$X_{\Phi(u)}^{1}(z_{1})=\int_{M}\mathcal{X}_{1}(z_{1},y)u(y,\eta)\mathrm{d}{\mathrm{L}}(y,\eta),$

where $L$ is the Liouville measure on $M$. Hence, the semi-norms of the
corresponding symbols $b_{1}(z_{1};\zeta_{1},u)$ and $\Phi_{1}(u)(z_{1})$ can
be controlled in terms of certain integrals of $u$ over $M$.

####  4.6.2. Proof of Proposition 4.5

Recall that we aim at estimating

$\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{-\tau}^{*}\mathbf{P}\\{\Phi,\chi
u\\}\right\|_{L^{2}},$

which, thanks to (4.17), can be rewritten as

$\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{-\tau}^{*}\mathbf{P}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi
u\right\|_{L^{2}},$

where $a_{\Phi}$ (resp. $Y_{\Phi}$) is a smooth function (resp. vector field)
on $M_{1}$. Recall from (4.19) and (4.20) that their $\mathscr{C}^{N}$ norms
are controlled by the $\mathscr{C}^{N+1}$ norm of $\Phi$.

We now fix $\chi_{1}$ which is compactly supported in $I$ and which is
identically equal to $1$ on the support of $\chi$. Thanks to (4.14), one can
then write, for every $N^{\prime}\geqslant 1$,

$\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{-\tau}^{*}\mathbf{P}\\{\Phi,\chi
u\\}\right\|_{L^{2}}\leqslant
C_{N^{\prime}}\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\mathbf{P}\varphi_{-\tau}^{*}\chi_{1}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\right\|_{H^{N^{\prime}}\rightarrow
L^{2}}\|u\|_{L^{2}}\\\
+\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{-\tau}^{*}\mathbf{P}\chi_{1}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\mathbf{Op}\left(b_{N^{\prime}}e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}\right)\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}\chi
u\right\|_{L^{2}}.$

Thanks to Lemmas 4.9, 4.13 and 4.14, there exists and $\vartheta_{0}>0$ (that
is independent of the choice of the interval $I$) such that, for every
$\mathfrak{S}_{0}\geqslant 3$, $m_{1}\geqslant 2$, $N_{1}\geqslant
10\mathfrak{S}_{0}+m_{1}+4$ and $-2\leqslant\sigma_{1}\leqslant 0$, one can
find $N^{\prime},N\gg 1$ and $C>0$ such that

$\displaystyle\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\mathbf{P}\varphi_{-\tau}^{*}\chi_{1}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\right\|_{H^{N^{\prime}}\rightarrow
L^{2}}$
$\displaystyle\qquad\qquad\leqslant\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\mathbf{P}\varphi_{-\tau}^{*}\chi_{1}\mathbf{B}_{-\mathfrak{S}_{0},-\sigma_{1}}^{(-m_{1}+2)}\|_{L^{2}\rightarrow
L^{2}}\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1}-2)}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\right\|_{H^{N^{\prime}}\rightarrow
L^{2}}$ $\displaystyle\qquad\qquad\leqslant Ce^{-\vartheta_{0}\tau\min
I}\|\Phi\|_{\mathscr{C}^{N}}.$

Hence, we are left with estimating the term

$\displaystyle\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{-\tau}^{*}\mathbf{P}\chi_{1}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\mathbf{Op}\left(b_{N^{\prime}}e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}\right)\right\|_{L^{2}\rightarrow
L^{2}}$
$\displaystyle\qquad\leqslant\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{-\tau}^{*}\mathbf{P}\chi_{1}\mathbf{B}_{-\mathfrak{S}_{0},-\sigma_{1}}^{(-m_{1}+2)}\right\|_{L^{2}\rightarrow
L^{2}}\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1}-2)}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\mathbf{B}_{-\mathfrak{S}_{0},-\sigma_{1}-1}^{(-m_{1}+5)}\right\|_{L^{2}\rightarrow
L^{2}}$
$\displaystyle\qquad\qquad\times\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}+1}^{(m_{1}-5)}\mathbf{Op}\left(b_{N^{\prime}}e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}\right)\right\|_{L^{2}\rightarrow
L^{2}}.$

We can therefore again apply Lemmas 4.9, 4.13 and 4.14 in addition with Lemma
4.16. This yields the existence of $\vartheta_{0}>0$ (that is still
independent of the choice of the interval $I$) such that, for every
$\mathfrak{S}_{0}\geqslant 3$, $m_{1}\geqslant 50\mathfrak{S}_{0}$,
$N_{1}\geqslant 50\mathfrak{S}_{0}+m_{1}$, and
$\sigma_{1}+1<\sigma_{0}\leqslant 0$ with $-2\leqslant\sigma_{1}\leqslant-1$,
one can find $N,C>0$ such that

$\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1},N_{1})}\varphi_{\tau}^{*}\mathbf{P}\chi_{1}\left(a_{\Phi}\partial_{r}+r^{-1}Y_{\Phi}\right)\chi_{1}\mathbf{Op}\left(b_{N^{\prime}}e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}\right)\right\|_{L^{2}\rightarrow
L^{2}}\leqslant Ce^{-\vartheta_{0}\tau\min I}\|\Phi\|_{\mathscr{C}^{N}}.$

The proof of Proposition 4.5 is finally complete.

##  5\. Global existence

The goal of this short section is to prove the following result.

######  Theorem 5.1.

Let $N_{0}\in\mathbb{N}$. Let $u_{0}\in\mathscr{C}_{c}^{N_{0}}(M).$ Then there
exists a unique
$u\in\mathscr{C}^{1}\left(\mathbb{R},L^{1}(M,\mathrm{d}{\mathrm{L}})\right)$
satisfying (1.1) on $\mathbb{R}$. In fact
$u\in\mathscr{C}^{k}\left(\mathbb{R},\mathscr{C}^{N_{0}-k}_{c}(M)\right)$ for
all $k=0,\ldots,N_{0}$.

All along this section, we will use the Liouville measure
$\mathrm{d}{\mathrm{L}}(r,z_{1})=r^{n-1}\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}(z_{1})$
rather than $\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}(z_{1})$. We start with a
local existence result.

######  Lemma 5.2.

Let $N_{0}\in\mathbb{N}$. Let $u_{0}\in\mathscr{C}_{c}^{N_{0}}(M).$ Then there
exist $\varepsilon>0$ and a unique
$u\in\mathscr{C}^{1}\left([-\varepsilon,\varepsilon],L^{1}(M,\mathrm{d}{\mathrm{L}})\right)$
satisfying (1.1) on $[-\varepsilon,\varepsilon]$. In fact
$u\in\mathscr{C}^{k}\left([-\varepsilon,\varepsilon],\mathscr{C}^{N_{0}-k}_{c}(M)\right)$
for all $k=0,\ldots,N_{0}$. Moreover, $\varepsilon$ depends only on
$\|u_{0}\|_{L^{1}(M)}$, $\|\nabla u_{0}\|_{L^{\infty}(M)}$ and on the support
of $u_{0}$.

###### Proof.

The proof is based on a classical argument involving the Banach theorem. Note
that $u\in\mathscr{C}^{1}\left([-\varepsilon,\varepsilon],L^{1}(M)\right)$
satisfies (1.1) if and only if one has

(5.1)
$u(t)=\varphi_{-t}^{*}u_{0}-\int_{0}^{t}\varphi_{s-t}^{*}\\{\Phi(u(s)),u(s)\\}~\mathrm{d}s,\quad|t|\leqslant\varepsilon.$

If $v\in\mathscr{C}^{0}([-\varepsilon,\varepsilon],L^{1}(M))$, we let
$u=\Psi(v)\in\mathscr{C}^{1}([-\varepsilon,\varepsilon],L^{1}(M))$ be the
unique solution of the equation

$\partial_{t}u+(X+X_{\Phi(v)})u=0,\quad u|_{t=0}=u_{0}.$

where $X_{\Phi(v)}\in\mathscr{C}^{\infty}(M,TM)$ is the Hamiltonian vector
field generated by $\Phi(v)$, that is $X_{\Phi(v)}f=\\{\Phi(v),f\\}$ for
$f\in\mathscr{C}^{1}(M)$. We shall explain in a few lines why it is indeed
defined on $[-\varepsilon,\varepsilon]$. Equivalently, one can write

$\Psi(v)(t)=u_{0}\circ\psi^{v}_{t,0}$

where $\psi^{v}_{t,s}:M\to M$ is the non-autonomous flow associated to the
vector field $X+X_{\Phi(v)}(t)$, defined by

(5.2)
$\partial_{t}\psi^{v}_{t,s}(z)+[X+X_{\Phi(v)(t)}](\psi^{v}_{t,s}(z))=0,\quad\psi^{v}_{s,s}(z)=z,\quad
z\in M.$

Recall from the Cauchy–Lipschitz Theorem (see [Har82, Theorem V.4.1] for
example) that the map $(t,z)\in[-\varepsilon,\varepsilon]\times
M\mapsto\psi_{t,0}^{v}(z)\in M$ is smooth with respect to the $z$-variable and
of class $\mathscr{C}^{1}$ with respect to the time variable (as $\Phi(v)$ is
$\mathscr{C}^{\infty}$). In (4.17), we already gave an exact expression of
this vector field. In particular, given an orbit $(r(t),z_{1}(t))$ of this
vector field, one has

$r(t)=r(0)+\int_{0}^{t}\Phi_{1}(v)(s)\,\mathrm{d}s,$

so that every orbit remains in a compact set of $M$. In particular, if $u_{0}$
is supported in the region $\\{|\xi|\leqslant R_{0}\\}$, then the solution $u$
is indeed defined on $[-\varepsilon,\varepsilon]$ and supported in
$\\{|\xi|\leqslant R_{0}+C_{K}\varepsilon|v|_{L^{1}(M)}\\}$, where $C_{K}>0$
depends only on the interaction kernel $K$.

We want to find a fixed point of $\Psi$. Let $B$ be the ball of radius
$|u_{0}|_{L^{1}}$ in the space
$\mathscr{C}^{0}([-\varepsilon,\varepsilon],L^{1}(M))$, for the norm

$\|v\|_{1}=\sup_{|t|\leqslant\varepsilon}\|v(t)\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}.$

Note that $\Psi:B\to B$. Let $v_{1},v_{2}\in B$ and set $u_{j}=\Psi(v_{j})$.
We write

$\partial_{t}(u_{2}-u_{1})=-\left(X+X_{\Phi(v_{2})}\right)(u_{2}-u_{1})+X_{\Phi(v_{2})-\Phi(v_{1})}u_{1},$

from which we infer that

$u_{2}(t)-u_{1}(t)=\int_{0}^{t}(\psi_{t,s}^{v_{2}})^{*}X_{\Phi(v_{2})(s)-\Phi(v_{1})(s)}u_{1}(s)\mathrm{d}s.$

As $\psi_{t,s}^{v_{2}}$ is volume preserving, one finds that

$\displaystyle\left\|u_{2}(t)-u_{1}(t)\right\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}$
$\displaystyle\leqslant\int_{0}^{t}\left|(\psi_{t,s}^{v_{2}})^{*}X_{\Phi(v_{2}-v_{1})}u_{2}(s)\right|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}\mathrm{d}s$
$\displaystyle\leqslant\varepsilon\sup_{|t|\leqslant\varepsilon}\|X_{\Phi(v_{2}-v_{1})}(t)\|_{\infty}\|u_{1}(t)\|_{\mathscr{C}^{1}(M)}.$

Here the norm on $\mathscr{C}^{1}(M)$ is taken with respect to any smooth
metric on $M$. One has $\|X_{\Phi(v_{2}-v_{1})}\|_{\infty}\leqslant
C\|\Phi(v_{2}-v_{1})\|_{\mathscr{C}^{1}}\leqslant\widetilde{C}\|v_{2}-v_{1}\|_{1}$,
and

$\|u_{1}\|_{\mathscr{C}^{1}(M)}=\sup_{|t|\leqslant\varepsilon}\|u_{0}\circ\psi^{v_{1}}_{t,0}\|_{\mathscr{C}^{1}(M)}\leqslant\|\nabla
u_{0}\|_{\infty}\sup_{|t|\leqslant\varepsilon}|\mathrm{d}\psi_{t,0}^{v_{1}}|_{L^{\infty}(Q)}$

where $Q=\\{|\xi|\leqslant R_{0}+C_{K}\|u_{0}\|_{L^{1}}\varepsilon\\}$. Now,
by differentiating (5.2), we find that $\mathrm{d}\psi_{t,s}^{v_{1}}$ solves a
linear differential equation, namely

$\partial_{t}\mathrm{d}\psi_{t,s}^{v_{1}}(z)=\mathrm{d}\left(X+X_{\Phi(v_{1})(t)}\right)\left(\psi_{t,s}^{v_{1}}(z)\right)\mathrm{d}\psi_{t,s}^{v_{2}}(z),\quad\mathrm{d}\psi_{s,s}^{v_{2}}(z)=\text{Id}.$

Applying Gronwall’s Lemma, one finds that
$|\mathrm{d}\psi_{t,0}^{v_{1}}|_{L^{\infty}(Q)}$ is bounded by

$Ce^{C\varepsilon(R_{0}+C_{K}(\varepsilon+1)|v|_{L^{1}})}\leqslant
Ce^{\widetilde{C}\varepsilon(R_{0}+|u_{0}|_{L^{1}})},$

where $C,\widetilde{C}>0$ are constants that depend only on the interaction
kernel. This implies the existence of a constant $C_{\mathrm{g},K}$ (depending
only on the interaction kernel $K$ and on the metric $\mathrm{g}$) such that,
for all $|t|\leqslant\varepsilon$,

$\|u_{2}(t)-u_{1}(t)\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}\leqslant
C_{\mathrm{g},K}\varepsilon e^{\varepsilon
C_{\mathrm{g},K}(R_{0}+\|u_{0}\|_{L^{1}})}\|\nabla
u_{0}\|_{L^{\infty}}\|v_{2}-v_{1}\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}.$

In particular, $\Psi$ is a contraction as soon as $\varepsilon$ is small
enough. Now the Banach theorem yields the existence of a unique fixed point
$u\in B$ of $\Psi$. Next, we write that

(5.3) $u(t)=u_{0}\circ\psi^{{\mathrm{u}}}_{t,0},\quad|t|\leqslant\varepsilon.$

We already saw that $(t,z)\in[-\varepsilon,\varepsilon]\times
M\mapsto\psi^{{\mathrm{u}}}_{t,0}(z)\in M$ is smooth with respect to $z$,
$\mathscr{C}^{1}$ with respect to $t$ and proper (so that $u(t)$ has compact
support). Applying (5.3) inductively, one obtains the expected regularity of
$u$. ∎

On the other hand, exploiting the Hamiltonian structure of the Vlasov
equation, we have

######  Lemma 5.3.

Let $J$ be an interval containing $0$. Let $u$ be a solution to (1.1) on $J$.
Then for all $p\geqslant 1$ and all $t\in J$,

$\|u(t)\|_{L^{p}(M,\mathrm{d}{\mathrm{L}})}=\|u_{0}\|_{L^{p}(M,\mathrm{d}{\mathrm{L}})},$

where the $L^{p}$ norm is taken with respect to the Liouville measure.

Finally let us proceed with the proof of Theorem 5.1.

