# Gevrey WKB method for Pseudodifferential Operators of real principal type

Richard Lascar  LJAD, Université Côte-d’-Azur, 28 Parc Valrose, 06028 Nice
cedex, France<EMAIL_ADDRESS>and  Iván Moyano  LJAD,
Université Côte-d’-Azur, 28 Parc Valrose, 06028 Nice cedex, France
<EMAIL_ADDRESS>
###### Abstract.

In this paper we investigate the conjugation of Fourier Integral Operators
(FIOs) associated to Gevrey phases and symbols and the corresponding
semiclassical pseudodifferential operators (pdos) in the Gevrey class. We
obtain an Egorov theorem compatible with Gevrey FIOs and real principal part
pdos with Gevrey symbols. As a consequence, we obtain a justification of the
usual microlocal WKB expansion for Gevrey pdos which are of real principal
part at a point in the phase space, with the natural Gevrey subexponential
asymptotics with respect to the semiclassical parameter.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations, hypothesis and main results
      1. 1.1.1 Main results: Gevrey WKB expansion and Gevrey Egorov’s theorem
    2. 1.2 Strategy and outline
    3. 1.3 Notation
  2. 2 Stationary phase and symbolic calculus in the Gevrey setting
    1. 2.1 Non-stationary and Stationary phase method in the Gevrey setting
      1. 2.1.1 Non-stationary phase lemma
      2. 2.1.2 Stationary phase lemma
    2. 2.2 Symbolic calculus with Gevrey symbols
    3. 2.3 Formal symbols and Borel lemma in the Gevrey setting
    4. 2.4 Formal quasinorms in the class of Gevrey symbols
  3. 3 Fourier Integral Operators in the Gevrey setting
    1. 3.1 FIO with Grevey symbols and phases
    2. 3.2 Composition of FIO and PDOs in the Gevrey setting
    3. 3.3 Gevrey wavefront and Lagrangian distributions
  4. 4 Proof of Theorem 2
  5. 5 A Gevrey FBI version of Egorov’s theorem: Proof of Theorem 3
    1. 5.1 Some complex symplectic geometry
      1. 5.1.1 FBI transforms
    2. 5.2 WKB Heuristics
    3. 5.3 Construction of the canonical transformation $\kappa$
    4. 5.4 Construction of the phase $\varphi$ on $\Gamma$
    5. 5.5 Construction of the symbol $a$ on $\Gamma$

##  1\. Introduction

The use of Gevrey functions in the study of some linear and non-linear partial
differential equations has experienced a significant developement in recent
years. In particular, the use of Gevrey norms have been instrumental in the
context of some non-linear Cauchy problems that may be ill-posed in the usual
Sobolev context [9] or the study of some singular limits difficult to describe
rigorously using other topologies [21, 1].

Let us recall that, as usual, for $N\in{\mathbb{N}}$ and $s>0$ a function
$f\in C^{\infty}({\mathbb{R}}^{N};{\mathbb{C}})$ is Gevrey of order $s$
(Gevrey-$s$ for short, or simply ${\mathcal{G}}^{s}$) whenever for every
compact $K\subset{\mathbb{R}}^{N}$ the following holds:

(1.1) $\exists
C_{K}>0,\quad\forall\alpha\in{\mathbb{N}}^{N},\qquad|\partial^{\alpha}f|_{L^{\infty}(K)}\leq
C_{K}^{|\alpha|+1}\alpha!^{s}.$

As can be seen from (1.1), the case $s=1$ coincides with the usual class of
analytic functions, but the cases $s>1$ may contain smooth functions which are
not analytic. Hence, the Gevrey class may be understood as a class of
functions in between the analytic and the smooth context.

The use of pseudodifferential calculus associated to particular Gevrey
classes, started by Boutet de Monvel and Kree in their seminal paper, [2,
Sect. 1], has also been important in the past, for example in connexion with
the study of propagation of Gevrey singularities of these pseudodifferential
operators [14, 16] or the diffraction of waves around an obstacle [17]. More
recently, the study of the FBI transform in the Gevrey context has risen some
interesting questions related to the quantization in the complex plane of
symbols admitting only quasi-holomorphic extensions which in fact are not
unique [11] and thus introduces new aspects with respect to the analytic
framework [22, 12, 20].

In this work we are concerned with the study of Gevrey pseudodifferential
operators and the resulting WKB expansion in the Gevrey class. In what
follows, we introduce some definitions of these Gevrey symbols and the related
pseudodifferential operators and then state our main results. We work in a
semiclassical framework (cf. [23]) involving a possibly small parameter $h>0$.

Let us note, for $n\in{\mathbb{N}}$, $m\in{\mathbb{R}}$ and $s>0$, the set
${\mathcal{S}}_{s}^{m}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ of
semiclassical Gevrey $s$ symbols of order $m$. Following [2, Sect. 1] we write
$a\in{\mathcal{S}}^{m}_{s}$ if and only if for some $C>0$

(1.2) $|\partial_{x}^{\alpha}\partial_{\theta}^{\beta}a(x,\theta,h)|\leq
C^{1+|\alpha|+|\beta|}\alpha!^{s}\beta!^{s}h^{-m}$

for all $(x,\theta)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ and
$\alpha,\beta\in{\mathbb{N}}^{n}$. Observe that this condition is more precise
than the usual definition of the class ${\mathcal{S}}^{m}_{1,0}$ (cf. for
example (cf. [23, Chapter 4])) because the constant $C$ in (1.2) is uniform
with respect to the multi-indices $\alpha,\beta$. As a consequence, we shall
need a specific version of symbolic calculus adapted to the class
${\mathcal{S}}^{m}_{s}$ (in particular Proposition 2.3 below). Finally, recall
that a pdo $A$ is elliptic at a point
$(x_{0},\theta_{0})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ whenever its
symbol $a$ satisfies $|a(x_{0},\theta_{0})|>0$.

We consider as usual (cf. [23, Chapter 4]) semiclassical pseudo-differential
operators (pdo for short) defined as suitable extensions of

$a(x,hD,h)u(x)=\frac{1}{(2\pi h)^{n}}\iint
a(x,\theta,h)e^{\frac{i}{h}\theta\cdot(x-y)}u(y)\,\mathrm{d}y\,\mathrm{d}\theta,\qquad
u\in\mathscr{S}({\mathbb{R}}^{n}),$

for a given symbol $a\in{\mathcal{S}}^{m}_{s}$. One has
$a(x,hD,h)={\operatorname{Op}_{h}}(a)$ and also
${\operatorname{Op}_{h}}(a)={\operatorname{Op}_{1}}(a_{h})$ with
$a_{h}(x,\theta)=a(x,h\theta)$. We shall use the notation
${\operatorname{Op}_{h}}(a)\in\Psi^{m}_{s}$ for such a pdo.

###  1.1. Motivations, hypothesis and main results

The motivation of our work is twofold. On the one hand, the WKB method is
known to hold true in the smooth ($\mathcal{C}^{\infty}$) and analytic
categories, which raises the natural question of its vaidity with respect to
the specific Gevrey asymptotics required in the Gevrey framework (in
particular of the form (1.3)). Our Theorem 1 makes this point explicit at
least microlocally. On the other hand, we obtain this microlocal WKB expansion
as a consequence of Theorem 2, which is a Gevrey version of the Egorov
theorem, a central result in microlocal analysis, first published in [5],
having its own independent interest. We state next our main hypothesis and the
context of our results.

Let $P=P(x,hD_{x},h)$ a semiclassic Gevrey$-s$ PDO of order zero in
${\mathbb{R}}^{n}$, with $n\geq 2$. Let $p=p(x,\xi)$ be the principal symbol
of $P$ and let $H_{p}$ be the associated Hamiltonian. We assume the following
hypothesis.

(H1):

$p$ is real,

(H2):

$dp(x_{0},\xi_{0})\not=0$,

(H3):

and $p(x_{0},\xi_{0})=0$.

If $P=P(x,hD_{x},h)$ satisfies hypothesis (H1), (H2) and (H3), it is customary
to say that $P$ is of _real principal type_ at the point
$(x_{0},\xi_{0})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ (cf. [7, Definition
3.1]).

The class of real principal type operators enjoys an important property
allowing to reduce a general pdo to a canonical form. This is known as Egorov
theorem, which in the $C^{\infty}$ class states roughly the following (cf. [7,
Proposition 3.1] for instance, among other references as [23, Chapter 8], [8,
Section 62], [19]): If $A=A(x,D_{x})$ and $B=B(x,D_{x})$ are two pdo of real
principal part with the same principal symbol, then there exists elliptic pdo
$R=R(x,D_{x})$ and $S=S(x,D_{x})$ of order zero such that the conjugation $AR-
SB$ is a smoothing operator. Of course, that this theorem is valid for a large
class of pseudo-differential (possibly semiclassical) operators, including
Gevrey symbols (see below for a precise definition). On the other hand, if we
assume further regularity properties on the class of symbols at hand, for
instance Gevrey regularity (see 1.1 for a definition), we may expect to get
more precise information on the canonical transformations and the pseudo-
differential calculus involved. In [10] gave a first Gevrey version, but the
result is not adapted to the semiclassical pdos and in particular to the WKB
asymptotics. In this paper we aim at proving a Gevrey version of Egorov
theorem which is sensitive to a small parameter $h$. We do this by using the
classical WKB approach.

####  1.1.1. Main results: Gevrey WKB expansion and Gevrey Egorov’s theorem

Our main result is a microlocal semiclassical WKB expansion for real principal
part operators in the Gevrey setting compatible with the usual required
asymptotics in the Gevrey framework.

######  Theorem 1 (Gevrey WKB expansion).

Let $n\geq 2$. Let $P=P(x,hD_{x},h)$ be a semiclassical ${\mathcal{G}}^{s}$
pdo of order zero in ${\mathbb{R}}^{n}$ of symbol $p=p(x,\xi)$ and assume that
$P$ is of real principal type at a point
$(x_{0},\xi_{0})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$. Let
$S\subset{\mathbb{R}}^{n}$ be a real ${\mathcal{G}}^{s}-$hypersurface of
${\mathbb{R}}^{n}$ transversal to $H_{p}$ at $(x_{0},\xi_{0})$ . Let
$\varphi\in{\mathcal{G}}^{s}({\mathbb{R}}^{n})$ be given. Assume that

$p(x,\varphi_{x}^{\prime})=0\qquad\textrm{and}\qquad\xi_{0}=\varphi_{x}^{\prime}(x_{0}).$

Under these hypothesis, one may solve the WKB problem near $x_{0}$, i.e.: If
$a_{0}$ and $b$ are given symbols in
${\mathcal{S}}^{0}_{s}({\mathbb{R}}^{n})$, one may find some
$a\in{\mathcal{S}}^{0}_{s}({\mathbb{R}}^{n})$ such that

$\left\\{\begin{array}[]{cc}\frac{1}{h}e^{-i\frac{\varphi}{h}}P\left(ae^{i\frac{\varphi}{h}}\right)-b={\mathcal{O}}_{{\mathcal{G}}^{s}}(h^{\infty}),&\textrm{close
to }x_{0},\\\ a|_{S}=a_{0}.&\end{array}\right.$

The smallness condition in Theorem 1 is defined in (1.3). As mentioned before,
we shall obtain Theorem 1 as a consequence of a particular version of Egorov’s
theorem in the Gevrey setting, according to our next result.

