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# On hyperbolicity and Gevrey well-posedness.
Part three: a model of weakly hyperbolic systems.

Baptiste Morisse School of Mathematics, Cardiff University -
<EMAIL_ADDRESS>The author is supported by the EPSRC grant
"Quantitative Estimates in Spectral Theory and Their Complexity"
(EP/N020154/1). The author thanks his advisor Benjamin Texier and Jean-
Francois Coulombel for all the remarks on this work. The author thanks warmly
the anonymous referee for all the detailed comments on this work.

###### Abstract

We consider a model of weakly hyperbolic systems of first-order, nonlinear
PDEs. Weak hyperbolicity means here that the principal symbol of the system
has a crossing of real-valued eigenvalues, and is not uniformly
diagonalizable. We prove the well-posedness of the Cauchy problem in the
Gevrey regularity for all Gevrey indices $\sigma$ in $(1/2,1)$. The proof is
based on the construction of a suitable approximate symmetrizer of the
principal symbol and an energy estimate in Gevrey spaces. We discuss both the
generality of the assumption on the structure of the principal symbol and the
sharpness of the lower bound of the Gevrey index.

###### Contents

  1. 1 Introduction
    1. 1.1 Background: on weakly hyperbolic systems
    2. 1.2 Background: on systems transitioning away from hyperbolicity
    3. 1.3 Generic time transitions
    4. 1.4 Current result
  2. 2 Main assumptions and results
  3. 3 Proof of the energy estimate
    1. 3.1 Time derivative of the energy
  4. 4 Pseudo-differential tools
    1. 4.1 Study of symbol $b$
    2. 4.2 Properties of class of symbols
    3. 4.3 pseudo-differential calculus
    4. 4.4 Energy estimate
  5. 5 Appendices: two lemmas of real analysis and metrics in the phase space
    1. 5.1 Glaeser-type inequalities
    2. 5.2 Metrics in the phase space and pseudo-differential calculus

##  1 Introduction

In this paper we prove an energy estimate for systems of the form

$\partial_{t}u=\begin{pmatrix}0&1\\\
(t+|x-x_{0}|^{2})e(t,x)&0\end{pmatrix}\partial_{x}u+F(t,x,u)u$ (1.1)

where $x\in{\mathbb{R}}$, $F(t,x,u)$ is nonlinear in $u$, and $e$ is a Gevrey
function that is bounded away from zero and compactly supported around
$(t,x)=(0,x_{0})$. This result translates by classical arguments into a local-
in-time well-posedness result in Gevrey spaces for the Cauchy problem for
(1.1).

This result could also be extended into a general well-posedness result for a
wider class of systems in several spatial dimensions:

$\partial_{t}u=\sum_{1\leq j\leq d}A_{j}(t,x)\partial_{x_{j}}u+f(t,x,u)$ (1.2)

where $x$ in ${\mathbb{R}}^{d}$, the $A_{j}$ are in ${\mathbb{R}}^{2\times
2}$, $f$ in ${\mathbb{R}}^{2}$, the $A_{j}$ have some smoothness in time and
are Gevrey regular in $x$, the nonlinearity $f$ is analytic in all variables,
and the principal symbol $A=\sum_{j}A_{j}(t,x)\xi_{j}$ experiences a
transition from hyperbolicity to ellipticity. Precisely, in order to extend
our result for (1.1) into a well-posedness result for (1.2), we assume

  * $\bullet$

hyperbolicity of the principal symbol $A$, that is the spectrum of
$A(t,x,\xi)$ is real.

  * $\bullet$

At a distinguished point
$(0,x_{0},\xi_{0})\in{\mathbb{R}}\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$,
the existence of a real and non semi-simple eigenvalue (semi-simplicity means
simplicity as a zero of the minimal polynomial of $A(t,x,\xi)$).

  * $\bullet$

And finally we assume that $A$ transitions from hyperbolicity to ellipticity
at $(0,x_{0},\xi_{0})$ for negative times. By transition from hyperbolicity to
ellipticity we mean the phenomenon studied in [Mor18]. Here this transition is
not degenerate, we will go back to this point in Section 1.3.

In a forthcoming version of this paper, we expound on these Assumptions, and
handle the general case of weakly hyperbolic systems of the form (1.2).

In the present version of this paper, we work exclusively with the model
(1.1). The fact that (1.1) is one-dimensional ($x\in{\mathbb{R}}$) does not
play any role in our analysis.

Further simplifying into $e\equiv 1$, $F(u)=\begin{pmatrix}0&0\\\
0&u_{1}\end{pmatrix}$, we find the system

$\partial_{t}\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix}=\begin{pmatrix}0&1\\\
t+x^{2}&0\end{pmatrix}\partial_{x}\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}+\begin{pmatrix}0\\\ u_{1}^{2}\end{pmatrix},$

which reduces to the wave-like equation in $u_{1}\in{\mathbb{R}}$:

$\partial_{t}^{2}u_{1}=\partial_{x}\left((t+x^{2})\partial_{x}u_{1}\right)+\partial_{x}(u_{1}^{2})$
(1.3)

The wave operator in (1.3) is singular at $(t,x)=(0,0)$, and elliptic for
$t+x^{2}<0$ – in particular for negative times.

Our interest is in the Cauchy problem at $t=0$, for forward times. Our present
result has a double background: first in well-posedness for weakly hyperbolic
systems, a line of research popularized in particular by Colombini and
collaborators [CJS83], [CN07] and [CNR], and in systems transitioning from
hyperbolic to ellipticity, a line of research initiated by Lerner, Morimoto
and Xu in [LMX10].

###  1.1 Background: on weakly hyperbolic systems

####  1.1.1 The classical result of Colombini, Janelli and Spagnolo

We consider here the following second-order, linear scalar equation

$\partial_{t}^{2}v=\partial_{x}\left(a\partial_{x}v\right)$ (1.4)

with $a=a(t)$ a nonnegative, $C^{k}([0,T])$ function for some $k\geq 1$. Such
weakly hyperbolic, second-order scalar equations have long been studied by in
Gevrey regularity.

A cornerstone of the domain is Colombini, Janelli and Spagnolo’s paper
[CJS83], which proved Gevrey well-posedness in the case of spatially-
independent symbol $a(t)$. The work of Colombini, Janelli and Spagnolo is
based on an energy estimate, which uses the particular structure of the wave
equation (1.4) and a lemma of real analysis which extends the classical
Glaeser’s inequality 111 In fact, Lemma 1 in [CJS83] is a weaker version of
Glaeser inequality: Lemma 1 gives a bound on the $L^{1}$ norm of $a^{1/k}$,
whereas the Glaeser inequality is pointwise for $a(t)^{1/2}$. , namely that if
$a(t)$ is a $C^{k}$ nonnegative function on $[0,T]$, then $a(t)^{1/k}$ is
absolutely continuous on $[0,T]$ (see Lemma 1 in [CJS83], and [Gla63] for
Glaeser’s inequality).

In the case when $a=a(t)$, equation (1.4) transforms into the scalar ODE

$\partial_{t}^{2}w(t,\xi)=-a(t)|\xi|^{2}w(t,\xi)$

thanks to the Fourier transform, and where we denote
$w(t,\xi)=\widehat{v}(t,\xi)\in{\mathbb{C}}$. As $a(t)$ is supposed to be only
nonnegative (weak hyperbolicity), we introduce a small parameter
$\varepsilon>0$ (later on $\varepsilon=\varepsilon(\xi)$) and the approximate
energy

$E_{\varepsilon}(t,\xi)=|\partial_{t}w(t,\xi)|^{2}+\left(a(t)+\varepsilon\right)|\xi|^{2}|w(t,\xi)|^{2}$

whose time derivative is

$\partial_{t}E_{\varepsilon}=a^{\prime}(t)|\xi|^{2}|w|^{2}+2\varepsilon|\xi|^{2}{\rm
Re}\,w\partial_{t}w.$

Having in mind a Gårding-type inequality to fulfil an energy estimate, we
bound the previous equality by

$\partial_{t}E_{\varepsilon}\leq|a^{\prime}(t)||\xi|^{2}|w|^{2}+\varepsilon^{1/2}|\xi|\,E_{\varepsilon}$

thanks to Cauchy-Schwarz’s inequality. To bound the term
$|a^{\prime}(t)||\xi|^{2}|w|^{2}$, we need here to link $|a^{\prime}|$ to
$a+\varepsilon$ in order to bound $|a^{\prime}(t)||\xi|^{2}|w|^{2}$ by the
term $\left(a(t)+\varepsilon\right)|\xi|^{2}|w(t,\xi)|^{2}$ of the energy (up
to a multiplicative constant). As
$\left((a+\varepsilon)^{1/k}\right)^{\prime}=\frac{1}{k}a^{\prime}/(a+\varepsilon)^{1-1/k}$,
we write

$\displaystyle|a^{\prime}(t)||\xi|^{2}|w|^{2}$ $\displaystyle=$
$\displaystyle\left|\frac{a^{\prime}}{(a+\varepsilon)^{1-1/k}}\right|\frac{1}{(a+\varepsilon)^{1/k}}\,(a+\varepsilon)|\xi|^{2}|w|^{2}$
$\displaystyle=$ $\displaystyle
k\left|\left((a+\varepsilon)^{1/k}\right)^{\prime}\right|\frac{1}{(a+\varepsilon)^{1/k}}\,(a+\varepsilon)|\xi|^{2}|w|^{2}.$

As $a$ is nonnegative, there holds

$\displaystyle\partial_{t}E_{\varepsilon}$ $\displaystyle\leq$
$\displaystyle\left|\left((a+\varepsilon)^{1/k}\right)^{\prime}\right|\frac{1}{(a+\varepsilon)^{1/k}}\,E_{\varepsilon}+\varepsilon^{1/2}|\xi|\,E_{\varepsilon}$
$\displaystyle\leq$
$\displaystyle\left|\left((a+\varepsilon)^{1/k}\right)^{\prime}\right|\varepsilon^{-1/k}\,E_{\varepsilon}+\varepsilon^{1/2}|\xi|\,E_{\varepsilon}$

hence

$\displaystyle E_{\varepsilon}(t,\xi)$ $\displaystyle\lesssim$
$\displaystyle\exp\left(\varepsilon^{-1/k}\int_{0}^{t}\left|\left((a+\varepsilon)^{1/k}\right)^{\prime}(s)\right|ds+t\varepsilon^{1/2}|\xi|\right)E_{\varepsilon}(0,\xi)$
$\displaystyle\lesssim$
$\displaystyle\exp\left(\varepsilon^{-1/k}|a|_{C^{k}}^{1/k}+T\varepsilon^{1/2}|\xi|\right)E_{\varepsilon}(0,\xi)$

for all $t\leq T$ thanks to Lemma 1 in [CJS83]. In order to optimize the
exponential term, we put $\varepsilon=|\xi|^{-2/(k+2)}$ to get finally

$E_{\varepsilon}(t,\xi)\lesssim e^{c|\xi|^{2/(k+2)}}E_{\varepsilon}(0,\xi)$

for some constant $c>0$.

Thanks to this (pointwise in frequency) energy estimate, the authors of
[CJS83] proved that the Cauchy problem associated to (1.4) is well-posed in
Gevrey spaces $\mathcal{G}^{\sigma}_{c}$ (see Definition 2.2 in [Mor17]) with
$\sigma>2/(k+2)$, where $k$ is the regularity of the coefficient of equation
(1.4). Note that, as the regularity of $a$ grows, the range of Gevrey indices
for which well-posedness holds grows as such.

####  1.1.2 Beyond the 1983 article of Colombini, Janelli and Spagnolo

The work of [CJS83] has been followed and extended notably by Colombini and
Nishitani in [CN07] and by Colombini, Nishitani and Rauch in [CNR].

In [CN07], Colombini and Nishitani study the case when $a$ depends also in
$x$, that is, $a(t,x)$ is assumed to be nonnegative and in
$C^{2}([0,T],G^{s}_{R})$ where $s$ stands as usual for $1/\sigma$ (see
Definition 2.1 in [Mor17] for Gevrey spaces defined from the spatial
viewpoint, and Proposition 2.1 therein for its link with
$\mathcal{G}^{\sigma}_{\tau}$). Note that, as it is made explicit in Theorem
1.3 in the paper of Colombini and Nishitani, it is assumed that $a(t,x)$ is in
fact nonnegative in $[-\delta,T+\delta]$ for some $\delta>0$. This additional
assumption on $a$ is crucial in the course of the proof of [CN07]. Indeed, in
order to extend the energy-based study in [CJS83], the authors of [CN07] use a
pseudo-differential calculus. In the context of symbols, Lemma 1 in [CJS83] is
no longer helpful, as it leads to an $L^{1}$ estimate of the time derivative
of $a$; instead, a pointwise inequality in $(t,x)$ is needed, hence the use of
Glaeser’s inequality. For Glaeser’s inequality to hold in a compact subspace
of ${\mathbb{R}}\times{\mathbb{R}}^{d}$, the nonnegativity condition on $a$
has to hold on a larger subspace containing the compact, see Appendix 5.1.
Well-posedness is then proved for any $1\leq s<2$ – that is for any
$1/2<\sigma\leq 1$ thanks to Proposition 2.1 in [Mor17] – extending the work
of [CJS83].

More recently, Colombini and Nishitani have pursued their line of research in
[CN17]. The authors are interested in wave equations with coefficients with
independent variables $t$ and $x$. Using an exponential weight and the same
metric in the phase space as we use in this paper, the authors of [CN17] prove
again well-posedness for $1/2<\sigma\leq 1$.

In [GJR18], Garetto, Jäh and Rhuzansky prove well-posedness in anisotropic
Sobolev spaces for a large class of linear systems of first-order PDEs. The
authors consider triangular principal symbols and source terms whose order are
sufficiently low compared to the dimension of the systems. Their method is
based on representation of solutions of triangular systems.

The work of Colombini, Nishitani and Rauch in [CNR] explores a different
method. Generic weakly hyperbolic systems (1.1) are considered, not only
second-order scalar equations (1.4) as in [CJS83] or [CN07], i.e. the
principal symbol $A(t,x,\xi)$ is there a $N\times N$ matrix with real spectrum
but with potential eigenvalue crossings. To study such general symbols, the
authors introduce a block size barometer $\theta=m-1$, which roughly measures
the extent to which $A(t,x,\xi)$ can be smoothly block diagonalized by blocks
of size $m$. For smoothly diagonalizable symbols, $\theta=0$ ; on the other
hand, $\theta=N-1$ if the symbol is not block diagonalizable at all - which is
typically our framework, for $N=2$. In order to get a general result on well-
posedness in Gevrey spaces, regardless of the spectral details of the
principal symbol of (1.1), a suitable Lyapunov symmetrizer is studied. In
exchange for a general statement, the range of Gevrey indices for which well-
posedness holds is quite reduced, and depends on $\theta$. Precisely, well-
posedness for (1.1) is proved for any

$\sigma\geq\min\left\\{\frac{1+6\theta}{2+6\theta},\frac{2+4\theta}{3+4\theta}\right\\}.$

Note that in our framework there holds $\theta=1$ which leads the lower bound
$6/7$ for the Gevrey index.