###### Proof.

Let $T_{\mathrm{max}}$ be the maximal (forward) time of existence associated
to the initial condition $u_{0}$, that is

$T_{\mathrm{max}}=\sup\left\\{T>0,\exists
u\in\mathscr{C}^{1}([0,T],L^{1}(M,\mathrm{d}{\mathrm{L}}))\text{ solution to
}\eqref{eq:vlasov}\text{ with initial condition }u_{0}\right\\}.$

Thanks to Lemma 5.2, $T_{\mathrm{max}}>0$. If $T_{\mathrm{max}}=+\infty$, then
we may conclude as in the end of the proof of Lemma 5.2. Let us show that the
case $T_{\mathrm{max}}<+\infty$ is impossible. Would it be the case, we could
proceed as in the proof of Lemma 5.2 to control $\|\nabla
u(t)\|_{L^{\infty}(M)}$ in the following way. First, using Lemma 5.3 with
$p=1$, if $u_{0}$ is supported in $\\{|\xi|\leqslant R_{0}\\}$, then $u(t)$ is
supported in $Q_{\mathrm{max}}=\\{|\xi|\leqslant
R_{0}+C_{K}T_{\mathrm{max}}\|u_{0}\|_{L^{1}}\\}$. Then, we can write

$\|\nabla
u(t)\|_{L^{\infty}}=\|\nabla\left(u_{0}\circ\psi^{u}_{t,0}\right)\|_{L^{\infty}}\leqslant\|\nabla
u_{0}\|_{L^{\infty}}\|\mathrm{d}\psi_{t,0}^{u}\|_{L^{\infty}(Q_{\mathrm{max}})}.$

One thus finds that $|\mathrm{d}\psi_{t,0}^{u}|_{L^{\infty}}$ is bounded by
$Ce^{CT_{\mathrm{max}}(R_{0}+(1+T_{\mathrm{max}})|u_{0}|_{L^{1}})}$, where
$C>0$ is a constant that depends only on the interaction kernel $K$ and on the
metric $\mathrm{g}$. In particular, there exists a constant $C_{\mathrm{max}}$
with a dependence with respect to the solution only through
$\|u_{0}\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}$, $\|\nabla
u_{0}\|_{L^{\infty}(M)}$, the support of $u_{0}$ and $T_{\mathrm{max}}$, such
that for all $t\in[0,T_{\mathrm{max}})$, we have $\|\nabla
u(t)\|_{L^{\infty}(M)}\leqslant C_{\mathrm{max}}$.

Now let $\eta>0$ to be fixed later and consider the Cauchy problem

(5.4)
$\partial_{t}\widetilde{u}+\left\\{\mathrm{H}+\Phi(\widetilde{u}),\,\widetilde{u}\right\\}=0,\quad\widetilde{u}|_{t=T_{\mathrm{max}}-\eta}=u(T_{\mathrm{max}}-\eta).$

Owing to Lemma 5.2 (using the invariance by translation in time of the
equation), there exists $\varepsilon>0$ depending only on
$\|u(T_{\mathrm{max}}-\eta)\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}$, $\|\nabla
u(T_{\mathrm{max}}-\eta)\|_{L^{\infty}(M)}$, and the support of
$u(T_{\mathrm{max}}-\eta)$, and therefore only on
$\|u_{0}\|_{L^{1}(M,\mathrm{d}{\mathrm{L}})}$, $\|\nabla
u_{0}\|_{L^{\infty}(M)}$, the support of $u_{0}$ and $T_{\mathrm{max}}$, such
that (5.4) is uniquely solvable on
$[T_{\mathrm{max}}-\eta-\varepsilon,T_{\mathrm{max}}-\eta+\varepsilon]$. We
can therefore pick $\eta=\varepsilon/2$, which shows by a uniqueness argument
that the solution to (1.1) can be continued beyond $T_{\mathrm{max}}$. It
leads to the expected contradiction. We argue identically for the backward
times of existence, which concludes the proof of the theorem. ∎

##  6\. Nonlinear exponential stability

This final Section is dedicated to the proof of Theorem 1.1. To this purpose,
we argue with a bootstrap argument, which we initiate in Subsection 6.1.
Loosely speaking, the idea is to work on the largest interval of nonnegative
times $\mathcal{J}_{k}(\varepsilon)$ on which the potential $\Phi(u)$ decays
exponentially fast for a well-chosen rate (with a control by a small enough
constant $\varepsilon$), with the aim to show that
$\mathcal{J}_{k}(\varepsilon)$ is actually $\mathbb{R}_{+}$: that this is the
case corresponds precisely to the main result of this Section, namely Theorem
6.1. Theorem 1.1 then follows.

Subsection 6.2 corresponds to a preliminary step in which we show that on
$\mathcal{J}_{k}(\varepsilon)$, the support of the solution to the Vlasov
equation (1.1) remains far from the null section. In Subsection 6.3, we reduce
the proof of Theorem 1.1 to two statements, Lemmas 6.4 and 6.5. Lemma 6.4
states that a refined control of $\Phi(u)$ (with exponential decay) can be
obtained on $\mathcal{J}_{k}(\varepsilon)$, modulo the uniform control of a
sliced anisotropic Sobolev norm. This uniform control correspond precisely to
Lemma 6.5. The proof of Lemma 6.4 is performed in Subsection 6.4 and relies
crucially on many results from Section 4. Subsection 6.5 is then about the
proof of Lemma 6.5. We again reduce it to the proof of two other statements,
Lemmas 6.6 and 6.7. Lemma 6.6 is a refined estimate of the sliced anisotropic
Sobolev norm, which shows that it is sufficient to prove an estimate of a
global anisotropic Sobolev norm with a polynomial growth in time. To prove
Lemma 6.6, we use crucially the bilinear estimate of Proposition 4.5 from
Section 4. This polynomial growth is precisely the purpose of Lemma 6.7.
Everything therefore finally boils down to an estimate of the global
anisotropic Sobolev norm, that is associated with our global escape function.
This is obtained thanks to a microlocal energy estimate for the Vlasov
equation (1.1), which we perform in Subsection 6.7. We conclude with
Subsection 6.8 in which we finally prove the weak convergence result (with
exponential speed) of Theorem 1.1.

###  6.1. The bootstrap argument

We are now in position to prove Theorem 1.1. Let $u_{0}$ be a smooth function
in $\mathscr{C}^{k_{0}}_{c}(\mathbb{R}_{+}^{*}\times M_{1})$ (with $k_{0}\gg
1$) and denote by $u(t,r,z_{1})$ the solution to (1.1) given by Theorem 5.1.
Let $r_{0}\in\left]0,1\right[$ be such that

$\operatorname{supp}u_{0}\subset\\{(x,v)\in\mathrm{T}^{*}\Sigma\leavevmode\nobreak\
:|v|\in[r_{0},r_{0}^{-1}]\\}.$

Let $0<\varepsilon\leqslant 1$ be a (small) parameter to be fixed later on and
take also $\vartheta=\vartheta_{0}r_{0}/4$, where $\vartheta_{0}>0$ is the
rate of convergence appearing in Section 4, namely the one given by
Proposition 4.5, Lemma 4.9, Corollaries 4.11, and 4.12, and (4.8). This
parameter $\vartheta$ is fixed once and for all. For every $k\geqslant 1$, we
introduce the following subinterval of $\mathbb{R}_{+}$:

(6.1) $\mathcal{J}_{k}(\varepsilon,u_{0})=\Bigg{\\{}T\geqslant 0:\ \forall
t\in[0,T],\ e^{\vartheta
t}\|\Phi(u(t))\|_{\mathscr{C}^{k}(\Sigma)}\leqslant\varepsilon\Bigg{\\}}.$

Imposing $\|u_{0}\|_{L^{1}(M)}$ small enough with respect to $\varepsilon$ (in
terms of $\|K\|_{\mathscr{C}^{k}}$), this defines by continuity a nonempty
interval which is closed by continuity of the solution with respect to the
time variable $t$. Our goal is to show that this interval is also open for
appropriate choices of the parameters $\varepsilon$ and $k$. This will imply
that this interval is $\mathbb{R}_{+}$. We will then be able to complete the
proof of Theorem 1.1 in Section 6.8. Hence, the main statement of this Section
reads:

######  Theorem 6.1.

There exist $k,k_{0}\geqslant 0$ such that, for all $r_{0}\in(0,1)$, one can
find $\varepsilon\in(0,1)$ and $c_{0}>0$ such that, for all
$u_{0}\in\mathscr{C}^{k_{0}}_{c}(M^{\times})$ that is supported in
$[r_{0},r_{0}^{-1}]\times M_{1}$ and satisfying
$\|u_{0}\|_{\mathscr{C}^{k_{0}}}\leqslant c_{0}\varepsilon$, one has

$\mathcal{J}_{k}(\varepsilon,u_{0})=\mathbb{R}_{+},$

where $\mathcal{J}_{k}(\varepsilon,u_{0})$ is defined by (6.1).

In order to alleviate notations, we will drop the dependence in $u_{0}$ and
write $\mathcal{J}_{k}(\varepsilon)$ instead of
$\mathcal{J}_{k}(\varepsilon,u_{0})$.

###  6.2. Uniform control of the support

We begin our proof with the following observation.

######  Lemma 6.2.

There exists a constant $c_{K,\mathrm{g}}>0$ depending only on the metric
$\mathrm{g}$ and on the interaction kernel $K$ such that, if
$\varepsilon<c_{K,\mathrm{g}}r_{0}\vartheta$, then, for every
$t\in\mathcal{J}_{k}(\varepsilon)$,

(6.2)
$\operatorname{supp}u(t)\subset\\{(x,v)\in\mathrm{T}^{*}\Sigma\leavevmode\nobreak\
:\leavevmode\nobreak\ ~|v|\in[r_{0}/2,2r_{0}]\\}.$

######  Remark 6.3.

In the proof of Theorem 6.1, $\varepsilon$ will first of all be chosen small
enough so that Lemma 6.2 holds; another smallness constraint of similar nature
will also appear in the subsequent Lemma 6.5.

###### Proof.

Let $T$ be an element of $\mathcal{J}_{k}(\varepsilon)$. Denote by
$\psi_{t,s}:M\to M$ the flow associated to the vector field
$X+X_{\Phi(u)}(t)$, defined by

$\partial_{t}\psi_{t,s}(z)=[X+X_{\Phi(u)}(t)](\psi_{t,s}(z)),\quad\psi_{s,s}(z)=z,\quad
z\in M.$

Recall that $u(t)$ is the solution to (1.1) with initial data $u_{0}$
compactly supported in $[r_{0},r_{0}^{-1}]\times M_{1}$. Thanks to (4.23), one
has, for every $t\in[0,T]$ and for every initial data $z=(r(0),z_{1}(0))$
lying in $[r_{0},r_{0}^{-1}]\times M_{1}$, one has

$r(t)=r(0)+\int_{0}^{t}\Phi_{1}(u(s))(z_{1}(s))\mathrm{d}s,$

where we set $(r(t),z_{1}(t))=\psi_{t,0}(z)$. According to the expression of
$\Phi_{1}$, one gets that there exists a constant depending only the
interaction kernel $K$ and the metric $\mathrm{g}$ such that, for every
$t\in[0,T]$,

$|r(t)-r(0)|\leqslant
C_{K,g}\int_{0}^{t}\|\Phi(u(s))\|_{\mathscr{C}^{1}}ds\leqslant\varepsilon
C_{K,\mathrm{g}}\vartheta^{-1},$

where we use that $t\in\mathcal{J}_{k}(\varepsilon)$ in the last inequality.
Hence, if we pick
$\displaystyle\varepsilon<\frac{r_{0}\vartheta}{2C_{K,\mathrm{g}}}$, we get
the expected conclusion. ∎

###  6.3. Proof of Theorem 6.1

We now explain how to prove Theorem 6.1. We start with the following key
statement. Let $\mathbf{B}_{3,-2}^{(m_{1},N_{1})}$ be the sliced anisotropic
operator defined in (4.4) with the parameters $(m_{1},N_{1})$ chosen so that
Proposition 4.5, Lemma 4.9 and Corollaries 4.11 and 4.15 hold true.

######  Lemma 6.4.

There exist $k,k_{0}\geqslant 1$ and $C_{K,r_{0}}>0$ (independent of
$\varepsilon$ and $u_{0}$) such that, for all
$t\in\mathcal{J}_{k}(\varepsilon)$,

$\|\Phi(u(t))\|_{\mathscr{C}^{k}}\leqslant
C_{K,r_{0}}\|u_{0}\|_{\mathscr{C}^{k_{0}}}e^{-\vartheta t}\\\
+C_{K,r_{0}}\int_{0}^{t}e^{-\frac{r_{0}\vartheta_{0}}{2}(t-s)}\|\Phi(u(s))\|_{\mathscr{C}^{k}(\Sigma)}\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(s))\|_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s.$

Observe that a control of
$\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(s))\|_{L^{2}(\mathbb{R}\times
M_{1})}$ for all $s\in\mathcal{J}_{k}(\varepsilon)$ is needed for Lemma 6.4 to
be relevant. This is the purpose of the next statement.

######  Lemma 6.5.

There exist a constant $\widetilde{c}_{K,\mathrm{g}}>0$ depending only on
$\mathrm{g}$ and $K$, as well as $k_{0}\in\mathbb{N}$ and $C>0$ (independent
of $\varepsilon$ and $u_{0}$) such that, for
$\varepsilon<\widetilde{c}_{K,\mathrm{g}}r_{0}\vartheta$, for all
$s\in\mathcal{J}_{k}(\varepsilon)$,

$\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(s))\|_{L^{2}(\mathbb{R}\times
M_{1})}\leqslant C\|u_{0}\|_{\mathscr{C}^{k_{0}}}.$

The reason for taking $\varepsilon$ small enough will come from the subsequent
Lemma 6.7. Hence, recalling that $\vartheta=\vartheta_{0}r_{0}/4$ and up to
increasing the value of $k_{0}$ (for the regularity of the initial data), one
gets that, for every $t\in\mathcal{J}_{k}(\varepsilon)$, for every $u_{0}$
(supported in $[r_{0},r_{0}^{-1}]\times M_{1}$) with
$\|u_{0}\|_{\mathscr{C}^{k_{0}}}$ small enough,

(6.3)
$C_{K,r_{0}}\int_{0}^{t}e^{-\frac{r_{0}\vartheta_{0}}{2}(t-s)}\|\Phi(u(s))\|_{\mathscr{C}^{k}(\Sigma)}\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(s))\|_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s\leqslant\frac{\varepsilon}{4}e^{-\vartheta t}.$

Gathering Lemmas 6.4 and 6.5, we find that
$\|\Phi(u(t))\|_{\mathscr{C}^{k}(\Sigma)}\leqslant\frac{\varepsilon}{2}e^{-\vartheta
t}$ from which we infer that $\mathcal{J}_{k}(\varepsilon)$ is open. As
already explained, this allows to conclude the proof of Theorem 6.1.

We are therefore left with the proofs of Lemmas 6.4 and 6.5, which is
precisely the purpose of the next two subsections.

