# Nonlinear chaotic Vlasov equations

Yann Chaubet  ,  Daniel Han-Kwan  and  Gabriel Rivière  Laboratoire de
Mathématiques Jean Leray, Nantes Université, UMR CNRS 6629, 2 rue de la
Houssinière, 44322 Nantes Cedex 03, France<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this article, we study nonlinear Vlasov equations with a smooth interaction
kernel on a compact manifold without boundary where the geodesic flow exhibits
strong chaotic behavior, known as the Anosov property. We show that, for small
initial data with finite regularity and supported away from the null section,
there exist global solutions to the nonlinear Vlasov equations which weakly
converge to an equilibrium of the free transport equation, and whose potential
strongly converges to zero, both with exponential speed. Central to our
approach are microlocal anisotropic Sobolev spaces, originally developed for
studying Pollicott-Ruelle resonances, that we further refine to deal with the
geometry of the full cotangent bundle, which paves the way to the analysis of
nonlinear Vlasov equations.

###### Contents

  1. 1 Introduction
  2. 2 An heuristic scheme of proof for a toy model
  3. 3 Sliced and global escape functions
  4. 4 Anisotropic Sobolev norms adapted to the free dynamics
  5. 5 Global existence
  6. 6 Nonlinear exponential stability
  7. A Toolbox on microlocal analysis
  8. B Proof of Theorem 1.3

##  1\. Introduction

Let $(\Sigma,\mathrm{g})$ be a smooth, compact, connected, oriented and
Riemannian manifold of dimension $n\geqslant 2$ which has no boundary. All
along the article, we make the assumption that $(\Sigma,\mathrm{g})$ is an
_Anosov_ manifold [Ano67a], which means that the geodesic flow displays
chaotic features, for instance strong sensitivity to initial conditions (we
refer to Section 3.2 below for a proper definition). In particular, this
condition holds as soon as $(\Sigma,\mathrm{g})$ has negative sectional
curvature; this includes the special case of compact quotients of the
hyperbolic plane (see e.g. [FH19]). We are interested in kinetic mean-field
equations, namely in the _nonlinear_ Vlasov equation

(1.1) $\left\\{\begin{aligned}
&\partial_{t}u+\left\\{\mathrm{H}+\Phi(u),\,u\right\\}=0,\\\
&u|_{t=0}=u_{0}\in L^{1}(\mathrm{T}^{*}\Sigma).\end{aligned}\right.$

Here $\\{\cdot,\,\cdot\\}$ denotes the Poisson bracket in
$\mathrm{T}^{*}\Sigma$, $\mathrm{H}$ stands for the classical Hamiltonian
associated with the geodesic flow, that is
$\mathrm{H}(x,\xi)=\displaystyle{\frac{1}{2}|\xi|^{2}_{x}},$ and the
interaction potential $\Phi(u)\in\mathscr{C}^{\infty}(\Sigma)$ is given by

(1.2)
$\Phi(u)(x)=\mathrm{K}\pi_{*}u(x)=\int_{\mathrm{T}^{*}\Sigma}K(x,y)u(y,\eta)\mathrm{d}\mathrm{L}(y,\eta),$

where $\mathrm{L}$ is the Liouville measure on $\mathrm{T}^{*}\Sigma$,
$\pi:(x,\xi)\in\mathrm{T}^{*}\Sigma\mapsto x\in\Sigma$ is the natural
projection and $\mathrm{K}$ is an integral operator acting on $L^{2}(\Sigma)$.
Its kernel $K\in\mathscr{C}^{\infty}(\Sigma\times\Sigma,\mathbb{R})$ is
assumed to satisfy

$\int_{\Sigma}K(x,y)\mathrm{d}\text{vol}_{\mathrm{g}}(y)=0\quad\text{for each
}x\in\Sigma,$

with $\text{vol}_{\mathrm{g}}$ the Riemannian volume induced by the metric
$\mathrm{g}$. From the physical point of view, when $u_{0}\geqslant 0$, the
function $u(t)$ can be interpreted as the distribution function in phase space
of a population of particles, whose dynamics in $\Sigma$ is steered by a self-
induced potential $\Phi(u)$, created by their non-trivial spatial density
$\pi_{*}u(t,x)=\int_{\mathrm{T}_{x}^{*}\Sigma}u(t,x,\xi)\,\mathrm{d}\mathrm{L}_{x}(\xi)$.

The main objective of the present paper is to prove the following result.

######  Theorem 1.1.

There are $N_{0}\in\mathbb{N}_{\geqslant 1}$ and $\vartheta_{0}>0$ depending
only on $(\Sigma,\mathrm{g})$ (and not on the interaction kernel $K$) such
that the following holds.

For any $0<r_{0}<1$, there are $C,\varepsilon>0$ such that, if $u_{0}$
satisfies

$\|u_{0}\|_{\mathscr{C}^{N_{0}}}\leqslant\varepsilon\quad\text{and}\quad\operatorname{supp}u_{0}\subset\\{(x,\xi)\in\mathrm{T}^{*}\Sigma\leavevmode\nobreak\
:\leavevmode\nobreak\ ~|\xi|_{x}\in(r_{0},r_{0}^{-1})\\},$

then (1.1) admits a unique solution
$u\in\mathscr{C}^{1}(\mathbb{R},L^{1}(\mathrm{T}^{*}\Sigma))$. Moreover, there
exist functions $h_{\pm\infty}\in L^{\infty}_{\mathrm{comp}}(\mathbb{R}_{>0})$
such that, for all $\psi\in\mathscr{C}^{\infty}_{c}(\mathrm{T}^{*}\Sigma)$,

$\left|\int_{\mathrm{T}^{*}\Sigma}u(\pm
t)\,\psi\,\mathrm{d}{\mathrm{L}}-\int_{\mathrm{T}^{*}\Sigma}\left(h_{\pm\infty}\circ\mathrm{H}\right)\,\psi\,\mathrm{d}{\mathrm{L}}\right|\leqslant
Ce^{-t\vartheta_{0}r_{0}}\|\psi\|_{\mathscr{C}^{N_{0}}}\|u_{0}\|_{\mathscr{C}^{N_{0}}},\qquad
t\geqslant 0.$

In addition, for every $N\geqslant 0$, there is $C_{N}>0$ such that

$\|\Phi(u(\pm t))\|_{\mathscr{C}^{N}}\leqslant
C_{N}e^{-t\vartheta_{0}r_{0}}\|u_{0}\|_{\mathscr{C}^{N_{0}}},\qquad t\geqslant
0.$

Theorem 1.1 provides a fine description of the large time behavior of small
solutions with finite regularity, initially compactly supported away from the
null section $\\{\xi=0\\}$, to the nonlinear Vlasov equation (1.1). Namely, it
shows that the distribution function $u(t)$ weakly converges to an equilibrium
of the the free transport equation

$\partial_{t}f+\left\\{\mathrm{H},\,f\right\\}=0,$

and the potential $\Phi(u(t))$ strongly converges to $0$, both with
exponential speed. A direct consequence of this theorem, by a rescaling
argument, is the following statement which states convergence to equilibrium
for weakly nonlinear perturbations of the free Vlasov equation.

######  Corollary 1.2.

There exists $N_{0}\geqslant 1$ such that, for any $0<r_{0}<1$ and any
$M_{0}>0$, one can find $\varepsilon_{0}>0$ so that if
$\|u_{0}\|_{\mathscr{C}^{N_{0}}}\leqslant M_{0}$ and

$\operatorname{supp}u_{0}\subset\\{(x,\xi)\in\mathrm{T}^{*}\Sigma\leavevmode\nobreak\
:\leavevmode\nobreak\ ~|\xi|_{x}\in(r_{0},r_{0}^{-1})\\},$

then, for every $0\leqslant\varepsilon\leqslant\varepsilon_{0}$ the solution
of the problem

(1.3) $\left\\{\begin{aligned}
&\partial_{t}u+\left\\{\mathrm{H}+\varepsilon\Phi(u),\,u\right\\}=0,\\\
&u|_{t=0}=u_{0},\end{aligned}\right.$

satisfies the same properties as in the conclusion of Theorem 1.1.

Under this form, our main result can be understood as an analogue for the
Vlasov equation of the recent results of Bahsoun, Liverani and Sélley on
globally coupled Anosov diffeomorphisms, acting discretely on distribution
functions [BLS23] (see also [ST21, Gal22, Tan23] and references therein). More
precisely, they consider weakly nonlinear mean-field perturbations of transfer
operators associated with Anosov diffeomorphisms. They prove that such
nonlinear transfer operators admit a unique physical (or Sinai-Ruelle-Bowen)
measure and that convergence to this equilibrium measure holds at an
exponential rate. The strategy of [BLS23] is based on a fixed point theorem
argument which is tailored for discrete time models.

###  1.1. Exponential mixing for the geodesic flow with the Anosov property

In the free case (that is when $K\equiv 0$), Theorem 1.1 is a consequence of
by now classical results. The weak convergence of $u$ without explicit speed
of convergence is a consequence of Anosov’s seminal work [Ano67a] following
earlier contributions by Hopf on the ergodicity of the Liouville measure
[Hop39, Hop40]. Anosov’s result was refined for manifolds of _constant_
negative curvature by providing speed of convergence using tools from harmonic
analysis on Lie groups [Rat87, Moo87, Pol92]. The extension of these works to
variable negative curvature was obtained in dimension $2$ using methods from
Markov partitions by Dolgopyat [Dol98] and in any dimension using Banach
spaces with anisotropic regularity adapted to Anosov flows by Liverani [Liv04]
(see also [Che98] for earlier results showing subexponential decay when
$r_{0}>0$). The outcome of the results in these references is that, when
$K\equiv 0$, we have

(1.4)
$\left|\int_{\mathrm{S}^{*}\Sigma}u(t)\psi\,\mathrm{d}{\mathrm{L}}_{1}-\int_{\mathrm{S}^{*}\Sigma}\left(\int_{\mathrm{S}^{*}\Sigma}u_{0}\mathrm{d}{\mathrm{L}}_{1}\right)\psi\,\mathrm{d}{\mathrm{L}}_{1}\right|\leqslant
Ce^{-\vartheta_{0}|t|}\|u_{0}\|_{\mathscr{C}^{N_{0}}}\|\psi\|_{\mathscr{C}^{N_{0}}},$

where $\mathrm{S}^{*}\Sigma=\\{(x,\xi)\in\mathrm{T}^{*}\Sigma:|\xi|_{x}=1\\}$
and ${\mathrm{L}}_{1}$ is the desintegration of the Liouville measure on
$\mathrm{S}^{*}\Sigma$. Integrating this expression over
$\mathrm{T}^{*}\Sigma$ yields the following theorem for the _free_ Vlasov
equation:

######  Theorem 1.3 (Dolgopyat-Liverani).

Let $K\equiv 0$. There exist an integer $N_{0}\geqslant 1$ and two constants
$C,\vartheta_{1}>0$ such that the following holds. For any $0\leqslant
r_{0}<1$, for any $u_{0}\in\mathscr{C}^{N_{0}}(\mathrm{T}^{*}\Sigma)$ such
that

$\operatorname{supp}u_{0}\subset\left\\{(x,\xi)\in\mathrm{T}^{*}\Sigma\leavevmode\nobreak\
:\leavevmode\nobreak\ ~|\xi|_{x}\in[r_{0},\infty[\right\\},$

and for any $\psi\in\mathscr{C}^{\infty}_{c}(\mathrm{T}^{*}\Sigma)$, one has

$\forall t\geqslant
0,\quad\left|\int_{\mathrm{T}^{*}\Sigma}u({\pm}t)\,\psi\,\,\mathrm{d}{\mathrm{L}}-\int_{\mathrm{T}^{*}\Sigma}\left(h_{\operatorname{lin}}\circ\mathrm{H}\right)\psi\,\mathrm{d}{\mathrm{L}}\right|\leqslant
C\frac{e^{-t\vartheta_{1}r_{0}}}{(1+t)^{n}}\|\psi\|_{\mathscr{C}^{N_{0}}}\|u_{0}\|_{\mathscr{C}^{N_{0}}},$

where

$h_{\operatorname{lin}}(r)=\int_{\mathrm{S}^{*}\Sigma}u_{0}(x,r\xi_{1})\,\mathrm{d}{\mathrm{L}}_{1}(x,\xi_{1}).$

Note that, in this free setting, the case $r_{0}=0$ is allowed but only
algebraic decay is reached in that case. For the reader’s convenience, see
Appendix B for a short proof of this theorem based on the mixing property
(1.4). Let us emphasize that we will not be able to use this theorem directly
and that we will rather rely on microlocal refinements of this result (see
Theorem 4.6 below). Indeed, (1.4) was subsequently recovered and improved
using methods from microlocal analysis [Tsu10, Tsu12, NZ15, FT17, FT21], with
a strategy initiated in [BT07, FRS08, FS11] and which can display analogies
with the study of quantum resonances [AC71, BC71, HS86]. These alternative
approaches allowed to give sharp expressions on the high frequency value of
the decay rate $\vartheta_{0}$ and to obtain results on the distribution of
the so-called Pollicott–Ruelle resonances. On top of that, the Fourier
analysis machinery behind these works turns out to be relevant for
applications to nonlinear Vlasov equations as we shall see in this article. In
fact, in all the above references, the upper bound in (1.4) involves the norm
of $u_{0}$ and $\psi$ in some Banach space of distributions with anisotropic
Hölder or Sobolev regularity adapted to the free dynamics. For the application
to nonlinear Vlasov equations, the microlocal anisotropic Sobolev norms of
Faure and Sjöstrand [FS11] appearing in the upper bound obtained by
Nonnenmacher and Zworski in [NZ15, §9] will be appropriate and this microlocal
approach will allow us to show that

(1.5)
$h_{\pm\infty}(r)=h_{\text{lin}}(r)-\int_{\mathrm{S}^{*}\Sigma}\left(\int_{0}^{\pm\infty}\\{\Phi(u(s)),u(s)\\}(x,r\xi_{1})ds\right)\mathrm{d}{\mathrm{L}}_{1}(x,\xi_{1})$

is well defined and that it is indeed the limiting distribution of the
solution to (1.1) provided $\|u_{0}\|_{\mathscr{C}^{N}}$ is small enough. Yet,
it may happen that other types of anistropic norms could be used modulo some
careful adaptations of our nonlinear arguments. For instance, the
aforementioned recent work [BLS23] (see also [BL24]) makes use of geometric
anisotropic Banach spaces of distributions as developped in [BL22] to study
nonlinear mean-field perturbations of Anosov diffeomorphisms, which can be
somehow viewed as discrete analogues of the Vlasov equation.

###  1.2. Other Vlasov equations of geometric origin

Other geometric Vlasov equations were previously studied with motivations from
physics and with an emphasis on the fine large time behavior of solutions.
Such equations on manifolds are in fact natural in the context of general
relativity. See for instance [Rin13], [ACGS22] or the recent works about the
stability of the Minkowski space-time for Einstein–Vlasov [Tay17, LT20, FJS21,
BFJ+21] and references therein. Compared with the present work, the free
Hamiltonian is given in these references by
$\mathrm{H}(x,\xi)=|\xi|_{x}^{2}/\sqrt{|\xi|^{2}_{x}+m}$ (with $m\geqslant
0$), the potential $\Phi$ is taken to be identically $0$ and the nonlinearity
comes from the fact that the (time dependent) metric $\mathrm{g}$ is given by
the Einstein equation with a source term depending linearly on certain
averages of $u$. We also refer to [ABJ17, Big23, VR22] which establish various
decay estimates of spacetime observables for the (linear) massless Vlasov
equation outside of black hole geometries, whose associated flows display
hyperbolic features. In other directions, we may finally mention [Mos23] for
the instability of the Anti-de Sitter space-time or [HL24b, HL24a, Tou23,
Tou24] for works related to derivations of Einstein–Vlasov from the Einstein
vacuum equation in the high frequency regime, the so-called Burnett
conjecture.

More closely related to our geometric framework, Velozo Ruiz and Velozo Ruiz
[VRVR23] treated recently the case of the free Vlasov equation set on a
_noncompact_ asymptotically hyperbolic manifold $\Sigma$ whose trapped set is
empty and established exponential convergence to $0$ of $\pi_{*}(u)(t)$ when
$r_{0}>0$ and polynomial convergence when $r_{0}\equiv 0$ and
$(\Sigma,\mathrm{g})=(\mathbb{H}^{2},y^{-2}(\mathrm{d}x^{2}+\mathrm{d}y^{2}))$
(see also [Sal07] which considers the case of a general metric on
$\mathbb{R}^{n}$). In this non-compact and non-trapping setting, dispersion
prevails, leading to decay for the spatial density $\pi_{*}(u)(t)$. Finally,
together with Bigorgne, they studied the stability of vacuum for the
Vlasov–Poisson equation on $\mathrm{T}^{*}\mathbb{R}^{n}$, in the presence of
an external potential $-\frac{1}{2}|x|^{2}$ which induces hyperbolic features
of the free Hamiltonian flow. Indeed, the linear flow has then a non-empty
trapped set which is the _hyperbolic fixed point_ $(x,\xi)=(0,0)$. For these
models and for the Poisson interaction $\mathrm{K}=\pm\Delta^{-1}$,
exponential decay to $0$ for $\pi_{*}u(t)$ was proved in [VRVR24] while, after
proper renormalization of the solution along the unstable manifold, the first
term in the asymptotic expansion was identified in [BVRVR23] when $n=2$
together with a fine asymptotic description of the limit distribution.

###  1.3. The flat torus case

Besides these models of geometric flavour, it is also instructive to compare
our work to what is known for Vlasov equations set on the cotangent bundle of
the flat torus $\mathbb{T}^{n}$, which has been, as far as we know, the unique
compact manifold where progress has been made concerning fine large time
behavior of solutions to (1.1). In this simple geometric framework, the Vlasov
equation reads

(1.6)
$\partial_{t}u+\xi\cdot\nabla_{x}u-\nabla_{x}\Phi(u)\cdot\nabla_{\xi}u=0,\quad
u|_{t=0}=u_{0}\in\mathscr{C}^{N}(\mathrm{T}^{*}\mathbb{T}^{n},\mathbb{R})\cap
L^{1}(\mathrm{T}^{*}\mathbb{T}^{n}),$

with the identification
$\mathrm{T}^{*}\mathbb{T}^{n}\simeq\mathbb{T}^{n}\times\mathbb{R}^{n}$. In a
breakthrough work [MV11], Mouhot and Villani have proved that small,
_analytic_ (or at least _Gevrey_ with high regularity index) solutions to
(1.6) weakly converges to a stationary state to free transport, and the
potential $\Phi(u)$ strongly converges to $0$, both with exponential speed
(sub-exponential in the Gevrey case), a result referred to in the mathematical
literature as _Landau Damping_. Prior to that, we mention that [Deg86] studied
the linear spectral problem, and in [CM98, HV09] the authors were able to
construct examples of solutions that display this behavior. It is also
important to note that these references do not restrict to smooth interaction
kernels $K$, and in particular treat Vlasov–Poisson equations; note though
that they impose a convolution kernel, that is $K(x,y)$ must be of the form
$k(x-y)$.

