{w\in
T^{*}\mathbb{R}^{4}\setminus o:|x_{s}(w)|\rightarrow\infty\text{ as
}s\rightarrow\infty\right\\}\cap\text{Char}(P),$
$\displaystyle\Lambda_{\infty}$ $\displaystyle=\left\\{w\in
T^{*}\mathbb{R}^{4}\setminus o:|x_{-s}(w)|\rightarrow\infty\text{ as
}s\rightarrow\infty\right\\}\cap\text{Char}(P).$

The trapped and non-trapped sets are defined as

$\Omega^{p}_{tr}=\Gamma_{tr}\cap\Lambda_{tr}\mbox{\quad
and\quad}\Omega^{p}_{\infty}=\Gamma_{\infty}\cap\Lambda_{\infty},$

respectively.

######  Definition 2.1.

The flow is said to be non-trapping if $\Omega^{p}_{tr}{\,=\,}\emptyset$.

Now, we may state the geometric control condition precisely. Recall that our
damping function was denoted $a$.

######  Definition 2.2.

We say that geometric control holds if

(2.1) $\displaystyle(\forall w\in\Omega^{p}_{tr})(\exists s\in\mathbb{R})\ \
a(x_{s}(w))>0.$

In contrast to the definition in [RT74] (given in Assumption (A)), we apply
this condition specifically to the trapped null bicharacteristic rays (since
all null bicharacteristic rays are trapped when the manifold is compact, such
a specification was unnecessary in [RT74]). We will assume that (2.1) holds.
Note that if (2.1) holds and $a\equiv 0$, then $\Omega^{p}_{tr}$ must be
empty, meaning that the flow is non-trapping. In this case, we are back in the
setting of [mst20]. For this reason, we will assume that $a>0$ on an open set.

It will be beneficial to utilize a scaling property of $P$. Given a solution
$u$ to $Pu=f$, consider

$\tilde{v}(t,x):=\gamma^{-2}u(\gamma t,\gamma x),\qquad\gamma>0.$

If we call

$\tilde{P}=D_{\alpha}\tilde{g}^{\alpha\beta}D_{\beta}+i\gamma\tilde{a}D_{t},\qquad\tilde{g}^{\alpha\beta}(x)=g^{\alpha\beta}(\gamma
x),\qquad\tilde{a}(x)=a(\gamma x),$

then $\tilde{v}$ solves

$\tilde{P}\tilde{v}=\tilde{f},\qquad\tilde{f}(t,x)=f(\gamma t,\gamma x)$

if and only if $u$ solves $Pu=f$ (we can similarly undo the scaling to move
between the frameworks). Notice that the scaled problem allows for an
arbitrarily large constant $\gamma$ in front of the damping.

Analogous Hamiltonian systems and trapped sets exist for the principal symbol
$\tilde{p}$ of $\tilde{P}$, and this amounts to simply replacing $g$ by
$\tilde{g}$. If we assume that geometric control holds for the flow generated
by $p$, then we must check that it holds for the scaled problem.

######  Proposition 2.3.

Assume that (2.1) holds. Then, for any $\gamma>0$, (2.1) holds for the flow
generated by $\tilde{p}$, with $a$ replaced by $\tilde{a}$.

Note that since $g^{00}\equiv-1,$ it follows that $\tilde{g}^{00}\equiv-1.$

###### Proof.

The flow generated by $\tilde{p}$ solves the system

$\left\\{\begin{aligned}
\frac{d}{ds}\tilde{t}_{s}&=-2\tilde{\tau}_{s}+2\tilde{g}^{0j}(\tilde{x}_{s})[\tilde{\xi}_{s}]_{j},\\\
\frac{d}{ds}\tilde{\tau}_{s}&=0,\\\
\frac{d}{ds}(\tilde{x}_{s})_{k}&=2\tilde{\tau}_{s}\tilde{g}^{0k}(\tilde{x}_{s})+2\tilde{g}^{kj}(\tilde{x}_{s})[\tilde{\xi}_{s}]_{j},\\\
\frac{d}{ds}(\tilde{\xi}_{s})_{k}&=-2\tilde{\tau}_{s}\partial_{x_{k}}\tilde{g}^{0j}(\tilde{x}_{s})[\tilde{\xi}_{s}]_{j}-\partial_{x_{k}}\tilde{g}^{ij}(\tilde{x}_{s})[\tilde{\xi}_{s}]_{i}[\tilde{\xi}_{s}]_{j},\\\
(\tilde{t}_{s},\tilde{\tau}_{s},\tilde{x}&{}_{s},\tilde{\xi}_{s})\big{|}_{s=0}=(t,\tau,x,\xi).\end{aligned}\right.$

Applying the chain rule and multiplying through by $\gamma$ provides us with
the system

$\left\\{\begin{aligned}
\frac{d}{ds}(\gamma\tilde{t}_{s})&=-2(\gamma\tilde{\tau}_{s})+2{g}^{0j}(\gamma\tilde{x}_{s})[\gamma\tilde{\xi}_{s}]_{j},\\\
\frac{d}{ds}(\gamma\tilde{\tau}_{s})&=0,\\\
\frac{d}{ds}(\gamma\tilde{x}_{s})_{k}&=2(\gamma\tilde{\tau}_{s}){g}^{0k}(\gamma\tilde{x}_{s})+2{g}^{kj}(\gamma\tilde{x}_{s})[\gamma\tilde{\xi}_{s}]_{j},\\\
\frac{d}{ds}(\gamma\tilde{\xi}_{s})_{k}&=-2(\gamma\tilde{\tau}_{s})[(\partial_{x_{k}}{g}^{0j})(\gamma\tilde{x}_{s})][\gamma\tilde{\xi}_{s}]_{j}-[(\partial_{x_{k}}{g}^{ij})(\gamma\tilde{x}_{s})][\gamma\tilde{\xi}_{s}]_{i}[\gamma\tilde{\xi}_{s}]_{j},\\\
\big{(}(\gamma\tilde{t})_{s},(\gamma&\tilde{\tau})_{s},(\gamma\tilde{x})_{s},(\gamma\tilde{\xi})_{s}\big{)}\big{|}_{s=0}=(\gamma
t,\gamma\tau,\gamma x,\gamma\xi).\end{aligned}\right.$

This is the same system that is solved by the Hamiltonian flow generated by
$p$ with initial data $(\gamma t,\gamma\tau,\gamma x,\gamma\xi).$ By
uniqueness, we can conclude that

$\left\\{\begin{array}[]{rlrl}\gamma\tilde{t}_{s}(t,\tau,x,\xi)&=t_{s}(\gamma
t,\gamma\tau,\gamma
x,\gamma\xi),&\qquad\gamma\tilde{\tau}_{s}(t,\tau,x,\xi)&=\tau_{s}(\gamma
t,\gamma\tau,\gamma x,\gamma\xi)\\\
\gamma\tilde{x}_{s}(t,\tau,x,\xi)&=x_{s}(\gamma t,\gamma\tau,\gamma
x,\gamma\xi),&\qquad\gamma\tilde{\xi}_{s}(t,\tau,x,\xi)&=\xi_{s}(\gamma
t,\gamma\tau,\gamma x,\gamma\xi).\end{array}\right.$

Now, let $w=\Omega^{\tilde{p}}_{tr}.$ From the above, we have that

$\tilde{x}_{s}(w)=\gamma^{-1}x_{s}(\tilde{w}),\qquad\tilde{w}=\gamma w.$

Since

$\sup_{s\in\mathbb{R}}|x_{s}(\tilde{w})|=\gamma\sup_{s\in\mathbb{R}}|\tilde{x}_{s}(w)|<\infty,$

it follows that $\tilde{w}\in\Omega^{p}_{tr}$. By (2.1), there exists
$s^{\prime}\in\mathbb{R}$ so that $a(x_{s^{\prime}}(\tilde{w}))>0,$ and so

$\tilde{a}(\tilde{x}_{s^{\prime}}(w))=a(\gamma\tilde{x}_{s^{\prime}}(w))=a(x_{s^{\prime}}(\tilde{w}))>0,$

which completes the proof. ∎

Now that we have shown that geometric control is invariant under scaling, we
will fix a large $\gamma>0$ and study the problem from the scaled perspective
(where our damping is now multiplied by $\gamma$) while reverting back to our
original notation ($x$ and $\xi$, no tildes, etc.). More precise conditions on
the size of $\gamma$ will come in Section 2.4. It is readily seen that it is
equivalent to prove Theorem 1.7 for the scaled problem, where we now have a
large constant in front of the damping term.

Our proof of Theorem 1.7 is a positive commutator argument. At the symbolic
level, this requires the construction of an escape function (as well as a
lower-order correction). We must consider the skew-adjoint contribution of
$P$, which will be a purely beneficial term due to the presence of the
damping. Let $p$ and $s_{skew}$ represent the principal symbols of the self
and skew-adjoint parts of $P$, respectively. Namely,

$\displaystyle{p}(\tau,x,\xi)$
$\displaystyle=-(\tau^{2}-2\tau{g}^{0j}(x)\xi_{j}-{g}^{ij}(x)\xi_{i}\xi_{j})$
$\displaystyle{s}_{skew}(\tau,x,\xi)$ $\displaystyle=i\gamma\tau a(x).$

The multiplication by $\gamma$ in $s_{skew}$ will prove advantageous for a
bootstrapping argument, which is precisely why we implement the
$\gamma$-scaling. Now, we are ready to state our escape function result, which
we will prove in Section 2.4.

######  Lemma 2.4.

For all $\lambda>1$, there exist symbols $q_{j}\in S^{j}(T^{*}\mathbb{R}^{3})$
and $m\in S^{0}(T^{*}\mathbb{R}^{3})$, all supported in $|\xi|\geq\lambda$, so
that

$H_{{p}}q-2i{s}_{skew}q+{p}m\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}\left\langle
x\right\rangle^{-2}\left(\tau^{2}+|\xi|^{2}\right),$

where $q=\tau q_{0}+q_{1}$.

Here, $S^{m}(T^{*}\mathbb{R}^{n})$ denotes the $m^{\operatorname{th}}$-order
Kohn-Nirenberg symbol class. To each $q\in S^{m}(T^{*}\mathbb{R}^{n})$, we
will associate the pseudodifferential operator
$q^{\operatorname{w}}(x,D):\mathcal{S}(\mathbb{R}^{n})\rightarrow\mathcal{S}(\mathbb{R}^{n}),$
namely the Weyl quantization of $p$, which is defined via the action

$q^{\operatorname{w}}(x,D)u(x)=(2\pi)^{-n}\int\limits_{\mathbb{R}^{n}}\int\limits_{\mathbb{R}^{n}}e^{i(x-y)\cdot\xi}q\left(\frac{x+y}{2},\xi\right)u(y)\,dyd\xi.$

In the proof of Lemma 2.4, it will be useful to work with the half-wave
decomposition, which allows us to avoid the cross terms in the principal
symbol. To that end, we factor $p$ as

$p(\tau,x,\xi)=-(\tau-b^{+}(x,\xi))(\tau-b^{-}(x,\xi)),$

where

$b^{\pm}(x,\xi)={g^{0j}\xi_{j}\pm\sqrt{\left(g^{0j}\xi_{j}\right)^{2}+g^{ij}\xi_{i}\xi_{j}}}.$

Observe that $b^{\pm}$ are both homogeneous of degree 1 in $\xi$.
Additionally, they are both signed.

######  Proposition 2.5.

For any $(x,\xi)\in T^{*}\mathbb{R}^{3}\setminus o$, we have that
$b^{+}(x,\xi)>0>b^{-}(x,\xi).$

###### Proof.

Let $\xi\neq 0.$ First, we show that $b^{+}>b^{-}$. Indeed, observe that

$b^{+}-b^{-}=2\sqrt{\left(g^{0j}\xi_{j}\right)^{2}+g^{ij}\xi_{i}\xi_{j}}>0$

using the ellipticity (see 1.2). Using ellipticity again, we have that

$\sqrt{\left(g^{0j}\xi_{j}\right)^{2}+g^{ij}\xi_{i}\xi_{j}}>|g^{0j}\xi_{j}|.$

Thus,

$b^{+}>g^{0j}\xi_{j}+|g^{0j}\xi_{j}|\geq 0,\qquad
b^{-}<g^{0j}\xi_{j}-|g^{0j}\xi_{j}|\leq 0.$

∎

We will call $p^{\pm}=\tau-b^{\pm},$ so that $p=-p^{+}p^{-}.$ In particular,
$p=0$ if and only if $p^{+}=0$ or $p^{-}=0$; due to Proposition 2.5, it cannot
be the case that $p^{+}(w)=p^{-}(w)=0$ for any $w\in
T^{*}\mathbb{R}^{4}\setminus o.$ The Hamiltonians $p^{\pm}$ also generate
flows
$\varphi_{s}^{\pm}(w){\,=\,}\left(t_{s}^{\pm}(w),\tau_{s}^{\pm}(w),x_{s}^{\pm}(w),\xi_{s}^{\pm}(w)\right)$
on $\mathbb{R}\times T^{*}\mathbb{R}^{4}$ which solve the Hamiltonian systems

$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}^{\pm}&=\partial_{\tau}p^{\pm}(\varphi_{s}^{\pm}(w)),&\dot{\tau}_{s}^{\pm}&=-\partial_{t}p^{\pm}(\varphi_{s}^{\pm}(w)),\\\
\dot{x}_{s}^{\pm}&=\nabla_{\xi}p^{\pm}(\varphi_{s}^{\pm}(w)),&\dot{\xi}_{s}^{\pm}&=-\nabla_{x}p^{\pm}(\varphi_{s}^{\pm}(w)),\end{array}\right.$

with initial data $w\in T^{*}\mathbb{R}^{4}.$ Note that

$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}^{\pm}&=1,&\dot{\tau}_{s}^{\pm}&=0,\\\
\dot{x}_{s}^{\pm}&=-\nabla_{\xi}b^{\pm}(\varphi_{s}^{\pm}(w)),&\dot{\xi}_{s}^{\pm}&=\nabla_{x}b^{\pm}(\varphi_{s}^{\pm}(w)).\end{array}\right.$

There is a direct correspondence between null bicharacteristics for
$\varphi_{s}$ and null bicharacteristics for $\varphi_{s}^{\pm}$.

######  Proposition 2.6.

Every null bicharacteristic for the flow generated by $p$ is a null
bicharacteristic for the flow generated by either $p^{+}$ or $p^{-}$. The
converse is also true.

###### Proof.

Recall that for any
$(t^{\prime},\tau^{\prime},x^{\prime},\xi^{\prime}){\,=:\,}w\in
T^{*}\mathbb{R}^{4}\setminus o,$ we have that $p(w){\,=\,}0$ if and only if
either $p^{+}(w){\,=\,}0$ or $p^{-}(w){\,=\,}0$. Without loss of generality,
suppose that $p^{+}(w){\,=\,}0.$ The Hamiltonians $p$ and $p^{+}$ generate the
systems

(2.2)
$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}&=-p^{+}(\varphi_{s})-p^{-}(\varphi_{s}),&\dot{\tau}_{s}&=0,\\\
(\dot{x}_{s})_{k}&=-p^{+}(\varphi_{s})p^{-}_{\xi_{k}}(\varphi_{s})-p^{-}(\varphi_{s})p^{+}_{\xi_{k}}(\varphi_{s}),&(\dot{\xi}_{s})_{k}&=p^{+}(\varphi_{s})p_{x_{k}}^{-}(\varphi_{s})+p^{-}(\varphi_{s})p_{x_{k}}^{+}(\varphi_{s}),\end{array}\right.$

and

(2.3)
$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}^{+}&=1,&\dot{\tau}_{s}^{+}&=0,\\\
(\dot{x}_{s}^{+})_{k}&=p_{\xi_{k}}^{+}(\varphi_{s}^{+}),&(\dot{\xi}_{s}^{+})_{k}&=-p_{x_{k}}^{+}(\varphi_{s}^{+}),\end{array}\right.$

respectively. We will take both systems to have initial data $w.$

We claim that since $p^{+}(w){\,=\,}0,$ we must have that
$p^{+}(\varphi_{s}(w)){\,=\,}0$ for all $s$. If not, then there would exist
$s^{\prime}$ so that $p^{-}(\varphi_{s^{\prime}}(w)){\,=\,}0,$ i.e.
$\tau_{s^{\prime}}^{-}(w)=b^{-}(x_{s^{\prime}}(w),\xi_{s^{\prime}}(w)){\,<\,}0$.
However, $\tau^{-}$ is constant and $p^{+}(w)=0,$ which implies that
$\tau_{s}^{-}(w)=\tau^{\prime}>0$ for all $s$.