###  6.4. Proof of Lemma 6.4

In this section, we prove Lemma 6.4. We first use Duhamel formula to write

(6.4)
$u(t)=\varphi_{-t}^{*}(u_{0})-\int_{0}^{t}\varphi_{s-t}^{*}\\{\Phi(u(s)),u(s)\\}\mathrm{d}s,$

which yields

(6.5)
$\Phi(u(t))=\Phi\left(\varphi_{-t}^{*}(u_{0})\right)-\int_{0}^{t}\Phi\left(\varphi_{s-t}^{*}\\{\Phi(u(s)),u(s)\\}\right)\mathrm{d}s.$

We pick $t\in\mathcal{J}_{k}(\varepsilon)$ and our goal is to estimate
$\|\Phi(u(t))\|_{\mathscr{C}^{k}(\Sigma)}$. We begin with the linear term in
(6.5). One has

$\partial_{x}^{\alpha}\left(\Phi\left(\varphi_{-t}^{*}(u_{0})\right)\right)=\int_{\mathrm{T}^{*}\Sigma}\partial_{x}^{\alpha}K(x,y)u_{0}(r,\varphi_{-tr}(y,\eta_{1}))\mathrm{d}{\mathrm{L}}(y,\eta_{1}).$

Thanks to Corollary 4.12 and as
$\int_{\Sigma}K(x,y)\mathrm{d}\text{vol}_{\mathrm{g}}(y)=0$ for every
$x\in\Sigma$, there exists some $k_{0}\geqslant 1$ (depending only on
$(\Sigma,\mathrm{g})$) and a constant $C_{k,r_{0}}>0$ (depending also on $k$
and $r_{0}$) such that

(6.6) $\|\Phi(\varphi_{-t}^{*}(u_{0}))\|_{\mathscr{C}^{k}(\Sigma)}\leqslant
C_{k,r_{0}}e^{-\vartheta
t}\|K\|_{\mathscr{C}^{k_{0}+k}}\|u_{0}\|_{\mathscr{C}^{k_{0}}}.$

Note that this step did not rely on the fact that $t$ belongs to
$\mathcal{J}_{k}(\varepsilon)$. We will now deal with the nonlinear term for
which this assumption will be crucially used. We now write, for
$\chi\in\mathscr{C}^{\infty}_{c}(\mathbb{R}_{+}^{*})$ which is equal to $1$ on
$[r_{0}/2,2r_{0}^{-1}]$ (where $u$ is supported thanks to Lemma 6.2),

(6.7)
$\int_{0}^{t}\partial_{x}^{\alpha}\Phi\left(\varphi_{s-t}^{*}\\{\Phi(u(s)),u(s)\\}\right)\mathrm{d}s\\\
=\int_{0}^{t}\left\langle\partial_{x}^{\alpha}K(x,\pi_{1}\circ\varphi_{(t-s)r}(z_{1}))\chi(r)r^{n-1},\chi(r)\\{\Phi(u(s)),u(s)\\}(r,z_{1})\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s,$

where $\pi_{1}:(y,\eta_{1})\in M_{1}=S^{*}\Sigma\mapsto y\in\Sigma$ and where
the $L^{2}$ scalar product is taken with respect to the measure
$\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}(z_{1})$. Using again
$\int_{\Sigma}K(x,y)\mathrm{d}\text{Vol}_{\mathrm{g}}(y)=0$, one can rewrite
(6.7) as

(6.8)
$\int_{0}^{t}\partial_{x}^{\alpha}\Phi\left(\varphi_{s-t}^{*}\\{\Phi(u(s)),u(s)\\}\right)\mathrm{d}s\\\
=\int_{0}^{t}\left\langle\varphi_{t-s}^{*}\mathbf{P}\left(\partial_{x}^{\alpha}K(x,\pi_{1}(z_{1}))\chi(r)r^{n-1}\right),\chi(r)\\{\Phi(u(s)),u(s)\\}(r,z_{1})\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s.$

Next we define $\mathbf{Q}=\mathrm{Id}-\mathbf{P}$ so that

$\mathbf{Q}(v)=\int_{M_{1}}v(z_{1},r)\mathrm{d}{\mathrm{L}}_{1}(z_{1}),\quad
v\in\mathscr{C}^{\infty}_{c}(\mathbb{R}_{+}^{*}\times M_{1}).$

We now decompose $u(s)=\mathbf{Q}(u(s))+\mathbf{P}(u(s))$ and use (4.23) to
rewrite

$\\{\Phi(u(s)),\mathbf{Q}(u(s))\\}(r,z_{1})=\Phi_{1}(u(s))(z_{1})\partial_{r}\mathbf{Q}(u(s)).$

Recall now that $X=rX_{1}$ is the geodesic vector field generating
$\varphi_{\tau}$ so that

$\partial_{r}\varphi_{\tau}^{*}=\varphi_{\tau}^{*}\left(\partial_{r}+\tau
X_{1}\right).$

Hence, since $\chi$ is identically equal to $1$ on the support of $u(s)$ (as
$s\in\mathcal{J}_{k}(\varepsilon)$), one has

(6.9)
$\displaystyle\int_{0}^{t}\partial_{x}^{\alpha}\Phi\left(\varphi_{s-t}^{*}\\{\Phi(u(s)),\mathbf{Q}(u(s))\\}\right)\mathrm{d}s=$
$\displaystyle-\int_{0}^{t}(t-s)\left\langle\varphi_{t-s}^{*}\mathbf{P}\left(\partial_{x}^{\alpha}K(x,\pi_{1}(z_{1}))\chi(r)\right),\chi(r)r^{n-1}\mathbf{Q}(u(s))(r)X_{1}\Phi_{1}(u(s))\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s$
$\displaystyle-(n-1)\int_{0}^{t}\left\langle\varphi_{t-s}^{*}\mathbf{P}\left(\partial_{x}^{\alpha}K(x,\pi_{1}(z_{1}))\chi(r)\right),\chi(r)r^{n-2}\mathbf{Q}(u(s))(r)\Phi_{1}(u(s))\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s.$

We will bound the first term on the right-hand side of (6.9), the second one
can be handled similarly. It writes, after integrating by parts with respect
to the $r$ variable,

$\int_{\mathbb{R}_{+}}\chi^{2}(r)\mathbf{Q}(u(s))(r)\left(\int_{M_{1}}\partial_{x}^{\alpha}K(x,\varphi_{(t-s)r}(z_{1}))X_{1}(\Phi_{1}(u(s)))(z_{1})\mathrm{d}{\mathrm{L}}_{1}(z_{1})\right)r^{n-1}\mathrm{d}r.$

Applying (4.8) together with (4.20), there exist a constant
$C_{K,\alpha,r_{0}}>0$ (depending on the convolution kernel, $r_{0}$ and
$(\Sigma,\mathrm{g})$) such that, for every
$t\in\mathcal{J}_{k}(\varepsilon)$, the modulus of the above integral is
bounded from above by

(6.10) $\displaystyle
C_{K,\alpha,r_{0}}\|u_{0}\|_{L^{1}}\int_{0}^{t}(t-s)e^{-\frac{\vartheta_{0}r_{0}}{2}(t-s)}\|\Phi_{1}(u(s))\|_{\mathcal{C}^{k_{0}+1}}\mathrm{d}s$
$\displaystyle\hskip 85.35826pt\leqslant
C_{K,\alpha,r_{0}}\|u_{0}\|_{L^{1}}\int_{0}^{t}(t-s)e^{-\frac{\vartheta_{0}r_{0}}{2}(t-s)}\|\Phi(u(s))\|_{\mathcal{C}^{k_{0}+2}}\mathrm{d}s$
$\displaystyle\hskip
85.35826pt\leqslant\widetilde{C}_{K,\alpha,r_{0}}\|u_{0}\|_{L^{1}}\varepsilon
e^{-\vartheta t},$

where we used that
$\|u(s)\|_{L^{1}(r^{n-1}\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1})}=\|u_{0}\|_{L^{1}(r^{n-1}\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1})}$
for every $s\geqslant 0$ and that $\vartheta=\vartheta_{0}r_{0}/4$. A similar
estimate holds for the second term of the right-hand side of (6.9) and thus we
are left with analyzing

$\int_{0}^{t}\partial_{x}^{\alpha}\Phi\left(\varphi_{s-t}^{*}\\{\Phi(u(s)),\mathbf{P}(u(s))\\}\right)\mathrm{d}s\\\
=-\int_{0}^{t}\left\langle\varphi_{t-s}^{*}\mathbf{P}\left(\partial_{x}^{\alpha}K(x,\pi_{1}(z_{1}))r^{n-1}\chi(r)\right),\\{\Phi(u(s)),\mathbf{P}(u(s))\\}(r,z_{1})\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s.$

After integration by parts and recalling that $\chi$ is identically equal to
$1$ on the support of $u$, this is equal to

(6.11) $\displaystyle\int_{0}^{t}\left\langle
r^{n-1}\chi(r)X_{\Phi(u(s))}\varphi_{t-s}^{*}\mathbf{P}\chi(r)\left(\partial_{x}^{\alpha}K(x,\pi_{1}(z_{1}))\right),\mathbf{P}(u(s))(r,z_{1})\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s$ $\displaystyle=\int_{0}^{t}\left\langle
T_{\alpha}(x),\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(s))(r,z_{1})\right\rangle_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s,$

where we set

$\displaystyle T_{\alpha}(x)$
$\displaystyle=\mathbf{B}_{-3,2}^{(-m_{1})}(X_{1}+C_{1})^{N_{1}}X_{\Phi(u(s))}\varphi_{t-s}^{*}\mathbf{P}\left(\chi(r)r^{n-1}\left(\partial_{x}^{\alpha}K(x,\pi_{1}(z_{1}))\right)\right).$

Thanks to Corollary 4.15 applied in positive time, there exist
$N^{\prime}\geqslant 1$ (depending only on the choice of the operator
$\mathbf{B}$) and a constant $C_{K,k,r_{0}}>0$ such that, for all
$|\alpha|\leqslant k$ and for all $x\in\Sigma$

$\|T_{\alpha}(x)\|_{L^{2}(\mathbb{R}\times M_{1})}\leqslant
C_{K,k,r_{0}}e^{-\frac{r_{0}\vartheta_{0}}{2}(t-s)}\|\Phi(u(s))\|_{\mathscr{C}^{N^{\prime}}(\Sigma)}.$

As a result (up to modifying $C_{K,k,r_{0}}>0$), (6.11) can be controlled by

(6.12)
$\left|\int_{0}^{t}\partial_{x}^{\alpha}\Phi\left(\varphi_{s-t}^{*}\\{\Phi(u(s)),\mathbf{P}(u(s))\\}\right)\mathrm{d}s\right|\\\
\leqslant
C_{K,k,r_{0}}\int_{0}^{t}e^{-\frac{r_{0}\vartheta_{0}}{2}(t-s)}\|\Phi(u(s))\|_{\mathscr{C}^{N^{\prime}}(\Sigma)}\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(s))\|_{L^{2}(\mathbb{R}\times
M_{1})}\mathrm{d}s.$

The proof of Lemma 6.4 is complete if we pick $k\geqslant N^{\prime}$ in the
definition of $\mathcal{J}_{k}(\varepsilon)$.

###  6.5. Proof of Lemma 6.5

In order to ease readability, we reduce Lemma 6.5 to the following two
statements.

######  Lemma 6.6.

There exist $k_{0}\in\mathbb{N}$, $C>0$ (independent of $\varepsilon$ and
$u_{0}$) such that, for all $t\in\mathcal{J}_{k}(\varepsilon)$,

$\left\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(t))\right\|_{L^{2}(\mathbb{R}\times
M_{1})}\leqslant C\|u_{0}\|_{\mathscr{C}^{k_{0}}}+C\varepsilon e^{-2\vartheta
t}\int_{0}^{t}e^{\vartheta s}\left\|\mathbf{A}_{3,0}\chi
u(s)\right\|_{L^{2}}ds.$

######  Lemma 6.7.

There exist a constant $\widetilde{c}_{K,\mathrm{g}}>0$ depending only on the
metric $\mathrm{g}$ and on the interaction kernel $K$, $p\in\mathbb{N}$ and
$C>0$ (independent of $\varepsilon$ and $u_{0}$) such that, for
$\varepsilon<\widetilde{c}_{K,\mathrm{g}}r_{0}\vartheta$, for every
$s\in\mathcal{J}_{k}(\varepsilon)$,

$\|\mathbf{A}_{3,0}\chi u(s)\|_{L^{2}(\mathbb{R}\times M_{1})}\leqslant
C\langle s\rangle^{p}\|u_{0}\|_{L^{2}}.$

######  Remark 6.8.

In Lemma 6.7, $\varepsilon$ must be small as follows. We set $C(r_{0},\mu)>0$
to be a positive constant such that, for all $u\in
L^{1}([r_{0}/4,4r_{0}]\times M_{1})$ and for all $(r,z_{1};\rho,\zeta_{1})$ in
$\mathrm{T}^{*}([r_{0}/4,4r_{0}]\times M_{1})$,

$\left|\\{\zeta(X_{\Phi(u)}),\mu(z_{1};\rho,\zeta_{1})\\}\right|\leqslant
C(r_{0},\mu)\|\Phi(u)\|_{\mathscr{C}^{2}}.$

Recall from Remark 4.17 that $\zeta(X_{\Phi(u)})$ is in fact of the form
$\frac{1}{r}\zeta_{1}(X_{\Phi(u)}^{1}(z_{1}))+\Phi_{1}(u)\rho,$ with
$\Phi_{1}(u)$ and $X_{\Phi(u)}^{1}$ defined through linear combination of
derivatives of $\Phi(u)$. We shall fix $\varepsilon>0$ small enough to ensure
that

(6.13) $3C(r_{0},\mu)\varepsilon\leqslant\frac{\vartheta}{8}.$

Assuming Lemmas 6.6 and 6.7, we obtain Lemma 6.5, as the possible polynomial
growth of $\|\mathbf{A}_{3,0}\chi u(s)\|_{L^{2}(\mathbb{R}\times M_{1})}$ is
compensated by the exponential factor $e^{-2\vartheta t}$. We provide the
proofs of Lemmas 6.6 and 6.7 in the next subsections.

###  6.6. Proof of Lemma 6.6

We start with the Duhamel formula to write

(6.14)
$\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(t))=\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(\varphi_{-t}^{*}(u_{0}))-\int_{0}^{t}\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\varphi_{s-t}^{*}\mathbf{P}\\{\Phi(u(s)),u(s)\\}\mathrm{d}s.$

Thanks to Lemma 4.9, there exists a constant $C>0$ and an integer
$N^{\prime}\geqslant 1$ such that, for all $t\geqslant 0$,

$\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(\varphi_{-t}^{*}(u_{0}))\|_{L^{2}(\mathbb{R}\times
M_{1})}\leqslant Ce^{-\vartheta t}\|u_{0}\|_{\mathscr{C}^{N^{\prime}}}.$

Hence, in order to prove Lemma 6.6, we just have to obtain the expected upper
bound on the integral remainder in the right-hand side of (6.14), which we
denote by

$R(t)=\int_{0}^{t}\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\varphi_{s-t}^{*}\mathbf{P}\\{\Phi(u(s)),u(s)\\}\mathrm{d}s.$

Since $\operatorname{supp}u(t)\subset[r_{0}/2,2r_{0}]$ by Lemma 6.2, this is
also equal to

$R(t)=\int_{0}^{t}\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\varphi_{s-t}^{*}\chi\mathbf{P}\\{\Phi(u(s)),u(s)\\}\mathrm{d}s,$

where $\chi\in\mathscr{C}^{\infty}_{c}(\mathbb{R}_{+}^{*})$ is identically
equal to $1$ on $[r_{0}/2,2r_{0}]$.