The linear mechanism at work in the torus is the so-called phase mixing, which
can be briefly summarized as follows. On the torus, the free motion is
_integrable_ , and the solution to the free transport equation has an explicit
formula, namely we have $u(t,x,\xi)=u(0,x-t\xi,\xi)$ and

$\displaystyle\int_{\mathrm{T}^{*}\mathbb{T}^{n}}u(t)\psi\mathrm{d}{\mathrm{L}}$
$\displaystyle=\int_{\mathrm{T}^{*}\mathbb{T}^{n}}\left(\int_{\mathbb{T}^{n}}u_{0}\,\mathrm{d}x\right)\psi\mathrm{d}x\mathrm{d}\xi+\sum_{k\neq
0}\int_{\mathbb{R}^{n}}\mathcal{F}_{x}u_{0}(k,\xi)\mathcal{F}_{x}\psi(-k,\xi)e^{itk\cdot\xi}\,\mathrm{d}\xi$
$\displaystyle=\int_{\mathrm{T}^{*}\mathbb{T}^{n}}\left(\int_{\mathbb{T}^{n}}u_{0}\,\mathrm{d}x\right)\psi\mathrm{d}x\mathrm{d}\xi+\sum_{k\neq
0}\int_{\mathbb{R}^{n}}\mathcal{F}_{x,\xi}{u_{0}}(k,v-tk)\mathcal{F}_{x,\xi}\psi(-k,-v)\,\mathrm{d}v,$

where $\mathcal{F}_{x}$ (resp. $\mathcal{F}_{x,\xi}$) refers to the Fourier
coefficients in $x$ (resp. Fourier coefficients in $x$ and Fourier transform
in $\xi$). Therefore, $u$ converges weakly to its space average as a
consequence of Riemann–Lebesgue’s theorem. Furthermore, the speed of
convergence can be quantified, depending on the regularity with respect to
$\xi$ of $u_{0}$. In particular, we have

$\left|\int_{\mathrm{T}^{*}\mathbb{T}^{n}}u(t)\psi\,\mathrm{d}{\mathrm{L}}-\int_{\mathrm{T}^{*}\mathbb{T}^{n}}\left(\int_{\mathbb{T}^{n}}u_{0}\mathrm{d}x\right)\psi\,\mathrm{d}x\mathrm{d}\xi\right|\leqslant
Ce^{-tR}\|u_{0}\|_{\operatorname{analytic}}\|\psi\|_{\operatorname{analytic}},$

where $\|\cdot\|_{\operatorname{analytic}}$ stands for some appropriate
analytic norm and $R$ is related to the radius of analyticity [Deg86]. This
exponential convergence can be compared to the one appearing in (1.4). The
linear mechanism we exploit in (1.4), i.e. the strong _chaotic_ features of
the geodesic flow in compact Anosov manifolds, is thus of very different
nature to that of the torus, which relies on the (strong) regularity of the
solution to the Vlasov equation. In the nonlinear setting, it results into the
fact that $\Phi(u(t))$ converges exponentially fast to $0$ in analytic
topology. For what concerns solutions $u(t,x,\xi)$ to (1.6), they weakly
converge to some limit distribution $h_{\pm\infty}(\xi)$, still with an
exponential speed, where $h_{\pm\infty}(\xi)$ is an analytic function on
$\mathbb{R}^{n}$. The main difficulty in the nonlinear analysis on
$\mathbb{T}^{n}$ is due to resonances referred to as plasma echoes which can
lead to catastrophic growth phenomena; they are the reason why the result of
[MV11] is restricted to solutions with infinite regularity. Heuristically,
these echoes give a threshold for the Gevrey regularity of the initial data,
which is related to the regularity of the interaction kernel $K$.

Moreover, [MV11] went beyond the small data regime, as they identify a class
of stable homogeneous equilibria $h(\xi)$, namely satisfying the so-called
Penrose stability condition. In particular, in the repulsive case, this is
automatically satisfied for smooth positive radial symmetric $h$ in dimension
$n\geqslant 3$. Large time behavior is then established for analytic data in a
small neighborhood of such equilibria. Shortly after, Lin and Zeng proved the
existence of low regularity periodic in time objects, the so-called BGK waves,
in the vicinity of Penrose stable equilibria [LZ11, LZ12], thus implying that
the main result of [MV11] does not hold in such a low regularity. In [Bed21],
Bedrossian showed that in high but finite regularity, the scenario of [MV11]
cannot hold, precisely due to the aforementioned plasma echoes (see also
[Zil21, Zil23]). Over the years, the seminal work [MV11] was sharpened and the
general strategy got simplified, first by Bedrossian, Masmoudi and Mouhot in
[BMM16] where the presumed optimal Gevrey regularity index was almost reached,
and more recently by Grenier, Nguyen and Rodnianski in [GNR21]. Following
these works, Ionescu, Pausader, Wang and Widmayer did obtain the optimal
Gevrey index [IPWW24]. We finally refer to [Bed22] for a more complete
discussion and review of the relevant mathematical literature.

In another direction, for the so-called Vlasov-HMF (Hamiltonian Mean Field)
equation, that is when the potential $\Phi(u)$ only depends on a _finite_
number of Fourier modes of the density $\pi_{*}u$, Faou and Rousset showed in
[FR16] that Gevrey regularity can be dispensed with, as they obtained
stability results in finite Sobolev regularity. This is possible because
“infinite chains” of plasma echoes cannot occur in this setting. See also
[BOY11, FHR21]. On Anosov manifolds, the linear mixing mechanism is strong
enough so that, even if we do not restrict to an interaction potential
involving a finite number of modes, resonances such as plasma echoes do not
seem to appear in the nonlinear analysis, which allows to obtain finite
regularity results.

###  1.4. Further questions

Whereas on the torus spaces of analytic functions were the right spaces to
capture a spectral gap for the dynamics, on Anosov manifolds, the right spaces
are _anisotropic Sobolev spaces_ (equipped with the aforementioned anisotropic
Sobolev norms as first constructed by Faure and Sjöstrand [FS11]) which we
shall introduce soon enough. This ultimately leads to the finite regularity
result of Theorem 1.1. However it is fair to acknowledge two shortcomings of
our analysis:

  * •

the kernel $K$ needs to be smooth (at least of class $\mathscr{C}^{k}$ with
$k$ large enough); in particular our work does not handle the Poisson case
$\mathrm{K}=\pm\Delta_{g}^{-1}$;

  * •

the initial support of the distribution function needs to be away from the
null section; this is because the aforementioned mixing property of the
geodesic flow degenerates and we have not been able to cope with this
degeneracy in the nonlinear analysis. In the free case, the decay results from
[Rat87, Dol98, Liv04, Tsu10, Tsu12, NZ15, FT21], ensure algebraic decay to
equilibrium when the initial data are non trivial near the null section
(recall Theorem 1.3) but it is not clear how to exploit this weaker mixing
property for the nonlinear problem.

Hopefully, they will be overcome in the future, but this likely will require
substantial work. For what concerns results in a small vicinity of a non-
trivial equilibrium $f_{0}(H)$ (where $f_{0}$ is not necessarily identically
$0$), our strategy of proof should in principle allow to write an abstract
Penrose stability condition as in the Euclidean setting but we still need to
understand in which practical situations it would be indeed relevant.

###  1.5. Organization of the article

Section 2 mainly serves pedagogical purposes: we describe an heuristic scheme
of proof for a toy model, namely a nonlinear perturbation of a contact Anosov
vector field. The main example is that of the geodesic flow on
$\mathrm{S}^{\star}\Sigma$. Some of the difficulties of the analysis already
appear at this level; yet, the main technical issues actually occur when
dealing with nonlinear Vlasov equations set on the full cotangent bundle
$\mathrm{T}^{*}\Sigma$.

With this perspective in mind, in Section 3, we proceed to geometric
constructions which are motivated by the way we will design the anisotropic
Sobolev spaces adapted to our nonlinear equations. More precisely, these
geometric preliminaries consist in building nice enough functions which decay
along the flow lines of the symplectic lift on
$\mathrm{T}^{*}\mathrm{T}^{*}\Sigma$ of the free motion, the so-called escape
functions. This kind of geometric approach is natural when dealing with the
study of linear operators having continuous spectrum on standard $L^{2}$
spaces and it originates in the study of quantum resonances [AC71, BC71, HS86]
whose fine description is related to the long time dynamics of Schrödinger
type equations. In the context of hyperbolic dynamical systems, these
geometric constructions appeared in [FRS08, FS11, DZ16, FT17] for the study of
the free motion on $\mathrm{S}^{*}\Sigma$. Here, we will build new escape
functions adapted to the free dynamics on the full cotangent bundle
$\mathrm{T}^{*}\Sigma$, in the spirit of these references. Namely, we
introduce, study and compare to each other what we call _sliced_ and _global_
escape functions. These two kinds of escape functions fulfill different
purposes which cannot be achieved at the same time. Loosely speaking, the
sliced escape functions enjoy appropriate properties when restricted to
$\mathrm{S}^{\star}\Sigma$, whereas the global escape functions are rather
adapted to the full dynamics on $\mathrm{T}^{*}\Sigma$.

Equipped with these objects, we can introduce, in Section 4, Sobolev spaces
with anisotropic regularity that are adapted to the free dynamics, referred to
as _sliced_ and _global_ anisotropic Sobolev spaces. As suggested by the
names, these spaces are associated with the sliced or global escape functions.
We study their main properties, most notably the exponential decay properties
they induce on the free transport. The most technical statement is a bilinear
estimate, adapted to the structure of the Vlasov equation, which displays
exponential decay and relates sliced and global anisotropic Sobolev norms;
this turns out to be one of the keys of the proof.

The end of the paper is devoted to the actual nonlinear analysis. First, in a
short Section 5, we prove the existence and uniqueness of a solution to (1.1).
In Section 6, we gather all pieces together to prove Theorem 1.1, using a
bootstrap argument. In particular, all the abstract functional estimates of
Section 4 are specifically tailored to this aim.

Finally, we provide in Appendix A a detailed toolbox on the classical
microlocal tools at the heart of our proofs: these are mostly used in Section
4 and in §6.7 and alluded to in the heuristic scheme of Section 2. Appendix B
presents a derivation of Theorem 1.3 from (1.4).

To conclude, we point out that, all along the article, for the sake of
simplicity, we consider the Vlasov equation (1.1) in positive times but the
argument can be straightforwardly adapted to negative times by considering the
negative geodesic vector field which still has the Anosov property.

### Acknowledgements

We thank Matthieu Léautaud for early discussions about the matter of this work
back in 2019 and Yannick Guedes Bonthonneau for his insights related to the
construction of escape functions. The three authors acknowledge the support of
the Centre Henri Lebesgue (ANR-11-LABX-0020-01). GR is also partially
supported by the Institut Universitaire de France and the PRC grant ADYCT
(ANR-20-CE40-0017).

##  2\. An heuristic scheme of proof for a toy model

Due to the use of microlocal methods involving anisotropic symbols (adapted to
the Anosov dynamics), the proof of our main theorem is fairly technically
involved. Yet, its general scheme is rather natural from the perspective of
nonlinear PDEs. In this section, for expository purposes, we aim at explaining
the general idea on a related toy model, without paying attention to technical
details, that are left to the following of the paper. Several difficulties
already appear at this level; however, specific issues also arise for
nonlinear Vlasov equations such as (1.1) – they will be discussed in the end
of the section.

###  2.1. The toy model: nonlinear perturbations of contact Anosov vector
fields

Let $M_{1}$ be a smooth compact manifold endowed with a _contact_ vector field
$X_{1}$ that has the _Anosov_ property111As already raised in the
introduction, this is in particular the case for the geodesic vector field on
the unit cotangent bundle $\mathrm{S}^{*}\Sigma$ of a negatively curved
surface $(\Sigma,\mathrm{g})$.. Denote by $(\varphi_{t})_{t\in\mathbb{R}}$ the
associated flow. Let ${\mathrm{L}}_{1}$ be the (contact) volume measure
preserved by $X_{1}$ and we fix a smooth vector field $V$ that also preserves
${\mathrm{L}}_{1}$. Let us consider the toy model

(2.1) $\partial_{t}u=X_{1}u+\omega(u)Vu,\quad u|_{t=0}=u_{0},$

where

$\omega(u)=\int_{M_{1}}u(t,z_{1})\phi(z_{1})\mathrm{d}{\mathrm{L}}_{1}(z_{1})$

with $\phi\in\mathscr{C}^{\infty}(M_{1})$ being a smooth function that is
fixed once and for all. The function $\phi$ is assumed to satisfy

$\int_{M_{1}}\phi(z_{1})\mathrm{d}{\mathrm{L}}_{1}(z_{1})=0.$

By Duhamel’s principle, the solution to the Cauchy problem (2.1) can be
written as

(2.2)
$u(t)=\varphi_{t}^{*}u_{0}+\int_{0}^{t}\omega(u(s))\varphi_{t-s}^{*}V(u(s))\mathrm{d}s.$

Note that we put aside existence issues and focus on the description of the
large time behavior of $u$. In order to prove weak convergence, we take a
smooth test function $\psi$ and write

$\langle
u(t),\psi\rangle=\langle\varphi_{t}^{*}u_{0},\psi\rangle+\int_{0}^{t}\omega(u(s))\left\langle\varphi_{(t-s)}^{*}V(u(s)),\psi\right\rangle\mathrm{d}s.$

The first term on the right-hand side converges (exponentially fast) to

$\int_{M_{1}}u_{0}\,\mathrm{d}{\mathrm{L}}_{1}\int_{M_{1}}\psi\,\mathrm{d}{\mathrm{L}}_{1}$

thanks to the decay of correlations result for contact Anosov flows [Liv04]
(cf. (1.4) in the case of geodesic flows). The second term splits in two parts
by decomposing $\psi$ as

$\psi=\left(\psi-\int_{M_{1}}\psi\,\mathrm{d}{\mathrm{L}}_{1}\right)+\int_{M_{1}}\psi\,\mathrm{d}{\mathrm{L}}_{1}=\mathrm{P}_{1}(\psi)+(\operatorname{Id}-\mathrm{P}_{1})(\psi)$

Since the vector field $V$ preserves ${\mathrm{L}}_{1}$, the contribution
coming from $\int_{M_{1}}\psi\mathrm{d}{\mathrm{L}}_{1}$ yields a term equal
to $0$. Hence we just need to understand if the integral

(2.3) $\int_{0}^{t}\omega(u(s))\left\langle
u(s),V\varphi_{-(t-s)}^{*}\mathrm{P}_{1}{\psi}\right\rangle\mathrm{d}s$

converges when $t\to\infty$.

Suppose now that we are given an essentially self-adjoint invertible operator
$B$ acting on $\mathscr{C}^{\infty}(M_{1})$ and $C,N_{0},\vartheta>0$ such
that for each $0\leqslant s<\infty$ we have

(2.4) $\|B\varphi_{{s}}^{*}\mathrm{P}_{1}B^{-1}\|_{L^{2}\to L^{2}}\leqslant
Ce^{-\vartheta
s}\quad\text{and}\quad\|B^{-1}V\varphi_{{-s}}^{*}\mathrm{P}_{1}\|_{\mathscr{C}^{N_{0}}\to
L^{2}}\leqslant Ce^{-\vartheta s}.$

Such an operator exists thanks to [FS11, NZ15], up to the fact that a loss of
derivative occurs, which can be fixed by the introduction of an appropriate
resolvent of the form $(X_{1}+1)^{-N}$, with $N$ large enough, in the
estimates. In the present discussion, this technical aspect will be put aside
for clarity of exposure but we will have to deal with this issue in the actual
proof of our main results.

The second bound of (2.4) allows to bound the integrand of (2.3) as

$\left|\omega(u(s))\left\langle
u(s),V\varphi_{-(t-s)}^{*}\mathrm{P}_{1}{\psi}\right\rangle\right|\leqslant
C|\omega(u(s))|\,\|\varphi\|_{\mathscr{C}^{N_{0}}}\|Bu(s)\|_{L^{2}}e^{-(t-s)\vartheta}.$

In particular, if we are able to show that for some $N>0$ and
$\vartheta^{\prime}\in\left]0,\vartheta\right[$, one has

(2.5) $\|Bu(s)\|_{L^{2}}\leqslant
C\|u_{0}\|_{\mathscr{C}^{N}}\quad\text{and}\quad|\omega(u(s))|=\mathcal{O}(e^{-\vartheta^{\prime}s})$

then we would obtain that the integral (2.3) would of size
$\mathcal{O}(e^{-\vartheta^{\prime}t})$ hence converging exponentially fast
towards zero.

To estimate $\omega(u(s))$, we proceed with a bootstrap argument, introducing
the interval

$\mathcal{J}=\left\\{T\in\mathbb{R}_{+}:\ \forall t\in[0,T],\
|\omega(u(s))|\leqslant e^{-\vartheta^{\prime}t}\right\\}.$

If $u_{0}$ is chosen small enough, this is a closed and nonempty interval. In
particular, to prove the second part of (2.5), it suffices to show that
$\mathcal{J}$ is open which would imply that $\mathcal{J}=\mathbb{R}_{+}$.
With this aim in mind, we write

$\omega(u(t))=\omega(\varphi_{t}^{*}u_{0})+\int_{0}^{t}\omega(u(s))\left\langle
u(s),V\varphi_{s-t}^{*}\phi\right\rangle\mathrm{d}s.$

Then, arguing as above, one can derive the bound, for all $t\in\mathcal{J}$,

$|\omega(u(t))|\leqslant C(e^{-\vartheta
t}\|u_{0}\|_{\mathscr{C}^{N_{0}}}+e^{-\vartheta^{\prime}t}\sup_{s\in\mathcal{J}}\|Bu(s)\|_{L^{2}}),$

since $\omega(\varphi_{t}^{*}u_{0})$ converges exponentially fast towards $0$,
by the decay of correlation result (1.4). Therefore if one can show the bound

(2.6) $\sup_{s\in\mathcal{J}}\|Bu(s)\|_{L^{2}}\leqslant
C\|u_{0}\|_{\mathscr{C}^{N}}$

this would imply that $\mathcal{J}$ is open, provided that

$\|u_{0}\|_{\mathscr{C}^{N}}\ll 1,$

hence we would get the bound (2.5), and the desired convergence would follow.

To obtain (2.6), it is natural to use energy estimates. However, if the
operator $B$ is not pseudo-differential (which will be the case when dealing
with the general problem on $\mathrm{T}^{*}\Sigma$), such estimates are
difficult to obtain as we cannot rely on symbolic calculus. It is nevertheless
possible to circumvent this problem, as follows. Suppose one can find an
invertible self-adjoint pseudo-differential operator $A$, that satisfies the
following properties:

  * (i)

the principal symbol of $A$ is of the form $e^{\lambda}$, with
$\lambda\in\mathscr{C}^{\infty}(\mathrm{T}^{*}M_{1})$ satisfying

(2.7) $\\{\langle\zeta_{1},X_{1}\rangle,\lambda\\}\leqslant 0,$

where $\langle\zeta_{1},X_{1}\rangle$ is the principal symbol associated with
$X_{1}$;

  * (ii)

one has the comparison estimate

(2.8) $\|BVv\|_{L^{2}}\leqslant\|Av\|_{L^{2}},\quad
v\in\mathscr{C}^{\infty}(M_{1}).$

The existence of such an operator $A$ is related to the construction of the
function $\lambda$, which is referred to as an escape function for the
operator $X_{1}$. As we shall see below, the decay estimate (2.7) will allow
to obtain polynomial bounds on $\|Au(t)\|$. Heuristically,
$\|A\cdot\|_{L^{2}}$ can be seen as a “top-order” norm, while
$\|B\cdot\|_{L^{2}}$ is “low-order”. As is often the case in nonlinear PDEs,
the low-order norm will be bounded, while we allow a growth for the top-order
one.

Since $A$ is pseudo-differential, we may now use a microlocal energy estimate,
which yields

(2.9) $\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|Au(t)\|_{L^{2}}^{2}$
$\displaystyle=2\operatorname{Re}\langle
AX_{1}u(t),Au(t)\rangle+2\omega(u(t))\operatorname{Re}\langle
AVu(t),Au(t)\rangle$
$\displaystyle=2\operatorname{Re}\langle[A,X_{1}]A^{-1}Au(t),Au(t)\rangle$
$\displaystyle\qquad\qquad\qquad+2\omega(u(t))\operatorname{Re}\langle[A,V]A^{-1}Au(t),Au(t)\rangle,$

where we used that $V$ and $X_{1}$ are volume preserving in the second line.
By the composition rule for pseudo-differential operators, we see that
$[A,V]A^{-1}$ is (almost) a pseudo-differential operator of order $0$. Indeed,
there is a logarithmic loss here and that will lead us to consider operators
$A_{t}$ depending on time. Again, we will not discuss all this in detail to
keep the presentation simple. Using the Calderón-Vaillancourt theorem and the
Cauchy-Schwarz inequality, we obtain

$\frac{\mathrm{d}}{\mathrm{d}t}\|Au(t)\|_{L^{2}}^{2}\leqslant
2\operatorname{Re}\langle[A,X_{1}]A^{-1}Au(t),Au(t)\rangle+C\omega(u(t))\|Au(t)\|_{L^{2}}^{2}.$

If the operator $[A,X_{1}]A^{-1}$ were nonpositive this would yield

$\frac{\mathrm{d}}{\mathrm{d}t}\left(e^{-\int_{0}^{t}C\omega(u(s))ds}\|Au(t)\|_{L^{2}}^{2}\right)\leqslant
0,$

and hence, integrating the above expression, we would get
$\|Au(t)\|_{L^{2}}\leqslant\widetilde{C}\|Au_{0}\|_{L^{2}}$ for
$t\in\mathcal{J}$, for some $\widetilde{C}>0$. Unfortunately, the
nonpositivity of $[A,X_{1}]A^{-1}$ cannot be exactly achieved, but almost,
thanks to the Gårding inequality, using (2.7) satisfied by $A$. The outcome is
that the operator is nonpositive, but only up to the addition of a remainder
with lower Sobolev regularity. Hence, by an induction scheme, one may obtain a
polynomial bound on $\|Au(t)\|_{L^{2}}$, that is

(2.10) $\|Au(t)\|_{L^{2}}\lesssim\langle
t\rangle^{p}\|u_{0}\|_{\mathscr{C}^{N}},\quad t\in\mathcal{J},$

for some $p\in\mathbb{N}$.

