# Integrated Local Energy Decay for Damped Magnetic Wave Equations on
Stationary Space-Times

Collin Kofroth  Johns Hopkins University Applied Physics Lab, 11100 Johns
Hopkins Road, Laurel, MD 20723-6099<EMAIL_ADDRESS>
###### Abstract.

We establish local energy decay for damped magnetic wave equations on
stationary, asymptotically flat space-times subject to the geometric control
condition. More specifically, we allow for the addition of time-independent
magnetic and scalar potentials, which negatively affect energy coercivity and
may add in unwieldy spectral effects. By asserting the non-existence of
eigenvalues in the lower half-plane and resonances on the real line, we are
able to apply spectral theory from the work of Metcalfe, Sterbenz, and Tataru
and combine with a generalization of prior work by the present author to
extend the latter work and establish local energy decay, under one additional
symmetry hypothesis. Namely, we assume that either the imaginary part of the
magnetic potentials are uniformly small or, more interestingly and novelly,
that the damping term is the dominant principal term in the skew-adjoint part
of the damped wave operator within the region where the metric perturbation
from that of Minkowski space is permitted to be large. We also obtain an
energy dichotomy if we do not prohibit non-zero real resonances. In order to
make the structure of the argument more cohesive, we contextualize the present
work within requisite existing theory.

###### Key words and phrases:

Local energy estimates; Asymptotically flat; Damped wave equation; Geometric
control; Trapping

###### 2020 Mathematics Subject Classification:

⟨35L05, 58J45, 35P25, 35Q93⟩

###### Contents

  1. 1 Introduction
    1. 1.1 Background and Problem Set-Up
    2. 1.2 Local Energy Spaces and Estimates
    3. 1.3 Past Results
    4. 1.4 Statement of Present Results
  2. 2 Frequency Analyses and Two-Point Local Energy Decay
    1. 2.1 High Frequency Analysis
    2. 2.2 Remaining Frequency Analyses and Two-Point Local Energy Decay
    3. 2.3 An Energy Dichotomy
  3. 3 Resolvent Theory and Local Energy Decay

##  1\. Introduction

###  1.1. Background and Problem Set-Up

In this work, we will establish (integrated) local energy decay for a broad
class of damped wave equations on stationary, asymptotically flat space-times,
including those which possess magnetic and scalar potential terms. This
generalizes the result of [Kof22], which lacked potential terms and assumed
that the damping was compactly-supported. The presence of the potentials allow
for interaction with the damping and the existence of both complex eigenvalues
and real resonances embedded in the continuous spectrum, all of which stand to
inhibit local energy decay. We will discuss each of these obstructions
carefully. Along the way, we also establish an energy dichotomy when non-zero
real resonances are permissible.

To set up the problem which we will study, let $(\mathbb{R}^{4},g)$ be a
Lorentzian manifold with coordinates $(t,x)\in\mathbb{R}\times\mathbb{R}^{3}$
and metric signature $(-+++).$ We will consider damped wave operators of the
form

$P=\Box_{A,g}+iaD_{t}+V,\qquad\Box_{A,g}=(D_{\alpha}+A_{\alpha})g^{\alpha\beta}(D_{\beta}+A_{\beta})$

where $a,V,$ and the components of $A$ are smooth and complex-valued; here, we
are utilizing the notation $D_{\alpha}=-i\partial_{\alpha}$, which should be
interpreted as an operator. We will call $A$ the magnetic potential, $V$ the
scalar potential, and $a$ the damping. If $a\equiv 0$, then we will simply
refer to $P$ as a wave operator. We will require that $P$ is asymptotically
flat. More precisely, we first define the family of norms

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|h\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{k}=\sum\limits_{|\alpha|\leq
k}\left\lVert\left\langle
x\right\rangle^{|\alpha|}\partial^{\alpha}h\right\rVert_{\ell^{1}_{j}L^{\infty}([0,T]\times
A_{j})},$

where $A_{j}=\\{\left\langle x\right\rangle\approx 2^{j}\\}$ for $j\geq 0$
denote inhomogeneous dyadic annuli, $\partial=(\partial_{t},\nabla_{x})$
denotes the space-time gradient, and $\left\langle
x\right\rangle=(1+|x|^{2})^{1/2}$ denotes the Japanese bracket of $x$. The
notation $A\lesssim B$ indicates 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 $\mathbb{R}^{3}.$ We may now define the $AF$ norm topology via

$\left\lVert(h,A,a,V)\right\rVert_{AF}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|h\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\left\langle
x\right\rangle
A\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{1}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\left\langle
x\right\rangle
a\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{1}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\left\langle
x\right\rangle^{2}V\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{0}.$

######  Definition 1.1.

We say that $P$ is asymptotically flat if
$\left\lVert(g-m,A,a,V)\right\rVert_{AF}<\infty$, and

$\displaystyle\left\lVert\left\langle
x\right\rangle^{|\alpha|}\partial^{\alpha}g\right\rVert_{\ell_{j}^{1}L^{\infty}([0,T]\times
A_{j})}$ $\displaystyle\lesssim_{\alpha}1,\qquad|\alpha|\geq 3,$
$\displaystyle\left\lVert\left\langle
x\right\rangle^{|\alpha|+1}\partial^{\alpha}A\right\rVert_{\ell_{j}^{1}L^{\infty}([0,T]\times
A_{j})}+\left\lVert\left\langle
x\right\rangle^{|\alpha|+1}\partial^{\alpha}a\right\rVert_{\ell_{j}^{1}L^{\infty}([0,T]\times
A_{j})}$ $\displaystyle\lesssim_{\alpha}1,\qquad|\alpha|\geq 2,$
$\displaystyle\left\lVert\left\langle
x\right\rangle^{|\alpha|+2}\partial^{\alpha}V\right\rVert_{\ell_{j}^{1}L^{\infty}([0,T]\times
A_{j})}$ $\displaystyle\lesssim_{\alpha}1,\qquad|\alpha|\geq 1,$

where $m$ denotes the Minkowski metric.

The latter conditions are placed to work in standard symbol classes, as
opposed to those with limited regularity. They extend the flavor of the
estimate present in the definition of asymptotic flatness, but one does not
require summability over the differentiation indices.

We will primarily be interested in when $P$ is stationary, which occurs when
$(g,A,a,V)$ are independent of $t$. In particular, $\partial_{t}$ is a Killing
field for $g$ when $P$ is stationary. When $P$ is a damped wave operator, we
will always assume that $a$ is independent of $t$, $a$ is non-negative for
$|x|\leq 2R_{0}$, and $a$ is positive on an open subset of $\\{|x|\leq
R_{0}\\}$. For $|x|\leq 2R_{0}$, the damping term in $P$ behaves as typical
viscous damping, and we consider it as a general time-independent first-order
perturbation term for $|x|>2R_{0}$. Further, we will assume throughout
$\partial_{t}$, while $dtt$ the constant time slices are uniformly space-like
(i.e. $dt$ is uniformly time-like). The former condition implies an
ellipticity condition on the terms in $P$ which are independent of $D_{t}$
(see Section 1.2 for more), while the latter guarantees that
$g^{00}\gtrsim-1$, allowing $P$ to be reduced to a normal form of $g^{00}=-1$.

Next, we introduce parameters that quantify various aspects of asymptotic
flatness. Namely, we instantiate

  * •

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

$\displaystyle\left\lVert(g-m,A,a,V)\right\rVert_{AF}$ $\displaystyle\leq
M_{0},\qquad\qquad\left\lVert(g-m,A,a,V)\right\rVert_{AF_{>R_{0}}}\leq\textbf{c}\ll
1,$

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 a
suitable $R_{0}$ for which the above holds.

  * •

the sequence $(c_{j})_{j\geq\log_{2}R_{0}}$ of positive real numbers
satisfying that

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

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.

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

The integers $0\leq j<\log_{2}R_{0}$ index finitely many dyadic regions. Since
the union of such regions is compact, we can extend $(c_{j})$ to such indices
in an arbitrary manner, although we will assume that the appended terms in the
sequence are positive and allow $(c_{j})$ to remain slowly-varying.

######  Remark 1.2.

The implicit constants in our inequalities throughout this work are permitted
to (and often will) depend on c (which is universal), $M_{0}$ (also
universal), and $R_{0}$ (only depends on c). However, they cannot depend on
$T$; the negation would be problematic and not allow one to e.g. take the
limit as $T\rightarrow\infty$. ∎

Next, we introduce definitions regarding the skewness of $P-iaD_{t}$ and the
size of the damping relative to the magnetic potential. We will generically
refer to these as symmetry conditions.

######  Definition 1.3.

We say that $P$ is

  * •

of symmetric wave type if $A$ and $V$ are real-valued.

  * •

$\varepsilon$-almost symmetric wave type if
$\left\lVert(0,\operatorname{Im}A,0,\operatorname{Im}V)\right\rVert_{AF}\leq\varepsilon$.

  * •

$\varepsilon$-weakly magnetic if
$\left\lVert(0,\operatorname{Im}A,0,0)\right\rVert_{AF}\leq\varepsilon$.

  * •

strongly $\varepsilon$-damping dominant if

$a(x)\geq\varepsilon^{-1}|\operatorname{Im}A(x)|,\qquad|x|\leq 2R_{0}.$

  * •

weakly $\varepsilon$-damping dominant if

$\frac{{g^{0j}\xi_{j}\pm\sqrt{\left(g^{0j}\xi_{j}\right)^{2}+g^{ij}\xi_{i}\xi_{j}}}}{\pm
2\sqrt{\left(g^{0j}\xi_{j}\right)^{2}+g^{ij}\xi_{i}\xi_{j}}}(a+g^{0\alpha}\operatorname{Im}A_{\alpha})\pm\frac{\xi_{k}}{{2\sqrt{\left(g^{0j}\xi_{j}\right)^{2}+g^{ij}\xi_{i}\xi_{j}}}}g^{k\alpha}\operatorname{Im}A_{\alpha}\geq\varepsilon
a,\qquad|x|\leq 2R_{0},\ \ \xi\neq 0.$

######  Remark 1.4.

The parameter $\varepsilon$ in the definitions of the $\varepsilon$-almost
symmetric wave type, $\varepsilon$-weakly magnetic condition, and strongly
$\varepsilon$-damping dominant condition should be viewed as being fixed after
$R_{0},M_{0}$. The same will not be true for the weakly $\varepsilon$-damping
dominant condition, which we will assume holds for some $\varepsilon>0$
independent of the previously-discussed parameters. ∎

######  Remark 1.5.

The weakly $\varepsilon$-damping dominant condition, while seemingly esoteric,
will arise naturally in our high frequency analysis and constitutes a
sharpening of the more concrete and directly-verifiable strongly
$\varepsilon$-damping dominant condition. First, we remark that we will need
the latter to hold for sufficiently small $\varepsilon>0$ in order to obtain
our results, whereas we only need the former to hold for some $\varepsilon>0$.
The range of $\varepsilon$ which are sufficient for our results to hold for
strongly $\varepsilon$-damping dominant $P$ will guarantee that the weaker
condition holds (for a different $\varepsilon$). We will now discuss their
relationship more explicitly.

In the language of Section 2.1, the weakly $\varepsilon$-damping dominant
condition may be written as

$\frac{b^{\pm}}{b^{\pm}-b^{\mp}}(a+g^{0\alpha}\operatorname{Im}A_{\alpha})+\frac{\xi_{k}}{b^{\pm}-b^{\mp}}g^{k\alpha}\operatorname{Im}A_{\alpha}\geq\varepsilon
a,\qquad\qquad|x|\leq 2R_{0},\qquad\xi\neq 0,$

where

$b^{+}(x,\xi)>0>b^{-}(x,\xi),\qquad
b^{\pm}(x,\xi)\approx\pm|\xi|,\qquad\xi\neq 0.$

Hence, the weakly $\varepsilon$-damping dominant condition may be recast (up
to a fixed constant coefficient on the lower-bound side) in more simple terms
as

$a+g^{0\alpha}\operatorname{Im}A_{\alpha}\pm\frac{\xi_{k}}{|\xi|}g^{k\alpha}\operatorname{Im}A_{\alpha}\geq\varepsilon
a.$

If $P$ is strongly $\varepsilon$-damping dominant, then

$\displaystyle
a+g^{0\alpha}\operatorname{Im}A_{\alpha}\pm\frac{\xi_{k}}{|\xi|}g^{k\alpha}\operatorname{Im}A_{\alpha}$
$\displaystyle\geq a-2(\left\lVert
g-m\right\rVert_{L^{\infty}}+1)|\operatorname{Im}A|$
$\displaystyle\geq\left(1-2\varepsilon(\left\lVert
g-m\right\rVert_{L^{\infty}}+1)\right)a,\ \ \ \ |x|\leq 2R_{0},\ \ \xi\neq 0.$

If $\varepsilon<(2\left\lVert g-m\right\rVert_{L^{\infty}}+2)^{-1},$ then the
weakly $\varepsilon^{\prime}$-damping dominant condition holds with

$\varepsilon^{\prime}=1-2\varepsilon(\left\lVert
g-m\right\rVert_{L^{\infty}}+1)>0.$

In particular, $0<\varepsilon^{\prime}<1$.