######  Theorem 2 (Gevrey Egorov theorem).

Let $P=P(x,hD_{x},h)$ be a ${\mathcal{G}}^{s}$ pdo with principal symbol
$p=p(x,\xi)$. Assume that $P$ is of real principal type at some point
$(x_{0},\xi_{0})\in T^{*}{\mathbb{R}}^{n}.$ If

$p(x_{0},\xi_{0})=0,\quad\frac{\partial p}{\partial\xi}(x_{0},\xi_{0})\not=0,$

then $P$ is microlocally conjugate to $hD_{x_{1}}$ by ${\mathcal{G}}^{s}$
FIOs.

We give a precise definition of microlocal conjugation in Definition 3.3,
after having introduced the necessary objects for the reader’s convenience.

######  Theorem 3 (Egorov’s theorem in the Gevrey class using FBI transforms).

Let $P=P(y,D_{y})$ be a $G^{s}-$differential operator of degree $m$ such that
near $(y_{0},\eta_{0})\in T^{*}{\mathbb{R}}^{n}\setminus\left\\{0\right\\}$,
the symbol $p(y,\eta)$ is of real principal type. Then, one can find a
$G^{s}$-FBI transform as (5.1) and a canonical transform $\kappa$ such that

$hD_{\mathop{\rm Re}\nolimits
x_{1}}\mathcal{T}u-h^{m}\mathcal{T}P(y,D_{y})u={\mathcal{O}}_{{\mathcal{G}}^{2s-1}}(h^{\infty}),$

with $\kappa(y_{0},\eta_{0})=(x_{0},\xi_{0})$,
$\xi_{0}=\frac{2}{i}\partial_{x}\phi(x_{0})$.

Observe that this theorem is valid for a large class of pseudo-differential
(possibly semiclassical) operators, including Gevrey symbols (see below for a
precise definition).

###  1.2. Strategy and outline

In Section 2 we recall some non-stationary and stationary asymptotics with
complex Gevrey phase and Gevrey symbols and review the symbolic calculus
adapted to Gevrey pdo. We also discuss formal symbols and Carleson’s theorem.

In Section 3 we introduce the class of Fourier integral operators associated
to Gevrey phases and symbols that will be crucial in the proof of our main
theorems.

In Section 4 we prove Theorem 2 by using a microlocal WKB expansion, involving
the usual steps: reduction to an evolution equation, construction of a
suitable phase as a solution to an eikonal equation and finally the
construction of a suitable symbol by imposing a hierarchy of transport
equations.

###  1.3. Notation

According to [11], in this work we shall say that, for $s\geq 1$ a given
function $g$, depending on a small parameter $h>0$, is a
${\mathcal{G}}^{s}$-small remainder, and we note
$g={\mathcal{O}}_{{\mathcal{G}}^{s}}(h^{\infty})$, if there exists $C>0$ such
that for all $\beta\in{\mathbb{N}}^{m}$,

(1.3) $|\partial^{\beta}g|\leq
C^{1+|\beta|}\beta!^{s}\exp\left(-\frac{h^{-\frac{1}{s}}}{C}\right).$

As usual, we use $\mathscr{S}({\mathbb{R}}^{n})$ to denote the space of
Schwartz functions in ${\mathbb{R}}^{n}$.

##  2\. Stationary phase and symbolic calculus in the Gevrey setting

In this section we review the some results concerning the symbolic calculus in
with Gevrey pseudodifferential operators, as some of them are difficult to
find in the literature. We first review the non-stationary and stationary
lemmas for Gevrey phases and symbols. As usual, this allows to give a sense to
the composition of pseudodifferential operators and yield suitable asymptotic
expansions for the resulting symbols. Of course, all the results reviewed
below are classical in the $\mathscr{C}^{\infty}$ framework, according to
classical references as [13, 18, 8, 23], but in the Gevrey context we need to
justify that all small remainders have the particular form given in (1.3),
which sometimes necessitates some modifications with respect to the smooth or
analytic cases.

###  2.1. Non-stationary and Stationary phase method in the Gevrey setting

We consider a symbol $a=a(x,y;h)$ in
${\mathcal{S}}^{m_{0}}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{m})$,
$m_{0}\in{\mathbb{R}}$, for a small parameter $0<h\leq 1$. Let $f=f(x,y)$ be a
phase function of class
${\mathcal{G}}^{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{m})$. Consider

(2.1)
$\mathcal{I}_{f}(h,y)=\int_{{\mathbb{R}}^{n}}e^{\frac{i}{h}f(x,y)}a(x,y;h)\,\mathrm{d}x,\qquad
y\in{\mathbb{R}}^{m},$

for a compactly-supported (in $x$) symbol $a$. The goal of this section is to
adapt the usual non-stationary and stationary phase asymptotics to the Gevrey
setting.

####  2.1.1. Non-stationary phase lemma

The following result is a non-stationary phase lemma adapted to the Gevrey
asymptotics in $h$, which has an independent interest.

######  Lemma 2.1 (Non-stationary phase).

Let $a=a(x,y;h)$ in
${\mathcal{S}}^{m_{0}}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{m})$ be
compactly supported and let $\operatorname{supp}a$ be its support. Assume
$f\in{\mathcal{G}}^{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{m})$ is such that

  1. (1)

$\mathop{\rm Im}\nolimits f(x,y)\geq 0$,

  2. (2)

$f_{x}^{\prime}(x,y)\not=0$ for every $(x,y)\in\operatorname{supp}a$.

Then, $\mathcal{I}_{f}(h,\cdot)$ is a ${\mathcal{G}}^{s}$-small remainder,
i.e., there exists $C>0$ such that for all $\beta\in{\mathbb{N}}^{m}$,

$|\partial_{y}^{\beta}\mathcal{I}_{f}(h,\cdot)|\leq
C^{1+|\beta|}\beta!^{s}\exp\left(-\frac{h^{-\frac{1}{s}}}{C}\right).$

###### Proof.

As usual, as $\operatorname{supp}a$ is compact, upon using a suitable
partition of the unity, we may reduce the result to a purely local situation.
Thanks to the hypothesis on $f$, we may assume that $\operatorname{supp}a$ is
a sufficiently small neighbourhood of a point around which it is possible to
use a diffeomorphism $\kappa$ straightening the phase $f$. Indeed, if
$\operatorname{supp}a$ is sufficiently small, let
$(\tilde{x},\tilde{y})=\kappa(x,y)$ with
$\tilde{x}=(\tilde{x}_{1},\tilde{x}^{\prime})$ and let $\kappa$ be such that

$\tilde{x}_{1}=f(x,y),\qquad\tilde{x}^{\prime}=x^{\prime},\qquad\tilde{y}=y,\qquad\qquad\textrm{in
}\operatorname{supp}a.$

Observe that, as $\mathop{\rm Im}\nolimits f(x,y)\geq 0$ and
$f_{x_{1}}(x,y)\not=0$, this mapping is indeed a local diffeomorphism. Upon
changing variables, we find

$\mathcal{I}_{f}(h,y)=\int_{{\mathbb{R}}^{n}}e^{\frac{i}{h}f(x,y)}a(x,y;h)\,\mathrm{d}x=\int_{{\mathbb{R}}^{n}}e^{\frac{i}{h}\tilde{x}_{1}}\tilde{a}(\tilde{x}_{1},\tilde{x}^{\prime},y;h)\,\mathrm{d}\tilde{x}_{1}\,\mathrm{d}\tilde{x}^{\prime}.$

Now, taking derivatives of any order with respect to $y$, the usual non-
stationary phase lemma gives exponential decay on $h$ and the result follows
(cf. for instance [23, Lemma 3.14]).

∎

####  2.1.2. Stationary phase lemma

We turn now to the stationary phase asymptotics in the Gevrey setting. Upon
using a suitable partition of unity, we may consider a purely local situation.
Let us pick a point $(x_{0},y_{0})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}$
such that the following

(2.2) $\displaystyle f\textrm{ is real-valued},$ (2.3) $\displaystyle
f_{x}^{\prime}(x_{0},y_{0})=0,$ (2.4)
$\displaystyle\det\left(f_{xx}^{\prime\prime}(x_{0},y_{0})\right)\not=0,$

hold. Consider $(x,y)$ sufficiently close to $(x_{0},y_{0})$ in
${\mathbb{R}}^{n}\times{\mathbb{R}}^{m}$. Then, there exists a function
$y\mapsto\chi(y)$ of class ${\mathcal{G}}^{s}$ in a neighbourhood of $y_{0}$
such that

(2.5) $f_{x}^{\prime}(\chi(y),y)=0,\qquad\chi(y_{0})=x_{0}.$

We have the following stationary phase asymptotics for locally defined symbols
that will be used in Proposition 2.3.

######  Lemma 2.2 (Stationary phase).

Assume that $\operatorname{supp}a$ is close to $x_{0}$ so that (2.5), (2.2),
(2.3) and (2.4) hold. Then, the symbol

(2.6) $b(y,h)=e^{-(i/h)f(\chi(y),y)}\int a(x,y;h)e^{(i/h)f(x,y)}\,\mathrm{d}x$

is of class ${\mathcal{G}}^{s}$ near $y_{0}$ and enjoys the asymptotic
expansion

(2.7) $b(y,h)\sim\left|\det\left(\frac{1}{2i\pi
h}f_{xx}^{\prime\prime}(\chi(y),y)\right)\right|^{-\frac{1}{2}}e^{i\pi\frac{\sigma}{4}}\sum_{j\geq
0}h^{j}L_{f,j,y}[a](\chi(y),y),$

where each $L_{f,j,y}[a]$ is a differential operator of degree $2j$ in $x$ and
$\sigma$ is the signature of the matrix $D^{2}_{x}f(x_{0},y_{0})$.

###### Proof.

Since $f$ is real-valued, we can use Morse lemma. By Taylor expansion of order
$2$ around $(x_{0},y_{0})$ we find

$f(x,y)=f(\chi(y),y)+\frac{1}{2}\langle Q(x,y)(x-\chi(y)),x-\chi(y)\rangle,$

whenever $(x,y)$ is close to $(x_{0},y_{0})$, where

$Q(x,y)=2\int_{0}^{1}(1-t)f_{xx}^{\prime\prime}\left(\chi(y)+t(x-\chi(y)),y\right)\,\mathrm{d}t.$

For $(x,y)$ close to $(x_{0},y_{0})$, $Q(x,y)$ is symmetric and invertible.
Setting

$Q_{0}=f_{xx}^{\prime\prime}(x_{0},y_{0}),$

we observe there is a map $(x,y)\mapsto R(x,y)$ such that

$Q_{0}=R(x,y)^{t}Q(x,y)R(x,y),\qquad\textrm{ with }R(x_{0},y_{0})=Id.$

Consider the substitution

$(x,y)\mapsto(\tilde{x}(x,y),y),$

with

$\tilde{x}(x,y)=R^{-1}(x,y)(x-\chi(y)),\qquad\textrm{near }(x_{0},y_{0}).$

Write

$b(y,w)=e^{-\frac{i}{h}f(\chi(y),y)}I(y,\omega)=\int e^{\frac{i}{2h}\langle
Q_{0}\tilde{x},\tilde{x}\rangle}\tilde{a}(\tilde{x},y,\omega)\,\mathrm{d}\tilde{x},$

for a ${\mathcal{G}}^{s}$ symbol $\tilde{a}$ with
$\tilde{x}-\operatorname{supp}$ close to zero. Next, using Plancherel’s
formula

$b(y,w)=|\det(\frac{1}{2i\pi h}Q_{0})|^{-\frac{1}{2}}e^{-\frac{ih}{2}\langle
Q_{0}^{-1}D,D\rangle}\tilde{a}(0,y).$

Now, the asymptotic expansion (2.7) follows from [13, 7.6.7] as a consequence
of the usual asymptotics for quadratic phases with nonnegative real part.