This polynomial bound implies the boundedness of $\|Bu(t)\|_{L^{2}}$. Indeed,
use (2.2) to get

(2.11)
$B\mathrm{P}_{1}u(t)={B}\varphi_{t}^{*}\mathrm{P}_{1}u_{0}+\int_{0}^{t}\omega(u(s))B\varphi_{(t-s)}^{*}\mathrm{P}_{1}B^{-1}BVA^{-1}Au(s)ds.$

Eventually, we can plug in the polynomial bound (2.10) for $\|Au(s)\|_{L^{2}}$
into (2.11). Then, using the first part of (2.4) and (2.8), we obtain the
bound (2.6) with $Bu(s)$ replaced by $\mathrm{B}P_{1}u(s)$. However $X$ and
$V$ preserve $\mathrm{L}_{1}$ and hence $B(\mathrm{Id}-\mathrm{P}_{1})u(s)$ is
constant equal to $(\mathrm{Id}-P_{1})u_{0}$. Thus we get (2.6) and this
completes the proof of the convergence result.

###  2.2. The case of the Vlasov equation on $\mathrm{T}^{*}\Sigma$

This scheme of proof for Anosov vector fields lays the foundation of the
strategy that we will follow when dealing with the Vlasov equation (1.1). The
main additional difficulty that we shall encounter lies in the fact that the
Vlasov equation is posed on the full cotangent bundle $M=\mathrm{T}^{*}\Sigma$
on which the geodesic flow does not have the Anosov property. This property
indeed only holds on each energy layer $M_{r}=\\{(x,\xi):\|\xi\|=r\\}$, for
$r>0$. This leads to the following issues when carrying out the above
strategy.

  1. (1)

It would be natural to consider integrated versions (over the $r$-variable) of
the anisotropic pseudo-differential operators $B$ (as constructed in [FS11]).
This would result into operators $\mathbf{B}$ defined on $M$. Proceeding like
this indeed allows to obtain exponential decay estimates in the spirit of
(2.4). However, doing so would have a major drawback coming from the fact that
the vector fields $V$ appearing in (1.1) involve derivatives with respect to
the variable $r$. In particular, as the integrated version of $B$ does not
have good pseudo-differential properties on the full cotangent bundle $M$,
and, as mentioned above, we would not be able to carry out the commutator
argument in (2.9). In order to overcome this problem, we define a new family
of pseudo-differential operators $\mathbf{A}$ adapted to the dynamics on $M$.
This relies on the construction of a global escape function for the dynamics
lifted on $\mathrm{T}^{*}M$ in the spirit of [FS11]. This dynamical issue is
tackled in Section 3 and then implemented in Section 4 to define pseudo-
differential operators which are tailored for the microlocal arguments used to
deal with (2.9).

  2. (2)

A crucial point in the above strategy was the comparison property (2.8)
between the norms defined by $A$ and $B$. We will construct sliced escape
functions, in the spirit of [FS11, NZ15], giving rise to an operator $B$
acting on functions of $\mathrm{S}^{*}\Sigma$. The sliced escape functions are
defined in such a way to be comparable with the previous global escape
functions. This in turn allows to compare the anisotropic norms induced by
$\mathbf{A}$ and the integrated version $\mathbf{B}$ (on the full cotangent
bundle) of $B$. This procedure is also discussed in Section 3.

Once we will have gathered these elements, we will be in position to apply the
strategy of §2.1 to the Vlasov equation on $M=\mathrm{T}^{*}\Sigma$. This is
achieved in Section 6 after showing in Section 5 the existence and uniqueness
of global solutions to the nonlinear problem (1.1). As a matter of fact, the
above strategy can be implemented at the expense of paying attention to two
more issues. First, the Anosov assumption degenerates on $\\{\xi=0\\}$ and we
need to ensure that the solution to (1.1) avoids this null section. This is
why we consider initial data supported away from the null section. Second, the
nonlinear perturbations are much more general than the $\omega(u(t))V$
considered in the above toy problem. This yields some extra (mostly technical)
complications in implementing the argument.

##  3\. Sliced and global escape functions

In this section, we build escape functions in the spirit of [FS11], which are
adapted to the free flow of the Vlasov equation (1.1). We start in Subsection
3.1 by explaining why the escape functions of [FS11] cannot be directly used
and why new constructions are required. Subsection 3.2 is then dedicated to
geometric preliminaries related to the geodesic flow and its Anosov property;
several useful notations are introduced along the way. In Subsections 3.3 and
3.4, we introduce the key notions of global and sliced order functions
together with the associated notions of global and sliced escape functions.
The actual construction of relevant sliced and global escape functions $F$ and
$G$ is performed in Subsections 3.6 to 3.8. The central result of this section
is Proposition 3.11, which gathers the properties of $F$ and $G$ and explains
how they compare to each other. This result is at the heart of our analytical
arguments in the upcoming Sections.

###  3.1. Differences with the escape functions of Faure and Sjöstrand

As already alluded to in the previous sections, we will introduce, in Section
4, functional spaces that are well suited to study the solutions of (1.1). The
presentation of this analytical set-up will be close to the one given by Faure
and Sjöstrand in [FS11] to which we refer for more details and motivations.
Recall that the key point in this reference compared with earlier related
works on the spectral theory of Anosov flows [Liv04, BL07, BL13] is the use of
microlocal methods. Such methods turn out to be well suited for the nonlinear
problems we study (see also [BT07, FRS08] for earlier use of microlocal tools
for the study of hyperbolic dynamical systems in discrete times). The main
difference with [FS11] is that we have to deal with functions involving the
radial variable $r=\sqrt{2H}$ in $M={\mathrm{T}}^{*}\Sigma$ and that we do not
only work with test functions in $M_{1}={\mathrm{S}}^{*}\Sigma$ where the flow
is Anosov. In some sense, this situation is reminiscent of what happens when
studying Anosov actions [GBGHW20], meaning that there is an extra natural
vector field $\partial_{r}$ which commutes with $r^{-1}X$ where $X$ is the
Hamiltonian vector field generated by $\mathrm{H}$.

A crucial ingredient in the work [FS11] is the construction of an _escape
function_ $\mathrm{T}^{*}M_{1}\to\mathbb{R}$ which decreases along the flow
lines of the symplectic lift

$\Phi_{t,1}:\mathrm{T}^{*}M_{1}\to\mathrm{T}^{*}M_{1}$

of the geodesic flow $\varphi_{t}:M_{1}\to M_{1}$. Since we are dealing here
with the global cotangent space $M=\mathrm{T}^{*}\Sigma$, a first natural idea
would be to extend an escape function $\mathrm{T}^{*}M_{1}\to\mathbb{R}$
chosen as in [FS11] to a map $G:\mathrm{T}^{*}M\to\mathbb{R}$, called a sliced
escape function, simply by saying that it does not depend on $r$ while it
depends on $\rho$ in a trivial way (here $\rho$ is the dual variable of $r$).
Proceeding like this indeed allows to build a space $\mathcal{G}$ for which
the results of [NZ15] about the exponential decay of correlations can be used
(at least far from the null section $\\{r=0\\}$).

However, dealing with nonlinear terms using such an approach requires an a
priori estimate on the norm of the solution of (1.1) in a space related to
$\mathcal{G}$. Ideally, one would like to obtain such a control by energy
estimates. The latter estimates require mainly two features : first, the
escape function $G$ needs to be a good symbol in $\mathrm{T}^{*}M$ ; second,
it needs to decrease along the flow lines of the symplectic lift

$\Phi_{t}:\mathrm{T}^{*}M\to\mathrm{T}^{*}M$

of the geodesic flow $\varphi_{t}:M\to M$. Both properties unfortunately
cannot be achieved for the sliced escape function $G$ introduced above — for
example, the decreasing property fails due to the fact that the Liouville form
$\alpha$ is preserved by $\Phi_{t,1}$ but not by $\Phi_{t}$.

Therefore, we are led to construct a global escape function
$F:\mathrm{T}^{*}M\to\mathbb{R}$ which indeed satisfies these two properties.
Nervertheless, although the function $F$ allows to build a space $\mathcal{F}$
on which one can derive appropriate energy estimates, there is no hope that
one can directly apply the results of [NZ15] on this space as it turns out
that the function $F$ cannot be seen as an escape function on
$\mathrm{T}^{*}M_{1}$. As a consequence, one is invited to make use of both
spaces $\mathcal{G}$ and $\mathcal{F}$ and combine the estimates obtained on
these spaces. To that aim, it will be important to be able to compare the two
associated norms, that is, to compare $G$ and $F$. In particular, the
construction of the two escape functions needs to be handled simultaneously in
a careful way, which is one of the main difficulties of the analysis of this
section.

###  3.2. Geometrical setting and notations

We denote $M=\mathrm{T}^{*}\Sigma$ and for each $r\geqslant 0$ we set

$M_{r}=\\{z=(x,\xi)\in M\leavevmode\nobreak\ :\leavevmode\nobreak\
|\xi|_{x}=r\\}.$

Let $\varphi_{t}:M\to M$ be the geodesic flow associated to $(\Sigma,g)$. We
let $X=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\varphi_{t}$ be the
generator of the flow. In what follows we will denote
$M^{\times}=M\setminus\underline{0}$ for simplicity, where $\underline{0}$
denotes the null section in $M$. In our development, it will be useful to use
polar coordinates

(3.1)
$(r,z_{1}):M^{\times}\overset{\sim}{\longrightarrow}\mathbb{R}_{>0}\times
M_{1}$

and where we set

$r(x,\xi)=|\xi|_{x},\quad
z_{1}(x,\xi)=\left(x,\frac{\xi}{|\xi|_{x}}\right),\quad(x,\xi)\in M^{\times}.$

Of course the flow $\varphi_{t}$ preserves the energy layers $\\{r=c\\}$ for
each $c\geqslant 0$. In particular we may see $X$ as a vector field $X_{r}$ on
$M_{r}$ for each $r$ and, in the polar coordinates (3.1), one has
$X_{r}(z_{1})=rX_{1}(z_{1})$.

######  Remark 3.1.

Observe that $X=rX_{1}(z_{1})$ extends smoothly to a vector field defined on
$\mathbb{R}\times M_{1}$.

We assume that the geodesic flow $\varphi_{t}:M_{1}\to M_{1}$ has the Anosov
property [Ano67a, FH19]. This means that for each $z_{1}\in M_{1}$ we have a
decomposition

$\mathrm{T}^{*}_{z_{1}}M_{1}=E_{\mathrm{u}}^{*}(z_{1})\oplus
E_{\mathrm{s}}^{*}(z_{1})\oplus E_{0}^{*}(z_{1})$

depending continuously on $z_{1}$, such that

$\left|\mathrm{d}\varphi_{t}(z_{1})^{-\top}\zeta_{1}\right|\leqslant\left\\{\begin{matrix}C\exp\left(-\vartheta_{1}t\right)|\zeta_{1}|,&\zeta_{1}\in
E_{\mathrm{s}}^{*}(z_{1}),&t\geqslant 0,\vspace{0.2cm}\\\
C\exp\left(-\vartheta_{1}|t|\right)|\zeta_{1}|,&\zeta_{1}\in
E_{\mathrm{u}}^{*}(z_{1}),&t\leqslant 0,\end{matrix}\right.$

for some constants $C,\vartheta_{1}>0$ not depending on $(z_{1};\zeta_{1})$.
Here we have

$E_{0}^{*}(z_{1})=\mathbb{R}\alpha(z_{1})\quad\text{and}\quad
E_{\mathrm{u}}^{*}(z_{1})\oplus
E_{\mathrm{s}}^{*}(z_{1})=\text{ker}\left(\alpha(z_{1})\right)$

where $\alpha\in\Omega^{1}(M)$ is the Liouville form restricted to $M_{1}$,
which is a contact form. We could in fact pick any smooth norm $|\cdot|$ on
$\mathrm{T}^{*}M_{1}$ but we will assume that it is the Sasaki induced by $g$
in view of having orthogonality between $E_{0}^{*}$ and $\text{ker}(\alpha)$.
The space $E_{\mathrm{u}}^{*}(z_{1})$ (resp. $E_{\mathrm{s}}^{*}(z_{1})$) is
called the co-unstable (resp. co-stable) bundle at $z_{1}$.

For each $r\geqslant 0$ we denote by
$\Psi_{r}:\mathrm{T}^{*}\Sigma\to\mathrm{T}^{*}\Sigma$ the map
$(x,\xi)\mapsto(x,r\xi)$ and we denote by $\partial_{r}$ the vector field on
$M^{\times}$ given by

$\partial_{r}=\left.\frac{\mathrm{d}}{\mathrm{d}r}\right|_{r=0}\Psi_{r}.$

For each $r>0$ and $z\in M_{r}$ we have
$T_{z}M=T_{z}M_{r}\oplus\mathbb{R}\partial_{r}$ and this decomposition yields
an inclusion map

$T_{z}^{*}M_{r}\hookrightarrow T_{z}^{*}M.$

Next, note that we have the commutation relation

$\varphi_{t}\circ\Psi_{r}=\Psi_{r}\circ\varphi_{tr},\quad r\geqslant 0,\quad
t\in\mathbb{R}.$

In particular, for each $r>0$, the flow $\varphi_{t}|_{M_{r}}:M_{r}\to M_{r}$
is also Anosov. Thus $E_{\mathrm{u}}^{*}(z)$ and $E_{\mathrm{s}}^{*}(z)$ are
well defined for each $z=(x,\xi)\in M^{\times}$ and for such $z$ we have the
decomposition

$\mathrm{T}^{*}_{z}M^{\times}=E_{\mathrm{r}}^{*}(z)\oplus
E_{\mathrm{u}}^{*}(z)\oplus E_{\mathrm{s}}^{*}(z)\oplus E_{0}^{*}(z),$

where $E_{\mathrm{r}}^{*}(z)$ is the co-radial bundle defined by

$E_{\mathrm{r}}^{*}(z)=\left\\{(x,\xi;\zeta)\in{\mathrm{T}}^{*}M^{\times}\leavevmode\nobreak\
:\leavevmode\nobreak\
{\langle\zeta,{\mathrm{T}}^{*}M_{|\xi|_{x}}\rangle=0}\right\\}.$

For simplicity we will denote

$E_{0,\mathrm{r}}^{*}=E^{*}_{\mathrm{r}}\oplus E_{0}^{*}.$

In the polar coordinates $z=(r,z_{1})$ given by (3.1), this splitting reads

(3.2) $\mathrm{T}_{z}^{*}M^{\times}\simeq\mathbb{R}\mathrm{d}r\oplus
E_{\mathrm{u}}^{*}(z_{1})\oplus E_{\mathrm{s}}^{*}(z_{1})\oplus
E_{0}^{*}(z_{1})=\mathbb{R}\mathrm{d}r\oplus\text{ker}(\alpha(z_{1}))\oplus\mathbb{R}\alpha(z_{1}).$

Recall that the last splitting is smooth while the first one is only Hölder in
general [Ano67b, FH19]. The corresponding coordinates are denoted by
$\zeta=(\rho;\zeta_{\perp};\zeta_{0})\in\mathbb{R}^{1+2(n-1)+1}$ and we will
use the metric

(3.3) $|\zeta|^{2}=\rho^{2}+|\zeta_{\perp}|^{2}+\zeta_{0}^{2}$

on ${\mathrm{T}}^{*}_{r,z_{1}}M^{\times}$.

######  Remark 3.2.

Note that this is not exactly the Sasaki metric induced by $g$ on $M$. The
standard Sasaki metric on $M$ would depend on $r$ and would rather write as

$|\zeta|_{\mathrm{Sasaki}}^{2}=\rho^{2}+r^{2}|\zeta_{\perp}|^{2}+r^{2}\zeta_{0}^{2}.$

Finally, we record the expression of the geodesic flow in the polar
coordinates (3.1),

(3.4)
$\varphi_{t}(r,z_{1})=\left(r,\varphi_{rt}(z_{1})\right),\quad(r,z_{1})\in\mathbb{R}_{>0}\times
M_{1}.$

Again, this flow can be extended to $\mathbb{R}\times M_{1}$.

###  3.3. A global order function

Recall that $\Phi_{t}$ is the flow on $\mathrm{T}^{*}M$ given by the
symplectic lift of $\varphi_{t}$. In other words,

$\Phi_{t}(z,\zeta)=\left(\varphi_{t}(z),\mathrm{d}\varphi_{t}(z)^{-\top}\zeta\right),\quad(z,\zeta)\in\mathrm{T}^{*}M,$

or, in the polar coordinates
$(z,\zeta)\simeq(r,z_{1};\rho,\zeta_{\perp},\zeta_{0})\in{\mathrm{T}}^{*}(\mathbb{R}_{>0}\times
M_{1})$,

(3.5)
$\Phi_{t}(z,\zeta)=\left(r,\varphi_{rt}(z_{1});\,\rho,\,\mathrm{d}\varphi_{rt}(z_{1})^{-\top}\zeta_{\perp},\,\zeta_{0}-rt\rho\right)$

We denote by
$\mathfrak{X}\in\mathscr{C}^{\infty}(\mathrm{T}^{*}M,{\mathrm{T}}\mathrm{T}^{*}M)$
the associated vector field. Our first step consists in constructing escape
functions for $\Phi_{t}$ (at least outside compact subsets of
${\mathrm{T}}^{*}M$).

Let $\mathrm{S}^{*}M$ be the co-sphere bundle of $M$, that is

${\mathrm{S}}^{*}M=({\mathrm{T}}^{*}M\setminus\underline{0})/\,\mathbb{R}_{>0},$

where $\underline{0}$ denotes the null section in ${\mathrm{T}}^{*}M$. For
$\zeta\in{\mathrm{T}}^{*}M\setminus\underline{0}$ we denote by $[\zeta]$ its
image in $\mathrm{S}^{*}M$. The flow $(\Phi_{t})$ induces a flow
$(\widetilde{\Phi}_{t})$ on $\mathrm{S}^{*}M$ and we denote by
$\widetilde{\mathfrak{X}}$ the associated vector field. If $E$ is a nontrivial
vector subbundle of $\mathrm{T}^{*}M$, we denote by
$\widetilde{E}\subset\mathrm{S}^{*}M$ the image of $E\setminus\underline{0}$
under the projection
${\mathrm{T}}^{*}M\setminus\underline{0}\to\mathrm{S}^{*}M$. Also we denote

$\mathrm{T}^{*}_{M_{1}}M=\\{(z,\zeta)\in\mathrm{T}^{*}M\leavevmode\nobreak\
:\leavevmode\nobreak\ z\in
M_{1}\\}\quad\text{and}\quad\mathrm{S}^{*}_{M_{1}}M=\\{(z,\zeta)\in\mathrm{S}^{*}M\leavevmode\nobreak\
:\leavevmode\nobreak\ z\in M_{1}\\}.$

######  Definition 3.3 (Global order function).

A function $\mu:\mathrm{T}^{*}M^{\times}\to[-1,1]$ is called a global order
function if it is of the form

(3.6)
$\mu(z,\zeta)=\widetilde{\mu}([z_{1};\rho,\zeta_{1}])\,\chi\left(|\zeta|\right),\quad(z,\zeta)=(r,z_{1};\rho,\zeta_{1})\in\mathrm{T}^{*}M^{\times},$

for some smooth functions $\widetilde{\mu}:\mathrm{S}^{*}_{M_{1}}M\to[-1,1]$
and $\chi:\mathbb{R}_{+}\to[0,1]$ such that $\chi(t)=0$ for $t$ small.