Thus, we can re-write (2.2) as

(2.4)
$\left\\{\begin{array}[]{rlrl}\dot{t}_{s}&=-p^{-}(\varphi_{s}),&\dot{\tau}_{s}&=0,\\\
(\dot{x}_{s})_{k}&=-p^{-}(\varphi_{s})p^{+}_{\xi_{k}}(\varphi_{s}),&(\dot{\xi}_{s})_{k}&=p^{-}(\varphi_{s})p_{x_{k}}^{+}(\varphi_{s}),\end{array}\right.$

with initial data $(t_{s},\tau_{s},x_{s},\xi_{s})\big{|}_{s=0}{=w}.$ Notice
that $t^{+}{\,=\,}t^{\prime}+s,$ and so we may re-parameterize (2.3) in terms
of $t^{+}$:

$\left\\{\begin{array}[]{rlrl}\dfrac{d}{dt^{+}}{t}_{t^{+}-t^{\prime}}^{+}&=1&\dfrac{d}{dt^{+}}{\tau}_{t^{+}-t^{\prime}}^{+}&=0,\\\
\left(\dfrac{d}{dt^{+}}{x}_{t^{+}-t^{\prime}}^{+}\right)_{k}&=p_{\xi_{k}}^{+}(\varphi_{t^{+}-t^{\prime}}^{+}),&\left(\dfrac{d}{dt^{+}}{\xi}_{t^{+}-t^{\prime}}^{+}\right)_{k}&=-p_{x_{k}}^{+}(\varphi_{t^{+}-t^{\prime}}^{+}),\end{array}\right.$

with initial data
$\Big{(}t^{+}_{t^{+}-t^{\prime}},\tau^{+}_{t^{+}-t^{\prime}},x^{+}_{t^{+}-t^{\prime}},\xi^{+}_{t^{+}-t^{\prime}}\Big{)}\big{|}_{t^{+}=t^{\prime}}{=w}.$

Next, we re-parameterize (2.4) to change the flow variable from $s$ to $t$
(which can be done since $t_{s}$ is strictly increasing and hence invertible),
generating the system

$\left\\{\begin{array}[]{rlrl}\dfrac{d}{dt}{t}_{s(t)}&=1,&\dfrac{d}{dt}{\tau}_{s(t)}&=0,\\\
\left(\dfrac{d}{dt}{x}_{s(t)}\right)_{k}&=p^{+}_{\xi_{k}}(\varphi_{s(t)}),&\left(\dfrac{d}{dt}{\xi}_{s(t)}\right)_{k}&=-p_{x_{k}}^{+}(\varphi_{s(t)}),\end{array}\right.$

which has initial data
$\left(t_{s(t)},\tau_{s(t)},x_{s(t)},\xi_{s(t)}\right)\big{|}_{t=t^{\prime}}{=w}.$
An application of uniqueness theory yields that
$\varphi_{s(t)}(w)=\varphi_{t^{+}-t^{\prime}}^{+}(w)$. The converse is similar
by reversing the above process. ∎

When working with the factored flow, the decoupling of $(t,\tau)$ and
$(x,\xi)$ allows us to project onto the $(x,\xi)$ components of the flow
without worrying about loss of information. For this reason, we will write
$\Pi_{x,\xi}\circ\varphi^{\pm}$ as simply $\varphi^{\pm}$, where
$\Pi_{x,\xi}(t,\tau,x,\xi){\,=\,}(x,\xi)$. Notice that when we project, we are
no longer looking at null bicharacteristics but, rather, bicharacteristics
with initial data having non-zero $\xi$ component.

Now, we may define all of the corresponding forward and backward trapped and
non-trapped sets for the half-wave flows as

$\displaystyle\Gamma_{tr}^{\pm}$ $\displaystyle=\left\\{w\in
T^{*}\mathbb{R}^{3}\setminus o:\sup_{s\geq
0}|x^{\pm}_{s}(w)|<\infty\right\\},$ $\displaystyle\Lambda_{tr}^{\pm}$
$\displaystyle=\left\\{w\in T^{*}\mathbb{R}^{3}\setminus o:\sup_{s\geq
0}|x^{\pm}_{-s}(w)|<\infty\right\\},$ $\displaystyle\Gamma_{\infty}^{\pm}$
$\displaystyle=\left\\{w\in T^{*}\mathbb{R}^{3}\setminus
o:|x^{\pm}_{s}(w)|\rightarrow\infty\ \operatorname{as}\
s\rightarrow\infty\right\\},$ $\displaystyle\Lambda_{\infty}^{\pm}$
$\displaystyle=\left\\{w\in T^{*}\mathbb{R}^{3}\setminus
o:|x^{\pm}_{-s}(w)|\rightarrow\infty\ \operatorname{as}\
s\rightarrow\infty\right\\}.$

The trapped and non-trapped sets are

$\displaystyle{\Omega}_{tr}^{\pm}=\Gamma_{tr}^{\pm}\cap\Lambda_{tr}^{\pm}$
$\displaystyle\mbox{\quad
and\quad}{\Omega}_{\infty}^{\pm}=\Gamma_{\infty}^{\pm}\cap\Lambda_{\infty}^{\pm},$
$\displaystyle{\Omega}_{tr}={\Omega}_{tr}^{+}\cup{\Omega}_{tr}^{-}$
$\displaystyle\mbox{\quad
and\quad}{\Omega}_{\infty}={\Omega}_{\infty}^{+}\cup{\Omega}_{\infty}^{-}.$

Note that the identities

${\Omega}_{tr}=\Pi_{x,\xi}(\Omega^{p}_{tr})\mbox{\quad
and\quad}{\Omega}_{\infty}=\Pi_{x,\xi}(\Omega^{p}_{\infty})$

hold as an immediate consequence of the factoring. Additionally, the factoring
allows us to re-state the geometric control condition as

$\left(w\in\Omega_{tr}^{+}\implies(\exists s\in\mathbb{R})\
\left(a(x^{+}_{s}(w))>0\right)\right)\mbox{\quad
and\quad}\left(w\in\Omega_{tr}^{-}\implies(\exists s\in\mathbb{R})\
\left(a(x^{-}_{s}(w))>0\right)\right).$

If $w\in\Omega_{tr}$, then it is either trapped with respect the flow
generated by $p^{+}$ or $p^{-}$ by Proposition 2.6. If it is trapped with
respect to $p^{+},$ then there is a time so that $w$ is flowed along a
$p^{+}$-bicharacteristic ray to a place where the damping is positive, and
similarly if it is trapped with respect to $p^{-}$.

###  2.3. Results on the Flow

Here, we establish results regarding the trapped/non-trapped sets and scalings
for the flows, culminating in an extension of geometric control to
bicharacteristic rays bounded either forward or backward in time. These
results largely follow the path outlined in [BR14], although we require
certain scaling results in order to utilize homogeneity arguments in later
proofs (which were unnecessary in [BR14] due to their use of semiclassical
rescaling). In particular, Lemma 2.10 and Propositions 2.11 and 2.12 are
analogous to results in Chapter 8 of [BR14] (namely, Lemma 8.2 and
Propositions 8.3 and 8.4, respectively).

We will start with a scaling result on the flow.

######  Proposition 2.7.

The flows generated by $p^{\pm}$ satisfy the scalings

$\left\\{\begin{aligned} x^{\pm}_{s}(x,\xi)&=x^{\pm}_{s}(x,\lambda\xi),\\\
\lambda\xi^{\pm}_{s}(x,\xi)&=\xi^{\pm}_{s}(x,\lambda\xi)\end{aligned}\right.$

for any $\lambda>0.$

###### Proof.

Label the functions on the right-hand side as $x_{s,\lambda}^{\pm}$ and
$\xi_{s,\lambda}^{\pm}$, respectively. Using the homogeneity of $b^{\pm}$, the
left-hand side $(x^{\pm}_{s},\lambda\xi^{\pm}_{s})$ solves the system

$\left\\{\begin{aligned}
\dfrac{d}{ds}x_{s}^{\pm}&=\nabla_{\xi}p^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})=\nabla_{\xi}p^{\pm}(x_{s}^{\pm},\lambda\xi_{s}^{\pm}),\\\
\dfrac{d}{ds}(\lambda\xi_{s}^{\pm})&=-\lambda\nabla_{x}p^{\pm}(x_{s}^{\pm},\xi^{\pm}_{s})=-\nabla_{x}p^{\pm}(x_{s}^{\pm},\lambda\xi_{s}^{\pm}),\\\
(x_{s}^{\pm},\lambda\xi_{s}^{\pm})\big{|}_{s=0}&=(x,\lambda\xi),\end{aligned}\right.$

while the right-hand side solves

$\left\\{\begin{aligned}
\frac{d}{ds}x_{s,\lambda}^{\pm}&=\nabla_{\xi}p^{\pm}(x_{s,\lambda}^{\pm},\xi_{s,\lambda}^{\pm}),\\\
\frac{d}{ds}(\xi_{s,\lambda}^{\pm})&=-\nabla_{x}p^{\pm}(x_{s,\lambda}^{\pm},\xi_{s,\lambda}^{\pm}),\\\
(x_{s,\lambda}^{\pm},\xi_{s,\lambda}^{\pm})\big{|}_{s=0}&=(x,\lambda\xi).\end{aligned}\right.$

Applying uniqueness theory completes the proof. ∎

This scaling implies that the trapped/non-trapped sets, and hence geometric
control, are entirely determined by unit speed null bicharacteristics, i.e. by
what happens on the unit cosphere bundle $S^{*}\mathbb{R}^{3}=\\{(x,\xi)\in
T^{*}\mathbb{R}^{3}:|\xi|=1\\}.$ Indeed, observe that

$x_{s}^{\pm}(x,\xi)=x^{\pm}_{s}(x,\xi/|\xi|).$

The forward/backward trapped sets are defined in terms of supremums of the
above over $s$, while the forward/backward non-trapped sets are defined via
limits in $s$, and the prior equation shows that all of these are unaffected
by the scaling in the $\xi$ component of the initial data. A more pertinent
scaling is given by the function

$\Phi^{\pm}(x,\xi)=\left(x,\frac{\xi}{|b^{\pm}(x,\xi)|}\right).$

The utility of this scaling comes from the fact that $b^{\pm}$ is a constant
of motion under the corresponding projected Hamiltonian flows and that

$\left|{\frac{\xi}{b^{+}(x,\xi)}}\right|\approx 1,$

which we now prove.

######  Proposition 2.8.

For any $(x,\xi)\in T^{*}\mathbb{R}^{3}\setminus o,$

$\left|{\frac{\xi}{b^{\pm}(x,\xi)}}\right|\approx 1.$

###### Proof.

By homogeneity,

$\left|{\frac{\xi}{b^{\pm}(x,\xi)}}\right|=\frac{1}{\left|b^{\pm}\left(x,\frac{\xi}{|\xi|}\right)\right|}.$

Write

$\left|b^{\pm}\left(x,\frac{\xi}{|\xi|}\right)\right|=\left|g^{0j}\frac{\xi_{j}}{|\xi|}\pm\sqrt{\left(g^{0j}\frac{\xi_{j}}{|\xi|}\right)^{2}+(g^{ij}-m^{ij})\frac{\xi_{i}}{|\xi|}\frac{\xi_{j}}{|\xi|}+m^{ij}\frac{\xi_{i}}{|\xi|}\frac{\xi_{j}}{|\xi|}}\right|.$

Since $\left\lVert g-m\right\rVert_{AF(|x|>R_{0})}\ll 1$, asymptotic flatness
guarantees that $g^{0j}$ and $g^{ij}-m^{ij}$ are small in the exterior region
$\\{|x|>R_{0}\\}$. Hence,

$\left|b^{\pm}\left(x,\frac{\xi}{|\xi|}\right)\right|\approx\sqrt{m^{ij}\frac{\xi_{i}}{|\xi|}\frac{\xi_{j}}{|\xi|}}=1$

when $|x|>R_{0}.$

In the interior region, we are considering $b^{\pm}$ on the compact set
$\\{|x|\leq R_{0}\\}\times\\{|\xi|=1\\}.$ Since we know that $|b^{\pm}|>0$ for
all $\xi\neq 0$ from Proposition 2.5, continuity guarantees the desired
boundedness here. ∎

In view of Proposition 2.8, it follows that

$\frac{\xi_{j}}{b^{\pm}(x,\xi)}\in
S^{0}_{\operatorname{hom}}(T^{*}\mathbb{R}^{3}\setminus o),\qquad j=1,2,3,$

where $S^{0}_{\operatorname{hom}}(T^{*}\mathbb{R}^{3}\setminus o)$ denotes the
$0^{\operatorname{th}}$-order homogeneous symbol class.

######  Remark 2.9.

As a consequence of the scaling, the sets

$\dot{\Gamma}_{tr}^{\pm}=\Gamma_{tr}^{\pm}\cap\Phi^{\pm}(T^{*}\mathbb{R}^{3}\setminus
o)\mbox{\quad
and\quad}\dot{\Lambda}_{tr}^{\pm}=\Lambda_{tr}^{\pm}\cap\Phi^{\pm}(T^{*}\mathbb{R}^{3}\setminus
o)$

are invariant under the flow. Indeed, it is readily seen that the (semi)
trapped nature is preserved. Further, since $b^{\pm}$ is constant along the
flow, it follows from Proposition 2.7 that

$\xi_{s}^{\pm}\left(x,\frac{\xi}{|b^{\pm}(x,\xi)|}\right)=\frac{1}{|b^{\pm}(x,\xi)|}\xi_{s}^{\pm}(x,\xi)=\frac{1}{|b^{\pm}(x_{s}^{\pm}(x,\xi),\xi_{s}^{\pm}(x,\xi))|}\xi_{s}^{\pm}(x,\xi).$

∎

Now, we prove a key result on non-trapped trajectories.

######  Lemma 2.10.

If $R\geq R_{0}$ and

$|x^{\pm}_{\pm s^{\prime}}(x,\xi)|\geq\max\\{2R,|x|+\delta\\}$

for some $(x,\xi)\in T^{*}\mathbb{R}^{3}\setminus o,$ $\delta>0,$ and
$s^{\prime}>0,$ then it holds for all $s\geq s^{\prime}$, and

$|x^{\pm}_{\pm s}(x,\xi)|\rightarrow\infty$

as $s\rightarrow\infty.$

That is, if we can get sufficiently far away from the origin and move radially
outward from the initial position, then the trajectories are necessarily non-
trapped. This can be proven directly, but the computations are simpler if one
uses the correspondence between null bicharacteristics for $p$ and $p^{\pm}$.