###  1.2 Background: on systems transitioning away from hyperbolicity

The question of the instability of systems transitioning away from
hyperbolicity has been first raised in [LMX10], extending the work [Mét05] on
initially elliptic systems. In [LMX10] quasilinear scalar equations are
considered, with analytic coefficients. It is assumed that these equations
experience a transition from initial hyperbolicity to ellipticity for positive
times. For such equations, it is proved in [LMX10] that the Cauchy problem
with initial analytic data is strongly unstable with respect to $C^{\infty}$
perturbation.

A similar instability result is established in [LNT17], in which quasilinear
systems with smooth coefficients are considered. In various cases of
transitions from initial hyperbolicity to ellipticity, the Cauchy problem in
Sobolev spaces is proved to be unstable, in the sense of Hadamard. That is,
hypothetical flow of the system fails to be Hölder from Sobolev spaces to
$L^{2}$. The article [Lu16] explores a similar theme in the context of high-
frequency solutions of singularly perturbed symmetric hyperbolic systems.

In a previous work [Mor18], we considered first order quasi-linear system
(1.1) experiencing a transition from hyperbolicity to ellipticity. A typical
example of symbols which falls into the class studied in Section 2.3 in
[Mor18] is

$A(t,x,\xi)=\begin{pmatrix}0&1\\\ -(t-t_{\star}(x,\xi))&0\end{pmatrix}$ (1.5)

in a neighborhood of
$(0,0,\xi_{0})\in{\mathbb{R}}_{t}\times{\mathbb{R}}^{d}_{x}\times{\mathbb{R}}^{d}_{\xi}$,
with

$t_{\star}(x,\xi)=|x|^{4}+|\xi-\xi_{0}|^{2}$ (1.6)

In such a case, we proved in Theorem 2.11 in [Mor18] that (1.1) is not well-
posed in Gevrey spaces for $\sigma\in(0,2/13)$. As explained in Section 2,3
therein, the term $|x|^{4}$ corresponds to a degenerate time transition. As we
see in Figure 1, the hyperbolic domain $\left\\{(t,x)\in[0,T]\times
B_{r}(x_{0})\,:\,t\leq|x|^{4}\right\\}$ for $|x|^{4}$ is thinner than the
hyperbolic domain $\left\\{(t,x)\in[0,T]\times
B_{r}(x_{0})\,:\,t\leq|x|^{2}\right\\}$ for $|x|^{2}$. This observation
allowed us to treat the term $|x|^{4}$ as a remainder term. Having treated the
case of degenerate transitions in our paper [Mor18], we now wish to handle
generic transitions. These involve, as explained in [LNT17], time-transition
functions of the form $t_{\star}(x)=x^{2}$, in one spatial dimension, and a
Jordan block for the principal symbol, that is (1.1) with
$t_{\star}(x)=x^{2}$.

Figure 1: Comparison between degenerate $x^{4}$ and non-degenerate $x^{2}$

###  1.3 Generic time transitions

The proof of [Mor18] in the case $t_{\star}(x)=|x|^{2}$ fails essentially due
to the size of the hyperbolic domain $\left\\{(t,x)\in[0,T]\times
B_{r}(x_{0})\,:\,t\leq|x|^{2}\right\\}$ in the setting developed therein. The
term $|x|^{2}$ may not be considered as a remainder term.

Thus in order to prove ill-posedness in the generic configuration, we have to
handle the not so small hyperbolic region under the transition curve. This
means proving a form of well-posedness for $t<t_{\star}$. At $t=t_{\star}$ the
unstable modes are turned on and the analysis of [Mor18] should apply. For the
analysis of [Mor18] to go through, we must find suitable analytic data
$\left(h_{\varepsilon}\right)_{\varepsilon>0}$ such that the Cauchy problem at
$t=t_{\star}$ is ill-posed (with the difficulty that $t_{\star}$ is a function
of $x$ in $1$d and of $(x,\xi)$ in multi-d).

The outstanding question is then to find suitable initial (at $t=0$, for all
$x$) data which give rise to the suitable unstable data $h_{\varepsilon}(x)$
at $t=t_{\star}(x)$. In other words, we want to solve the backward-in-time
Cauchy problem, in the hyperbolic zone, from $t=t_{\star}(x)$ to $t=0$. This
motivates the form of the principal symbol under consideration here, as we
describe in the next Section.

###  1.4 Current result

As mentioned above, generic transitions from hyperbolic to ellipticity involve
in one spatial dimension principal symbols of the form (1.5) with
$t_{\star}(x)=x^{2}$. In order to study these transitions, we must understand
the backward-in-time Cauchy problem for such operators. This motivates the
form of our principal symbol in (1.1). The function $e$ is assumed to be
bounded away from zero and Gevrey (see Assumption 2.1). Under this assumption,
we prove an energy estimate for solutions with compact support with regularity
$\mathcal{G}^{\sigma}_{\tau}$ for any $\sigma\geq 1/2$ and $\tau>0$ small.
This is Theorem 1.

The proof relies on the construction of a suitable symmetrizer $S={\rm
op}(\text{diag}(1,b))$ with symbol
$b(t,x,\xi)=(t+x^{2}+\langle\xi\rangle^{-c})^{-1/2}$ and a Gevrey energy
estimate. An important observation is that the symbol $b$ does not belong to a
standard class of symbols. Indeed, $b(0,0,\xi)=\langle\xi\rangle^{c/2}$
whereas $b(t,x,\xi)\in S^{0}_{1,0}$ when $t\geq\underline{t}$ and $|x|>r$. To
reconcile both point of views, we make use of class of symbols defined with
respect to a metric of the phase space, as described in [Ler11]. In Lemma 4.8,
we prove that $b\in S(b,g)$ where the time-dependent metric $g$ is defined in
(4.6). This metric has already been used in [CN07] and [CN17]. Our paper
relies also on our paper [Mor17] which contains our work on pseudo-
differential operators with symbols which are Gevrey regular in the spatial
variable.

We note that the symmetrizer $S$ is anisotropic, as it stresses out more
(Sobolev) regularity for the second component $u_{2}$ than for the first one
$u_{1}$. This observation is closely related to [GJR18], in which well-
posedness is proved in anisotropic Sobolev spaces for a certain class of
weakly hyperbolic systems. In [GJR18], the additional assumption on the order
of the force terms compared to the dimension of the systems can be read as
$c=2$ in our settings, where $c$ is the order of the perturbation. In the
present paper, without such a strong assumption on the source term $F$ we
cannot expect to reach $c=2$ but rather $c\in(0,1]$. See also Remark 4.16,
which gives a hint on how to reach $c=4/3$.

######  Remark 1.1.

Our result is outside the range of the article [CN07]. The symbol $a(t,x)$,
which is in our case similar to $t+x^{2}$, does not satisfy Glaeser’s
inequality for negative times. This result is also an improvement of the
result given in [CNR], as we attain in our paper the lower bound $1/2$ for the
Gevrey indices, compared to the lower bound $6/7$ as described above. The main
difference is that, in our paper, we take care of the spectral details of the
principal symbol, as we assume it is a $2$ by $2$ matrix, with a specific
crossing of eigenvalues.

##  2 Main assumptions and results

We consider the Cauchy problem (1.1) which we rewrite in a more compact way as

$\partial_{t}u=\begin{pmatrix}0&1\\\
a(t,x)&0\end{pmatrix}\partial_{x}u+F(t,x,u)u$ (2.1)

where $x$ is in ${\mathbb{R}}$, $u$ in ${\mathbb{R}}^{2}$ and $F(t,x,u)$ is a
$2\times 2$ matrix. We describe first our assumptions on the regularity and
the structure of both $a$ and $F$.

######  Assumption 2.1 (Structure and regularity for $a$).

We assume that

$a(t,x)=\left(t+|x-x_{0}|^{2}\right)e(t,x)$

where $e(t,x)$ has compact support $[0,T^{\prime}]\times
B_{r^{\prime}}(x_{0})$ for some $T^{\prime}>0$ and $r^{\prime}>0$. Besides,
$e$ is in $C^{2}([0,T^{\prime}],G^{1/\sigma}_{R})$, that is there is $C>0$
such that

$\left|\partial_{x}^{\alpha}e(t,x)\right|\leq
CR^{|\alpha|}\alpha!^{1/\sigma}\quad,\quad\forall\,\alpha\in{\mathbb{N}}\;,\;\forall\,(t,x)\in[0,T^{\prime}]\times
B_{r^{\prime}}(x_{0}).$

There is also $0<T<T^{\prime}$ and $0<r<r^{\prime}$ such that

$1/2\leq e(t,x)\leq 2\quad,\quad\forall\,(t,x)\in[0,T]\times B_{r}(x_{0}).$
(2.2)

We denote

$\underline{\tau}=R^{-\sigma}/\sigma$ (2.3)

the Gevrey regularity of $a$, in the Fourier point of view (see Definition 2.2
and Proposition 2.1 in [Mor17] with $s=1/\sigma$). Concerning the force term
$F(t,x,u)$, we make the following

######  Assumption 2.2 (Regularity for $F$).

The function $F$ is entire in $u$ in a neighborhood of $u=0$ and there holds

$F(t,x,u)=\sum_{k\in{\mathbb{N}}^{2}}F_{k}(t,x)u^{k}$ (2.4)

where coefficients $F_{k}(t,x)$ are in $\mathcal{G}^{\sigma}_{\tau_{0}}$,
uniformly in $t$ and $k\in{\mathbb{N}}^{2}$.

As spaces $\mathcal{G}^{\sigma}_{\tau}$ are algebra, if $u$ is controlled in
$\mathcal{G}^{\sigma}_{\tau}$ the same holds for all powers $u^{k}$. We could
lighten the assumption of analyticity in the variable $u$ for $F$ by assuming
some Gevrey regularity. This would only add technicalities, which we choose to
avoid at this stage.

The main result of our paper is an energy estimate in Gevrey space
${\rm\mathcal{G}}^{\sigma}_{\tau}$ for any $\sigma\geq 1/2$ and for small
$\tau$. The lower Gevrey index $1/2$ is the expected lower bound for the
Gevrey regularity in the presence of a source term $F(t,x,u)u$. With
additional assumption on $F$, the same analysis may lead to a lower bound
$\sigma\geq 1/3$ (see Remark 4.16). To obtain such a result, we define a
suitable symmetriser for $A$, introducing first the symbol

$b(t,x,\xi)=\left(a(t,x)+\langle\xi\rangle^{-c}\right)^{-1/2}$ (2.5)

for some $c\in(0,2]$ and denoting

$\langle\xi\rangle=\left(1+|\xi|^{2}\right)^{1/2}$ (2.6)

Defining

$S(t,x,\xi)=\begin{pmatrix}1&0\\\ 0&b(t,x,\xi)\end{pmatrix}$ (2.7)

one key point is that

$S^{2}\begin{pmatrix}0&1\\\
a+\langle\cdot\rangle^{-c}\end{pmatrix}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$ (2.8)

is real symmetric. The perturbation by a lower order term implies working in
Gevrey regularity, but in exchange allows for an approximate symmetrization of
the principal symbol $A$. This is closely related to the work of Colombini and
Métivier [CM17] for uniformly diagonalizable symbols, depending only on time.

Section 4.1 will be devoted to prove that $b$ is in the class of symbols
$S(b,g)$, defined in (5.11) and the metric $g$ defined in (4.6). This is done
principally thanks to the non-negativity of $a$ and Glaeser’s inequality (see
Lemma 4.2 and Section 5.1 below).

In all the following, we denote

$D={\rm op}\left(\langle\xi\rangle\right)\quad\text{and}\quad D^{\sigma}={\rm
op}\left(\langle\xi\rangle^{\sigma}\right).$ (2.9)

Let $\sigma\in(0,1)$, $\tau>0$ and $u$ in ${\rm\mathcal{G}}^{\sigma}_{\tau}$.
We introduce the Gevrey energy

$E(\tau,u(t))=\frac{1}{2}\left|{\rm op}(S)e^{\tau
D^{\sigma}}u(t)\right|^{2}_{L^{2}}.$ (2.10)

Thanks to the result of sharp finite speed propagation for (2.1) under
assumptions of "constant outside a compact set", the result of [CR10] can be
used. We look for solutions with compact support in $(t,x)$ included in
$[0,T]\times B_{r}(x_{0})$, which can be done if the initial datum $u_{0}$ has
sufficiently small compact support (with respect to $T$ and the finite speed
propagation of (2.1)). The existence of such solutions with regularity in
$\mathcal{G}^{\sigma}_{\tau}$ is assured by our main result and by standard
results on local well-posedness in Gevrey for such systems.

######  Theorem 1.

For any $\tau_{0}<\underline{\tau}$ with $\underline{\tau}$ defined in (2.3),
there is ${\bm{\tau}}>0$ such that

$E\left(\tau_{0}-{\bm{\tau}}t,u(t)\right)\lesssim
E(\tau_{0},u(0))\quad,\quad\forall\,t\in\left[0,\min\left(T,\frac{\tau_{0}}{{\bm{\tau}}}\right)\right].$

Section 3 is devoted to the proof of Theorem 1. Here are some remarks
concerning our result:

  * •

Concerning the case $x\in{\mathbb{R}}^{d}$ for $d\geq 2$, that is for

$\partial_{t}u=\sum_{1\leq j\leq d}A_{j}(t,x)\partial_{x_{j}}u+F(t,x,u)u$

our method described in the present paper may also apply. Considering the
principal symbol $A(t,x,\xi)=\sum_{1\leq j\leq d}A_{j}(t,x)i\xi_{j}$, the
analogous of (2.1) is for the principal symbol to have normal form

$A(t,x,\xi)=i|\xi|\begin{pmatrix}0&1\\\
a(t,x,\omega)&0\end{pmatrix}\quad\text{with}\quad\omega=\frac{\xi}{|\xi|}$

with $a(t,x,\omega)\sim t+x^{2}$.

  * •

For higher dimensions for the system, our method may also apply for normal
forms

$\begin{pmatrix}0&1&0&{3}\\\ 0&0&1&0&{2}\\\ {6}\\\
a_{1}&a_{2}&{2}&a_{N-1}&0\end{pmatrix}$

We would need to adapt consequently our symmetrizer $S$ as

$\begin{pmatrix}0&1&0&{3}\\\ 0&0&1&0&{2}\\\ {6}\\\
a_{1}+\langle\xi\rangle^{-c}&a_{2}&{2}&a_{N-1}&0\end{pmatrix}$

and the expected lower bound for the Gevrey index is then $1-1/N$ (again, in
the presence of source term $Fu$).

  * •

The general case $a(t,x)\geq 0$, with $C^{2}$ regularity, could be treated by
our method, even in the case where $\partial_{t}a(0,x_{0})=0$. The symmetrizer
$S$ could be defined in the same way. The symbol $b$ is still in $S(b,g)$,
thanks to Glaeser inequality. The only main difference would be the care of
the term $\partial_{t}S$ in the energy estimate. In the case
$\partial_{t}a(0,x_{0})=0$, a time Glaeser inequality holds which allows to
control $\partial_{t}a$ in terms of $b^{-1}$.

Finally we note that the main case of fully quasilinear systems $A=A(u)$, such
as Euler equations with Van der Waals laws or other physical meaningful
systems, are out of reach of our current analysis and understanding. The study
of such systems should be a main topic in the future of our research.