We are now in position to apply the bilinear estimate of Proposition 4.5.
Indeed, since $s\in\mathcal{J}_{k}(\varepsilon)$ and up to taking $k\geqslant
N$ (with $N$ the regularity parameter appearing in this proposition), one has

$\displaystyle\|{R}(t)\|_{L^{2}}$ $\displaystyle\leqslant
C\int_{0}^{t}e^{-\frac{\vartheta_{0}r_{0}}{2}(t-s)}\|\Phi(u(s))\|_{\mathscr{C}^{k}}\left(\|u(s)\|_{L^{2}(\mathbb{R}\times
M_{1})}+\left\|\mathbf{A}_{3,0}\chi u(s)\right\|_{L^{2}}\right)\mathrm{d}s$
$\displaystyle\leqslant C\varepsilon e^{-2\vartheta t}\int_{0}^{t}e^{\vartheta
s}\left(\|u(s)\|_{L^{2}(\mathbb{R}\times M_{1})}+\left\|\mathbf{A}_{3,0}\chi
u(s)\right\|_{L^{2}}\right)\mathrm{d}s.$

By Lemma 5.3, one has666We only have an upper bound as we took the measure
$\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}$ for the $L^{2}$ scalar product.
$\left\|u(s)\right\|_{L^{2}}\leqslant C_{r_{0}}\|u_{0}\|_{L^{2}}$ for every
$s\in\mathcal{J}_{k}(\varepsilon)$. Hence, we can ensure that the contribution
of this term to the upper bound for
$\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\mathbf{P}(u(t))\|_{L^{2}(\mathbb{R}\times
M_{1})}$ is controlled by $Ce^{-\vartheta t}\varepsilon\|u_{0}\|_{L^{2}}$ for
every $t\in\mathcal{J}_{k}(\varepsilon)$. This concludes the proof of Lemma
6.6 provided that we pick $k_{0}\geqslant N^{\prime}$.

###  6.7. Proof of Lemma 6.7

In this section, we use microlocal energy estimates to prove Lemma 6.7. This
is precisely here that our construction of a global escape function is
crucial.

Due to a logarithmic loss of derivatives that will occur in the energy
estimates, we will not study directly $\|\mathbf{A}_{3,0}\chi
u(s)\|_{L^{2}(\mathbb{R}\times M_{1})}$ but rather

$\|\mathbf{A}_{3,\sigma_{0}(s)}\chi u(s)\|_{L^{2}(\mathbb{R}\times M_{1})}$

where $\sigma_{0}(s)=\frac{1}{8}e^{-\vartheta s}$ is a time-dependent weight.
This will yield some extra dissipation in the energy estimates that will
precisely compensate the aforementioned logarithmic loss of derivatives. This
idea of considering a time-dependent regularity is somehow reminiscent of the
time-dependent weights often used for Cauchy-Kowalevskaya type theorems in
analytic regularity (see e.g. [Caf90] and the proofs of Landau damping in the
torus of [BMM16] and [GNR21]). The fact that $\sigma_{0}$ remains non-negative
for all times is important and, as we shall see at the end of the proof,
estimating $\|\mathbf{A}_{3,\sigma_{0}(s)}\chi u(s)\|_{L^{2}(\mathbb{R}\times
M_{1})}$ will be sufficient to bound $\|\mathbf{A}_{3,0}\chi
u(s)\|_{L^{2}(\mathbb{R}\times M_{1})}$.

Recall that $\mathbf{A}_{3,\sigma_{0}(t)}$ is of the form

$\mathbf{A}_{3,\sigma_{0}(t)}v(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i(r-s)\rho}{\operatorname{Op}^{M_{1}}}\left(e^{(3\mu+\sigma_{0}(t))\log
f}\right)v(s,z_{1})\mathrm{d}\rho\mathrm{d}s,$

where $\mu$ and $f$ are given by Proposition 3.11. Let $\chi$ be a smooth
function that is compactly supported in $]r_{0}/4,4r_{0}[$ and that is
identically equal to $1$ on $[r_{0}/2,2r_{0}]$. Thanks to Lemma A.10, there
exists a constant $C>0$ such that

$\left\|\mathbf{A}_{3,\sigma_{0}(t)}u(t)\right\|_{L^{2}}\leqslant\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|_{L^{2}}+C\|u_{0}\|_{L^{2}}.$

Hence, we can restrict our attention to estimating the norm of
$\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\|_{L^{2}}$. To this
purpose, we write

(6.15)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|_{L^{2}}^{2}=-2\operatorname{Re}$
$\displaystyle\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
Xu(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle$
$\displaystyle-2\operatorname{Re}\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
X_{\Phi(u(t))}u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle$
$\displaystyle-\frac{1}{4}\vartheta
e^{-t\vartheta}\operatorname{Re}\left\langle\chi\widetilde{\mathbf{A}}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle,$

where

$\widetilde{\mathbf{A}}_{3,\sigma_{0}(t)}v(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i(r-s)\rho}{\operatorname{Op}^{M_{1}}}\left(\log
fe^{(3\mu+\sigma_{0}(t))\log f}\right)v(s,z_{1})\mathrm{d}\rho\mathrm{d}s.$

Now using the composition rule for pseudo-differential operators on
$\mathbb{R}\times M_{1}$ (see Remark A.9 for a brief reminder), one has

$\widetilde{\mathbf{A}}_{3,\sigma_{0}(t)}\chi^{2}=\mathbf{Op}\left(\log
f\right)\chi\mathbf{Op}\left(e^{(3\mu+\sigma_{0}(t))\log f}\right)\chi\\\
+\mathbf{Op}\left(b_{-1}e^{(3\mu+\sigma_{0}(t))\log f}\right)+\mathbf{R}_{N},$

where $b_{-1}$ is an element in
$\overline{S}^{-1+0}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ and the
remainder $\mathbf{R}_{N}$ is a bounded operator from
$H^{-N}_{\text{comp}}(\mathbb{R}\times M_{1})$ to
$H^{N}_{\text{loc}}(\mathbb{R}\times M_{1})$ (with the semi-norm of $R_{N}$
that is uniformly bounded in terms of $t$). Applying the composition rule for
pseudo-differential operators, one finds that the last term in the right-hand
side of (6.15) satisfies

(6.16)
$\displaystyle\left\langle\chi\widetilde{\mathbf{A}}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\rangle=\left\langle\chi\mathbf{Op}(\log
f)\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle$
$\displaystyle+\mathcal{O}\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right),$

where the constant in the remainder depends on $\vartheta$, and on the choice
of $f$ and $\mu$ (but not on $t$). Note that we used one more time that
$\|u(t)\|_{L^{2}}\leqslant 2^{n}r_{0}^{-n+1}\|u_{0}\|_{L^{2}}$ for every
$t\in\mathcal{J}_{k}(\varepsilon)$ as the $L^{2}$-scalar product is taken with
respect to $\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}(z_{1})$ rather than
$r^{n-1}\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}(z_{1})$ .

Next we deal with the first term in (6.15). As $X$ preserves the measure
$\mathrm{d}r\mathrm{d}{\mathrm{L}}_{1}$ on $\mathbb{R}\times M_{1}$, one has

$\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
Xu(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\rangle=\left\langle\left[\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi,X\right]u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\rangle$

We can now insert the almost inverse
$\mathbf{W}_{3,\sigma_{0}(t)}^{(N)}=\mathbf{Op}\left(e^{-(3\mu+\sigma_{0}(t)\log
f)}b_{N}\right)$ of $\mathbf{A}_{3,\sigma_{0}(t)}$, as given by (4.14), with
$N\geqslant 5$. Applying the composition rule together with Calderón-
Vaillancourt one more time, it yields

(6.17) $\displaystyle\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
Xu(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle$
$\displaystyle\hskip
17.64056pt=\left\langle\left[\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi,X\right]{\mathbf{W}}_{3,\sigma_{0}(t)}^{(N)}\mathbf{A}_{3,\sigma_{0}(t)}u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\rangle$ $\displaystyle\hskip
199.16928pt+\mathcal{O}\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right),$

where the constant in the remainder depends again on $\vartheta$ and the
choice of $f$ and $\mu$ (but not on $t$). Using Lemmas 6.2 and A.10, we can
(up to increasing the size of the constant in the remainder) insert a cutoff
$\chi$ so that (6.17) becomes

$\left\langle\left(\left[\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi,X\right]\mathbf{W}_{3,\sigma_{0}(t)}^{(N)}\chi\right)\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\\\
+\mathcal{O}\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right).$

Another application of the composition rule together with the Calderón-
Vaillancourt Theorem allows us to write the above term as

$\left\langle\mathbf{Op}\left(\chi^{3}{\mathfrak{X}}((3\mu+\sigma_{0}(t))\log
f)\right)\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\\\
+\mathcal{O}\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right).$

By Proposition 3.11, we know that
$\chi(r)\mathfrak{X}((3\mu+\sigma_{0}(t))\log f)(z,\zeta)\geqslant 0$ for
$\langle\zeta\rangle\geqslant R_{0}$. Hence, up to adding a pseudo-
differential that is compactly supported in the variable $\zeta$ (and thus
smoothing), we are in position to apply the sharp Gårding inequality [DZ19,
Prop. E.23] (see Section A.1.4 and Remark A.9 for a brief reminder). It allows
to end up with

(6.18) $-2\operatorname{Re}\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
Xu(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\leqslant
C\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right),$

where $C>0$ is a constant that is independent of $t$.

It remains to deal with the second term in the right-hand side of (6.15).
Using that $X_{\Phi(u)}$ is an Hamiltonian vector field (and thus preserves
the Liouville measure), we can write

$\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
X_{\Phi(u(t))}u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\\\
=\left\langle\left[\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi,X_{\Phi(u(t))}\right]u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\rangle\\\ +\left\langle
X_{\Phi(u(t))}(r^{-n+1})\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle.$

As $t\in\mathcal{J}_{k}(\varepsilon)$, we find

$-2\operatorname{Re}\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
X_{\Phi(u(t))}u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\\\
\leqslant-2\operatorname{Re}\left\langle\left[\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi,X_{\Phi(u(t))}\right]u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\rangle\\\ +n(2r_{0})^{-n}\varepsilon e^{-\vartheta
t}\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\|^{2}.$

Proceeding as for the case of the vector field $X$ (i.e. through several
applications of the composition rule together with the Calderón-Vaillancourt
Theorem), we find that there exists a constant $C>0$ (depending on
$\vartheta$, $r_{0}$ and on the choices of $f$ and $\mu$) such that

$-2\operatorname{Re}\left\langle\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
X_{\Phi(u(t))}u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\\\
\leqslant-2\operatorname{Re}\left\langle\mathbf{Op}\left(3\log
f\\{\zeta(X_{\Phi(u(t))}),\mu\\}\right)\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle\\\ +C\varepsilon
e^{-\vartheta t}\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|^{2}+C\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right).$

Combining this inequality with (6.15), (6.16) and (6.18), we are led to study
the term

$-2\left\langle\chi\mathbf{Op}\left(\left[\frac{1}{8}\vartheta
e^{-t\vartheta}+3\\{\zeta(X_{\Phi(u(t))}),\mu\\}\right]\log
f\right)\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t),\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\right\rangle.$

Thanks to (6.13), for $\varepsilon$ small enough, one finds that, for all
$t\in\mathcal{J}_{k}(\varepsilon)$,

$\frac{1}{8}\vartheta
e^{-t\vartheta}+3\\{\zeta(X_{\Phi(u(t))}),\mu\\}\geqslant 0.$

We can thus apply the sharp Gårding inequality one more time (together with
the composition rule and the Calderón-Vaillancourt Theorem). This eventually
yields the upper bound

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|_{L^{2}}^{2}$ $\displaystyle\leqslant C\varepsilon e^{-\vartheta
t}\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|^{2}+C\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right)$
$\displaystyle\leqslant C\varepsilon e^{-\vartheta
t}\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|^{2}+C\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{8}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right),$

which implies

(6.19)
$\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)}\chi
u(t)\right\|_{L^{2}}^{2}e^{-C\varepsilon\int_{0}^{t}e^{-\vartheta
s}\mathrm{d}s}\right)\leqslant
C\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{1}{8}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right),$

where the constant $C>0$ depends on $r_{0}$, $\vartheta$ and the choice of the
function $f$ and $\mu$.

Iterating this procedure $p$ times (where $p$ is such that
$0<3+\frac{1}{8}-\frac{p}{8}\leqslant\frac{1}{4}$), recalling that
$3(1-2\delta)>3-1/8$, so that Proposition 3.11 can be indeed applied at each
step, we find, according to (6.21), that for every $0\leqslant q\leqslant p$
and for every $t\in\mathcal{J}_{k}(\varepsilon)$,

(6.20)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\left\|\chi\mathbf{A}_{3,\sigma_{0}(t)-\frac{q}{8}}\chi
u(t)\right\|_{L^{2}}^{2}e^{-C\varepsilon\int_{0}^{t}e^{-\vartheta
s}\mathrm{d}s}\right)$ $\displaystyle\leqslant
C\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{q}{8}-\frac{1}{4}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right)$
(6.21)
$\displaystyle\leqslant\widetilde{C}\left(\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{q+1}{8}}u(t)\|_{L^{2}}^{2}+\|u_{0}\|_{L^{2}}^{2}\right)$

By definition of $p$, using the Calderón-Vaillancourt (together with the
composition rule) one more time, there holds

$\|\mathbf{A}_{3,\sigma_{0}(t)-\frac{p}{8}-\frac{1}{4}}\chi
u(t)\|_{L^{2}}\leqslant
C\|u(t)\|_{L^{2}}\leqslant(r_{0}/2)^{1-n}\|u_{0}\|_{L^{2}}.$

By induction, using (6.20) for $q=p$ and (6.21) for $q<p$, we deduce that for
every $t\in\mathcal{J}_{k}(\varepsilon)$,

$\|\mathbf{A}_{3,\sigma_{0}(t)}\chi u(t)\|_{L^{2}}\leqslant C\langle
t\rangle^{p}\|u_{0}\|_{L^{2}},$

and by the composition rule and the Calderón-Vaillancourt Theorem, we conclude
that, for every $t\in\mathcal{J}_{k}(\varepsilon)$,

$\|\mathbf{A}_{3,0}\chi u(t)\|_{L^{2}}\leqslant C\langle
t\rangle^{p}\|u_{0}\|_{L^{2}},$

where the constant depends only $\vartheta$ and the choice of the escape
function $F$ used to define $\mathbf{A}$. We have therefore proved the final
Lemma 6.7.

###  6.8. End of the proof

We are now ready to complete the proof of Theorem 1.1. Let $\psi$ be a smooth
test function. We have for all $t\in\mathbb{R}_{+}$,

(6.22) $\langle
u(t),\psi\rangle=\langle\varphi_{-t}^{*}u_{0},\psi\rangle+\int_{0}^{t}\langle\varphi_{-(t-s)}^{*}X_{\Phi(u(s))}u(s),\psi\rangle~\mathrm{d}s.$

By Lemma 6.2, $u(t)$ is supported away from the null section, and since the
geodesic flow preserves this property, we can assume without loss of
generality that this is also the case for $\psi$. By Corollary 4.12, the first
term in the right-hand side of (6.22) converges exponentially fast to

$\int_{M^{\times}}\left(\int_{M_{1}}\psi(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime})\right)u_{0}(r,z_{1})\mathrm{d}{\mathrm{L}}(r,z_{1})=\left\langle\left(\int_{M_{1}}u_{0}(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime})\right),\psi\right\rangle.$

For what concerns the second one, we write

(6.23)
$\int_{0}^{t}\langle\varphi_{(t-s)}^{*}X_{\Phi(u(s))}u(s),\psi\rangle~\mathrm{d}s=\int_{0}^{t}\langle
X_{\Phi(u(s))}u(s),\mathbf{P}\psi\circ\varphi_{-(t-s)}\rangle~\mathrm{d}s\\\
+\int_{0}^{t}\langle X_{\Phi(u(s))}u(s),\mathbf{Q}\psi\rangle~\mathrm{d}s.$

In order to analyze the two terms above, recall from Remark 4.17 that
$X_{\Phi(u)}$ is of the form
$\frac{1}{r}X_{\Phi(u)}^{1}(z_{1})+\Phi_{1}(u)\partial_{r},$ with
$\Phi_{1}(u)$ and $X_{\Phi(u)}^{1}$ defined through the smooth kernels given
in (4.21) and (4.22). From (4.19) and (4.20), we also have that their semi-
norms can be controlled by $\|\Phi(u(t))\|_{\mathscr{C}^{k}}$.