∎

###  2.2. Symbolic calculus with Gevrey symbols

The calculus of Gevrey PDO has been established previously. Let
$Q={\operatorname{Op}_{h}}(q)$, $A={\operatorname{Op}_{h}}(a)$,
$q\in{\mathcal{S}}^{m_{0}}_{s}$, $a\in{\mathcal{S}}^{m}_{s}$. The composition
$Q\circ A={\operatorname{Op}_{h}}(q\circ a)$ can be written

(2.8) $(q\circ a)(x,\xi)=\frac{1}{(2\pi h)^{n}}\iint
e^{-\frac{i}{h}(x-y)\cdot(\xi-\eta)}q(x,\eta)a(y,\xi)\,\mathrm{d}y\,\mathrm{d}\eta.$

######  Proposition 2.3.

Let $Q={\operatorname{Op}_{h}}(q)$, $A={\operatorname{Op}_{h}}(a)$,
$q\in{\mathcal{S}}^{m_{0}}_{s}$, $a\in{\mathcal{S}}^{m}_{s}$ be given. Then,
the composition $Q\circ A={\operatorname{Op}_{h}}(q\circ a)$ satisfies, for
arbitrary $N\in{\mathbb{N}}$,

(2.9) $(q\circ
a)(x,\xi)=\sum_{|\alpha|<N}\frac{h^{|\alpha|}}{\alpha!}D_{\xi}^{\alpha}q(x,\xi)\partial_{x}^{\alpha}a(x,\xi)+r_{N}(q,a)(x,\xi),$

where $r_{N}(q,a)$ is a ${\mathcal{G}}^{s}$ symbol of order $m_{0}+m-N$ given
by

$r_{N}(q,a)(x,\xi)=\frac{h^{N}}{i(2\pi
h)^{n}}\frac{1}{i^{N}}\sum_{|\alpha|=N}\int_{0}^{1}\frac{(1-\theta)^{(N-1)}}{(N-1)!}\iint_{{\mathbb{R}}^{2n}}e^{\frac{y\cdot\eta}{ih}}(\partial_{\xi}^{\alpha}q)(x,\xi+\eta)\partial_{x}^{\alpha}a(x+\theta
y,\xi)\,\mathrm{d}\theta\,\mathrm{d}y\,\mathrm{d}\eta.$

We shall use (2.9) with $N=1$ in the following sections. The proof relies on
the following Lemma, which folllows the lines in [18, Lemmas 4.1.2 and 4.1.5].

######  Lemma 2.4.

For every $t\in{\mathbb{R}}^{*}$, let $J^{t}$ be the operator defined by

$J^{t}b(x,\xi)=|t|^{-n}\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}b(x+z,\xi+\zeta)e^{\frac{z\cdot\xi}{it}}\frac{\,\mathrm{d}z\,\mathrm{d}\zeta}{(2\pi)^{n}},\qquad
b\in{\mathcal{S}}^{m}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}).$

Then, $J^{t}$ maps
${\mathcal{S}}_{s}^{m^{\prime}}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ into
itself. Moreover, for any $N\in{\mathbb{N}}$,

$(J^{t}b)(x,\xi)=\sum_{|\alpha|<N}\frac{t^{|\alpha|}}{\alpha!}D_{\xi}^{\alpha}\partial_{x}^{\alpha}b(x,\xi)+r_{N}(t)(x,\xi)$

with $r_{N}(t)\in{\mathcal{S}}^{m-N}_{s}$ and

$r_{N}(t)(x,\xi)=t^{N}\int_{0}^{1}\frac{(1-\theta)^{N-1}}{(N-1)!}J^{\theta
t}\left((D_{\xi}\cdot\partial_{x})^{N}b\right)(x,\xi)\,\mathrm{d}\theta.$

###### Proof.

Let $N\in{\mathbb{N}}$. By Taylor’s expansion up to order $N$ in the
exponential we have

$J^{t}=\sum_{k<N}\frac{t^{k}}{k!}(D_{\xi}\partial_{x})^{k}+\int_{0}^{1}\frac{(1-\theta)^{N-1}}{(N-1)!}J^{\theta
t}\left((tD_{\xi}\partial_{x})^{N}\right)\,\mathrm{d}\theta.$

Hence, $J^{t}$ maps
${\mathcal{S}}_{s}^{m^{\prime}}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ into
itself and one has

$(J^{t}b)(x,\xi)=\sum_{|\alpha|<N}\frac{t^{|\alpha|}}{\alpha!}(D_{\xi}^{\alpha}\partial_{x}^{\alpha}b(x,\xi)+r_{N}(t)(x,\xi)$

with

$r_{N}(t)(x,\xi)=t^{N}\int_{0}^{1}\frac{(1-\theta)^{N-1}}{(N-1)!}J^{\theta
t}\left((tD_{\xi}\partial_{x})^{N}b\right)(x,\xi)\,\mathrm{d}\theta.$

∎

###### Proof of Proposition 2.3.

From (2.8), we may write

$(q\circ a)(x,\xi)=\frac{1}{(2\pi h)^{n}}\iint
e^{-\frac{i}{h}z\cdot\zeta}q(x,\xi+\zeta)a(x+z,\xi)\,\mathrm{d}z\,\mathrm{d}\zeta,$

thanks to the changes of variables $x+z=y$ and $\xi-\eta=\zeta$. For
$t\in{\mathbb{R}}^{*}$ set

$J_{0}^{t}b=\frac{1}{(2\pi|t|)^{n}}\iint
b(z,\zeta)e^{\frac{1}{it}z\cdot\zeta}\,\mathrm{d}z\,\mathrm{d}\zeta,\qquad
b=b(z,\zeta)\in{\mathcal{S}}^{m^{\prime}}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}).$

Then, setting $C_{x,\xi}=q(x,\xi+\zeta)a(x+\zeta,\xi)$ one has

$(q\circ a)(x,\xi)=J_{0}^{h}C_{x,\xi}.$

So thanks to Lemma 2.4 we may write

$(q\circ
a)(x,\xi)=J^{h}C_{x,\xi}(0,0)=\sum_{|\alpha|<N}\frac{h^{|\alpha|}}{\alpha!}D_{\xi}^{\alpha}q(x,\xi)\partial_{x}^{\alpha}a(x,\xi)+r_{N}(h)(q,a)(x,\xi)$

where

$r_{N}(h)(q,a)(x,\xi)=\int_{0}^{1}\frac{(1-\theta)^{N-1}}{(N-1)!}J^{\theta
h}\left((tD_{\xi}\partial_{x})^{N}C_{x,\xi}(0,0)\right)\,\mathrm{d}\theta.$

Now, expanding the remainder

$r_{N}(q,a)(x,\xi)=\frac{1}{(2\pi
h)^{n}}\sum_{|\gamma|=N}\frac{h^{N}N!}{\gamma!}\int_{0}^{1}\iint_{{\mathbb{R}}^{2n}}\frac{(1-\theta)^{N-1}}{(N-1)!}e^{\frac{z\cdot\zeta}{ih}}D_{\xi}^{\gamma}q(x,\xi+\zeta)\partial_{x}^{\gamma}a(x+\theta
z,\xi)\,\mathrm{d}z\,\mathrm{d}\zeta\,\mathrm{d}\theta.$

Using Gevrey stationary phase lemma one may prove that $r_{N}(q,a)$ is a
Gevrey symbol of order $m_{0}+m-N$ and thus,

$(q\circ
a)(x,\xi)=\sum_{|\alpha|<N}\frac{h^{|\alpha|}}{\alpha!}D_{\xi}^{\alpha}q(x,\xi)\partial_{x}^{\alpha}a(x,\xi)+r_{N}(q,a)(x,\xi),$

which ends the proof.

∎

###  2.3. Formal symbols and Borel lemma in the Gevrey setting

In this section we introduce a notion of Gevrey formal symbol of a given order
and give conditions ensuring the existence of a Gevrey symbol realizing a
given sequence of formal symbols, based on Carleson’s theorem [3].

######  Definition 2.5 (Formal symbols).

Let $m\in{\mathbb{R}}$. Let $(a_{j})_{j\geq 0}$ be a sequence of symbols such
that

$a_{j}\in{\mathcal{S}}^{-j+m}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}),\qquad\forall
j\in{\mathbb{N}}.$

The sequence of $(a_{j})_{j\geq 0}$ is said to be a formal ${\mathcal{G}}^{s}$
symbol of degree $m$ if there exists $C>0$ such that

(2.10) $|\partial_{x}^{\alpha}\partial_{\theta}^{\beta}a_{j}(x,\theta)|\leq
C^{1+|\alpha|+|\beta|+j}j!^{s}\alpha!^{s}\beta!^{s}h^{j-m},\qquad\forall(x,\theta)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n},$

for all $\alpha,\beta\in{\mathbb{N}}^{n}$ and $j\in{\mathbb{N}}$.

Carleson’s theorem [3] allows us to realise Gevrey $s$ formal symbols into
symbols. This result is a Gevrey Borel theorem as follows.

######  Theorem 4.

Let $m\in{\mathbb{R}}$ and let $(a_{j})_{j\geq 0}\in{\mathcal{S}}^{-j+m}$ be a
formal ${\mathcal{G}}^{s}$ symbol of degree $m$. Then, there exists
$a\in{\mathcal{S}}_{s}^{m}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ such that
for any $N>0$, $\alpha,\beta\in{\mathbb{N}}^{n}$ one has uniformly

$|\partial_{x}^{\alpha}\partial_{\theta}^{\beta}(a-\sum_{j<N}a_{j})|\leq
C^{1+\alpha+\beta+N}N!^{s}\alpha!^{s}\beta!^{s}h^{N-m}.$

###### Proof.

Consider the sequence $(a_{j}(x,\theta,h)h^{-j}j!)_{j\geq 0}$. This is a
$(s+1)$-sequence in the Boutet-Kree sense, according to [2]. Using Carleson
theorem we have that there exists
$g(t,x,\theta,h)\in{\mathcal{G}}^{s+1}(\overline{R}_{+},{\mathcal{S}}_{s}^{m})$
such that for all $j\geq 0$

$\partial_{t}^{j}g(0,x,\eta,h)=j!a_{j}(x,\theta,h)h^{-j}.$

Setting $a(x,\theta,h)=g(t,x,\theta,h)|_{t=h>0}$ for $h$ small, by Taylor’s
formula

$|\partial_{x}^{\alpha}\partial_{\theta}^{\beta}(a-\sum_{j<N}a_{j})|\leq\sup_{\theta^{\prime}\in[0,1]}||\partial_{x}^{\alpha}\partial_{\theta}^{\beta}\partial_{t}^{j}g(\theta^{\prime}h,x,\theta,h)|\frac{h^{N}}{N!}.$

As $g$ is a function in ${\mathcal{G}}^{s+1}$ function w.r.t. $t$ and in
${\mathcal{G}}^{s}$ w.r.t. $(x,\theta)$, one has for some $C^{\prime}>0$

$|\partial_{x}^{\alpha}\partial_{\theta}^{\beta}(a-\sum_{j<N}a_{j})|\leq
C^{\prime 1+|\alpha|+|\beta|+}N!^{s}\alpha!^{s}\beta!^{s}h^{N-m},$

so the theorem is proven. ∎

###  2.4. Formal quasinorms in the class of Gevrey symbols

In the set ${\mathcal{S}}_{s}^{m}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ we
may define, following [15], a family of quasinorms (cf. [2]) of the form :

(2.11)
$N_{m}(a,T)(x,\theta)=\sum_{(\alpha,\beta)\in{\mathbb{N}}^{d}}\frac{h^{m}T^{|\alpha|+|\beta|}}{\alpha!^{s}\beta!^{s}}|\partial_{x}^{\alpha}\partial_{\theta}^{\beta}a(x,\theta)|,$

where $T>0$ is a fixed parameter. We also set

$\overline{N_{m}}(a,T):=\sup_{(x,\theta)}N_{m}(a,T)(x,\theta).$

By Leibniz rule, one has

$\overline{N}_{m+m^{\prime}}(aa^{\prime},T)\leq\overline{N}_{m}(a,T)\overline{N}_{m^{\prime}}(a^{\prime},T),$

for any $a\in{\mathcal{S}}_{s}^{m}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$
and
$a^{\prime}\in{\mathcal{S}}_{s}^{m^{\prime}}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$.