On the other hand, if $g=m$, then the weakly $\varepsilon$-damping dominant
condition stipulates that

$\frac{1}{2}(a-\operatorname{Im}A_{0})\pm\frac{1}{2|\xi|}\sum\limits_{k=1}^{3}\xi_{k}\operatorname{Im}A_{k}\geq\varepsilon
a,\qquad|x|\leq 2R_{0},\qquad\xi\neq 0.$

By considering when $\xi$ is a standard basis vector $e_{j}$ and summing the
results over $j$, we obtain that

$3(a-\operatorname{Im}A_{0})\pm\sum\limits_{k=1}^{3}\operatorname{Im}A_{k}\geq
6\varepsilon a,\qquad|x|\leq 2R_{0}.$

Focusing the minus case, this yields that

$a\geq(3(1-2\varepsilon))^{-1}\left(2\operatorname{Im}A_{0}+|\operatorname{Im}A|\right),\qquad|x|\leq
2R_{0}.$

If $\operatorname{Im}A_{0}\geq 0,$ then $P$ is $\varepsilon^{\prime}$-strongly
damping dominant, with $\varepsilon^{\prime}=3(1-2\varepsilon).$ Hence, the
conditions are equivalent in the flat case for non-negative
$\operatorname{Im}A_{0}$ and appropriate (and different) $\varepsilon$. ∎

We note that $\operatorname{Im}A$ is already guaranteed to be small for
$|x|>R_{0}$ by asymptotic flatness, so the $\varepsilon$-weakly magnetic
condition is more so a condition for $|x|\leq R_{0}$. This is a weakening of
the more standard $\varepsilon$-almost symmetry condition, which will arise
briefly in Section 2.3. The $\varepsilon$-damping dominant conditions are, to
our knowledge, more original and seemingly more natural; we will only discuss
the strong version for now since it is simpler to parse and easier to verify
(as well as stronger). It stipulates that the damping must be the dominant
skew-adjoint term (at the principal level) for $|x|\leq 2R_{0}.$ Thus, the
magnetic potential is permitted to have large imaginary components for
$|x|\leq 2R_{0},$ so long as the damping is more significant, in the sense
stated above. Moreover, the $\varepsilon$ in the strongly
$\varepsilon$-damping dominant condition must satisfy that
$\varepsilon<(2\left\lVert g-m\right\rVert_{L^{\infty}}+2)^{-1}$ for our
results to hold in this work (in which case the weakly $\varepsilon$-damping
dominant condition holds with a different $\varepsilon$, as discussed in the
previous remark). This indicates that if the metric perturbation is
sufficiently large within $\\{|x|\leq R_{0}\\}$, then $|\operatorname{Im}A|$
must be sufficiently smaller than $a$. If the perturbation is small, then
$|\operatorname{Im}A|$ and $a$ are permitted to be close for $|x|\leq 2R_{0}$.

Finally, we remark that neither the $\varepsilon$-weakly magnetic nor the
strongly $\varepsilon$-damping dominant condition is stronger than the other,
although working with the conditions is somewhat similar in the high frequency
analysis; the same is true for the $\varepsilon$-weakly magnetic condition and
the weakly $\varepsilon$-damping dominant condition, for similar reasons. One
should view the damping condition as allowing $|\operatorname{Im}A|$ to be
potentially quite large for $|x|\leq 2R_{0}$, so long as $a$ is sufficiently
larger (although, of course, both are inherently limited by the $M_{0}$
parameter, as well). However, let us say that $a$ is zero outside of a ball of
radius $\delta\ll 1$ and is small within the ball. Then, the strongly
$\varepsilon$-damping dominant condition would require that
$\operatorname{Im}A$ be zero for $\delta\leq|x|\leq 2R_{0}$ and small
everywhere. The $\varepsilon$-weakly magnetic condition allows for
$|\operatorname{Im}A_{\alpha}|>0$ for $|x|\leq 2R_{0},$ so long as we control
the $AF$ size uniformly.

###  1.2. Local Energy Spaces and Estimates

The study of localized energy estimates dates back to the work of [Mora66,
Mora68, Mora75], [MRS77] on Minkowski space-time. Local energy estimates
constitute a powerful measure of dispersion, implying Strichartz estimates
([BT07, BT08], [JSS90, JSS91], [MMT08], [MMTT10], [MT09, MT12], [Tat08],
[Toh12]) and pointwise decay estimates (e.g. [MTT12], [Tat13], [Morg20],
[MW21], [Looi21], [Hi23]). The latter has been used to tackle various
generalizations of Price’s law which, in its simplest form, conjectured a
$t^{-3}$ pointwise decay rate of waves on non-rotating black hole space-times;
see the listed references on pointwise decay and [Hi22] (which settles the
conjecture affirmatively in full generality) for more. The particular local
energy estimate of interest in this work, integrated local energy decay, is a
powerful quantitative statement given in the form of an inequality which may
be qualitatively interpreted as expressing that the energy of a wave must
disperse quickly enough to be time-integrable within any compact region of
space.

In order to define the relevant energy inequalities explicitly, we will first
define the local energy norms

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

The predual norm to the $LE$ norm is the $LE^{*}$ norm, which is defined as

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

If we wish for the time interval to be e.g. $[0,T]$ in the above norms, then
we will use the notation e.g. $\left\lVert u\right\rVert_{LE[0,T]}$. A
subscript of $c$ on any of these spaces denotes compact spatial support,
whereas a subscript of $0$ denotes the closure of $C_{c}^{\infty}$ in the
relevant norm.

Analogous to the local energy spaces, we define the spatial local energy
spaces $\mathcal{LE},\mathcal{LE}^{1},\mathcal{LE}^{*}$ when the time variable
is fixed (and there is no time derivative nor integral involved in the norms,
either). We will also require spaces which allow us to track dependence on a
spectral parameter $\omega$, namely

$\displaystyle\mathcal{LE}_{\omega}^{1}=\mathcal{LE}^{1}\cap|\omega|^{-1}\mathcal{LE},\qquad\qquad\dot{H}^{1}_{\omega}=\dot{H}^{1}\cap|\omega|^{-1}L^{2}.$

These spaces are equipped with the norms

$\displaystyle\left\lVert
u\right\rVert_{\mathcal{LE}_{\omega}^{1}}=\left\lVert
u\right\rVert_{\mathcal{LE}^{1}}+|\omega|\left\lVert
u\right\rVert_{\mathcal{LE}},\qquad\qquad\left\lVert
u\right\rVert_{\dot{H}_{\omega}^{1}}=\left\lVert
u\right\rVert_{\dot{H}^{1}}+|\omega|\left\lVert u\right\rVert_{L^{2}},$

respectively. They will become relevant in this work when we introduce the
resolvent formalism in Section 3. Now, we will define local energy decay.

######  Definition 1.6.

We say that (integrated) local energy decay holds for an asymptotically flat
damped wave operator if

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

with the implicit constant being independent of $T$.

The next estimate is a weaker variant of local energy decay which does not see
complex eigenvalues nor non-zero real resonances (to be defined in Section 3).

######  Definition 1.7.

We say that two-point local energy decay holds for an asymptotically flat
damped wave operator if

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

with the implicit constant being independent of $T$.

The last estimate is a standard uniform energy bound, which embodies the
behavior that one observes when the energy of the system is non-increasing.

######  Definition 1.8.

We say that uniform energy bounds hold for an asymptotically flat damped wave
operator if

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

with the implicit constant being independent of $T$.

Notice that the uniform energy bounds provide a link between the two-point
local energy decay estimate and the local energy decay estimate. In order to
discuss the uniform estimate further, let $Pu=f$, and assume that $P$ is
stationary, asymptotically flat, and of symmetric wave type. Consider the
sesquilinear map $\boldsymbol{E}$ on the energy space
$\mathcal{E}:=\dot{H}^{1}\times L^{2}$ defined by

$\boldsymbol{E}[\boldsymbol{u},\boldsymbol{v}](t)=\int\limits_{\mathbb{R}^{3}}P_{0}u\bar{v}-g^{00}\partial_{t}u\overline{\partial_{t}v}\,dx,\qquad
P_{0}=P\big{|}_{D_{t}=0},\qquad\boldsymbol{w}=(w,\partial_{t}w)\in\mathcal{E}.$

This induces a quadratic energy functional, i.e. an energy form

$E[\boldsymbol{u}](t):=\boldsymbol{E}[\boldsymbol{u},\boldsymbol{u}](t).$

When $\partial_{t}$ is a uniformly time-like vector field (which we will
assume throughout), $P_{0}$ is guaranteed to be elliptic (in the principal
sense). Direct integration by parts gives that

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

If $a$ is real-valued and non-negative, then it follows that

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

If $A,V\equiv 0$, then $P_{0}$ is uniformly elliptic (in the sense of the full
symbol), which implies that the energy is coercive, i.e.

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

Energy coercivity applied to the above now provides the uniform energy bound
(1.3).

When $A$ and $V$ are non-zero, one is not guaranteed energy coercivity (and
thus not guaranteed uniform energy bounds), even if $a$ is non-negative.
However, one does get an almost-coercive energy statement from the ellipticity
of $P_{0}$ and the asymptotic flatness assumption:

(1.4) $\left\lVert\partial u(t)\right\rVert_{L^{2}}^{2}\lesssim
E[\boldsymbol{u}](t)+\left\lVert u(t)\right\rVert_{L^{2}_{c}}^{2}.$

When $P$ is not of symmetric wave type, then one redefines the energy by
replacing $P_{0}$ with its symmetric part (see Section 2.3); the estimate
(1.4) still holds for the energy associated to $P_{0}$ by ellipticity
regardless of the symmetry. The uniform bound (1.3) still need not hold when
$P_{0}$ is replaced by its symmetric part. Thus, it is not straightforward to
transition from (1.2) to (1.1) when $A,V\not\equiv 0$, even for well-signed
dampings; this is true even in the symmetric wave type case.

###  1.3. Past Results

In [MST20], the authors proved (amongst other results) that local energy decay
for stationary $AF$ wave operators is equivalent to an absence of geodesic
trapping, negative eigenfunctions, and real resonances. We will describe
trapping in Section 2.1 and the spectral objects in Section 3. In short,
trapping occurs when there exist bicharacteristic rays which live within a
compact set for all time. The spectral obstructions correspond to singular
behavior of the resolvent \- negative eigenfunctions live in $L^{2}$ and have
corresponding eigenvalue in the lower half-plane, whereas real resonances lie
on the real line and have a corresponding resonant state which lives in a
local energy space (one must distinguish between zero and non-zero
resonances). In [MST20], the authors also did not necessarily possess a
coercive energy; they proved local energy decay by establishing (1.2) using
local energy estimates in different frequency regimes, then they utilized
resolvent estimates.

The work [BR14] utilized dissipative Mourre commutator methods to establish
that if the space-time is stationary and asymptotically Euclidean (i.e.
$(\mathbb{R}^{4},g)$ is a product manifold and hence possesses no non-trivial
metric cross terms $dt\otimes dx^{j}$), then one has local energy decay for
$AF$ stationary damped wave operators with $A,V\equiv 0$ and $a$ being a non-
negative short-range potential, provided that $a$ satisfied a dynamical
hypothesis called geometric control. This condition requires that all trapped
null bicharacteristic rays intersect where $a>0$ (see Definition 2.1 for a
precise definition), although the authors of [BR14] only required this for
trapped geodesics due to the product manifold structure. Geometric control
dates back to [RT74], which utilized it to obtain exponential energy decay
(i.e. uniform stabilization) for dissipative problems on compact manifolds. In
[Kof22], we generalized the work of [BR14] to the asymptotically flat case
(i.e. allowed the metric to possess cross terms). To be precise, we will
record this result, which is Theorem 1.9 in [Kof22] (adding in the missing
assumption of uniformly space-like time slices).

######  Theorem 1.9.

Let $P$ be a stationary, asymptotically flat damped wave operator satisfying
the geometric control condition with $A,V\equiv 0$ and
$\operatorname{supp}a\subset\\{|x|\leq R_{0}\\}$, and suppose that
$\partial_{t}$ is uniformly time-like while the constant time slices are
uniformly space-like. Then, local energy decay holds, with the implicit
constant in (1.1) independent of $T$.

In [Kof22], uniform energy bounds held as a result of the conditions on the
damping and lack of potentials, which made it sufficient to prove the two-
point bound in order to establish local energy decay. In order to establish
the former estimate, [Kof22] followed the strategy set forth in [MST20]:

  1. (1)

Establish local energy estimates that imply local energy decay for Schwartz
functions, whose corresponding function space we denote as $\mathcal{S}$,
which are cutoff to high, medium, and low frequency regimes.

  2. (2)

Utilize a time frequency partition of unity to prove the estimate

$\left\lVert u\right\rVert_{LE^{1}}\lesssim\left\lVert
Pu\right\rVert_{LE^{*}},\qquad u\in\mathcal{S}.$

  3. (3)

Apply an extension procedure to add back in the energy at times $0$ and $T$.

###  1.4. Statement of Present Results

We will generalize the work of [Kof22] to include the lower-order magnetic and
scalar potentials, along with more general damping functions. Our first result
is an extension of the high frequency estimate present in [MST20] and [Kof22]
to our setting.

######  Theorem 1.10.

Let $P$ be a stationary, asymptotically flat damped wave operator which
satisfies the geometric control condition and is either weakly
$\varepsilon$-damping dominant for some $\varepsilon>0$ or
$\varepsilon$-weakly magnetic with $\varepsilon\ll_{R_{0},M_{0}}1$.
Additionally, assume that $\partial_{t}$ uniformly time-like while the
constant time slices are uniformly space-like. Then, the high frequency
estimate

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

holds with an implicit constant which is independent of $T$.

This theorem is the one of the primary results whose proof must be adapted
from [Kof22] to account for the lower-order terms. It is also where one
requires the most substantial deviation from [MST20], since trapping is high
frequency. We utilize the symmetry-based assumptions in the theorem so that we
may deal with the additional lower-order terms. In particular, this is where
the conditions on the interaction between the damping and magnetic potentials
come into play. In order to leverage the sign of the damping to mitigate the
harmful effects of the trapping, we must limit the magnetic potential
appropriately.

The medium and low frequencies are not affected by the damping (which may
simply be viewed as a general sub-principal $AF$ term, as opposed to being
leveraged like in the high frequency setting) and follow directly from the
work in [MST20]. This allows us to establish the two-point local energy
estimate under the hypothesis that zero is not a resonance, which is needed in
the low frequency regime.

######  Theorem 1.11.

Let $P$ be a stationary, asymptotically flat damped wave operator which
satisfies the zero non-resonance and geometric control conditions and is
either weakly $\varepsilon$-damping dominant for some $\varepsilon>0$ or
$\varepsilon$-weakly magnetic with $\varepsilon\ll_{R_{0},M_{0}}1$.
Additionally, assume that $\partial_{t}$ is uniformly time-like while the
constant time slices are uniformly space-like. Then, two-point local energy
decay holds, with the implicit constant in (1.2) independent of $T$.