##  3 Proof of the energy estimate

In order to study (2.1) in Gevrey spaces, a classical approach is to introduce
a Gevrey radius $\tau(t)$ which decreases linearly in time. Let
$\tau_{0}<\underline{\tau}$. We define

$\tau(t)=\tau_{0}-{\bm{\tau}}t.$ (3.1)

with ${\bm{\tau}}>0$ to be determined in the course of the proof.

###  3.1 Time derivative of the energy

We compute here the time derivative of the energy $E$ defined in (2.10). The
energy $E$ depends on time through the symbol $b$, the Gevrey radius $\tau(t)$
and $u$.

We introduce

$v(t)=e^{\tau(t)D^{\sigma}}u(t)$ (3.2)

with $\tau(t)$ defined in (3.1) and $D^{\sigma}$ in (2.9). There holds

$\partial_{t}v(t)=-{\bm{\tau}}D^{\sigma}v(t)+e^{\tau(t)D^{\sigma}}\partial_{t}u(t).$

As $u$ solves system (2.1), $v$ solves

$\partial_{t}v=-{\bm{\tau}}D^{\sigma}v+e^{\tau
D^{\sigma}}\left(A\partial_{x}u+F(u)u\right).$

In order to work with $v$, we use Notation (3.1) in [Mor17] for the
conjugation operator of a Gevrey function, writing

$\partial_{t}v=-{\bm{\tau}}D^{\sigma}v+\left(A^{(\tau)}\partial_{x}v+F(u)^{(\tau)}v\right)$
(3.3)

where the coefficients of matrices $A^{(\tau)}$ and $F^{(\tau)}$ are the
Gevrey conjugated coefficients of $A$ and $F(u)$.

We compute the time derivative of the energy $E(\tau(t),u(t))$ defined in
(2.10). Using notation $v$ defined in (3.2), the energy is

$E(t,u(t))=\frac{1}{2}\left|{\rm op}(S)v\right|^{2}_{L^{2}}$

where the symbol $S$ is defined in (2.7). Denoting here $\langle\cdot\rangle$
the $L^{2}({\mathbb{R}}^{d})$ scalar product, we compute

$\displaystyle\partial_{t}E$ $\displaystyle=$ $\displaystyle{\rm
Re}\,\langle{\rm op}(S)\partial_{t}v,{\rm op}(S)v\rangle+{\rm Re}\,\langle{\rm
op}(\partial_{t}S)v,{\rm op}(S)v\rangle$

Using (3.3) there holds

$\partial_{t}E={}-{\bm{\tau}}{\rm E_{1}}+{\rm E_{2}}+{\rm E_{3}}+{\rm E_{4}}$
(3.4)

where

${\rm E_{1}}={\rm Re}\,\langle{\rm op}(S)D^{\sigma}v,{\rm op}(S)v\rangle$
(3.5)

is the time-derivative of the Gevrey weight ;

${\rm E_{2}}={\rm Re}\,\langle{\rm op}(S)A^{(\tau)}\partial_{x}v,{\rm
op}(S)v\rangle$ (3.6)

are linear terms in the equations ;

${\rm E_{3}}={\rm Re}\,\langle{\rm op}(\partial_{t}S)v,{\rm op}(S)v\rangle$
(3.7)

is the time-derivative of the symmetrizer ;

${\rm E_{4}}={\rm Re}\,\langle{\rm op}(S)F(u)^{(\tau)}v,{\rm op}(S)v\rangle$
(3.8)

are the non-linear terms in the equation. The term ${\rm E_{1}}$ is of higher
order than the energy, thanks to the $D^{\sigma}$ term coming from the time
derivative of the Gevrey weight. The minus sign in front of ${\rm E_{1}}$ is
crucial in order to control the remainder terms ${\rm E_{2}}$, ${\rm E_{3}}$
and ${\rm E_{4}}$. We focus now on each of those terms.

####  3.1.1 The term ${\rm E_{1}}$

The term ${\rm E_{1}}$ controls more than the $L^{2}$-norm of ${\rm op}(S)v$.
Component-wise, we get

${\rm E_{1}}={\rm Re}\,\langle D^{\sigma}v_{1},v_{1}\rangle+{\rm
Re}\,\langle{\rm op}(b)D^{\sigma}v_{2},{\rm op}(b)v_{2}\rangle.$

The anisotropy of the symetriser $S$ reads in this equality. The first term is
simply equals to $\left|D^{\sigma/2}v_{1}\right|^{2}_{L^{2}}$: we have a
control of the $H^{\sigma/2}$-norm of $v_{1}$.

Concerning the component $v_{2}$, we compute

$\displaystyle{\rm Re}\,\langle{\rm op}(b)D^{\sigma}v_{2},{\rm
op}(b)v_{2}\rangle$ $\displaystyle=$ $\displaystyle{\rm Re}\,\langle
D^{\sigma}{\rm op}(b)v_{2},{\rm op}(b)v_{2}\rangle+{\rm Re}\,\langle\left[{\rm
op}(b),D^{\sigma}\right]v_{2},{\rm op}(b)v_{2}\rangle$ $\displaystyle=$
$\displaystyle\left|D^{\sigma/2}{\rm op}(b)v_{2}\right|^{2}_{L^{2}}+{\rm
Re}\,\langle\left[{\rm op}(b),D^{\sigma}\right]v_{2},{\rm op}(b)v_{2}\rangle.$

Focusing on the commutator term on the right-hand side, we write

$\displaystyle{\rm Re}\,\langle\left[{\rm op}(b),D^{\sigma}\right]v_{2},{\rm
op}(b)v_{2}\rangle$ $\displaystyle=$ $\displaystyle{\rm Re}\,\langle
D^{-\sigma/2}\left[{\rm op}(b),D^{\sigma}\right]{\rm
op}(b)^{-1}D^{-\sigma/2}\,D^{\sigma/2}{\rm op}(b)v_{2},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle$ $\displaystyle=$ $\displaystyle{\rm Re}\,\langle{\rm
op}\left(S\left(\lambda^{-2},g\right)\right)\,D^{\sigma/2}{\rm
op}(b)v_{2},D^{\sigma/2}{\rm op}(b)v_{2}\rangle$

thanks to Lemma 4.11. As $c\in(0,2]$, there holds $\lambda^{-2}\leq 1$ hence
operators ${\rm op}\left(S\left(\lambda^{-2},g\right)\right)$ are bounded in
$L^{2}$ using Lemma 4.13. We conclude thus by

${\rm
E_{1}}\sim\left|D^{\sigma/2}v_{1}\right|^{2}_{L^{2}}+\left|D^{\sigma/2}{\rm
op}(b)v_{2}\right|^{2}_{L^{2}}.$ (3.9)

####  3.1.2 The term ${\rm E_{2}}$

The crucial cancellations take place here. They rely on our choice of $b$
defined in (2.5). As $a$ is in ${\rm\mathcal{G}}^{\sigma}_{\underline{\tau}}$
with $\underline{\tau}$ defined in (2.3) and by the results of Section 5 in
[Mor17] (see also [CNR]), there is a symbol $\tilde{a}$ in $S^{0}_{1,0}$ such
that

$a^{(\tau)}={\rm op}(\tilde{a})$ (3.10)

for all $\tau=\tau(t)$, as $\tau(t)\leq\tau_{0}<\underline{\tau}$ by
definition (3.1). Denoting

$\tilde{A}=\begin{pmatrix}0&1\\\ \tilde{a}&0\end{pmatrix}$

we may then write

$\displaystyle A^{(\tau)}$ $\displaystyle=$ $\displaystyle{\rm
op}(\widetilde{A})$ $\displaystyle=$ $\displaystyle A+{\rm
op}\left(\widetilde{A}-A\right)$ $\displaystyle=$ $\displaystyle{\rm
op}\left(A_{\natural}\right)-\begin{pmatrix}0&0\\\ D^{-c}&0\end{pmatrix}+{\rm
op}\left(\widetilde{A}-A\right)$

where $D$ is defined in (2.9).

We make use of this decomposition to write

${\rm E_{2}}={\rm Re}\,\langle{\rm op}(S){\rm
op}\left(A_{\natural}\right)\partial_{x}v,{\rm op}(S)v\rangle+R_{2}$ (3.11)

where $R_{2}$ comprises remainder terms:

$R_{2}=-{\rm Re}\,\langle{\rm op}(S)\left(\begin{pmatrix}0&0\\\
D^{-c}&0\end{pmatrix}+{\rm
op}\left(\widetilde{A}-A\right)\right)\partial_{x}v,{\rm op}(S)v\rangle.$

As $S$ is defined as a microlocal symmetriser for $A_{\natural}$, we write

${\rm E_{2}}={\rm Re}\,\langle{\rm op}(S)^{2}{\rm
op}(A_{\natural})\partial_{x}v,v\rangle+R_{2},$

as ${\rm op}(S)^{*}={\rm op}(S)$ in Weyl quantization for diagonal matrices
with real symbols.

Next, applying equality (4.12) of Lemma 4.11 on composition of operators,
there holds

${\rm op}(b)^{2}={\rm op}\left(b^{2}\right)+{\rm
op}\left(S\left(b^{2}\lambda^{-2},g\right)\right)$

as $\\{b,b\\}=0$. Thus

${\rm E_{2}}={\rm Re}\,\langle{\rm
op}\left(S^{2}A_{\natural}\right)\partial_{x}v,v\rangle+\widetilde{R}_{2}$

where

$\widetilde{R}_{2}=R_{2}+{\rm Re}\,\langle\left({\rm
op}\left(S^{2}A_{\natural}\right)-{\rm op}(S)^{2}{\rm
op}(A_{\natural})\right)\partial_{x}v,v\rangle.$

By definition of $S$, the leading term of ${\rm E_{2}}$ cancels and there
holds finally

$\displaystyle{\rm E_{2}}$ $\displaystyle=$ $\displaystyle-{\rm
Re}\,\langle{\rm op}(S)\begin{pmatrix}0&0\\\
D^{-c}&0\end{pmatrix}\partial_{x}v,{\rm op}(S)v\rangle$ (3.14)
$\displaystyle\quad+{\rm Re}\,\langle\left({\rm
op}\left(S^{2}A_{\natural}\right)-{\rm op}(S)^{2}{\rm
op}(A_{\natural})\right)\partial_{x}v,v\rangle$ $\displaystyle\quad+{\rm
Re}\,\langle{\rm op}(S){\rm op}(\widetilde{A}-A)\partial_{x}v,{\rm
op}(S)v\rangle.$

####  3.1.3 The term ${\rm E_{3}}$

We first note that

$\partial_{t}S=\begin{pmatrix}0&0\\\ 0&\partial_{t}b\end{pmatrix}$

and that $\partial_{t}b=-\frac{1}{2}\partial_{t}a\,b^{3}$. Thanks to
Assumption 2.1, function $\partial_{t}a(t,x)$ is positive. We may then write

$\partial_{t}b=-\frac{1}{2}\left(\sqrt{\partial_{t}a}\,b\right)^{2}b$ (3.15)

and aim at getting a Gårding-type estimate. As $\sqrt{\partial_{t}a}$ depends
only on $(t,x)$ variables, it is in $S(1,g)$, hence $\sqrt{\partial_{t}a}\,b$
is in $S\left(b,g\right)$ by Lemma 4.10. First, applying equality (4.12) of
Lemma 4.11 and as $\\{\sqrt{\partial_{t}a}\,b,\sqrt{\partial_{t}a}\,b\\}=0$,
there holds

${\rm op}\left(\left(\sqrt{\partial_{t}a}\,b\right)^{2}\right)={\rm
op}\left(\sqrt{\partial_{t}a}\,b\right)^{2}+{\rm
op}\left(S\left(b^{2}\lambda^{-2},g\right)\right).$

Second, using again Lemma 4.11 there holds

${\rm op}\left(\partial_{t}b\right)=-\frac{1}{2}{\rm
op}\left(\left(\sqrt{\partial_{t}a}\,b\right)^{2}\right){\rm op}(b)+{\rm
op}\left(i\\{\partial_{t}a\,b^{2},b\\}\right)+{\rm
op}\left(S\left(b^{3}\lambda^{-2},g\right)\right).$

The subprincipal symbol $i\\{\partial_{t}a\,b^{2},b\\}$ is a priori in
$S(b^{3}\lambda^{-1},g)$. A careful computation gives however

$\displaystyle i\\{\partial_{t}a\,b^{2},b\\}$ $\displaystyle=$ $\displaystyle
i(\partial_{tx}a)b^{2}\,\partial_{\xi}b$ $\displaystyle=$
$\displaystyle-\frac{i}{2}(\partial_{tx}a)b^{5}\partial_{\xi}\langle\cdot\rangle^{-c}$

which is in $S(b^{5}\langle\cdot\rangle^{-c-1},g)$. We conclude then that

${\rm op}\left(\partial_{t}b\right)=-\frac{1}{2}{\rm
op}\left(\left(\sqrt{\partial_{t}a}\,b\right)\right)^{2}{\rm op}(b)+{\rm
op}\left(S(b^{5}\langle\cdot\rangle^{-c-1},g)\right)+{\rm
op}\left(S\left(b^{3}\lambda^{-2},g\right)\right).$ (3.16)

Note that both terms are not comparable, as $c$ may go from $0$ to $2$.

This implies

$\displaystyle{\rm E_{3}}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}{\rm
Re}\,\langle\left({\rm op}\left(\sqrt{\partial_{t}a}\,b\right)\right)^{2}{\rm
op}(b)v_{2},{\rm op}(b)v_{2}\rangle$ $\displaystyle\quad+{\rm Re}\,\langle{\rm
op}\left(S(b^{5}\langle\cdot\rangle^{-c-1},g)\right)v_{2},{\rm
op}(b)v_{2}\rangle+{\rm Re}\,\langle{\rm
op}\left(S\left(b^{3}\lambda^{-2},g\right)\right)v_{2},{\rm
op}(b)v_{2}\rangle.$

The first term in the above right-hand side satisfies

$\displaystyle{\rm Re}\,\langle\left({\rm
op}\left(\sqrt{\partial_{t}a}\,b\right)\right)^{2}{\rm op}(b)v_{2},{\rm
op}(b)v_{2}\rangle$ $\displaystyle=$ $\displaystyle\left|{\rm
op}\left(\sqrt{\partial_{t}a}\,b\right){\rm op}(b)v_{2}\right|^{2}$
$\displaystyle\geq$ $\displaystyle 0.$

Thus

${\rm E_{3}}\leq{\rm Re}\,\langle{\rm
op}\left(S(b^{5}\langle\cdot\rangle^{-c-1},g)\right)v_{2},{\rm
op}(b)v_{2}\rangle+{\rm Re}\,\langle{\rm
op}\left(S\left(b^{3}\lambda^{-2},g\right)\right)v_{2},{\rm
op}(b)v_{2}\rangle.$ (3.17)

Note that the term ${\rm E_{3}}$ does not depend on $v_{1}$ as the symbol $S$
is anisotropic.