With that conventions at hand, we can rewrite (6.23) as

$\int_{0}^{t}\langle
X_{\Phi(u(s))}u(s),\varphi_{(t-s)}^{*}\mathbf{P}\psi\rangle~\mathrm{d}s-\int_{0}^{t}\left\langle\Phi_{1}(u(s))\mathbf{P}u(s),\left(\partial_{r}\mathbf{Q}\psi+\frac{n-1}{r}\mathbf{Q}\psi\right)\right\rangle~\mathrm{d}s\\\
-\int_{0}^{t}\left\langle\Phi_{1}(u(s))\mathbf{Q}u(s),\left(\partial_{r}\mathbf{Q}\psi+\frac{n-1}{r}\mathbf{Q}\psi\right)\right\rangle~\mathrm{d}s.$

For the fist term, we can argue as in the proof of Lemma 6.4 and use Theorem
6.1 and Lemma 6.5 to show that it converges exponentially fast to $0$ as
$t\to\infty$. For the second, we use Lemma 6.5 one more time together with the
fact that

$\left\|\mathbf{B}_{3,-2}^{(m_{1},N_{1})}\Phi_{1}(u(s))(\mathbf{B}_{3,-2}^{(m_{1},N_{1})})^{-1}\right\|_{L^{2}\rightarrow
L^{2}}\leqslant C\|\Phi(u(s))\|_{\mathscr{C}^{k}}.$

This last property follows from the composition rule for pseudo-differential
on $M_{1}$ together with the Calderón-Vaillancourt (up to increasing the value
of $k$). Thanks to Theorem 6.1, this shows that the second term converges
exponentially fast to

$-\left\langle\int_{0}^{\infty}\Phi_{1}(u(s))\mathbf{P}u(s)\mathrm{d}s,\left(\partial_{r}\mathbf{Q}\psi+\frac{n-1}{r}\mathbf{Q}\psi\right)\right\rangle.~$

The third term also converges converges exponentially to some limit thanks to
Theorem 6.1 together with the fact that the $L^{p}$ norms of $u(t)$ are
preserved. In summary, it yields that the Duhamel remainder term converges
exponentially fast to

$\left\langle\underbrace{\int_{0}^{+\infty}X_{\Phi(u(s))}u(s)\mathrm{d}s}_{=w_{\infty}},\int_{M_{1}}\psi(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime})\right\rangle=\left\langle\left(\int_{M_{1}}w_{\infty}(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime})\right),\psi\right\rangle.$

All in all, we deduce that $u(t)$ weakly converges exponentially fast to the
distribution

$u_{\infty}=\int_{M_{1}}u_{0}(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime})+\int_{M_{1}}w_{\infty}(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime}).$

By Lemma 5.3, we deduce that $u_{\infty}\in L^{1}\cap L^{\infty}(M)$ and that
the support of $u_{\infty}$ avoids the null section. Hence we can write it as
$h_{\infty}\circ\mathrm{H}$ for some $h_{\infty}\in
L^{\infty}_{\operatorname{comp}}(\mathbb{R}_{>0})$.

To conclude, the exponential decay estimate of
$\|\Phi(u(t))\|_{\mathscr{C}^{N}}$ for all $N\geqslant 0$ follows from the
previous calculation by replacing $\psi$ by $\partial_{x}^{\alpha}K(x,\cdot)$
which has zero average.

##  Appendix A Toolbox on microlocal analysis

In this appendix, we review material from microlocal analysis with a special
emphasis on pseudo-differential operators on manifolds. These tools are used
all along the article and we refer to [Hör07, Ch. 18], [Zwo12, Ch. 4, 9, 14]
and [DZ19, App. E] – that we closely follow – for more details and background
on this theory. A notable difference with these references is the use of an
isochore atlas and of the Weyl quantization which simplify some aspects of the
exposition as it was pointed to us by Guedes Bonthonneau [GB].

###  A.1. Pseudodifferential operators on $\mathbb{R}^{N}$

Let $a$ be a function in the Schwartz class $\mathscr{S}(\mathbb{R}^{2N})$.
One can define the Weyl quantization of the symbol $a$ as

$\forall
u\in\mathscr{S}(\mathbb{R}^{N}),\quad{\operatorname{Op}^{\mathrm{w}}}(a)u(x)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{2N}}e^{i\langle
x-y,\xi\rangle}a\left(\frac{x+y}{2},\xi\right)u(y)dyd\xi.$

One can verify that ${\operatorname{Op}^{\mathrm{w}}}(a)u$ belongs to
$\mathscr{S}(\mathbb{R}^{2N})$. Let us observe that the Schwartz kernel of
${\operatorname{Op}^{\mathrm{w}}}(a)$ is given by

$K_{a}(x,y)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}e^{i\langle
x-y,\xi\rangle}a\left(\frac{x+y}{2},\xi\right)d\xi.$

One can also recover the symbol from the kernel through the formula

(A.1)
$a(x,\xi)=\int_{\mathbb{R}^{N}}K_{a}\left(x+\frac{\eta}{2},x-\frac{\eta}{2}\right)e^{-i\langle\eta,\xi\rangle}d\eta,$

which is often referred to as the Wigner transform.

We can also introduce the class of Kohn-Nirenberg (or classical) symbols
[Zwo12, §9.3] $a$, for every $m\in\mathbb{R}$,

$S^{m}(\mathrm{T}^{*}(\mathbb{R}^{N}))=\left\\{a\in\mathscr{C}^{\infty}(\mathrm{T}^{*}\mathbb{R}^{N}):\
\forall(\alpha,\beta),\
\sup_{x,\xi}\,\langle\xi\rangle^{|\beta|-m}|\partial^{\alpha}_{x}\partial_{\xi}^{\beta}a|<\infty\right\\},$

where we used the standard convention
$\langle\xi\rangle=\left(1+|\xi|^{2}\right)^{\frac{1}{2}}$. One can verify
that the definition of $\operatorname{Op}^{\mathrm{w}}(a)$ extends to symbols
lying in $S^{m}(\mathrm{T}^{*}(\mathbb{R}^{N}))$ and that, for every
$m\in\mathbb{R}$ and every $a\in S^{m}(\mathrm{T}^{*}(\mathbb{R}^{N}))$,

${\operatorname{Op}^{\mathrm{w}}}(a):\mathscr{S}(\mathbb{R}^{N})\rightarrow\mathscr{S}(\mathbb{R}^{N})$

is a bounded operator [Zwo12, Th. 4.16]. We denote by
$\Psi^{m}(\mathbb{R}^{n})$ the set of operators of the form
${\operatorname{Op}^{\mathrm{w}}}(a)$ with $a$ belonging to
$S^{m}(\mathrm{T}^{*}(\mathbb{R}^{N}))$. This is what we will refer to the set
of pseudo-differential operators of order $m$ on $\mathbb{R}^{N}$. The useful
extra classes of operators are

$\Psi^{-\infty}(\mathbb{R}^{N})=\bigcap_{m\in\mathbb{R}}\Psi^{m}(\mathbb{R}^{N}),\qquad\Psi^{m+}(\mathbb{R}^{N})=\bigcap_{m^{\prime}>m}\Psi^{m^{\prime}}(\mathbb{R}^{N})$

and their subset $\Psi_{\text{comp}}(\mathbb{R}^{N})$ consisting of pseudo-
differential operators whose symbol $a$ is compactly supported. Recall from
[Zwo12, Th. 9.6] that, if $a\in S^{m}(\mathrm{T}^{*}(\mathbb{R}^{N}))$, then
the Schwartz kernel $K_{a}$ of ${\operatorname{Op}^{\mathrm{w}}}(a)$ belongs
to $\mathscr{S}^{\prime}(\mathbb{R}^{2N})$ and is smooth outside the diagonal

$\Delta=\left\\{(x,x)\leavevmode\nobreak\ :\leavevmode\nobreak\
x\in\mathbb{R}^{N}\right\\}.$

Moreover, for every $N_{0}>|\alpha|+|\beta|+m+N$, one has

$|\partial_{x}^{\alpha}\partial_{y}^{\beta}K_{a}(x,y)|\leqslant
C_{N_{0}}|x-y|^{-N_{0}},\quad(x,y)\in\mathbb{R}^{2N}.$

Conversely, if the Schwartz kernel $K\in\mathscr{C}^{\infty}(\mathbb{R}^{2N})$
of a smoothing operator $A$ [Hör15, Th.5.2.6] verifies, for all $\alpha,\beta$
and for all $N_{0}$, $\langle
x-y\rangle^{N_{0}}\partial_{x}^{\alpha}\partial_{y}^{\beta}K(x,y)\in
L^{\infty}(\mathbb{R}^{2N})$ and one can show that
$A={\operatorname{Op}^{\mathrm{w}}}(a)$ for some $a\in
S^{-\infty}(\mathrm{T}^{*}\mathbb{R}^{N})$ [Zwo12, Th. 9.6].

######  Remark A.1.

The key observation in view of defining pseudo-differential operators on
manifolds is that, given a smooth diffeomorphism
$\gamma:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ (with all the derivatives of
$\gamma$ and $\gamma^{-1}$ being bounded on $\mathbb{R}^{N}$), one can define
its symplectic lift

$\widetilde{\gamma}:\mathrm{T}^{*}\mathbb{R}^{N}\rightarrow\mathrm{T}^{*}\mathbb{R}^{N},\quad(x,\xi)\mapsto\left(\gamma^{-1}(x),\mathrm{d}\gamma(x)^{\top}\xi\right),$

and verify that $a\in S^{m}$ implies that $a\circ\widetilde{\gamma}\in S^{m}$
[Zwo12, Th. 9.4].

Let us now review a few standard properties of these operators.

####  A.1.1. Adjoint

One can check the following, for every $a\in
S^{m}(\mathrm{T}^{*}(\mathbb{R}^{N}))$,

(A.2) $\langle{\operatorname{Op}^{\mathrm{w}}}(a)u,v\rangle_{L^{2}}=\langle
u,{\operatorname{Op}^{\mathrm{w}}}(\overline{a})v\rangle_{L^{2}},\quad
u,v\in\mathscr{S}(\mathbb{R}^{N}).$

In particular, if $a$ is real-valued, the operator is formally selfadjoint.

####  A.1.2. Composition formula

Given $a\in S^{m_{1}}(\mathrm{T}^{*}(\mathbb{R}^{N}))$ and $b\in
S^{m_{2}}(\mathrm{T}^{*}(\mathbb{R}^{N}))$, one has

(A.3)
${\operatorname{Op}^{\mathrm{w}}}(a)\circ{\operatorname{Op}^{\mathrm{w}}}(b)={\operatorname{Op}^{\mathrm{w}}}(a\operatorname{\sharp}b),$

where $a\operatorname{\sharp}b$ is the Moyal product of $a$ and $b$. It can be
shown [Zwo12, Th. 9.5] that $a\operatorname{\sharp}b$ belongs to
$S^{m_{1}+m_{2}}(\mathrm{T}^{*}(\mathbb{R}^{N}))$ and that, for every
$N_{0}\geqslant 0$,

(A.4)
$(a\operatorname{\sharp}b)(x,\xi)-\sum_{k=0}^{N_{0}}\frac{\left(\langle\partial_{\xi},\partial_{y}\rangle-\langle\partial_{\eta},\partial_{x}\rangle\right)^{k}}{(2i)^{k}k!}\left(a(x,\xi)b(y,\eta)\right)|_{x=y,\xi=\eta}\in
S^{m_{1}+m_{2}-N_{0}-1}(\mathrm{T}^{*}(\mathbb{R}^{N})),$

where one can verify that each term in the sum belongs to
$S^{m_{1}+m_{2}-k}(\mathrm{T}^{*}(\mathbb{R}^{N}))$. In particular, one can
observe that the first term in the sum is equal to $ab$ while the second one
is $\frac{1}{2i}\\{a,b\\}$. Going through the proof of this result, one can
verify that the semi-norms of the remainder of order $N_{0}$ are controlled by
the semi-norms of
$\partial_{x}^{\alpha_{1}}\partial_{\xi}^{\beta_{1}}a\,\partial_{x}^{\alpha_{2}}\partial_{\xi}^{\beta_{2}}b$
with $|\alpha_{1}|+|\alpha_{2}|=|\beta_{1}|+|\beta_{2}|=N_{0}$.

These properties immediately lead to the formula for the bracket

(A.5)
$[{\operatorname{Op}^{\mathrm{w}}}(a),{\operatorname{Op}^{\mathrm{w}}}(b)]={\operatorname{Op}^{\mathrm{w}}}(a\operatorname{\sharp}b-b\sharp
a),$

and one can verify that the symbol $c=a\operatorname{\sharp}b-b\sharp a$ has
the following asymptotic form:

(A.6)
$c(x,\xi)-2\sum_{k=0}^{N_{0}}\frac{\left(\langle\partial_{\xi},\partial_{y}\rangle-\langle\partial_{\eta},\partial_{x}\rangle\right)^{2k+1}}{(2i)^{2k+1}(2k+1)!}\left(a(x,\xi)b(y,\eta)\right)|_{x=y,\xi=\eta}\in
S^{m_{1}+m_{2}-2N_{0}-3}(\mathrm{T}^{*}(\mathbb{R}^{N})).$

Again, one can check that each term in the sum belongs to
$S^{m_{1}+m_{2}-2k-1}(\mathrm{T}^{*}(\mathbb{R}^{N}))$.

####  A.1.3. Action on Sobolev spaces

In order to make these formulas of interest, one needs to understand the
action of these operators on standard Sobolev spaces

$H^{s}(\mathbb{R}^{N})=\left\\{u\in\mathscr{S}^{\prime}(\mathbb{R}^{N}):\langle\xi\rangle^{s}\widehat{u}(\xi)\in
L^{2}(\mathbb{R}^{N})\right\\},\quad s\in\mathbb{R},$

where $\widehat{u}(\xi)$ is the Fourier transform of $u$. The Calderón-
Vaillancourt Theorem reads as follows777Technically speaking, this section
deals with the $s=m=0$ but, using the composition rule, one can derive this
more general version. [Zwo12, §4.5]. For every $s\in\mathbb{R}$, there exists
$C_{s}>0$ and $N_{s}\in\mathbb{Z}_{+}$ such that, for every $a\in
S^{m}(\mathrm{T}^{*}\mathbb{R}^{N})$,

(A.7) $\left\|{\operatorname{Op}^{\mathrm{w}}}(a)\right\|_{H^{s}\rightarrow
H^{s-m}}\leqslant C_{s}\sum_{|\alpha|+|\beta|\leqslant
N_{s}}\left\|\langle\xi\rangle^{-m+|\beta|}\partial^{\alpha}_{x}\partial_{\xi}^{\beta}a\right\|_{\infty}.$

Combined with (A.3), this yields asymptotic formulas for
${\operatorname{Op}^{\mathrm{w}}}(a)\circ{\operatorname{Op}^{\mathrm{w}}}(b)$
with terms that are “more and more smoothing”.