We will define the zero non-resonance condition in Section 2.2 and its
relation to zero resonant states in Section 3. If we additionally impose that
$P$ is of $\varepsilon$-almost symmetric wave type or that $P$ satisfies the
$\varepsilon$-damping dominant as well as the analogous estimate with $A$
replaced by $V$, then we obtain a straightforward energy dichotomy (just as in
[MST20], where it is Theorem 2.16) as a consequence of the two-point local
energy estimate and a uniform energy relation. The given symmetry conditions
fulfill a similar role to a condition on the absence of non-zero embedded
resonances, and it is needed to obtain the aforementioned uniform energy
relation (i.e. an almost-conserved energy property if $Pu=0$) which will
appear in the proof.

######  Theorem 1.12.

Let $P$ be a stationary, asymptotically flat damped wave operator which is
either

  1. (a)

$\varepsilon$-almost symmetric wave type for sufficiently small
$\varepsilon\ll_{R_{0},M_{0}}1$

  2. or

  3. (b)

strongly $\varepsilon$-damping dominant with respect to the magnetic
potential, the gradient of the magnetic potential, and the scalar potential
for sufficiently small $\varepsilon\ll_{R_{0},M_{0}}1$, equivalently

$a(x)\geq\varepsilon^{-1}(|A(x)|+|\nabla A(x)|+|V(x)|),\qquad|x|\leq 2R_{0},$

and satisfies the zero non-resonance and geometric control conditions.
Additionally, assume that $\partial_{t}$ is uniformly time-like while the
constant time slices are uniformly space-like. Then, there exists an
$\alpha>0$ so that any solution to

$Pu=f,\qquad u[0]\in\mathcal{E},\qquad f\in LE^{*}+L^{1}L^{2}$

satisfies one of the following two properties:

  1. (1)

Exponential growth asymptotics in terms of the data and forcing:

$\left\lVert\partial u(t)\right\rVert_{L^{2}}\gtrsim e^{\alpha
t}\left(\left\lVert\partial u(0)\right\rVert_{L^{2}}+\left\lVert
f\right\rVert_{LE^{*}+L^{1}L^{2}[0,\infty)}\right),\qquad t\gg 1.$

  2. (2)

Local energy decay:

$\left\lVert u\right\rVert_{LE^{1}[0,\infty)}+\left\lVert\partial
u\right\rVert_{L^{\infty}L^{2}[0,\infty)}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\left\lVert
f\right\rVert_{LE^{*}+L^{1}L^{2}[0,\infty)}.$

######  Remark 1.13.

Condition (b) on the damping can be readily weakened to assuming that $P$ is
$\varepsilon$-weakly damping dominant (in order for Theorem 1.11 to apply) and
that, for $\varepsilon$ small enough and all $t>0$, the estimate

$\displaystyle-2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P^{a}u}\,dxds-2i\operatorname{Im}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P_{0}^{a}u}\,dx\,ds\leq-C(\varepsilon,\textbf{c})\int\limits_{0}^{t}\int\limits_{B_{2R_{0}}(0)}a|\partial_{s}u|^{2}\,dxds+\mathcal{O}(\textbf{c})\left\lVert
u\right\rVert_{LE^{1}}^{2}$

holds for some $C(\varepsilon,\textbf{c})>0$, where $P^{a}$ and $P_{0}^{a}$
are the skew-adjoint parts of $P$ and $P_{0}$, respectively. The assumption
given in the dichotomy was stated as such merely for tractability, as it will
both satisfy the relevant hypothesis in Theorem 1.11 and allows the above
bound to hold as a consequence of Hölder’s inequality, Young’s inequality for
products, and asymptotic flatness. ∎

At least at a heuristic level, the solutions which exhibit case (1) behavior
stem from eigenvalues in the lower half-plane of the corresponding stationary
problem (see Section 3), which represent poles of the resolvent. The resolvent
has meromorphic continuation to the entire lower half-plane, and the poles
must occur within a relatively compact subset of frequencies. In particular,
there are only finitely many such eigenvalues, and each generalized eigenspace
has finite dimension by Fredholm theory. We also remark that we do not obtain
improvements from the corresponding result in [MST20] here, nor do we obtain
versions of their non-stationary results since our high frequency work
exploits the stationarity. While more refined energy space decompositions are
likely (and are also present in [MST20] within the non-damped setting), the
proper statements and results in the context of damped waves are not clear to
us at this time.

Finally, we have local energy decay. Here, we must further assume that $P$
satisfies various spectral hypotheses, which are defined in Section 3.

######  Theorem 1.14.

Let $P$ be a stationary, asymptotically flat damped wave operator which
satisfies the zero non-resonance and geometric control conditions and is
either weakly $\varepsilon$-damping dominant for some $\varepsilon>0$ or
$\varepsilon$-weakly magnetic with $\varepsilon\ll_{R_{0},M_{0}}1$. Suppose
further that $P$ has no negative eigenfunctions nor real resonances and that
$\partial_{t}$ is uniformly time-like while the constant time slices are
uniformly space-like. Then, local energy decay holds, with the implicit
constant in (1.1) independent of $T$.

In order to prove this result, we cannot necessarily rely on uniform energy
bounds to pass from the two-point local energy estimate to local energy decay
like in [Kof22] and instead rely on spectral theory as in [MST20]. The
structure of proving Theorem 1.14, and hence of the overall paper, is as
follows and is motivated by [MST20]:

  1. (1)

Section 2.1: Extend the high frequency analysis from [Kof22] to the present
context. One must take care to consider how the damping interacts with the
remaining principal skew-adjoint terms; this is the purpose of the
$\varepsilon$-weakly magnetic and $\varepsilon$-damping dominant conditions.
We will explain the results that we borrow and why they apply here.

  2. (2)

Section 2.2: Provide an overview of the low and medium frequency analysis;
these do not require change from [MST20], since the damping is simply treated
as a first-order $AF$ perturbation term. From here, Theorem 1.11 follows
readily from the work in [MST20], [Kof22]. As an immediate consequence, we
will establish Theorem 1.12 in Section 2.3, although this has no bearing on
the proof of Theorem 1.14.

  3. (3)

Section 3: Summarize the necessary resolvent theory in [MST20] required to
prove Theorem 1.14. This will require Theorem 1.10 and Theorem 1.11. Once one
is armed with the relevant frequency estimates, the work in [MST20] applies
rather directly. We will summarize and/or provide many (but not all) of their
arguments for the required results, especially where we believe that further
elucidation would be beneficial for the sake of exposition. We require little
deviation from their theory in our present work.

######  Remark 1.15.

As a consequence of Remark 1.5, Theorems 1.10, 1.11, and 1.14 hold if the
symmetry conditions are replaced by the strongly $\varepsilon$-damping
dominant condition with $2\varepsilon<(\left\lVert
g-m\right\rVert_{L^{\infty}}+1)^{-1}$. ∎

######  Remark 1.16.

As opposed to writing the d’Alembertian in divergence form (i.e.
$\Box_{g}=D_{\alpha}g^{\alpha\beta}D_{\beta}$) and utilizing the volume form
$dV=dxdt$ for our analysis, one could work with the geometric d’Alembertian
(that is, the Laplace-Beltrami form)

$\widetilde{\Box}_{g}=|g|^{-1/2}D_{\alpha}|g|^{1/2}g^{\alpha\beta}D_{\beta},\qquad|g|=|\det
g^{\alpha\beta}|$

with the volume form $dV=|g|^{1/2}dxdt.$ Each d’Alembertian is symmetric with
respect to the associated volume form on $L^{2}(dV)$. One can transition from
the latter framework to the former by conjugating the operator by $|g|^{1/4}$
(see e.g. [Tat13], [Morg20]); lower-order terms arise, but they are
permissible in view of the magnetic and scalar potential terms already
allowable in $P$. For this reason, we are working with the former case. ∎

Acknowledgements. The author would like to thank Jason Metcalfe for helpful
discussions. He would also like to thank both anonymous reviewers for their
feedback, including the previously-missing assumption of uniformly space-like
time-slices, which is also needed in [Kof22]. Declarations. The author reports
there are no competing interests to declare.

##  2\. Frequency Analyses and Two-Point Local Energy Decay

In this section, we will prove Theorem 1.11 using high, low, and medium
frequency analyses. The low and medium frequency work follows directly from
that in [MST20], while the high frequency work is a variation on [Kof22] and
requires the symmetry assumptions as described in Section 1.1.

To start, we will define cutoff notation which we will use throughout the
duration of the paper. Namely, we will fix $\chi\in C_{c}^{\infty}\text{ non-
increasing},\ \chi\equiv 1\text{ for }|x|\leq 1,\ \chi\equiv 0\text{ for
}|x|>2$ and define $\chi_{<R}(|x|)=\chi(|x|/R),$ $\chi_{>R}=1-\chi_{<R}.$ We
will assume further that $\chi$ is the square of a smooth function for
notation convenience (as otherwise, we could work with $\chi^{2}$). We will
occasionally add the variable into the subscript to make the specific
dependence clear (e.g. write $\chi_{|\xi|<\lambda}$ when working truncating
the spatial frequencies $\xi$ below a threshold $\lambda$).

###  2.1. High Frequency Analysis

We will recall the relevant framework and results from [Kof22] needed to prove
the high frequency estimate. When deviation occurs, we will proceed carefully
and explicitly. The need for the $\varepsilon$-damping dominant and
$\varepsilon$-weakly magnetic conditions only come up in one place, in the
proof of Lemma 2.3. Throughout this section, we will assume that $P$ is
stationary. This work is motivated by [Kof22], [MST20], and [BR14].

The high frequency analysis is rooted in the behavior of the bicharacteristic
flow generated by the principal symbol of $P$. First, we make a minor
simplification. Since the constant time slices are assumed uniformly space-
like, it follows that $g^{00}\lesssim-1$. Dividing through by $-g^{00}$
preserves the assumptions on the operator coefficients (see e.g. [MT12]);
hence, we may assume that $g^{00}=-1$.

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

$p(x,\tau,\xi)=-(\tau^{2}-2\tau
g^{0j}(x)\xi_{j}-g^{ij}(x)\xi_{i}\xi_{j}),\qquad(t,x,\tau,\xi)\in
T^{*}\mathbb{R}^{4}\setminus o.$

Notice that since $P$ is stationary, $p$ is independent of $t$. This symbol
generates a bicharacteristic flow on $\mathbb{R}\times T^{*}\mathbb{R}^{4}$
given by
$\varphi_{s}(w)=\left(t_{s}(w),x_{s}(w),\tau_{s}(w),\xi_{s}(w)\right)$ 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 $w\in T^{*}\mathbb{R}^{4}.$ 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 $p$, denoted $\operatorname{Char}(P)$.
We remark that there is no distinguishing between $\pm p$ on
$\operatorname{Char}(P)$, hence the minus sign in front of $p$ is somewhat
inconsequential on it.

Using the flow $\varphi_{s}$, we define the forward and backward trapped and
non-trapped sets with respect to $\varphi_{s}$, respectively, as

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

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. The flow is said to be non-trapping if
$\Omega^{p}_{tr}{\,=\,}\emptyset$. Otherwise, the flow is said to possess
trapping. Now, we may precisely state the geometric control condition.

######  Definition 2.1.

We say that the geometric control condition holds if

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

In order to leverage the sign of the damping, [Kof22] utilized a scaling
argument which we will also exploit. Namely, if $u$ solves $Pu=f$, we consider

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

For this discussion, a tilde over a function will denote dilation by $\gamma$
in each coordinate, as done to define $\tilde{u}$. If we call

$\tilde{P}=(D_{\alpha}+\gamma\tilde{A}_{\alpha})\tilde{g}^{\alpha\beta}(D_{\beta}+\gamma\tilde{A}_{\beta})+i\gamma\tilde{a}D_{t}+\gamma^{2}\tilde{V},$

then

$\tilde{P}(\gamma^{-2}\tilde{u})=\tilde{f}\qquad\text{ if and only if }\qquad
Pu=f.$

The benefit of this scaling is that we obtain an arbitrarily large constant in
front of the damping function. However, we underscore that such a large
constant is also inherited by the magnetic potential.

Analogous Hamiltonian systems and trapped/non-trapped sets exist for the
principal symbol $\tilde{p}$ of $\tilde{P}$, which amounts to simply dilating
the coordinates of $g$. If we assume that geometric control holds for the flow
generated by $p$, then it is proven in [Kof22] that it holds for the flow
generated by $\tilde{p}$. Since the proof is straightforward, we will omit it
here and only record the result, which is Proposition 2.6 in [Kof22].

######  Proposition 2.2.

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

Since the lower-order terms $A$ and $V$ are not at the principal level, they
do not affect $p$ nor $\tilde{p}$. Henceforth, we will fix a large $\gamma>0$
and study the problem from the scaled perspective while reverting back to our
original notation (e.g. no tildes). It is readily seen that it is equivalent
to prove Theorem 1.10 for the scaled problem, where we now have a large
constant in front of the damping term.

The proof of the version of Theorem 1.10 present in [Kof22] (i.e. $A,V\equiv
0,\ \operatorname{supp}a\subset\\{|x|\leq R_{0}\\}$) is a positive commutator
argument. At the symbolic level, this requires the construction of an escape
function and a lower-order correction term. Let $p$ and $s_{skew}$ represent
the principal symbols of the self and skew-adjoint parts of $P$, respectively.
Namely,

$\displaystyle{p}(x,\tau,\xi)$
$\displaystyle=-(\tau^{2}-2\tau{g}^{0j}(x)\xi_{j}-{g}^{ij}(x)\xi_{i}\xi_{j})$
$\displaystyle{s}_{skew}(x,\tau,\xi)$
$\displaystyle=i\gamma\left(\operatorname{Im}A_{\alpha}(x)(g^{\alpha
0}(x)\tau+g^{\alpha k}(x)\xi_{k})+\tau\operatorname{Re}a(x)\right).$

The multiplication by $\gamma$ in $s_{skew}$ will prove advantageous for a
bootstrapping argument, which is precisely why we implement the
$\gamma$-scaling. However, the imaginary part of $A$ has interaction with the
damping and also features multiplication by $\gamma$. The
$\varepsilon$-damping dominant and $\varepsilon$-weakly magnetic conditions
are applied in order to retain the positivity effects of the damping.