##  4 Pseudo-differential tools

This Section aims to remind a few tools of pseudo-differential calculus and of
symbols associated to a general metric of the phase space. We start first by
study the symbol $b$, which leads naturally to a specific metric $g$ that
encodes the specific dynamics of our system. We then define classes of symbols
$S(M,g)$, and the properties of pseudo-differential operators associated to
such symbols.

###  4.1 Study of symbol $b$

We define the symbol

$a_{\natural}=a_{\natural}(t,x,\xi)=a(t,x)+\langle\xi\rangle^{-c}$ (4.1)

where the additional term $\langle\xi\rangle^{-c}$ makes the symbol
$a_{\natural}$ positive. This is a standard approach when dealing with weakly
hyperbolic equations, see [CJS83]. Thanks to this notation, we may write the
symbol $b$ defined by (2.5) as $b=a_{\natural}^{-1/2}$.

######  Lemma 4.1 (Bounds for $b$).

The symbol $b$ satisfies the upper bound

$b(t,x,\xi)\leq\langle\xi\rangle^{c/2}\quad,\quad\forall\,(t,x,\xi)\in[0,T]\times
B_{r}(x_{0})\times{\mathbb{R}}$ (4.2)

and is bounded from below

$b(t,x,\xi)\gtrsim 1\quad,\quad\forall\,(t,x,\xi)\in[0,T]\times
B_{r}(x_{0})\times{\mathbb{R}}$ (4.3)

###### Proof.

The proof of the upper bound (4.2) is immediate as $a$ is non negative. For
the lower bound, there holds

$b(t,x,\xi)\geq\left(\sup a+1\right)^{-1/2}$

where the $\sup$ of $a$ is over $[0,T]\times B_{r}(x_{0})$.

∎

In order to compute carefully some estimates on the derivatives of $b$, we
prove first a local Glaeser inequality for $a$, as it is non-negative locally
around $x=x_{0}$.

######  Lemma 4.2 (Glaeser inequality for $a$).

Under Assumption 2.1, there is a neighborhood $[0,T]\times B_{r}(x_{0})$ of
$(0,x_{0})\in{\mathbb{R}}_{t}\times{\mathbb{R}}_{x}$ and a constant
$C_{T,r}>0$ for which there holds

$\left(\partial_{x}a(t,x)\right)^{2}\leq
C_{T,r}\,a(t,x)\quad,\quad\forall\,(t,x)\in[0,T]\times B_{r}(x_{0}).$ (4.4)

The proof is postponed to Appendix 5.1. The following Lemma gives precise
estimates on the derivatives of $b$.

######  Lemma 4.3 (Derivatives of the symbol $b$).

There is a bounded sequence of constants $C_{\alpha,\beta}>0$ for which there
holds

$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}b(t,x,\xi)|\leq
C_{\alpha,\beta}\widetilde{R}^{|\alpha|}2^{|\beta|}\,\alpha!^{1/\sigma}\beta!\,b(t,x,\xi)\,b(t,x,\xi)^{|\alpha|}\,\langle\xi\rangle^{-|\beta|}\quad,\quad\forall\,(\alpha,\beta)\in{\mathbb{N}}\times{\mathbb{N}}$
(4.5)

for all $(t,x)$ in $[0,T]\times B_{r}(x_{0})$ and $\xi$ in ${\mathbb{R}}$, and
where $\widetilde{R}=R(1+|a|_{1/\sigma,R}|)>R$.

The proof is postponed in Appendix 5.1. It relies on the Faà di Bruno formula
(see Lemma 5.1) and the Glaeser inequality for $a$ proved in Lemma 4.2. We
follow through with some remarks on this result.

######  Remark 4.4.

Thanks to inequality (4.2), Lemma 4.3 implies that $b\in
S^{c/2}_{1,c/2}G^{1/\sigma}_{\widetilde{R}}$, as defined in [Mor17]. Without
the Glaeser inequality described in Lemma 4.2, we would only prove that $b\in
S^{c/2}_{1,c}G^{1/\sigma}_{\widetilde{R}}$, whereas $c$ may be in $[1,2]$.

The importance of the Glaeser inequality explains why we do not define $b$ as
$\left(\widetilde{a}+\langle\xi\rangle^{-c}\right)^{-1/2}$ where
$\widetilde{a}$ is defined in (3.10) as the symbol of operator $a^{(\tau)}$,
the Gevrey conjugation of $a$. Indeed the symbol $\widetilde{a}$ does not
satisfy a priori the Glaeser inequality, as it is not real.

######  Remark 4.5.

As $a$ has compact support, $b(\cdot,\xi)$ is constant outside a compact set
of ${\mathbb{R}}_{t}\times{\mathbb{R}}_{x}$ which does not depend on $\xi$.

The bounds (4.5) show in particular that the symbol $b$ has a variable order
and a varying "class" with respect to time and space. Indeed, for
$(t,x)=(0,x_{0})$, symbol $a_{\natural}$ is equal to $\langle\xi\rangle^{-c}$,
hence $b(t=0)$ is likely to be in the class of classical symbols
$S^{c/2}_{1,c/2}$. But as time goes, the order of $b$ decreases. In fact, for
$t\geq\underline{t}>0$, there holds simply $a_{\natural}\geq
t\geq\underline{t}$, hence

$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}b(t,x,\xi)|\leq
C_{\alpha,\beta}\widetilde{R}^{|\alpha|}\,\alpha!^{1/\sigma}\beta!\,\underline{t}^{-1/2}\,\underline{t}^{-|\alpha|/2}\,\langle\xi\rangle^{-|\beta|}$

for all $t\geq\underline{t}$. Then $b$ is in the classical space of symbols
$S^{0}_{1,0}$ for all $t\geq\underline{t}$.

A way to reconcile both points of view is to introduce the following time-
dependent, non-flat metric in the phase space

$g_{(x,\xi)}({\rm d}x,{\rm d}\xi)=\frac{|{\rm
d}x|^{2}}{a_{\natural}(t,x,\xi)}+\frac{|{\rm
d}\xi|^{2}}{\langle\xi\rangle^{2}}.$ (4.6)

Lemma 4.3 reads now as

$b\in S(b,g)$ (4.7)

where $S(b,g)$ is defined in Definition 5.11. Both the weight and the metric
are time-dependent, hence encoding precisely the dynamic of the system.

###  4.2 Properties of class of symbols

Properties of pseudo-differential calculus come directly from properties of
the metric and the weights associated to the metric. We give here the
fundamental statements about the metric and the weights, which are necessary
for a pseudo-differential calculus to be coherent (see Lemma 4.11). For the
sake of simplicity and completeness, we choose to postpone definitions and
proofs in the Appendix 5.2.

######  Lemma 4.6.

The metric $g$ defined in (4.6) is an admissible metric.

See Lemma 5.9 in the Appendix and its proof for further details.

######  Lemma 4.7.

For all $k\in{\mathbb{Z}}$, symbols $b^{k}$ are admissible with respect to the
metric $g$. For all $m\in{\mathbb{R}}$, symbols $\langle\xi\rangle^{m}$ are
admissible with respect to the metric $g$.

In particular, Lemma 4.3 implies

######  Lemma 4.8.

For any $k$ in ${\mathbb{Z}}$, the symbol $b^{k}$ is in $S(b^{k},g)$.

###### Proof.

The case $k=1$ is just Lemma 4.3. Hence the result for any $k\geq 1$, thanks
to Lemma 4.10. The case $k=-1$ is proved by the same proof as Lemma 4.3, as
$b^{-1}=a_{\natural}^{1/2}$. ∎

Those preliminary lemmas on basic properties of the metric and the weights
will be used in the next Section. We continue by linking spaces of symbols
$S(M,g)$ for weights $M$ admissible for $g$ and classical spaces of symbols
$S^{m}_{\rho,\delta}$.

######  Lemma 4.9 (Embeddings).

For all $m\in{\mathbb{R}}$, the following embedding holds

$S^{m}_{1,0}\,\subset\,S\left(\langle\cdot\rangle^{m},g\right).$ (4.8)

Let $M$ be an admissible weight satisfying $M(x,\xi)\leq\langle\xi\rangle^{m}$
for all $(x,\xi)\in{\mathbb{R}}\times{\mathbb{R}}$ for some
$m\in{\mathbb{R}}$. Then the following embedding holds

$S(M,g)\,\subset\,S^{m}_{1,c/2}.$ (4.9)

The proof is postponed in the Appendix.

###  4.3 pseudo-differential calculus

We use here the Weyl quantization, which we recall

${\rm op}(a)u(x)={\rm op}_{1/2}(a)u(x)=\int e^{2\pi
i(x-y)\cdot\xi}a\left(\frac{x+y}{2},\xi\right)u(y)dyd\xi.$

We recall the algebra property of general classes of symbols $S(M,g)$. Let
$M_{1}$ and $M_{2}$ be both admissible weights for the metric $g$.

######  Lemma 4.10.

For any $p_{j}\in S(M_{j},g)$ with $j=1,2$, there holds

$p_{1}p_{2}\in S(M_{1}M_{2},g).$

The proof is straightforward, using Leibniz formula and Definition 5.11.

We now state Theorem 2.3.7 in [Ler11], concerning the composition of operators
with symbols in $S(M,g)$. For two symbols $p_{1}$ and $p_{2}$, we denote
$p_{1}\sharp p_{2}$ the symbol satisfying

${\rm op}(p_{1}){\rm op}(p_{2})={\rm op}(p_{1}\sharp p_{2}).$

We denote also

$\lambda(t,x,\xi)=b^{-1}(t,x,\xi)\langle\xi\rangle.$ (4.10)

######  Lemma 4.11 (Composition).

Let $M_{1}$ and $M_{2}$ two admissible weights for $g$, and $p_{j}\in
S(M_{j},g)$. Then for all $\nu$ in ${\mathbb{N}}$ there holds

$p_{1}\sharp p_{2}-\left(\sum_{0\leq
k<\nu}2^{-k}\sum_{|\alpha|+|\beta|=k}\frac{(-i)^{|\beta|}}{\alpha!\beta!}\partial_{\xi}^{\beta}\partial_{x}^{\alpha}p_{1}\partial_{\xi}^{\alpha}\partial_{x}^{\beta}p_{2}\right)\,\in
S(M_{1}M_{2}\lambda^{-\nu},g)$ (4.11)

where $\lambda$ is defined by (4.10). In particular, there holds

$p_{1}\sharp p_{2}-\frac{1}{2i}\\{p_{1},p_{2}\\}\,\in
S(M_{1}M_{2}\lambda^{-2},g)$ (4.12)

where $\\{\cdot,\cdot\\}$ denotes the usual Poisson bracket on
${\mathbb{R}}_{x}\times{\mathbb{R}}_{\xi}$. About commutators:

$[{\rm op}(p_{1}),{\rm op}(p_{2})]\in{\rm
op}\left(S\left(M_{1}M_{2}\lambda^{-2},g\right)\right).$ (4.13)

In the previous Lemma, we see that the powers of the symbol $\lambda$ act as a
gradation for the remainder term in composition of operators. In the case of
usual flat metrics on the phase space, the symbol $\lambda$ is simply equal to
$\langle\xi\rangle$, and the previous Lemma reads (for instance) as
$p_{1}\sharp p_{2}-p_{1}p_{2}\in S^{m_{1}+m_{2}-1}_{1,0}$ for $p_{1}$ and
$p_{2}$ in $S^{m_{j}}_{1,0}$.

We pursue by giving a result on the inversion of ${\rm op}(b)$ up to any low
order remainder term.

######  Lemma 4.12 (Inversion of ${\rm op}(b)$).

For any $\nu\in{\mathbb{N}}$, there is a symbol $c_{\nu}$ in $S(b^{-1},g)$
such that

${\rm op}(b){\rm op}(c_{\nu})-{\rm Id}\in{\rm
op}\left(S(\lambda^{-(\nu+1)},g)\right).$ (4.14)

###### Proof.

Our aim is to solve the equation

$b\sharp c_{\nu}=1+S(\lambda^{-(\nu+1)},g).$

We proceed by induction on $\nu$, using equality (4.11). Denoting

$\omega_{k}(p_{1},p_{2})=2^{-k}\sum_{|\alpha|+|\beta|=k}\frac{(-i)^{|\beta|}}{\alpha!\beta!}\partial_{\xi}^{\beta}\partial_{x}^{\alpha}p_{1}\partial_{\xi}^{\alpha}\partial_{x}^{\beta}p_{2}$

equality (4.11) states

$p_{1}\sharp p_{2}=\sum_{0\leq
k<\nu}\omega_{k}(p_{1},p_{2})+S(M_{1}M_{2}\lambda^{-\nu},g).$

Note that in particular

$\omega_{0}(p_{1},p_{2})=p_{1}p_{2}\quad,\quad\omega_{1}(p_{1},p_{2})=-\frac{i}{2}\\{p_{1},p_{2}\\}.$

For $\nu=0$, there holds

$b\sharp c_{0}=bc_{0}+S(\lambda^{-1},g)$

so that

$c_{0}=b^{-1}.$

Let $\nu\geq 0$, and assume $c_{\nu-1}\in S(\lambda^{-(\nu-1)},g)$ solves

$b\sharp c_{\nu-1}=1+S(\lambda^{-\nu},g).$

Then on one side

$\displaystyle b\sharp\left(c_{\nu}-c_{\nu-1}\right)$ $\displaystyle=$
$\displaystyle\sum_{0\leq
k<\nu+1}\omega_{k}(b,c_{\nu}-c_{\nu-1})+S(\lambda^{-(\nu+1)},g)$
$\displaystyle=$ $\displaystyle b(c_{\nu}-c_{\nu-1})+S(\lambda^{-(\nu+1)},g)$

as $c_{\nu}-c_{\nu-1}\in S(\lambda^{-\nu},g)$, and on the other side thanks to
the equation there holds

$b\sharp\left(c_{\nu}-c_{\nu-1}\right)=\left(1-b\sharp
c_{\nu-1}\right)+S(\lambda^{-(\nu+1)},g)$

which leads to

$c_{\nu}-c_{\nu-1}=b^{-1}\left(1-b\sharp c_{\nu-1}\right)$

which is in $S(b^{-1},g)$. ∎

Finally, we recall Theorem 2.5.1 of [Ler11].

######  Lemma 4.13 (Action).

Let $p$ be in $S(1,g)$. Then ${\rm op}(p)$ acts continuously on $L^{2}$.

###  4.4 Energy estimate

In Section 3.1, we observed cancellations in $\partial_{t}E$. The next step is
to bound the remainder terms in ${\rm E_{2}}$, ${\rm E_{3}}$ and ${\rm E_{4}}$
by a fraction of the negative term ${\rm E_{1}}$ . This is done thanks to the
properties of the pseudo-differential calculus described in Appendix 5.2 and
Lemma 4.9 ; by choice of the exponent $c>0$ of the correction term
$\langle\xi\rangle^{-c}$ which appears in the definition (4.1) of
$a_{\natural}$ ; and by a lower bound on the Gevrey index $\sigma$.

####  4.4.1 Estimate of ${\rm E_{2}}$

The term ${\rm E_{2}}$, defined in (3.6) is equal, thanks to the previous
computations, to the sum of (3.14), (3.14) and (3.14).