###### Proof.

Without loss of generality, we will work with the $x^{+}$ bicharacteristic
ray. By Proposition 2.6, it suffices to prove the result for the null
bicharacteristic ray $x_{\pm s}$ with initial data $\tilde{w}$, where
$\tilde{w}$ is the lift of $w$ to $T^{*}\mathbb{R}^{4}\setminus o$ which is
consistent with the comment immediately following the aforementioned
proposition (in particular, the $\tau$ component is strictly positive). For
any $z\in T^{*}\mathbb{R}^{4}\setminus o$, we explicitly calculate that

$\frac{1}{2}\frac{\partial^{2}}{\partial s^{2}}|x_{\pm
s}(z)|^{2}=\left|\frac{\partial}{\partial s}x_{\pm s}(z)\right|^{2}+x_{\pm
s}(z)\cdot\frac{\partial^{2}}{\partial s^{2}}x_{\pm s}(z),$

where

$\displaystyle\left|\frac{\partial}{\partial s}x_{\pm
s}(z)\right|^{2}=4\tau_{\pm s}^{2}(z)\left(\sum\limits_{k=1}^{3}g^{0k}(x_{\pm
s}(z))\right)^{2}$ $\displaystyle+4\sum\limits_{k=1}^{3}[(g^{ki}(x_{\pm
s}(z))(\xi_{\pm s}(z))_{i}][(g^{kj}(x_{\pm s}(z))(\xi_{\pm s}(z))_{j}]$
$\displaystyle+8\sum\limits_{k=1}^{3}\tau_{\pm s}(z)g^{0k}(x_{\pm
s}(z))g^{kj}(x_{\pm s}(z))(\xi_{j}(z))_{\pm s},$

and

$\displaystyle x_{\pm s}(z)\cdot\frac{\partial^{2}}{\partial s^{2}}x_{\pm
s}(z)$ $\displaystyle=4\tau_{\pm s}(z)(x_{\pm
s}(z))_{k}[\partial_{\ell}g^{0k}(x_{\pm s}(z))]\left(\tau_{\pm
s}(z)g^{0\ell}(x_{\pm s}(z))+g^{\ell j}(x_{\pm s}(z))(\xi_{\pm
s}(z))_{j}\right)$ $\displaystyle\qquad+4(x_{\pm
s}(z))_{j}[\partial_{\ell}g^{kj}(x_{\pm s}(z))]\left(\tau_{\pm
s}(z)g^{0\ell}(x_{\pm s}(z))+g^{\ell j}(x_{\pm s}(z))(\xi_{\pm
s}(z))_{j}\right)(\xi_{\pm s}(z))_{j}$ $\displaystyle\qquad-2(x_{\pm
s}(z))_{k}g^{kj}(x_{\pm s}(z))\left(2\tau_{\pm s}(z)\partial_{j}g^{0i}(x_{\pm
s}(z))(\xi_{\pm s}(z))_{i}+\partial_{j}g^{i\ell}(x_{\pm s}(z))(\xi_{\pm
s}(z))_{i}(\xi_{\pm s}(z))_{\ell}\right).$

Since $\tau_{s}$ is constant for stationary $P$, it follows that

$\tau_{\pm s}(z)=\tau_{0}=b^{+}(\Pi_{x,\xi}(z))=b^{\pm}\left(x^{+}_{\pm
s}(\Pi_{x,\xi}(z)),\xi^{+}_{\pm s}(\Pi_{x,\xi}(z))\right)\approx|\xi^{+}_{\pm
s}(\Pi_{x,\xi}(z))|,$

and so

$\frac{\partial^{2}}{\partial s^{2}}\left|x_{\pm
s}(z)\right|^{2}\gtrsim|\xi^{+}_{\pm
s}(\Pi_{x,\xi}(z))|^{2}\left(1-\left\lVert g-m\right\rVert_{AF_{>R}}\right)$

provided that $|x_{\pm s}(z)|>R$. In such a case, we have that $\left\lVert
g-m\right\rVert_{AF_{>R}}\ll 1,$ and thus

$\frac{\partial^{2}}{\partial s^{2}}|x_{\pm s}(z)|^{2}>0.$

Using the condition that $|x^{\pm}_{\pm
s^{\prime}}(x,\xi)|\geq\max\\{2R,|x|+\delta$} in a typical mean value theorem
argument (elementary and, thus, omitted) establishes the existence of an
$s^{\prime\prime}\in[0,s^{\prime}]$ such that

$|x_{\pm s^{\prime\prime}}(\tilde{w})|^{2}>R^{2}\mbox{\quad
and\quad}\left(\frac{\partial}{\partial s}|x_{\pm
s}(\tilde{w})|^{2}\right)\Big{|}_{s=s^{\prime\prime}}>0.$

All together, we have that $|x_{\pm s}(\tilde{w})|^{2}$ has positive
derivative at $s=s^{\prime\prime}$, and its derivative is increasing for all
$s\geq s^{\prime\prime}$. In particular, $|x_{\pm s}(\tilde{w})|^{2}$ is
increasing for all $s\geq s^{\prime\prime},$ which implies the result. ∎

As a consequence, we can use the trapped and non-trapped sets to partition
phase space.

######  Proposition 2.11.

  1. (a)

We can partition $T^{*}\mathbb{R}^{3}\setminus o$ as

$\displaystyle T^{*}\mathbb{R}^{3}\setminus o$
$\displaystyle=\Gamma_{tr}^{\pm}\sqcup\Gamma_{\infty}^{\pm}=\Lambda_{tr}^{\pm}\sqcup\Lambda_{\infty}^{\pm},$
$\displaystyle T^{*}\mathbb{R}^{3}\setminus o$
$\displaystyle=\Gamma_{tr}^{\pm}\cup\Lambda_{tr}^{\pm}\cup\Omega_{\infty}^{\pm}.$

  2. (b)

$\Gamma^{\pm}_{\infty},\Lambda^{\pm}_{\infty},\Omega^{\pm}_{\infty}$ are open
in $T^{*}\mathbb{R}^{3}\setminus o$, and
$\Gamma_{tr}^{\pm},\Lambda_{tr}^{\pm},\Omega^{\pm}_{tr}$ are closed.

  3. (c)

If $K\subset\Omega_{\infty}^{\pm}$ is compact, then for every $R\geq R_{0}$,
there exists $T^{\prime}\geq 0$ so that

$|x^{\pm}_{s}(v)|>R$

for every $|s|\geq T^{\prime}$ and ${v\in K}.$ Also,

$\bigcup\limits_{s\in\mathbb{R}}\varphi_{s}^{\pm}(K)$

is closed in $T^{*}\mathbb{R}^{3}\setminus o.$

We omit the proof of this result, as it follows directly from using Lemma 2.10
and continuity of the flow in both the evolution parameter and the data, as in
[BR14] (see Proposition 8.3 in the aforementioned work).

Finally, we show that if one assumes geometric control for bounded
bicharacteristic rays, then it holds for semi-bounded bicharacteristic rays
(that is, those which are bounded forward or backward in time). Although the
proof is similar to that given in [BR14], we include in here due to its
importance in our work and the seemingly increased complexity of our sets (at
least notationally).

######  Proposition 2.12.

Assume that the geometric control condition (2.1) holds. If
$w\in\dot{\Gamma}_{tr}^{\pm}$, then there exists $s_{\pm}\geq 0$ so that
$a\left(x_{s_{\pm}}^{\pm}(w)\right)>0.$ The same is true for
$w\in\dot{\Lambda}_{tr}^{\pm},$ but with $s_{\pm}\leq 0.$

###### Proof.

We will only demonstrate this for $\dot{\Gamma}_{tr}^{+}$, as the work to
establish the remaining cases is similar. If $w\in\dot{\Gamma}_{tr}^{+},$ then

$\alpha:=\sup_{s\geq 0}|x^{+}_{s}(w)|<\infty.$

According to Remark 2.9, $|\xi_{s}^{+}(w)|\approx 1$ for all $s\in\mathbb{R}$.
Thus,

$\sup_{s\geq 0}|\varphi_{s}^{+}(w)|<\infty.$

Then, there exists a point $w^{\prime}\in T^{*}\mathbb{R}^{3}$ and a sequence
$(s_{n})$ of non-negative real numbers such that
$\varphi^{+}_{s_{n}}(w)\rightarrow w^{\prime}$ as $s_{n}\rightarrow\infty.$
For any $s\in\mathbb{R},$ the group law for the flow tells us that
$\varphi^{+}_{s+s_{n}}(w)=\varphi_{s}^{+}(\varphi^{+}_{s_{n}}(w)),$ and so

$x^{+}_{s+s_{n}}(w)=\Pi_{x}\circ\varphi_{s}^{+}(\varphi^{+}_{s_{n}}(w))\rightarrow
x_{s}^{+}(w^{\prime})\text{ as }s_{n}\rightarrow\infty.$

Since $s+s_{n}\geq 0$ for large enough $n$, it follows that
$|x_{s}^{+}(w^{\prime})|\leq\alpha$ for all $s\in\mathbb{R}.$ By (2.1), there
exists $s^{\prime}\in\mathbb{R}$ for which
$a(x^{+}_{s^{\prime}}(w^{\prime}))>0.$ Recall that
$x_{s^{\prime}+s_{n}}^{+}(w)\rightarrow x^{+}_{s^{\prime}}(w^{\prime})$ as
$n\rightarrow\infty$. Since $a$ is continuous and $s^{\prime}+s_{n}\geq 0$ for
$n$ large enough, we conclude that
$a\left(x_{s^{\prime}+s_{N}}^{+}(w)\right)>0$ for some large $N$. ∎

###  2.4. Escape Function Construction

We will construct our symbols in multiple steps:

  1. (1)

On the characteristic set. Since we are utilizing the half-wave decomposition,
working on the characteristic set amounts to working on each individual light
cone, then combining together. There are three regions of interest, two sub-
regions of the interior region $\\{|x|\leq R\\}$ and the exterior region
$\\{|x|>R\\}$. Here, $R\geq R_{0}$.

    1. (a)

Interior, semi-bounded null bicharacteristics. As opposed to working with the
trapped and non-trapped sets, we will first work with the semi-bounded null
bicharacteristics with initial data living in the interior region $\\{|x|\leq
R\\}$. Working with the trapped and non-trapped sets can be difficult, since
one can have non-trapped trajectories which are bounded forward or backward in
time (but not both). Heuristically, these trajectories constitute the boundary
of the non-trapped set. Instead, we will explicitly work with trajectories
which are bounded forward or backward in time. This is where geometric control
is used. This step is inspired by the work in [BR14].

    2. (b)

The remainder of the interior region. Since there is no trapping here, we
construct a symbol similar to the one constructed in [BR14], [Doi96], and
[mst20]. We will need to make an appropriate modification to avoid trapped
trajectories while working with the half-wave symbols.

    3. (c)

The exterior region. As a consequence of asymptotic flatness, there are no
trapped trajectories here. Hence, this follows from a similar multiplier to
that used to prove local energy decay for the flat wave equation, although the
multiplier must be appropriately adapted to the geometry. Here, we are
motivated by prior work in [MMT08] and [mst20].

  2. (2)

On the elliptic set. Here, we construct a correction term. That is, we will
construct a lower-order symbol which provides no contribution on the
characteristic set and provides positivity off of it. This is based on the
work in [mst20].

We will break this construction up into a sequence of lemmas, starting with
(1a). While our construction follows that of [BR14], we reason differently.
Their argument utilizes semiclassical rescaling, which provides compactness
for their interior, semi-trapped set. Since we are sticking with the
microlocal framework, we instead utilize homogeneity arguments to obtain this
compactness. This is one of the reasons to work with the half-wave
decomposition (the other being related to step (1b), which we will outline
once we get there).

With this in mind, we will utilize the sets

$\displaystyle\Omega_{R}^{\pm}$
$\displaystyle:=\left(\Gamma_{tr}^{\pm}\cup\Lambda_{tr}^{\pm}\right)\cap\\{|x|\leq
R\\},$ $\displaystyle\dot{\Omega}_{R}^{\pm}$
$\displaystyle:=\Omega_{R}^{\pm}\cap\Phi^{\pm}(T^{*}\mathbb{R}^{3}\setminus
o).$

As a consequence of Proposition 2.8 and Proposition 2.11(b), the latter set is
compact.

######  Lemma 2.13 (Semi-bounded Escape Function Construction).

There exist $q^{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$, an open
set $V_{R}^{\pm}\supset\Omega_{R}^{\pm}$, and $C^{\pm}\in\mathbb{R}_{+}$ so
that

$H_{p^{\pm}}q^{\pm}+C^{\pm}a\gtrsim_{R}\mathbbm{1}_{V_{R}^{\pm}}.$

Further, $q^{\pm}=q_{1}^{\pm}\circ\Phi^{\pm},$ where $q_{1}^{\pm}\in
C_{c}^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$.

Here, $\Phi^{\pm}\in S^{0}_{\operatorname{hom}}(T^{*}\mathbb{R}^{3}\setminus
o)$ is the scaling function introduced in Section 2.3. The fact that we omit
the zero section is unavoidable, but it is non-problematic; we will introduce
high-frequency cutoffs to our symbols later on which allow for smooth
extensions to all of phase space.

###### Proof.

We will first construct a symbol $q_{1}^{\pm}$ and an open set
$\dot{V}_{R}^{\pm}\supset\dot{\Omega}_{R}^{\pm}$ such that

$H_{p^{\pm}}q_{1}^{\pm}+C^{\pm}a\gtrsim_{R}\mathbbm{1}_{\dot{V}_{R}^{\pm}}.$

To that end, let $w^{\pm}\in\dot{\Omega}_{R}^{\pm}.$ By Proposition 2.12,
there exists $s_{w^{\pm}}\in\mathbb{R}$ for which
$a(x^{\pm}_{s_{w^{\pm}}}(w^{\pm}))>0$. Say that
$2\alpha_{w^{\pm}}:=a(x^{\pm}_{s_{w^{\pm}}}(w^{\pm})).$ By the continuity of
the flow in the initial data, there exists a neighborhood $U_{w^{\pm}}$ of
$w^{\pm}$ so that $a(x^{\pm}_{s_{w^{\pm}}}(z))>\alpha_{w^{\pm}}$ for all $z\in
U_{w^{\pm}}$. Select a smooth cutoff $\chi_{w^{\pm}}\in
C_{c}^{\infty}(T^{*}\mathbb{R}^{3})$ so that $\operatorname{supp}\
\chi_{w^{\pm}}\subset U_{w^{\pm}}$ and $\chi_{w^{\pm}}\equiv 1$ on a smaller
neighborhood $V_{w^{\pm}}$ of ${w^{\pm}}$. Now, we define a symbol on
$T^{*}\mathbb{R}^{3}\setminus o$ given by

$q_{w^{\pm}}(x,\xi)=\int\limits_{0}^{s_{w^{\pm}}}\left(\chi_{w^{\pm}}\circ\varphi^{\pm}_{-s}\right)(x,\xi)\,ds.$

Such a symbol is readily seen to be well-defined, and it is smooth by the
aforementioned smooth flow dependence on data. Next, we demonstrate its
symbolic nature. By continuity of the flow,
$\varphi^{\pm}_{[0,s_{w^{\pm}}]}(\overline{U_{w^{\pm}}}):=\varphi^{\pm}([0,s_{w^{\pm}}]\times\overline{U_{w^{\pm}}})$
is compact. If
$(x,\xi)\notin\varphi^{\pm}_{[0,s_{w^{\pm}}]}(\overline{U_{w^{\pm}}}),$ then
$(x,\xi)\notin\varphi^{\pm}_{s}(\overline{U_{w^{\pm}}})$ for any
$s\in[0,s_{w}]$. Then, $\varphi_{-s}^{\pm}(x,\xi)\notin\overline{U_{w^{\pm}}}$
for any $s\in[0,s_{w^{\pm}}]$, implying that $q_{w^{\pm}}(x,\xi)=0.$ Hence,
$q_{w^{\pm}}\in C_{c}^{\infty}(T^{*}\mathbb{R}^{3}\setminus o).$

Applying the Hamiltonian vector field $H_{p^{\pm}}$ gives us

$H_{p^{\pm}}q_{w^{\pm}}=\int\limits_{0}^{s_{w^{\pm}}}H_{p^{\pm}}(\chi_{w^{\pm}}\circ\varphi_{-s}^{\pm})\,ds=-\int\limits_{0}^{s_{w^{\pm}}}\partial_{s}\left(\chi_{w^{\pm}}\circ\varphi_{-s}^{\pm}\right)\,ds=\chi_{w^{\pm}}-\chi_{w^{\pm}}\circ\varphi_{-s_{w^{\pm}}}^{\pm}.$