The precise escape function construction is as follows.

######  Lemma 2.3.

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

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

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

Here, $H_{p}$ is the Hamiltonian vector field induced by $p$, and
$S^{m}(T^{*}\mathbb{R}^{3})$ denotes the standard Kohn-Nirenberg symbol class
of order $m$. To each symbol $b\in S^{m}(T^{*}\mathbb{R}^{3})$, we have the
associated Weyl quantization of $b$, denoted $b^{\operatorname{w}}$, which is
a pseudodifferential operator of order $m$ defined by the action

$b^{\operatorname{w}}(x,D)u(x)=(2\pi)^{-3}\int\limits_{\mathbb{R}^{3}}\int\limits_{\mathbb{R}^{3}}e^{i(x-y)\cdot\xi}\
b\left(\frac{x+y}{2},\xi\right)u(y)\,dyd\xi,\qquad
u\in\mathcal{S}(\mathbb{R}^{3}).$

We will use $\Psi^{m}(\mathbb{R}^{3})$ to denote the space of
pseudodifferential operators on $\mathbb{R}^{3}$ of order $m$, and write

$\Psi^{-\infty}(\mathbb{R}^{3}):=\bigcap\limits_{m\in\mathbb{R}}\Psi^{m}(\mathbb{R}^{3})$

for the space of smoothing operators on $\mathbb{R}^{3}.$

Lemma 2.3 was proven in [Kof22] in the special case of $A,V\equiv 0$ and
$\operatorname{supp}a\subseteq\\{|x|\leq R_{0}\\}$, leading to a simplified
$s_{skew}$. As in the aforementioned work, we will work with the half-wave
decomposition; it is proven in [Kof22] that the null bicharacterstics are
equivalent through a reparameterization argument, and this fact continues to
hold here without any change (lower-order terms do not affect $p$ and hence
will not affect its induced bicharacteristic flow). That is, we factor $p$ as

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

where

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

Using that $\partial_{t}$ is uniformly time-like, it is readily seen that
$b^{\pm}$ are both positively homogeneous of degree 1 in $\xi$, and

$b^{+}(x,\xi)>0>b^{-}(x,\xi)$

whenever $\xi\neq 0$. The Hamiltonians $p^{\pm}:=\tau-b^{\pm}$ also generate
bicharacteristic flows

$\varphi_{s}^{\pm}(w){\,=\,}\left(t_{s}^{\pm}(w),x_{s}^{\pm}(w),\tau_{s}^{\pm}(w),\xi_{s}^{\pm}(w)\right)$

on $\mathbb{R}\times T^{*}\mathbb{R}^{4}$ which solve the Hamiltonian systems

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

with initial data $w\in T^{*}\mathbb{R}^{4}.$ Observe that the $(t,\tau)$ and
$(x,\xi)$ systems are decoupled, allowing us to project onto the $(x,\xi)$
components of the flow without losing information. Notice that, after 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 the forward and backward (denoted by the $\pm$ notation)
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

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

As a consequence of the factoring, we have the identities

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

where $\Pi_{x,\xi}(t,x,\tau,\xi)=(x,\xi).$ Additionally, we may re-state
geometric control in terms of the factored flow. If $w\in\Omega_{tr}$, then it
is either trapped with respect the flow generated by $p^{+}$ or $p^{-}$. 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 point where the damping is positive,
and similarly if it is trapped with respect to $p^{-}$.

Since $A$ and $V$ do not occur at the principal level for $P$, they will not
affect the individual components of the escape function construction. Hence,
the construction in [Kof22] applies directly. The methodology in [Kof22],
motivated by a combination of [BR14], [MST20], and [MMT08], is performed in
the following steps:

  1. (1)

On the characteristic set. We will refer to $\\{|x|\leq R_{0}\\}$ as the
interior and $\\{|x|>R_{0}\\}$ as the exterior.

    1. (a)

Interior, semi-bounded null bicharacteristics. Here, one considers semi-
bounded null bicharacteristics with initial data living in the interior
region. Working with semi-bounded trajectories is favorable since they include
both trajectories that are trapped and those which escape slowly.
Additionally, geometric control extends to such trajectories. This is where
geometric control is used.

    2. (b)

The remainder of the interior region. In this region, all of the trajectories
escape both forward and backward in time. This region is more classical, but
care must be taken both to avoid the trapping and incorporate the half-wave
structure.

    3. (c)

The exterior region. As a consequence of asymptotic flatness, there are no
trapped trajectories in this region. Here, one appeals to geometrically-
adapted flat wave theory. The multiplier also allows for the absorption of an
error term which arises in the prior region.

  2. (2)

On the elliptic set. Here, one requires a lower-order symbol which provides no
contribution on the characteristic set and provides positivity off of it. This
essentially follows from the minimization of an appropriate quadratic in the
dual time variable $\tau$.

Now, we cite the specific results from [Kof22] (namely, Lemmas 2.13 and 2.16,
respectively). First, we define the interior, semi-trapped set

$\Omega_{R_{0}}^{\pm}:=\left(\Gamma_{tr}^{\pm}\cup\Lambda_{tr}^{\pm}\right)\cap\\{|x|\leq
R_{0}\\}.$

This is related to step (1a). Next, we define the function

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

It can be shown directly that

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

for any $\lambda>0$ due to the homogeneity of $b^{\pm}$. Since
$|\xi/b^{\pm}(x,\xi)|\approx 1$ for $\xi\neq 0$ and $b^{\pm}$ is a constant of
motion for the flow generated by $p^{\pm}$ (which explicitly utilizes that $g$
is stationary), the function $\Phi^{\pm}$ provides a lifting which is useful
to pair with scaling arguments. The first portion of the construction (1a) is
contained in the following lemma, which was motivated by [BR14]. This is where
geometric control is utilized. (More precisely, geometric control also applies
to semi-trapped trajectories; see [Kof22].)

######  Lemma 2.4 (Semi-bounded Escape Function Construction).

There exist $q_{1}^{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$, an
open set $V_{R_{0}}^{\pm}\supset\Omega_{R_{0}}^{\pm}$, and
$C^{\pm}\in\mathbb{R}_{+}$ so that

$H_{p^{\pm}}q_{1}^{\pm}+C^{\pm}a{\chi_{<R_{0}}}\gtrsim_{R_{0}}\mathbbm{1}_{V_{R_{0}}^{\pm}}.$

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

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

######  Lemma 2.5 (Non-trapped Escape Function Construction).

There exist $q_{2}^{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$ and
$W^{\pm}\subset\Omega_{\infty}^{\pm}$ so that $V_{R_{0}}^{\pm}\cup
W^{\pm}=T^{*}\mathbb{R}^{3}\setminus o,\ W^{\pm}\supset\\{|x|>R_{0}\\}$ and

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

Further, $q_{2}^{\pm}=q_{in}^{\pm}+q_{out}^{\pm},$ where
$q_{in}^{\pm}=\tilde{q}_{in}^{\pm}\circ\Phi^{\pm}$, with
$\tilde{q}_{in}^{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{3}\setminus o)$ being
supported in $\\{|x|\leq 4R\\}$ and $q_{out}^{\pm}\in
S^{0}_{\operatorname{hom}}(T^{*}\mathbb{R}^{3}\setminus o)$.

We remark that the behavior of $(c_{j})$ does not matter so much for $0\leq
j<\log_{2}R_{0},$ as long as each corresponding $c_{j}$ is positive.

To complete the remaining steps and prove Lemma 2.3, we proceed similarly to
the work in [Kof22], with special attention paid to the new contributions of
$s_{skew}.$ The presence of $\operatorname{Im}A$ in $s_{skew}$ did not occur
in [Kof22] and must be dealt with here. The symmetry conditions arise when one
must balance the ability to leverage the sign on the damping for $|x|\leq
2R_{0}$ with the necessity to absorb the unsigned magnetic terms.

###### Proof of Lemma 2.3.

First, we truncate the symbols to stay away from $\xi=0$:

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

where $\sigma_{1},\ \sigma_{2}\gg 1$. Unlike [Kof22], we need two parameters
$\sigma_{1}$ and $\sigma_{2}$, as opposed to just one parameter; the
additional parameter is needed to deal with unsigned first-order errors. The
exponentiation is implemented for bootstrapping: Taking derivatives of the
exponentials will provide multiplication by $\sigma_{1}$ and $\sigma_{2}$. It
is readily seen that $q^{\pm}_{j,>\lambda}\in S^{0}(T^{*}\mathbb{R}^{3})$ via
the chain rule. We combine the symbols constructed on the individual light
cones together as

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

Calling

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

we can see that

$\displaystyle\left(H_{p}q+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha k}\xi_{k})\right)q\right)\big{|}_{\tau=b^{\pm}}$
$\displaystyle=\left(H_{p}q_{1}+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha
k}\xi_{k})\right)q_{1}\right)\big{|}_{\tau=b^{\pm}}+\left(H_{p}q_{2}+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha k}\xi_{k})\right)q_{2}\right)\big{|}_{\tau=b^{\pm}}$
$\displaystyle=H_{p}q_{1}\big{|}_{\tau=b^{\pm}}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{1,>\lambda}^{\pm}$
$\displaystyle\qquad+H_{p}q_{2}\big{|}_{\tau=b^{\pm}}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{2,>\lambda}^{\pm}.$

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

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

By making $\sigma_{1},\sigma_{2}$ sufficiently large, we get that

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

and so

$\displaystyle\left(H_{p}q+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha k}\xi_{k})\right)q\right)\big{|}_{\tau=b^{\pm}}$
$\displaystyle\gtrsim\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}H_{p^{\pm}}q_{1}^{\pm}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{1,>\lambda}^{\pm}$
$\displaystyle\qquad+\sigma_{2}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}H_{p^{\pm}}q_{2}^{\pm}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{2,>\lambda}^{\pm}.$

We will consider each line in the above lower bound separately. First, observe
that

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

Case 1: $P$ is weakly $\varepsilon$-damping dominant. By using the weakly
$\varepsilon$-damping dominant condition and choosing $\gamma$ sufficiently
larger than $\varepsilon^{-1}\sigma_{1}$, we may apply Lemma 2.4 to obtain
that, for $|x|\approx 2^{\ell}$,

(2.2)
$\displaystyle\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}H_{p^{\pm}}q_{1}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{1,>\lambda}^{\pm}$
$\displaystyle=\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{1}^{\pm}+\left(\frac{2\gamma}{\sigma_{1}}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\left(\operatorname{Re}a+g^{0\alpha}\operatorname{Im}A_{\alpha}\right)+\left(\frac{2\gamma}{\sigma_{1}}\right)\frac{\xi_{k}}{b^{\pm}-b^{\mp}}\operatorname{Im}A_{\alpha}g^{\alpha
k}\right)$
$\displaystyle\gtrsim\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{1}^{\pm}+\frac{2\gamma}{\sigma_{1}}(\varepsilon\operatorname{Re}a\chi_{<R_{0}}-c_{\ell}2^{-\ell}\chi_{>R_{0}})\right)$
$\displaystyle\gtrsim\sigma_{1}\chi_{|\xi|>\lambda}|\xi|^{2}\left(\mathbbm{1}_{{V_{R_{0}}^{\pm}}}-\frac{2\gamma}{\sigma_{1}}c_{\ell}2^{-\ell}\chi_{>R_{0}}\right).$

Next, we use the weakly $\varepsilon$-damping dominant, use that the damping
is non-negative for $|x|\leq 2R_{0}$, use Lemma 2.5, and choose $\sigma_{2}$
sufficiently larger than $\gamma$ to give that, for $|x|\approx 2^{\ell}$,

(2.3)
$\displaystyle\sigma_{2}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}H_{p^{\pm}}q_{2}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{2,>\lambda}^{\pm}$
$\displaystyle=\sigma_{2}\chi_{<R_{0}}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{2}^{\pm}+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\left(\operatorname{Re}a+g^{0\alpha}\operatorname{Im}A_{\alpha}\right)+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{\xi_{k}}{b^{\pm}-b^{\mp}}\operatorname{Im}A_{\alpha}g^{\alpha
k}\right)$
$\displaystyle\qquad+\sigma_{2}\chi_{>R_{0}}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{2}^{\pm}+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\left(\operatorname{Re}a+g^{0\alpha}\operatorname{Im}A_{\alpha}\right)+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{\xi_{k}}{b^{\pm}-b^{\mp}}\operatorname{Im}A_{\alpha}g^{\alpha
k}\right)$
$\displaystyle\gtrsim\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|<R_{0}}|\xi|^{2}\mathbbm{1}_{W^{\pm}}+\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|>R_{0}}c_{\ell}2^{-\ell}|\xi|^{2}\left(1-\frac{2\gamma}{\sigma_{2}}\right)$
$\displaystyle\gtrsim\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|<R_{0}}|\xi|^{2}\mathbbm{1}_{W^{\pm}}+\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|>R_{0}}c_{\ell}2^{-\ell}|\xi|^{2}.$

Case 2: $P$ is $\varepsilon$-weakly magnetic. By using the
$\varepsilon$-weakly magnetic condition and choosing $\gamma$ sufficiently
larger than $\sigma_{1}$, we may apply Lemma 2.4 to obtain that, for
$|x|\approx 2^{\ell}$,

(2.4)
$\displaystyle\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}H_{p^{\pm}}q_{1}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{1,>\lambda}^{\pm}$
$\displaystyle=\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{1}^{\pm}+\left(\frac{2\gamma}{\sigma_{1}}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\left(\operatorname{Re}a+g^{0\alpha}\operatorname{Im}A_{\alpha}\right)+\left(\frac{2\gamma}{\sigma_{1}}\right)\frac{\xi_{k}}{b^{\pm}-b^{\mp}}\operatorname{Im}A_{\alpha}g^{\alpha
k}\right)$
$\displaystyle\gtrsim\sigma_{1}(b^{+}-b^{-})^{2}q_{1,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{1}^{\pm}+\frac{2\gamma}{\sigma_{1}}(\operatorname{Re}a\chi_{<R_{0}}-\varepsilon(1+\left\lVert
g-m\right\rVert_{L^{\infty}})\chi_{<R_{0}}-c_{\ell}2^{-\ell}\chi_{>R_{0}})\right)$
$\displaystyle\gtrsim\sigma_{1}\chi_{|\xi|\geq\lambda}|\xi|^{2}\left(\mathbbm{1}_{{V_{R_{0}}^{\pm}}}-\frac{2\gamma}{\sigma_{1}}\left(\varepsilon(1+\left\lVert
g-m\right\rVert_{L^{\infty}})\chi_{<R_{0}}+c_{\ell}2^{-\ell}\chi_{>R_{0}}\right)\right).$

The poorly-signed interior term above is somewhat problematic and will dictate
the choice of $\varepsilon$.