$\bullet$ First we focus on (3.14), using the results of Section 3.1.1. We
write

$\displaystyle-{\rm Re}\,\langle{\rm op}(S)\begin{pmatrix}0&0\\\
D^{-c}&0\end{pmatrix}\partial_{x}v,{\rm op}(S)v\rangle$ $\displaystyle=$
$\displaystyle-{\rm Re}\,\langle{\rm op}(b)D^{-c}\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle$ $\displaystyle=$ $\displaystyle-{\rm Re}\,\langle
D^{-\sigma/2}{\rm
op}(b)D^{-c-\sigma/2}\partial_{x}D^{\sigma/2}v_{1},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle.$

The operator $D^{-\sigma/2}{\rm op}(b)D^{-c-\sigma/2}\partial_{x}$ has a
symbol in the class $S\left(b\langle\cdot\rangle^{1-c-\sigma},g\right)$, which
is embedded in $S(1,g)$ as soon as

$1-c/2-\sigma\leq 0.$ (4.15)

If the constraint is satisfied, the operator acts continuously on $L^{2}$,
hence

$\left|\eqref{4.FII.1}\right|\lesssim{\rm E_{1}}.$ (4.16)

$\bullet$ Second, we focus on (3.14). Here, equality (4.12) of Lemma 4.11 and
cancellations of brackets $\\{b,b\\}=0$ and $\\{b,a_{\natural}\\}=0$ imply
that

${\rm op}\left(S^{2}A_{\natural}\right)-{\rm op}(S)^{2}{\rm
op}(A_{\natural})=\begin{pmatrix}0&0\\\ {\rm
op}\left(S(\lambda^{-2},g)\right)&0\end{pmatrix}.$

Hence

$\displaystyle{\rm Re}\,\langle\left({\rm
op}\left(S^{2}A_{\natural}\right)-{\rm op}(S)^{2}{\rm
op}(A_{\natural})\right)\partial_{x}v,v\rangle$ $\displaystyle=$
$\displaystyle{\rm Re}\,\langle\begin{pmatrix}0&0\\\ {\rm
op}\left(S(\lambda^{-2},g)\right)&0\end{pmatrix}\partial_{x}v,v\rangle$
$\displaystyle=$ $\displaystyle{\rm Re}\,\langle{\rm
op}\left(S(\lambda^{-2},g)\right)\partial_{x}v_{1},v_{2}\rangle.$

Next, we proceed as we did in the previous point for (3.14). As we control
$D^{\sigma/2}{\rm op}(b)v_{2}$ in $L^{2}$ norm, we use Lemma 4.12 to make
appear ${\rm op}(b)$ up to a remainder in $S(\lambda^{-(\nu-1)},g)$ for $\nu$
to be chosen later. Hence there holds

$\displaystyle{\rm Re}\,\langle{\rm
op}\left(S(\lambda^{-2},g)\right)\partial_{x}v_{1},v_{2}\rangle$
$\displaystyle={\rm Re}\,\langle{\rm op}(c_{\nu}){\rm
op}\left(S(\lambda^{-2},g)\right)\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle+{\rm Re}\,\langle{\rm
op}\left(S(\lambda^{-2-\nu-1},g)\right)\partial_{x}v_{1},v_{2}\rangle$
$\displaystyle={\rm Re}\,\langle D^{-\sigma/2}{\rm op}(c_{\nu}){\rm
op}\left(S(\lambda^{-2},g)\right)D^{-\sigma/2}\partial_{x}D^{\sigma/2}v_{1},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle+R_{2,2}$

with

$R_{2,2}={\rm Re}\,\langle{\rm
op}\left(S(\lambda^{-3-\nu},g)\right)\partial_{x}v_{1},v_{2}\rangle.$

As $c_{\nu}$ is in $S(b^{-1},g)$, we get

$D^{-\sigma/2}{\rm op}(c_{\nu}){\rm
op}\left(S(\lambda^{-2},g)\right)D^{-\sigma/2}\partial_{x}={\rm
op}\left(S\left(\langle\cdot\rangle^{-1-\sigma}b,g\right)\right).$

Thanks to inequality (4.2), the symbol satisfies
$\langle\cdot\rangle^{-1-\sigma}b\leq 1$ which implies the boundedness in
$L^{2}$ of the previous operator, hence

$\left|{\rm Re}\,\langle D^{-\sigma/2}{\rm op}(c_{\nu}){\rm
op}\left(S(\lambda^{-2},g)\right)D^{-\sigma/2}\partial_{x}D^{\sigma/2}v_{1},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle\right|\lesssim{\rm E_{1}}.$

We consider now the remainder term $R_{2,2}$, writing

$R_{2,2}={\rm Re}\,\langle{\rm
op}\left(S\left(\langle\cdot\rangle^{1-\sigma}\lambda^{-3-\nu},g\right)\right)D^{\sigma/2}v_{1},D^{\sigma/2}v_{2}\rangle$
(4.17)

and both definition (4.10) of $\lambda$ and inequality (4.2) imply that

${\rm
op}\left(S\left(\langle\cdot\rangle^{1-\sigma}\lambda^{-3-\nu},g\right)\right)\subset{\rm
op}\left(S\left(\langle\cdot\rangle^{-2-\nu-\sigma+c(3+\nu)/2},g\right)\right).$

As soon as the constraint

$\frac{c}{2}\leq 1-\frac{1-\sigma}{3+\nu}$ (4.18)

is satisfied, operators ${\rm
op}\left(S\left(\langle\cdot\rangle^{1-\sigma}\lambda^{-3-\nu},g\right)\right)$
act thus continuously on $L^{2}$ thanks to Lemma 4.13. Then there holds

$\left|R_{2,2}\right|\lesssim{\rm E_{1}}$

and finally

$\left|\eqref{4.FII.2}\right|\lesssim{\rm E_{1}}.$ (4.19)

######  Remark 4.14.

Constraint (4.18) is essentially technical. Note that, as $\nu$ goes to
infinity, constraint (4.18) becomes $c/2<1$. This is the uncertainty principle
for the metric $g$.

$\bullet$ Third, we focus on (3.14). Thanks to Lemma 5.2 in [Mor17], the
symbol $\widetilde{a}$ satisfies

$\tilde{a}-a=i\partial_{x}a\,\partial_{\xi}\langle\xi\rangle^{\sigma}+S^{-2(1-\sigma)}_{1,0}$
(4.20)

and we write thus

$\displaystyle{\rm Re}\,\langle{\rm op}(S){\rm
op}(\widetilde{A}-A)\partial_{x}v,{\rm op}(S)v\rangle$ $\displaystyle=$
$\displaystyle{\rm Re}\,\langle{\rm op}(S)\begin{pmatrix}0&0\\\ {\rm
op}(\widetilde{a})-a&0\end{pmatrix}\partial_{x}v,{\rm op}(S)v\rangle$ (4.21)
$\displaystyle=$ $\displaystyle{\rm Re}\,\langle{\rm op}(b){\rm
op}(\widetilde{a}-a)\partial_{x}v_{1},{\rm op}(b)v_{2}\rangle$
$\displaystyle=$ $\displaystyle{\rm Re}\,\langle{\rm op}(b){\rm
op}\left(i\partial_{x}a\,\partial_{\xi}\langle\cdot\rangle^{\sigma}\right)\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle+R_{2,3}$

where

$R_{2,3}={\rm Re}\,\langle{\rm op}(b){\rm
op}\left(S^{-2(1-\sigma)}_{1,0}\right)\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle.$

The sub-principal symbol
$i\partial_{x}a\,\partial_{\xi}\langle\xi\rangle^{\sigma}$ is a priori in
$S^{-(1-\sigma)}_{1,0}$, which would be insufficient to counterbalance both
${\rm op}(b)$ and $\partial_{x}$. Indeed, by Lemma 4.8 and Lemma 4.9, there
holds ${\rm op}(b)\partial_{x}\in{\rm op}\left(S^{1+c/2}_{1,c/2}\right)$
versus the straigthforward estimate ${\rm
op}\left(i\partial_{x}a\,\partial_{\xi}\langle\xi\rangle^{\sigma}\right)\in{\rm
op}\left(S^{-(1-\sigma)}_{1,0}\right)$. But using the Glaeser inequality for
$a$ described in Lemma 4.2 and definition (4.6) of the metric $g$, we prove
that in fact

$i\partial_{x}a\,\partial_{\xi}\langle\xi\rangle^{\sigma}\in
S(b^{-1}\langle\cdot\rangle^{\sigma-1},g).$ (4.22)

Indeed for any $\alpha$, $\beta$ in ${\mathbb{N}}$, there holds

$\displaystyle\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left(i\partial_{x}a(t,x)\,\partial_{\xi}\langle\xi\rangle^{\sigma}\right)\right|$
$\displaystyle=$
$\displaystyle\left|\partial_{x}^{\alpha+1}a(t,x)\,\partial_{\xi}^{\beta+1}\langle\xi\rangle^{\sigma}\right|$
$\displaystyle\lesssim$
$\displaystyle\left|\partial_{x}^{\alpha+1}a(t,x)\right|\,\langle\xi\rangle^{\sigma-1-|\beta|}.$

The lower bound (4.3) for $b$ implies

$\left|\partial_{x}^{\alpha+1}a(t,x)\right|\leq\left|\partial_{x}^{\alpha+1}a\right|_{L^{\infty}([0,T]\times
B_{r}(x_{0})}b^{-1+|\alpha|}\quad,\quad\forall\,(t,x)\in[0,T]\times
B_{r}(x_{0}).$

For $|\alpha|=0$, Glaeser’s Lemma 4.2 and definition (2.5) of $b$ lead to

$\displaystyle\left|\partial_{x}^{\alpha+1}a(t,x)\right|$
$\displaystyle\lesssim$
$\displaystyle\left|\partial_{x}^{2}a\right|_{L^{\infty}([0,T]\times
B_{r}(x_{0})}^{1/2}a(t,x)^{1/2}$ $\displaystyle\lesssim$
$\displaystyle\left|\partial_{x}^{2}a\right|_{L^{\infty}([0,T]\times
B_{r}(x_{0})}^{1/2}\left(a(t,x)+\langle\xi\rangle^{-c}\right)^{1/2}$
$\displaystyle\lesssim$
$\displaystyle\left|\partial_{x}^{2}a\right|_{L^{\infty}([0,T]\times
B_{r}(x_{0})}^{1/2}b(t,x,\xi)^{-1+|\alpha|}$

for any $(t,x,\xi)\in[0,T]\times B_{r}(x_{0})\times{\mathbb{R}}$. Thus, for
any $\alpha\in{\mathbb{N}}$, there is $C_{\alpha}>0$ such that

$\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left(i\partial_{x}a(t,x)\,\partial_{\xi}\langle\xi\rangle^{\sigma}\right)\right|\lesssim
C_{\alpha}\,b(t,x,\xi)^{-1+|\alpha|}\,\langle\xi\rangle^{\sigma-1-|\beta|}\quad,\quad\forall\,(t,x,\xi)\in[0,T]\times
B_{r}(x_{0})\times{\mathbb{R}}$

combining both cases, hence the proof of (4.22).

For the first term in the right-hand side of (4.21), we follow the same path
as in the above treatment of (3.14) and (3.14), writing

$\displaystyle{\rm Re}\,\langle{\rm op}(b){\rm
op}\left(i\partial_{x}a\,\partial_{\xi}\langle\cdot\rangle^{\sigma}\right)\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle$ $\displaystyle={\rm Re}\,\langle D^{-\sigma/2}{\rm
op}(b){\rm
op}\left(i\partial_{x}a\,\partial_{\xi}\langle\cdot\rangle^{\sigma}\right)D^{-\sigma/2}\partial_{x}\,D^{\sigma/2}v_{1},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle$ $\displaystyle={\rm Re}\,\langle{\rm
op}\left(S\left(1,g\right)\right)\,D^{\sigma/2}v_{1},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle.$

Hence, by Lemma 4.13,

$\left|{\rm Re}\,\langle{\rm op}(b){\rm
op}\left(i\partial_{x}a\,\partial_{\xi}\langle\cdot\rangle^{\sigma}\right)\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle\right|\lesssim{\rm E_{1}}.$

For the remainder term $R_{2,3}$, there holds

${\rm Re}\,\langle{\rm op}(b){\rm
op}\left(S_{1,0}^{-2(1-\sigma)}\right)\partial_{x}v_{1},{\rm
op}(b)v_{2}\rangle={\rm Re}\,\langle{\rm
op}\left(S\left(b\langle\cdot\rangle^{-(1-\sigma)},g\right)\right)D^{\sigma/2}v_{1},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle.$

Thanks to inequality (4.2) on $b$, we prove
$b\langle\cdot\rangle^{-(1-\sigma)}\leq\langle\cdot\rangle^{c/2+\sigma-1}$
which implies

${\rm
op}\left(S\left(b\langle\cdot\rangle^{-(1-\sigma)},g\right)\right)\subset{\rm
op}\left(S\left(\langle\cdot\rangle^{c/2+\sigma-1},g\right)\right).$

Hence, as soon as

$c/2+\sigma-1\leq 0$ (4.23)

holds, operators ${\rm
op}\left(S\left(b\langle\cdot\rangle^{-(1-\sigma)},g\right)\right)$ act on
$L^{2}$ thanks to Lemma 4.13, thus

$\left|\eqref{4.FII.3}\right|\lesssim{\rm E_{1}}$ (4.24)

using again Lemma 4.13.

Putting together estimates (4.16), (4.19) and (4.24), there is a constant
$C_{{\rm 2}}>0$ such that

$\left|{\rm E_{2}}\right|\leq C_{{\rm 2}}\,{\rm E_{1}}.$ (4.25)

######  Remark 4.15.

The discussion before estimate (4.22) on the subprincipal symbol of
$\tilde{a}$ stresses out the importance of a careful study of subprincipal
symbols when dealing with weakly hyperbolic systems. It echoes the
considerations and computations which lead to equality (3.16) for
$\partial_{t}b$.