These quasinorms will be important in the proof of Theorem 2.

##  3\. Fourier Integral Operators in the Gevrey setting

In this section we present some basic results concerning the specific class of
Fourier Integral Operators (FIO for short) associated to Gevrey symbols and
classes and how they relate to the class of Gevrey pdo.

###  3.1. FIO with Grevey symbols and phases

We work with a local class of Gevrey semiclassical Fourier integral operator
defined in an analogous way to [8, 61.1] in the $h$-independent case and also
to [23] for the $h$ dependent case in the smooth class. These are operators of
the form

(3.1) $Fu(x,h)=\frac{1}{(2\pi
h)^{n}}\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}e^{\frac{i}{h}(S(x,\eta)-y\cdot\eta)}a(x,\eta)u(y,h)\,\mathrm{d}y\,\mathrm{d}\eta,$

where the phase function
$S=S(x,\eta)\in{\mathcal{G}}^{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ is
such that $\det S^{\prime\prime}_{x,\eta}\not=0$ and the symbol $a=a(x,\eta)$
is in some class
$a\in{\mathcal{S}}^{m}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$. We will
sometimes use the notation $F\in\mathcal{I}^{m}_{s}({\mathbb{R}}^{n})$ for
such an operator.

For a FIO $F$ as above, define (cf. [8, Section 61.1]) its adjoint $F^{*}$ by

$\langle Fu,v\rangle=\langle u,F^{*}v\rangle,$

so that

$F^{*}u(x,h)=\frac{1}{(2\pi
h)^{n}}\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}e^{\frac{i}{h}(S(y,\eta)-x\cdot\eta))}\overline{a(y,\eta)}u(y,h)\,\mathrm{d}y\,\mathrm{d}\eta.$

Then, we have a Gevrey version of [8, Lemma 62.4].

######  Lemma 3.1.

Let $F\in\mathcal{I}^{m}_{s}$ and let $F^{*}$ be its adjoint. Then,

$FF^{*}u(x,h)=\frac{1}{(2\pi
h)^{n}}\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}e^{\frac{i}{h}(S(x,\xi)-S(y,\xi))}a(x,\xi)\overline{a(y,\eta)}u(y,h)\,\mathrm{d}y\,\mathrm{d}\xi,\qquad\forall
u\in\mathscr{D}^{\prime}({\mathbb{R}}^{n}),$

we can write

$FF^{*}=K_{1}+K_{2},$

where $K_{1}\in\Psi^{2m}_{s}$ and $K_{2}={\mathcal{O}}_{s}(h^{\infty})$.

###### Proof.

Using a Gevrey$-s$ partition of the unity we may assume that
$\operatorname{supp}(a)$ is small enough so that the mapping

$\Sigma:(y,\eta)\mapsto(x,\xi),\qquad\xi=S^{\prime}_{x}(x,\eta),\quad
y=S^{\prime}_{\eta}(x,\eta)$

is a canonical transformation on $\operatorname{supp}(a)$ of class
${\mathcal{G}}^{s}$. Let $\chi\in{\mathcal{G}}^{s}_{0}({\mathbb{R}}^{n})$ and
$r>0$ small enough so that $supp(\chi)\subset B(0,r)$ and $\chi=1$ on
$B(0,\frac{r}{2})$. We can write

$S(x,\xi)-S(y,\xi)=\Sigma(x,y,\xi)(x-y).$

Then, we have

$FF^{*}=K_{1}+K_{2}$

with

(3.2) $\displaystyle K_{1}u(x,h)$
$\displaystyle=\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}\mathfrak{a}(x,y,\eta)\chi\left(\frac{x-y}{\delta}\right)e^{\frac{i}{h}(x-y)\cdot\eta}|J(x,y,\eta)|u(y,h)\frac{\,\mathrm{d}y\,\mathrm{d}\xi}{(2\pi
h)^{n}},$ (3.3) $\displaystyle K_{2}u(x,h)$
$\displaystyle=\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}\mathfrak{a}(x,y,\eta)\left(1-\chi\left(\frac{x-y}{\delta}\right)\right)e^{\frac{i}{h}(x-y)\cdot\eta}|J(x,y,\eta)|u(y,h)\frac{\,\mathrm{d}y\,\mathrm{d}\xi}{(2\pi
h)^{n}},$

where

(3.4)
$\mathfrak{a}(x,y,\eta):=a(x,\Sigma^{-1}(x,y,\eta))\overline{a(y,\Sigma^{-1}(x,y,\eta))},\qquad
J=\partial_{\eta}\Sigma^{-1}.$

Using Kuranishi’s trick (cf. [23, Chapter 8]), we deduce that $K_{1}$ is a
${\mathcal{G}}^{s}$ PDO of order $2m$.

Next, using Gevrey non stationary phase we obtain that $K_{2}$ is a Gevrey $s$
negligeable reminder, while it is easy to compute the principal symbol of
$K_{1}$, which is of order $2m$.

∎

###  3.2. Composition of FIO and PDOs in the Gevrey setting

The following result is an adaptation of [8, Lemmas 62.2 and 62.3] to the
Gevrey stting, which is possible with only minor modifications thanks to Lemma
2.1.

######  Proposition 3.2.

Let $F$ be a ${\mathcal{G}}^{s}$-FIO of the form (3.1) for some phase
$S\in{\mathcal{G}}^{s}$ and symbol $a\in{\mathcal{S}}^{m_{0}}_{s}$. Then, for
every $A=A(x,D)\in\Psi_{s}^{m}$ one has

(3.5) $AFu(x)=\frac{1}{(2\pi h)^{n}}\int
c(x,\eta)e^{\frac{i}{h}S(x,\eta)}\hat{u}(\eta)\,\mathrm{d}\eta,\qquad
u\in\mathscr{D}^{\prime}({\mathbb{R}}^{n}),$

for some $c\in{\mathcal{S}}_{s}^{m+m_{0}}$ whose principal symbol satisfies

$c_{0}(x,\eta)=a_{0}(x,S^{\prime}_{x}(x,\eta))a(x,\eta),$

where $a_{0}$ is the principal symbol of $A$. Also,

(3.6) $FAu(x)=\frac{1}{(2\pi
h)^{n}}\int_{{\mathbb{R}}^{n}}c^{\prime}(x,\eta)e^{\frac{i}{h}S(x,\eta)}\hat{u}(\eta)\,\mathrm{d}\eta+R^{\prime}u,\qquad
u\in\mathscr{D}^{\prime}({\mathbb{R}}^{n}),$

for some $c^{\prime}\in{\mathcal{S}}^{m+m_{0}}$ whose principal symbol
satisfies

(3.7) $c_{0}^{\prime}(x,\eta)=a_{0}S^{\prime}_{\eta}(x,\eta)a(x,\eta),$

and $R^{\prime}\in\Psi_{s}^{-\infty}$.

###  3.3. Gevrey wavefront and Lagrangian distributions

We end this section with some details on the wavefront set adapted to the
Gevrey setting.

######  Definition 3.3 (Microlocal conjugation by Gevrey FIOs).

Given $P,Q\in\Psi^{m}_{s}$, we say that $P$ and $Q$ are microlocal conjugate
in the ${\mathcal{G}}^{s}$ sense around a point $(x_{0},\xi_{0})\in
T^{*}{\mathbb{R}}^{n}$ if there exists a FIO $R\in\mathcal{I}_{s}^{m}$ such
that $RP$ equals $QR$ microlocally around $(x_{0},\xi_{0})$.

Consider a Lagrangian distribution of the form

(3.8) $u(x,h)=\int
e^{\frac{i}{h}\phi(x,\theta)}a(x,\theta,h)\,\mathrm{d}\theta$

for a phase $\phi$ and amplitude $a$ satisfying

$\displaystyle\phi\in{\mathcal{G}}^{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{N}),$
$\displaystyle\mathop{\rm Im}\nolimits\phi\geq 0,\qquad d\phi\not=0,$
$\displaystyle
a\in{\mathcal{S}}^{m_{0}}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{N}).$

Assume $\phi$ homogeneous of degree 1 in $\theta$ and
$a\in{\mathcal{S}}^{m_{0},k_{s}}$ or
$a\in{\mathcal{G}}^{s}_{0}\cap{\mathcal{S}}^{m_{0}}_{s}$ (see [15])

If $\operatorname{WF}_{h}(u)$ is the semiclassical wavefront of
$(u(\cdot,h))_{h}$, then it is standard that

$\operatorname{WF}_{h}(u)\subset\left\\{(x,\phi_{x}^{\prime})\,|\quad(x,\theta)\in
F,\,\phi_{\theta}^{\prime}(x,\theta)=0\right\\},$

with $supp(a)\subset F$. The next result is a analogue version of this fact
for the Gevrey semiclassical wavefront $\operatorname{WF}_{s,h}(u)$.

######  Lemma 3.4.

Let $u$ as in (3.8). Then, one has

$\operatorname{WF}_{s,h}(u)\subset\Lambda=\left\\{(x,\phi_{x}^{\prime})\,|\quad(x,\theta)\in
F,\,\phi_{\theta}^{\prime}(x,\theta)=0\right\\},$

###### Proof.

This is a consequence of the results in [13, Ch. 25] with minor modifications.
The Gevrey asymptotic comes from the non-stationary phase lemma in Lemma 2.1.

∎

##  4\. Proof of Theorem 2

The goal of this section is to prove Theorem 2. We divide the proof into
several steps, according to the usual WKB method.

Step 1. Setting of the problem.

We may assume w.l.g. that $(x_{0},\xi_{0})=(0,0)$. Let us choose once for all
a particular real coordinate and write as usual
$x=(x_{1},x^{\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{n-1}$ and
$\xi=(\xi_{1},\xi^{\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{n-1}$. As $P$ is
of real principal type by hypothesis, we can assume that the principal symbol
of $P$ writes (cf. for instance [23, Section 12.2])

(4.1) $p(x,\xi_{1},\xi^{\prime})=\xi_{1}-\lambda(x,\xi^{\prime}),$

for some real symbol
$\lambda\in{\mathcal{S}}_{s}^{0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n-1})$
real. If $\lambda(x,\xi^{\prime})$ stands for the principal symbol of $Q$ let
us write

(4.2) $P(x,hD_{x_{1}},hD_{x^{\prime}})=hD_{x_{1}}+Q(x,hD_{x^{\prime}}).$

Step 2. Suitable phase using the eikonal equation.