####  A.1.4. Gårding inequality

A key property is the sharp Gårding inequality [Zwo12, Th. 9.11] which states
that one can find $C_{m},N_{m}>0$ such that, for every $a\in
S^{m}(\mathrm{T}^{*}\mathbb{R}^{N})$ satisfying $a\geqslant 0$, one has

(A.8) $\langle{\operatorname{Op}^{\mathrm{w}}}(a)u,u\rangle\geqslant-
C_{m}\|u\|^{2}_{H^{m-\frac{1}{2}}}\sum_{|\alpha|+|\beta|\leqslant
N_{m}}\|\langle\xi\rangle^{-m+|\beta|}\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a\|_{\infty},\quad
u\in\mathscr{S}(\mathbb{R}^{N}).$

####  A.1.5. Change of variables

Finally, if we let $\gamma:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ be a
smooth diffeomorphism which is equal to the identity outside a compact set and
$a$ be an element in $S^{m}(\mathrm{T}^{*}\mathbb{R}^{N})$, then there exists
[Zwo12, Th. 9.10] $a_{\gamma}$ in $S^{m}(\mathrm{T}^{*}\mathbb{R}^{N})$ such
that

(A.9)
$(\gamma^{-1})^{*}{\operatorname{Op}^{\mathrm{w}}}(a)\gamma^{*}={\operatorname{Op}^{\mathrm{w}}}(a_{\gamma}).$

Following the proof in this reference, in the case where $a$ belongs to
$\mathscr{S}(\mathbb{R}^{2N})$, the kernel of $K_{a_{\gamma}}$ writes

$K_{a_{\gamma}}(x,y)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}a\left(\frac{\gamma(x)+\gamma(y)}{2},\xi\right)e^{i\langle\gamma(x)-\gamma(y),\xi\rangle}d\xi.$

Thanks to (A.1), we can write an asymptotic expression for $a_{\gamma}(x,\xi)$
following the lines of [Zwo12, Ch. 9]. In fact, a remarkable feature of the
Weyl quantization is that, in the case of a smooth _volume preserving_
diffeomorphism (which is the identity outside a compact set), one can find a
sequence $(a_{\gamma,k})_{k\geqslant 2}$ such that, for every $k\geqslant 2$,
$a_{\gamma,k}\in S^{m-k}(\mathrm{T}^{*}\mathbb{R}^{N})$ and for every
$N_{0}\geqslant 2$,

(A.10)
$a_{\gamma}(x,\xi)-a\circ\widetilde{\gamma}(x,\xi)-\sum_{k=2}^{N_{0}}a_{\gamma,k}(x,\xi)\in
S^{m-N_{0}-1}(\mathrm{T}^{*}\mathbb{R}^{N}),$

where each $a_{\gamma,k}$ depends linearly on derivatives of order $k$ of $a$
with respect to $\xi$. See e.g. [GB] or [Zwo12, Th. 9.3].

###  A.2. Operator-valued symbols

Due to the product structure of the phase space $\mathbb{R}\times M_{1}$, it
is convenient to introduce pseudo-differential operators on $\mathbb{R}$ with
operator-valued symbols. For the sake of simplicity, we restrict ourselves to
$N=1$ but the presentation can be extended to any $N\geqslant 1$ as in the
previous subsection. We fix two Hilbert spaces $\mathscr{H}_{1}$ and
$\mathscr{H}_{2}$ and the space of continuous linear maps
$\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2})$ between these two spaces. As
above, the space of (Kohn-Nirenberg) operator-valued symbols is given, for
every $m\in\mathbb{R}$, by

$S^{m}(\mathrm{T}^{*}(\mathbb{R}),\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2}))\\\
=\left\\{A\in\mathscr{C}^{\infty}(\mathrm{T}^{*}\mathbb{R},\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2})):\
\forall(\alpha,\beta),\
\sup_{r,\rho}\langle\rho\rangle^{|\beta|-m}\|\partial^{\alpha}_{r}\partial_{\rho}^{\beta}A\|_{\mathscr{H}_{1}\rightarrow\mathscr{H}_{2}}<\infty\right\\}.$

As before, one can define the Weyl quantization of such a symbol as follows

$\forall
u\in\mathscr{S}(\mathbb{R},\mathscr{H}_{1}),\quad{\operatorname{Op}^{\mathrm{w}}}(A)u(r)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i\langle
r-s,\rho\rangle}A\left(\frac{r+s}{2},\rho\right)u(s)dsd\rho,$

and, along the same lines, one can verify that

${\operatorname{Op}^{\mathrm{w}}}(A):\mathscr{S}(\mathbb{R},\mathscr{H}_{1})\rightarrow\mathscr{S}(\mathbb{R},\mathscr{H}_{2})$

is a continuous linear map. Again, one can introduce the space of pseudo-
differential operators $\Psi^{m}(\mathbb{R},\mathscr{H}_{1},\mathscr{H}_{2})$
of order $m$ with values in $\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2})$ as
the space of all operators of the previous form
${\operatorname{Op}^{\mathrm{w}}}(A)$.

The formula for the adjoint remains true in this setting, i.e. for every $u$
in $\mathscr{S}(\mathbb{R},\mathscr{H}_{1})$ and every $v$ in
$\mathscr{S}(\mathbb{R},\mathscr{H}_{2})$

(A.11)
$\int_{\mathbb{R}}\langle{\operatorname{Op}^{\mathrm{w}}}(A)u(r),v(r)\rangle_{\mathscr{H}_{2}}dr=\int_{\mathbb{R}}\langle
u(r),{\operatorname{Op}^{\mathrm{w}}}(A^{*})v(r)\rangle_{\mathscr{H}_{1}}dr.$

Similarly, one can write a composition formula for this quantization:

(A.12)
${\operatorname{Op}^{\mathrm{w}}}(A)\circ{\operatorname{Op}^{\mathrm{w}}}(B)={\operatorname{Op}^{\mathrm{w}}}(a\operatorname{\sharp}b),$

where $a\operatorname{\sharp}b$ is the Moyal product of the two operators
$A\in
S^{m_{1}}(\mathrm{T}^{*}(\mathbb{R}),\mathscr{L}(\mathscr{H}_{2},\mathscr{H}_{3}))$
and $B\in
S^{m_{2}}(\mathrm{T}^{*}(\mathbb{R}),\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2}))$.
Again, it can be shown that $a\operatorname{\sharp}b$ belongs to
$S^{m_{1}+m_{2}}(\mathrm{T}^{*}(\mathbb{R}),\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{3}))$
and that, for every $N_{0}\geqslant 0$

(A.13)
$a\operatorname{\sharp}b(r,\rho)-\sum_{k=0}^{N_{0}}\frac{\left(\partial_{\rho}\partial_{s}-\partial_{\sigma}\partial_{r}\right)^{k}}{(2i)^{k}k!}\left(A(r,\rho)B(s,\sigma)\right)|_{r=s,\rho=\sigma},$

belongs to
$S^{m_{1}+m_{2}-N_{0}-1}(\mathrm{T}^{*}(\mathbb{R}^{N}),\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{3}))$
where one can verify that each term in the sum belongs to
$S^{m_{1}+m_{2}-k}(\mathrm{T}^{*}(\mathbb{R}),\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{3}))$.
A notable difference with the scalar case is that there is no particular
simplification when writing
$[{\operatorname{Op}^{\mathrm{w}}}(A),{\operatorname{Op}^{\mathrm{w}}}(B)]$.
In particular, we do not have in general that
$[{\operatorname{Op}^{\mathrm{w}}}(A),{\operatorname{Op}^{\mathrm{w}}}(B)]\in\Psi^{m_{1}+m_{2}-1}$.
Finally, the analogue of the Calderón-Vaillancourt Theorem reads (on $H^{s}$
spaces): for every $s\in\mathbb{R}$, there exist $C_{s}>0$ and
$N_{s}\in\mathbb{Z}_{+}$ such that, for every $a\in
S^{m}(\mathrm{T}^{*}\mathbb{R},\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2}))$,

(A.14)
$\left\|{\operatorname{Op}^{\mathrm{w}}}(A)\right\|_{H^{s}(\mathbb{R},\mathscr{H}_{1})\rightarrow
H^{s-m}(\mathbb{R},\mathscr{H}_{2})}\leqslant
C_{0}\sum_{|\alpha|+|\beta|\leqslant
N_{0}}\sup_{r,\rho}\left\|\langle\rho\rangle^{-m+|\beta|}\partial^{\alpha}_{r}\partial_{\rho}^{\beta}A(r,\rho)\right\|_{\mathscr{L}(\mathscr{H}_{1},\mathscr{H}_{2})}.$

###  A.3. Pseudodifferential operators on $\mathbb{R}\times M_{1}$

####  A.3.1. The case of $M_{1}$

As in the case of $\mathbb{R}^{N}$, we can introduce the class of Kohn-
Nirenberg symbols:

$S^{m}(\mathrm{T}^{*}M_{1})=\left\\{a\in\mathcal{C}^{\infty}(\mathrm{T}^{*}M_{1}):\forall(\alpha,\beta),\
\|\langle\zeta_{1}\rangle^{|\beta|-m}\partial_{z_{1}}^{\alpha}\partial_{\zeta_{1}}^{\beta}a\|_{\infty}<\infty\right\\},$

where the derivatives are understood in local coordinate charts. Again, we use
the convention

$S^{m+}(\mathrm{T}^{*}M_{1})=\bigcap_{m^{\prime}>m}S^{m^{\prime}}(\mathrm{T}^{*}M_{1}),$

We will now proceed as in [Zwo12, §14.2.3] to associate to each $a$ in
$S^{m}(\mathrm{T}^{*}M_{1})$ a pseudo-differential operator in
$\Psi^{m}(M_{1})$ (see $\S 14.2.2$ in this reference for a precise definition
of pseudo-differential operators on a compact manifold). For later purposes,
we introduce the semi-norms associated to this class of symbols

$\forall(\alpha,\beta)\in\mathbb{Z}_{+}^{2},\quad
p_{m,\alpha,\beta}(a)=\sum_{j\in
J}\|\langle\zeta_{1}\rangle^{|\beta|-m}\partial_{z_{1}}^{\alpha}\partial_{\zeta_{1}}^{\beta}(\widetilde{\gamma}_{j}^{-1})^{*}(a)\|_{\infty},$

where we have fixed a finite atlas $(U_{j},\gamma_{j})_{j\in J}$ on $M_{1}$
made of smooth diffeomorphism $\gamma_{j}:U_{j}\subset M_{1}\rightarrow
V_{j}\subset\mathbb{R}^{2n-1}$ such that
$\gamma_{j}^{*}(\text{Vol}_{g_{S}})=\text{Leb}_{\mathbb{R}^{2n-1}}$. Here
$\text{Vol}_{g_{S}}$ denotes the volume form induced by the Sasaki metric
$g_{S}$ on $M_{1}$. The existence of an isochore atlas follows from an
argument due to Moser [Mos65]. We then fix a smooth partition of unity
$(\chi_{j})_{j\in J}$, i.e.

$\forall z_{1}\in M_{1},\quad\sum_{j\in
J}\chi_{j}^{2}(z_{1})=1,\quad\chi_{j}\in\mathscr{C}^{\infty}_{c}(U_{j},[0,1]).$

For every $j\in J$, we also fix a smooth function
$\widetilde{\chi}_{j}\in\mathscr{C}^{\infty}_{c}(U_{j},[0,1])$ such that
$\widetilde{\chi}_{j}\chi_{j}=\chi_{j}$. We can then set, for every $a\in
S^{m}(\mathrm{T}^{*}M_{1})$,

(A.15) ${\operatorname{Op}^{M_{1}}}(a)=\sum_{j\in
J}\chi_{j}\gamma_{j}^{*}{\operatorname{Op}^{\mathrm{w}}}\left((\widetilde{\gamma}_{j}^{-1})^{*}(\widetilde{\chi_{j}}a)\right)(\gamma_{j}^{-1})^{*}\chi_{j}:\mathscr{C}^{\infty}(M_{1})\rightarrow\mathscr{C}^{\infty}(M_{1}).$

######  Remark A.2.

For any $\ell\in J$, we fix two smooth functions
$(\psi_{\ell,j})_{j=1,2}\in\mathscr{C}^{\infty}_{c}(U_{\ell},[0,1])$ such that
$\psi_{\ell,j}\widetilde{\chi}_{\ell}=\widetilde{\chi}_{\ell}$ and
$\psi_{\ell,1}\psi_{\ell,2}=\psi_{\ell,2}$. One can then write

$(\gamma_{\ell}^{-1})^{*}\psi_{\ell,1}{\operatorname{Op}^{M_{1}}}(a)\psi_{\ell,2}\gamma_{\ell}^{*}\\\
=\sum_{j\in
J}(\psi_{\ell,1}\chi_{j})\circ\gamma_{\ell}^{-1}(\gamma_{j}\circ\gamma_{\ell}^{-1})^{*}{\operatorname{Op}^{\mathrm{w}}}\left((\widetilde{\gamma}_{j}^{-1})^{*}(\widetilde{\chi_{j}}a)\right)(\chi_{j}\psi_{\ell,2})\circ\gamma_{j}^{-1}(\gamma_{\ell}\circ\gamma_{j}^{-1})^{*}$

Thanks to (A.3) and to (A.9), there exists $\widetilde{a}_{\ell}\in
S^{m}(\mathrm{T}^{*}\mathbb{R}^{2n-1})$ such that

$(\gamma_{\ell}^{-1})^{*}\psi_{\ell,1}{\operatorname{Op}^{M_{1}}}(a)\psi_{\ell,2}\gamma_{\ell}^{*}={\operatorname{Op}^{\mathrm{w}}}(\widetilde{a}_{\ell}).$

Moreover, thanks to (A.10) and to the fact that we picked an isochore atlas,
$a_{\ell}$ has an asymptotic expansion of the form

$\widetilde{a}_{\ell}=(\psi_{\ell,1}\psi_{\ell,2})\circ\gamma_{\ell}^{-1}a\circ\widetilde{\gamma}_{\ell}^{-1}+\sum_{k=2}^{N_{0}}a_{k,\ell}+\mathcal{O}_{S^{m-N_{0}-1}}(1),$

where each $a_{k,l}\in S^{m-k}(\mathrm{T}^{*}\mathbb{R}^{N})$ depends linearly
on derivatives of order $k$ of $a$ with respect to $\xi$ and is supported in
$V_{\ell}\times\mathbb{R}^{N}$. Multiplying this expression by
$\chi_{\ell}^{2}$ and making the sum over $\ell$, we define
$\sigma({\operatorname{Op}^{M_{1}}}(a))$ the principal symbol of
${\operatorname{Op}^{M_{1}}}(a)$. Hence, one finds that the principal symbol
of ${\operatorname{Op}^{M_{1}}}(a)$ is well defined modulo
$S^{m-2}(\mathrm{T}^{*}M_{1})$. This is a specific feature of the fact that we
picked an isochore atlas combined with the Weyl quantization.

######  Remark A.3.

With our choice of quantization, we have the following simplified expressions.
If $b(z_{1})$ is a function that is independent of $\zeta_{1}$, then
${\operatorname{Op}^{M_{1}}}(b)u=bu$. If $Y$ is a smooth vector field on $M$,
then $Y={\operatorname{Op}^{M_{1}}}(\zeta_{1}(Y(z_{1}))+r(z_{1}))$ [Zwo12, Th.
4.5], where $r(z_{1})$ depends on the choice of coordinate charts and linearly
on $Y$. Indeed, one can write

$Yu=\sum_{j\in J}Y(\chi_{j}^{2}u)=\sum_{j\in J}\chi_{j}Y(\chi_{j}u)=\sum_{j\in
J}\chi_{j}\gamma_{j}^{*}\left((\gamma_{j}^{-1})^{*}Y\gamma_{j}^{*}\right)(\gamma_{j}^{-1})^{*}\chi_{j}u,$

and apply [Zwo12, Th. 4.5] to express the vector field
$(\gamma_{j}^{-1})^{*}Y\gamma_{j}^{*}$ as
${\operatorname{Op}^{\mathrm{w}}}(\zeta(Y)+r_{j})$. Note also that, for a
volume preserving vector field, the remainder is equal to $0$ thanks to our
choice of isochore charts.