####  4.4.2 Estimate of ${\rm E_{3}}$

We proceed as before, focusing first on the first term of the right-hand side
of inequality (3.17). Using Lemma 4.12, we write

$\displaystyle{\rm Re}\,\langle{\rm
op}\left(S(b^{5}\langle\cdot\rangle^{-c-1},g)\right)v_{2},{\rm
op}(b)v_{2}\rangle$ $\displaystyle={\rm Re}\,\langle D^{-\sigma/2}{\rm
op}\left(S(b^{5}\langle\cdot\rangle^{-c-1},g)\right){\rm
op}\left(c_{\nu}\right)D^{-\sigma/2}\,D^{\sigma/2}{\rm
op}(b)v_{2},D^{\sigma/2}{\rm op}(b)v_{2}\rangle$ $\displaystyle\quad+{\rm
Re}\,\langle D^{-\sigma/2}{\rm
op}\left(S(b^{5}\langle\cdot\rangle^{-c-1},g)\right){\rm
op}\left(S\left(\lambda^{-\nu-1},g\right)\right)v_{2},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle$ $\displaystyle={\rm Re}\,\langle{\rm
op}\left(S(b^{4}\langle\cdot\rangle^{-\sigma-c-1},g)\right)\,D^{\sigma/2}{\rm
op}(b)v_{2},D^{\sigma/2}{\rm op}(b)v_{2}\rangle$ $\displaystyle\quad+{\rm
Re}\,\langle{\rm
op}\left(S\left(b^{5}\langle\cdot\rangle^{-\sigma-c-1}\lambda^{-\nu-1},g\right)\right)D^{\sigma/2}v_{2},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle.$

Following the scheme developed in the previous points, we use inequality (4.3)
to get both bounds

$b^{4}\langle\cdot\rangle^{-\sigma-c-1}\leq\langle\cdot\rangle^{c-\sigma-1}$

and

$b^{5}\langle\cdot\rangle^{-\sigma-c-1}\lambda^{-\nu-1}=b^{6+\nu}\langle\cdot\rangle^{-\sigma-c-2-\nu}\leq\langle\cdot\rangle^{c(4+\nu)/2-\sigma-\nu-2}.$

To use Lemma 4.13 for $L^{2}$-boundedness of the associated operators, we need
constraints

$c-\sigma-1\leq 0$ (4.26)

and

$c(4+\nu)/2-\sigma-\nu-2\leq 0.$ (4.27)

As soon as both constraints are satisfied, by inequality (3.17), there is
$C_{{\rm 3}}>0$ such that

$\left|{\rm E_{3}}\right|\leq C_{{\rm 3}}\,{\rm E_{1}}.$ (4.28)

####  4.4.3 Estimate of ${\rm E_{4}}$

We consider now the non-linear, 0th order term ${\rm E_{4}}$. Without any
additional assumption on the (matrix) structure of the non-linearity, the
control of the source term may lead to another constraint linking $c$ and
$\sigma$. In particular, however we control $D^{\sigma/2}{\rm op}(b)v_{2}$ in
an $L^{2}$ norm, the term ${\rm op}(b)\left(F(u)^{(\tau)}v\right)_{2}$ in
${\rm E_{4}}$ may not be controlled in the same way. We thus have to bound the
operator ${\rm op}(b)$ using $D^{-\sigma}$.

We write first, as before,

$\displaystyle{\rm Re}\,\langle{\rm op}(b)\left(F(u)^{(\tau)}v\right)_{2},{\rm
op}(b)v_{2}\rangle+{\rm
Re}\,\langle\left(F(u)^{(\tau)}v\right)_{1},v_{1}\rangle$ $\displaystyle={\rm
Re}\,\langle D^{-\sigma/2}{\rm
op}(b)D^{-\sigma/2}\,D^{\sigma/2}\left(F(u)^{(\tau)}v\right)_{2},D^{\sigma/2}{\rm
op}(b)v_{2}\rangle+{\rm
Re}\,\langle\left(F(u)^{(\tau)}v\right)_{1},v_{1}\rangle$ (4.29)

Next, by Lemma 4.9 and Lemma 4.11, there holds

$D^{-\sigma/2}{\rm op}(b)D^{-\sigma/2}\in{\rm
op}\left(S\left(b\langle\cdot\rangle^{-\sigma},g\right)\right).$

Thanks to the upper bound (4.2) for $b$, we get

${\rm
op}\left(S\left(b\langle\cdot\rangle^{-\sigma},g\right)\right)\subset{\rm
op}\left(S\left(\langle\cdot\rangle^{c/2-\sigma},g\right)\right).$

As soon as the constraint

$c/2-\sigma\leq 0$ (4.30)

is satisfied, operators ${\rm
op}\left(S\left(\langle\cdot\rangle^{c/2-\sigma},g\right)\right)$ act
continuously on $L^{2}$ thanks to Lemma 4.13. This implies

$\left|D^{-\sigma/2}{\rm
op}(b)D^{-\sigma/2}\,D^{\sigma/2}\left(F(u)^{(\tau)}v\right)_{2}\right|_{L^{2}}\lesssim\left|D^{\sigma/2}\left(F(u)^{(\tau)}v\right)_{2}\right|_{L^{2}}.$

######  Remark 4.16.

Constraint (4.30) comes from control of the term ${\rm
op}(b)\left(F(u)^{(\tau)}v\right)_{2}$, which decomposes into

${\rm op}(b)\left(F(u)^{(\tau)}v\right)_{2}={\rm
op}(b)F(u)_{21}^{(\tau)}v_{1}+{\rm op}(b)F(u)_{22}^{(\tau)}v_{2}.$

At a first level of approximation, there holds

${\rm op}(b)\left(F(u)^{(\tau)}v\right)_{2}\approx F(u)_{21}^{(\tau)}{\rm
op}(b)v_{1}+F(u)_{22}^{(\tau)}{\rm op}(b)v_{2}.$

The second term of the right-hand side may be controlled directly by the term
${\rm E_{1}}$, but not the first term, as ${\rm op}(b)v_{1}$ is not a priori
controlled in a $H^{\sigma/2}$ norm.

Adding the structural assumption that $F(u)_{21}\equiv 0$ may then help loosen
the constraint (4.30), and in the end the lower bound on the Gevrey index.
This is typical of weakly hyperbolic systems: a perturbation by a lower order
term may induce a Gevrey loss of regularity. A careful analysis of
subprincipal symbol involving the approximated symbol $a_{\natural}$ is thus
of great importance.

The control of non-linearity is made thanks to the property of algebra of
Gevrey spaces, and the analytical structure of $F(u)$. As $u$ is in
${\rm\mathcal{G}}^{\sigma}_{\tau}$, Assumption 2.2 and the property that
$H^{\sigma/2}{\rm\mathcal{G}}^{\sigma}_{\tau}$ is an algebra thanks to Remark
3 in [Mor17], Proposition 3.2 therein implies that $F(u)^{(\tau)}$ acts
continuously in $H^{\sigma/2}$, hence

$\left|D^{\sigma/2}\left(F(u)^{(\tau)}v\right)_{2}\right|_{L^{2}}\lesssim\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(H^{\sigma/2})}\left|D^{\sigma/2}v\right|_{L^{2}}.$

Using Cauchy-Schwarz’ inequality to get an estimate of (4.29), there holds

$\displaystyle\left|\eqref{4.local.E4}\right|$
$\displaystyle\lesssim\left|D^{-\sigma/2}{\rm
op}(b)\left(F(u)^{(\tau)}v\right)_{2}\right|_{L^{2}}\,\left|{\rm
op}(b)D^{\sigma/2}v_{2}\right|_{L^{2}}+\left|\left(F(u)^{(\tau)}v\right)_{1}\right|_{L^{2}}\,\left|v_{1}\right|_{L^{2}}$
$\displaystyle\lesssim\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(H^{\sigma/2})}\left|D^{\sigma/2}v\right|_{L^{2}}\,\left|{\rm
op}(b)D^{\sigma/2}v_{2}\right|_{L^{2}}+\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(L^{2})}\,\left|v_{1}\right|_{L^{2}}^{2}$
$\displaystyle\lesssim\left(\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(H^{\sigma/2})}+\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(L^{2})}\right){\rm
E_{1}}.$

We conclude by

$\left|{\rm E_{4}}\right|\leq C_{{\rm 4}}\,{\rm E_{1}}$ (4.31)

for some $C_{{\rm 4}}>0$ depending essentially on
$\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(H^{\sigma/2})}+\left\|F(u)^{(\tau)}\right\|_{\mathcal{L}(L^{2})}$.

####  4.4.4 Conclusion

In order to complete the proof of Theorem 1, we put together the different
constraints between $c$ and $\sigma$ that appear in the estimates of the
energy. First, combining constraint (4.15) and constraint (4.23), there holds

$\frac{c}{2}=1-\sigma.$

This equality between the parameter $c$, used to regularize the weakness in
the hyperbolicity of the system, and the Gevrey index $\sigma$ already
appeared in the seminal paper [CJS83].

Next, we gather constraints (4.18), (4.26), (4.27) and (4.30). The last
constraint implies immediately the expected lower bound for the Gevrey index

$\sigma\geq 1/2.$ (4.32)

The three other constraints are weaker, hence do not interfere in the lower
bound. However, as soon as constraint (4.30) breaks down (see Remark 4.16),
the lower bound for the Gevrey index is better. Indeed, constraint (4.26)
implies the inequality $1-\sigma\leq\frac{1}{2}(1+\sigma)$, equivalent to

$\sigma\geq\frac{1}{3}.$ (4.33)

Note that both constraints (4.18) and (4.27) are equivalent, as $\nu$ tends to
infinity, to $c/2\leq 1$ – which is the limitation of $c$ imposed by the
uncertainty principle for the metric $g$.

We prove Theorem 1 by taking ${\bm{\tau}}>C_{{\rm 2}}+C_{{\rm 3}}+C_{{\rm
4}}$, where the constants are defined respectively in (4.25), (4.28) and
(4.31).

##  5 Appendices: two lemmas of real analysis and metrics in the phase space

###  5.1 Glaeser-type inequalities

We start by recalling the Faà di Bruno formula on iterated derivatives of
composition of functions:

######  Lemma 5.1 (Faà di Bruno formula).

Let $f:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ and
$g:{\mathbb{R}}\to{\mathbb{R}}$ be two $C^{\infty}$ functions. Then

$\forall\,\alpha,\beta\in\mathbb{N}^{d}\quad,\quad\frac{\partial_{x}^{\alpha}\partial_{\xi}^{\beta}(g\circ
f)}{\alpha!\beta!}=\sum_{1\leq k\leq|\alpha+\beta|}\frac{g^{(k)}\circ
f}{k!}\sum_{(\alpha_{1},\beta_{1})+\cdots+(\alpha_{k},\beta_{k})=(\alpha,\beta)\atop(\alpha_{j},\beta_{j})\neq(0,0)}\prod_{1\leq
j\leq
k}\frac{\partial_{x}^{\alpha_{j}}\partial_{\xi}^{\beta_{j}}f}{\alpha_{j}!\beta_{j}!}.$
(5.1)

We recall that for a $d$-tuple
$\alpha_{j}=(\alpha_{j}(1),\ldots,\alpha_{j}(d))$, we denote
$\alpha_{j}!=\prod_{1\leq p\leq d}\alpha_{j}(p)!$, and
$\partial_{x}^{\alpha_{j}}$ means
$\partial_{x_{1}}^{\alpha_{j}(1)}\circ\cdots\circ\partial_{x_{d}}^{\alpha_{j}(d)}$.
For further use, we denote

$N(\alpha,k)=\left|\left\\{(\alpha_{1},\ldots,\alpha_{k})\,\Big{|}\,\alpha_{1}+\cdots+\alpha_{k}=\alpha\,,\,\alpha_{j}\geq
1\right\\}\right|.$ (5.2)

By combinatorial arguments, for all $\alpha\in{\mathbb{N}}^{d}$ and $k\geq 1$
there holds

$N(\alpha,k)=\prod_{1\leq j\leq d}\binom{\alpha(j)-1}{k-1}.$

By putting $f(y)=y^{n}$ and $g(x)=e^{x}$ in the Faà di Bruno formula, we
obtain

$n^{\alpha}=\sum_{1\leq
k\leq|\alpha|}\binom{n}{k}\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\binom{\alpha}{\alpha_{1},\ldots,\alpha_{k}}.$

Next we recall the classical Glaeser inequality (see [Gla63]):

######  Lemma 5.2 (Global Glaeser inequality).

Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ be a non negative $C^{2}$ function,
such that $\partial_{x}^{2}f$ is bounded. Then

$\forall x\in{\mathbb{R}}^{n}\;,\quad|\partial_{x}f(x)|^{2}\leq
2|\partial_{x}^{2}f|_{L^{\infty}({\mathbb{R}}^{n})}f(x).$ (5.3)

The local result (inequality holds at any point) comes from a global
assumption on $f$ (non negativity of $f$, boundedness of $\partial_{x}^{2}f$).
The constant $2|\partial_{x}^{2}f|_{L^{\infty}({\mathbb{R}}^{n})}$ is optimal.
The proof of the Lemma is classical, and is based on the integral Taylor
expansion formula.

Local versions of the previous statement, that is with assumptions valid only
in an open set of ${\mathbb{R}}^{n}$, also exist. For any
$x_{0}\in{\mathbb{R}}^{d}$ and $r>0$, we denote

$B_{r}(x_{0})=\left\\{x\in{\mathbb{R}}^{d}\,:\,|x-x_{0}|<r\right\\}.$

In all the following, we consider $f:B_{r}(x_{0})\to{\mathbb{R}}$ a
nonnegative, $C^{2}$ function. We give first a sharp version of a local
Glaeser’s inequality, used in the present paper.

######  Lemma 5.3 (Sharp local Glaeser inequality).

Assuming that

$\min_{x\in\overline{B}_{r}(x_{0})}f(x)>0$

then, for any $p>0$ and any $r^{\prime}<r$, there holds

$\forall
x\in\overline{B}_{r^{\prime}}(x_{0})\;,\quad|\partial_{x}f(x)|^{p}\leq\frac{\left(|\partial_{x}f|_{L^{\infty}(\overline{B}_{r^{\prime}}(x_{0}))}\right)^{p}}{\min_{\overline{B}_{r^{\prime}}(x_{0})}f}f(x).$
(5.4)

###### Proof.

The inequality (5.4) is straightforward as there holds both

$\forall
x\in\overline{B}_{r^{\prime}}(x_{0})\;,\quad\min_{\overline{B}_{r^{\prime}}(x_{0})}f\leq
f(x)$

and

$\forall
x\in\overline{B}_{r^{\prime}}(x_{0})\;,\quad|\partial_{x}f(x)|^{p}\leq\left(|\partial_{x}f|_{L^{\infty}(\overline{B}_{r^{\prime}}(x_{0}))}\right)^{p}.$

∎

######  Remark 5.4.

Note that in this case, the Glaeser constant does not depend a priori of the
$L^{\infty}$ norm of the second order derivatives of $f$. We may indeed think
of polynomials of degree $2$ which are locally bounded from below by a
positive constant and have a positive discriminant.

Using Lemma 5.3, we prove here Lemmas 4.2 and 4.3.

######  Lemma 5.5 (Glaeser inequality for $a$).

Under Assumption 2.1, there is a neighborhood $[0,T]\times B_{r}(x_{0})$ of
$(0,x_{0})\in{\mathbb{R}}_{t}\times{\mathbb{R}}_{x}$ and a constant
$C_{T,r}>0$ for which there holds

$\forall\,(t,x)\in[0,T]\times
B_{r}(x_{0})\quad,\quad\left(\partial_{x}a(t,x)\right)^{2}\leq
C_{T,r}\,a(t,x).$ (5.5)

###### Proof of Lemma 4.2.

Thanks to Assumption 2.1, there holds

$\left(\partial_{x}a\right)^{2}=a(t,x)\left(\frac{4x^{2}}{t+x^{2}}e+(t+x^{2})\frac{\left(\partial_{x}e\right)^{2}}{e}+4x\partial_{x}e\right)$

and the term
$\left(\frac{4x^{2}}{t+x^{2}}e+(t+x^{2})\frac{\left(\partial_{x}e\right)^{2}}{e}+4x\partial_{x}e\right)$
is locally bounded thanks to Lemma 5.3.