Notice that the term $-\chi_{w^{\pm}}\circ\varphi_{-s_{w^{\pm}}}^{\pm}$ is
non-positive and that

$\operatorname{supp}\left(\chi_{w^{\pm}}\circ\varphi_{-s_{w^{\pm}}}^{\pm}\right)\subset\left\\{v:\varphi_{-s_{w^{\pm}}}^{\pm}(v)\in
U_{w^{\pm}}\right\\}=\left\\{v:v\in\varphi_{s_{w^{\pm}}}^{\pm}(U_{w^{\pm}})\right\\}\subset\\{x:a(x)>\alpha_{w^{\pm}}\\}.$

Using this support property, we can use the damping to absorb the poorly-
signed term and obtain non-negativity of $H_{p^{\pm}}q_{w^{\pm}}$. Indeed, if
we call $C_{w^{\pm}}=2(\alpha_{w^{\pm}})^{-1}$, then we have

$\chi_{w^{\pm}}\circ\varphi_{-s_{w^{\pm}}}+C_{w^{\pm}}a(x)\geq 0.$

Thus,

$H_{p^{\pm}}q_{w^{\pm}}+C_{w^{\pm}}a\gtrsim\mathbbm{1}_{V_{w^{\pm}}}.$

Since $\dot{\Omega}_{R}^{\pm}$ is compact, we can reduce the open cover
$\\{V_{w^{\pm}}\\}_{{w^{\pm}}\in\dot{\Omega}_{R}^{\pm}}$ to a finite subcover
$\\{V_{w_{j}^{\pm}}\\}_{j=1}^{m},$ with each
$w_{j}^{\pm}\in\dot{\Omega}_{R}^{\pm}.$ Call

$\dot{V}_{R}^{\pm}=\bigcup\limits_{j=1}^{m}V_{w^{\pm}_{j}},\qquad
q^{\pm}_{1}=\sum\limits_{j=1}^{m}q_{w_{j}^{\pm}},\qquad\text{and}\qquad{C}^{\pm}=\sum\limits_{j=1}^{m}C_{w_{j}^{\pm}}.$

This provides us with a symbol $q_{1}^{\pm}\in
C_{c}^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$ so that

$H_{p^{\pm}}q_{1}^{\pm}+{C}^{\pm}a\gtrsim\mathbbm{1}_{\dot{V}_{R}^{\pm}},\qquad\dot{V}_{R}^{\pm}\supset\dot{\Omega}_{R}^{\pm}.$

Finally, we will extend the above estimate from an indicator on
$\dot{V}_{R}^{\pm}$ to an indicator on a neighborhood
$V_{R}^{\pm}\supset\Omega_{R}^{\pm}$. Consider the function
$q^{\pm}:T^{*}\mathbb{R}^{3}\setminus o\rightarrow\mathbb{R}$ given by

$q^{\pm}=q_{1}^{\pm}\circ\Phi^{\pm}.$

Since geometric control is invariant under $\Phi^{\pm},$ we can see that
$q^{\pm}\neq 0.$ By definition,

$H_{p^{\pm}}q^{\pm}\big{|}_{(x,\xi)}=\frac{d}{ds}\left(q^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})\right)\big{|}_{s=0}.$

Since $b^{\pm}$ is a constant of motion for the Hamiltonian system generated
by $p^{\pm}$, it follows that

$(\nabla_{x}b^{\pm})(x_{s}^{\pm},\xi_{s}^{\pm})\dot{x}_{s}^{\pm}+(\nabla_{\xi}b^{\pm})(x_{s}^{\pm},\xi_{s}^{\pm})\dot{\xi}_{s}^{\pm}=0$

for all $s$. Using this, we calculate that

$\displaystyle\frac{d}{ds}\left(q^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})\right)$
$\displaystyle=\frac{d}{ds}\left(q_{1}^{\pm}\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\right)$
$\displaystyle=(\nabla_{x}q_{1}^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\cdot\left(\dot{x}_{s}^{\pm}\right)$
$\displaystyle+(\nabla_{\xi}q_{1}^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\cdot\frac{(|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|\dot{\xi}_{s}^{\pm}-\xi_{s}^{\pm}\left((\nabla_{x}b^{\pm})(x_{s}^{\pm},\xi_{s}^{\pm})\dot{x}_{s}^{\pm}+(\nabla_{\xi}b^{\pm})(x_{s}^{\pm},\xi_{s}^{\pm})\dot{\xi}_{s}^{\pm}\right)}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|^{2}}$
$\displaystyle=(\nabla_{x}q_{1}^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\cdot(\nabla_{\xi}p^{\pm})(x_{s}^{\pm},\xi_{s}^{\pm})$
$\displaystyle\qquad-\frac{1}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}(\nabla_{\xi}q_{1}^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\cdot(\nabla_{x}p^{\pm})(x_{s}^{\pm},\xi_{s}^{\pm})$
$\displaystyle=(\nabla_{x}q_{1}^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\cdot(\nabla_{\xi}p^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)$
$\displaystyle\qquad-(\nabla_{\xi}q_{1}^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)\cdot(\nabla_{x}p^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)$
$\displaystyle=H_{p^{\pm}}q_{1}^{\pm}\big{|}_{\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)},$

where we have used homogeneity to obtain that

$(\nabla_{\xi}p^{\pm})\left(x_{s}^{\pm},\xi_{s}^{\pm}\right)=(\nabla_{\xi}p^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right)$

and

$\frac{1}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}(\nabla_{x}p^{\pm})\left(x_{s}^{\pm},\xi_{s}^{\pm}\right)=(\nabla_{x}p^{\pm})\left(x_{s}^{\pm},\frac{\xi_{s}^{\pm}}{|b^{\pm}(x_{s}^{\pm},\xi_{s}^{\pm})|}\right).$

If we define $V_{R}^{\pm}=(\Phi^{\pm})^{-1}\left(\dot{V}_{R}^{\pm}\right),$
then we have an open neighborhood of $\Omega_{R}^{\pm}$ such that

$H_{p^{\pm}}q^{\pm}\big{|}_{(x,\xi)}+C^{\pm}a(x)=H_{p^{\pm}}q_{1}^{\pm}\big{|}_{\left(x,\frac{\xi}{|b^{\pm}(x,\xi)|}\right)}+C^{\pm}a(x)\gtrsim\left(\mathbbm{1}_{\dot{V}_{R}^{\pm}}\circ\Phi^{\pm}\right)(x,\xi)\geq\mathbbm{1}_{V_{R}^{\pm}},$

since $\Phi^{\pm}(V_{R}^{\pm})\subset\dot{V}_{R}^{\pm}$. ∎

Now that we have completed step (1a), we move on to parts (1b) and (1c). Step
(1b) pertains to non-trapped null bicharacteristics in the interior region.
The symbol that we produce follows the construction appearing in [Doi96] and
utilized in many other works, such as [BR14] and [mst20]. Like in [mst20], we
perform a factoring argument. The reason for studying the half-wave
decomposition is due to the presence of a cutoff needed to make our
constructed “symbol” genuinely symbolic. In the unfactored setting, cross
terms in the metric arise when differentiating the cutoff in the computation
of the Poisson bracket, generating an error term that is difficult to control.
In the factored setting, this error can be handled straightforwardly.

Step (1c) takes place in the exterior region. This is of little concern, as we
possess robust exterior estimates. We utilize this symbol as a means of
bootstrapping the aforementioned error term, which will be compactly-supported
in the region where the exterior symbol has strictly positive Poisson bracket
with $p^{\pm}$.

To these ends, we will analyze both half-waves simultaneously (as in Lemma
2.13). While this portion of the argument follows the one given in [mst20], it
does require a modification; the escape function on interior, non-trapped null
bicharacteristics needs an appropriate adjustment to ensure that it avoids
trapped trajectories. We start with a proposition where we construct a
function that will be used for the previously-described error absorption. The
construction of this function comes from e.g. [mst20], [Tat08].

######  Proposition 2.14.

Let $\sigma>0.$ Then, there exists $f\in C^{\infty}$ satisfying
$f(r)\approx_{\sigma}1$ when $r>R_{0}$ and $f^{\prime}(r)\approx\sigma
c_{j}2^{-j}f(r)$ when $r\approx 2^{j}>R_{0}$.

Here, $(c_{j})$ is the slow-varying sequence introduced in Section 1.2.

######  Remark 2.15.

Although the sequence $(c_{j})$ is not defined for all natural numbers, the
indices where it is not defined index finitely many dyadic regions (in
particular, they omit where the operator $P$ need not be a small $AF$
perturbation). Since this region is compact, we can extend the sequence to
such indices in an arbitrary manner. The typical way that this sequence is
extended is by choosing $c_{j}$ so that $\left\lVert
g-m\right\rVert_{AF(A_{j})}\lesssim c_{j}$ for the previously-undefined
indices $j$. ∎

###### Proof.

As in [Tat08], we can construct a smooth function $c(s)$ from the sequence
$(c_{j})$ such that $c(s)\in(c_{j},2c_{j})$ for each $s\in(2^{j},2^{j+1})$ and
$|c^{\prime}(s)|\leq\delta s^{-1}c(s).$ Since $(c_{j})$ is a positive sequence
which converges to zero, it has a positive maximum, say $c_{N}$. Then, we
observe that

$\textbf{c}\lesssim c_{N}\leq
c(2^{N}+2^{N-1})=\left|\int_{2^{N}+2^{N-1}}^{\infty}c^{\prime}(s)\,ds\right|\lesssim\int\limits_{1}^{\infty}\frac{c(s)}{s}\,ds$

and

$\int\limits_{1}^{\infty}\frac{c(s)}{s}\,ds\leq\sum_{j=0}^{\infty}\int\limits_{2^{j}}^{2^{j+1}}\frac{2c_{j}}{2^{j}}\,ds=2\sum_{j=0}^{\infty}c_{j}\lesssim\textbf{c}.$

That is,

$\int\limits_{1}^{\infty}\frac{c(s)}{s}\,ds\approx\textbf{c}.$

Now, set

$f(r)=\exp\left(\sigma\int\limits_{1}^{r}\frac{c(s)}{s}\,ds\right).$

From our prior estimate, it is immediate that

$f(r)\approx e^{\sigma\textbf{c}}\approx_{\sigma}1$

for $r>R_{0},$ and

$f^{\prime}(r)=\sigma\frac{c(r)}{r}f(r)\approx\sigma c_{j}2^{-j}f(r)$

for $r\approx 2^{j}.$ ∎

Now, we complete steps (1b) and (1c).

######  Lemma 2.16 (Non-trapped Escape Function Construction).

Let $R\geq R_{0}$. Then, there exist
$q^{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$ and
$W^{\pm}\subset\Omega_{\infty}^{\pm}$ so that $V_{R}^{\pm}\cup
W^{\pm}=T^{*}\mathbb{R}^{3}\setminus o$ and

$H_{p^{\pm}}q^{\pm}\gtrsim c_{j}2^{-j}\mathbbm{1}_{W^{\pm}},\qquad|x|\approx
2^{j}.$

Further, $q^{\pm}=\varepsilon q_{in}^{\pm}+q_{out}^{\pm},$ where
$q_{in}^{\pm}=\tilde{q}_{in}^{\pm}\circ\Phi^{\pm}$ with
$\tilde{q}_{in}^{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$ is
supported in $\\{|x|\leq 4R\\}$, $q_{out}^{\pm}\in
S^{0}_{\operatorname{hom}}(T^{*}\mathbb{R}^{3}\setminus o)$, and
$\varepsilon>0$ is sufficiently small.

The inclusion of the sequence $(c_{j})$ is necessitated by the prior
proposition, which is used for bootstrapping purposes in the exterior region.
Its slowly varying nature allows one to work in the weight $\left\langle
x\right\rangle^{-2}$ from the powers $|x|\approx 2^{-j}$ which will arise in
the exterior (there is no trouble working in the weight $\left\langle
x\right\rangle^{-2}$ in the interior region by compactness).

###### Proof.

Recall from Proposition 2.8 that $|\xi|\approx|b^{\pm}(x,\xi)|$ on
$T^{*}\mathbb{R}^{3}\setminus o;$ let $c^{\pm},C^{\pm}>0$ denote the
respective lower and upper bound implicit constants in the inequalities and
take $\delta^{\pm}$ such that $c^{\pm}-\delta^{\pm}>0.$ Now, choose
$\psi^{\pm}\in C_{c}^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$ such that

$\displaystyle\operatorname{supp}\psi^{\pm}$
$\displaystyle\subset\Omega_{\infty}^{\pm}\cap\\{|x|\leq
R\\}\cap\\{c^{\pm}-\delta^{\pm}<|\xi|<C^{\pm}+1\\},$
$\displaystyle\psi^{\pm}\equiv 1\text{ on }U_{R}^{\pm}$
$\displaystyle:=\left(\Omega_{\infty}^{\pm}\cap\\{|x|\leq
R\\}\cap\Phi^{\pm}(T^{*}\mathbb{R}^{3}\setminus
o)\right)\setminus\dot{V}_{R},$

where $R\geq R_{0}.$ Now, we define the function

$\tilde{q}_{in}^{\pm}(x,\xi)=-\chi_{<2R}(|x|)\int\limits_{0}^{\infty}\psi^{\pm}\circ\varphi_{s}^{\pm}(x,\xi)\,ds,\qquad(x,\xi)\in
T^{*}\mathbb{R}^{3}\setminus o.$

Since non-trapped null bicharacteristic rays must exit any compact set after a
finite amount of time, this integral is well-defined for each $(x,\xi)\in
T^{*}\mathbb{R}^{3}\setminus o,$ which establishes $\tilde{q}_{in}^{\pm}$ as a
well-defined function. It takes more work to show that $\tilde{q}_{in}^{\pm}$
is smooth. Similar to [BR14], we will begin by establishing a maximal amount
of time that bicharacteristic rays can remain in the support of the integrand.
We already know that $\operatorname{supp}\psi^{\pm}$ is compact. Let $V^{\pm}$
be an open neighborhood of $\operatorname{supp}\psi^{\pm}$ such that
$\overline{V^{\pm}}\subset\Omega_{\infty}^{\pm}$. Take $\overline{V^{\pm}}=K$
in Proposition 2.11(c), and let $T^{\prime}$ be as given in the proposition.
We claim that every point $w^{\pm}\in T^{*}\mathbb{R}^{3}\setminus o$ has a
neighborhood $U_{w^{\pm}}$ of $w^{\pm}$ and a time $s_{w^{\pm}}\geq 0$ such
that $(\psi^{\pm}\circ\varphi_{s}^{\pm})(z)=0$ for every $z\in U_{w^{\pm}}$
and $s\in\mathbb{R}_{+}\setminus[s_{w^{\pm}},s_{w^{\pm}}+T^{\prime}].$ That
is, all bicharacteristics (with speed $\approx 1$) can spend no more than time
$T^{\prime}$ within $\operatorname{supp}\psi^{\pm}.$ The time $s_{w^{\pm}}$
bears no similarity to the variable of the same name in the proof of
Proposition 2.13. As a direct consequence of Proposition 2.11, we may take
$U_{w^{\pm}}=V^{\pm}$ and $s_{w^{\pm}}=0$ whenever
$w^{\pm}\in\operatorname{supp}\psi^{\pm}\subset V^{\pm}.$ If