From our choice of picking an isochore atlas, we can immediately verify that
the following properties hold, for every $a\in S^{m}(\mathrm{T}^{*}M_{1})$

(A.16) $\forall
u,v\in\mathscr{C}^{\infty}(M_{1}),\quad\langle{\operatorname{Op}^{M_{1}}}(a)u,v\rangle_{L^{2}(M_{1})}=\langle
u,{\operatorname{Op}^{M_{1}}}(\overline{a})v\rangle_{L^{2}(M_{1})},$

and there exists $C_{m},N_{m}>0$ such that, if $a\geqslant 0$, then

(A.17) $\forall u\in\mathscr{C}^{\infty}(M_{1}),\
\langle{\operatorname{Op}^{M_{1}}}(a)u,u\rangle_{L^{2}(M_{1})}\geqslant-
C_{m}\|u\|^{2}_{H^{m-\frac{1}{2}}}\sum_{|\alpha|+|\beta|\leqslant
N_{m}}\|\langle\zeta_{1}\rangle^{-m+|\beta|}\partial_{z_{1}}^{\alpha}\partial_{\zeta_{1}}^{\beta}a\|_{\infty}.$

We also find that the Calderón-Vaillancourt Theorem remains true for these
operators. In other words, for every $s\in\mathbb{R}$, one can find
$C_{s},N_{s}>0$ such that

(A.18) $\|{\operatorname{Op}^{M_{1}}}(a)\|_{H^{s}(M_{1})\rightarrow
H^{s-m}(M_{1})}\leqslant C_{s}\sum_{|\alpha|+|\beta|\leqslant
N_{s}}\|\langle\zeta_{1}\rangle^{-m+|\beta|}\partial_{z_{1}}^{\alpha}\partial_{\zeta_{1}}^{\beta}a\|_{\infty}.$

######  Remark A.4.

More generally, one can verify that every pseudo-differential operator
$A\in\Psi^{m}(M_{1})$ verifies

(A.19) $\|A\|_{H^{s}(M_{1})\rightarrow H^{s-m}(M_{1})}<\infty.$

Similarly, following the proof of [Zwo12, Th. 14.1], we can give asymptotic
expansions for the operator
${\operatorname{Op}^{M_{1}}}(a){\operatorname{Op}^{M_{1}}}(b)$ when $a\in
S^{m_{1}}(\mathrm{T}^{*}M_{1})$ and $b\in S^{m_{2}}(\mathrm{T}^{*}M_{1})$. In
fact, for any $\ell\in J$, we fix a smooth function
$\psi_{\ell}\in\mathscr{C}^{\infty}_{c}(U_{\ell},[0,1])$ such that
$\psi_{\ell}\widetilde{\chi}_{\ell}=\widetilde{\chi}_{\ell}$. Proceeding and
keeping the same notations as in Remark A.2, we can then determine the symbol

$(\gamma_{\ell}^{-1})^{*}\psi_{\ell,1}{\operatorname{Op}^{M_{1}}}(a){\operatorname{Op}^{M_{1}}}(b)\psi_{\ell,2}\gamma_{\ell}^{*}$

in view of determining the principal symbol of
${\operatorname{Op}^{M_{1}}}(a){\operatorname{Op}^{M_{1}}}(b)$:

$(\gamma_{\ell}^{-1})^{*}\psi_{\ell,1}{\operatorname{Op}^{M_{1}}}(a){\operatorname{Op}^{M_{1}}}(b)\psi_{\ell,2}\gamma_{\ell}^{*}\\\
=\sum_{j,k\in
J}(\gamma_{j}\circ\gamma_{\ell}^{-1})^{*}(\psi_{\ell,1}\chi_{j})\circ\gamma_{j}^{-1}{\operatorname{Op}^{\mathrm{w}}}((\widetilde{\gamma}_{j}^{-1})^{*}(\widetilde{\chi}_{j}a))(\chi_{j}\widetilde{\psi}_{\ell})\circ\gamma_{j}^{-1}(\gamma_{\ell}\circ\gamma_{j}^{-1})^{*}\\\
\sum_{k\in
J}(\gamma_{k}\circ\gamma_{\ell}^{-1})^{*}(\widetilde{\psi}_{\ell}\chi_{k})\circ\gamma_{k}^{-1}{\operatorname{Op}^{\mathrm{w}}}((\widetilde{\gamma}_{k}^{-1})^{*}(\widetilde{\chi}_{k}b))(\chi_{k}\psi_{\ell,2})\circ\gamma_{k}^{-1}(\gamma_{\ell}\circ\gamma_{k}^{-1})^{*}+{\operatorname{Op}^{\mathrm{w}}}(r_{\ell}),$

where $\widetilde{\psi}_{\ell}\in\mathscr{C}^{\infty}_{c}(U_{\ell})$ is
identically equal to $1$ on the support of $\psi_{\ell,j}$, $j=1,2$, and where
$r_{\ell}$ is an element in $S^{-\infty}(\mathrm{T}^{*}\mathbb{R}^{N})$ thanks
to the kernel representation of pseudo-differential operators recalled in
§A.1. The exact same calculation as in Remark A.2 shows that

$c_{\ell}=(ab)\circ\widetilde{\gamma}_{\ell}^{-1}(\psi_{\ell,1}\psi_{\ell,2})^{2}\circ\gamma_{\ell}^{-1}+\frac{1}{2i}\left\\{a\psi_{\ell,1},b\psi_{\ell,2}\right\\}\circ\widetilde{\gamma}_{\ell}^{-1}+\sum_{k=2}^{N_{0}}c_{k,\ell}+\mathcal{O}_{S^{m_{1}+m_{2}-N_{0}-1}}(1).$

Multiplying by $\chi_{\ell}^{2}$ and summing over $\ell$, we find that

(A.20)
${\operatorname{Op}^{M_{1}}}(a){\operatorname{Op}^{M_{1}}}(b)={\operatorname{Op}^{M_{1}}}\left(ab+\frac{1}{2i}\\{a,b\\}\right)+\mathcal{O}_{\Psi^{m_{1}+m_{2}-2}(M_{1})}(1).$

More precisely, there exist bilinear differential operators888It means that
$L_{k}(a,b)=\sum_{|\alpha|+|\beta|\leqslant
2k}c_{\alpha,\beta}(z_{1};\zeta_{1})\partial^{\alpha}a\,\partial^{\beta}b$.
$(L_{k})_{k\geqslant 0}$ of order $\leqslant 2k$ (depending on our choice of
coordinate charts) such that, for every $a\in S^{m_{1}}(\mathrm{T}^{*}M_{1})$
and for every $b\in S^{m_{2}}(\mathrm{T}^{*}M_{1})$, one has, for every
$N_{0}\geqslant 0$,

(A.21)
$R_{N_{0}}(a,b)={\operatorname{Op}^{M_{1}}}(a){\operatorname{Op}^{M_{1}}}(b)-\sum_{k=0}^{N_{0}}{\operatorname{Op}^{M_{1}}}(L_{k}(a,b))\in\Psi^{m_{1}+m_{2}-N_{0}-1}(M_{1}),$

where

  * •

$L_{0}(a,b)=ab$,

  * •

${\mathrm{L}}_{1}(a,b)=\frac{1}{2i}\\{a,b\\}$,

  * •

for every $k\geqslant 2$ and for every
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})\in\mathbb{Z}^{8n-4}$,
there exist smooth functions $c_{\alpha,k}$ on $M_{1}$ (depending on our
various choices of coordinate charts and cutoff functions) such that

(A.22) $L_{k}(a,b)=\sum_{|\alpha|\leqslant
2k,|\alpha_{2}|+|\alpha_{4}|=k}c_{\alpha,k}(z_{1})\partial_{z_{1}}^{\alpha_{1}}\partial_{\zeta_{1}}^{\alpha_{2}}a\,\partial_{z_{1}}^{\alpha_{3}}\partial_{\zeta_{1}}^{\alpha_{4}}b\in
S^{m_{1}+m_{2}-k}(\mathrm{T}^{*}M_{1}).$

We observe that the formulas for the bracket
$[{\operatorname{Op}^{M_{1}}}(a),{\operatorname{Op}^{M_{1}}}(b)]$ also
translate to the case of manifolds thanks to the above discussion.

######  Remark A.5.

From the proof of (A.20), we can estimate the size of the remainder as a
continuous operator from $H^{s}(M_{1})$ to
$H^{s+N_{0}+1-(m_{1}+m_{2})}(M_{1})$ in terms of the semi-norms of $a$ and
$b$. Indeed, in view of estimating these norms, one can conjugate
$R_{N_{0}}(a,b)$ by $(\gamma_{\ell}^{-1})^{*}\psi_{\ell,1}$ (on the left) and
$(\gamma_{k})^{*}\psi_{k,2}$ (on the right) and verify that all the involved
terms can be expressed in terms of a finite number of derivatives of $a$ and
$b$ (with at least $N_{0}$ derivatives with respect to $\zeta_{1}$). More
precisely, one can verify that, for every $N_{0},N_{1}\geqslant 0$,

$\|R_{N_{0}+N_{1}}(a,b)\|_{H^{s}\rightarrow
H^{s+N_{0}+1-(m_{1}+m_{2})}}\leqslant C_{N_{0},N_{1},s,m_{1},m_{2}}\\\
\sup_{|\alpha_{0}|+|\alpha_{1}|=N_{1}}\sup_{|\beta_{0}|+|\beta_{1}|\leqslant
K_{N_{0},N_{1},s,m_{1},m_{2}}}\left\\{p_{m_{1},\beta_{0}}(\partial_{\zeta_{1}}^{\alpha_{0}}a)p_{m_{2},\beta_{1}}(\partial_{\zeta_{1}}^{\alpha_{1}}b))\right\\}.$

Here, we took two indices $N_{0}$ and $N_{1}$ as, when working on
$\mathbb{R}\times M_{1}$, we will have an extra parameter $\rho$ that we will
consider as a kind of semiclassical gain through the small parameter
$\langle\rho\rangle^{-1}$. For these symbols, this gain will occur with each
derivative with respect to $\zeta_{1}$ so that the above remainder will be of
size $\langle\rho\rangle^{-N_{1}}$. See for instance the proof of Lemma A.12
below.

####  A.3.2. Exponential of pseudo-differential operators on manifolds

We consider a smooth function $\lambda$ verifying the following assumptions:

(A.23) $\lambda\in
S^{0+}(\mathrm{T}^{*}M_{1},\mathbb{R})\quad\text{and}\quad\exists m,R\geqslant
0\ \text{such that}\ |\lambda(z_{1};\zeta_{1})|\leqslant
m\log(R+\langle\zeta_{1}\rangle).$

Then, according to [Zwo12, Th. 8.6] (adapted to the case of compact manifolds,
see also [BC94, Th. 6.4] for general Hörmander-Weyl symbols [Hör07, §18.4]),
one finds that, the equation

$\partial_{t}B(t)={\operatorname{Op}^{M_{1}}}(\lambda)B(t),\quad
B(0)=\text{Id}$

has a solution that we denote by
$\exp(t{\operatorname{Op}^{M_{1}}}(\lambda))$. More precisely, it is a smooth
function of $t$ with values in continuous linear mapping from
$\mathscr{C}^{\infty}(M_{1})$ to itself. In fact, the proof in [Zwo12]
(adapted to the case of compact manifolds) which is of semiclassical nature
shows that, for every $R_{0}\geqslant 1$, one can find $0<h_{0}<1$ such that,
for every $t\in[-R_{0},R_{0}]$, for every $s\in\mathbb{R}$ and for every
$0<h<h_{0}$, the following holds:

  * •

there exists a bounded operator $Q_{h,1}(t):H^{s}\rightarrow H^{s}$ (with a
norm that is bounded independently of $t\in[-R_{0},R_{0}]$ and $0<h<h_{0}$)
such that one can find a symbol $b_{h,1}$ which is equal to $1$ modulo in
$S^{-1+}(\mathrm{T}^{*}M_{1})$ with the following property

(A.24)
$\exp\left(t{\operatorname{Op}^{M_{1}}}(\lambda(z_{1},h\zeta_{1}))\right)=Q_{h,1}(t){\operatorname{Op}^{M_{1}}}\left(e^{t\lambda(z_{1},h\zeta_{1})}b_{h,1}(z_{1},h\zeta_{1})\right),$

where we note that $b_{h,1}$ is independent of $t$ and has all its semi-norms
(in $S^{0+}(\mathrm{T}^{*}M_{1})$) uniformly bounded in terms of $0<h<h_{0}$.
Recall that the operator
${\operatorname{Op}^{M_{1}}}\left(e^{t\lambda(z_{1},h\zeta_{1})}b_{h,1}(z_{1},h\zeta_{1})\right)$
appears has a pseudo-inverse in the proof of [Zwo12, Lemma 8.4], namely

(A.25)
${\operatorname{Op}^{M_{1}}}\left(e^{t\lambda(z_{1},h\zeta_{1})}b_{h,1}(z_{1},h\zeta_{1})\right){\operatorname{Op}^{M_{1}}}\left(e^{-t\lambda(z_{1},h\zeta_{1})}\right)=\text{Id}+\mathcal{O}_{H^{s}\rightarrow
H^{s}}(h).$

Observe that the involved symbols lie in the class
$S^{R_{0}m}(\mathrm{T}^{*}M_{1})$ of standard Kohn-Nirenberg symbols.

  * •

Similarly, there exists a bounded operator $Q_{h,2}(t):H^{s}\rightarrow H^{s}$
(with a norm that is bounded independently of $t\in[-R_{0},R_{0}]$ and
$0<h<h_{0}$) such that one can find a symbol $b_{h,2}$ which is equal to $1$
modulo $S^{-1+}(\mathrm{T}^{*}M_{1})$ with the following property

(A.26)
$\exp\left(t{\operatorname{Op}^{M_{1}}}(\lambda(z_{1},h\zeta_{1}))\right)={\operatorname{Op}^{M_{1}}}\left(e^{t\lambda(z_{1},h\zeta_{1})}b_{h,2}(z_{1},h\zeta_{1})\right)Q_{h,2}(t),$

with the same properties for $b_{h,2}$.