∎

######  Lemma 5.6 (Derivatives of the symbol $b$).

We recall first definition (2.5) of $b$:

$b(t,x,\xi)=\left(a(t,x)+\langle\xi\rangle^{-c}\right)^{-1/2}.$

There is a bounded sequence of constants $C_{\alpha,\beta}>0$ for which there
holds

$\forall\,(\alpha,\beta)\in{\mathbb{N}}\times{\mathbb{N}}\;,\quad|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}b(t,x,\xi)|\leq
C_{\alpha,\beta}\widetilde{R}^{|\alpha|}2^{|\beta|}\,\alpha!^{s}\beta!\,b(t,x,\xi)\,b(t,x,\xi)^{|\alpha|}\,\langle\xi\rangle^{-|\beta|}$
(5.6)

for all $(t,x)$ in $[0,T]\times B_{r}(x_{0})$ and $\xi$ in ${\mathbb{R}}$. We
introduce the standard notation $s=1/\sigma$. And $\widetilde{R}$ satisfies

$\widetilde{R}=R(1+|a|_{s,R})>R.$ (5.7)

###### Proof of Lemma 4.3.

By the Faà di Bruno formula (Lemma 5.1) on iterated derivatives of composition
of functions, using the fact that
$\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a_{\natural}\equiv 0$ as soon as
$|\alpha|>0$ and $|\beta|>0$, we deduce

$\displaystyle\frac{1}{\alpha!\beta!}\,\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left(a_{\natural}^{-1/2}\right)$
$\displaystyle=\sum_{1\leq k\leq|\alpha|\atop 1\leq
k^{\prime}\leq|\beta|}\frac{c_{k}c_{k^{\prime}}}{k!k^{\prime}!}\,a_{\natural}^{-1/2-k-k^{\prime}}\left(\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\partial_{x}^{\alpha_{j}}a\right)\times\left(\sum_{\beta_{1}+\cdots+\beta_{k^{\prime}}=\beta\atop\beta_{j}\geq
1}\prod_{j=1}^{k^{\prime}}\frac{1}{\beta_{j}!}\partial_{\xi}^{\beta_{j}}\langle\cdot\rangle^{-c}\right)$

where coefficients $c_{k+k^{\prime}}$ are defined by
$\left(y^{-1/2}\right)^{(k)}=c_{k}y^{-1/2-k}$.

Next, there holds

$\left|\frac{1}{\beta_{j}!}\partial_{\xi}^{\beta_{j}}\langle\xi\rangle^{-c}\right|\leq\langle\xi\rangle^{-c-|\beta_{j}|}$

as $c\leq 2$, hence

$\left|\sum_{\beta_{1}+\cdots+\beta_{k^{\prime}}=\beta\atop\beta_{j}\geq
1}\prod_{j=1}^{k^{\prime}}\frac{1}{\beta_{j}!}\partial_{\xi}^{\beta_{j}}\langle\cdot\rangle^{-c}\right|\leq
N(\beta,k^{\prime})\,\langle\cdot\rangle^{-k^{\prime}c-|\beta|}$

where we denote

$N(\beta,k^{\prime})=\left|\big{\\{}(\beta_{1},\ldots,\beta_{k^{\prime}})\,|\,\beta_{1}+\cdots+\beta_{k^{\prime}}=\beta\,,\,\beta_{j}\geq
1\big{\\}}\right|.$

Thanks to the the bound (4.2), there holds
$a_{\natural}^{-1}\leq\langle\cdot\rangle^{c}$, hence

$a_{\natural}^{-k^{\prime}}\left|\sum_{\beta_{1}+\cdots+\beta_{k^{\prime}}=\beta\atop\beta_{j}\geq
1}\prod_{j=1}^{k^{\prime}}\frac{1}{\beta_{j}!}\partial_{\xi}^{\beta_{j}}\langle\cdot\rangle^{-c}\right|\leq
N(\beta,k^{\prime})\,\langle\cdot\rangle^{-|\beta|}.$

We focus now on the sum

$\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\partial_{x}^{\alpha_{j}}a.$

If $|\alpha_{j}|=1$, we may use Lemma 4.2 to bound
$\partial_{x}^{\alpha_{j}}$. We introduce then

$I_{1}(\alpha_{1},\ldots,\alpha_{k})=\\{j:|\alpha_{j}|=1\\}$

and there holds

$\forall\,j\in
I_{1}\;,\quad|\partial_{x}^{\alpha_{j}}a|\leq\left(C_{T,r}^{1/2}\,a_{\natural}^{1/2}\right)^{\alpha_{j}}$

thus

$\displaystyle\left|\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\partial_{x}^{\alpha_{j}}a\right|$
$\displaystyle\leq$
$\displaystyle\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\,\prod_{j\in
I_{1}}\frac{1}{\alpha_{j}!}|\partial_{x}^{\alpha_{j}}a|\,\prod_{j\notin
I_{1}}\frac{1}{\alpha_{j}!}|\partial_{x}^{\alpha_{j}}a|$ $\displaystyle\leq$
$\displaystyle\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\,\prod_{j\in
I_{1}}\frac{1}{\alpha_{j}!}\left(C_{T,r}^{1/2}\,a_{\natural}^{1/2}\right)^{\alpha_{j}}\,\prod_{j\notin
I_{1}}\frac{1}{\alpha_{j}!}|\partial_{x}^{\alpha_{j}}a|.$

For indices not in $I_{1}$, that is for $|\alpha_{j}|\geq 2$, we use the fact
that $a$ is in $G^{s}_{R}$, hence

$\displaystyle\left|\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\,\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\partial_{x}^{\alpha_{j}}a\right|$
$\displaystyle\leq$
$\displaystyle\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j\in
I_{1}}\frac{1}{\alpha_{j}!}\left(C_{T,r}^{1/2}\,a_{\natural}^{1/2}\right)^{\alpha_{j}}\,\prod_{j\notin
I_{1}}\frac{1}{\alpha_{j}!}|a|_{s,R}\,R^{|\alpha_{j}|}\alpha_{j}!^{s}.$

As the $k$-tuple $(\alpha_{1},\ldots,\alpha_{k})$ satisfies
$\alpha_{1}+\cdots+\alpha_{k}=\alpha$, there holds
$|\alpha_{1}|+\cdots+|\alpha_{k}|=|\alpha|$ hence

$\displaystyle|I_{1}(\alpha_{1},\ldots,\alpha_{k})|$ $\displaystyle=$
$\displaystyle|\alpha|-\sum_{j\notin I_{1}}|\alpha_{j}|$ $\displaystyle\leq$
$\displaystyle|\alpha|-2(k-|I_{1}|)$

which leads to $|I_{1}|\geq 2k-|\alpha|$. As $a_{\natural}\leq 1$, we get

$\displaystyle\left|\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\,\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\partial_{x}^{\alpha_{j}}a\right|$
$\displaystyle\leq$ $\displaystyle
a_{\natural}^{k-|\alpha|/2}\,\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j\in I_{1}}\frac{1}{\alpha_{j}!}C_{T,r}^{1/2}\,\prod_{j\notin
I_{1}}\frac{1}{\alpha_{j}!}\alpha_{j}!^{s}|a|_{s,R}\,R^{|\alpha_{j}|}.$

We need then to compare $C_{T,r}^{1/2}$ with $|a|_{s,R}R$. Up to shrinking
$T$, we may assume that

$C_{T,r}^{1/2}\leq|a|_{s,R}R.$ (5.8)

There holds

$\displaystyle\left|\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\partial_{x}^{\alpha_{j}}a\right|$
$\displaystyle\leq$ $\displaystyle
a_{\natural}^{k-|\alpha|/2}|a|_{s,R}^{k}R^{|\alpha|}\,\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\frac{1}{\alpha_{j}!}\alpha_{j}!^{s}$ $\displaystyle\leq$
$\displaystyle
a_{\natural}^{k-|\alpha|/2}\,\alpha!^{s-1}|a|_{s,R}^{k}R^{|\alpha|}\,\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\binom{\alpha}{\alpha_{1},\ldots,\alpha_{k}}^{1-s}.$

Denote

$C_{s}(\alpha,k)=\sum_{\alpha_{1}+\cdots+\alpha_{k}=\alpha\atop\alpha_{j}\geq
1}\prod_{j=1}^{k}\binom{\alpha}{\alpha_{1},\ldots,\alpha_{k}}^{1-s}.$

As $\binom{\alpha}{\alpha_{1},\ldots,\alpha_{k}}\geq 1$ and $s\geq 1$, there
holds

$C_{s}(\alpha,k)\leq N(\alpha,k).$

We put altogether all the inequalities:

$\displaystyle\left|\frac{1}{\alpha!\beta!}\,\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left(a_{\natural}^{-1/2}\right)\right|$
$\displaystyle\leq\sum_{1\leq k\leq|\alpha|\atop 1\leq
k^{\prime}\leq|\beta|}\frac{|c_{k}|\,|c_{k^{\prime}}|}{k!k^{\prime}!}\,a_{\natural}^{-1/2-k}\,a_{\natural}^{k-|\alpha|/2}\,\alpha!^{s-1}|a|_{s,R}^{k}R^{|\alpha|}\,C_{s}(\alpha,k)\,N(\beta,k^{\prime})\,\langle\cdot\rangle^{-|\beta|}$
$\displaystyle\leq
a_{\natural}^{-1/2-|\alpha|/2}\alpha!^{s-1}R^{|\alpha|}\,\langle\cdot\rangle^{-|\beta|}\,\sum_{1\leq
k\leq|\alpha|\atop 1\leq
k^{\prime}\leq|\beta|}\frac{|c_{k}|\,|c_{k^{\prime}}|}{k!k^{\prime}!}\,N(\beta,k^{\prime})|a|_{s,R}^{k}\,C_{s}(\alpha,k)$
$\displaystyle\leq
b(t,x,\xi)\,a_{\natural}^{-|\alpha|/2}\alpha!^{s-1}R^{|\alpha|}\,\langle\cdot\rangle^{-|\beta|}\,\left(\sum_{1\leq
k\leq|\alpha|}\frac{|c_{k}|}{k!}|a|_{s,R}^{k}\,N(\alpha,k)\right)\,\left(\sum_{1\leq
k^{\prime}\leq|\beta|}\frac{|c_{k^{\prime}}|}{k^{\prime}!}\,N(\beta,k^{\prime})\right).$

By definition of the $c_{k}$ there holds

$\displaystyle c_{k}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{k-1}\left(-1/2-j\right)$ $\displaystyle=$
$\displaystyle(-1/2)^{k}\prod_{j=0}^{k-1}\left(2j+1\right)$ $\displaystyle=$
$\displaystyle\left(\frac{-1}{2}\right)^{k}\frac{(2k)!}{\prod_{j=0}^{k-1}(2(j+1))}$
$\displaystyle=$ $\displaystyle\left(\frac{-1}{4}\right)^{k}\frac{(2k)!}{k!}.$

In particular there holds $|c_{k}|\leq k!$ by Stirling’s inequality. This
implies that

$\sum_{1\leq
k\leq|\alpha|}\frac{|c_{k}|}{k!}|a|_{s,R}^{k}\,N(\alpha,k)\leq(1+|a|_{s,R})^{|\alpha|}$

and

$\sum_{1\leq
k^{\prime}\leq|\beta|}\frac{|c_{k^{\prime}}|}{k^{\prime}!}\,N(\beta,k^{\prime})\leq
2^{|\beta|}.$

Finally there holds

$\displaystyle\left|\frac{1}{\alpha!\beta!}\,\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left(a_{\natural}^{-1/2}\right)\right|$
$\displaystyle\leq
b(t,x,\xi)\,b^{|\alpha|}\alpha!^{s-1}\left(R(1+|a|_{s,R})\right)^{|\alpha|}2^{|\beta|}\,\langle\cdot\rangle^{-|\beta|}$

which suffices to end the proof.

∎

We do not use Lemma 5.7 here, but include it since it may prove useful in
further work on weakly hyperbolic systems. We note that a local statement can
be deduced from Lemma 5.2, using a $C^{\infty}$ nonnegative function $\varphi$
with compact support $\overline{B}_{r}(x_{0})$, and equals to $1$ in
$\overline{B}_{r^{\prime}}(x_{0})$ for some $r^{\prime}<r$. We may then extend
any locally defined, nonnegative function into a globally defined, nonnegative
one.

We first introduce some notations. For any domain $\mathcal{D}\subset
B_{r}(x_{0})$ and $j\in{\mathbb{N}}$, we denote

$M_{j}(f;\mathcal{D})=\sup\left\\{|\partial_{x}^{\alpha}f|\,:\,x\in\mathcal{D}\,,\,|\alpha|=j\right\\}.$

For any $0<r^{\prime}<r$ we define

$\mathcal{C}_{r^{\prime},r}(x_{0})=\left\\{x\in{\mathbb{R}}^{d}\,:\,r^{\prime}<|x-x_{0}|<r\right\\}.$

######  Lemma 5.7 (Local Glaeser inequality).

Let $f:B_{r}(x_{0})\to{\mathbb{R}}$ be a nonnegative $C^{2}$ function. Then

$\forall
x\in\overline{B}_{r^{\prime}}(x_{0})\;,\quad|\partial_{x}f(x)|^{2}\leq
G(f;x_{0},r^{\prime},r)f(x)$ (5.9)

for any $r^{\prime}<r$. The local Glaeser’s constant $G(f;x_{0},r^{\prime},r)$
is defined by

$G(f;x_{0},r^{\prime},r)=2M_{2}(f;B_{r}(x_{0}))+\frac{4}{r-r^{\prime}}M_{1}\left(f;\mathcal{C}_{r^{\prime},r}(x_{0})\right)+\frac{4}{(r-r^{\prime})^{2}}M_{0}\left(f;\mathcal{C}_{r^{\prime},r}(x_{0})\right).$
(5.10)

###### Proof.

Let $\varphi$ be a $C^{\infty}$ function with compact support
$\overline{B}_{r}(x_{0})$, satisfying also $0\leq\varphi\leq 1$ and
$\varphi(x)=1$ for all $x\in\overline{B}_{r^{\prime}}(x_{0})$. Then the
function $f\varphi$ satisfies the conditions for applying Lemma 5.2. Hence
(5.3) leads to

$|(f\varphi)^{\prime}(x)|^{2}\leq
2M_{2}(f\varphi;{\mathbb{R}}^{n})f(x)\varphi(x)$

for all $x$ in ${\mathbb{R}}^{n}$. As $\varphi$ is identically one in
$\overline{B}_{r^{\prime}}(x_{0})$, there holds

$|f^{\prime}(x)|^{2}\leq 2M_{2}(f\varphi;{\mathbb{R}}^{n})f(x)$

for all $x\in\overline{B}_{r^{\prime}}(x_{0})$.