Next, we use the $\varepsilon$-weakly magnetic condition, apply Lemma 2.5, and
choose $\sigma_{2}$ sufficiently larger than $\gamma$ to give that, for
$|x|\approx 2^{\ell}$,

(2.5)
$\displaystyle\sigma_{2}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}H_{p^{\pm}}q_{1}\pm
2\gamma(b^{+}-b^{-})\left(b^{\pm}(\operatorname{Re}a+\operatorname{Im}A_{\alpha}g^{\alpha
0})+\operatorname{Im}A_{\alpha}g^{\alpha
k}\xi_{k}|_{\tau=b^{\pm}}\right)q_{2,>\lambda}^{\pm}$
$\displaystyle=\sigma_{2}\chi_{<R_{0}}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{2}^{\pm}+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\left(\operatorname{Re}a+g^{0\alpha}\operatorname{Im}A_{\alpha}\right)+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{\xi_{k}}{b^{\pm}-b^{\mp}}\operatorname{Im}A_{\alpha}g^{\alpha
k}\right)$
$\displaystyle\qquad+\sigma_{2}\chi_{>R_{0}}(b^{+}-b^{-})^{2}q_{2,>\lambda}^{\pm}\left(H_{p^{\pm}}q_{2}^{\pm}+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{b^{\pm}}{b^{\pm}-b^{\mp}}\left(\operatorname{Re}a+g^{0\alpha}\operatorname{Im}A_{\alpha}\right)+\left(\frac{2\gamma}{\sigma_{2}}\right)\frac{\xi_{k}}{b^{\pm}-b^{\mp}}\operatorname{Im}A_{\alpha}g^{\alpha
k}\right)$
$\displaystyle\gtrsim\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|<R_{0}}|\xi|^{2}\left(\mathbbm{1}_{W^{\pm}}-\frac{2\gamma\varepsilon(1+\left\lVert
g-m\right\rVert_{L^{\infty}})}{\sigma_{2}}\right)+\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|>R_{0}}c_{\ell}2^{-\ell}|\xi|^{2}\left(1-\frac{2\gamma}{\sigma_{2}}\right)$
$\displaystyle\gtrsim\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|<R_{0}}|\xi|^{2}\left(\mathbbm{1}_{W^{\pm}}-\frac{2\gamma\varepsilon(1+\left\lVert
g-m\right\rVert_{L^{\infty}})}{\sigma_{2}}\right)+\sigma_{2}\chi_{|\xi|>\lambda}\chi_{|x|>R_{0}}c_{\ell}2^{-\ell}|\xi|^{2}.$

We claim that, in both cases, the estimate

(2.6)
$\left(H_{p}q+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha
k}\xi_{k})\right)q\right)\big{|}_{\tau=b^{\pm}}\gtrsim\mathbbm{1}_{|\xi|\geq\lambda}c_{\ell}2^{-\ell}|\xi|^{2},\qquad|x|\approx
2^{\ell}$

holds. Indeed, recall that $V_{R_{0}}^{\pm}\cup
W^{\pm}=T^{*}\mathbb{R}^{3}\setminus o$. In Case 1, we combine (2.2) and (2.3)
and, if necessary, further increase $\sigma_{2}$ to directly get (2.6). In
Case 2, we obtain (2.6) by combining (2.4) and (2.5) together, then choosing
$\varepsilon\ll_{R_{0},M_{0}}1$ and, if necessary, further increase
$\sigma_{2}$. From (2.6), we use that $(c_{\ell})$ is a slowly-varying,
summable sequence to conclude that

$\left(H_{p}q+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha
k}\xi_{k})\right)q\right)\big{|}_{\tau=b^{\pm}}\gtrsim\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}|\xi|^{2}.$

The work on the elliptic set in e.g. [MST20], [Kof22] applies without
modification. Summarily, if we write

(2.7)
$H_{p}q+2\gamma\left(\tau\operatorname{Re}a+\operatorname{Im}A_{\alpha}(g^{\alpha
0}\tau+g^{\alpha
k}\xi_{k})\right)q+pm=(a_{0}-m)\tau^{2}+\left(a_{1}+(b^{+}+b^{-})m\right)\tau+(a_{2}-b^{+}b^{-}m),$

where $a_{j}\in S^{j}$, then we choose

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

Such an $m$ ensures that the quadratic polynomial (2.7) in $\tau$ is concave
up and has no real zeros. ∎

The proof of Theorem 1.10 is highly similar to the proof in [Kof22] for the
analogous result. There are a few additional terms to deal with, but the
aforementioned work demonstrates how to deal with them, as they are lower-
order. We will briefly summarize the argument in [Kof22] and add in additional
details for the new terms. First, we remark that one can readily reduce the
theorem to a simplified estimate, which we state as a proposition.

######  Proposition 2.6.

In order to prove Theorem 1.10, it suffices to prove that

(2.8) $\left\lVert v_{>\lambda}\right\rVert_{LE^{1}_{<2R_{0}}}\lesssim
C(\lambda,\gamma)\left(\left\lVert
Pv\right\rVert_{LE^{*}_{c}}^{1/2}\left\lVert
v\right\rVert_{LE^{1}}^{1/2}+\left\lVert
v\right\rVert_{L^{2}L^{2}}\right)+\gamma\lambda^{-1/2}\left\lVert
v\right\rVert_{LE^{1}}$

for all $v$ supported in $\\{|x|\leq 2R_{0}\\},$ where
$v_{\lambda}=\chi_{|\xi|>\lambda}(D_{x})v.$

This reduction was shown in [MST20] and [Kof22]. We will not reproduce the
proof here, but the idea is as follows:

  1. (1)

Use asymptotic flatness to reduce to the case of $Pu$ and $u[0]$ compactly-
supported in $\\{|x|\leq 2R_{0}\\}$.

  2. (2)

Utilize a unit time interval and Duhamel’s theorem to reduce to $u[0]=0$ and
$Pu\in LE^{*}_{c}$.

  3. (3)

Remove the upper bound on the time integrals using a cutoff argument, making
it sufficient to integrate in $t$ from $-\infty$ to $\infty.$

  4. (4)

Reduce to solutions supported in $\\{|x|\leq 2R_{0}\\}$ via standard exterior
wave estimates.

  5. (5)

Add back in the low frequencies, then take $\lambda\gg\gamma$ and apply
Young’s inequality for products in order to bootstrap the $LE^{1}$ terms on
the right into the left.

Now, we prove Theorem 1.10. During the proof, we will use
$\left\langle\cdot,\cdot\right\rangle$ to denote the $L^{2}L^{2}$ inner
product. We also recall the $\gamma$-scaling that was introduced after
Definition 2.1.

###### Proof of Theorem 1.10.

We compute that

(2.9) $\displaystyle 2\text{Im}$ $\displaystyle\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v\right\rangle+\frac{i\gamma}{2}\left\langle[(\operatorname{Im}A_{\alpha}g^{\alpha\beta}D_{\beta}+D_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\alpha}),m^{\operatorname{w}}]v,v\right\rangle+\frac{i\gamma}{2}\left\langle[aD_{t},m^{\operatorname{w}}]v,v\right\rangle$
$\displaystyle\qquad\qquad-i\gamma\left\langle[(\operatorname{Re}A_{\alpha}g^{\alpha\beta}D_{\beta}+D_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta}),q^{\operatorname{w}}],v,v\right\rangle$
$\displaystyle\qquad\qquad-\frac{\gamma}{2}\left\langle\left(m^{\operatorname{w}}(\operatorname{Re}A_{\alpha}g^{\alpha\beta}D_{\beta}+D_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta})+(\operatorname{Re}A_{\alpha}g^{\alpha\beta}D_{\beta}+D_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta})m^{\operatorname{w}}\right)v,v\right\rangle$
$\displaystyle\qquad\qquad+i\gamma^{2}\left\langle[\operatorname{Im}A_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\beta}-\operatorname{Re}A_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta},q^{\operatorname{w}}]v,v\right\rangle$
$\displaystyle\qquad\qquad-2\gamma^{2}\left\langle\left(q^{\operatorname{w}}(\operatorname{Im}A_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta})+(\operatorname{Im}A_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta})q^{\operatorname{w}}\right)v,v\right\rangle$
$\displaystyle\qquad\qquad-\frac{\gamma^{2}}{2}\left\langle\left(m^{\operatorname{w}}(\operatorname{Re}A_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta})+(\operatorname{Re}A_{\alpha}g^{\alpha\beta}\operatorname{Re}A_{\beta})m^{\operatorname{w}}\right)v,v\right\rangle$
$\displaystyle\qquad\qquad+\frac{\gamma^{2}}{2}\left\langle\left(m^{\operatorname{w}}(\operatorname{Im}A_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\beta})+(\operatorname{Im}A_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\beta})m^{\operatorname{w}}\right)v,v\right\rangle$
$\displaystyle\qquad\qquad+\gamma^{2}\left\langle[\operatorname{Re}A_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\beta},m^{\operatorname{w}}]v,v\right\rangle+i\gamma^{2}\left\langle[q^{\operatorname{w}},\operatorname{Re}V]v,v\right\rangle+\frac{i\gamma}{2}\left\langle[\operatorname{Im}V,m^{\operatorname{w}}]v,v\right\rangle$
$\displaystyle\qquad\qquad-\frac{\gamma^{2}}{2}\left\langle(m^{\operatorname{w}}\operatorname{Re}V+\operatorname{Re}Vm^{\operatorname{w}})v,v\right\rangle-{\gamma^{2}}\left\langle(q^{\operatorname{w}}\operatorname{Im}V+\operatorname{Im}Vq^{\operatorname{w}})v,v\right\rangle$
$\displaystyle\qquad\qquad+\gamma\left\langle\operatorname{Im}aD_{t}m^{\operatorname{w}}v,v\right\rangle+\gamma\left\langle[\operatorname{Im}a,q^{\operatorname{w}}]D_{t}v,v\right\rangle$
$\displaystyle=\left\langle
i[\Box_{g},q^{\operatorname{w}}]v,v\right\rangle+\frac{1}{2}\left\langle(\Box_{g}m^{\operatorname{w}}+m^{\operatorname{w}}\Box_{g})v,v\right\rangle$
$\displaystyle\qquad\qquad+\gamma\left\langle\left(q^{\operatorname{w}}(\operatorname{Re}aD_{t}+\operatorname{Im}A_{\alpha}g^{\alpha\beta}D_{\beta}+D_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\beta})+(\operatorname{Re}aD_{t}+\operatorname{Im}A_{\alpha}g^{\alpha\beta}D_{\beta}+D_{\alpha}g^{\alpha\beta}\operatorname{Im}A_{\beta})q^{\operatorname{w}}\right)v,v\right\rangle.$

Notice that the right-hand side of (2.9) may be written as

$\displaystyle\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v,v\right\rangle+\left\langle{R}_{0}v,v\right\rangle,\qquad\operatorname{where}\qquad{R}_{0}\in\Psi^{0}(\mathbb{R}^{3}).$

Split $v$ as

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

and recall that $q,m$ are both supported at frequencies $|\xi|\geq\lambda.$ By
choosing $\gamma$ large enough, the sharp Gårding inequality yields that

$\displaystyle\left\langle(H_{p}q-2is_{skew}q+mp)^{\operatorname{w}}v,v\right\rangle$
$\displaystyle\gtrsim\left\langle\left(\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\right)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle$
$\displaystyle\qquad-\left\lVert
v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}+\left\langle
Sv,v\right\rangle,\qquad\operatorname{where}\qquad
S\in\Psi^{-\infty}(\mathbb{R}^{3}).$

Integrating by parts one time gives that

$\displaystyle\left\langle\left(\chi_{|\xi|>\lambda}\langle
x\rangle^{-2}(|\xi|^{2}+\tau^{2})\right)^{\operatorname{w}}v_{>>\lambda},v_{>>\lambda}\right\rangle$
$\displaystyle\gtrsim\left\lVert\partial
v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}^{2}+\left\langle{R}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle,\qquad\operatorname{where}\qquad{R}_{1}\in\Psi^{1}(\mathbb{R}^{3}).$

All together, the right-hand side of (2.9) is bounded below by a multiple of

$\displaystyle\left\lVert\partial
v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}^{2}-\left|\left\langle{R}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right|-\left\lVert
v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}-\left|\left\langle{R}_{0}v,v\right\rangle\right|.$

Bounding the errors is performed using Plancherel’s theorem. With more
specificity, one may utilize the standard Sobolev mapping properties of
pseudodifferential operators, the frequency localization, and the compact
spatial support of $v$ to obtain that

$\displaystyle\left|\left\langle{R}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right|+\left\lVert
v_{>>\lambda}\right\rVert_{H^{1/2}_{t,x}}^{2}\lesssim\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}.$

One shows that

$\left|\left\langle{R}_{0}v,v\right\rangle\right|\lesssim
C(\lambda)\left\lVert v\right\rVert_{L^{2}L^{2}}^{2}$

in a similar manner, except that one cannot leverage frequency localization
and may incur an implicit constant which depends on $\lambda$. Such a term
will appear on the upper bound side of (2.8), so having such an implicit
constant is permissible.