We consider now the following Cauchy problem: Find $\varphi$ such that

(4.3) $\left\\{\begin{array}[]{l}\frac{\partial\varphi}{\partial
x_{1}}-\lambda\left(x,\varphi_{x}^{\prime}\right)=0,\\\
\varphi|_{x_{1}=0}=x^{\prime}\cdot\eta^{\prime},\end{array}\right.$

where
$\lambda\in{\mathcal{S}}_{s}^{0}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n-1})$
given by (4.1). The Cauchy problem (4.3) has a solution $\varphi$ of class
${\mathcal{G}}^{s}$. Moreover, since
$\det\varphi_{x^{\prime},\xi^{\prime}}^{\prime\prime}\not=0$ the phase
function $\varphi$ generates a canonical transform.

Our goal in the following sections is to find suitable $h$-FIOs $F,G$ such
that

(4.4) $GPF=hD_{x_{1}}+{\mathcal{O}}_{s}(h^{\infty}),$

Step 3. Reducing the order of the remainder by conjugation.

Let us introduce a $h-$FIO of the form

$Fu(x,h)=\frac{1}{(2\pi h)^{n-1}}\int
e^{\frac{i}{h}\varphi(x,\eta^{\prime})}a(x,\eta^{\prime})\hat{u}(x_{1},\frac{\eta^{\prime}}{h})d\eta^{\prime},\qquad
u\in\mathscr{S}({\mathbb{R}}^{n}),$

where $a\in{\mathcal{S}}_{s}^{\mu}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n-1})$,
$\hat{u}$ denotes the partial Fourier transform in the variables $x^{\prime}$
only and $\varphi$ is given as before. Assume that $a$ is elliptic when
$x_{1}=0,x^{\prime}=0,\xi_{1}=0,\xi^{\prime}=0$.

Now, $F$ defined above is a h-FIO of the form (3.1) associated to a
${\mathcal{G}}^{s}$ phase and symbol. As $P\in\Psi_{s}^{0}$ by hypothesis, we
can compose $F$ and $P$ using Proposition 3.2. Indeed, from (4.2) we get, for
every $u\in\mathscr{S}({\mathbb{R}}^{n})$,

$PFu=(hD_{x_{1}}+Q)Fu=hD_{x_{1}}Fu+QFu$

Now, as the symbol $a$ is supposed to be elliptic at the point
$(x_{1},x^{\prime},\xi_{1},\xi^{\prime})=0$, we may find a microlocal inverse
of $F$ near this point and thus write

$PFu=F(hD_{x_{1}})+F_{-1}u,$

for some $F_{-1}\in I_{s}^{\mu-1}$. Moreover, the symbol of $F_{-1}$, namely
$\mathfrak{f}$, admits the expansion (as in [13, Theorem 7.7.7]):

$\mathfrak{f}=\sum_{\alpha\geq
0}\frac{h^{|\alpha|}}{\alpha!}p^{(\alpha)}(x,\varphi_{x}^{\prime})D_{y}^{\alpha}(e^{\frac{i}{h}\rho}a)|_{y=x},$

with

$\rho(x,y,\eta^{\prime})=\varphi(y,\eta^{\prime})-\varphi(x,\eta^{\prime})-(y-x)\varphi_{x}^{\prime}(x,\eta^{\prime}).$

We observe that the first term of the expansion vanishes, for

$p(x,\varphi_{x}^{\prime})(e^{\frac{i}{h}\rho}a)|_{y=x}=p(x,\varphi_{x}^{\prime})(e^{\frac{i}{h}(\varphi(y,\eta^{\prime})-\varphi(x,\eta^{\prime})-(y-x))}a)|_{y=x}=hD_{x_{1}}\varphi-\lambda(x,\varphi_{x^{\prime}}^{\prime})=0,$

thanks to (4.3). Hence, $F_{-1}\in\Psi_{s}^{-1}$.

Let now $G$ be the microlocal inverse of the elliptic $h$-FIO $F$ near zero.
Then, we can write

$GF=I+r,\qquad\textrm{microlocally near }(0,0,0,0),$

for $r\in\Psi_{s}^{-\infty}$ and hence

$GPFu=GF(hD_{x_{1}})+GF_{-1}=hD_{x_{1}}+R,$

where $R=rhD_{x_{1}}+GF_{-1}$. Now, as $G$ and $F_{-1}$ are associated to the
same generating function, we have in particular that $GF_{-1}$ is a pdo and
moreover $GF_{-1}\in\Psi_{s}^{-1}$. Furthermore, as $r$ is a regularising pdo,
we also have $rhD_{x_{1}}\in\Psi_{s}^{-1}$, which guarantees that

$GPFu=hD_{x_{1}}+R,\qquad R\in\Psi_{s}^{-1},$

microlocally near zero. This means that $P$ is microlocally conjugate by
${\mathcal{G}}^{s}$ FIOs to $hD_{x_{1}}$ up to a remainder of degree $-1$.

Step 4. Reducing arbitrarily the order of the remainder by conjugation.

We aim now to iterate the scheme of the previous step in order to get (4.4)
for a remainder of arbitrary order. We do this thanks to the following
iterative scheme, which produces a formal Gevrey symbol.

######  Lemma 4.1.

Let $q\in{\mathcal{S}}^{-1}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$.
Then, there is a sequence $(a_{j})_{j\geq 0}$ of symbols such that
$a_{j}\in{\mathcal{S}}^{-j}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$
satisfying

$\frac{h}{i}\partial_{x_{1}}a_{0}+qa_{0}=0,\qquad\textrm{and}\qquad
a_{0}|_{x_{1}=0}\textrm{ is elliptic},$

and for all $j\geq 1$,

(4.5)
$\frac{h}{i}\partial_{x_{1}}a_{j}+qa_{j}=r_{1}(q,a_{j-1}),\qquad\textrm{and}\qquad
a_{j}|_{x_{1}=0}=0.$

Moreover, if

$A_{j}={\operatorname{Op}_{h}}(a_{j}),\qquad
A^{(N)}=A_{0}+\dots+A_{N-1},\,\,N\geq 1,$

one has

$(hD_{x_{1}}+Q)A^{(N)}=hA^{(N)}D_{x_{1}}+R_{N},$

with $R_{N}$ a ${\mathcal{G}}^{s}\in\Psi_{s}^{-(N+1)}$. Finally, if $q$ is of
order $m_{0}$ small enough the above sequence of $(a_{j})_{j\geq 0}$ is a
formal Gevrey $s$ symbol in the sense of Defintion 2.10.

###### Proof.

Equations (4.5) are solved by induction setting for $j\geq 1$

$a_{j}=-i\frac{a_{0}}{h}\int_{0}^{x_{1}}\frac{r_{1}}{a_{0}}(q,a_{j-1})\,\mathrm{d}t.$

One has
$a_{j}\in{\mathcal{S}}_{s}^{-j}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, in
view of (2.9) and (4.5) we have $R_{N}$ is of order $-N-1$.

We now show that if $q$ is of order $m_{0}$ small enough the above sequence of
$(a_{j})_{j\geq 0}$ is a formal Gevrey $s$ symbol.

In order to prove (2.10) we use the formal quasinorms for Gevrey symbols
introduced in (2.11). Upon repeating the scheme of the previous step, we may
assume that $P=hD_{1}+Q$, with $Q\in\Psi_{s}^{m_{0}}$ of order $m_{0}$ small,
is a microlocal conjugate to $hD_{x_{1}}$ by ${\mathcal{G}}^{s}$ FIOs, modulo
negligeable remainders. If $q$ is the symbol of $Q$, recall that

${\operatorname{Op}_{h}}(q)\circ{\operatorname{Op}_{h}}(a)={\operatorname{Op}_{h}}(qa+R),\qquad
R=R(a,q)$

with a remainder $R$ as in Lemma 2.3. Let us define next
$\chi\in{\mathcal{G}}^{s}_{0}({\mathbb{R}}^{2n})$ such that

$\operatorname{supp}\chi\subseteq\left\\{(y,\eta);\,|y|+|\eta|\leq
r\right\\}\qquad\textrm{ and }\qquad\chi=1\textrm{ on
}\left\\{(y,\eta);\,|y|+|\eta|\leq\frac{r}{2}\right\\}$

and let us write

$R(a,q)=R_{\chi}(a,q)+S_{\chi},\qquad R_{\chi}=\chi R(a,q),\quad
S_{\chi}=(1-\chi)R(a,q).$

We deduce that $S_{\chi}$ is Gevrey $s$ remainder by the non-stationary phase
asymptotics in Lemma 2.1. Now if $\alpha,\beta\in{\mathbb{N}}^{n}$, if
${\epsilon}>0$ small enough, we also have

(4.6)
$\overline{N_{m}}(\partial_{x}^{\alpha}\partial_{\theta}^{\alpha}a,T)\leq
T^{-(|\alpha|+|\beta|)}{\epsilon}^{-s(|\alpha|+|\beta|)}\alpha!^{s}\beta!^{s}\overline{N_{m}}(a,T(1+{\epsilon})^{s}).$

Now, if
$q\in{\mathcal{S}}^{m_{0}}_{s}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ with
$m_{0}\leq-n$, then from the previous equations,

$\overline{N}_{m-1}(R_{\chi}(a),T)\leq\tau^{2n}C_{n}{\epsilon}^{-s}M_{0}(q)T^{-1}\overline{N}_{m}(a,T(1+{\epsilon})^{s}).$

If $(a_{j})_{j\geq 0}$ is the sequence defined by (4.5). We claim that
estimate (4.6) implies, for $j\geq 0$, ${\epsilon}>0$ small that for some
$M_{0}^{\prime}=M_{0}^{\prime}(r,q,n)>0$ one has

(4.7) $\overline{N}_{-j}(a_{j},T)\leq M_{0}^{\prime
j}{\epsilon}^{-sj}j^{sj}\overline{N}_{0}(a_{0},2T)T^{-j}.$

Indeed, this is true if $j=1$ from (4.6). Assuming that (4.7) holds for $j-1$
with $j\geq 2$, we have for ${\epsilon}$, $\tilde{{\epsilon}}>0$ small

$\overline{N}_{-j}(a_{j},T)\leq{\epsilon}^{-sj}\frac{M_{0}^{\prime
j}}{T^{j}}\frac{(j-1)^{s(j-1)}}{(1+{\epsilon})^{s(j-1)}\tilde{{\epsilon}}^{s(j-1)}}\overline{N}_{0}\left(a_{0},T(1+{\epsilon})^{s}\left(1+\frac{\tilde{{\epsilon}}}{j-1}\right)^{s(j-1)}\right).$

We fix $\delta>0$ small and choose
$\frac{\tilde{{\epsilon}}}{j-1}=\frac{\delta}{j}$ and
${\epsilon}=\frac{\delta}{j}$ in the previous inequality, so that

$(1+{\epsilon})^{s}\left(1+\frac{\tilde{{\epsilon}}}{j-1}\right)^{s(j-1)}=\left(1+\frac{\delta}{j}\right)^{sj}.$

It remains to check that

$\frac{(1+{\epsilon})^{s(j-1)}}{(j-1)^{s(j-1)}}\tilde{{\epsilon}}^{-s(j-1)}\delta^{sj}j^{?sj}\leq\frac{\delta^{s}}{j^{s}},$

which is $(1+\frac{\delta}{j})^{-s(j-1)}\leq 1$, which is true since $j\geq
1$. So this allows to prove (4.7).