$w^{\pm}\notin\bigcup\limits_{s\in\mathbb{R}}\varphi_{-s}^{\pm}\left(\operatorname{supp}\psi^{\pm}\right)=:\mathcal{X}^{\pm},$

then the fact that $\mathcal{X}^{\pm}$ is closed provides an open neighborhood
$U_{w^{\pm}}$ of $w^{\pm}$ such that $\mathcal{X}^{\pm}\cap
U_{w^{\pm}}=\emptyset.$ For each $z\in U_{w^{\pm}}$, we have that
$\varphi^{\pm}_{s}(z)\notin\operatorname{supp}\psi^{\pm}$ for all
$s\in\mathbb{R}$, i.e. $(\psi^{\pm}\circ\varphi_{s}^{\pm})(z)=0$ for
$s\in\mathbb{R}$. Hence, this case holds with $U_{w^{\pm}}$ as defined and
$s_{w^{\pm}}=0$. Finally, let
$w^{\pm}\in\mathcal{X}^{\pm}\setminus\operatorname{supp}\psi^{\pm}.$ Then,
$\varphi_{s^{\prime}}^{\pm}(w^{\pm})\in\operatorname{supp}\psi^{\pm}$ for some
$s^{\prime}\in\mathbb{R}\setminus\\{0\\}.$ If $s^{\prime}>0,$ then we can
combine this with the fact that
$\varphi_{0}^{\pm}(w)\notin\operatorname{supp}\psi^{\pm}$ and the continuity
of the flow to obtain $s_{w^{\pm}}>0$ such that
$\varphi_{s_{w^{\pm}}}^{\pm}(w)\in V^{\pm}$ and
$\varphi_{s}^{\pm}(w)\notin\operatorname{supp}\psi^{\pm}$ for all
$s\in[0,s_{w^{\pm}}]$. By continuity of the flow in the data, we can extend
the above to a neighborhood $U_{w^{\pm}}$. That is, there exists a
neighborhood $U_{w^{\pm}}$ of $w^{\pm}$ so that for all $z\in U_{w^{\pm}}$, we
have that $\varphi^{\pm}_{s_{w^{\pm}}}(z)\in V^{\pm}$ and
$(\psi^{\pm}\circ\varphi_{s}^{\pm})(z)=0$ for all $s\in[0,s_{w^{\pm}}].$
Applying Proposition 2.11 to $K=V^{\pm}$ implies that
$(\psi^{\pm}\circ\varphi_{s}^{\pm})(z)=0$ for all $z\in U_{w^{\pm}}$ and
$s\in[0,s_{w^{\pm}}]\cup[s_{w^{\pm}}+T^{\prime},\infty).$ It remains to
consider if we cannot assume that $s^{\prime}>0$. In this case,

$w^{\pm}\notin\bigcup\limits_{s\in\mathbb{R}_{+}}\varphi_{-s}^{\pm}(\operatorname{supp}\psi^{\pm})=:\mathcal{X}^{\pm}_{-}$

Note that $\mathcal{X}^{\pm}_{-}$ is closed by the same logic which showed
that $\mathcal{X}^{\pm}$ is closed (see the proof in Proposition 2.11(c)).
From here, one can simply proceed as in the case where
$w^{\pm}\notin\mathcal{X}^{\pm}.$

Using this result, we know that the integral present in $\tilde{q}_{in}^{\pm}$
is always over an interval of maximal length $T^{\prime}$. Hence,
differentiation under the integral sign is non-problematic and in view of the
regularity of the flow map, we conclude that $\tilde{q}_{in}^{\pm}\in
C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$. Additionally, it is supported in
$\\{|x|\leq 4R\\}$. In particular, it is smooth and bounded in all derivatives
on the compact set

$\\{|x|\leq 4R\\}\cap\Phi^{\pm}(T^{*}\mathbb{R}^{3}\setminus o).$

Now, consider the smooth function

$q_{in}^{\pm}=\tilde{q}_{in}^{\pm}\circ\Phi^{\pm}$

defined on $T^{*}\mathbb{R}^{3}\setminus o.$ As in the proof of Lemma 2.13, we
get that

$H_{p^{\pm}}q_{in}^{\pm}\big{|}_{(x,\xi)}=H_{p^{\pm}}\tilde{q}_{in}^{\pm}\big{|}_{\Phi^{\pm}(x,\xi)}.$

Now, we calculate that

$\displaystyle H_{p^{\pm}}\tilde{q}_{in}^{\pm}\big{|}_{\Phi^{\pm}(x,\xi)}$
$\displaystyle=\chi_{<2R}(|x|)\psi^{\pm}\left(x,\frac{\xi}{|b^{\pm}(x,\xi)|}\right)$
$\displaystyle\qquad+\frac{1}{2R}b_{\xi_{k}}^{\pm}\left(x,\frac{\xi}{|b^{\pm}(x,\xi)|}\right)\frac{x_{k}}{|x|}\chi^{\prime}\left(\frac{|x|}{2R}\right)\int\limits_{0}^{\infty}\psi^{\pm}\circ\varphi^{\pm}_{s}\left(x,\frac{\xi}{|b^{\pm}(x,\xi)|}\right)\,ds.$

The first term is non-negative, supported in
$\Omega_{\infty}^{\pm}\cap\\{|x|\leq R\\}$, and equal to $1$ on
$U^{\pm}:=\Phi^{-1}(\dot{U}_{R}^{\pm})$. The second term is an error term
which is supported in $\\{2R\leq|x|\leq 4R\\}.$ The primary purpose of the
exterior multiplier is to absorb this error term. To that end, let

$q_{out}^{\pm}=-\chi_{>R}(|x|)f(|x|)b_{\xi_{k}}^{\pm}\frac{x_{k}}{|x|},$

where $f$ is the function constructed in Proposition 2.14. It is easy to see
that $q_{out}^{\pm}\in S^{0}_{\operatorname{hom}}(T^{*}\mathbb{R}^{3}\setminus
o)$, as it is smooth, bounded in all $x$ derivatives due to asymptotic
flatness, homogeneous of degree $0$, and satisfies the appropriate symbol
estimate. One can readily compute that

$\displaystyle H_{p^{\pm}}q^{\pm}_{out}$
$\displaystyle=b^{\pm}_{\xi_{k}}\frac{x_{k}}{|x|}\chi_{>R}(|x|)f^{\prime}(|x|)b^{\pm}_{\xi_{j}}\frac{x_{j}}{|x|}+b_{\xi_{k}}^{\pm}\left(\delta_{jk}-\frac{x_{j}x_{k}}{|x|^{2}}\right)\chi_{>R}(|x|)\frac{f(|x|)}{|x|}\left(\delta_{jl}-\frac{x_{j}x_{l}}{|x|^{2}}\right)b_{\xi_{l}}^{\pm}$
$\displaystyle\qquad+R^{-1}\chi^{\prime}\left(\frac{|x|}{R}\right)b^{\pm}_{\xi_{k}}\frac{x_{k}}{|x|}f(|x|)b^{\pm}_{\xi_{j}}\frac{x_{j}}{|x|}+\mathcal{O}(\left\langle
x\right\rangle|\partial g|)\chi_{>R}(|x|)|x|^{-1}.$

We remark that the last term is small for $|x|>R$ by asymptotic flatness (and
it is localized to this region due to the cutoff), while the remaining terms
are all non-negative. The third term is non-negative and supported in the
annulus $\\{R\leq|x|\leq 2R\\}$ due to the support of $\chi^{\prime}$. Making
$\sigma$ large enough and using asymptotic flatness provides that, for any
$|x|\approx 2^{j}$,

$\displaystyle H_{p^{\pm}}q_{out}^{\pm}$
$\displaystyle>\frac{\sigma}{2}c_{j}2^{-j}f(|x|)\chi_{>R}(|x|)\frac{|x\cdot\nabla_{\xi}b^{\pm}|^{2}}{|x|^{2}}+\chi_{>R}(|x|)\frac{f(|x|)}{|x|}\left(|\nabla_{\xi}b^{\pm}|^{2}-\frac{|x\cdot\nabla_{\xi}b^{\pm}|^{2}}{|x|^{2}}\right)$
$\displaystyle\gtrsim c_{j}2^{-j}\chi_{>R}(|x|)|\nabla_{\xi}b^{\pm}|^{2}$
$\displaystyle\gtrsim c_{j}2^{-j}\chi_{>R}(|x|).$

Thus, $H_{p^{\pm}}q_{out}^{\pm}$ is non-negative, strictly positive for
$|x|>R$, and

$H_{p^{\pm}}q_{out}^{\pm}\gtrsim c_{j}2^{-j}\chi_{>R},\qquad|x|\approx 2^{j}.$

Recall that the error term in $H_{p^{\pm}}q_{in}^{\pm}$ is bounded and
supported in $\\{2R\leq|x|\leq 4R\\}$, and $H_{p^{\pm}}q_{out}^{\pm}$ is
strictly positive on the support of this error (with a uniform bound from
below on this set).

Define

$q^{\pm}=\varepsilon q_{in}^{\pm}+q_{out}^{\pm}\in
C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o),$

where $0<\varepsilon\ll 1$. By choosing $\varepsilon$ sufficiently small, we
may absorb the aforementioned error due to our prior discussion, obtaining
that $H_{p^{\pm}}q^{\pm}$ is non-negative everywhere and positive on

$W^{\pm}:=U^{\pm}\cup\\{(x,\xi)\in T^{*}\mathbb{R}^{3}\setminus o:|x|>R\\}.$

By Proposition 2.11(a),

$\displaystyle V_{R}^{\pm}\cup U^{\pm}$
$\displaystyle=V_{R}^{\pm}\cup\left((\Omega_{\infty}^{\pm}\cap\\{|x|\leq
R\\})\setminus V_{R}^{\pm}\right)$
$\displaystyle\supset\left(\Omega_{R}^{\pm}\cup\Omega_{\infty}\right)\cap\\{|x|\leq
R\\}$ $\displaystyle=\left(T^{*}\mathbb{R}^{3}\setminus o\right)\cap\\{|x|\leq
R\\},$

and so

$\displaystyle V_{R}^{\pm}\cup U^{\pm}$
$\displaystyle=\left(T^{*}\mathbb{R}^{3}\setminus o\right)\cap\\{|x|\leq
R\\},$ $\displaystyle V_{R}^{\pm}\cup W^{\pm}$
$\displaystyle=T^{*}\mathbb{R}^{3}\setminus o.$

We have already shown that

$H_{p^{\pm}}q^{\pm}\approx 1\qquad(x,\xi)\in U^{\pm}$

and

$H_{p^{\pm}}q^{\pm}\gtrsim c_{j}2^{-j}\chi_{>R},\qquad|x|\approx 2^{j}.$

The latter estimate readily extends to

$H_{p^{\pm}}q^{\pm}\gtrsim c_{j}2^{-j}\mathbbm{1}_{W^{\pm}},\qquad|x|\approx
2^{j}$

by the compactness of the interior region $\\{|x|\leq R\\}$. ∎

Now, we combine on the light cones to get our desired symbol $q$, as well as
obtain positivity on the elliptic set (step (2)). This largely follows the
steps present in [mst20], although we have additional technicalities resulting
from the damping.

###### (Proof of Lemma 2.4).

Let $q_{1}^{\pm}$ denote the symbol $q^{\pm}$ constructed in Lemma 2.13 (not
the symbol $q_{1}^{\pm}$ from the same lemma) and $q_{2}^{\pm}$ denote the
symbol $q^{\pm}$ constructed in Lemma 2.16. We remark that, as a consequence
of the chain rule, both symbols satisfy the standard $S^{0}$ bounds for
$|\xi|\geq 1.$ First, we truncate to the high-frequency regime via the symbols

$q^{\pm}_{j,>\lambda}=e^{-\sigma q_{j}^{\pm}}\chi_{>\lambda}(|b^{\pm}|),\
j=1,2,$

where $\sigma$ is the parameter in Proposition 2.14. We assume that
$\lambda>1.$ The exponentiation is implemented for bootstrapping: taking
derivatives of the exponential will provide multiplication by $\sigma\gg 1$.
Since $|b^{\pm}|\approx|\xi|,$ these cutoffs genuinely truncate to high
frequencies when $\lambda$ is large. Further, the truncation to $|\xi|\gtrsim
1$ eliminates the singularities of $q^{\pm}_{j},$ i.e.
$q_{j}^{\pm}\chi_{>\lambda}(|b^{\pm}|)$ smoothly extends to an element of
$S^{0}(T^{*}\mathbb{R}^{3}).$

We claim that exponentiation preserves the symbol class, so that
$q_{j,>\lambda}^{\pm}\in S^{0}(T^{*}\mathbb{R}^{3}).$ We can immediately see
that $q_{j,>\lambda}^{\pm}$ is smooth. Note that for $|\xi|\geq\lambda,$ the
exponentials $e^{-\sigma q_{j}^{\pm}}$ are bounded since $q_{j}^{\pm}$ are
bounded, and for $|\xi|<\lambda$, we immediately have that
$q_{j,>\lambda}^{\pm}\equiv 0$. When checking the symbolic nature of
$q_{j,>\lambda}^{\pm}$, we only need to study the boundedness of the $\xi$
derivatives since our symbols $q_{j}^{\pm}$ are bounded in all derivatives in
$x$. Taking a partial derivative in $\xi$ provides that

$\partial_{\xi_{k}}q^{\pm}_{j,>\lambda}=-\sigma(\partial_{\xi_{k}}q_{j}^{\pm})q^{\pm}_{j,>\lambda}\mp\frac{e^{-\sigma
q^{\pm}_{j}}}{\lambda}(\partial_{\xi_{k}}b^{\pm})\chi^{\prime}\left(\frac{|b^{\pm}|}{\lambda}\right).$

The first term is $\mathcal{O}(\left\langle\xi\right\rangle^{-1})$, and the
second term is compactly-supported in $\xi$. Due to the aforementioned compact
support, we only need consider further $\xi$ differentiation of
$\sigma(\partial_{\xi_{k}}q_{j}^{\pm})q^{\pm}_{j,>\lambda}$. If the $\xi$
derivative lands on the exponential, then the result is
$\mathcal{O}(\left\langle\xi\right\rangle^{-2})$ by the prior argument. If the
derivative lands on $\partial_{\xi}q_{j}^{\pm}$, then the same asymptotics
hold since $\partial_{\xi}q_{j}^{\pm}\in S^{-1}(T^{*}\mathbb{R}^{3}).$ If the
derivative lands on the cutoff, then the result is compactly-supported in
$\xi$. Inducting establishes that $q^{\pm}_{j,>\lambda}\in
S^{0}(T^{*}\mathbb{R}^{3})$.

Now, we combine the symbols constructed on each light cones together as

$q(\tau,x,\xi)=(\tau-b^{+})(q^{-}_{1,>\lambda}+q^{-}_{2,>\lambda})+(\tau-b^{-})(q^{+}_{1,>\lambda}+q^{+}_{2,>\lambda}).$

Calling

$q_{j}=(\tau-b^{+})q^{-}_{j,>\lambda}+(\tau-b^{-})q^{+}_{j,>\lambda},$

we can see that

$\displaystyle(H_{p}q+2\gamma\tau aq)\big{|}_{\tau=b^{\pm}}$
$\displaystyle=(H_{p}q_{1}+2\gamma\tau
aq_{1})\big{|}_{\tau=b^{\pm}}+(H_{p}q_{2}+2\gamma\tau
aq_{2})\big{|}_{\tau=b^{\pm}}$
$\displaystyle=H_{p}q_{1}\big{|}_{\tau=b^{\pm}}\pm 2\gamma
b^{\pm}(b^{+}-b^{-})aq_{1,>\lambda}^{\pm}$
$\displaystyle\qquad+H_{p}q_{2}\big{|}_{\tau=b^{\pm}}\pm 2\gamma
b^{\pm}(b^{+}-b^{-})aq_{2,>\lambda}^{\pm}.$

We will work with each term in the last equality separately. First, we compute
that

$\displaystyle H_{p}q_{j}\big{|}_{\tau=b^{\pm}}$
$\displaystyle=-(b^{+}-b^{-})^{2}H_{p^{\pm}}q_{j,>\lambda}^{\pm}$
$\displaystyle\qquad-(b^{\pm}-b^{\mp})q_{j,>\lambda}^{\pm}(b_{\xi_{j}}^{\pm}b_{x_{j}}^{\mp}-b^{\pm}_{x_{j}}b^{\mp}_{\xi_{j}})$
$\displaystyle=\sigma(b^{+}-b^{-})^{2}q_{j,>\lambda}^{\pm}H_{p^{\pm}}q_{j}^{\pm}$
$\displaystyle\qquad-(b^{\pm}-b^{\mp})q_{j,>\lambda}^{\pm}(b_{\xi_{j}}^{\pm}b_{x_{j}}^{\mp}-b^{\pm}_{x_{j}}b^{\mp}_{\xi_{j}}).$

By making $\sigma$ sufficiently large, we get that

$\displaystyle H_{p}q_{j}\big{|}_{\tau=b^{\pm}}$
$\displaystyle\geq\frac{1}{2}\sigma(b^{+}-b^{-})^{2}q_{j,>\lambda}^{\pm}H_{p^{\pm}}q_{j}^{\pm}+E_{j}^{\pm},$

where $E_{j}^{\pm}$ are error terms which are supported in a neighborhood of
the region where $H_{p^{\pm}}q_{j}^{\pm}=0$. These terms are non-problematic,
as they are readily absorbed into the above estimate with differing $j$ when
we combine the estimates together. Hence, we will drop the $E_{j}^{\pm}$’s for
ease of notation.