As mentioned above, the purpose of this subsection is to construct global
order functions that are adapted to the dynamics, in the following sense.

######  Proposition 3.4.

For any $0<\delta<1$ and any small conical neighborhoods
$\Upsilon_{{\mathrm{u}}}$, $\Upsilon_{\mathrm{s}}$ and
$\Upsilon_{0,\mathrm{r}}$ of $E_{\mathrm{u}}^{*}$, $E_{\mathrm{s}}^{*}$ and
$E_{0,\mathrm{r}}^{*}$ respectively, there exist a global order function $\mu$
of the form (3.6) and $\eta>0$ such that in the region $\\{|\zeta|\geqslant
1\\}$, we have the following properties.

  1. _(i)_

$\mu=1$ (resp. $0,-1$) near $E_{\mathrm{s}}^{*}$ (resp.
$E_{0,\mathrm{r}}^{*}$, $E_{\mathrm{u}}^{*}$) ;

  2. _(ii)_

$\mu>1-2\delta$ on $\Upsilon_{\mathrm{s}}$ ;

  3. _(iii)_

$\mu<-1+2\delta$ on $\Upsilon_{\mathrm{u}}$ ;

  4. _(iv)_

${\mathfrak{X}}{\mu}<-\eta$ outside
$\Upsilon_{\mathrm{u}}\cup\Upsilon_{\mathrm{s}}\cup\Upsilon_{0,\mathrm{r}}$;

  5. _(v)_

${\mathfrak{X}}{\mu}\leqslant 0$ everywhere.

Any global order function satisfying the above properties for some
$0<\delta<1$ and some neighborhoods $\Upsilon_{{\mathrm{u}}}$,
$\Upsilon_{\mathrm{s}}$ and $\Upsilon_{0,\mathrm{r}}$ as above will be labeled
as _good_.

To prove this result we will proceed with the method of [FS11] to which we
refer if more details are needed. The main difference is that we need to pay
attention to the extra variable $\rho$. We begin by constructing the function
$\widetilde{\mu}$.

######  Lemma 3.5.

Let $0<\delta<1$. Then there are $\eta>0$ and arbitrarily small neighbourhoods
$\widetilde{\Gamma}_{\mathrm{u}}$, $\widetilde{\Gamma}_{\mathrm{s}}$,
$\widetilde{\Gamma}_{{\mathrm{u}}{\mathrm{u}}}$ and
$\widetilde{\Gamma}_{{\mathrm{s}}{\mathrm{s}}}$ of
$\widetilde{E^{*}_{\mathrm{u}}\oplus E^{*}_{0,\mathrm{r}}}$,
$\widetilde{E^{*}_{\mathrm{s}}\oplus E^{*}_{0,\mathrm{r}}}$,
$\widetilde{E}^{*}_{\mathrm{u}}$ and $\widetilde{E}^{*}_{\mathrm{s}}$ in
$\mathrm{S}^{*}_{M_{1}}M$, and smooth functions
$\widetilde{\mu}_{\mathrm{u}},\widetilde{\mu}_{\mathrm{s}}:\mathrm{S}^{*}_{M_{1}}M\to[0,1]$
such that

  1. _(i)_

$\mathrm{\widetilde{m}}_{{\mathrm{s}}}\equiv 1$ near
$\widetilde{E}_{\mathrm{s}}^{*}$ and
$\mathrm{\widetilde{m}}_{{\mathrm{u}}}\equiv 1$ near
$\widetilde{E}_{\mathrm{u}}^{*}$ ;

  2. _(ii)_

$\mathrm{\widetilde{m}}_{{\mathrm{s}}}\equiv 0$ near
$\widetilde{E^{*}_{\mathrm{u}}\oplus E^{*}_{0,\mathrm{r}}}$ and
$\mathrm{\widetilde{m}}_{{\mathrm{u}}}\equiv 0$ near
$\widetilde{E^{*}_{\mathrm{s}}\oplus E^{*}_{0,\mathrm{r}}}$ ;

  3. _(iii)_

$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{s}}\leqslant-\eta$ outside
$\widetilde{\Gamma}^{{\mathrm{u}}}\cup\widetilde{\Gamma}^{{\mathrm{s}}{\mathrm{s}}}$
and $\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{u}}\geqslant\eta$
outside
$\widetilde{\Gamma}^{{\mathrm{s}}}\cup\widetilde{\Gamma}^{{\mathrm{u}}{\mathrm{u}}}$;

  4. _(iv)_

$\widetilde{\mu}_{\mathrm{s}}<\delta$ on $\widetilde{\Gamma}_{\mathrm{u}}$ and
$\widetilde{\mu}_{\mathrm{u}}<\delta$ on $\widetilde{\Gamma}_{{\mathrm{s}}}$;

  5. _(v)_

$\widetilde{\mu}_{\mathrm{s}}>1-\delta$ on
$\widetilde{\Gamma}_{{\mathrm{s}}{\mathrm{s}}}$ and
$\widetilde{\mu}_{\mathrm{u}}>1-\delta$ on
$\widetilde{\Gamma}_{{\mathrm{u}}{\mathrm{u}}}$;

  6. _(vi)_

$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{s}}\leqslant 0$ and
$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{u}}\geqslant 0$ everywhere.

In the proof below, we will identify $\mathrm{T}^{*}_{M_{1}}M$ with
$\mathbb{R}_{\rho}\times\mathrm{T}^{*}M_{1}$ so that a point
$(1,z_{1};\rho,\zeta_{1})\in\mathrm{T}^{*}_{M_{1}}M$ is identified with
$(z_{1};\rho,\zeta_{1})\in\mathbb{R}_{\rho}\times\mathrm{T}^{*}M_{1}$.

###### Proof of Lemma 3.5.

We introduce the strongly stable cone and the weakly unstable cone of aperture
$\theta>0$ as

$\displaystyle C^{{\mathrm{s}}{\mathrm{s}}}(\theta)$
$\displaystyle=\left\\{(z_{1};\rho,\zeta_{1})\in(\mathbb{R}\times{\mathrm{T}}^{*}M_{1})\setminus\underline{0}:\
\theta|\zeta_{\mathrm{s}}|\geqslant|\rho\mathrm{d}r+\zeta_{\mathrm{u}}+\zeta_{0}\alpha|\right\\},$
$\displaystyle C^{{\mathrm{u}}}(\theta)$
$\displaystyle=\left\\{(z_{1};\rho,\zeta_{1})\in(\mathbb{R}\times{\mathrm{T}}^{*}M_{1})\setminus\underline{0}:\
\theta|\rho\mathrm{d}r+{\zeta_{\mathrm{u}}}+\zeta_{0}\alpha|\geqslant|{\zeta_{\mathrm{s}}}|\right\\}.$

Here, $\underline{0}$ denotes the set $\\{(z_{1};0)\\}$ inside
$\mathbb{R}_{\rho}\times{\mathrm{T}}^{*}M_{1}$. Both sets are disjoint for
$0<\theta<1$ by construction and they can be identified with subsets of
${\mathrm{T}}^{*}M^{\times}\setminus\underline{0}$. Moreover we denote by
$\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta)$ and
$\widetilde{C}^{{\mathrm{u}}}(\theta)$ their images in
$\mathrm{S}^{*}_{M_{1}}M.$

Before going further, observe that, from the Anosov assumption and (3.5), the
limit sets $\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(0)$ and
$\widetilde{C}^{{\mathrm{u}}}(0)$ are respectively repulsive and attractive
for the flow $\widetilde{\Phi}_{t}$ on $\mathrm{S}^{*}_{M_{1}}M$; more
precisely, for each $\theta\in\left]0,1\right[$, there exists $\tau>0$ such
that for each $t\geqslant\tau$, it holds

(3.7)
$\widetilde{\Phi}_{t}\left(\complement\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta)\right)\subset\widetilde{C}^{\mathrm{u}}(\theta/2)\quad\text{and}\quad\widetilde{\Phi}_{-t}\left(\complement\widetilde{C}^{{\mathrm{u}}}(\theta)\right)\subset\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta/2).$

Now let $\delta>0$ small and $T>\tau$ (to be determined in a few lines in
terms of $\delta$). Let $\widetilde{\mu}_{0}$ be a function in
$\mathscr{C}^{\infty}({\mathrm{S}}_{M_{1}}^{*}M,[0,1])$ that is equal to $1$
(resp. $0$) on $\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta)$ (resp.
$\widetilde{C}^{{\mathrm{u}}}(\theta)$). Now observe that for $T>\tau$ large
enough, the sets

$\widetilde{\Gamma}^{{\mathrm{s}}{\mathrm{s}}}=S_{M_{1}}^{*}M\setminus\left(\bigcup_{t\in[0,T]}\widetilde{\Phi}_{-t}(\widetilde{C}^{{\mathrm{u}}}(\theta))\right)\quad\text{and}\quad\widetilde{\Gamma}^{{\mathrm{u}}}=S_{M_{1}}^{*}M\setminus\left(\bigcup_{t\in[0,T]}\widetilde{\Phi}_{t}(\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta))\right)$

are respectively included in
$\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta/2)$ and
$\widetilde{C}^{{\mathrm{u}}}(\theta/2)$. Let us define

$\widetilde{\mu}_{\mathrm{s}}=\frac{1}{2(T+\tau)}\int_{-T-\tau}^{T+\tau}\widetilde{\mu}_{0}\circ\widetilde{\Phi}_{s}\,\mathrm{d}s.$

First, note that __(i)__ and __(ii)__ are automatically satisfied for
$\widetilde{\mu}_{\mathrm{s}}$. Indeed, set

$N_{\mathrm{u}}=\bigcap_{|t|\leqslant
T+\tau}\Phi_{t}(\widetilde{C}^{\mathrm{u}}(\theta))\quad\text{and}~\quad
N_{{\mathrm{s}}{\mathrm{s}}}=\bigcap_{|t|\leqslant
T+\tau}\Phi_{t}(\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta)).$

Then $N_{\mathrm{u}}$ and $N_{{\mathrm{s}}{\mathrm{s}}}$ are neighborhoods of
$\widetilde{E_{\mathrm{u}}^{*}\oplus\widetilde{E}_{0,\mathrm{r}}^{*}}$ and
$\widetilde{E}_{\mathrm{s}}^{*}$ respectively and we have
$\widetilde{\mu}_{\mathrm{s}}([\zeta])=0$ for $[\zeta]\in N_{\mathrm{u}}$ and
$\widetilde{\mu}_{\mathrm{s}}([\zeta])=1$ for $[\zeta]~\in
N_{{\mathrm{s}}{\mathrm{s}}}$.

Next, we write

(3.8)
$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{s}}=\frac{1}{2(T+\tau)}\left(\widetilde{\mu}_{0}\circ\widetilde{\Phi}_{T+\tau}-\widetilde{\mu}_{0}\circ\widetilde{\Phi}_{-T-\tau}\right).$

Then by (3.7) we have, for each
$[\zeta]\notin(\widetilde{\Gamma}^{\mathrm{u}}\cup\widetilde{\Gamma}^{{\mathrm{s}}{\mathrm{s}}})$,

$\widetilde{\Phi}_{T+\tau}([\zeta])\in\widetilde{C}^{\mathrm{u}}(\theta)\quad\text{and}\quad\widetilde{\Phi}_{-T-\tau}([\zeta])\in\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta),$

as it follows from the definitions of $\widetilde{\Gamma}^{\mathrm{u}}$ and
$\widetilde{\Gamma}^{{\mathrm{s}}{\mathrm{s}}}$. Hence we get

$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{s}}=-\frac{1}{2(T+\tau)}\quad\text{
on
}\complement(\widetilde{\Gamma}^{\mathrm{u}}\cup\widetilde{\Gamma}^{{\mathrm{s}}{\mathrm{s}}}),$

and __(iii)__ is satisfied for $\widetilde{\mu}_{\mathrm{s}}$.

Next we turn to __(iv)__ and __(v)__. If
$[\zeta]\in\widetilde{\Gamma}^{{\mathrm{u}}}$, then
$\widetilde{\Phi}_{-t}([\zeta])$ does not belong to
$\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta)$ for $t\in[0,T]$. Hence
$\widetilde{\Phi}_{-T}([\zeta])\notin\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta)$
and thus by (3.7) we have
$\widetilde{\Phi}_{t}([\zeta])\in\widetilde{C}^{\mathrm{u}}(\theta/2)$ for
each $t\geqslant-T+\tau$. This gives
$\widetilde{\mu}_{\mathrm{s}}([\zeta])\leqslant{\tau}/{2(T+\tau)}$. Similarly
if $[\zeta]\in\widetilde{\Gamma}^{{\mathrm{s}}{\mathrm{s}}}$ one can show that
$\widetilde{\mu}_{\mathrm{u}}([\zeta])\geqslant 1-\tau/2(\tau+T)$. Therefore
$\widetilde{\mu}_{\mathrm{s}}$ verifies __(iv)__ and __(v)__ by picking $T>0$
large enough.

Finally, point __(vi)__ is a consequence of (3.7) and (3.8). Indeed, if
$[\zeta]\in\widetilde{\Gamma}_{{\mathrm{s}}{\mathrm{s}}}$, then
$\widetilde{\Phi}_{-T-\tau}([\zeta])\in\widetilde{C}^{{\mathrm{s}}{\mathrm{s}}}(\theta/2)$
by (3.7) and thus
$(\widetilde{\mu}_{0}\circ\widetilde{\Phi}_{-T-\tau})([\zeta])=1$. In
particular (3.8) yields
$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{s}}([\zeta])\leqslant 0$. We
prove similarly that
$\widetilde{\mathfrak{X}}\widetilde{\mu}_{\mathrm{s}}\leqslant 0$ on
$\widetilde{\Gamma}_{\mathrm{u}}$.

Considering the cones $\widetilde{C}^{{\mathrm{u}}{\mathrm{u}}}(\theta)$ and
$\widetilde{C}^{{\mathrm{s}}}(\theta)$ (by taking the flow in backward times),
one can construct a function $\widetilde{\mu}_{\mathrm{u}}$ satisfying __(i)_—
_(vi)__.

Finally note that all the neighborhoods constructed above can be made
arbitrarily small, by taking $\theta$ arbitrarily small. Also $\eta$ depends
only on $T$ and $\tau$, and thus on $\delta$ and $\theta$. ∎

With Lemma 3.5, Proposition 3.4 becomes at hand.

###### Proof of Proposition 3.4.

Consider $\delta>0$ small and
$\eta,\widetilde{\mu}_{\mathrm{b}},\widetilde{\Gamma}_{\mathrm{b}}$ and
$\widetilde{\Gamma}_{\mathrm{b}\mathrm{b}}$ as in Lemma 3.5 for
$\mathrm{b}=\mathrm{u},\mathrm{s}$. Then, set

(3.9)
$\widetilde{\Upsilon}_{\mathrm{u}}=\widetilde{\Gamma}_{\mathrm{u}}\cap\widetilde{\Gamma}_{\mathrm{u}\mathrm{u}},\quad\widetilde{\Upsilon}_{\mathrm{s}}=\widetilde{\Gamma}_{\mathrm{s}}\cap\widetilde{\Gamma}_{\mathrm{s}\mathrm{s}}\quad\text{and}\quad\widetilde{\Upsilon}_{0,\mathrm{r}}=\widetilde{\Gamma}_{\mathrm{u}}\cap\widetilde{\Gamma}_{\mathrm{s}}.$

Define the function
$\widetilde{\mu}\in\mathscr{C}^{\infty}(\mathrm{S}^{*}_{M_{1}}M,[-1,1])$ by

(3.10)
$\widetilde{\mu}=\widetilde{\mu}_{\mathrm{s}}-\widetilde{\mu}_{\mathrm{u}}$

and let $\chi\in\mathscr{C}^{\infty}(\mathbb{R}_{+},[0,1])$ such that
$\chi(t)=0$ for $t\leqslant 1/2$ and $\chi(t)=1$ for $t\geqslant 1$. Finally,
define the function ${\mu}$ by (3.6). That ${\mu}$ satisfies points __(i)_—
_(v)__ of Proposition 3.5 is an immediate consequence of Lemma 3.5 ; indeed it
suffices to take conical sets $\Upsilon_{{\mathrm{u}}}$,
$\Upsilon_{\mathrm{s}}$ and $\Upsilon_{0,\mathrm{r}}$ in
$\mathrm{T}^{*}M^{\times}$ that project on
$\widetilde{\Upsilon}_{{\mathrm{u}}}$, $\widetilde{\Upsilon}_{\mathrm{s}}$ and
$\widetilde{\Upsilon}_{0,\mathrm{r}}$ (the latter are defined in (3.9)). ∎

######  Remark 3.6.

Let us record that the proof of Proposition 3.4 shows that any function
$\mu:\mathrm{T}^{*}M^{\times}\to[-1,1]$ given by

(3.11)
$\mu(r,z_{1};\rho,\zeta_{1})=\widetilde{\mu}([z_{1};\rho,\zeta_{1}])\chi(|\zeta|),\quad\widetilde{\mu}=\widetilde{\mu}_{\mathrm{s}}-\widetilde{\mu}_{\mathrm{u}},$

where $\widetilde{\mu}_{\mathrm{s}}$ and $\widetilde{\mu}_{\mathrm{u}}$
satisfy the conditions of Lemma 3.5 and
$\chi\in\mathscr{C}^{\infty}(\mathbb{R}_{+},[0,1])$ is such that $\chi(t)=0$
for $t\leqslant 1/2$ and $\chi(t)=1$ for $t\geqslant 1$, is a good global
order function.

###  3.4. A sliced order function

In view of applying the exponential mixing properties from [NZ15], we now
proceed to the construction of an order function which is closer to the one
used in this reference. Namely, we want to find a fonction
$\nu:\mathrm{T}^{*}M^{\times}\to[-1,1]$ that is decreasing along the sliced
flow $\Phi_{t,1}:\mathrm{T}^{*}M\to\mathrm{T}^{*}M$. The latter is given by

$\Phi_{t,1}(z;\zeta)=\left(r,\varphi_{rt}(z_{1});\,\rho,\,\mathrm{d}(\varphi_{rt}|_{M_{1}})(z_{1})^{-\top}\zeta_{1}\right),\quad(z;\zeta)=(r,z_{1};\rho,\zeta_{1})\in\mathrm{T}^{*}M^{\times}.$

We denote by
$\mathfrak{X}_{1}\in\mathscr{C}^{\infty}(\mathrm{T}^{*}M^{\times},\mathrm{T}\mathrm{T}^{*}M^{\times})$
its generator — note that
$\mathfrak{X}_{1}=\mathfrak{X}+\rho\,\partial_{\zeta_{0}}$ in the coordinates
$(r,z_{1};\rho,\zeta_{\perp},\zeta_{0}).$

######  Definition 3.7 (Sliced order function).

A sliced order function is a map $\nu:\mathrm{T}^{*}M^{\times}\to[-1,1]$ which
is a linear combination of functions of the form

$(r,z_{1};\rho,\zeta_{1})\mapsto\widetilde{\mu}_{1}([z_{1};\zeta_{1}])\,\chi\left(\frac{|\zeta_{1}|}{\langle\rho\rangle}\right)$

for some functions
$\widetilde{\mu}_{1}\in\mathscr{C}^{\infty}(\mathrm{S}^{*}M_{1},[0,1])$ and
$\chi\in\mathscr{C}^{\infty}(\mathbb{R},[0,1])$ such that $\chi(t)=0$ for $t$
small.

Given a sliced order function $\nu$, we can define the corresponding function
on $\mathrm{T}^{*}M_{1}$ by letting

(3.12) $\nu_{1}(z_{1};\zeta_{1})=\nu\left(z_{1},0,\zeta_{1}\right).$

As a consequence of Proposition 3.4, one has the following result.

######  Corollary 3.8.

There exist $\eta>0$ and a sliced order function
$\nu:\mathrm{T}^{*}M^{\times}\to[-1,1]$ such that the function
$\nu_{1}=\nu|_{\mathrm{T}^{*}M_{1}}:\mathrm{T}^{*}M_{1}\to[-1,1]$ satisfies
the following properties, in the region $\\{|\zeta_{1}|\geqslant 1\\}$.