Summarizing, we have shown that the right-hand side of (2.9) is bounded below
by a multiple of

(2.10) $\displaystyle\left\lVert\partial
v_{>\lambda}\right\rVert_{LE_{<2R_{0}}}^{2}-C(\lambda)\left\lVert
v\right\rVert_{L^{2}L^{2}}^{2}-\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}.$

Next, we consider the left-hand side of (2.9). By the Cauchy-Schwarz and
Plancherel’s theorem,

$\displaystyle\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v\right\rangle$
$\displaystyle=\left\langle
Pv,\left(q^{\operatorname{w}}-\frac{i}{2}m^{\operatorname{w}}\right)v_{>>\lambda}\right\rangle+\left\langle{S}v,v\right\rangle,\qquad\qquad{S}\in\Psi^{-\infty}(\mathbb{R}^{3})$
$\displaystyle\lesssim C(\lambda)\left(\left\lVert
Pv\right\rVert_{LE^{*}_{c}}\left\lVert v\right\rVert_{LE^{1}}+\left\lVert
v\right\rVert_{L^{2}L^{2}}^{2}\right).$

By once again applying frequency splitting, the remaining inner products on
the left-hand side of (2.9) are of the form

$(\gamma+\gamma^{2})\left(\left\langle\widetilde{R}_{0}v,v\right\rangle+\left\langle\widetilde{R}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right),\qquad\widetilde{R}_{j}\in\Psi^{j}(\mathbb{R}^{3}).$

We have discussed how to bound both of these terms; namely,

$\left|\left\langle\widetilde{R}_{0}v,v\right\rangle\right|+\left|\left\langle\widetilde{R}_{1}v_{>>\lambda},v_{>>\lambda}\right\rangle\right|\lesssim
C(\lambda)\left\lVert v\right\rVert_{L^{2}L^{2}}^{2}+\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}.$

Factoring in the scalar coefficients of these inner products, we have
demonstrated that the left-hand side of (2.9) is bounded above by a multiple
of

(2.11) $\displaystyle C(\lambda,\gamma)\left(\left\lVert
Pv\right\rVert_{LE^{*}_{c}}\left\lVert v\right\rVert_{LE^{1}}+\left\lVert
v\right\rVert_{L^{2}L^{2}}^{2}\right)+\gamma^{2}\lambda^{-1}\left\lVert
v\right\rVert_{LE^{1}}^{2}$

Combining (2.10)-(2.11) in application to (2.9) and completing the $LE^{1}$
norm on the lower-bound side provides (2.8). ∎

###  2.2. Remaining Frequency Analyses and Two-Point Local Energy Decay

In order to establish Theorem 1.11, we require similar estimates in the low
and medium frequency regimes. The damping does not play a meaningful role in
either regime, as it may be treated as a lower-order perturbation term. Like
in [Kof22], the relevant estimates from [MST20] carry through. We will briefly
summarize why this is the case, in lieu of full proofs.

At low frequencies, the obstruction to local energy decay arises when $P$ has
a resonance at frequency zero (see Section 3 for a precise definition and
further discussion on spectral obstructions to local energy decay). A
quantitative condition on the existence of corresponding zero resonant states
is the following zero non-resonance condition.

######  Definition 2.7.

$P$ is said to satisfy a zero non-resonance condition if there exists some
$K_{0}$, independent of $t$, such that

(2.12) $\displaystyle\left\lVert u\right\rVert_{\dot{H}^{1}}\leq
K_{0}\left\lVert P_{0}u\right\rVert_{\dot{H}^{-1}}\qquad\forall
u\in\dot{H}^{1}.$

The elliptic operator

$P_{0}=P\big{|}_{D_{t}=0}=(D_{j}+A_{j})g^{jk}(D_{k}+A_{k})+A_{0}g^{0j}(D_{j}+A_{j})+(D_{j}+A_{j})g^{j0}A_{0}+(A_{0})^{2}g^{00}+V$

represents $P$ at time frequency zero, and we underscore that the damping does
not appear. Hence, the damping has no bearing on whether or not the zero non-
resonance condition holds. For example, if $P$ is stationary and
asymptotically flat with $\operatorname{Im}A\equiv 0$, $V>0,$ and $A_{0}=0$,
then $P$ satisfies the zero non-resonance condition. This follows from Lemma
6.2iii in [MST20], which also features a more general condition. The relevant
low frequency estimate is the following, and the corresponding theorem in
[MST20] is Theorem 6.1.

######  Theorem 2.8.

Let $P$ be an asymptotically flat damped wave operator which satisfies the
zero non-resonance condition, and suppose that $\partial_{t}$ is uniformly
time-like. Then, the bound

(2.13) $\left\lVert
u\right\rVert_{LE^{1}}\lesssim\left\lVert\partial_{t}u\right\rVert_{LE^{1}_{c}}+\left\lVert
Pu\right\rVert_{LE^{*}}$

for all $u\in\mathcal{S}(\mathbb{R}^{4})$.

The proof of Theorem 2.8 leverages weighted elliptic estimates for the flat
Laplacian $\Delta$ in order to get similar estimates for $AF$ perturbations.
Once again, the damping does not play a meaningful role. At frequency zero, it
provides no contribution, and near frequency zero, it is absorbed by the error
term in (2.13); the damping arises when estimating $P_{0}u$ by $Pu$ within a
compact spatial set.

At medium frequencies, we require 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 Carleman estimates. The
Carleman weights which we need 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. We remark that the theorem does not imply an
absence of embedded eigenvalues/resonances on the real line.

######  Theorem 2.9.

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

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

with the implicit constant independent of $\delta$.

The proof of this theorem utilizes two intermediate Carleman estimates within
two different regions of space, which may be combined using a cutoff argument
in order to prove Theorem 2.9.

  1. (1)

Within a large compact set: The damping term is well-signed and readily
absorbable as a perturbation due to the conditions on the weight $\varphi,$
which will be convex.

  2. (2)

Outside of a large compact set: Here, the damping is a small $AF$
perturbation, so the proof in [MST20] follows through without any
modification. Within this region, the authors of [MST20] bend the weight to be
constant near infinity in order to apply exterior wave estimates. This leads
to breaking this case into three sub-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.

The proofs of the Carleman estimates in the above 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 high, low, and medium frequency estimates are the key ingredients needed
to establish Theorem 1.11. As in [MST20], [Kof22], it sufficient to remove the
Cauchy data at times $0$ and $T$ in order to prove Theorem 1.11; we will
elaborate on this momentarily. This makes it significantly easier to perform
frequency localization. The pertinent result in [MST20] is Theorem 7.1.

######  Theorem 2.10.

Let $P$ be a stationary, asymptotically flat damped wave operator satisfying
the geometric control condition (2.1), and suppose that $\partial_{t}$ is
uniformly time-like. Then, the estimate

(2.15) $\displaystyle\left\lVert u\right\rVert_{LE^{1}}\lesssim\left\lVert
Pu\right\rVert_{LE^{*}}$

holds for all $u\in\mathcal{S}(\mathbb{R}^{4})$.

###### Proof.

We will utilize a time-frequency partition of unity. Let $0<\tau_{0}\ll 1$ and
$\tau_{1}\gg 1$, which will be chosen with more precision shortly. Then, we
can write

$\displaystyle u$
$\displaystyle=\chi_{|\tau|<\tau_{0}}(D_{t})u+\chi_{\tau_{0}<|\tau|<\tau_{1}}(D_{t})u+\chi_{|\tau|>\tau_{1}}(D_{t})u=:Q_{1}u+Q_{2}u+Q_{3}u.$

Since $P$ is stationary, it commutes with each $Q_{j},$ and so it suffices to
show that

(2.16) $\left\lVert Q_{j}u\right\rVert_{LE^{1}}\lesssim\left\lVert
Pu\right\rVert_{LE^{*}},\qquad j=1,2,3.$

First, we apply Theorem 2.13 to $Q_{1}u$ and appeal to Plancherel’s theorem in
order to obtain that

$\displaystyle\left\lVert
Q_{1}u\right\rVert_{LE^{1}}\lesssim\left\lVert\partial_{t}(Q_{1}u)\right\rVert_{LE^{1}_{c}}+\left\lVert
P(Q_{1}u)\right\rVert_{LE^{*}}\lesssim\tau_{0}\left\lVert
Q_{1}u\right\rVert_{LE^{1}_{c}}+\left\lVert Pu\right\rVert_{LE^{*}}.$

If $\tau_{0}$ is sufficiently small, then we may absorb the error term on the
upper-hand side into lower-bound side, which provides (2.16) for $j=1.$

We proceed similarly with $Q_{2}u$ via Theorem 2.9:

$\displaystyle\left\lVert(1+\varphi_{+}^{\prime\prime})^{1/2}e^{\varphi}\left(\nabla
Q_{2}u,\left\langle
r\right\rangle^{-1}(1+\varphi^{\prime})Q_{2}u\right)\right\rVert_{LE}+\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}\partial_{t}Q_{2}u\right\rVert_{LE}$
$\displaystyle\lesssim\left\lVert
e^{\varphi}P(Q_{2}u)\right\rVert_{LE^{*}}+\delta\left(\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}Q_{2}u\right\rVert_{LE}+\left\lVert\left\langle
r\right\rangle^{-1}(1+\varphi_{+}^{\prime\prime})^{1/2}(1+\varphi^{\prime})e^{\varphi}\partial_{t}Q_{2}u\right\rVert_{LE}\right)$
$\displaystyle\lesssim\left\lVert
e^{\varphi}Pu\right\rVert_{LE^{*}}+\frac{\delta}{\tau_{0}}\left\lVert(1+\varphi^{\prime})^{1/2}e^{\varphi}\partial_{t}Q_{2}u\right\rVert_{LE}+\delta\tau_{1}\left\lVert\left\langle
r\right\rangle^{-1}(1+\varphi_{+}^{\prime\prime})^{1/2}(1+\varphi^{\prime})e^{\varphi}Q_{2}u\right\rVert_{LE}.$

By choosing $\delta$ sufficiently small, last two terms absorb into the left-
hand side. Since $\varphi$ is bounded and $\varphi^{\prime}\geq 0$, we obtain
(2.16) for $j=2.$

Finally, we apply Theorem 1.10 to $Q_{3}u(t-T/2)$:

$\left\lVert
Q_{3}u\right\rVert_{LE^{1}[-T/2,T/2]}\lesssim\left\lVert\partial(Q_{3}u)(-T/2)\right\rVert_{L^{2}}+\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{LE[-T/2,T/2]}+\left\lVert
P(Q_{3}u)\right\rVert_{LE^{*}[-T/2,T/2]}.$

Taking the limit as $T\rightarrow\infty$ and then applying Plancherel’s
theorem give that

$\left\lVert Q_{3}u\right\rVert_{LE^{1}}\lesssim\tau_{1}^{-1}\left\lVert
Q_{3}u\right\rVert_{LE^{1}}+\left\lVert Pu\right\rVert_{LE^{*}}.$

If $\tau_{1}$ is large enough, then the error term on the right absorbs into
the left, giving (2.16) for $j=3.$ ∎

We underscore how important it is that $\delta$ may be chosen arbitrarily in
Theorem 2.9: It allowed for compatibility with the high and low frequency
estimates regardless of how high or low the frequency thresholds became,
respectively (so long as they were away from zero and infinity).

As in Section 7 of [MST20], one proves that Theorem 2.10 implies Theorem 1.11
by fixing $u$ and constructing a function $v$ which matches the Cauchy data of
$u$ at times $0$ and $T$ (and satisfies an appropriate bound) which allows one
to apply (2.15) to $u-v.$ This construction is performing using a partition of
unity on the support of $u[0]$, $u[T]$, and $Pu.$ In particular, one splits
into an interior region $\\{|x|<4R_{0}\\}$ and an exterior region
$\\{|x|>2R_{0}\\}$. The damping is non-problematic in the interior and is
small in the exterior (as with the other lower-order terms). The latter fact
is important since a time reversal argument is used, which makes the damping a
driving force.

In the interior, one utilizes a unit time interval partition of unity
$\\{\chi_{j}\\}$ and analyzes the equations
$Pv_{j}=\chi_{j}f,$ matching the data at times $0$ and $T$ with the first and
last of elements in the partition, respectively. One then generates the
desired approximate of $u$ via $\sum\chi_{j}v_{j}.$ In the exterior region,
one chooses an appropriate small $AF$ perturbation of $\Box$ which matches $P$
in the exterior. If one considers the same differential equation (same data
and forcing) but replaces $P$ by the perturbation, one obtains good bounds via
local energy decay. By truncating the solution appropriately to $|x|>R_{0}$
and $t<T,$ one obtains the desired approximate in the exterior.

###  2.3. An Energy Dichotomy

Here, we apply Theorem 1.11 in order to prove the energy dichotomy present in
Theorem 1.12. Very little deviation is needed from the strategy given in
[MST20], although we must take advantage of the damping in the damping-
dominant case. In particular, we must use the damping to absorb time
derivative error terms for $|x|\leq 2R_{0}$, outside of which we may use
asymptotic flatness (for space derivative terms, one can use Young’s
inequality for products, at the expense of shrinking $\varepsilon$). In the
$\varepsilon$-almost symmetric case, such interior errors are automatically
small.