Observe that

$\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\approx 1.$

By choosing $\gamma$ large enough, we may apply Lemma 2.13 to obtain that

(2.5) $\displaystyle(H_{p}q_{1}+2\gamma\tau aq_{1})\big{|}_{\tau=b^{\pm}}$
$\displaystyle\geq\frac{1}{2}\sigma(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}H_{p^{\pm}}q_{1}^{\pm}\pm
2\gamma b^{\pm}(b^{+}-b^{-})aq_{1,>\lambda}^{\pm}$
$\displaystyle=\frac{1}{2}\sigma(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{1}^{\pm}+\left(\frac{4\gamma}{\sigma}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}a\right)$
$\displaystyle\gtrsim|\xi|^{2}q_{1,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{1}^{\pm}+\frac{\gamma}{\sigma}a\right)$
$\displaystyle\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}\mathbbm{1}_{{V_{R}^{\pm}}}|\xi|^{2}.$

Notice that $\gamma$ depends on c (and $\sigma)$. For the $j=2$ term, we use
the prior computation, the fact that the damping term has positive sign, and
Lemma 2.16:

(2.6) $\displaystyle(H_{p}q_{2}+2\gamma\tau aq_{2})\big{|}_{\tau=b^{\pm}}$
$\displaystyle\geq\frac{1}{2}\sigma(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}H_{p^{\pm}}q_{2}^{\pm}\pm
2\gamma b^{\pm}(b^{+}-b^{-})aq_{2,>\lambda}^{\pm}$
$\displaystyle\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}\mathbbm{1}_{W^{\pm}}c_{j}2^{-j}|\xi|^{2},\qquad|x|\approx
2^{j}.$

Recall that $V_{R}^{\pm}\cup W^{\pm}=T^{*}\mathbb{R}^{3}\setminus o$.
Combining (2.5) and (2.6) together, we conclude that

$(H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{\pm}}\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}\langle
x\rangle^{-2}|\xi|^{2},$

where we have used the slowly-varying, summable nature of $(c_{j})$.

This provides the desired bound over the characteristic set. To extend it to
all of phase space, we must construct a lower-order correction term.
Explicitly, we seek an $m\in S^{0}$ so that

$H_{p}q+2\gamma\tau aq+mp\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}\left\langle
x\right\rangle^{-2}|\xi|^{2}.$

If we write

$H_{p}q+2\gamma a\tau q=a_{0}\tau^{2}+a_{1}\tau+a_{2},$

where $a_{j}\in S^{j}$, then we have already established that

(2.7) $\displaystyle
a_{0}(x,\xi)(b^{\pm}(x,\xi))^{2}+a_{1}(x,\xi)b^{\pm}(x,\xi)+a_{2}(x,\xi)\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}\left\langle
x\right\rangle^{-2}|\xi|^{2}$

So, we must analyze the quantity

$a_{0}\tau^{2}+a_{1}\tau+a_{2}+p{m}=(a_{0}-{m})\tau^{2}+(a_{1}+(b^{+}+b^{-}){m})\tau+(a_{2}-b^{+}b^{-}{m}).$

If we choose ${m}$ so that

(2.8) $\displaystyle a_{0}-{m}>0,\qquad|\xi|\geq\lambda$

and

(2.9)
$\displaystyle(a_{1}+(b^{+}+b^{-}){m})^{2}-4(a_{0}-{m})(a_{2}-b^{+}b^{-}{m})<0,\qquad|\xi|\geq\lambda,$

then we will have that $a_{0}\tau^{2}+a_{1}\tau+a_{2}+mp$ is positive for
$|\xi|\geq\lambda$ (the first condition on ${m}$ guarantees that this
polynomial in $\tau$ is concave up, and the second guarantees that there are
no real zeros).

Let us begin by focusing on (2.9). The function

$\displaystyle P({m})$
$\displaystyle=(a_{1}+(b^{+}+b^{-}){m})^{2}-4(a_{0}-{m})(a_{2}-b^{+}b^{-}{m})$
$\displaystyle=(b^{+}-b^{-})^{2}{m}^{2}+(2a_{1}(b^{+}+b^{-})+4a_{0}b^{+}b^{-}+4a_{2}){m}+(a_{1}^{2}-4a_{0}a_{2})$

is a quadratic polynomial in ${m}$ with a positive coefficient on the
quadratic term, so it will achieve a minimal value at

${m}=-\frac{a_{1}(b^{+}+b^{-})+2(a_{0}b^{+}b^{-}+a_{2})}{(b^{+}-b^{-})^{2}}.$

It is readily seen that ${m}\in S^{0}$ and that $m$ is supported where
$|\xi|\geq\lambda$. This minimal value is

$\displaystyle P({m})$
$\displaystyle=\left(a_{1}-(b^{+}-b^{-})\frac{a_{1}(b^{+}+b^{-})+2(a_{0}b^{+}b^{-}+a_{2})}{(b^{+}-b^{-})^{2}}\right)^{2}$
$\displaystyle\qquad-4\left(a_{0}+\frac{a_{1}(b^{+}+b^{-})+2(a_{0}b^{+}b^{-}+a_{2})}{(b^{+}-b^{-})^{2}}\right)\left(a_{2}+b^{+}b^{-}\frac{a_{1}(b^{+}+b^{-})+2(a_{0}b^{+}b^{-}+a_{2})}{(b^{+}-b^{-})^{2}}\right)$
$\displaystyle=-4\frac{(a_{0}(b^{+})^{2}+a_{1}b^{+}+a_{2})(a_{0}(b^{-})^{2}+a_{1}b^{-}+a_{2})}{(b^{+}-b^{-})^{2}}$
$\displaystyle=-4(b^{+}-b^{-})^{-2}\left((H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{+}}\right)\left(H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{-}}\right)$ $\displaystyle<0,$

where we have used (2.7). So, (2.9) is satisfied. To establish (2.8), one can
readily check that

$\displaystyle a_{0}-{m}$
$\displaystyle=a_{0}+\frac{a_{1}(b^{+}+b^{-})+2(a_{0}b^{+}b^{-}+a_{2})}{(b^{+}-b^{-})^{2}}$
$\displaystyle=(b^{+}-b^{-})^{-2}\left((H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{+}}+(H_{p}q+2\gamma\tau aq)\big{|}_{\tau=b^{-}}\right)$
$\displaystyle>0$

for $|\xi|\geq\lambda$.

This gives us that

$H_{p}q+2\gamma\tau aq+mp>0$

for $|\xi|\geq\lambda$. In fact, we can check that the minimal value of the
above in $\tau$ for $|\xi|\geq\lambda$ is

$\frac{(a_{0}(b^{+})^{2}+a_{1}b^{+}+a_{2})(a_{0}(b^{+})^{2}+a_{1}b^{+}+a_{2})}{(a_{0}(b^{+})^{2}+a_{1}b^{+}+a_{2})+(a_{0}(b^{-})^{2}+a_{1}b^{-}+a_{2})}=\frac{\left((H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{+}}\right)\left(H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{-}}\right)}{(H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{+}}+(H_{p}q+2\gamma\tau aq)\big{|}_{\tau=b^{-}}}.$

The numerator is bounded below by $\left\langle x\right\rangle^{-4}|\xi|^{4}.$
In view of the support and symbolic properties of $q$, the denominator
satisfies the bounds

$(H_{p}q+2\gamma\tau aq)\big{|}_{\tau=b^{+}}+(H_{p}q+2\gamma\tau
aq)\big{|}_{\tau=b^{-}}\approx\left\langle x\right\rangle^{-2}|\xi|^{2}.$

Since $|b^{\pm}(x,\xi)|\approx|\xi|$ and $|\tau|=|b^{\pm}(x,\xi)|$ in the
above, we conclude the desired result. ∎

###  2.5. Starting Energy Estimates and Case Reductions

In this section, we will establish various useful energy estimates, then
reduce the proof of Theorem 1.7 to a simpler problem. Our starting point is a
standard uniform energy inequality.

######  Proposition 2.17.

Let $P$ be a stationary damped wave operator, $\partial_{t}$ be uniformly
time-like, and $T>0$. Then, we have the estimate

(2.10) $\displaystyle\left\lVert\partial
u(t)\right\rVert_{L^{2}}^{2}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\int\limits_{0}^{T}\int\limits_{\mathbb{R}^{3}}|Pu\
\partial_{t}u|\,dxdt,\qquad 0\leq t\leq T$

for all $u\in\mathcal{W}_{T}$.

###### Proof.

Call $Pu=f$, and define the energy functional

$E[u](t)=\int\limits_{\mathbb{R}^{3}}D_{i}g^{ij}D_{j}u\overline{u}-g^{00}|\partial_{t}u|^{2}\,dx.$

After integrating the first term by parts, this functional is readily seen to
be coercive due to the uniformly time-like nature of $\partial_{t},$ i.e.

$E[u](t)\approx\left\lVert\partial u(t)\right\rVert_{L^{2}}^{2}.$

Differentiating in $t$ and integrating by parts, gives that

$\displaystyle\frac{d}{dt}E[u](t)$
$\displaystyle=-\int\limits_{\mathbb{R}^{3}}g^{00}(\partial_{t}^{2}u\partial_{t}\bar{u}+\partial_{t}u\partial_{t}^{2}\bar{u})\,dx+\int\limits_{\mathbb{R}^{3}}D_{i}g^{ij}D_{j}\partial_{t}u\bar{u}+D_{i}g^{ij}D_{j}u\partial_{t}\bar{u}\,dx$
$\displaystyle=\int\limits_{\mathbb{R}^{3}}(g^{00}D_{t}^{2}+D_{i}g^{ij}D_{j})u\partial_{t}\bar{u}+\partial_{t}u\overline{(g^{00}D_{t}^{2}+D_{i}g^{ij}D_{j})u}\,dx$
$\displaystyle=\int\limits_{\mathbb{R}^{3}}\left(-(g^{0j}D_{j}D_{t}+D_{j}g^{0j}D_{t}+iaD_{t})u+f\right)\partial_{t}\bar{u}$
$\displaystyle\qquad+\partial_{t}u\overline{\left(-(g^{0j}D_{j}D_{t}+D_{j}g^{0j}D_{t}+iaD_{t})u+f\right)}\,dx$
$\displaystyle=2\text{Re}\int\limits_{\mathbb{R}^{3}}\bar{f}\partial_{t}u\,dx-2\int\limits_{\mathbb{R}^{3}}a|\partial_{t}u|^{2}\,dx.$

Dropping the damping term and integrating the resulting estimate in time
yields the inequality

$E[u](t)\lesssim
E[u](0)+\int\limits_{0}^{T}\int\limits_{\mathbb{R}^{3}}|{f}\partial_{t}u|\,dxdt,\qquad
0\leq t\leq T.$

Applying the coercivity allows us to conclude. ∎

######  Remark 2.18.

Note that when $Pu=0,$ we have the energy dissipation statement

$\frac{d}{dt}E[u](t)=-2\int\limits_{\mathbb{R}^{3}}a|\partial_{t}u|^{2}\,dx\leq
0.$

∎

Applying the Schwarz inequality to Proposition 2.17 provides us with estimates
which will prove useful throughout this work.

######  Corollary 2.19.

Under the same assumptions as Proposition 2.17, the uniform energy estimates

$\left\lVert\partial
u\right\rVert_{L^{\infty}_{t}L^{2}_{x}}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\left\lVert Pu\right\rVert_{L^{1}_{t}L^{2}_{x}},$
$\left\lVert\partial
u\right\rVert_{L^{\infty}_{t}L^{2}_{x}}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\left\lVert Pu\right\rVert_{LE^{*}}^{1/2}\left\lVert
u\right\rVert_{LE^{1}}^{1/2},$

and

$\left\lVert\partial
u\right\rVert_{L^{\infty}_{t}L^{2}_{x}}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\varepsilon^{-1}\left\lVert
Pu\right\rVert_{LE^{*}+L^{1}_{t}L^{2}_{x}}+\varepsilon\left\lVert
u\right\rVert_{LE^{1}},\qquad\forall\varepsilon>0$

hold.

###### Proof.

By Proposition 2.17, we have the estimate

$\left\lVert\partial u(t)\right\rVert_{L^{2}}^{2}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\int\limits_{0}^{T}\int\limits_{\mathbb{R}^{3}}|Pu\partial_{t}u|\,dxdt,\qquad
0\leq t\leq T.$

To obtain the first estimate that we claimed, one applies the Schwarz
inequality, takes a supremum in time of $\partial_{t}u$, and uses Young’s
inequality for products. To obtain the second estimate, write

$|Pu\ \partial_{t}u|=\left(\left\langle
x\right\rangle^{1/2}|Pu|\right)\left(\left\langle
x\right\rangle^{-1/2}|\partial_{t}u|\right)$

in (2.10) and apply the Schwarz inequality and Hölder’s inequality applied to
$\ell^{1}$ with conjugate exponents $(p,q)=(1,\infty)$. ∎

Next, we cite an exterior estimate from [mst20] (see [MMT08] for a similar
result for the Schrödinger equation). Outside of a large enough compact
spatial set, the operator $P$ is a small $AF$ perturbation of $\Box$, in which
case we obtain good energy estimates with a necessary truncation error (here,
we must measure the energy at time $T$, which is non-problematic in view of
Corollary 2.19).

######  Proposition 2.20 (Proposition 3.2 in [mst20]).

If $P$ is asymptotically flat and $R\geq R_{0}$, then

(2.11) $\displaystyle\left\lVert
u\right\rVert_{LE^{1}_{>R}}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}_{>R}}+\left\lVert\partial
u(T)\right\rVert_{L^{2}_{>R}}+R^{-1}\left\lVert
u\right\rVert_{LE_{R}}+\left\lVert Pu\right\rVert_{LE^{*}_{>R}}.$

Since the damping is identically zero in this region, the result holds without
any modification to the proof given in [mst20]. For this reason, we will omit
the details here, although we will provide the overarching idea: Their proof
is a positive commutator argument using the multiplier $Q_{1}+Q_{2}$, where

$Q_{1}=\chi_{>2R}(|x|)f(|x|)\frac{x_{j}}{|x|}g^{jk}D_{k}+D_{k}\chi_{>2R}(|x|)f(|x|)\frac{x_{j}}{|x|}g^{jk}$

is the principal term, and

$Q_{2}=\chi_{>2R}(|x|)f^{\prime}(|x|)$

is the lower-order correction term. Here,
$f(|x|)=\displaystyle{\frac{|x|}{|x|+2^{j}}}$, and $j$ is chosen so that
$2^{j}\geq R.$

Before moving on to proving Theorem 1.7, we will simplify its proof through
case reductions. As in Section 4 of [mst20], one can readily reduce to the
case of $u$ having zero Cauchy data at times $0$ and $T$, as well as $f\in
LE^{*}_{c}$. To do this, one constructs an approximate solution to a problem
with the same data and forcing with a wave operator which is a small $AF$
perturbation of $\Box$ and agrees with $P$ for $|x|>R_{0},$ then considers
approximations using a unit time interval partition of unity and matching the
initial (respectively final time) Cauchy data on the first (respectively last)
solution granted by the partition. In view of these reductions, it is enough
to establish

(2.12) $\left\lVert u\right\rVert_{LE^{1}[0,T]}\lesssim\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{LE[0,T]}+\left\lVert
Pu\right\rVert_{LE_{c}^{*}[0,T]}$

for all $u\in\mathcal{W}_{T}$ satisfying that $u[0]=u[T]=0$ in order to prove
Theorem 1.7. The implicit constant in (2.12) is still independent of $T$.