As a consequence, the sequence of symbols $(a_{j})_{j\in{\mathbb{N}}}$ defined
by (4.5) is a formal ${\mathcal{G}}^{s}$ symbol as desired.

∎

Step 5. Conclusion.

Thanks to Lemma 4.1, there exists a sequence of symbols
$(a_{j})_{j\in{\mathbb{N}}}$ defined by (4.5) of order zero. Using Theorem 4
we deduce that there exists $a\in{\mathcal{S}}_{s}^{m}$ realising
$(a_{j})_{j\in{\mathbb{N}}}$. As a consequence, the $h$-FIO associated to the
symbol $a$ and the phase $\varphi$ satisfying (4.3) satisfies that
$PF=F(hD_{x_{1}})$ microlocally near zero up to negligeable remainders. This
ends the proof of Theorem 1.

##  5\. A Gevrey FBI version of Egorov’s theorem: Proof of Theorem 3

Let us describe a Gevrey$-s$ transform of the type

(5.1) $\mathcal{T}u(x,h)=\int e^{\frac{i}{h}\varphi(x,y)}a(x,y,h)u(y)dy,\qquad
u\in\mathcal{S}^{\prime},$

with some phase $\varphi$ of Gevrey class.

###  5.1. Some complex symplectic geometry

For $n\in{\mathbb{N}}$, we use coordinates $z=(z_{1},\dots,z_{n})$ for a point
$z\in{\mathbb{C}}^{n}$. As usual, we note $z_{j}=x_{j}+iy_{j}$, for
$x_{j},y_{j}\in{\mathbb{R}}$ and $j=1,\dots,n$, so that the holomorphic and
antiholomorphic derivatives write

$\frac{\partial}{\partial z_{j}}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{j}}-i\frac{\partial}{\partial
y_{j}}\right),\qquad\frac{\partial}{\partial\overline{z}_{j}}=\frac{1}{2}\left(\frac{\partial}{\partial
x_{j}}+i\frac{\partial}{\partial y_{j}}\right).$

Let us denote

${\mathcal{B}}=\left\\{\frac{\partial}{\partial
z_{j}},\frac{\partial}{\partial\overline{z_{j}}}\right\\}_{j=1,\dots,n},\qquad{\mathcal{B}}^{*}=\left\\{\,\mathrm{d}z_{1},\dots,\,\mathrm{d}z_{n},\,\mathrm{d}\overline{z}_{1},\dots\,\mathrm{d}\overline{z}_{n}\right\\},$

where ${\mathcal{B}}$ is the canonical complex basis of ${\mathbb{C}}^{n}$ and
${\mathcal{B}}^{*}$ is the associated basis of 1-forms. As usual, the
canonical volume form is

$\,\mathrm{d}z\wedge\,\mathrm{d}\overline{z}=\left(\frac{2}{i}\right)^{n}\,\mathrm{d}m(z),$

where $\,\mathrm{d}m$ is the Lebesgue measure and

$\,\mathrm{d}z=\,\mathrm{d}z_{1}\dots\,\mathrm{d}z_{n},\qquad\,\mathrm{d}\overline{z}=\,\mathrm{d}\overline{z}_{1}\dots\,\mathrm{d}\overline{z}_{n},$

If $\sigma$ be the canonical symplectic form on
${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$, given by

$\sigma=\,\mathrm{d}\xi\wedge\,\mathrm{d}x.$

the canonical form $\sigma_{{\mathbb{C}}}$ in
${\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ such that

$\sigma_{{\mathbb{C}}}=\,\mathrm{d}\zeta\wedge\,\mathrm{d}\eta,$

is an extension of $\sigma$ satisfying

$\mathop{\rm
Re}\nolimits\sigma_{{\mathbb{C}}}=\,\mathrm{d}\xi\wedge\,\mathrm{d}x-\,\mathrm{d}\eta\wedge\,\mathrm{d}y,\qquad\mathop{\rm
Im}\nolimits\sigma_{{\mathbb{C}}}=\,\mathrm{d}\xi\wedge\,\mathrm{d}y+\,\mathrm{d}\eta\wedge\,\mathrm{d}x.$

Recall that a submanifold
$\Lambda\subset{\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ is totally real if
$\Lambda\cap i\Lambda=\left\\{0\right\\}$, I-Lagrangian if $\mathop{\rm
Im}\nolimits\sigma_{{\mathbb{C}}}|_{\Lambda}=0$ and R-Lagrangian if
$\mathop{\rm Re}\nolimits\sigma_{{\mathbb{C}}}|_{\Lambda}=0$. If
$A\in{\mathcal{M}}_{n\times n}({\mathbb{C}})$ and
$f:{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}$ is a quadratic function given by

$f(z)=\frac{1}{2}\langle Az,z\rangle,\qquad z\in{\mathbb{C}}^{n},$

then the submainfold
$\Lambda_{f}\subset{\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ defined by

$\Lambda_{f}=\left\\{\left(z,\frac{\,\mathrm{d}f}{\,\mathrm{d}z}(z)\right);\,z\in{\mathbb{C}}^{n}\right\\}$

is Lagrangian for $\sigma_{{\mathbb{C}}}$ (i.e., both I-Lagrangian and
R-Lagrangian).

####  5.1.1. FBI transforms

Let $\varphi=\varphi(z,x)$ be a holomorphic quadratic function on
${\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ satisfying that

$\mathop{\rm Im}\nolimits\left(\frac{\partial^{2}\varphi}{\partial
x^{2}}\right)\textrm{ is positive
definite}\qquad\textrm{and}\qquad\det\left(\frac{\partial^{2}\varphi}{\partial
x\partial z}\right)\not=0.$

Let $\kappa_{\varphi}$ the associated complex transformation
$\kappa_{\varphi}:{\mathbb{C}}^{n}\times{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$
defined implicitly by

$\kappa_{\varphi}\left(x,-\frac{\partial\varphi}{\partial
x}(z,x)\right)=\left(z,\frac{\partial\varphi}{\partial z}(z,x)\right),\qquad
z,x\in{\mathbb{C}}^{n}.$

According to ([23]), the transformation $\kappa_{\varphi}$ is canonical for
$\sigma_{{\mathbb{C}}}$. The FBI operator associated to $\varphi$ is

(5.2)
${\mathcal{T}}_{\varphi}u(z)=\frac{c_{\varphi}}{h^{\frac{3n}{4}}}\int_{{\mathbb{R}}^{n}}e^{\frac{i}{h}\varphi(z,x)}u(x)\,\mathrm{d}x,\qquad
u\in\mathscr{S}({\mathbb{R}}^{n}),$

where

$c_{\varphi}:=\frac{|\det\frac{\partial^{2}\varphi}{\partial x\partial
z}|}{2^{\frac{n}{2}}\pi^{\frac{3n}{4}}|\det\mathop{\rm
Im}\nolimits\frac{\partial^{2}\varphi}{\partial x^{2}}|^{\frac{1}{4}}}.$

For the particular choices

$\varphi_{0}(z,x)=\frac{i}{2}(z-x)^{2},\qquad\Phi_{0}(z)=\frac{1}{2}|\mathop{\rm
Im}\nolimits z|^{2},\qquad z,x\in{\mathbb{C}}.$

we call ${\mathcal{T}}_{0}$ the associated FBI transform defined by (5.2),
which is called Bargman transformation.

We state the following corollary.

######  Corollary 5.1 (Gevrey singular solutions).

Instead of performing directly a WFB construction on $P$ we prefer to
conjugate $P$ to $hD_{1}$ to solve the Cauchy problem

$\left\\{\begin{array}[]{l}hD_{1}u=0,\\\
u|_{x_{1}=0}=u_{0},\end{array}\right.$

for which obviously we have $u=u_{0}\otimes 1_{x_{1}}$.

Choosing a family $(u_{0}(x^{\prime},h))$ with a Grevey
$\operatorname{WF}_{h}$ set reduced to one point in $T^{*}{\mathbb{R}}^{n-1}$
we obtain a family $(u(x,h))$ of solutions to $Pu=0$ with a Gevrey
$\operatorname{WF}_{h}$ set reduced to a segment of the null bicharachteristic
strip of $P$ through $(x_{0},\xi_{0})$.

###  5.2. WKB Heuristics

Set formally

$\mathcal{T}u(x,h)=\int e^{\frac{i}{h}\varphi(x,y)}a(x,y,h)u(y)dy.$

The condition

$hD_{\mathop{\rm Re}\nolimits
x_{1}}\mathcal{T}u-h^{m}\mathcal{T}P(y,D_{y})u\sim 0\qquad\textrm{in
}G^{2s-1},$

is equivalent to solving an eikonal equation for the phase $\varphi$ of the
form

$\varphi_{x_{1}}^{\prime}(x,y)=p(y,-\varphi_{y}^{\prime}(x,y))$

and a transport equation for the symbol $a=a(x,y,h)$ of the form

$P^{\prime}a=e^{-\frac{i}{h}\varphi}\left((hD_{\mathop{\rm Re}\nolimits
x_{1}}-h^{m}P^{t}(y,D_{y}))(ae^{\frac{i}{h}\varphi})\right)=0,\quad\textrm{in
}{\mathcal{O}}_{s}(h^{\infty}),\textrm{close to }(x_{0},y_{0}),$

###  5.3. Construction of the canonical transformation $\kappa$

######  Proposition 5.2.

Let $P=P(y,D_{y})$ be a $G^{s}-$differential operator of degree $m$ such that
near $(y_{0},\eta_{0})\in T^{*}{\mathbb{R}}^{n}\setminus\left\\{0\right\\}$,
the symbol $p(y,\eta)$ is of real principal type. Then, there exists a
canonical transform $\kappa$ such that

$\kappa(y_{0},\eta_{0})=(x_{0},\xi_{0}),\qquad\textrm{where}\quad\xi_{0}=\frac{2}{i}\partial_{x}\phi(x_{0}).$

Moreover, if $y=\pi\circ\kappa^{-1}$, the submanifold defined by

$\Gamma=\left\\{(x,y(x))\in{\mathbb{C}}^{n}\times{\mathbb{R}}^{n};\,\,x\textrm{
close to }x_{0}\right\\}$

is totally real in ${\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ and of maximal
dimension $2n$.

###### Proof.

Let $\chi$ be a real $G^{s}$ canonical map defined near $(y_{0},\eta_{0})$
mapping $p(y,\eta)$ to $\eta_{1}$. Such a $\chi$ can be obtained by applying
Darboux Lemma.

Now let $\mathcal{T}_{0}$ be the Bargman transform. $\mathcal{T}_{0}$ has
phase functions $\varphi_{0}(x,y)=\frac{i}{2}(x-y)^{2}$, and
$\Phi_{0}(x)=\frac{1}{2}(\mathop{\rm Im}\nolimits x)^{2}$.