To end the proof we have to give an upper bound of
$M_{2}(f\varphi;{\mathbb{R}}^{n})$, with respect to the distance
$r-r^{\prime}$. First there holds

$M_{2}(f\varphi;{\mathbb{R}}^{n})\leq
M_{2}(f;B_{r}(x_{0}))+2M_{1}\left(f;\mathcal{C}_{r^{\prime},r}(x_{0})\right)M_{1}(\varphi;{\mathbb{R}}^{n})+M_{0}\left(f;\mathcal{C}_{r^{\prime},r}(x_{0})\right)M_{2}(\varphi;{\mathbb{R}}^{n}).$

Second, for any $x_{r^{\prime}}$ such that $|x_{r^{\prime}}|=r^{\prime}$ we
denote

$x_{r}=x_{0}+\frac{r}{r^{\prime}}(x_{r^{\prime}}-x_{0})$

the only point of $\overline{B}_{r}(x_{0})$ such that $|x_{r}-x_{0}|=r$ and
$x_{r^{\prime}}$ is in the interval $[x_{0},x_{r}]$. By the mean value theorem
there is $s\in[0,1]$ such that

$\varphi(x_{r})-\varphi(x_{r^{\prime}})=(x_{r}-x_{r^{\prime}})\cdot\partial_{x}\varphi\left(x_{0}+s\frac{r}{r^{\prime}}(x_{r^{\prime}}-x_{0})\right)$

thus

$\left|\partial_{x}\varphi\left(x_{0}+s\frac{r}{r^{\prime}}(x_{r^{\prime}}-x_{0})\right)\right|=\frac{1}{r-r^{\prime}}$

as $\varphi(x_{r})=0$, $\varphi(x_{r^{\prime}})=1$ and
$|x_{r}-x_{r^{\prime}}|=r-r^{\prime}$ and then

$M_{1}(\varphi)\geq\frac{1}{r-r^{\prime}}.$

By the same way we can prove also that

$M_{2}(\varphi)\geq\frac{2}{(r-r^{\prime})^{2}}.$

To end the proof, it suffices to construct $\varphi$ such that the previous
lower bound are equalities. ∎

######  Remark 5.8.

In the estimate (5.10) appears the distance $r-r^{\prime}$. In the worst case,
it is the distance between the neighborhood of $x_{0}$ such that the Glaeser
inequality holds, and the possible point $\tilde{x}$ such that
$f(\tilde{x})=0$ and $\partial_{x}f(\tilde{x})\neq 0$, at which Glaser
inequality fails.

For example, let take $f(x)=x$ in $[0,+\infty[$. Then $f^{\prime}(x)=1$ and
there holds, for any $x_{0}>0$:

$\forall\,x\in[x_{0},x_{0}+1]\;,\quad(f^{\prime}(x))^{2}\leq C(x_{0})f(x)$

with $C(x_{0})=1/x_{0}$. By comparison, the constant $G(f;r^{\prime},r)$ of
the previous Lemma verifies

$\displaystyle G(f;r^{\prime},r)$ $\displaystyle\leq$ $\displaystyle
M_{2}(f;[x_{0},x_{0}+1])+\frac{2}{x_{0}}M_{1}(f;[0,x_{0}])+\frac{2}{x_{0}^{2}}M_{0}(f;[0,x_{0}])$
$\displaystyle\lesssim$ $\displaystyle\frac{1}{x_{0}}$

as $M_{0}(f;[0,x_{0}])\leq x_{0}$.

###  5.2 Metrics in the phase space and pseudo-differential calculus

We refer to the Chapter 2 of [Ler11] for the basic definitions and expected
properties of metrics in the phase space and associated symbols. As we wish
this paper to be self-contained, we give the few needed definitions, and prove
that our metric $g$ and symbol $b$ satisfy them. Hence we may use all
properties of pseudo-differential calculus with symbols in $S(b,g)$ and other
related classes.

######  Lemma 5.9 (Admissibility of the metric).

The metric $g$ defined by (4.6) is admissible, that is:

  1. 1.

The metric $g$ is slowly varying (see Definition $2.2.1$ in [Ler11]), as there
are $C>0$ and $r>0$ such that for all $X,Y,T\in{\mathbb{R}}\times{\mathbb{R}}$
there holds

$g_{X}(X-Y)\leq r^{2}\implies C^{-1}g_{Y}(T)\leq g_{X}(T)\leq Cg_{Y}(T).$
(5.11)

  2. 2.

The metric $g$ satisfies the uncertainty principle (see Section $2.2.3$ and
specifically $2.2.12$ in [Ler11]), that is

$\lambda(t,x,\xi)\geq 1\quad,\quad\forall\,t,x,\xi.$ (5.12)

where $\lambda$ is defined by (4.10).

  3. 3.

The metric $g$ is temperate (see Lemma $2.2.14$ in [Ler11]), that is there are
$C>0$ and $N>0$ such that

$\frac{g_{X}(T)}{g_{Y}(T)}\leq
C\left(1+g_{X}^{\sigma}(X-Y)\right)^{N}\quad,\quad\forall\,X,Y,T\text{ in
}{\mathbb{R}}\times{\mathbb{R}}.$ (5.13)

The metric $g_{X}^{\sigma}$ is defined by

$g_{X}^{\sigma}(Y)=\langle
X_{2}\rangle^{2}|Y_{1}|^{2}+a_{\natural}(t,X)|Y_{2}|^{2}.$

###### Proof.

We follow here partially the proof of Lemma 3.1 in [CN07]. We remind that
$g_{X}(Y)$ reads

$g_{X}(Y)=\frac{|Y_{1}|^{2}}{a_{\natural}(t,X)}+\frac{|Y_{2}|^{2}}{\langle
X_{2}\rangle^{2}}$

with $X=(X_{1},X_{2})$ (the first component has to be seen as
$x\in{\mathbb{R}}$ and the second component as $\xi\in{\mathbb{R}}$).

$1.$ Assume that there holds

$\frac{|X_{2}-Y_{2}|^{2}}{\langle X_{2}\rangle^{2}}\leq r_{2}^{2}$ (5.14)

for some $r_{2}>0$. This implies in particular that

$\displaystyle|Y_{2}|^{2}$ $\displaystyle\leq$ $\displaystyle
2|X_{2}|^{2}+2|X_{2}-Y_{2}|^{2}$ $\displaystyle\leq$ $\displaystyle
2(1+r_{2}^{2})|X_{2}|^{2}$

hence $\langle Y_{2}\rangle\leq\sqrt{2}(1+r_{2}^{2})^{1/2}\langle
X_{2}\rangle$. The same way we prove

$|X_{2}|^{2}\leq 2r_{2}^{2}|X_{2}|^{2}+2|Y_{2}|^{2}$

which leads to $(1-2r_{2}^{2})^{1/2}\langle X_{2}\rangle\leq\sqrt{2}\langle
Y_{2}\rangle$ as soon as $r_{2}<1/\sqrt{2}$. We have just proved the following

$\frac{|X_{2}-Y_{2}|^{2}}{\langle X_{2}\rangle^{2}}\leq r_{2}^{2}\implies
C_{2}^{-1}\frac{|T_{2}|^{2}}{\langle
Y_{2}\rangle^{2}}\leq\frac{|T_{2}|^{2}}{\langle X_{2}\rangle^{2}}\leq
C_{2}\frac{|T_{2}|^{2}}{\langle
Y_{2}\rangle^{2}}\quad,\quad\forall\,T_{2}\in{\mathbb{R}}$

with $r_{2}<1/\sqrt{2}$ and $C_{2}>0$ depending only on $r_{2}$.

Next, we consider the part $\frac{|Y_{1}|^{2}}{a_{\natural}(t,X)}$ of the
metric $g$. Assume that there holds

$\frac{|X_{1}-Y_{1}|^{2}}{a_{\natural}(t,X)}\leq r_{1}^{2}$ (5.15)

for some $X$ and $Y$ in ${\mathbb{R}}\times{\mathbb{R}}$ and $r_{1}>0$, and
that (5.14) still holds with $r_{2}<1/\sqrt{2}$. We aim to compare
$a_{\natural}(t,X)$ and $a_{\natural}(t,Y)$. As $a(t,x)$ is smooth with
respect to $x$, there holds

$a(t,X_{1})=a(t,Y_{1})+(X_{1}-Y_{1})\partial_{x}a(t,Y_{1})+R$

with $|R|\leq|X_{1}-Y_{1}|^{2}\sup\left|\partial_{x}^{2}a(t,x)\right|$. Note
that, as $a$ has compact support in $x$, that
$\sup\left|\partial_{x}^{2}a(t,x)\right|$ is independent of $X_{1}$ and
$Y_{1}$. Then, using (5.15) we get

$|R|\leq C_{2}^{\prime}a_{\natural}(t,X)r_{1}^{2}$

and

$\left|(X_{1}-Y_{1})\partial_{x}a(t,Y_{1})\right|\leq
C_{2}^{\prime\prime}a_{\natural}(t,X)^{1/2}r_{1}a_{\natural}(t,Y)^{1/2}$

thanks to Lemma 4.2. Note that neither $C_{2}^{\prime}$ nor
$C_{2}^{\prime\prime}$ depend on $X$ or $Y$. Then, by positivity of $a$, there
holds

$\displaystyle a(t,X_{1})$ $\displaystyle\leq$ $\displaystyle
a_{\natural}(t,Y)+C_{2}^{\prime\prime}a_{\natural}(t,X)^{1/2}r_{1}a_{\natural}(t,Y)^{1/2}+C_{2}^{\prime}a_{\natural}(t,X)r_{1}^{2}$
$\displaystyle\leq$ $\displaystyle
2a_{\natural}(t,Y)+\left(C_{2}^{\prime}+C_{2}^{\prime\prime
2}/4\right)r_{1}^{2}a_{\natural}(t,X).$

As soon as $r_{1}$ satisfies $C_{2}^{\prime}+C_{2}^{\prime\prime
2}/4<r_{1}^{-2}$, we get

$a_{\natural}(t,X)\leq C_{2}a_{\natural}(t,Y)$

which suffices to prove finally (5.11).

2\. The uncertainty principle for general metrics in the phase space reads in
our case (5.12). As the inequality should hold for all times, in particular
for $t=0$ this leads to

$\lambda(0,x,\xi)=\langle\xi\rangle^{-c/2}\langle\xi\rangle\geq
1\quad,\quad\forall\,\xi\in{\mathbb{R}}$

hence $c\leq 2$. As $\lambda$ is decreasing as time goes by, this inequality
is sufficient to ensure (5.12) for all times.

3\. As $g$ is slowly varying, inequality (5.13) is satisfied if
$g_{X}(X-Y)\leq r^{2}$. Assume then that $g_{X}(X-Y)>r^{2}$. As we note that
$g^{\sigma}_{X}(X-Y)=a_{\natural}(t,X)\langle X_{2}\rangle^{2}g_{X}(X-Y)$,
this implies that

$\displaystyle g^{\sigma}_{X}(X-Y)$ $\displaystyle\geq$ $\displaystyle
a_{\natural}(t,X)\langle X_{2}\rangle^{2}r^{2}$ (5.16) $\displaystyle\geq$
$\displaystyle\langle X_{2}\rangle^{2-c}r^{2}$

by nonnegativity of $a$.

For all $T\in{\mathbb{R}}\times{\mathbb{R}}$, there holds

$\frac{g_{X}(T)}{g_{Y}(T)}\leq\max\left(\frac{a_{\natural}(t,Y)}{a_{\natural}(t,X)}\,,\,\frac{\langle
Y_{2}\rangle^{2}}{\langle X_{2}\rangle^{2}}\right).$

To prove (5.13), it suffices then to prove that

$\frac{a_{\natural}(t,Y)}{a_{\natural}(t,X)}\leq
C^{\prime}\left(1+g_{X}^{\sigma}(X-Y)\right)^{N^{\prime}}$ (5.17)

for some $C^{\prime}>0$, $N^{\prime}>0$ and

$\frac{\langle Y_{2}\rangle^{2}}{\langle X_{2}\rangle^{2}}\leq
C^{\prime\prime}\left(1+g_{X}^{\sigma}(X-Y)\right)^{N^{\prime\prime}}$ (5.18)

for some $C^{\prime\prime}>0$, $N^{\prime\prime}>0$.

Consider first the former inequality. We proceed as for the point $1.$,
proving there is a constant $B>0$ such that

$a(t,Y_{1})=a(t,X_{1})+B\left(a(t,X_{1})+|X_{1}-Y_{1}|^{2}\right).$

Here, the control of $|X_{1}-Y_{1}|^{2}$ is done by $g_{X}^{\sigma}(X-Y)$, as
there holds

$\displaystyle|X_{1}-Y_{1}|^{2}$ $\displaystyle\leq$ $\displaystyle
g_{X}^{\sigma}(X-Y)$ $\displaystyle\leq$ $\displaystyle C\langle
X_{2}\rangle^{-c}g_{X}^{\sigma}(X-Y)^{1+c/(2-c)}$

thanks to (5.16). Then there is $B^{\prime}>0$ such that

$a(t,Y_{1})\leq
B^{\prime}a_{\natural}(t,X)\left(1+g_{X}^{\sigma}(X-Y)\right)^{1+c/(2-c)}.$

As there holds $\langle Y_{2}\rangle^{-c}\leq\langle
Y_{2}\rangle^{-c}(1+|X_{2}-Y_{2}|)^{c}$, we get

$\displaystyle\langle Y_{2}\rangle^{-c}$ $\displaystyle\leq$
$\displaystyle\langle Y_{2}\rangle^{-c}(1+|X_{2}-Y_{2}|)^{c}$
$\displaystyle\leq$ $\displaystyle\langle
Y_{2}\rangle^{-c}\left(1+g_{X}^{\sigma}(X-Y)^{1/2+c/2(2-c)}\right)^{c}$

hence

$a_{\natural}(t,Y)\leq
C^{\prime}a_{\natural}(t,X)\left(1+g_{X}^{\sigma}(X-Y)\right)^{N^{\prime}}.$

for some $C^{\prime}>0$ and $N^{\prime}>0$.

We may prove the second inequality (5.18) by the same way, and end the proof.
∎

######  Lemma 5.10 (Admissibility of the symbol $b$).

The symbol $b$ defined by (2.5) is an admissible weight for the metric $g$,
that is there are $C>0$ and $N>0$ such that

$\frac{b(t,X)}{b(t,Y)}\leq
C\left(1+g_{X}^{\sigma}(X-Y)\right)^{N}\quad,\quad\forall\,X,Y\text{ in
}{\mathbb{R}}\times{\mathbb{R}}.$ (5.19)

###### Proof.

In the course of the previous Lemma, we prove inequality (5.17). In view of
the definition of an admissible weight, this means exactly that $a_{\natural}$
is an admissible weight for $g$. Lemma $2.2.22$ with $f(t)=t^{-1/2}$ and the
fact that $a_{\natural}\in S(a_{\natural},g)$ implies then that
$b=a_{\natural}^{-1/2}$ is also an admissible weight.

∎

For an admissible weight $M$ on ${\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$, we
introduce the classes of symbols $S(M,g)$ associated to the metric $g$:

######  Definition 5.11 (Definition of classes of symbols).

The space of symbols $S(M,g)$ is defined as the set of $C^{\infty}$ functions
$f(t,\cdot)$ on ${\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ such that, for all
$(\alpha,\beta)$ in ${\mathbb{N}}^{d}\times{\mathbb{N}}^{d}$, there is
$C_{\alpha,\beta}>0$ such that

$\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}f(t,x,\xi)\right|\leq
C_{\alpha,\beta}M(x,\xi)\,b(t,x,\xi)^{|\alpha|}\langle\xi\rangle^{-|\beta|}$

uniformly in $(t,x,\xi)$.

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