  1. _(i)_

$\nu_{1}$ is equal to $1$ (resp. $0,-1$) in a conical neighborhood of
$E_{\mathrm{s}}^{*}$ (resp. $E_{0}^{*},E_{\mathrm{u}}^{*}$) ;

  2. _(ii)_

$\mathfrak{X}_{1}\nu_{1}\leqslant 0$ ;

  3. _(iii)_

$\mathfrak{X}_{1}\nu_{1}<0$ outside a conical neighborhood of $E_{0}^{*}\cup
E_{\mathrm{u}}^{*}\cup E_{\mathrm{s}}^{*}$.

A sliced order function satisfying the properties above will be labeled as
_good_.

###### Proof.

Take $\mu:\mathrm{T}^{*}M\to[-1,1]$ a good global order function given by
Proposition 3.4 and any $\chi\in\mathscr{C}^{\infty}(\mathbb{R},[0,1])$ such
that $\chi(t)=0$ for $t$ small and $\chi(t)=1$ for $t\geqslant 1$. Then set

$\nu(r,z_{1};\rho,\zeta_{1})=\mu|_{\mathrm{T}^{*}M_{1}}(z_{1};\zeta_{1})\chi\left(\frac{|\zeta_{1}|}{\langle\rho\rangle}\right).$

Then $\nu$ is a sliced order function. Moreover, since
$\mathfrak{X}_{1}=\mathfrak{X}$ on $\mathrm{T}^{*}M_{1}=\\{\rho=0\\}$, it is
straightforward to check that $\nu$ is a good one. ∎

As a matter of fact, we will need to find good global and sliced order
functions which are comparable. This is the purpose of the next subsection.

###  3.5. Comparison of order functions

######  Lemma 3.9.

Let $0<\delta<1/2$. Then, one can find a global order function
$\mu=\mu_{{\mathrm{s}}}-\mu_{{\mathrm{u}}}$ as in Remark 3.6 and
$T_{1},\beta>1$ such that the following holds.

Letting
$\widetilde{\mu}_{{\mathrm{s}},1}=\mu_{{\mathrm{s}}}|_{\mathrm{S}^{*}M_{1}}$,
$\widetilde{\mu}_{{\mathrm{u}},1}=\mu_{{\mathrm{u}}}|_{\mathrm{S}^{*}M_{1}},$

(3.13)
$\nu_{\mathrm{s}}(z_{1};\rho,\zeta_{1})=(\widetilde{\mu}_{{\mathrm{s}},1}\circ\widetilde{\Phi}_{T_{1}})([z_{1};\zeta_{1}])\,\chi\left(\frac{|\zeta_{1}|}{\beta\langle\rho\rangle}\right),$

and

(3.14)
$\nu_{\mathrm{u}}(z_{1};\rho,\zeta_{1})=(\widetilde{\mu}_{{\mathrm{u}},1}\circ\widetilde{\Phi}_{T_{1}})([z_{1};\zeta_{1}])\,\chi\left(\frac{\beta|\zeta_{1}|}{\langle\rho\rangle}\right),$

one has the following properties:

  1. _(i)_

$\nu=\nu_{\mathrm{s}}-\nu_{\mathrm{u}}$ is a good sliced order function;

  2. _(ii)_

$\nu_{\mathrm{s}}\leqslant\mu_{\mathrm{s}}$ and
$\mu_{\mathrm{u}}\leqslant\nu_{\mathrm{u}}$ for $|\zeta|\geqslant 1$;

  3. _(iii)_

$\nu_{\mathrm{s}}>0$ implies that
$|\zeta_{\perp}|^{2}\geqslant\rho^{2}+\zeta_{0}^{2}.$

###### Proof.

By items __(i)_ _and_ _(ii)__ of Lemma 3.5, for $T_{1}>0$ large enough it
holds

$\operatorname{supp}(\widetilde{\mu}_{{\mathrm{s}},1}\circ\widetilde{\Phi}_{T_{1}})\subset\operatorname{int}\\{\widetilde{\mu}_{{\mathrm{s}},1}=1\\}.$

Hence for $T_{1}$ large,
$\operatorname{supp}(\widetilde{\mu}_{{\mathrm{s}},1}\circ\widetilde{\Phi}_{T_{1}})$
is contained in the interior of $\\{\widetilde{\mu}_{{\mathrm{s}}}=1\\}$. In
particular taking $\beta>0$ large enough, we have for $|\zeta|\geqslant 1$

$\nu_{\mathrm{s}}(z_{1};\rho,\zeta_{1})>0\quad\implies\quad{\mu}_{\mathrm{s}}(z_{1};\rho,\zeta_{1})=1.$

In particular this yields $\nu_{\mathrm{s}}\leqslant\mu_{\mathrm{s}}$ for
$|\zeta|\geqslant 1$.

Let
$\pi:\mathrm{S}^{*}_{M_{1}}M\setminus\widetilde{E_{\mathrm{r}}^{*}}\to\mathrm{S}^{*}M_{1}$
be the map $[(1,z_{1};\rho,\zeta_{1})]\mapsto(z_{1};\zeta_{1}/|\zeta_{1}|)$,
and let
$K=\overline{\pi(\operatorname{supp}(\widetilde{\mu}_{{\mathrm{u}}}))}\subset
S^{*}M_{1}$, which is well defined since
$\operatorname{supp}\widetilde{\mu}_{\mathrm{u}}$ does not intersect
$\widetilde{E_{\mathrm{r}}^{*}}$. Then
${K\cap\widetilde{E_{\mathrm{s}}^{*}\oplus E_{0}^{*}}=\emptyset}$ by __(ii)__
of Lemma 3.4 and thus for $T_{1}$ large enough, we have

$\\{\widetilde{\mu}_{{\mathrm{u}},1}\circ\widetilde{\Phi}_{T_{1}}=1\\}\supset
K.$

Therefore, if $\beta>0$ is chosen large enough, we have
$\nu_{{\mathrm{u}}}\geqslant{\mu}_{{\mathrm{u}}}$ on $\\{|\zeta|\geqslant
1\\}$. This completes the proof of the first two items. For the third item,
this follows by construction (up to increasing the values of $\beta$ and
$T_{1}$). ∎

###  3.6. Global and sliced escape functions

######  Definition 3.10.

A good global escape function is a smooth map
$F:\mathrm{T}^{*}M^{\times}\to\mathbb{R}$ independent of $r$, which is a
symbol in $\overline{S}^{0+}(\mathrm{T}^{*}M^{\times})$, and such that

$\mathfrak{X}F\leqslant 0$

in the region $\\{|\zeta|\geqslant 1\\}.$

Since $F$ does not depend on $r$, the condition
$F\in\overline{S}^{0+}(\mathrm{T}^{*}M^{\times})$ means that for every
$\varepsilon>0$ and $\beta,\gamma\in\mathbb{N}^{n}$, there exists
$C_{\beta,\gamma,\varepsilon}>0$ such that

$|\partial^{\beta}_{z}\partial^{\gamma}_{\zeta}F(z,\zeta)|\leqslant
C_{\beta,\gamma,\varepsilon}\langle\zeta\rangle^{\varepsilon-|\gamma|},\quad(z,\zeta)\in\mathrm{T}^{*}M^{\times}.$

Again these symbols can be viewed as symbols on
$\mathrm{T}^{*}(\mathbb{R}\times M_{1})$. Note that we only impose a
nonincreasing condition (and not a decreasing condition) along the flow lines
of $\Phi_{t}$ because it will be sufficient to show that the solution of (1.1)
grows at most polynomially in time in a space defined by a good global escape
function $F$ (see the proof of Lemma 6.7 below).

We can now state the main result of this section.

######  Proposition 3.11.

For each $\delta\in\left]0,1\right[$, there are good global and sliced order
functions $\mu$ and $\nu$, as well as positive symbols
$f\in\overline{S}^{2}(\mathrm{T}^{*}M^{\times})$ (independent of $r$) and
$g_{1}\in\overline{S}^{2}(\mathrm{T}^{*}M_{1})$, such that the following
holds.

Let $g:\mathrm{T}^{*}M^{\times}\to\mathbb{R}$ be defined by

$g(r,z_{1};\rho,\zeta_{1})=g_{1}(z_{1};\zeta_{1}/\langle\rho\rangle).$

Let also $\mathfrak{S}>0$ and $\sigma\in\mathbb{R}$ such that
$|\sigma|<(1-\delta)\mathfrak{S}$ and denote

$F_{\mathfrak{S},\sigma}=(\mathfrak{S}\mu+\sigma)\log f\quad\text{and}\quad
G_{\mathfrak{S},\sigma}=(\mathfrak{S}\nu+\sigma)\log g.$

Then we have the following properties:

  1. _(i)_

there exists $c>0$ such that $g_{1}(z_{1};\zeta_{1})\geqslant
c\langle\zeta_{1}\rangle^{2}$;

  2. _(ii)_

$F_{\mathfrak{S},\sigma}$ is a global escape function ;

  3. _(iii)_

for $|\zeta_{1}|\geqslant 1$, we have

$\mathfrak{S}\nu(z_{1},0,\zeta_{1})=\mathfrak{S}\quad\text{near
}E_{\mathrm{s}}^{*}\qquad\text{and}\qquad\mathfrak{S}\nu(z_{1},0,\zeta_{1})=-\mathfrak{S}\quad\text{near
}E_{\mathrm{u}}^{*};$

  4. _(iv)_

for $|\zeta_{1}|\geqslant 1$, it holds
$\mathfrak{X}_{1}(\nu)(z_{1},0,\zeta_{1})\leqslant 0$;

  5. _(v)_

whenever $\sigma\leqslant 0$, we have

$G_{\mathfrak{S},\sigma}\leqslant
F_{\mathfrak{S},\sigma}+2(\mathfrak{S}+|\sigma|)\log\langle\rho\rangle$

in the region $\bigl{\\{}|\zeta|\geqslant 1\bigr{\\}}.$

By analogy with the definition of a good global escape function, we will say
in the following that $G_{\mathfrak{S},\sigma}$ is a _good sliced escape
function_ even if it is maybe not decreasing along the flow lines. The rest of
this section is devoted to the proof of Proposition 3.11.

###  3.7. Construction of a global pseudo-norm

Before going further, let us introduce some additional norms on
$\mathrm{T}^{*}M_{1}$. Given $T>1$, we set

$N_{T}(z_{1};\zeta_{1})=\int_{0}^{T}|\Phi_{t}(z_{1};\zeta_{\perp})|^{2}\mathrm{d}t.$

Then note that we have

$\mathfrak{X}_{1}N_{T}(z_{1};\zeta_{1})=|\Phi_{T}(z_{1};\zeta_{\perp})|^{2}-|\zeta_{\perp}|^{2}.$

Hence, thanks to the Anosov property, we can fix $T>1$ large enough to ensure
the existence of two conic neighborhoods
$\Gamma_{{\mathrm{u}}{\mathrm{u}}}^{1}$ and
$\Gamma_{{\mathrm{s}}{\mathrm{s}}}^{1}$ of $E_{\mathrm{u}}^{*}$ and
$E_{\mathrm{s}}^{*}$ respectively such that

(3.15) $\mathfrak{X}_{1}N_{T}\geqslant c_{\Sigma,\mathrm{g}}N_{T}\quad\text{in
}\Gamma_{{\mathrm{u}}{\mathrm{u}}}^{1}\qquad\text{and}\qquad\mathfrak{X}_{1}N_{T}\leqslant-
c_{\Sigma,\mathrm{g}}N_{T}\quad\text{in
}\Gamma_{{\mathrm{s}}{\mathrm{s}}}^{1}$

where $c_{\Sigma,\mathrm{g}}>0$ is a constant depending only on the Riemannian
manifold. The neighborhood and parameter $T>1$ appearing in the above
definition are fixed once and for all in this rest of this section.

Let $Q\geqslant 1$ be some parameter (to be determined later) and set

$\widetilde{g}_{1}(z_{1};\zeta_{1})=\chi\left(\frac{|\zeta_{0}|^{2}}{N_{T}(\zeta_{1})}\right)\zeta_{0}^{2}+\left(1-\chi\left(\frac{|\zeta_{0}|^{2}}{N_{T}(\zeta_{1})}\right)\right)N_{T}(\zeta_{1}),$

where we take $\chi(t)=1$ for $t\geqslant 1$ and $\chi(t)=0$ for $|t|\leqslant
1/2$. We also define

(3.16)
$f(r,z_{1};\rho,\zeta_{1})=Q+\chi\left(\frac{|\zeta_{\perp}|^{2}}{\zeta_{0}^{2}+\rho^{2}}\right)\widetilde{g}_{1}(z_{1};\zeta_{1}),\quad(r,z_{1};\rho,\zeta_{1})\in\mathrm{T}^{*}M^{\times}.$

One can check that this function belongs to
$\overline{S}^{2}(\mathrm{T}^{*}M^{\times})$; we also note that it is
independent of $r$. The next results says that $F_{\mathfrak{S},\sigma}$ is
indeed a good global escape function for an appropriate choice of $Q$.

######  Lemma 3.12.

Let $0<\delta<1/2$. Then, one can find a global order function $\mu$ as in
Lemma 3.9 and $Q_{0}>1$ such that for all $Q\geqslant Q_{0}$, and all
$\mathfrak{S}\geqslant 1$ and $\sigma\in\mathbb{R}$ verifying
$|\sigma|\leqslant(1-2\delta)\mathfrak{S}$, one has

(3.17) $\mathfrak{X}\bigl{(}(\mathfrak{S}\mathrm{m}+\sigma)\log
f\bigr{)}\leqslant 0$

in the region $\\{|\zeta|\geqslant 1\\}$.

###### Proof.

Let $\mu$ be a global order function as obtained from Lemma 3.9. By
definition, for $|\zeta|\geqslant 1$, one has

$\mu(r,z_{1};\rho,\zeta_{1})=\widetilde{\mu}\left([z_{1};\rho,\zeta_{1}]\right)=\widetilde{\mu}_{\mathrm{s}}\left([z_{1};\rho,\zeta_{1}]\right)-\widetilde{\mu}_{\mathrm{u}}\left([z_{1};\rho,\zeta_{1}]\right).$

Hence, we have

$\mathfrak{X}\bigl{(}(\mathfrak{S}{\mu}+\sigma)\log
f\bigr{)}=\mathfrak{S}(\mathfrak{X}\widetilde{\mu})\log
f+(\mathfrak{S}\widetilde{\mu}+\sigma)\frac{\mathfrak{X}f}{f}.$

Recall that by construction we have, in the region $\\{|\zeta|\geqslant 1\\}$,

(3.18) $\mathfrak{X}\widetilde{\mu}\leqslant
0\quad\text{and}\quad|\mathfrak{X}f/f|\leqslant r\,C_{T},$

for some constant $C_{T}>0$ depending only on $T$ and $(\Sigma,\mathrm{g})$.
In particular, this constant does not depend on $Q$ and $r$. By picking the
neighborhoods $\widetilde{\Gamma}_{{\mathrm{u}}{\mathrm{u}}}$,
$\widetilde{\Gamma}_{{\mathrm{s}}{\mathrm{s}}}$,
$\widetilde{\Gamma}_{\mathrm{u}}$ and $\widetilde{\Gamma}_{\mathrm{s}}$ in
Lemma 3.5 small enough, we can verify that, there exist small conic
neighborhoods $Y_{{\mathrm{u}}{\mathrm{u}}},Y_{{\mathrm{s}}{\mathrm{s}}}$ and
$Y_{0,\mathrm{r}}$ of $E_{\mathrm{u}}^{*}$, $E_{\mathrm{s}}^{*}$ and
$E_{0}^{*}\oplus E_{\mathrm{r}}^{*}$ such that

  * •

in $\\{|\zeta|\geqslant 1\\}\cap\complement(Y_{{\mathrm{u}}{\mathrm{u}}}\cup
Y_{{\mathrm{s}}{\mathrm{s}}}\cup Y_{0,\mathrm{r}})$, one has
$\mathfrak{X}\mu\leqslant-\eta r$;

  * •

the quantity
$\displaystyle\chi\left(\frac{|\zeta_{\perp}|^{2}}{\zeta_{0}^{2}+\rho^{2}}\right)$
is equal to $0$ in $Y_{0,\mathrm{r}}$ and equal to $1$ in
$Y_{{\mathrm{u}}{\mathrm{u}}}\cup Y_{{\mathrm{s}}{\mathrm{s}}}$;

  * •

the quantity
$\displaystyle\chi\left(\frac{|\zeta_{0}|^{2}}{N_{T}(\zeta_{1})}\right)$ is
equal to $0$ in $Y_{{\mathrm{u}}{\mathrm{u}}}\cup
Y_{{\mathrm{s}}{\mathrm{s}}}$;

  * •

there holds $\mu\geqslant 1-2\delta$ in $Y_{{\mathrm{s}}{\mathrm{s}}}$ and
$\mu\leqslant 2\delta$ in $Y_{{\mathrm{u}}{\mathrm{u}}}$.

In particular, thanks to (3.18), we can deduce on the one hand that
$\mathfrak{X}(\mathfrak{S}{\mu}+\sigma)\leqslant 0$ in $Y_{0,\mathrm{r}}$. On
the other hand one has, in $\complement(Y_{{\mathrm{u}}{\mathrm{u}}}\cup
Y_{{\mathrm{s}}{\mathrm{s}}}\cup Y_{0,\mathrm{r}})$,

$\mathfrak{X}(\mathfrak{S}{\mu}+\sigma)\leqslant-\mathfrak{S}\eta r\log
Q+C_{T}r(\mathfrak{S}+|\sigma|)\leqslant r\mathfrak{S}\left(-\eta\log
Q+2C_{T}\right).$

Hence, picking $Q>1$ large enough, we obtain (3.17) outside
$Y_{{\mathrm{u}}{\mathrm{u}}}\cup Y_{{\mathrm{s}}{\mathrm{s}}}$. In these two
regions, we use that $\mathfrak{X}\widetilde{\mu}\leqslant 0$ to obtain

$\mathfrak{X}\bigl{(}(\mathfrak{S}{\mu}+\sigma)\log
f\bigr{)}\leqslant(\mathfrak{S}\widetilde{\mu}+\sigma)\frac{\mathfrak{X}f}{f}.$

However by Lemma 3.5, one has
$\mathfrak{S}{\mu}+\sigma\geqslant\mathfrak{S}(1-2\delta)+\sigma$ in
$Y_{{\mathrm{s}}{\mathrm{s}}}$ while, in $Y_{{\mathrm{u}}{\mathrm{u}}}$, one
has
$\mathfrak{S}\widetilde{\mu}+\sigma\leqslant-\mathfrak{S}(1-2\delta)+\sigma$.
Finally, by (3.15), one finds that, in $Y_{{\mathrm{u}}{\mathrm{u}}}\cup
Y_{{\mathrm{s}}{\mathrm{s}}},$

$\mathfrak{X}((\mathfrak{S}{\mu}+\sigma)\log f)\leqslant-
rc_{\Sigma,\mathrm{g}}\left(\mathfrak{S}(1-2\delta)-|\sigma|\right).$

The right-hand side of the above inequality is indeed nonpositive as soon as
the condition $|\sigma|\leqslant\mathfrak{S}(1-2\delta)$ holds. ∎

###  3.8. Proof of Proposition 3.11

We are now in position to prove Proposition 3.11. We let $\mu$ be the order
function appearing in Lemma 3.12 and $Q>1$ be the parameter appearing in that
same Lemma. The second item of Proposition 3.11 follows from this Lemma. Now
letting $g_{1}=Q_{0}+\widetilde{g}_{1}$ and $\nu$ be the sliced order function
from Lemma 3.9, we directly obtain all the remaining items except for the last
one (that is the comparison between $G_{\mathfrak{S},\sigma}$ and
$F_{\mathfrak{S},\sigma}$). To deal with this point, note that we have

(3.19) $\log f-2\log\langle\rho\rangle\leqslant\log g$

everywhere, since $f\leqslant\langle\rho\rangle^{2}g$ by (3.16) and the
definition of $g_{1}$. In particular, one has, for $\sigma\leqslant 0$,

$\sigma\log g\leqslant\sigma\log f+2|\sigma|\log\langle\rho\rangle,$

and, in the region in the region $\\{\nu\leqslant 0\\}$,

$\mathfrak{S}\nu\log g\leqslant\mathfrak{S}\nu\log
f+2|\mathfrak{S}|\log\langle\rho\rangle\leqslant\mathfrak{S}\mu\log
f+2|\mathfrak{S}|\log\langle\rho\rangle,$

where we used Lemma 3.9 to write the last inequality. Recall that
$\nu=\nu_{\mathrm{s}}-\nu_{\mathrm{u}}$ where $\nu_{\mathrm{s}}$ and
$\nu_{\mathrm{u}}$ are defined in (3.13) and (3.14). According to the third
item in Lemma 3.9, one has

$\\{\nu_{\mathrm{s}}>0\\}\subset\left\\{(z;\zeta)\leavevmode\nobreak\
:\leavevmode\nobreak\
\chi\left({|\zeta_{\perp}|^{2}}/(\rho^{2}+\zeta_{0}^{2})\right)=1\right\\}.$

As $\\{\nu>0\\}\subset\\{\nu_{{\mathrm{s}}}>0\\}$, we obtain that, in
$\\{\nu>0\\}\cap\\{|\zeta|\geqslant 1\\}$,

$\mathfrak{S}\nu\log g\leqslant\mathfrak{S}\nu\log
f\leqslant\mathfrak{S}\mu\log f,$

simply because $\nu\leqslant\mu$ and $g\leqslant f$ in that region. This
completes the proof the last item and thus the proof of the Proposition.