Since $\xi|_{\Lambda_{\Phi_{0}}}$ is real, we will choose
$\kappa=K_{\mathcal{T}_{0}}\circ\chi$. $\kappa$ is parametrized by
$\in{\mathbb{C}}^{n}$ close to
$x_{0}\in(0,y_{0}^{\prime}-i\eta_{0}^{\prime})$, setting

$(y(x),\eta(x))=\kappa^{-1}\left(x,\frac{2}{i}\partial_{x}\Phi_{0}(x)\right).$

The map $x\in{\mathbb{C}}^{n}\mapsto y(x)\in{\mathbb{R}}^{n}$ is of class
$G^{s}$ and

$\partial y:T_{x}{\mathbb{C}}^{n}\rightarrow
T_{y(x)}{\mathbb{R}}^{n}\qquad\textrm{is surjective}.$

Moreover, $\overline{\partial}y$ is biejctive since $dx$ is canonical.

Setting now

$\Gamma=\left\\{(x,y(x))\in{\mathbb{C}}^{n}\times{\mathbb{R}}^{n};\,\,x\textrm{
close to }x_{0}\right\\}$

we get that $\Gamma$ is totally real in
${\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ and of maximal dimension $2n$.

∎

###  5.4. Construction of the phase $\varphi$ on $\Gamma$

In this section we obtain and solve an eikonal equation for the phase
$\varphi$. The condition

$hD_{\mathop{\rm Re}\nolimits
x_{1}}\mathcal{T}u-h^{m}\mathcal{T}P(y,D_{y})u\sim 0\qquad\textrm{in }G^{s},$

is equivalent to solving an eikonal equation for the phase $\varphi$. We first
construct $\varphi$ and $a$ on $\Gamma$. Next, we use an extension argument
providing a (formal) sequence $(a_{j})_{j\geq 0}$ of Gevrey symbols. We
conclude by Carleson’s moment method, which allow to construct a Gevrey symbol
$a$ on ${\mathbb{C}}^{n}\times{\mathbb{R}}^{n}$.

######  Proposition 5.3 (Eikonal equation).

There exists a Gevrey-$s$ phase $\varphi=\varphi(x,y)$ satisfying

$\varphi_{x_{1}}^{\prime}(x,y)=p(y,-\varphi_{y}^{\prime}(x,y))\qquad\textrm{
on }\Gamma$

for $\Gamma$ is given in Proposition 5.2. Moreover,
$\varphi_{y}(x_{0},y_{0})=-\eta_{0}\in{\mathbb{R}}^{n}$,
$\operatorname{Im}\varphi^{\prime\prime}_{y}>0$,
$\operatorname{det}\varphi^{\prime\prime}_{x,y}\neq 0$

###### Proof.

Let us choose $\varphi$ such that

$\frac{\partial\varphi}{\partial x}=\xi,\qquad\frac{\partial\varphi}{\partial
y}=-\eta.$

Observe that, since $\kappa$ is canonical, the $1-$form

$\omega=\left(\xi(x)-\frac{\partial y^{t}}{\partial
x}\eta(x)\right)dx-\frac{\partial
y^{t}}{\partial\overline{x}}\eta(x)d\overline{x}$

is closed for $x$ close to $x_{0}$. Now, $\varphi$ defined above satisfies

$\varphi_{x_{1}}^{\prime}=\xi_{1}(x)=-\mathop{\rm Im}\nolimits
x_{1}=p(y(x),\eta(x))\qquad\textrm{and}\qquad\eta(x)=-\varphi_{y}^{\prime}(x,y(x)).$

In order to get an FBI transform we may impose further

$\varphi(x,y)=\frac{i}{2}(x^{\prime}-y^{\prime})^{2}-y_{1,0}\eta_{1,0}+iC(y_{1}-y_{1,0})^{2},\qquad\mathop{\rm
Re}\nolimits C>0.$

∎

###  5.5. Construction of the symbol $a$ on $\Gamma$

######  Proposition 5.4 (Transport equation).

There exists a symbol $a=a(x,y,h)$ such that

$P^{\prime}a=e^{-\frac{i}{h}\varphi}\left((hD_{\mathop{\rm Re}\nolimits
x_{1}}-h^{m}P^{t}(y,D_{y}))(ae^{\frac{i}{h}\varphi})\right)=0,\qquad\textrm{close
to }(x_{0},y_{0}).$

Furthermore, $a$ is of class $G^{s}$ and elliptic at $(x_{0},y_{0})$.

###### Proof.

As $P^{\prime}$ is a $h-PDO$ of degree $0$ with symbol

$p^{\prime}(x,y,\xi,\eta)=\mathop{\rm
Re}\nolimits\xi_{1}+\varphi_{x_{1}}^{\prime}-P\varphi(x,y,\eta),$

where

$P\varphi(x,y,\eta)=p(y,-\eta-\varphi_{y}^{\prime}).$

According to Proposition 5.3 phase solves

$\varphi_{x_{1}}^{\prime}(x,y)=p(y,-\varphi_{y}^{\prime}(x,y))\qquad\textrm{
on }\Gamma.$

Then for $(x,y)\in\Gamma$ such that $\xi=\eta=0$ one has $p^{\prime}=0$.
Moreover,

$a=\frac{\partial p^{\prime}}{\partial\xi}|_{\xi=\eta=0}=e_{1},\qquad
b=\frac{\partial p^{\prime}}{\partial\eta}|_{\xi=\eta=0}=\frac{\partial
p}{\partial\eta}(y,-\varphi_{y}^{\prime}),\qquad\textrm{on }\Gamma.$

Consider the vector field on ${\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$ defined
by

$\mathcal{V}^{\prime}=a\partial_{x}+\overline{a}\partial_{\overline{x}}+b\partial_{y}+\overline{b}\partial_{\overline{y}}.$

One checks that $\mathcal{V}^{\prime}|_{\Gamma}$ is tangent to $\Gamma$ if and
only if

$b\in{\mathbb{R}}^{n}\qquad\textrm{and}\qquad b=\left(\frac{\partial
y}{\partial x}\right)a+\left(\frac{\partial
y}{\partial\overline{x}}\right)\overline{a}.$

Observe that if
$\mathcal{V}=\alpha\partial_{x}+\overline{\alpha}\partial_{\overline{x}}$ is a
vector field on ${\mathbb{C}}^{n}$ and $u(x)=u^{\prime}(x,y(x))$, one has
$\mathcal{V}^{\prime}u^{\prime}=\mathcal{V}u$ on $\Gamma$ if and only if

$\alpha=a,\qquad b=\left(\frac{\partial y}{\partial
x}\right)a+\left(\frac{\partial
y}{\partial\overline{x}}\right)\overline{a},\quad\textrm{with}\quad\partial_{\overline{x}}u^{\prime}=\partial_{\overline{y}}u^{\prime}=0\textrm{
on }\Gamma.$

We fulfill the tangency condition on $\Gamma$ by choosing

$b=\frac{\partial
p}{\partial\eta}(y,-\varphi_{y}^{\prime})\qquad\textrm{and}\qquad
a=\frac{\partial p^{\prime}}{\partial\xi}|_{\xi=\eta=0}.$

This follows by construction, as by setting

$\chi^{\prime}H_{p}=H_{\eta_{1}}=\begin{pmatrix}e_{1}\\\
0\end{pmatrix},\qquad(\chi^{\prime})^{-1}=\begin{pmatrix}P&Q\\\
R&S\end{pmatrix}\textrm{(real symplectic matrix)},$

with $P$ invertible, $P^{t}R$ and $Q^{t}S$ symmetric and $P^{t}S-R^{t}Q=I$, we
have

$\frac{\partial y}{\partial x}=\frac{1}{2}(P-iQ),\qquad\frac{\partial
y}{\partial\overline{x}}=\frac{1}{2}(P+iQ).$

Then,

$\left(\frac{\partial y}{\partial x}e_{1}+\frac{\partial
y}{\partial\overline{x}}e_{1}\right)=Pe_{1}=\frac{\partial
p}{\partial\eta}(y,-\varphi_{y}^{\prime}),\qquad\textrm{ on }\Gamma,$

since by construction

$\chi^{-1}(\mathop{\rm Re}\nolimits x,-\mathop{\rm Im}\nolimits
x)=(y(x),\eta(x)).$

∎

We will solve transport equations on $\Gamma$ ad extend solutions almost
holomorphycally from $\Gamma$. This yields a formal solution
$(a_{j}^{\prime})_{j\geq 0}$ with
$a_{j}^{\prime}\in{\mathcal{S}}_{s}^{-j}({\mathbb{C}}^{n}\times{\mathbb{C}}^{n})$.

We set

$b^{\prime}e^{\frac{i}{h}\varphi}=\frac{1}{h}\left(hD_{\mathop{\rm
Re}\nolimits x_{1}}-h^{m}P^{t}(y,D_{y})\right)(ae^{\frac{i}{h}\varphi}).$

The eikonal equation is satisfied since $\varphi_{\mathop{\rm Re}\nolimits
x_{1}}^{\prime}=p(y,-\varphi_{y}^{\prime})$. Now set

$e^{\frac{i}{h}\varphi}P_{\varphi}(x,y,hD_{y})=h^{m}P^{t}(y,D_{y})(\cdot
e^{\frac{i}{h}\varphi}),$

The principal symbol of $P_{\varphi}$, namely $p_{\varphi}$, satisfies

$p_{\varphi}|_{\eta=0}=p(y,-\varphi_{y}^{\prime}).$

If

$P^{\prime}=hD_{\mathop{\rm Re}\nolimits x_{1}}+\varphi_{\mathop{\rm
Re}\nolimits x_{1}}^{\prime}-P_{\varphi}(x,y,hD_{y})$

we have to solve a WKB problem for $P^{\prime}$ with $\varphi^{\prime}=0$,
which leads to some expansions. If $Q(y,hD_{y})$ is a $hPDO$ of order zero on
${\mathbb{R}}^{n}$, if $\varphi$ is a complex phase with $\mathop{\rm
Im}\nolimits\varphi\geq 0$ and $\,\mathrm{d}\varphi\not=0$ when $\mathop{\rm
Im}\nolimits\varphi=0$, one has

(5.3) $Q(ae^{\frac{i}{h}\varphi})\sim e^{\frac{i}{h}\varphi}\sum_{\alpha\geq
0}\frac{h^{|\alpha|}}{\alpha!}\tilde{q}^{(\alpha)}(y,\varphi_{y}^{\prime})D^{\alpha}_{z}(e^{\frac{i}{h}\rho}a)|_{y=z},$

where $\tilde{q}$ is an almost holomorphic extension of $q$ to the whole
${\mathbb{C}}^{n}\times{\mathbb{C}}^{n}$, $a$ is a symbol and $\rho$ is given
by

$\rho(y,z)=\varphi(z)-\varphi(y)-(z-y)\cdot\varphi_{y}^{\prime}(y).$

The expansion (5.3) was obtained by Melin and Sjöstrand in [20]. We deal with
the Gevrey case and explain how to adapt the original proofs to compute the
remainders in our case.

We deal with phases of the form

$a(y,X)=\varphi(z)+(y-z)\cdot\eta,$

where we have written $X=(z,\eta)\in{\mathbb{R}}^{2n}$, $\varphi(z)$ is a
phase function in ${\mathcal{G}}^{s}$ defined close to
$y_{0}\in{\mathbb{R}}^{n}$ and complex-valued with $\mathop{\rm
Im}\nolimits\varphi\geq 0$,
$\,\mathrm{d}\varphi(y_{0})=-\eta_{0}\in{\mathbb{R}}^{n}\setminus 0$. We shall
work close to $X_{0}:=(y_{0},-\eta_{0})$ and one has then

$\mathop{\rm Im}\nolimits a(y,X)\geq 0,\qquad
a_{X}^{\prime}(y_{0},X_{0})=0,\qquad\det
a_{X,X}^{\prime\prime}(y_{0},X_{0})\not=0.$

We note $u_{h}(X)$ some ${\mathcal{G}}^{s}$ symbol on ${\mathbb{R}}^{2n}$ of
degree zero compactly supported close to
$X_{0}=(y_{0},\varphi^{\prime}_{y}(y_{0}))$. We have the following lemma.