###### Proof.

Split $P$ and $P_{0}$ into self- and skew-adjoint parts

$P=P^{s}+P^{a},\qquad P_{0}=P_{0}^{s}+P_{0}^{a},$

respectively, and define the energy of the symmetric part of $P$ as

${E}^{s}[\boldsymbol{u}](t)=\int\limits_{\mathbb{R}^{3}}P_{0}^{s}u\bar{u}-g^{00}|\partial_{t}u|^{2}\,dx.$

Then,

$E^{s}[\boldsymbol{u}](t)=E^{s}[\boldsymbol{u}](0)+2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P^{s}u}\,dxds.$

Applying this to

$E[\boldsymbol{u}](t)=\int\limits_{\mathbb{R}^{3}}P_{0}u\bar{u}-g^{00}|\partial_{t}u|^{2}\,dx$

and utilizing the symmetry assumptions yield that

(2.17) $\displaystyle E[\boldsymbol{u}](t)$
$\displaystyle=E^{s}[\boldsymbol{u}](0)+2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P^{s}u}\,dxds+\int\limits_{\mathbb{R}^{3}}P_{0}^{a}u\bar{u}\,dx$
$\displaystyle=E[\boldsymbol{u}](0)-\int\limits_{\mathbb{R}^{3}}P_{0}^{a}u(0)\overline{u(0)}\,dx+2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{f}\,dxds-2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P^{a}u}\,dxds+\int\limits_{\mathbb{R}^{3}}P_{0}^{a}u\bar{u}\,dx$
$\displaystyle=E[\boldsymbol{u}](0)+2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{f}\,dxds-2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P^{a}u}\,dxds+\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}(P_{0}^{a}u\bar{u})\,dx\,ds$
$\displaystyle=E[\boldsymbol{u}](0)+2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{f}\,dxds-2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P^{a}u}\,dxds-2i\operatorname{Im}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{P_{0}^{a}u}\,dx\,ds$
$\displaystyle\leq
E[\boldsymbol{u}](0)+2\operatorname{Re}\int\limits_{0}^{t}\int\limits_{\mathbb{R}^{3}}\partial_{s}u\overline{f}\,dxds-C(\varepsilon,\textbf{c})\int\limits_{0}^{t}\int\limits_{B_{2R_{0}}(0)}\operatorname{Re}a|\partial_{s}u|^{2}\,dxds+D(\varepsilon,\textbf{c})\left\lVert
u\right\rVert_{LE^{1}}^{2}$

where we have used the assumed symmetry assumptions in the last line,
shrinking $\varepsilon$ if necessary. In the above,
$C(\varepsilon,\textbf{c})>0$, and
$D(\varepsilon,\textbf{c})=\mathcal{O}(\varepsilon)$ in the almost-symmetric
case (and $C(\varepsilon,\textbf{c})=2$), whereas
$D(\varepsilon,\textbf{c})=\mathcal{O}(\textbf{c})$ in the damping-dominant
case.

Dropping the damping term from the above and pairing the result with (1.4)
gives that

$\displaystyle\left\lVert\partial
u\right\rVert_{L^{\infty}L^{2}[0,T]}^{2}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert\partial^{\leq
1}u(T)\right\rVert_{L^{2}_{c}}^{2}+D(\varepsilon,\textbf{c})\left\lVert
u\right\rVert_{LE^{1}[0,T]}^{2}+\int\limits_{0}^{T}\int\limits_{\mathbb{R}^{3}}|\partial_{t}u||f|\,dxdt,$

where $\partial^{\leq 1}u:=\sum\limits_{|\alpha|\leq 1}\partial^{\alpha}u.$
The Schwarz inequality and Hölder’s inequality imply that

(2.18) $\displaystyle\left\lVert\partial
u\right\rVert_{L^{\infty}L^{2}[0,T]}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}+\left\lVert\partial^{\leq
1}u(T)\right\rVert_{L^{2}_{c}}+(\delta+D(\varepsilon,\textbf{c}))\left\lVert
u\right\rVert_{LE^{1}[0,T]}+\delta^{-1}\left\lVert
f\right\rVert_{LE^{*}+L^{1}L^{2}[0,T]},$

where $\delta>0$ is arbitrary. Now, we will make use of the two-point local
energy estimate (1.2). By combining (2.18) with (1.2) and choosing $\delta$
sufficiently small (and shrinking $\varepsilon$ if necessary in the almost-
symmetric case), we obtain that

$\left\lVert\partial u(T)\right\rVert_{L^{2}}^{2}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert\partial^{\leq
1}u(T)\right\rVert_{L^{2}_{c}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}.$

On the other hand, (1.2) directly gives that

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

and so

(2.19) $\displaystyle\left\lVert\partial
u\right\rVert_{L^{2}L^{2}[0,T]}^{2}\lesssim\left\lVert\partial
u(T)\right\rVert_{L^{2}}^{2}+(T+1)\left(\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}\right).$

Notice that if we call $E(T)=\left\lVert\partial
u\right\rVert_{L^{2}L^{2}[0,T]}^{2},$ then (2.19) gives that

(2.20) $\displaystyle E^{\prime}(T)=\left\lVert\partial
u(T)\right\rVert_{L^{2}}^{2}\geq\alpha E(T)-(T+1)\left(\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}\right),$

where $\alpha>0$ is a constant.

We will consider two cases. First, we assume that

$E(T)<2\alpha^{-1}(T+1)\left(\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}\right)$

for all $T>0$. In such a case, we have that, in particular,

$T^{-1}\left\lVert\partial
u\right\rVert_{L^{2}L^{2}[0,T]}^{2}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}.$

By the mean-value theorem for integrals, there exists a sequence $(T_{j})$ so
that $T_{j}\rightarrow\infty$ as $j\rightarrow\infty$, and

$\left\lVert\partial
u(T_{j})\right\rVert_{L^{2}}^{2}\lesssim\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}.$

We may use (1.2) and let $T_{j}\rightarrow\infty$ to conclude that local
energy decay holds.

Next, we consider if

(2.21) $\displaystyle E(T^{\prime})\geq
2\alpha^{-1}(T^{\prime}+1)\left(\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}\right)$

for some $T^{\prime}>0$. Then, $E(T)$ must bound (2.21) from above for all
$T\geq T^{\prime}$, since (2.20) implies that $E(T)$ is increasing for all
$T>T^{\prime}.$ By applying an integrating factor argument to the differential
inequality (2.20) and using (2.21), we get that

$E(T)\gtrsim_{T^{\prime}}e^{\alpha T}\left(\left\lVert\partial
u(0)\right\rVert_{L^{2}}^{2}+\left\lVert
f\right\rVert^{2}_{LE^{*}+L^{1}L^{2}[0,\infty)}\right),\qquad T\geq
T^{\prime},\qquad T^{\prime}\gg 1.$

Combining this with (2.19) gives the exponential growth for large enough $T$.
∎

##  3\. Resolvent Theory and Local Energy Decay

In this section, we will introduce the spectral theory required to prove
Theorem 1.14. This follows from the resolvent formalism introduced in [MST20]
and the corresponding scattering theory. We will summarize the relevant parts
of their work here and provide details that were either omitted or are
instructive to repeat. In particular, the results necessary to prove Theorem
1.14 are based on the same frequency estimates present in both [MST20] and
here, so our work requires little deviation. Throughout this section, we
assume that $P$ is stationary.

Consider $Pu=0.$ One arrives at the stationary problem by studying “mode
solutions” of the form $u(t,x)=u_{\omega}(x)e^{i\omega t}$, where
$\omega\in\mathbb{C}$ (equivalently, one replaces $D_{t}$ by $\omega$).
Plugging such $u$ into the given homogeneous equation generates the stationary
equation

$P_{\omega}u_{\omega}=0,\qquad\operatorname{where}\qquad
P_{\omega}=\Delta_{g,A}+W(x,D_{x})+\omega B(x,D_{x})+g^{00}\omega^{2},$

and

$\displaystyle\Delta_{g,A}$ $\displaystyle=(D_{j}+A_{j})g^{jk}(D_{k}+A_{k}),$
$\displaystyle W(x,D_{x})$
$\displaystyle=A_{0}g^{0j}(D_{j}+A_{j})+(D_{j}+A_{j})g^{j0}A_{0}+(A_{0})^{2}g^{00}+V,$
$\displaystyle B(x,D_{x})$
$\displaystyle=g^{0j}(D_{j}+A_{j})+(D_{j}+A_{j})g^{j0}+2A_{0}g^{00}+ia.$

The resolvent $R_{\omega}$ is defined as the inverse of $P_{\omega}$ when such
an inverse exists. More explicitly, if we consider the homogeneous Cauchy
problem

$Pu=0,\qquad u(0)=0,\qquad-g^{00}\partial_{t}u(0)=f,$

then we may formally define $R_{\omega}$ via the Fourier-Laplace transform of
$u$, i.e.

$R_{\omega}f=\int\limits_{0}^{\infty}e^{-i\omega
t}u(t)\,dt=:\mathcal{F}_{t\rightarrow\omega}(\mathbbm{1}_{[0,\infty)}(t)u),\qquad\omega\in\mathbb{C}.$

One may check via formal integration by parts that both definitions of
$R_{\omega}$ are consistent. In this section, we will take $f$ to be in either
$L^{2}$ or $\mathcal{L}\mathcal{E}^{*}$, and it will be clear from context
which is the case. It remains to make rigorous sense of $R_{\omega}$ as a
well-defined bounded operator.

From the global energy bounds and Gronwall’s inequality, it follows that $u$
satisfies the crude estimate

(3.1) $\left\lVert\partial u(t)\right\rVert_{L^{2}}\lesssim e^{ct}\left\lVert
f\right\rVert_{L^{2}},\qquad c\geq 0.$

Using (3.1) and the Minkowski integral inequality, we obtain that

(3.2) $\displaystyle\left\lVert
R_{\omega}f\right\rVert_{\dot{H}^{1}}\leq\int\limits_{0}^{\infty}e^{\operatorname{Im}\omega
t}\left\lVert\nabla
u(t,\cdot)\right\rVert_{L^{2}}\,dt\lesssim\int\limits_{0}^{\infty}e^{\operatorname{Im}\omega
t}e^{ct}\left\lVert
f\right\rVert_{L^{2}}\,dt\lesssim|\operatorname{Im}\omega+c|^{-1}\left\lVert
f\right\rVert_{L^{2}},\qquad\operatorname{Im}\omega+c<0.$

Meanwhile, integrating by parts once provides that $\omega
R_{\omega}f=-i\mathcal{F}_{t\rightarrow\omega}(\mathbbm{1}_{[0,\infty)}\partial_{t}u).$
Taking the $L^{2}$ norm of the above and performing the same work as in (3.2)
yields an identical upper bound. Combining these estimates together gives the
inequality

(3.3) $\displaystyle\left\lVert
R_{\omega}f\right\rVert_{\dot{H}_{\omega}^{1}}\lesssim|\operatorname{Im}\omega+c|^{-1}\left\lVert
f\right\rVert_{L^{2}},\qquad\operatorname{Im}\omega+c<0.$

Hence, we may validly define the resolvent as a bounded operator from $L^{2}$
to $\dot{H}_{\omega}^{1}$, provided that $\omega$ is in the range given in
(3.3). Notice that if the uniform energy bound (1.3) holds, then the resolvent
is holomorphic in the lower half-plane
$\mathcal{H}:=\\{\omega\in\mathbb{C}:\operatorname{Im}\omega<0\\}$ and
satisfies the bound

(3.4) $\displaystyle\left\lVert
R_{\omega}\right\rVert_{L^{2}\rightarrow\dot{H}^{1}_{\omega}}\lesssim|\operatorname{Im}\omega|^{-1},\qquad\omega\in\mathcal{H}.$

If the uniform energy bound does not hold, then one is only guaranteed
meromorphic continuation to $\mathcal{H}.$ Due to this tie-in with uniform
energy bounds, we will refer to (3.4) as the uniform energy resolvent bound.
There is also an analogous resolvent bound to local energy decay, which we
state as a theorem and will not prove here; see the proof of Theorem 2.3 in
[MST20] for more. It is largely a consequence of Plancherel’s theorem, along
with utilization of facts which we will discuss after the statement of the
theorem. The damping plays no meaningful role here.

######  Theorem 3.1.

Local energy decay holds for a stationary damped wave operator $P$ if and only
if $R_{\omega}$ satisfies the local energy resolvent bound

(3.5) $\displaystyle\left\lVert
R_{\omega}\right\rVert_{\mathcal{L}\mathcal{E}^{*}\rightarrow\mathcal{L}\mathcal{E}^{1}_{\omega}}\lesssim
1,\qquad\omega\in\mathcal{H}.$

We observe that if the uniform energy resolvent bound holds, then the local
energy resolvent bound holds for $\operatorname{Im}\omega\lesssim-1$, since

(3.6) $\displaystyle\left\lVert
R_{\omega}f\right\rVert_{\mathcal{LE}_{\omega}^{1}}$
$\displaystyle\lesssim\left\lVert
R_{\omega}f\right\rVert_{\dot{H}^{1}}+|\omega|\left\lVert
R_{\omega}f\right\rVert_{L^{2}}=\left\lVert
R_{\omega}f\right\rVert_{\dot{H}_{\omega}^{1}}\lesssim|\operatorname{Im}\omega|^{-1}\left\lVert
f\right\rVert_{L^{2}}\lesssim\left\lVert f\right\rVert_{\mathcal{LE}^{*}}.$

On the other hand (3.5) implies (3.4); Fredholm theory implies that
$R_{\omega}$ is bounded and holomorphic from $L^{2}$ to
$\dot{H}^{1}_{\omega}$, and the precise operator norm bound can by obtained by
splitting $f\in L^{2}$ as

$f=\chi_{<|\operatorname{Im}\omega|^{-1}}f+\chi_{>|\operatorname{Im}\omega|^{-1}}f,$

then using (3.5) and (3.3). Hence, the uniform energy and local energy
resolvent bounds have close relation. We previously alluded to two spectral
obstructions to local energy decay (and hence local energy resolvent bounds),
which we outline more precisely now.

Elements in the kernel of $P_{\omega}$ living in $L^{2}$ correspond to finite
rank poles $\omega$ of $R_{\omega}$. Given that $P_{\omega}$ is an elliptic
operator, such eigenfunctions live in $H^{s}$ for all $s\in\mathbb{R}$, yet
the corresponding mode solution to $Pu=0$ must possess exponential growth in
time since $\omega\in\mathcal{H}$. In particular, the initial energy is
finite, yet both terms on the lower-bound side of the local energy decay
estimate (1.1) are unbounded as $T\rightarrow\infty$. This behavior violates
both uniform energy bounds and local energy decay. Since the eigenvalues have
negative imaginary part, we will call these negative eigenfunctions.