##  4\. Anisotropic Sobolev norms adapted to the free dynamics

The global and sliced escape functions from Section 3 will now be used to
design anisotropic Sobolev norms that will be at the heart of the proof of
Theorem 1.1. In this Section, we define the associated anisotropic norms,
which are adapted to the free dynamics and establish some of their main
properties.

In Subsection 4.1, we start by explaining how the global (resp. sliced) escape
functions give rise to the class of pseudo-differential operators
$\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}$ (resp.
$\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}$), which in turn allow to
define the _global_ (resp. _sliced_) anisotropic norms, see Definition 4.1
(resp. Definition 4.3). In Subsection 4.2, we state the key result of this
section, that is the bilinear estimate of Proposition 4.5. This estimate is
tailored to handle the nonlinear part of the Vlasov equation (1.1); loosely
speaking, it shows an exponential decay (obtained from the strong mixing of
the geodesic flow) which shows that the nonlinear part might be treated in a
perturbative way. It also features an estimate of the action of the operators
$\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}$ by the anisotropic
Sobolev norm associated with $\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}$ (that
is, a control of a sliced anistropic Sobolev norm by a global anistropic
Sobolev norm).

The remainder of the section is then mostly dedicated to the proof of
Proposition 4.5; along the way, we also prove some estimates that will turn
out handy for the upcoming bootstrap analysis. The purpose of Subsections 4.2
and 4.3 is to show that our sliced escape functions (and their associated
pseudo-differential operators) are well adapted to the abstract framework of
[NZ15], in view of obtaining a refined microlocal version of the decay of
correlations of Theorem 1.3. This verification is first thoroughly performed
on $\mathrm{S}^{\star}\Sigma$ in Subsection 4.2 and then lifted on the
cotangent space (minus the null section) in Subsection 4.3.

Eventually, the proof of Proposition 4.5 is concluded in Subsections 4.4 and
4.6. In particular we need to establish a connection between the sliced and
global anisotropic Sobolev norms, using in a crucial way the comparison
properties between the global and sliced escape functions, that were obtained
in Proposition 3.11.

###  4.1. Definitions

Let $\mathfrak{S}_{0}>0$ and $|\sigma_{0}|<(1-\delta)\mathfrak{S}_{0}$. The
function $e^{F_{\mathfrak{S}_{0},\sigma_{0}}}$ built from our global escape
function $F_{\mathfrak{S}_{0},\sigma_{0}}$ given by Proposition 3.11 is a
function on $\mathbb{R}\times{\mathrm{T}}^{*}M_{1}$ that we identified with a
function on

${\mathrm{T}}^{*}M^{\times}\simeq{\mathrm{T}}^{*}(\mathbb{R}_{>0}\times
M_{1})\subset{\mathrm{T}}^{*}\mathbb{R}\times{\mathrm{T}}^{*}M_{1}.$

It defines a symbol in the standard class
$\overline{S}^{2\mathfrak{S}_{0}}({\mathrm{T}}^{*}(\mathbb{R}\times M_{1}))$
of functions amenable to pseudo-differential calculus. See Appendix A.3 for
the precise definition. We can then define

(4.1)
$\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}=\mathbf{Op}\left(e^{F_{\mathfrak{S}_{0},\sigma_{0}}}\right),$

where $\mathbf{Op}$ is a quantization procedure on
${\mathrm{T}}^{*}(\mathbb{R}\times M_{1})$ [Zwo12, Ch. 14]. In fact, due to
the product structure of our space, we can set

$\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}(u)(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i(r-s)\rho}{\operatorname{Op}^{M_{1}}}\left(\exp{F_{\mathfrak{S}_{0},\sigma_{0}}(\rho,\,\cdot)}\right)u(s,z_{1})\mathrm{d}s\mathrm{d}\rho,$

where ${\operatorname{Op}^{M_{1}}}$ is a quantization procedure on $M_{1}$.
Again refer to Appendix A.3 for more details. These operators are well defined
for every $u\in\mathscr{C}^{\infty}_{c}(\mathbb{R}\times M_{1})$ (hence
compactly supported functions on $M^{\times}$) and they map the set
$\mathscr{S}(\mathbb{R}\times M_{1})$ of smooth functions that are rapidly
decaying in $r$ to itself.

######  Definition 4.1.

The _global anisotropic Sobolev norms_ are, for
$\mathfrak{S}_{0},\sigma_{0}\in\mathbb{R}$, the norms

$\left\|\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}u\right\|_{L^{2}(\mathbb{R}\times
M_{1})},\quad u\in\mathscr{C}^{\infty}_{c}(\mathbb{R}\times M_{1}),$

where the $L^{2}$-norm is taken with respect to the measure
$\mathrm{d}r\,\mathrm{d}L_{1}(z_{1})$ rather than the Liouville measure
$r^{n-1}\mathrm{d}r\,\mathrm{d}L_{1}(z_{1})$ that we would extend to
$\mathbb{R}\times M_{1}$.

The associated global anisotropic Sobolev spaces are defined as the completion
of $\mathscr{C}^{\infty}_{c}(\mathbb{R}\times M_{1})$ for these norms in the
space of tempered distributions.

One key aspect of the analysis of (1.1) will be to use these global
anisotropic Sobolev norms, which are adapted to the dynamical properties of
the free flow, to perform microlocal energy estimates.

######  Remark 4.2.

This “renormalized” measure is still invariant by the vector field $X=rX_{1}$
and we will always pick it for the $L^{2}$-scalar product. The choice of this
measure slightly simplifies the use of a quantization procedure adapted to
$\mathbb{R}\times M_{1}$. Yet, at some points, it will add extra constants in
our upper bounds depending on the support of the test function $u$.

In order to make the connection with the exponential decay results from
[NZ15], we also introduce a family of slightly more exotic pseudo-differential
operators built from our sliced escape function. Namely, let
$\mathfrak{S}_{1}>0$ and $|\sigma_{1}|<(1-2\delta)\mathfrak{S}_{1}$. The
escape function given by

$G_{\mathfrak{S}_{1},\sigma_{1},1}(z_{1};\zeta_{1})=G_{\mathfrak{S}_{1},\sigma_{1}}(r,z_{1};0,\zeta_{1})$

is independent from $r$ and belongs to the class of symbols
$\overline{S}^{0+}(\mathrm{T}^{*}M_{1})=S^{0+}(\mathrm{T}^{*}M_{1})$ amenable
to pseudo-differential calculus on the _compact_ manifold $M_{1}$. In
particular, for every $s\in\mathbb{R}$, the operator
$\exp{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{1},\sigma_{1},1})$ defines a
bounded operator from $H^{s}(M_{1})$ to $H^{s-2\mathfrak{S}_{1}}(M_{1})$. See
Appendix A.3.2 for a brief reminder and references. This allows to define the
operator-valued pseudo-differential operator

(4.2) $\forall
m_{1}\in\mathbb{R},\quad\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}(u)(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\frac{e^{i(r-s)\rho}}{(1+\rho^{2})^{\frac{m_{1}}{2}}}\exp\left({\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)\right)u(s,z_{1})\mathrm{d}s\,\mathrm{d}\rho.$

See Appendix A.2 for a brief reminder for the pseudo-differential calculus of
operator-valued symbols. This operator admits an inverse which is given by

$\mathbf{B}_{-\mathfrak{S}_{1},-\sigma_{1}}^{(-m_{1})}(u)(r,z_{1})=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\frac{e^{i(r-s)\rho}}{(1+\rho^{2})^{-\frac{m_{1}}{2}}}\exp\left(-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)\right)u(s,z_{1})dsd\rho.$

Recall from [FS11, Lemma 3.3] that we can pick $C_{1}>0$ large enough
(depending on $\mathfrak{S}_{1}$) to ensure that, for every
$\sigma_{1}\in[-2,2]$ and for every $\operatorname{Re}(z)\geqslant C_{1}$,

(4.3)
$\left\|e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\left(X_{1}+z\right)^{-1}e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}<\infty.$

Hence, for every integer $N_{1}\geqslant 1$, we can set

(4.4)
$\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1},N_{1})}=\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}(X_{1}+C_{1})^{-N_{1}}.$

Here the operator $X_{1}$ is identified with an operator acting on
$\mathbb{R}\times M_{1}$.

######  Definition 4.3.

The _sliced anisotropic Sobolev norms_ are, for
$m_{1},\mathfrak{S}_{1},\sigma_{1}\in\mathbb{R}$, $N_{1}\geqslant 1$, the
norms

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1},N_{1})}u\right\|_{L^{2}},\quad
u\in\mathscr{C}^{\infty}_{c}(\mathbb{R}\times M_{1}).$

The associated sliced anisotropic Sobolev spaces are defined as the completion
of the space $\mathscr{C}^{\infty}_{c}(\mathbb{R}\times M_{1})$ for these
norms.

######  Remark 4.4.

Recall that $X=rX_{1}$ where
$X=\frac{\mathrm{d}}{\mathrm{d}t}\varphi_{t}^{*}|_{t=0}$. Theorem 1.4 in
[FS11] implies the meromorphic extension of the resolvent in (4.3) up to
$\text{Re}(z)\geqslant-c_{0}\mathfrak{S},$ where $c_{0}>0$ is a geometric
constant depending only on $(\Sigma,\mathrm{g})$ according to Proposition 3.11
and $\mathfrak{S}$ is the regularity parameter in their anisotropic spaces. In
fact, there is an additional constant $C$ (still independent of $(\nu,g)$
defining the escape function) in the statement of this reference which comes
from the fact the vector field may not be volume preserving. Here, due to the
particular form of our quantization procedure (with isochore charts) and to
the fact that $X_{1}$ is volume preserving, it holds in fact

$X_{1}={\operatorname{Op}^{M_{1}}}\bigl{(}\langle\zeta_{1},X_{1}\rangle\bigr{)}$

(see Remark A.3). Therefore the constant $C$ from [FS11, Th. 1.4] can be taken
equal to $0$ (see Equation $(3.8)$ in this reference). It should be noted that
the escape function in [FS11] satisfies slightly stronger requirements that
the ones satisfied by $G_{\mathfrak{S}_{1},\sigma_{1},1}$. Yet, using an
approach through radial type estimates, these requirements were simplified by
Dyatlov and Zworski in [DZ16, §3] and one only needs that
$G_{\mathfrak{S}_{1},\sigma_{1},1}$ satisfies the weaker properties stated in
Proposition 3.11 (namely the first, third and fourth item of this statement).
In that case, one can verify that the resolvent can be extended to the region

(4.5) $\text{Re}(z)>-c_{0}(\mathfrak{S}_{1}-|\sigma_{1}|),$

where $c_{0}>0$ is some geometric constant depending only on
$(\Sigma,\mathrm{g})$. We refer to [Dya23, §3.3.1, Th. 2] for more details on
the value of the constant $c_{0}$ in terms of the Lyapunov exponents of the
flow.

###  4.2. A bilinear estimate

The main result of this section is the following bilinear estimate, which will
be crucial in the upcoming nonlinear analysis.

######  Proposition 4.5.

There exists $\vartheta_{0}>0$ such that the following holds.

Let $\mathfrak{S}_{0},m_{1},N_{1}\geqslant 0$ such that

$\mathfrak{S}_{0}\geqslant 3,\quad m_{1}\geqslant
50\mathfrak{S}_{0}\quad\text{and}\quad N_{1}\geqslant
50\mathfrak{S}_{0}+m_{1}.$

Let also $I\subset\mathbb{R}$ be an interval with
$\overline{I}\subset\mathbb{R}_{+}^{*}$ and
$\chi\in\mathscr{C}^{\infty}_{c}(I)$.

Then there exists a constant $C>0$ such that, for all
$\Phi\in\mathscr{C}^{\infty}(\Sigma)$, and for all
$u\in\mathscr{C}^{\infty}_{c}(\mathbb{R}\times M_{1})$,

(4.6)
$\left\|\mathbf{B}_{\mathfrak{S}_{0},-2}^{(m_{1},N_{1})}\varphi_{-t}^{*}\mathbf{P}\\{\Phi,\chi
u\\}\right\|_{L^{2}}\leqslant Ce^{-\vartheta_{0}t\min
I}\|\Phi\|_{\mathscr{C}^{N}}\left(\left\|u\right\|_{L^{2}}+\left\|\mathbf{A}_{\mathfrak{S}_{0},0}\chi
u\right\|_{L^{2}}\right),\quad t\geqslant 0.$

Here
$\mathbf{P}:\mathscr{C}^{\infty}(M^{\times})\to\mathscr{C}^{\infty}(M^{\times})$
is given by

(4.7)
$\mathbf{P}v(x_{1},r\xi_{1})=v(x_{1},r\xi_{1})-\int_{M_{1}}v(y_{1},r\eta_{1})\mathrm{d}\mathrm{\mathrm{L}}_{1}(y_{1},\eta_{1}),\quad
v\in\mathscr{C}^{\infty}(M^{\times}),$

where $\mathrm{\mathrm{L}}_{1}$ is the normalized Liouville measure on the
unit cotangent bundle $M_{1}=\mathrm{S}^{*}\Sigma$.

This proposition will be relevant to ensure the convergence of the integral
remainder term in the Duhamel formula for the Vlasov equation (1.1). More
precisely, the estimate (4.6) is intended to be applied with $\Phi=\Phi(u(t))$
and $u=u(t)$ where $u(t)$ is the solution to the nonlinear Vlasov equation
(1.1). The right-hand side of (4.6) achieves two important goals:

  * •

exponential decay in time is obtained;

  * •

the sliced anisotropic norm
$\left\|\mathbf{B}_{\mathfrak{S}_{0},-2}^{(m_{1},N_{1})}(\cdots)\right\|_{L^{2}}$
in the left-hand side is controlled by the global anisotropic norm
$\left\|\mathbf{A}_{\mathfrak{S}_{0},0}\chi u\right\|_{L^{2}}$ on the right-
hand side.

The exponential decay will help the convergence of the integral remainder
(1.5) provided that the norms in the right-hand side of (4.6) do not grow too
fast with $t$. We can first observe that all $L^{p}$ norms of $u(t)$ are
preserved under the evolution by (1.1) so that $\left\|u(t)\right\|_{L^{2}}$
will not contribute to any growth in time. Similarly, the fact that
$\Phi(u(t))$ is expressed in terms of a smooth interaction kernel implies that
the term $\|\Phi(u(t))\|_{\mathscr{C}^{N}}$ remains bounded with time. Hence,
the only difficulty is to find an a priori upper bound on the global
anisotropic Sobolev norm $\|\mathbf{A}_{\mathfrak{S}_{0},0}\chi
u(t)\|_{L^{2}}$. This is exactly where the choice of the global escape
function will turn out to be crucial in view of getting a polynomial upper
bound. In fact, as we shall use it in the proof of Lemma 6.7 below, the
principal symbol of $\mathbf{A}_{\mathfrak{S}_{0},0}$ belongs to a nice enough
class of symbols on $\mathbb{R}\times M_{1}$, which is the key to performing
an energy estimate with the help of microlocal methods.

The rest of this section is devoted to the proof of Proposition 4.5 and, along
the way, we shall also obtain intermediate corollaries that will be helpful
for the upcoming nonlinear analysis. The proof is organized as follows. First,
we recall the exponential decay result for the geodesic flow restricted to
$M_{1}$. Then, we lift these results to $\mathbb{R}\times M_{1}$ and finally
combine them with the properties of our escape functions (together with
microlocal arguments) to get Proposition 4.5.

###  4.3. Exponential decay on $M_{1}$

Unfortunately, the decay estimate given by Theorem 1.3 does not seem
sufficient for our needs and we will rather rely on a microlocal refinement of
this result following from [NZ15]:

######  Theorem 4.6 (Nonnenmacher-Zworski).

There is $\widetilde{\vartheta}_{0}>0$ such that, for each
$\mathfrak{S}_{1}\geqslant 3$, one can find a constant222As we shall see in
the proof, this is in fact the same constant as in (4.3). $C_{1}>0$ so that
the following property holds.

Let $N>10\mathfrak{S}_{1}$ and $\sigma_{1}\in[-2,2]$. Then, there exists a
constant $C>0$ such that, for every $t\geqslant 0$,

$\left\|e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\left(X_{1}+C_{1}\right)^{-N}\varphi_{-t}^{*}\mathrm{P}_{1}e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}\leqslant Ce^{-t\widetilde{\vartheta}_{0}},$

where $\mathrm{P}_{1}v=v-\int_{M_{1}}v\,\mathrm{d}\mathrm{\mathrm{L}}_{1}.$

Recall that the original decay of correlations result in (1.4) is rather
written for smooth test functions on $M_{1}$, i.e. for every
$u,v\in\mathscr{C}^{k_{0}}(M_{1})$,

(4.8)
$\left|\int_{M_{1}}(u\circ\varphi_{-t})v\,\mathrm{d}\mathrm{L}-\int_{M_{1}}u\mathrm{d}\mathrm{\mathrm{L}}_{1}\int_{M_{1}}v\mathrm{d}\mathrm{\mathrm{L}}_{1}\right|\\\
\leqslant
Ce^{-t\vartheta_{0}}\|u\|_{\mathscr{C}^{k_{0}}}\|v\|_{\mathscr{C}^{k_{0}}},$

which can directly be recovered from the refined estimate of Theorem 4.6.

######  Remark 4.7.

For simplicity, we choose to work in Section 6 with a fixed regularity
parameter $\mathfrak{S}_{1}$ independent of the geometry, more specifically
$\mathfrak{S}_{1}=3$ to ensure that
$(1-2\delta)\mathfrak{S}_{1}>2\geqslant|\sigma_{1}|$ (provided $\delta>0$ is
small enough). Modulo some slightly more tedious work, this could probably be
optimized to pick any $\mathfrak{S}_{1}>1$ but we do not pursue this issue
here.

###### Proof.

The results in [NZ15] are not exactly stated under such a precise form. Yet,
this result is implicitely proved in $\S 9$ of this reference as a consequence
of their main result and, for the sake of completeness, we will briefly recall
the analytical context of this result and the argument to go from resolvent
estimates to exponential decay of the flow.

The main result (Theorem $2$) in [NZ15] is a result on semiclassical
selfadjoint pseudo-differential operators $P(h)$ of order $m>0$ such that the
following holds

  1. (1)

there exists a complex absorbing potential, i.e. a selfadjoint (semiclassical)
pseudo-differential $W(h)$ of order $0\leqslant k\leqslant m$ whose principal
symbol $W$ is nonnegative and satisfies certain growth assumptions (Eq. (1.9)
and (1.10) in that reference);

  2. (2)

the level set $p^{-1}(E)$ (where $p$ is the principal symbol of $P(h)$) is a
smooth submanifold with $\mathrm{d}p|_{p^{-1}(E)}\neq 0$;

  3. (3)

the trapped set at energy $p^{-1}((E-\delta,E+\delta))$ is a normally
hyperbolic and symplectic manifold. Recall that the trapped set consists in
all the Hamiltonian trajectories for the flow induced by $p$ that never enters
the damping region in forward and backward times.