######  Remark A.6.

Recall that the proof in [Zwo12] goes as follows. Suppose that there exists a
solution to

$\partial_{t}B_{h}(t)={\operatorname{Op}^{M_{1}}}_{h}(\lambda)B_{h}(t),\quad
B_{h}(0)=\text{Id},$

where
${\operatorname{Op}^{M_{1}}}_{h}(b(z_{1};\zeta_{1}))={\operatorname{Op}^{M_{1}}}(b(z_{1},h\zeta_{1}))$.
Then, letting $U_{h}(t)={\operatorname{Op}_{h}^{M_{1}}}(e^{t\lambda})$, one
has

$\partial_{t}\left(U_{h}(-t)B_{h}(t)\right)=\left(-{\operatorname{Op}_{h}^{M_{1}}}(\lambda
e^{-t\lambda})+{\operatorname{Op}_{h}^{M_{1}}}(e^{-t\lambda}){\operatorname{Op}_{h}^{M_{1}}}(\lambda)\right)B_{h}(t)=V_{h}(t)B_{h}(t).$

Hence, if one sets $C_{h}(t)=-V_{h}(t)U_{h}(-t)^{-1}$, one gets a smooth
family of bounded operator on $H^{s}(M_{1})$ using the composition rule and
the Calderón-Vaillancourt Theorem. Now the Cauchy-Lipschitz Theorem ensures
the existence and uniqueness of a smooth solution $Q_{h}(t):H^{s}\rightarrow
H^{s}$ to the problem

$\partial_{t}Q_{h}(t)=C_{h}(t)Q_{h}(t),\quad Q_{h}(0)=\text{Id},$

and $B_{h}(t)$ is then given by $U_{h}(-t)^{-1}Q_{h}(t)$. Using (A.25) with
$h>0$ small enough, one can find an expression for $U_{h}(-t)^{-1}$ and this
yields (A.26). The same reasoning yields (A.24).

######  Remark A.7.

Note that it would require slightly more work to check that $Q_{h,j}(t)$ is a
pseudo-differential operator but we do not discuss this issue as this is not
necessary for our analysis.

####  A.3.3. Pseudodifferential operators on $\mathbb{R}\times M_{1}$ as
operator valued pseudo-differential operators on $\mathbb{R}$

In view of our analysis, it is convenient to use the framework of §A.2 to
define operators associated with symbols lying in

$\overline{S}^{m}(\mathrm{T}^{*}(\mathbb{R}\times
M_{1}))=\left\\{a\in\mathcal{C}^{\infty}(\mathrm{T}^{*}(\mathbb{R}\times
M_{1})):\forall(\alpha,\beta),\
\|\langle(\rho,\zeta_{1})\rangle^{|\beta|-m}\partial_{rz_{1}}^{\alpha}\partial_{\rho\zeta_{1}}^{\beta}a\|_{\infty}<\infty\right\\}.$

Namely, we define

(A.27)
$\mathbf{Op}(a)u(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i(r-s)\rho}{\operatorname{Op}^{M_{1}}}\left(a\left(\frac{r+s}{2},\rho;.\right)\right)u(s,z_{1})dsd\rho,$

which can be identified with a pseudo-differential operator with symbol taking
values in $\mathscr{L}(H^{\sigma}(M_{1}),H^{\sigma-m}(M_{1}))$ (for every
$\sigma\in\mathbb{R}$). Note that compared with the class $S^{m}$ we require
the derivatives to be bounded uniformly on $\mathbb{R}$ (rather than on
compact sets).

Letting

$\mathscr{S}(\mathbb{R}\times
M_{1})=\left\\{u\in\mathscr{C}^{\infty}(\mathbb{R}\times M_{1}):\
\forall(\alpha,\beta),\
\|(1+|r|)^{\alpha}\partial^{\beta}_{rz_{1}}u(r,z_{1})\|_{\infty}<\infty\right\\},$

this quantization procedure induces a bounded operator

$\mathbf{Op}(a):\mathscr{S}(\mathbb{R}\times
M_{1})\rightarrow\mathscr{S}(\mathbb{R}\times M_{1}).$

######  Remark A.8.

The advantage of devising such a definition is that we can quantize more
exotic symbols than in the standard Kohn-Nirenberg classes
$\overline{S}^{m}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ or
$S^{m}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$. For instance, one can pick in
this definition functions of the form $(1+\rho^{2})^{m_{0}}a(z_{1};\zeta_{1})$
(with $m_{0}\in\mathbb{R}$ and $a\in S^{m}(\mathrm{T}^{*}M_{1})$) and view the
corresponding operator as an element of
$\Psi^{m_{0}}(\mathbb{R},\mathscr{L}(H^{\sigma}(M_{1}),H^{\sigma-m}(M_{1})))$.

######  Remark A.9.

When picking symbols $a$ in $\overline{S}^{m}(\mathrm{T}^{*}(\mathbb{R}\times
M_{1}))$ or in $S^{m}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$, one recover
using (A.9) that $\mathbf{Op}(a)$ is a pseudo-differential operator of order
$m$ on the noncompact manifold $\mathbb{R}\times M_{1}$ in the sense of [DZ19,
Prop. E.13, Rk p.551]. For instance, $\mathbf{Op}(a)$ verifies the Calderón-
Vaillancourt Theorem between $H^{s}_{\text{comp}}$ and $H^{s-m}_{\text{loc}}$
[DZ19, Prop. E.22]. Moreover, if we fix a smooth cutoff function
$\chi\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$, then $\mathbf{Op}(a)\chi$ is
properly supported in the sense of [DZ19, §A.7] and it is thus amenable to the
composition rule for pseudo-differential operators on noncompact manifold
[DZ19, Prop. E.17]. Similarly, if
$\widetilde{\chi}\in\mathcal{C}^{\infty}_{c}(\mathbb{R})$, then
$\chi\mathbf{Op}(a)\widetilde{\chi}$ is compactly supported. In that case,
$\chi\mathbf{Op}(a)\widetilde{\chi}$ verifies the Gårding inequality for
smooth $u$ that are compactly supported in $\mathbb{R}\times M_{1}$ [DZ19,
Prop. E.23].

####  A.3.4. Cutoff functions in the $\mathbb{R}$ variable

In the above class of symbols, one has the following lemma:

######  Lemma A.10.

Let $\chi_{1},\chi_{2}$ be two smooth functions on $\mathbb{R}$ that have
disjoint supports, all of whose derivatives are bounded and such that
$\chi_{2}$ has compact support. Let
$a\in\overline{S}^{m}(\mathrm{T}^{*}(M_{1}\times\mathbb{R}))$. Then, one has,
for every $N_{0},N_{1}\geqslant 0$,

$\|\chi_{1}\mathbf{Op}(a)\chi_{2}\|_{H^{-N_{1}}(\mathbb{R},H^{-N_{0}}(M_{1}))\rightarrow
H^{N_{1}}(\mathbb{R},H^{N_{0}}(M_{1}))}<\infty,$

and

$\|\chi_{1}\mathbf{Op}(a)X\chi_{2}\|_{H^{-N_{1}}(\mathbb{R},H^{-N_{0}}(M_{1}))\rightarrow
H^{N_{1}}(\mathbb{R},H^{N_{0}}(M_{1}))}<\infty,$

where $X=rX_{1}$ is the geodesic vector field.

######  Remark A.11.

Note that, for every $N_{0}\geqslant 0,$ one has

$\left\|u\right\|_{H^{N_{0}}(\mathbb{R}\times M_{1})}\leqslant
C_{N_{0}}\left\|u\right\|_{H^{N_{0}}(\mathbb{R},H^{N_{0}}(M_{1}))},$

and

$\left\|u\right\|_{H^{-N_{0}}(\mathbb{R},H^{-N_{0}}(\times M_{1}))}\leqslant
C_{N_{0}}\left\|u\right\|_{H^{-N_{0}}(\mathbb{R}\times M_{1})}.$

Hence, Lemma A.10 can be translated to the standard Sobolev spaces on
$\mathbb{R}\times M_{1}$.

###### Proof.

We treat the case with $X$ (the other case works analoguously). To that aim,
we write the composition formula (A.12) for $\chi_{1}\mathbf{Op}(a)X_{1}$ and
we find, for every $N_{2}\geqslant 0$,

$\chi_{1}\mathbf{Op}(a)X_{1}={\operatorname{Op}^{\mathrm{w}}}\left(\sum_{k=0}^{N_{2}}\frac{1}{(-2i)^{k}k!}\chi^{(k)}_{1}{\operatorname{Op}^{M_{1}}}(\partial_{\rho}^{k}a)X_{1}+R_{N_{2}}\right),$

where the remainder lies in
$S^{m-N_{2}-1}(\mathbb{R},\mathscr{L}(H^{s}(M_{1}),H^{s-m}(M_{1})))$ with the
semi-norm of $R_{N_{0}}$ (in that class) that depends on the semi-norms of
$\chi_{1}^{(N_{2})}(r){\operatorname{Op}^{M_{1}}}(\partial_{\rho}^{N_{2}}a)X_{1}$.
We apply the composition formula one more time to this expression after
multiplying this equality on the right by $r\chi_{2}(r)$. We find that

$\chi_{1}\mathbf{Op}(a)rX_{1}\chi_{2}={\operatorname{Op}^{\mathrm{w}}}\left(\widetilde{R}_{N_{2}}(r,\rho)\right)\widetilde{\chi}_{2},$

where $\widetilde{\chi}_{2}$ is compactly supported function which is equal to
$1$ on the support of $\chi_{2}$ and $\widetilde{R}_{N_{2}}$ is an operator-
valued symbol that can be expressed in terms
${\operatorname{Op}^{M_{1}}}(\partial_{\rho}^{N_{2}}a)$. As $a$ belongs to
$\overline{S}^{m}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$, this defines a
bounded operator from $H^{-(N_{0}+1)}(M_{1})$ to $H^{N_{0}}(M_{1})$ whose norm
is bounded by $\langle\rho\rangle^{-N_{1}^{\prime}}$ provided that $N_{2}\gg
N_{0}+N_{1}^{\prime}+m.$ Picking $N_{1}^{\prime}\gg N_{1}$, this yields the
expected upper bound. ∎

####  A.3.5. Pseudo inverse on $\mathbb{R}\times M_{1}$

We conclude this appendix with two observations showing that one can always
find an (almost) inverse for $\mathbf{Op}(a)$ when $a>0$ is a simple enough
symbol. More precisely, suppose that $\lambda(z_{1};\zeta_{1})$ verifies
(A.23) and that $m_{0}\in\mathbb{R}$. We define

$B_{\lambda,m_{0}}u(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i(r-s)\rho}(1+\rho^{2})^{m_{0}}\exp{\operatorname{Op}^{M_{1}}}(\lambda)u(s,z_{1})dsd\rho.$

With this notation at hand, one finds

(A.28) $\forall u\in\mathscr{S}(\mathbb{R}\times M_{1}),\quad
B_{\lambda,m_{0}}B_{-\lambda,-m_{0}}u=u.$

Finally, we discuss the case of the more general symbols in
$\overline{S}^{m}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$:

######  Lemma A.12.

Let $\lambda\in\overline{S}^{+0}(\mathrm{T}^{*}(M_{1}\times\mathbb{R}))$ that
is independent of the variable $r$ and such that
$|\lambda(z_{1},\rho,\zeta_{1})|\leqslant
m\ln(R+\langle(\rho,\zeta_{1})\rangle)$ for some given $m,R\geqslant 0$. Then,
$a=e^{\lambda}$ belongs to $\overline{S}^{m+}(\mathrm{T}^{*}(\mathbb{R}\times
M_{1}))$

Moreover, given $N_{0},N_{1}\geqslant 0$, one can find
$b_{N_{0},N_{1}}\in\overline{S}^{0}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$
which is independent of $r$ and equal to $1$ modulo
$\overline{S}^{-1+}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ such that, for
every $\chi\in\mathscr{C}^{\infty}_{c}(\mathbb{R})$, the operator

$\left(\mathbf{Op}(e^{-\lambda}b_{N_{0},N_{1}})\mathbf{Op}(e^{\lambda})-\operatorname{Id}\right)\chi:H^{-N_{1}}\left(\mathbb{R},H^{-N_{0}}(M_{1})\right)\rightarrow
H^{N_{1}}\left(\mathbb{R},H^{N_{0}}(M_{1})\right)$

is bounded.

###### Proof.

The fact that $e^{\lambda}$ belongs to
$\overline{S}^{m+}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ follows directly
from the definition of this class of symbols. We now fix some
$b\in\overline{S}^{0}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ which is equal
to $1$ modulo $\overline{S}^{-1+}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ and
which is independent of the variable $r\in\mathbb{R}$. From the composition
rule (A.12), one has that, for every $u\in\mathscr{S}(\mathbb{R}\times
M_{1})$,

$\mathbf{Op}(e^{\lambda})\mathbf{Op}(e^{-\lambda}b)u(r,z_{1}))=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i(r-s)\rho}{\operatorname{Op}^{M_{1}}}(e^{-\lambda(\rho)}b(\rho)){\operatorname{Op}^{M_{1}}}(e^{\lambda(\rho)})u(s,z_{1})d\rho
ds.$

Using the composition rule (A.20) on $M_{1}$ together with Remark A.5, we can
find
$b_{N_{0},N_{1}}(z_{1},\rho,\zeta_{1})\in\overline{S}^{0}(\mathrm{T}^{*}(\mathbb{R}\times
M_{1}))$ such that, for every $\rho\in\mathbb{R}$,

${\operatorname{Op}^{M_{1}}}(e^{-\lambda(\rho)}b_{N_{0},N_{1}^{\prime}}(\rho)){\operatorname{Op}^{M_{1}}}(e^{\lambda(\rho)})=\text{Id}+\mathbf{R}_{N_{0},N_{1}^{\prime}}(\rho),$

where
$\|\mathbf{R}_{N_{0},N_{1}^{\prime}}(\rho)\|_{H^{-N_{0}}\rightarrow{H}^{N_{0}}}\leqslant
C_{N_{0},N_{1}^{\prime}}\langle\rho\rangle^{-N_{1}^{\prime}}$. Applying this
identity with $N_{1}^{\prime}\gg N_{1}$, we obtain the expected result. ∎

##  Appendix B Proof of Theorem 1.3

In this short appendix, we briefly explain how to deduce Theorem 1.3 from the
upper bound (1.4),

$\left|\int_{\mathrm{S}^{*}\Sigma}u(t)\psi\mathrm{d}{\mathrm{L}}_{1}-\int_{\mathrm{S}^{*}\Sigma}\left(\int_{\mathrm{S}^{*}\Sigma}u_{0}\mathrm{d}{\mathrm{L}}_{1}\right)\psi\mathrm{d}{\mathrm{L}}_{1}\right|\leqslant
Ce^{-\vartheta_{0}|t|}\|u_{0}\|_{\mathscr{C}^{N_{0}}}\|\psi\|_{\mathscr{C}^{N_{0}}}.$

Write

$\int_{\mathrm{T}^{*}\Sigma}u(t)\psi\mathrm{d}{\mathrm{L}}=c_{n}\int_{0}^{\infty}r^{n-1}\int_{\mathrm{S}^{*}\Sigma}u_{0}(r,\varphi_{tr}(x,\xi_{1}))\psi(r,x,\xi_{1})\mathrm{d}{\mathrm{L}}_{1}(x,\xi_{1})\mathrm{d}r,$

where $(x,\xi)=(r,x,\xi_{1})$ are the spherical coordinates on
$\mathrm{T}^{*}\Sigma$ and $c_{n}$ is a normalizing constant depending only on
$n$. Applying (1.4) to the integral over $\mathrm{S}^{*}\Sigma$, one gets

$\int_{\mathrm{T}^{*}\Sigma}u(t)\psi\mathrm{d}{\mathrm{L}}=\int_{\mathrm{T}^{*}\Sigma}\left(\int_{\mathrm{S}^{*}\Sigma}u_{0}(r,x,\xi_{1})\mathrm{d}{\mathrm{L}}_{1}(x,\xi_{1})\right)\psi\mathrm{d}{\mathrm{L}}+\mathcal{O}_{\psi,u_{0}}\left(\int_{r_{0}}^{\infty}e^{-tr\vartheta_{0}}r^{n-1}dr\right),$

where the constant depends linearly on the $\mathscr{C}^{N_{0}}$ norms of
$u_{0}$ and $\psi$. This concludes the argument.

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