######  Definition 3.2.

A negative eigenfunction for $P$ is a non-zero $u_{\omega}\in L^{2}$ such that
$P_{\omega}u_{\omega}=0$, with $\omega\in\mathcal{H}.$

The corresponding eigenvalues are isolated and contained within a relatively
compact set in $\mathcal{H}$, which follows due to (3.3) and the high
frequency estimate (1.5).

Next, we notice that the local energy resolvent bound must hold uniformly up
to the real line in order to obtain local energy decay, per Theorem 3.1. To
that end, we have another potential obstruction - a failure to obtain the
continuous extension of $R_{\omega}$ to $\mathbb{R}$. First, we consider when
one takes the limit as the spectral parameter approaches a non-zero real
value.

######  Definition 3.3.

An embedded resonant state for $P$ is a non-zero
$u_{\omega}\in\mathcal{L}\mathcal{E}^{1}_{\omega}$, with
$\omega\in\mathbb{R}\setminus\\{0\\}$ being the corresponding embedded
resonance, satisfying the outgoing/Sommerfeld radiation condition

(3.7) $\left\lVert\left\langle
x\right\rangle^{-1/2}(\partial_{r}+i\omega)u_{\omega}\right\rVert_{L^{2}(A_{j})}\rightarrow
0\qquad\operatorname{as}j\rightarrow\infty$

such that

$P_{\omega}u_{\omega}=0.$

The outgoing radiation condition as stated here is a variant of the standard
one, adapted to the dyadic structure of our spaces. It acts as a boundary
condition at infinity to ensure unique solutions to the above problem for a
fixed $\omega$. Such states are subtle obstructions, since the corresponding
mode solution does not generally have finite energy (as $u_{\omega}$ is only
guaranteed to live in $\mathcal{L}\mathcal{E}^{1}_{\omega}$). However, as
explained in [MST20], one may perform a truncation procedure and utilize
normalized/Regge-Wheeler-type coordinates (see [Tat13]) to produce functions
$\chi_{>1}(t-r)e^{i\omega t}u_{\omega}$ whose energy exhibits a growth rate of
$t^{1/2}$ yet whose image under $P$ lives in $L^{1}L^{2}.$ Hence, they
represent an explicit obstruction to uniform and local energy bounds. The non-
existence of embedded resonant states is equivalent to (3.5) near the
punctured real line and to a limiting absorption principle. In our context,
the corresponding result is as follows.

######  Theorem 3.4.

Let $P$ be a stationary, asymptotically flat damped wave operator which
satisfies the geometric control condition and is either weakly
$\varepsilon$-damping dominant for some $\varepsilon>0$ or
$\varepsilon$-weakly magnetic with $\varepsilon\ll_{R_{0},M_{0}}1$.
Additionally, assume that $\partial_{t}$ is uniformly time-like. For any
$\omega_{0}\in\mathbb{R}\setminus\\{0\\}$, the following are equivalent:

  1. (1)

$\omega_{0}$ is not a resonance.

  2. (2)

The bound

$\left\lVert
u\right\rVert_{\mathcal{L}\mathcal{E}^{1}_{\omega_{0}}}\lesssim\left\lVert
P_{\omega_{0}}u\right\rVert_{\mathcal{L}\mathcal{E}^{*}}$

holds for all $u\in\mathcal{L}\mathcal{E}^{1}_{\omega_{0}}$ satisfying the
outgoing radiation condition (3.7).

  3. (3)

The local energy resolvent bound (3.5) holds uniformly for
$\omega\in\mathcal{H}$ near $\omega_{0}$, and the limit

$R_{\omega_{0}}f=\lim_{\mathcal{H}\ni\omega\rightarrow\omega_{0}}R_{\omega}f,\qquad
f\in\mathcal{L}\mathcal{E}^{*}$

converges strongly on compact sets and satisfies the outgoing radiation
condition (3.7).

We sketch the proof, as it is similar to that given in [MST20] for their
version of the result (which is Proposition 2.5 in their work); only minor
alterations are necessary to account for the damping and the lack of the non-
trapping hypothesis. The result is perturbative of the work in [Kof22], just
as the version in [MST20] was perturbative of the case where $P=\Box_{g}$.

###### Proof.

First, consider when $A,V\equiv 0$ and $a\equiv 0$ for $|x|>2R_{0}.$ In this
scenario, we are in the setting of [Kof22] and obtain full local energy decay.
By Theorem 3.1, we have the local energy resolvent bound (3.5), in which case
all three statements in Theorem 3.4 hold.

In general, let $\widetilde{P}$ denote the principal part of $P$, and consider
$\widetilde{Q}=\widetilde{P}+\chi_{<R_{0}}iaD_{t}$. Note that our starting
case applies to $\widetilde{Q}$, hence it satisfies local energy decay. Let
$\widetilde{R}_{\omega}$ denote the resolvent of $\widetilde{Q}_{\omega}$,
which is holomorphic in $\mathcal{H}$ and continuous up to the real line. We
seek a solution to $P_{\omega}u=f$ of the form $u=\widetilde{R}_{\omega}g$ for
some $g$ (that is, $u$ is in the range of the resolvent and hence outgoing).
If $Q_{\omega}=P_{\omega}-\widetilde{Q}_{\omega}$, then

$P_{\omega}u=f\qquad\text{ if and only if
}\qquad(I+Q_{\omega}\widetilde{R}_{\omega})g=f.$

The family of operators $Q_{\omega}\widetilde{R}_{\omega}$ is compact from
$\mathcal{L}\mathcal{E}^{*}$ to $\mathcal{L}\mathcal{E}^{*}$, holomorphic in
the lower-half plane, and continuous up to the real line. Further,
$I+Q_{\omega}\widetilde{R}_{\omega}$ is invertible for
$-\operatorname{Im}\omega\gg 1$ by a Neumann series argument; to see this,
note that $Q_{\omega}$ is first-order, then use a similar estimate to (3.6).
Thus, $I+Q_{\omega}\widetilde{R}_{\omega}$ is a family of zero-index Fredholm
operators. The conclusion of the theorem follows from the analytic Fredholm
theorem and its consequences. ∎

Next, we consider when the spectral parameter approaches 0. In this case, we
must replace the outgoing radiation condition (which is not meaningful when
$\omega=0$) with a new condition which still limits the asymptotics of the
functions.

######  Definition 3.5.

A zero resonant state for $P$ is a non-zero $u\in\mathcal{L}\mathcal{E}_{0}$
such that $P_{0}u=0$. If, in addition, $u\in L^{2},$ then we call $u$ a zero
eigenfunction.

Such resonant states are annihilated by $P$ while having finite energy.
However, they also possess an unbounded $LE^{1}$ norm as $T\rightarrow\infty$,
which violates local energy decay, as well as two-point local energy decay. As
shown in [MST20] (see Proposition 2.10 in their work), one also has an
analogous characterization of a resonance at zero to that of the non-zero real
resonances in Theorem 3.4. We record their result without proof, for reasons
which immediately follow the statement of the theorem.

######  Theorem 3.6.

Let $P$ be an asymptotically flat damped wave operator satisfying the
geometric control condition and $\partial_{t}$ be uniformly time-like. Then,
the following are equivalent:

  1. (1)

$0$ is not a resonance.

  2. (2)

The zero resolvent bound

(3.8) $\displaystyle\left\lVert
u\right\rVert_{\mathcal{L}\mathcal{E}^{1}}\lesssim\left\lVert
P_{0}u\right\rVert_{\mathcal{L}\mathcal{E}^{*}}$

holds for all $u\in\mathcal{L}\mathcal{E}^{1}_{0}$.

  3. (3)

The zero non-resonance condition (2.12) holds.

  4. (4)

The local energy resolvent bound (3.5) holds uniformly for
$\omega\in\mathcal{H}$ near $0$, and the limit

$R_{0}f=\lim_{\mathcal{H}\ni\omega\rightarrow 0}R_{\omega}f,\qquad
f\in\mathcal{L}\mathcal{E}^{*}$

converges strongly on compact sets.

  5. (5)

The stationary local energy decay estimate

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

with the implicit constant being independent of $T$.

The damping does not arise in $P_{0}$ and simply acts as an absorbable lower-
order term when transitioning back to $P$, so the added context of this paper
has absolutely no effect on the work present in [MST20]. In particular, we
recover all of the same low frequency work, which is what is utilized for this
theorem. Again, we refer the reader to the aforementioned work for more.

Theorem 2.8 gives the implication (3)$\implies(5)$, whereas repeating the
proof of Theorem 2.8 on the resolvent side produces (3)$\implies(4).$ Next, we
remark that the bound present in (3.8) is more directly related to zero not
constituting a resonance than the zero non-resonance condition (2.12).
However, their equivalence can be seen from the fact that $P_{0}$ is an
elliptic $AF$ perturbation of the Laplacian.

Now that we have outlined the obstructions to local energy decay, we proceed
with a proof of Theorem 1.14. By Theorem 3.1, it suffices to establish (3.5).
Once again, we may follow [MST20], as the proof only requires the resolvent
theory and frequency estimates.

###### Proof of Theorem 1.14.

Since there are no negative eigenfunctions, the resolvent is bounded and
holomorphic in $\mathcal{H}$, which implies that

$\left\lVert R_{\omega}f\right\rVert_{\dot{H}^{1}_{\omega}}\leq
C(\operatorname{Im}\omega)\left\lVert
f\right\rVert_{\mathcal{L}\mathcal{E}^{*}},\qquad\omega\in\mathcal{H},$

for some constant $C(\operatorname{Im}\omega)$ depending on
$\operatorname{Im}\omega$ which is bounded away from the real line. This is
not a uniform bound and a priori may become unbounded as
$\operatorname{Im}\omega\rightarrow 0$, but it does holds up to any
neighborhood of $\mathbb{R}$. Similar to the work in (3.6), this implies the
local energy resolvent bound (3.5) away from any neighborhood of the real
line. It remains to establish (3.5) within a strip in the lower half-plane
sufficiently close to $\mathbb{R}$. We break this strip into three exhaustive
cases. Case I: $|\omega|\gg 1$ This is in the high frequency regime,
motivating us to use the high frequency estimate (1.5), which is applicable by
hypothesis. Let $u$ solve $P_{\omega}u=f,$ and call $v=e^{i\omega t}u.$ Then,
$v$ solves $Pv=g$, where $g=e^{i\omega t}f.$ We will apply the high frequency
estimate to the interval $[-T,0]$. More precisely, if we call
$\tilde{v}(t,x)=v(t-T,x)$, then this solves $P\tilde{v}=\tilde{g},$ where
$\tilde{g}(t,x)=g(t-T,x).$ Applying Theorem 1.10 to $\tilde{v}$ provides that

$\left\lVert\tilde{v}\right\rVert_{LE^{1}[0,T]}+\left\lVert\partial\tilde{v}\right\rVert_{L^{\infty}L^{2}[0,T]}\lesssim\left\lVert\partial\tilde{v}(0)\right\rVert_{L^{2}}+\left\lVert\left\langle
x\right\rangle^{-2}\tilde{v}\right\rVert_{LE[0,T]}+\left\lVert\tilde{g}\right\rVert_{LE^{*}+L^{1}L^{2}[0,T]}.$

We immediately calculate that

$\displaystyle\left\lVert\tilde{v}\right\rVert_{LE^{1}[0,T]}$
$\displaystyle=\left(\frac{e^{2T\operatorname{Im}\omega}-1}{2\operatorname{Im}\omega}\right)^{1/2}\left\lVert
u\right\rVert_{\mathcal{LE}^{1}_{\omega}}$
$\displaystyle\left\lVert\partial\tilde{v}\right\rVert_{L^{\infty}L^{2}[0,T]}$
$\displaystyle=\left\lVert u\right\rVert_{\dot{H}^{1}_{\omega}}$
$\displaystyle\left\lVert\partial\tilde{v}(0)\right\rVert_{L^{2}}$
$\displaystyle=e^{T\operatorname{Im}\omega}\left\lVert
u\right\rVert_{\dot{H}^{1}_{\omega}}$ $\displaystyle\left\lVert\left\langle
x\right\rangle^{-2}\tilde{v}\right\rVert_{LE[0,T]}$
$\displaystyle=\left(\frac{e^{2T\operatorname{Im}\omega}-1}{2\operatorname{Im}\omega}\right)^{1/2}\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{\mathcal{LE}}$
$\displaystyle\left\lVert\tilde{g}\right\rVert_{LE^{*}+L^{1}L^{2}[0,T]}$
$\displaystyle=\left\lVert
f\right\rVert_{\left(\frac{\exp{(2T\operatorname{Im}\omega)}-1}{2\operatorname{Im}\omega}\right)^{-1/2}\mathcal{LE^{*}}+\left(\frac{\exp{(T\operatorname{Im}\omega)}-1}{\operatorname{Im}\omega}\right)^{-1}{L^{2}}}.$

Plugging these calculations into the high frequency bound for $\tilde{v}$ and
taking the limit as $T\rightarrow\infty$ yields

$\left\lVert
u\right\rVert_{\mathcal{LE}^{1}_{\omega}}+|\operatorname{Im}\omega|^{1/2}\left\lVert
u\right\rVert_{\dot{H}^{1}_{\omega}}\lesssim\left\lVert\left\langle
x\right\rangle^{-2}u\right\rVert_{\mathcal{LE}}+\left\lVert
f\right\rVert_{\mathcal{LE}^{*}+|\operatorname{Im}\omega|^{1/2}L^{2}}.$

For sufficiently large $\omega$, the first term on the right absorbs into the
first term on the left, which implies (3.5). Case II: $|\omega|\ll 1$ Since
zero is not a resonance, Theorem 3.6 implies that (3.5) holds for all $\omega$
sufficiently close to 0. Case III: $|\omega|\approx 1$ Since there are no real
resonances, Theorem 3.4 implies that (3.5) holds for all $\omega$ in this
region. Hence, (3.5) holds. ∎

## References