We will explicitly establish an additional reduction, namely to solutions with
compact spatial support, using Proposition 2.20. Claim. It suffices to prove
(2.12), and hence Theorem 1.7, for $u$ supported in $\\{|x|\leq 2R_{0}\\}$.

###### Proof.

Write $u=\chi_{<R_{0}}u+\chi_{>R_{0}}u.$ On the exterior piece
$\chi_{>R_{0}}u$, we apply Proposition 2.20 and Corollary 2.19 to get that

$\displaystyle\left\lVert\chi_{>R_{0}}u\right\rVert_{LE^{1}[0,T]}\lesssim
R_{0}^{-1}\left\lVert\chi_{>R_{0}}u\right\rVert_{LE_{R_{0}}[0,T]}+\left\lVert
P(\chi_{>R_{0}}u)\right\rVert_{LE^{*}_{>R_{0}}[0,T]}.$

The first term on the right is directly bounded by $\left\lVert
u\right\rVert_{LE^{1}_{R_{0}}[0,T]}.$ For the second term, we write

$P(\chi_{>R_{0}}u)=(\chi_{>R_{0}})Pu+[P,\chi_{>R_{0}}]u,$

and one can calculate that

$[P,\chi_{>R_{0}}]u(t,x)=\mathcal{O}(R_{0}^{-1})\chi^{\prime}\left(\frac{|x|}{R_{0}}\right)\partial
u(t,x)+\mathcal{O}(R_{0}^{-2})\chi^{\prime\prime}\left(\frac{|x|}{R_{0}}\right)u(t,x).$

In $LE^{*}$, this term bounded by $\left\lVert
u\right\rVert_{LE^{1}_{R_{0}\leq|\cdot|\leq 2R_{0}}[0,T]}$, and so we have

$\left\lVert\chi_{>R_{0}}u\right\rVert_{LE^{1}[0,T]}\lesssim\left\lVert
u\right\rVert_{LE^{1}_{R_{0}}[0,T]}.$

Suppose that (2.12) holds for $\chi_{<R_{0}}u.$ From this, we get the estimate

$\left\lVert\chi_{<R_{0}}u\right\rVert_{LE^{1}[0,T]}\lesssim\left\lVert\left\langle
x\right\rangle^{-2}\chi_{<R_{0}}u\right\rVert_{LE[0,T]}+\left\lVert
Pu\right\rVert_{LE^{*}_{c}[0,T]}+\left\lVert[P,\chi_{<R_{0}}]u\right\rVert_{LE^{*}_{c}[0,T]},$

and so

$\displaystyle\left\lVert u\right\rVert_{LE^{1}[0,T]}$
$\displaystyle\leq\left\lVert\chi_{<R_{0}}u\right\rVert_{LE^{1}[0,T]}+\left\lVert\chi_{>R_{0}}u\right\rVert_{LE^{1}[0,T]}$
$\displaystyle\lesssim\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{LE[0,T]}+\left\lVert
Pu\right\rVert_{LE^{*}_{c}[0,T]}+\left\lVert
u\right\rVert_{LE^{1}_{R_{0}\leq|\cdot|\leq 2R_{0}}[0,T]}.$

The last term is readily estimated via Proposition 2.20, which establishes
(2.12). ∎

We record the results of these case reductions in the following proposition.

######  Proposition 2.21.

In order to establish Theorem 1.7, it is sufficient to prove the estimate

(2.13) $\displaystyle\left\lVert
v\right\rVert_{LE^{1}[0,T]}\lesssim\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}[0,T]}+\left\lVert
Pv\right\rVert_{LE^{*}[0,T]}$

for $v$ supported in $\\{|x|\leq 2R_{0}\\}$ with $v[0]=v[T]=0.$

Again, this implicit constant is independent of $T$ but will depend on
$R_{0}$. Using the compact support of $v$ to transition between the weighted
and unweighted spaces will inherently generate multiplication by powers of
$R_{0}$, but this does not matter since the constant in the above may depend
on such a parameter.

###  2.6. Proof of the High Frequency Estimate

Armed with the established case reductions, we will proceed with a proof of
Theorem 1.7. Recall that it is equivalent to analyze the scaled problem.

###### Proof of Theorem 1.7.

We will break this proof into a sequence of steps.

Step 1: Setting up the Positive Commutator and the Frequency Decomposition.
First, we remark that in view of Proposition 2.21, it will suffice to prove
(2.13) for $v$ supported in $\\{|x|<2R_{0}\\}$ with $v[0]=v[T]=0.$ We can
extend $v$ by zero to be defined for $t\in\mathbb{R}$ and vanish for
$t\notin(0,T).$ Then,

(2.14) $\displaystyle 2\text{Im}\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v\right\rangle+\frac{i\gamma}{2}\left\langle[aD_{t},m^{\operatorname{w}}]v,v\right\rangle$
$\displaystyle=\left\langle
i[\Box_{g},q^{\operatorname{w}}]v,v\right\rangle+\gamma\left\langle(q^{\operatorname{w}}aD_{t}+aD_{t}q^{\operatorname{w}})v,v\right\rangle$
$\displaystyle\qquad\qquad+\frac{1}{2}\left\langle(\Box_{g}m^{\operatorname{w}}+m^{\operatorname{w}}\Box_{g})v,v\right\rangle.$

The right-hand side of (2.14) can be written as

$\displaystyle\left\langle i[\Box_{g},q^{\operatorname{w}}]v,v\right\rangle$
$\displaystyle+\gamma\left\langle(q^{\operatorname{w}}aD_{t}+aD_{t}q^{\operatorname{w}})v,v\right\rangle+\frac{1}{2}\left\langle(\Box_{g}m^{\operatorname{w}}+m^{\operatorname{w}}\Box_{g})v,v\right\rangle$
$\displaystyle=\left\langle(H_{p}q+2\gamma\tau
aq+mp)^{\operatorname{w}}v,v\right\rangle+\left\langle A_{0}v,v\right\rangle,$

where $A_{0}\in\Psi^{0}$. Recall that $2\gamma\tau a=-2is_{skew}.$ By choosing
$\gamma>0$ large enough, we can apply Lemma 2.4 to get

(2.15) $\displaystyle
H_{p}q-2is_{skew}q+mp-C\mathbbm{1}_{|\xi|\geq\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\geq 0,$

where $C>0$ is the implicit constant in Lemma 2.4. We can readily replace
$\mathbbm{1}_{|\xi|\geq\lambda}$ with $\chi_{|\xi|>\lambda}$.

We will split $v$ into frequency components via
$v=v_{>>\lambda}+v_{<<\lambda}$, where

$\displaystyle v_{>>\lambda}$
$\displaystyle=\chi_{|\xi|+|\tau|>\lambda}(\partial)v,$ $\displaystyle
v_{<<\lambda}$ $\displaystyle=\chi_{|\xi|+|\tau|<\lambda}(\partial)v.$

Since the desired estimate is a high frequency estimate, we will first analyze
the high frequency components of $v$.

Step 2a: High Frequencies - Applying the Gårding inequality. By (2.15), we may
apply the sharp Gårding inequality to obtain that

$\displaystyle\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle\gtrsim\left\langle\left(\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\right)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle-\left\lVert
v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}.$

We remark that the implicit constant may be chosen independently of $\lambda$
since there is a $\chi_{|\xi|>\lambda}$ cutoff embedded into $q$ and $m$, and
hence differentiation occurring in asymptotic expansion calculations possess
coefficients which are either independent of $\lambda$ or feature inverse
powers of $\lambda$ (one can also entirely ignore the potential $\lambda$
dependence and argue via Cauchy-Schwarz and Young’s inequality for products,
although this introduces more parameters to track).

Since $\chi_{|\xi|+|\tau|<\lambda}\in S^{-\infty}$, it follows that

$\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v,v\right\rangle=\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle+\left\langle
S_{0}v,v\right\rangle,$

where $S_{0}\in\Psi^{-\infty}.$ In particular,

(2.16)
$\displaystyle\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v,v\right\rangle\gtrsim\left\langle\left(\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\right)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle-\left\lVert
v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}+\left\langle
S_{0}v,v\right\rangle.$

Using the pseudodifferential composition formula, we compute that

(2.17) $\displaystyle\left(\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\right)^{\operatorname{w}}=(\chi_{|\xi|>\lambda}(D_{x}))^{1/2}D_{\alpha}\langle
x\rangle^{-2}D_{\alpha}(\chi_{|\xi|>\lambda}(D_{x}))^{1/2}+A_{1},$

where $A_{1}\in\Psi^{1}$ arises from non-principal terms in the asymptotic
expansion of the Moyal product (and the expansion features terms which are
either independent of $\lambda$ or involve inverse powers of $\lambda$).
Integrating by parts once gives that

(2.18) $\displaystyle\left\langle\left(\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\right)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle$
$\displaystyle=\left\lVert\left\langle x\right\rangle^{-1}\partial
v_{>\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}^{2}+\left\langle{A}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle$
$\displaystyle\gtrsim\left\lVert\partial
v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}^{2}+\left\langle{A}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle,$

where $v_{>\lambda}=\chi_{|\xi|>\lambda}(D_{x})v$ and $A_{1}\in\Psi^{1}$ is
now a modification of the previous version of the same variable in (2.17) (to-
be-explained momentarily). One might expect the term
$\partial\left((\chi_{|\xi|>\lambda}(D_{x}))^{1/2})v_{>>\lambda}\right)$ to
appear instead of $\partial v_{>\lambda}$, but it is readily seen that

$\left(\chi_{|\xi|>\lambda}(|\xi|)\right)^{1/2}\chi_{|\xi|+|\tau|>\lambda}(|(\tau,\xi)|)\approx\chi_{|\xi|>\lambda}(|\xi|)\chi_{|\xi|+|\tau|>\lambda}(|(\tau,\xi)|)=\chi_{|\xi|>\lambda}(|\xi|).$

In particular, the $\tau$ has no effect on the resulting cutoff, and
$\chi,\chi^{1/2}$ are both smooth, non-decreasing, and have the same support
properties (and only differ on a compact set). The only effect in exchanging
these terms is modifying $A_{1}$ in order the errors resulting from this
switch; hence, the $A_{1}$ in (2.18) is different than in (2.17). For this
reason, none of our analysis changes by working with $v_{>\lambda}$, and we
will stick with this for notational convenience.

After incorporating (2.18) into (2.16), we have that

(2.19)
$\displaystyle\left\langle(H_{p}q-2is_{skew}aq+mp)^{\operatorname{w}}v,v\right\rangle+\left\langle
A_{0}v,v\right\rangle$ $\displaystyle\gtrsim\left\lVert\partial
v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}^{2}-\left\lVert
v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}$
$\displaystyle\qquad-\left|\left\langle
A_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right|-\left|\left\langle
A_{0}v,v\right\rangle\right|-\left|\left\langle S_{0}v,v\right\rangle\right|.$

Step 2b: High Frequencies - Handling the Error Terms in (2.19). We will first
analyze the term $\left\langle A_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle.$
Since $A_{1}\in\Psi^{1},$ it is bounded from $H^{1}_{t,x}$ to
$L^{2}_{t}L^{2}_{x}$ (and the operator norm will yield no positive-power
$\lambda$ contributions due to the previous comment on the asymptotic
expansion of the symbol). By using the Schwarz inequality and this mapping
property, we have that

(2.20) $\displaystyle\left|\left\langle
A_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right|\lesssim\left\lVert
v_{>>\lambda}\right\rVert_{H^{1}_{t,x}}\left\lVert
v_{>>\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}.$

Using Plancherel’s theorem in $(t,x)$, the frequency localization, and the
compact support of $v$, we obtain the bounds

(2.21) $\displaystyle\left\lVert v_{>>\lambda}\right\rVert_{H^{1}_{t,x}}$
$\displaystyle\lesssim\left\lVert\left\langle(\tau,\xi)\right\rangle\chi_{|\xi|+|\tau|>\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}\lesssim\left\lVert\left\langle(\tau,\xi)\right\rangle\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}=\left\lVert
v\right\rVert_{H^{1}_{t,x}}\lesssim\left\lVert v\right\rVert_{LE^{1}},$

and

(2.22) $\displaystyle\left\lVert
v_{>>\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\approx\left\lVert\chi_{|\xi|+|\tau|>\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}\lesssim\left\lVert\frac{|\tau|+|\xi|}{\lambda}\chi_{|\xi|+|\tau|>\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}\lesssim\lambda^{-1}\left\lVert\partial
v\right\rVert_{L^{2}_{t}L^{2}_{x}}\lesssim\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}.$

Applying (2.21) and (2.22) to (2.20) yields that

$\left|\left\langle
A_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right|\lesssim\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}.$

For the term $\left\lVert v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}$, note
that

$\displaystyle\left\lVert v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}$
$\displaystyle\lesssim\left\lVert\left\langle(\tau,\xi)\right\rangle^{1/2}\chi_{|\xi|+|\tau|>\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}^{2}=\left\lVert\left\langle(\tau,\xi)\right\rangle^{-1/2}\left\langle(\tau,\xi)\right\rangle\chi_{|\xi|+|\tau|>\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}^{2}$
$\displaystyle\lesssim\lambda^{-1}\left\lVert\left\langle(\tau,\xi)\right\rangle\chi_{|\xi|+|\tau|>\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}^{2}\lesssim\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}.$

For the $\left\langle A_{0}v,v\right\rangle$ term, we can use
$L^{2}$-boundedness and the compact support of $v$ to get

(2.23) $\displaystyle\left|\left\langle
A_{0}v,v\right\rangle\right|\leq\left\lVert
A_{0}v\right\rVert_{L^{2}_{t}L^{2}_{x}}\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}\lesssim C(\lambda)\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}^{2}.$

While this bound is $\lambda$-dependent, such terms appear on the upper bound
side of the desired inequality, and hence can depend on $\lambda$ in an
arbitrary manner (as opposed to the $LE^{1}$ terms which need an inverse power
of $\lambda$ for bootstrapping). The meaning of $C(\lambda)$ will change
fluidly, just as one continuously re-notates a potentially-changing constant
by $C$ when calculating successive inequalities.