Then, Theorem $2$ in [NZ15] is a resolvent estimate for $(P(h)-iW(h)-z)^{-1}$
when $z$ lies in a small neighbodhood of size $\lambda_{0}h$ of $E$ (with
$\lambda_{0}>0$ that can be expressed in terms of purely dynamical
quantities). We also refer to [GS88, WZ11, Dya15, Dya16] for related results
when the incoming and outgoing trapped sets are regular enough.

It is important to note that the role of $W(h)$ is purely auxiliary in this
result and one has to find some concrete operators $\widetilde{P}(h)$ where
such a $W(h)$ exists and to deduce some applications on the spectral gap for
$\widetilde{P}(h)$. Among the several examples given in this reference, one
has the case where $P(h)=-ihX_{1}$ where $X_{1}$ is a contact Anosov flow on a
compact manifold (a geodesic flow is the main example of a contact flow). In
fact, as explained in Example $2$ and $\S 9.1$ of this reference, one can
design an operator $W$ with these requirements when $P(h)=-ihX_{1}$. This
follows from the fact that the trapped set333Here trapped set refers to the
Hamiltonian trajectories that stay in a bounded region of
$\mathrm{T}^{*}M_{1}$ in the energy window $(E-\delta,E+\delta)$ for this
operator is given by $\\{(z_{1};\zeta_{0},0):|\zeta_{0}-E|<\delta\\}\subset
E_{0}^{*}$. Recall that $p(z_{1};\zeta_{1})=\zeta_{1}(X_{1}(z_{1}))$ in this
setting. The contact assumption is thus here to ensure that our trapped set
$E_{0}^{*}$ has a symplectic structure and the Anosov assumption allows to
verify the normally hyperbolic property. Hence, Theorem 2 from this reference
can be applied and it yields a resolvent estimate for the corresponding
modified operator, i.e. $(-ihX_{1}-iW(h)-z)^{-1}$. Now, the strategy in [NZ15,
§9] is to convert this key resolvent estimate into a resolvent estimate for a
more relevant operator regarding the dynamical properties (see Theorem $4$ in
this reference).

The strategy is as follows. Define

$\widetilde{P}(h)=e^{{\operatorname{Op}^{M_{1}}}_{h}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}(-ihX_{1})e^{-{\operatorname{Op}_{h}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)},$

where ${\operatorname{Op}_{h}^{M_{1}}}$ is a semiclassical quantization
procedure [Zwo12, DZ19]. By construction, $\widetilde{P}(h)$ agrees
microlocally with $P(h)-iW(h)$ in a compact region of $\mathrm{T}^{*}M_{1}$.
According to Remark 4.4, the geometric properties of our sliced escape
function444This Remark was stated in a nonsemiclassical set-up but the proof
in [DZ16] allows $0<h\leqslant 1.$ ensures that the resolvent
$(\widetilde{P}(h)-z)^{-1}$ extends meromorphically to a strip

$\\{\text{Re}(z)\geqslant-(c_{0}((1-2\delta)\mathfrak{S}_{1}+|\sigma_{1}|)h\\}.$

Moreover, the principal symbol of $\widetilde{P}(h)$ is given by
$\zeta_{1}(X_{1})+ih\\{\zeta_{1}(X_{1}),G_{\mathfrak{S}_{1},\sigma_{1},1}\\}$
modulo terms lying in $h^{2}S^{-1+}(\mathrm{T}^{*}M_{1})$ (see Remark A.3).
Then, thanks to Proposition 3.11 and to (3.15), we can verify that the
imaginary part is not greater than $-ch<0$ (for some positive $c>0$) near
$E_{u}^{*}$ and $E_{s}^{*}$. In particular, using the Anosov property, only
points lying in $E_{0}^{*}$ does not end up in the damping region of our
nonselfadjoint operator (meaning in either backward or forward times).

######  Remark 4.8.

In [NZ15], the authors made the stronger assumptions of [FS11] on the escape
function used to define $\widetilde{P}(h)$ which ensures that the imaginary
part is $\leqslant-ch$ outside a small neighborhood of $E_{0}^{*}$. Yet, the
choice of this escape function was made to ensure the Fredholm property and
the meromorphic continuation of the resolvent beyond the real axis [NZ15,
Lemma 9.2] and they could as well have used the larger class of weights
functions as introduced in [DZ16] for which these properties remain true (see
Remark 4.4). See for instance [DZ15, §6] for resolvent estimates with these
weaker weights. As we have just explained it, the only difference is that the
damping only occurs near $E_{u}^{*}$ and $E_{s}^{*}$ (but all points outside
$E_{0}^{*}$ end up in these neighborhoods in either backward or forward
times).

Hence, one can define a (selfadjoint) pseudo-differential operator
$W_{\infty}(h)$ which is compactly supported (near $E_{0}^{*}$), whose
wavefront set does not intersect the wavefront of $W(h)$ and such that the
wavefront set of $\text{Id}-W_{\infty}(h)$ does not intersect the trapped set
of the Hamiltonian flow near the energy $E$. See [NZ15, §9.3] for more precise
statements. The key point is that
$P_{\infty}(h)=\widetilde{P}(h)-iW_{\infty}(h)$ has now an empty undamped set
and one can apply (nontrapping) resolvent estimate to this operator near
$z=E$. See [NZ15, Lemma 9.3] and the references therein. Then, with these two
resolvent estimates at hand and still following §9 from this reference, one
can make use of the gluing method due to Datchev and Vasy [DV12] in view of
getting a resolvent estimate for $(\widetilde{P}(h)-z)^{-1}$ near $z=E$ – see
[NZ15, Th. 4] for a precise statement of the semiclassical resolvent estimate.

Once we have these semiclassical resolvent estimates, we can turn to the decay
estimate of Theorem 4.6 for the induced flow. Recall that [NZ15, Th. 4] (when
translated in a nonsemiclassical form, see Equation (9.13) in this reference)
states the existence of a parameter $\vartheta_{1}>0$ such that, for every
$\mathfrak{S}_{1}\geqslant 3$ and for every $\sigma_{1}\in[-2,2]$, one can
find a constant $C_{\mathfrak{S}_{1},\sigma_{1}}>0$ with the following
property

(4.9)
$\displaystyle\left\|e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\left(X_{1}+z\right)^{-1}e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}$ $\displaystyle\hskip 227.62204pt\leqslant
C_{\mathfrak{S}_{1},\sigma_{1}}(1+|\operatorname{Im}(z)|)^{2(\mathfrak{S}_{1}+|\sigma_{1}|)+1},$

whenever $z\in\mathbb{C}$ verifies
$\operatorname{Re}(z)\in[-\vartheta_{1},\vartheta_{1}]$ and
$|\text{Im}(z)|\geqslant\vartheta_{1}$.

Such a resolvent estimate can be transferred into our expected exponentially
decaying upper bound. Let $v,w\in\mathscr{C}^{\infty}(M)$ ; we want to
estimate

(4.10)
$\int_{M_{1}}(\mathrm{P}_{1}(v)\circ\varphi_{-\tau})w\,\mathrm{d}{\mathrm{L}}_{1}=\int_{M_{1}}(\mathrm{P}_{1}(v)\circ\varphi_{-\tau})\mathrm{P}_{1}(w)\,\mathrm{d}{\mathrm{L}}_{1}.$

To do that, we proceed as [NZ15, §9, Proof of Cor. 5]. More precisely, we use
Stone’s formula to rewrite (4.10) as

(4.11) $\displaystyle\frac{1}{2\pi}\lim_{\varepsilon\to
0}\int_{\mathbb{R}}e^{i\tau\lambda}\left\langle\left((X_{1}-i\lambda+\varepsilon)^{-1}-(X_{1}-i\lambda-\varepsilon)^{-1}\right)\mathrm{P}_{1}(v),\overline{\mathrm{P}_{1}(w)}\right\rangle_{L^{2}(M_{1})}\mathrm{d}\lambda.$

From this, we infer that, for every $N\geqslant 0$, (4.10) writes

$\frac{1}{2\pi}\lim_{\varepsilon\to
0}\int_{\mathbb{R}}\frac{\mathrm{d}\lambda\,e^{i\tau\lambda}}{(C_{1}+iz)^{N}}\sum_{\pm}\pm\left\langle(X_{1}-iz\pm\varepsilon)^{-1}(X_{1}+C_{1})^{N}\mathrm{P}_{1}(v),\overline{\mathrm{P}_{1}(w)}\right\rangle_{L^{2}(M_{1})},$

where $C_{1}$ is chosen large enough to ensure that (4.3) holds true. Then, we
use the resolvent bound (4.9) (and its analogue in backward time) and pick
$N\geqslant 10\mathfrak{S}_{1}$ to ensure the convergence of the integral in
$\lambda$. It allows us make a deformation of contour from
$\lambda=\mathbb{R}$ to $\lambda=\mathbb{R}+i\vartheta_{0}$ where we choose
$\vartheta_{0}$ in such a way that there is no Ruelle-Pollicott resonances
(except for $0$) in the strip delimited by these two lines. As a consequence,
one gets that (4.10) is equal to

$\frac{1}{2\pi}\lim_{\varepsilon\to
0}\int_{\mathbb{R}+i\vartheta_{0}}\frac{\mathrm{d}\lambda\,e^{i\tau\lambda}}{(C_{1}+iz)^{N}}\sum_{\pm}\pm\left\langle(X_{1}-iz\pm\varepsilon)^{-1}(X_{1}+C_{1})^{N}\mathrm{P}_{1}(v),\overline{\mathrm{P}_{1}(w)}\right\rangle_{L^{2}(M_{1})}.$

Finally by (4.9) we obtain that for every $\tau\geqslant 0$,

$\left|\int_{M_{1}}(\mathrm{P}_{1}(v)\circ\varphi_{-\tau})w\,\mathrm{d}{\mathrm{L}}_{1}\right|\leqslant
Ce^{-\tau\vartheta_{0}}\left\|e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}(X_{1}+C_{1})^{N}\mathrm{P}_{1}(v)\right\|_{L^{2}}\\\
\times\|e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\mathrm{P}_{1}(w)\|_{L^{2}},$

which concludes the proof of the Theorem thanks to (4.3). ∎

###  4.4. Lifting Theorem 4.6 to $\mathbb{R}\times M_{1}$

Building on Theorem 4.6, we are able to derive the following lemma which lifts
this result to $\mathbb{R}\times M_{1}$:

######  Lemma 4.9.

There exists $\vartheta_{0}>0$ depending only on the choice of $(\nu,g_{1})$
defined in Proposition 3.11 such that the following holds.

Let $\mathfrak{S}_{1}\geqslant 3$, $\sigma_{1}\in[-2,2]$,
$N\in\mathbb{Z}_{+}^{*}$ and $m_{1},m_{2}\in\mathbb{Z}$ be such that

$m_{1}m_{2}\leqslant 0,\quad m_{1}+m_{2}>1,\quad\text{and}\quad N\geqslant
10\mathfrak{S}_{1}-\min\\{m_{1},m_{2}\\}+2.$

Let also $I$ be an open and bounded interval such that
$\overline{I}\subset\mathbb{R}_{+}^{*}$ and
$\chi\in\mathscr{C}^{\infty}_{c}(I)$. Then, there exists a constant $C>0$ such
that, for all $t\geqslant 0$,

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1},N)}\chi\varphi_{-t}^{*}\mathbf{P}\mathbf{B}_{-\mathfrak{S}_{1},-\sigma_{1}}^{(m_{2})}\right\|_{L^{2}\rightarrow
L^{2}}\leqslant Ce^{-t\vartheta_{0}\min I}.$

######  Remark 4.10.

By taking the flow in positive time, one obtains a similar statement by
remplacing $\mathfrak{S}_{1}$ by $-\mathfrak{S}_{1}<0$.

###### Proof.

Let $u$ be an element in $\mathscr{S}(\mathbb{R}\times M_{1})$. From the
definition of our sliced anisotropic norms (4.2) and (4.4), we want to compute
the norm of

(4.12)
$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}\chi\varphi_{t}^{*}\mathbf{P}(X_{1}+C_{1})^{-N}\mathbf{B}_{-\mathfrak{S}_{1},-\sigma_{1}}^{(m_{2})}u\right\|_{L^{2}},$

where $\chi\in\mathscr{C}^{\infty}_{c}(\mathbb{R}_{+}^{*})$ is compactly
supported inside the interval $I$. To this end, we use Plancherel’s formula to
write (4.12) as

$\displaystyle\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}}\frac{1}{(1+\rho^{2})^{m_{1}}}\times\Biggl{\|}\int_{I\times\mathbb{R}^{2}}$
$\displaystyle\frac{e^{is(\widetilde{\rho}-\rho)}e^{-i\widetilde{r}\widetilde{\rho}}}{(1+\widetilde{\rho}^{2})^{\frac{m_{2}}{2}}}e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}{e^{tsX_{1}}\mathrm{P}_{1}\chi(s)}$
$\displaystyle\quad\times{(X_{1}+C_{1})^{-N}}e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}u(\widetilde{r})\,\mathrm{d}s\,\mathrm{d}\widetilde{\rho}\,\mathrm{d}\widetilde{r}\Biggr{\|}_{L^{2}(M_{1})}^{2}\mathrm{d}\rho,$

or equivalently

$\displaystyle\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}}\frac{1}{(1+\rho^{2})^{m_{1}}}\times\Biggl{\|}\int_{I\times\mathbb{R}}$
$\displaystyle\frac{e^{is(\widetilde{\rho}-\rho)}}{(1+\widetilde{\rho}^{2})^{\frac{m_{2}}{2}}}e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}{e^{tsX_{1}}\mathrm{P}_{1}\chi(s)}$
$\displaystyle\quad\times{(X_{1}+C_{1})^{-N}}e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\widehat{u}(\widetilde{\rho})\,\mathrm{d}s\,\mathrm{d}\widetilde{\rho}\Biggr{\|}_{L^{2}(M_{1})}^{2}\mathrm{d}\rho,$

where $\widehat{u}$ is the Fourier transform of $u$ with respect to the radial
variable. Let us introduce the operator

$P_{\rho,\widetilde{\rho}}=\frac{1+i(\widetilde{\rho}-\rho)\partial_{s}}{1+(\rho-\widetilde{\rho})^{2}}.$

Then it holds

$P_{\rho,\widetilde{\rho}}\left(e^{is(\widetilde{\rho}-\rho)}\right)=e^{is(\widetilde{\rho}-\rho)},$

and integrating by parts yields, for $N_{2}\geqslant 0$,

$\displaystyle\left\|\mathbf{B}_{N_{1},\sigma_{1}}^{(m_{1})}\chi\varphi_{t}^{*}\mathbf{P}(X_{1}+C_{1})^{-N}\mathbf{B}_{-\mathfrak{S}_{1},-\sigma_{1}}^{(m_{2})}u\right\|_{L^{2}}^{2}$
$\displaystyle\qquad=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}}\frac{1}{(1+\rho^{2})^{m_{1}}}\left\|\int_{I\times\mathbb{R}}e^{-is\rho}\langle\widetilde{\rho}\rangle^{-m_{2}}\langle\widetilde{\rho}-\rho\rangle^{-N_{2}}W_{s,\rho,\widetilde{\rho}}\left[\widehat{u}(\widetilde{\rho})\right]\,\mathrm{d}s\,\mathrm{d}\widetilde{\rho}\,\right\|_{L^{2}(M_{1})}^{2}\mathrm{d}\rho,$

where the operator $W_{s,\rho,\widetilde{\rho}}$ is given by

$W_{s,\rho,\widetilde{\rho}}=e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\left(\frac{e^{is\widetilde{\rho}}(P_{\rho,\widetilde{\rho}})^{N_{2}}(e^{tsX_{1}}\mathrm{P}_{1}\chi(s))(X_{1}+C_{1})^{-N}}{\langle\widetilde{\rho}-\rho\rangle^{-N_{2}}}\right)e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}.$

By Cauchy-Schwarz inequality, one has

$\left\|\mathbf{B}_{N_{1},\sigma_{1}}^{(m_{1})}\chi\varphi_{t}^{*}\mathbf{P}(X_{1}+C_{1})^{-N}\mathbf{B}_{-\mathfrak{S}_{1},-\sigma_{1}}^{(m_{2})}u\right\|_{L^{2}}^{2}\leqslant
I_{m_{1},m_{2},N_{2}}\sup_{s\in
I,\,\rho\in\mathbb{R}}\int_{\mathbb{R}}\left\|W_{s,\rho,\widetilde{\rho}}\left[\widehat{u}(\widetilde{\rho})\right]\right\|_{L^{2}(M_{1})}^{2}\,\mathrm{d}\widetilde{\rho},$

where

$I_{m_{1},m_{2},N_{2}}=\frac{|I|^{2}}{(2\pi)^{4}}\int_{\mathbb{R}^{2}}\frac{1}{(1+\rho^{2})^{m_{1}}}\frac{1}{(1+\widetilde{\rho}^{2})^{m_{2}}}\frac{1}{(1+(\rho-\widetilde{\rho})^{2})^{N_{2}}}\mathrm{d}\rho\,\mathrm{d}\widetilde{\rho}$

is finite provided $N_{2}=-\min\\{m_{1},m_{2}\\}+1$ and $m_{1}+m_{2}>1$.
Hence, recalling that
$\|u\|_{L^{2}}=\int_{\mathbb{R}}\|\widehat{u}(\widetilde{\rho})\|_{L^{2}(M_{1})}^{2}\mathrm{d}\widetilde{\rho}$,
the proof boils down to estimating the operator norm of
$W_{s,\rho,\widetilde{\rho}}$, which can be bounded by a sum of terms of the
form

$t^{\ell_{1}}\left\|e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\chi^{(\ell_{2})}(s)e^{tsX_{1}}\mathrm{P}_{1}X_{1}^{\ell_{1}}(X_{1}+C_{1})^{-N}e^{-{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{1},\sigma_{1},1}\right)}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}$

with $\ell_{1}+\ell_{2}\leqslant N_{2}$ and $s\in I$. According to Theorem
4.6, the latter are bounded by $Ct^{\ell_{1}}e^{-t\vartheta_{0}\min I},$
provided that $N-N_{2}>10\mathfrak{S}_{1}$. The result follows. ∎

As a direct consequence of Lemma 4.9, we deduce the following decay estimate
for smooth test functions which will also be used in our nonlinear analysis:

######  Corollary 4.11.

There exists $\vartheta_{0}>0$ depending only on the choice of $(\nu,g_{1})$
defined in Proposition 3.11 such that the following holds.

Let $\mathfrak{S}_{1}\geqslant 3$, $\sigma_{1}\in[-2,2]$, $m_{1}\in\mathbb{Z}$
and $I$ be an open and bounded interval such that
$\overline{I}\subset\mathbb{R}_{+}^{*}$. Then, there exist a constant $C>0$
and an integer $N\in\mathbb{N}$ such that, for all $t\geqslant 0$ and for
every $u\in\mathscr{C}^{N}_{c}(I\times M_{1})$

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}\varphi_{-t}^{*}\mathbf{P}u\right\|_{L^{2}}\leqslant
Ce^{-t\vartheta_{0}\min I}\|u\|_{\mathscr{C}^{N}}.$

Again, we can get a similar statement for the flow in positive time by
replacing $\mathfrak{S}_{1}$ by $-\mathfrak{S}_{1}$. We also record that we
implicitely recover Theorem 1.3 from the introduction (away from the zero
section):

######  Corollary 4.12.

There exist $\vartheta_{0},k_{0}>0$ depending only on Riemannian manifold
$(\Sigma,\mathrm{g})$ such that, for every open and bounded interval $I$ such
that $\overline{I}\subset\mathbb{R}_{+}^{*}$, one can find a constant $C>0$
such that, for all $t\geqslant 0$ and for every
$u,v\in\mathscr{C}^{k_{0}}_{c}(I\times M_{1})$,

$\left|\int_{M^{\times}}(u\circ\varphi_{-t})v\,\mathrm{d}{\mathrm{L}}-\int_{M^{\times}}\left(\int_{M_{1}}u(r,z_{1}^{\prime})\mathrm{d}{\mathrm{L}}_{1}(z_{1}^{\prime})\right)v(r,z_{1})\mathrm{d}{\mathrm{L}}(r,z_{1})\right|\\\
\leqslant Ce^{-t\vartheta_{0}\min
I}\|u\|_{\mathscr{C}^{k_{0}}}\|v\|_{\mathscr{C}^{k_{0}}}.$

###  4.5. Comparison between the different norms

We now discuss how the different anisotropic norms we have introduced compare
to each other in various senses that are relevant to the proof of Proposition
4.5. We begin with a comparison of sliced anisotropic Sobolev norms.