The smoothing term $\left\langle S_{0}v,v\right\rangle$ can be bounded in the
same way as $\left\langle A_{0}v,v\right\rangle$ (in particular,
$S_{0}\in\Psi^{0}$). Thus, we have the lower bound

(2.24)
$\displaystyle\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v,v\right\rangle+\left\langle
A_{0}v,v\right\rangle$ $\displaystyle\gtrsim\left\lVert\partial
v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}^{2}-C(\lambda)\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}^{2}-\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}.$

Next, we look at the left-hand side of (2.14). Step 3: Bounding the Left-Hand
Side of (2.14). Since $[aD_{t},m^{\operatorname{w}}]\in\Psi^{0}$, performing
the same work as in (2.23) provides that

(2.25)
$\displaystyle\left|\frac{i\gamma}{2}\left\langle[aD_{t},m^{\operatorname{w}}]v,v\right\rangle\right|\lesssim
C(\lambda)\left\lVert v\right\rVert_{L^{2}_{t}L^{2}_{x}}^{2}.$

For remaining term on the left-hand side of (2.14), we split $v$ into high and
low frequency components once again to get that

$2\text{Im}\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v\right\rangle=2\text{Im}\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v_{>>\lambda}\right\rangle+\left\langle
S_{1}v,v\right\rangle,$

where $S_{1}\in\Psi^{-\infty}.$ We have already demonstrated how to bound
smoothing operator terms. For the other (primary) piece, we apply the Schwarz
inequality, use the $\Psi$DO mapping properties of
$q^{\operatorname{w}}\in\Psi^{1}$ and $m^{\operatorname{w}}\in\Psi^{0}$, and
leverage the compact support of $v$ (just as performed previously) to get

(2.26) $\displaystyle\left|2\text{Im}\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v_{>>\lambda}\right\rangle\right|\lesssim
C(\lambda)\left\lVert Pv\right\rVert_{L^{2}_{t}L^{2}_{x}}\left\lVert
v\right\rVert_{LE^{1}}\lesssim C(\lambda)\left\lVert
Pv\right\rVert_{LE^{*}_{c}}\left\lVert v\right\rVert_{LE^{1}}.$

Step 4: Combining the Established Bounds Into a High Frequency Bound. Putting
(2.14), (2.24), (2.25), and (2.26) together, we obtain that

$\displaystyle\left\lVert\partial v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}$
$\displaystyle\lesssim C(\lambda)\left(\left\lVert
Pv\right\rVert_{LE^{*}}^{1/2}\left\lVert
v\right\rVert_{LE^{1}}^{1/2}+\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}\right)+\lambda^{-1/2}\left\lVert
v\right\rVert_{LE^{1}}.$

Completing the $LE^{1}_{<2R_{0}}$ norm on the left-hand side of the above,

$\displaystyle\left\lVert v_{>\lambda}\right\rVert_{LE^{1}_{<2R_{0}}}$
$\displaystyle\lesssim C(\lambda)\left(\left\lVert
Pv\right\rVert_{LE^{*}}^{1/2}\left\lVert
v\right\rVert_{LE^{1}}^{1/2}+\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}\right)+\lambda^{-1/2}\left\lVert
v\right\rVert_{LE^{1}}+\left\lVert\left\langle
x\right\rangle^{-1}v_{>\lambda}\right\rVert_{LE}.$

We note that

$\left\lVert\left\langle
x\right\rangle^{-1}v_{>\lambda}\right\rVert_{LE}\lesssim\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}},$

once again using Plancherel’s theorem. Thus,

(2.27) $\displaystyle\left\lVert
v_{>\lambda}\right\rVert_{LE^{1}_{<2R_{0}}}\lesssim
C(\lambda)\left(\left\lVert Pv\right\rVert_{LE^{*}}^{1/2}\left\lVert
v\right\rVert_{LE^{1}}^{1/2}+\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}\right)+\lambda^{-1/2}\left\lVert
v\right\rVert_{LE^{1}}.$

This establishes an estimate on the high frequencies. We must add in the lower
frequencies to the left-hand side. That is, we must add $\left\lVert
v_{<\lambda}\right\rVert_{LE^{1}_{<2R_{0}}}$ to both sides of (2.27).

Step 5a: Lower Frequencies - Further Frequency Splitting and Bounding the Low-
Low Term. First, we get the bound

$\left\lVert\left\langle
x\right\rangle^{-1}v_{<\lambda}\right\rVert_{LE}\lesssim\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}$

via Plancherel’s theorem. For the term $\left\lVert\partial
v_{<\lambda}\right\rVert_{LE}$, we write

$v_{<\lambda}=v_{<>\sigma\lambda}+v_{<<\sigma\lambda},$

where

$\displaystyle v_{<>\sigma\lambda}$
$\displaystyle=\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|>\sigma\lambda}(D_{t})v,$
$\displaystyle v_{<<\sigma\lambda}$
$\displaystyle=\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|<\sigma\lambda}(D_{t})v,$

and $\sigma\gg 1$ will be chosen later (and does not denote the same $\sigma$
as used in the construction of the escape function). Applying Plancherel’s
theorem, frequency localization, and the compact support of $v$ again yields

$\displaystyle\left\lVert\partial
v_{<<\sigma\lambda}\right\rVert_{LE}\lesssim\left\lVert(|\tau|+|\xi|)\chi_{|\xi|<\lambda}\chi_{|\tau|<\sigma\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}\lesssim\sigma\lambda\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}.$

Step 5b: Lower Frequencies - Bounding the Low-High Term. For
$v_{<>\sigma\lambda},$ we compute that

(2.28) $\displaystyle\left\lVert\partial
v_{<>\sigma\lambda}\right\rVert_{LE}\lesssim\left\lVert(|\tau|+|\xi|)\chi_{|\xi|<\lambda}\chi_{|\tau|>\sigma\lambda}\hat{v}\right\rVert_{L^{2}_{\tau}L^{2}_{\xi}}\lesssim\lambda\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+(\sigma\lambda)^{-1}\left\lVert(\partial_{t}^{2}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}.$

For the last term on the right, we utilize the expression for $Pv$ to write

(2.29)
$\displaystyle\left\lVert(\partial_{t}^{2}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\left\lVert(Pv)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}+\left\lVert\left((g^{0j}D_{j}+D_{j}g^{0j})D_{t}v\right)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\qquad+\left\lVert(D_{i}g^{ij}D_{j}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}+\left\lVert(aD_{t}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}.$

One can readily check that

(2.30)
$\displaystyle\left\lVert(Pv)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}\lesssim\left\lVert
Pv\right\rVert_{LE^{*}},$

and

(2.31)
$\displaystyle\left\lVert(aD_{t}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}\lesssim\left\lVert\partial
v\right\rVert_{LE}.$

For the other terms, we note that as functions, one has that $g^{\alpha
j},D_{j}g^{\alpha j}\in S^{0}$ for all $\alpha\in\\{0,1,2,3\\}$ and
$j\in\\{1,2,3\\}$, and so

$[\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|>\sigma\lambda}(D_{t}),g^{\alpha
j}]\in\Psi^{-1},\qquad[\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|>\sigma\lambda}(D_{t}),D_{j}g^{\alpha
j}]\in\Psi^{-1}.$

In particular, the above two operators are bounded on $L^{2}_{t}L^{2}_{x}.$
Pairing this with the fact that Fourier multipliers commute, we have that

(2.32)
$\displaystyle\left\lVert\left((g^{0j}D_{j}+D_{j}g^{0j})D_{t}v\right)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\left\lVert(D_{j}g^{0j})\left(D_{t}v\right)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}+\left\lVert
g^{0j}\left(D_{j}D_{t}v\right)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\qquad+\left\lVert[\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|>\sigma\lambda}(D_{t}),(D_{j}g^{0j})]D_{t}v\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\qquad+\left\lVert[\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|>\sigma\lambda}(D_{t}),g^{0j}]D_{j}D_{t}v\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\lambda\left\lVert\partial
v\right\rVert_{LE}+C(\lambda)\left\lVert v\right\rVert_{L^{2}_{t}L^{2}_{x}},$

and

(2.33)
$\displaystyle\left\lVert(D_{i}g^{ij}D_{j}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\left\lVert(D_{i}g^{ij})(D_{j}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}+\left\lVert
g^{ij}(D_{i}D_{j}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\qquad+\left\lVert([\chi_{|\xi|<\lambda}(D_{x})\chi_{|\tau|>\sigma\lambda}(D_{t}),(D_{i}g^{ij})](D_{j}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\qquad+\left\lVert([\chi_{|\xi|<\lambda}(D_{x})\beta_{|\tau|\geq\sigma\lambda}(D_{t}),g^{ij}](D_{i}D_{j}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim C(\lambda)\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}.$

Applying (2.30)-(2.33) to (2.29) gives that

$\displaystyle\left\lVert(\partial_{t}^{2}v)_{<>\sigma\lambda}\right\rVert_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim C(\lambda)\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+\lambda\left\lVert\partial
v\right\rVert_{LE}+\left\lVert Pv\right\rVert_{LE^{*}}.$

Plugging the resulting estimate into (2.28) implies that

$\displaystyle\left\lVert\partial v_{<>\sigma\lambda}\right\rVert_{LE}$
$\displaystyle\lesssim C(\lambda)\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+(\sigma\lambda)^{-1}\left\lVert
Pv\right\rVert_{LE^{*}}+\sigma^{-1}\left\lVert\partial v\right\rVert_{LE}.$

Step 5c: Lower-Frequency - Combining All Lower Frequency Contributions. Thus,
the full low frequency contribution yields

(2.34) $\displaystyle\left\lVert\partial v_{<\lambda}\right\rVert_{LE}$
$\displaystyle\lesssim\max\\{C(\lambda),\sigma\lambda\\}\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+(\sigma\lambda)^{-1}\left\lVert
Pv\right\rVert_{LE^{*}}+\sigma^{-1}\left\lVert\partial v\right\rVert_{LE}$
$\displaystyle\lesssim\max\\{C(\lambda),\sigma\lambda\\}\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+(\sigma\lambda)^{-1}\left\lVert
Pv\right\rVert_{LE^{*}}+\sigma^{-1}\left\lVert v\right\rVert_{LE^{1}}.$

Step 6: Combining the High and Lower-Frequency Bounds. Now, we can combine the
high frequency work (2.27) with the low frequency work (2.34) and apply
Young’s inequality for products with parameter $\delta>0$ to obtain that

$\displaystyle\left\lVert v\right\rVert_{LE^{1}_{<2R_{0}}}$
$\displaystyle\lesssim C(\lambda)\left\lVert
Pv\right\rVert_{LE^{*}}^{1/2}\left\lVert
v\right\rVert_{LE^{1}}^{1/2}+\max\\{C(\lambda),\sigma\lambda\\}\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+(\sigma\lambda)^{-1}\left\lVert
Pv\right\rVert_{LE^{*}}+\left(\sigma^{-1}+\lambda^{-1/2}\right)\left\lVert
v\right\rVert_{LE^{1}}$
$\displaystyle\lesssim\max\\{C(\lambda),\sigma\lambda\\}\left\lVert
v\right\rVert_{L^{2}_{t}L^{2}_{x}}+\left([C(\lambda)]^{2}\delta^{-1}+(\sigma\lambda)^{-1}\right)\left\lVert
Pv\right\rVert_{LE^{*}}+\left(\delta+\sigma^{-1}+\lambda^{-1/2}\right)\left\lVert
v\right\rVert_{LE^{1}}$

Due to the support of $v$ in $x$, we know that $\left\lVert
v\right\rVert_{LE^{1}_{<2R_{0}}}=\left\lVert v\right\rVert_{LE^{1}}$. Picking
$\delta$ sufficiently small and $\lambda,\sigma$ sufficiently large (all of
which will depend on $R_{0}$) allows us to absorb the $\left\lVert
v\right\rVert_{LE^{1}}$ term on the right-hand side into the left-hand side,
providing (2.13) and completing the proof. ∎

##  3\. Local Energy Decay

In this section, we explain how recovering the high frequency estimate in
[mst20] allows us establish local energy decay by appealing to existing
estimates in their work (which makes it easier to perform time frequency
localization). By an extension argument outlined in Section 3.3, it is
sufficient to prove to reduce to the case of Schwartz functions, which allows
for the removal of data terms. The simplified version of local energy decay
for Schwartz functions can be readily proven by combining the proven high
frequency estimates with appropriate medium and low frequency estimates then
utilizing a time-frequency partition of unity. The medium and low frequency
estimates that we require come from the work in [mst20] and do not depend on
the trapping nor the damping. We remark that such analyses are also
independent of the stationarity of $P$.

###  3.1. Medium Frequencies

The goal of the medium frequency estimate is to establish a weighted estimate
which implies local energy decay for solutions supported at any range of time
frequencies bounded away from both zero and infinity. This is rooted in the
notion of a Carleman estimate, which is weighted $L^{2}_{t}L^{2}_{x}$
estimates where the weight is pseudoconvex. The constants in our inequalities
will depend on the parameter c introduced in Section 1.2, but they will (and
must) be independent of the parameters in $\varphi$; the Carleman weights for
our estimates are radial. The Carleman weights which we use here are
constructed in e.g. [Bo18], [KT01].

The main medium frequency estimate is the following, and the corresponding
theorem in [mst20] is Theorem 5.4.

######  Theorem 3.1.

Let $P$ be an asymptotically flat damped wave operator, and suppose that
$\partial_{t}$ is uniformly time-like. Then, for any $\delta>0$, there exists
a bounded, non-decreasing radial weight $\varphi=\varphi(\ln(1+r))$ so that
for all $u\in\mathcal{S}(\mathbb{R}^{4})$, we have the bound

(3.1) $\left\lVert(1+\varphi_{+}^{\prime\prime})^{1/2}e^{\varphi}(\nabla
u,\left\langle
r\right\rangle^{-1}(1+\varphi^{\prime})u\right\rVert_{LE}+\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}\partial_{t}u\right\rVert_{LE}\\\
\lesssim\left\lVert
e^{\varphi}Pu\right\rVert_{LE^{*}}+\delta\left(\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}u\right\rVert_{LE}+\left\lVert\left\langle
r\right\rangle^{-1}(1+\varphi_{+}^{\prime\prime})^{1/2}(1+\varphi^{\prime})e^{\varphi}\partial_{t}u\right\rVert_{LE}\right).$

######  Remark 3.2.

We will justify why this is an appropriate estimate on the medium frequencies.
To that end, suppose that $u$ is supported at time frequencies $\tau$ such
that $0<\tau_{0}\leq|\tau|\leq\tau_{1}$, where $\tau_{0}<\tau_{1}$. For
compatibility with the other frequency regimes, we will want $\tau_{0}\ll
1\ll\tau_{1}$. Plancherel’s theorem yields that

$\delta\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}u\right\rVert_{LE}\lesssim\frac{\delta}{\tau_{0}}\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}\partial_{t}u\right\rVert_{LE},$

while

$\delta\left\lVert\left\langle
r\right\rangle^{-1}(1+\varphi_{+}^{\prime\prime})^{1/2}(1+\varphi^{\prime})e^{\varphi}\partial_{t}u\right\rVert_{LE}\lesssim\delta\tau_{1}\left\lVert\left\langle
r\right\rangle^{-1}(1+\varphi_{+}^{\prime\prime})^{1/2}(1+\varphi^{\prime})e^{\varphi}u\right\rVert_{LE}.$

By choosing $\delta$ sufficiently small, both terms absorb into the left-hand
side of (3.1) in a direct fashion. We can translate our work immediately into
a local energy decay estimate for $u$, with an implicit constant which depends
on $\varphi$. Notice that since $\delta$ can be chosen arbitrarily, (3.1)
allows for any interval of frequencies bounded away from both zero and
infinity. ∎

The proof of this theorem is broken up into two Carleman estimates, one which
applies within a large compact set and one which applies outside of this
compact set. Within the compact set, the damping term is well-signed and
readily absorbable as a perturbation due to the conditions on the weight
$\varphi.$ Here, the weight will be convex. Outside of the compact set, the
damping is zero, so the proof in [mst20] follows through without any
modification. In this region, one desires to use Proposition 2.20, which
requires a constant weight. To that end, one breaks up the exterior into three
regions: one where the Carleman weight is convex, a transition region where
the conditions break in order to bend the weight to be constant near infinity,
and a region near infinity where the weight is constant.

In both regions (the compact set and its exterior), the work in [mst20] allows
for more general (e.g. unsigned) lower-order terms than a damping term. The
proofs of these estimates within the aforementioned regions are based on
positive commutator arguments utilizing the self- and skew-adjoint parts of
the conjugated operator $P_{\varphi}=e^{\varphi}Pe^{-\varphi}.$ The work is
then combined using a cutoff argument.

###  3.2. Low Frequencies

We will set

$P_{0}:=P\big{|}_{D_{t}=0}=D_{i}g^{ij}D_{j}.$

This represents $P$ at time frequency zero, and one can utilize it to obtain
information in a neighborhood of this frequency. Since $\partial_{t}$ is
uniformly time-like, $P_{0}$ is uniformly elliptic. The operator $P_{0}$ is a
special case of that found in [mst20], so all of the results in their work
apply here with almost no modification. Notice that the damping does not arise
in $P_{0}.$

At low frequencies, the obstruction to local energy decay arises when $P$ has