######  Lemma 4.13.

Let $\mathfrak{S}_{1},\sigma_{1},\sigma_{2}\in\mathbb{R}$ and
$m_{1},m_{2}\in\mathbb{Z}$ verifying

$\mathfrak{S}_{1}\geqslant 1,\quad m_{1}m_{2}\leqslant 0,\quad
m_{1}+m_{2}>2,\quad\text{and}\quad\sigma_{1}+\sigma_{2}\leqslant 0.$

Let also $I$ be an open and bounded interval such that
$\overline{I}\subset\mathbb{R}_{+}^{*}$ and
$\chi\in\mathscr{C}^{\infty}_{c}(I)$. Then, there exist a constant $C>0$ and
an integer $N\geqslant 1$ such that

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}(a\partial_{r})\chi\mathbf{B}_{-\mathfrak{S}_{1},\sigma_{2}}^{(m_{2})}\right\|_{L^{2}\rightarrow
L^{2}}\leqslant C\left\|a\right\|_{\mathscr{C}^{N}},\qquad
a\in\mathscr{C}^{\infty}(M_{1}).$

###### Proof.

We proceed as in the proof of Lemma 4.9 and we write that

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}(a(z_{1})\partial_{r})\chi\mathbf{B}_{-\mathfrak{S}_{1},\sigma_{2}}^{(m_{2})}u\right\|_{L^{2}}^{2}=\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}}\frac{\rho^{2}}{(1+\rho^{2})^{m_{1}}}\\\
\left\|\int_{I\times\mathbb{R}^{2}}\frac{e^{is(\widetilde{\rho}-\rho)}\chi(s)}{(1+\widetilde{\rho}^{2})^{\frac{m_{2}}{2}}}e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{1},\sigma_{1}})}ae^{{\operatorname{Op}^{M_{1}}}(G_{-\mathfrak{S}_{1},\sigma_{2}})}\widehat{u}(\widetilde{\rho})\,\mathrm{d}s\,\mathrm{d}\widetilde{\rho}\right\|^{2}_{L^{2}(M_{1})}\mathrm{d}\rho.$

As in the proof of Lemma 4.9, we can integrate by parts with respect to the
variable $s$ using the operator $P_{\rho,\widetilde{\rho}}$ in view of getting
the missing integrability in the $\rho$ (or $\widetilde{\rho}$) variable.
Hence, under the assumption that $m_{1}+m_{2}>2$, one can find some uniform
constant $C>0$ such that

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}(a\partial_{r})\chi\mathbf{B}_{-\mathfrak{S}_{1},\sigma_{2}}^{(m_{2})}u\right\|_{L^{2}}^{2}\leqslant
C\left\|e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{1},\sigma_{1},1})}ae^{{\operatorname{Op}^{M_{1}}}(G_{-\mathfrak{S}_{1},\sigma_{2},1})}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}^{2}\\\
\times\int_{\mathbb{R}}\left\|\widehat{u}(\widetilde{\rho})\right\|^{2}_{L^{2}(M_{1})}\mathrm{d}\widetilde{\rho}.$

Hence, applying Plancherel’s formula, everything boils down to an estimate on

$\left\|e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{1},\sigma_{1},1})}ae^{{\operatorname{Op}^{M_{1}}}(G_{-\mathfrak{S}_{1},\sigma_{2},1})}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}.$

Using the composition rule (A.21) in
$S^{2\mathfrak{S}_{1}}(\mathrm{T}^{*}M_{1})$ together with the approximate
expressions (A.26) and (A.24) for the exponential555Note that this may require
to rescale the symbol of $\mathbf{B}$ by a parameter $h>0$ that depends only
on $N_{1}$ and on the choice of the escape function. As our arguments only
care about what happens outside a compact subset of $\mathrm{T}^{*}M_{1}$,
this rescaling of the escape function has no importance. of a pseudo-
differential operator, one finds that this quantity is uniformly bounded as
soon as $\sigma_{1}+\sigma_{2}\leqslant 0$. Moreover, the upper bound (given
by the composition rule and the Calderón-Vaillancourt Theorem (A.18)) is
linear in terms of $\|a\|_{\mathscr{C}^{N}}$ for some large enough $N\geqslant
1$. ∎

Arguing similarly, one obtains

######  Lemma 4.14.

Let $\mathfrak{S}_{1},\sigma_{1},\sigma_{2}\in\mathbb{R}$ and
$m_{1},m_{2}\in\mathbb{Z}$ verifying

$\mathfrak{S}_{1}\geqslant 1,\quad m_{1}m_{2}\leqslant 0,\quad
m_{1}+m_{2}>1,\quad\text{and}\quad\sigma_{1}+\sigma_{2}+1\leqslant 0.$

Let also $I$ be an open and bounded interval such that
$\overline{I}\subset\mathbb{R}_{+}^{*}$ and
$\chi\in\mathscr{C}^{\infty}_{c}(I)$. Then, there exist a constant $C>0$ and
an integer $N\geqslant 1$ such that, for every smooth vector field $Y$ on
$M_{1}$, one has

$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}Y\chi\mathbf{B}_{-\mathfrak{S}_{1},\sigma_{2}}^{(m_{2})}\right\|_{L^{2}\rightarrow
L^{2}}\leqslant C\left\|Y\right\|_{\mathscr{C}^{N}}.$

From the two lemmas above and Corollary 4.11, we deduce the following result
on the semigroup $(\varphi_{t})_{t\geqslant 0}$:

######  Corollary 4.15.

There exists $\vartheta_{0}>0$ depending only on the choice of $(\nu,g_{1})$
defined in Proposition 3.11 such that the following holds.

Let $\mathfrak{S}_{1}\geqslant 3$, $\ell\geqslant 0$, $\sigma_{1}\in[-1,1]$
and $m_{1}\in\mathbb{Z}$. Let also $I$ be an open and bounded interval such
that $\overline{I}\subset\mathbb{R}_{+}^{*}$. Then, there exist $C>0$,
$N\in\mathbb{N}$ such that, for all $t\geqslant 0$, for every
$u\in\mathscr{C}^{N}_{c}(I\times M_{1})$, for every smooth function $a$ in
$M_{1}$ and for every smooth vector field $Y$ on $M_{1}$,

(4.13)
$\left\|\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}X_{1}^{\ell}\left(a\partial_{r}+Y\right)\varphi_{-t}^{*}\mathbf{P}u\right\|_{L^{2}}\leqslant
Ce^{-t\vartheta_{0}\min
I}\|u\|_{\mathscr{C}^{N}}\left(\|a\|_{\mathscr{C}^{N}}+\|Y\|_{\mathscr{C}^{N}}\right).$

Again, the same result holds true for positive time provided that we replace
$\mathfrak{S}_{1}$ by $-\mathfrak{S}_{1}$ in the definition of
$\mathbf{B}_{\mathfrak{S}_{1},\sigma_{1}}^{(m_{1})}$.

###### Proof.

First note that one has

$X_{1}^{\ell}(a\partial_{r}+Y)=X_{1}[X_{1}^{\ell-1},a\partial_{r}+Y]+X_{1}(a\partial_{r}+Y)X_{1}^{\ell-1}.$

Hence, by induction, we get

$X_{1}^{\ell}(a\partial_{r}+Y)=\sum_{k=0}^{\ell}(a_{\ell,k}\partial_{r}+Y_{\ell,k})X_{1}^{k},$

where $Y_{\ell,k}$ (resp. $a_{\ell,k}$) is a vector field (resp. a function)
on $M_{1}$ (whose semi-norms depend on a finite number of derivatives of $Y$).
In particular, to get the south result, it suffices to bound the quantity

$\max_{k=0,\dots,\ell}\left\|\mathbf{B}_{N_{1},\sigma_{1}}^{(m_{1})}\left(a_{\ell,k}\partial_{r}+Y_{\ell,k}\right)\varphi_{-t}^{*}\mathbf{P}X_{1}^{k}u\right\|_{L^{2}}.$

Such a bound is obtained as a direct consequence of Corollary 4.11 and Lemmas
4.13 and 4.14. ∎

We end this subsection by establishing the connection between the sliced and
the global anisotropic Sobolev norms. To this purpose, we recall that
$\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}$ admits an almost inverse thanks to
the composition formula. More precisely, according to Lemma A.12, Remark A.11
and given any $N^{\prime}\geqslant 1$, one can find
$b_{N^{\prime}}\in\overline{S}^{0}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$
which is independent of $r$ and equal to $1$ modulo
$\overline{S}^{-1+}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ such that, for
every $\chi\in\mathscr{C}^{\infty}_{c}(\mathbb{R})$, it holds

(4.14)
$\left\|\left(\mathbf{Op}\left(e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}b_{N^{\prime}}\right)\mathbf{A}_{\mathfrak{S}_{0},\sigma_{0}}-\text{Id}\right)\chi\right\|_{H^{-N^{\prime}}(\mathbb{R}\times
M_{1})\rightarrow H^{N^{\prime}}(\mathbb{R}\times M_{1})}<\infty.$

With this convention at hand, one has the following result.

######  Lemma 4.16.

For every $\mathfrak{S}_{0}\geqslant 1$, $\sigma_{1}<\sigma_{0}\leqslant 0$,
$m_{1}\geqslant 5(\mathfrak{S}_{0}+|\sigma_{1}|)$ and $N^{\prime}\geqslant 0$,
one has

$\left\|\mathbf{B}_{\mathfrak{S}_{0},\sigma_{1}}^{(m_{1})}\mathbf{Op}\left(e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}b_{N^{\prime}}\right)\right\|_{L^{2}\rightarrow
L^{2}}<\infty.$

###### Proof.

Arguing as in the proof of Lemma 4.9, owing to Plancherel’s identity, it is
sufficient to study, for all $\rho\in\mathbb{R}$, the quantity

(4.15)
$\left\|e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{0},\sigma_{1},1})}{\operatorname{Op}^{M_{1}}}(e^{-F_{\mathfrak{S}_{0},\sigma_{0}}(\rho)}b_{N^{\prime}}(\rho))\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}.$

The latter is bounded from above by

$\left\|e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{0},\sigma_{1},1})}e^{-{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{0},\sigma_{1},1}(z_{1},\langle\rho\rangle^{-1}\zeta_{1}))}\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}\\\
\times\left\|e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{0},\sigma_{1},1}(z_{1},\langle\rho\rangle^{-1}\zeta_{1}))}{\operatorname{Op}^{M_{1}}}(e^{-F_{\mathfrak{S}_{0},\sigma_{0}}}(\rho)b_{N^{\prime}}(\rho))\right\|_{L^{2}(M_{1})\rightarrow
L^{2}(M_{1})}.$

Using the composition rule (A.21) and relying on the approximate expressions
(A.26) and (A.24) for the exponential of a pseudo-differential operator, we
have on the one hand

$\left\|e^{{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{0},\sigma_{1},1})}e^{-{\operatorname{Op}^{M_{1}}}(G_{\mathfrak{S}_{0},\sigma_{1},1}(z_{1},\langle\rho\rangle^{-1}\zeta_{1}))}\right\|_{L^{2}\rightarrow
L^{2}}\leqslant C\langle\rho\rangle^{2(\mathfrak{S}_{0}+|\sigma_{1}|)}.$

On the other hand, we recall that by construction of the sliced and global
escape functions, we have
$G_{\mathfrak{S}_{0},\sigma_{1},1}(z_{1},\langle\rho\rangle^{-1}\zeta_{1})=G_{\mathfrak{S}_{0},\sigma_{1}}(z_{1},\rho,\zeta_{1})$
and Proposition 3.11 gives

(4.16) $G_{\mathfrak{S}_{0},\sigma_{0}}(z_{1},\rho,\zeta_{1})\leqslant
F_{\mathfrak{S}_{0},\sigma_{0}}(z_{1},\rho,\zeta_{1})+2(\mathfrak{S}_{0}+|\sigma_{0}|)\log\langle\rho\rangle$

in the region $\\{|\zeta|\geqslant 1\\}$. By the composition rule in
$S^{\mathfrak{S}_{0}+|\sigma_{1}|}(\mathrm{T}^{*}M_{1})$ (see Appendix A.3.1)
and the approximate expressions (A.26) and (A.24) for the exponential of a
pseudo-differential operator, we can, modulo a bounded operator on $L^{2}$ (of
norm $\mathcal{O}(\langle\rho\rangle^{2(\mathfrak{S}_{0}+|\sigma_{1}|)})$),
focus on estimating the following norm:

$\langle\rho\rangle^{\sigma_{0}-\sigma_{1}}\left\|{\operatorname{Op}^{M_{1}}}\left(b(\rho)e^{(\sigma_{1}-\sigma_{0})\log
g}e^{G_{\mathfrak{S}_{0},\sigma_{0}}(\rho)-F_{\mathfrak{S}_{0},\sigma_{0}}(\rho)}\right)\right\|_{L^{2}\rightarrow
L^{2}}$

where by assumption $\sigma_{1}-\sigma_{0}<0$ and where
$b\in\overline{S}^{0}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ is independent
of $r$ and equal to $1$ modulo
$\overline{S}^{-1+}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$. By the Calderón-
Vaillancourt theorem (see e.g. (A.18)) and by (4.16), we get the estimate

$\left\|{\operatorname{Op}^{M_{1}}}\left(b(\rho)e^{(\sigma_{1}-\sigma_{0})\log
g}e^{G_{\mathfrak{S}_{0},\sigma_{0}}(\rho)-F_{\mathfrak{S}_{0},\sigma_{0}}(\rho)}\right)\right\|_{L^{2}\rightarrow
L^{2}}\leqslant C\langle\rho\rangle^{2(\mathfrak{S}_{0}+|\sigma_{0}|)}$

We finally obtain the bound

$\left\|e^{{\operatorname{Op}^{M_{1}}}\left(G_{\mathfrak{S}_{0},\sigma_{1}}\right)}{\operatorname{Op}^{M_{1}}}\left(e^{-F_{\mathfrak{S}_{0},\sigma_{0}}(\rho)}b_{N^{\prime}}(\rho)\right)\right\|_{L^{2}\rightarrow
L^{2}}\leqslant C\langle\rho\rangle^{4(\mathfrak{S}_{0}+|\sigma_{1}|)},$

and we can conclude, relying on the fact that $m_{1}\geqslant
5(\mathfrak{S}_{0}+|\sigma_{1}|)$. ∎

###  4.6. Proof of Proposition 4.5

We are eventually in position to prove Proposition 4.5. Before getting to this
matter, we first give a description of the Hamiltonian vector field $X_{\Phi}$
generated by a smooth function $\Phi$ on $\Sigma$.

####  4.6.1. Hamiltonian vector fields

In what follows we will work with the coordinates
$(r,z_{1})=(r,x,\xi_{1})\in\mathbb{R}_{+}^{*}\times M_{1}$. Recall that the
connection map associated with the Riemannian metric $\mathrm{g}$ on $\Sigma$
is defined as follows. Let $z=(x,\xi)\in M=\mathrm{T}^{*}\Sigma$ and let
$z(t)=(x(t),\xi(t))$ be a curve in $M$ such that $z(0)=z$ and
$z^{\prime}(0)=Z$. Then, one sets

$\mathcal{C}_{z}:Z\in\mathrm{T}_{z}M\mapsto\nabla_{z^{\prime}(0)}\xi(0)\in\mathrm{T}_{x}^{*}\Sigma,$

where $\nabla$ is the Levi-Civita connection associated to
$(\Sigma,\mathrm{g})$. One can verify that $\mathcal{C}_{z}$ induces an
isomorphism between the horizontal space $\text{Ker}(D\pi(z)),$ with
$\pi:(x,\xi)\in M\mapsto x\in\Sigma$ the canonical projection [Rug07, Ch. 1].
Then, one has that $X_{\Phi}(z)=\mathcal{C}_{z}^{-1}(\mathrm{d}\Phi(x))$ with
$\mathrm{d}\Phi(x)$ that can be decomposed as follows

$\mathrm{d}\Phi(x)=\mathrm{\mathrm{missing}}{g}_{x}^{*}\left(\mathrm{d}\Phi(x),\xi_{1}\right)\xi_{1}+D^{\perp}_{\xi_{1}}\Phi(x).$

In the coordinates $(r,z_{1})=(r,x,\xi_{1})\in\mathbb{R}_{+}^{*}\times M_{1}$,
it finally yields the following decomposition for the vector field $X_{\Phi}$:

(4.17)
$X_{\Phi}=\frac{1}{r}\mathcal{C}_{z_{1}}^{-1}\left(D^{\perp}_{\xi_{1}}\Phi(x)\right)+\mathrm{g}_{x}^{*}\left(\mathrm{d}\Phi(x),\xi_{1}\right)\partial_{r},$

where the vector field

$X_{\Phi}^{1}(z_{1})=\mathcal{C}_{z_{1}}^{-1}\left(D^{\perp}_{\xi_{1}}\Phi(x)\right)$

can be identified with a vector field on $M_{1}$. This can be reexpressed in
terms of pseudo-differential operators on $\mathbb{R}\times M_{1}$ as defined
in Appendix A.2. Namely, one has

(4.18)
$X_{\Phi}=\frac{1}{r}\mathbf{Op}\left(b_{1}(z_{1};\zeta_{1})\right)+\mathbf{Op}\left(\rho\Phi_{1}(z_{1})\right),$

where $b_{1}(z_{1};\zeta_{1})$ is a polynomial of degree $1$ in the
$\zeta_{1}$ variable. Namely, one has

(4.19) $|\beta|\geqslant 2\quad\Longrightarrow\quad
p_{1,\alpha,\beta}(\rho\Phi_{1}(z_{1}))=p_{1,\alpha,\beta}(b_{1}(z_{1};\zeta_{1}))=0,$

and

(4.20) $|\beta|=0,1\quad\Longrightarrow\quad
p_{1,\alpha,\beta}(\rho\Phi_{1}(z_{1}))+p_{1,\alpha,\beta}(b_{1}(z_{1};\zeta_{1}))\leqslant
C_{\alpha,\beta}\|\Phi\|_{\mathscr{C}^{|\alpha|+1}(\Sigma)},$

where $p_{1,\alpha,\beta}$ are the semi-norms associated to the class of
symbols $\overline{S}^{1}(\mathrm{T}^{*}(\mathbb{R}\times M_{1}))$ (see
Appendix A.3).

######  Remark 4.17.

For later purposes, we record a few useful relations when
$\Phi=\Phi(u)=\mathrm{K}\pi_{*}u$, for
$\mathrm{K}f(x)=\int_{\Sigma}K(x,y)f(y)\,\mathrm{d}\text{vol}_{\mathrm{g}}(y)$
the smoothing interaction operator in the Vlasov equation (1.1). Namely, we
can set

(4.21)
$K_{1}(x,\xi_{1},y)=\mathrm{g}_{x}^{*}(\mathrm{d}_{x}K(x,y),\xi_{1})\in\mathscr{C}^{\infty}(M_{1}\times\Sigma,\mathbb{R}),$

and

(4.22)
$\mathcal{X}_{1}(x,\xi_{1},y)=\mathcal{C}_{z_{1}}^{-1}D_{\xi_{1}}^{\perp}K(x,y)\in\mathscr{C}^{\infty}(M_{1}\times\Sigma,TM_{1}).$

This allows to rewrite $X_{\Phi(u)}$ as

(4.23)
$X_{\Phi(u)}(r,z_{1})=\frac{1}{r}X_{\Phi(u)}^{1}(z_{1})+\Phi_{1}(u)(z_{1})\partial_{r},$

where

$\Phi_{1}(u)(z_{1})=\int_{M}K_{1}(z_{1},y)u(y,\eta)\mathrm{d}{\mathrm{L}}(y,\eta)$

and