######  Lemma 5.5.

Under the previous assumptions there is a stationary phase expansion of Gevrey
type $2s-1$ with the form

(5.4)
$\iint_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}e^{\frac{i}{h}a(y,X)}u_{h}(X)\,\mathrm{d}X\sim
e^{\frac{i}{h}a(y,Z_{y})}\sum_{\nu=0}^{\infty}h^{\nu+n}C_{y,\nu}(D)u_{h}(Z_{y}),$

where $\sim$ means modulo some small ${\mathcal{G}}^{2s-1}$ remainders.

Above we have noted $a(y,Z)$ an almost holomorphic extension of $a(y,X)$ to
the whole ${\mathbb{C}}^{2n}$ of the same class as $a$ and similarly for
$u_{h}$. Moreover, $y\mapsto Z_{y}$ is the function defined by
$a_{Z}^{\prime}(y,Z)=0$ close to $X_{0}$ with $Z_{y_{0}}=X_{0}$.

###### Proof.

We follow closely [20] and use Morse type coordinates. We also use the Stokes
formula and deform ${\mathbb{R}}^{2n}$ into complex contours in
${\mathbb{C}}^{2n}$.

The equations

$a_{Z}^{\prime}(y,Z)=0,\qquad Z_{y_{0}}=X_{0}$

defines a ${\mathcal{G}}^{s}$ function since $\det
a_{Z,Z}^{\prime\prime}(y_{0},X_{0})\not=0$. By Lemma [20, Lemma 2.1] one has

$\mathop{\rm Im}\nolimits a(y,Z_{y})\geq\frac{1}{C}|\mathop{\rm Im}\nolimits
Z_{y}|^{2}.$

Set

$h(y,Z)=a(y,Z+Z_{y})-a(y,Z_{y}),$

which is defined close to $(y_{0},0)$ and modulo a small error induces in $Z$
a quadratic form. One has

$\partial_{Z}h(y,0)=0,\qquad\partial_{\overline{Z}}h(y,Z)={\mathcal{O}}(1)\exp(-\frac{1}{C}|\mathop{\rm
Im}\nolimits(Z+Z_{y})|^{\frac{-1}{s-1}}).$

Set

$R(y,Z)=2\int_{0}^{1}(1-\theta)h_{Z,Z}^{\prime\prime}(y,\theta
Z)\,\mathrm{d}\theta.$

Since $u\mapsto\exp(-\frac{1}{C}u^{\frac{-1}{s-1}})$ is increasing one has

$\partial_{\overline{Z}}h(y,Z)={\mathcal{O}}(1)\exp(-\frac{1}{C}(|\mathop{\rm
Im}\nolimits Z|^{\frac{-1}{s-1}}+|\mathop{\rm Im}\nolimits
Z_{y}|^{\frac{-1}{s-1}}))$

Writing

$R(y,Z)=iQ^{t}(y,Z)Q(y,Z),\qquad Q(y_{0},0)=A^{-1},\quad
A^{t}R(y_{0},0)A=iId,$

we define a change of coordinates in ${\mathbb{C}}^{2n}$ by

$\overline{Z}(Z)=Q(y,Z-Z_{y})(Z-Z_{y}),$

since $Q(y_{0},0)\in GL(2n,{\mathbb{C}})$. In these coordinates one has

$a(y,Z)=a(y,Z_{y})+\frac{i}{2}\langle\overline{Z},\overline{Z}\rangle+\rho(y,Z),$

with

$\langle\overline{Z},\overline{Z}\rangle:=|\overline{X}|^{2}-|\overline{Y}|^{2}+2i\overline{X}\cdot\overline{Y},\qquad\overline{Z}=\overline{X}+i\overline{Y}\in{\mathbb{C}}^{2n}.$

The function $\rho$ is small and

$|\rho(y,Z)|={\mathcal{O}}(1)\exp(-\frac{1}{C}|\mathop{\rm
Im}\nolimits(Z-Z_{y})|^{-\frac{1}{s-1}}).$

The map $Z\mapsto\overline{Z}(Z)$ has an inverse
$\overline{Z}\mapsto\overline{Z}(Z)$ and these two maps depend on $y$ also.
Define for $\sigma\in[0,1]$ the integration paths

$\Gamma_{y,\sigma}:\overline{X}\mapsto
Z(\overline{Z}_{\sigma}),\qquad\overline{Z}_{\sigma}=\overline{X}+i\sigma
g(y,\overline{X}),\quad\overline{X}\in{\mathbb{R}}^{2n},$

where $g$ is smooth and such that ${\mathbb{R}}^{2n}$ is given by
$\overline{Y}=g(y,\overline{X})$ in the new coordinates. Now, using Stokes
formula we deform $\Gamma_{y,1}={\mathbb{R}}^{2n}$ into $\Gamma_{y,0}$.
Following [20, Lemma 2.4], we have

$\mathop{\rm Im}\nolimits
a(y,Z(\overline{Z}_{\sigma})\geq\frac{1}{C}(1-\sigma)(|\mathop{\rm
Im}\nolimits
Z_{y}|^{2}+|\overline{X}|^{2})\geq\frac{1}{C^{\prime}}|\mathop{\rm
Im}\nolimits Z(\overline{Z}_{\sigma}|^{2}.$

Define the $(2n,0)$ form in ${\mathbb{C}}^{2n}$ by

$\omega_{h}=f_{h}(Z)\,\mathrm{d}Z_{1}\wedge\cdots\wedge\,\mathrm{d}Z_{2n},\qquad
f_{h}(Z)=e^{\frac{i}{h}a(y,Z)}u_{h}(Z),$

since

$\partial_{\overline{Z}}f_{h}=e^{\frac{i}{h}a(y,Z)}\left(\partial_{\overline{Z}}u_{h}+\frac{i}{h}u_{h}\partial_{\overline{Z}}a\right),$

we have

$\iint_{{\mathbb{R}}^{2n}}e^{\frac{i}{h}a(y,X)}u_{h}(X)\,\mathrm{d}X=\iint_{\Gamma_{y,1}}e^{\frac{i}{h}a(y,Z)}u_{h}(X)\,\mathrm{d}Z_{1}\wedge\dots\wedge\,\mathrm{d}Z_{2n}$

and

(5.5)
$\iint_{\Gamma_{y,0}}e^{\frac{i}{h}a(y,Z)}u_{h}(X)\,\mathrm{d}Z_{1}\wedge\dots\wedge\,\mathrm{d}Z_{2n}=\iint_{V}e^{\frac{i}{h}a(y,Z_{y})}u_{h}(Z(\overline{X})e^{-\frac{1}{2h}|\overline{X}|^{2}+\frac{i}{h}\rho}\left|\frac{\partial
Z}{\partial\overline{X}}\right|\,\mathrm{d}\overline{X},$

where $V$ is a neighbourhood of the origin in ${\mathbb{R}}^{2n}$.

Using [20, Lemma 2.5] we may reduce the computation of the asymptotics of the
right-hand side of (5.5) to the case $\rho=0$. Indeed, if $Y\mapsto(AY,Y)$ is
a non-degenerate quadratic form with $\mathop{\rm Im}\nolimits A\geq 0$, and
$\chi$ is a ${\mathcal{G}}^{s}$ cut-off function close to the origin in
${\mathbb{R}}^{N}$ we can write

$h^{-\frac{N}{2}}\int
e^{\frac{i}{2h}(AY,Y)}u_{h}(X+Y)\,\mathrm{d}Y=\ell_{\chi}(X)+r_{\chi}(X),$

for

$\ell_{\chi}(X)=h^{-\frac{N}{2}}\int
e^{\frac{i}{2h}(AY,Y)}u_{h}(X+Y)\chi(Y)\,\mathrm{d}Y,\qquad
r_{\chi}(X)=h^{-\frac{N}{2}}\int
e^{\frac{i}{2h}(AY,Y)}u_{h}(X+Y)(1-\chi(Y))\,\mathrm{d}Y.$

Now,as

$|\partial_{X}^{\alpha}r_{\chi}(X)|\leq
C^{1+|\alpha|}\alpha!^{s}\exp(-\frac{1}{C}h^{-\frac{1}{s}}),$

we deduce that $r_{h}$ is a small ${\mathcal{G}}^{s}$ remainder. On the other
hand, $\ell_{X}$ is a ${\mathcal{G}}^{s}$ symbol of the same order as $u_{h}$
having the asymptotic expansion

$\ell_{\chi}(X)\sim
C_{A}\sum_{\nu=0}\frac{h^{\nu}}{\nu!(2i)^{\nu}}(A^{-1}D,D)^{\nu}u_{h}(X),\qquad
C_{A}=(\frac{1}{2\pi i}\det A)^{-\frac{1}{2}}.$

As a consequence,

$v_{h}(y):=e^{-\frac{i}{h}a(y,Z_{y})}\int_{{\mathbb{R}}^{2n}}e^{\frac{i}{h}a(y,X)}u_{h}(X)\,\mathrm{d}X$

is a ${\mathcal{G}}^{s}$ symbol of order $n$ admitting an expansion of the
form

$v_{h}(y)\sim\sum_{\nu=0}^{\infty}h^{\nu+n}C_{\nu,y}(D)u_{h}(Z_{y})$

with $C_{\nu,y}(D)$ differential operators of degree less or equal than
$2\nu$. The first term above rewrites as $h^{n}C_{0}(y)u_{h}(Z_{y})$ for a
function $C_{0}(y)(2\pi)^{-n}$ which is a suitable branch of the square root
of $\det(\frac{1}{i}a_{Z,Z}^{\prime\prime}(y,Z_{y}))^{-1}$. From [11] we
deduce that the term

$R_{N}(y):=\sum_{\nu=N}^{\infty}h^{\nu+n}C_{\nu,y}(D)u_{h}(Z_{y})$

satisfies

$|\partial_{y}^{\gamma}R_{N}(y)|\leq
C^{1+\gamma+N}\gamma!^{s}N!^{2s-1}h^{N+n},$

where the Gevrey loss is an standard observation. Moreover the last term
coming from Stokes formula is also a small ${\mathcal{G}}^{s}$ reminder at
least when $a(y,\cdot)$ depends analytically in $y$, which is the case for the
phase $a(y,X)=(y-z)\eta+\varphi(z)$, $X=(z,\eta)$. Indeed, writing this term
as

$R(y)=\iint_{\bigcup_{\sigma\in[0,1]}\Gamma_{y,\sigma}}e^{\frac{i}{h}a(y,Z)}\left(\partial_{\overline{Z}}u_{h}+\frac{iu_{h}}{h}\partial_{\overline{Z}}a\right)\wedge\,\mathrm{d}Z_{1}\wedge\dots\wedge\,\mathrm{d}Z_{2n}$

and computing $\partial_{y}^{\lambda}R(y)$ one has

(5.6)
$\partial_{y}^{\lambda}R(y)=\sum\frac{\lambda!}{\mu!\nu!}\iint_{\bigcup_{\sigma\in[0,1]}\Gamma_{y,\sigma}}e^{\frac{i}{h}a(y,Z)}\frac{\nu!}{\ell!\nu_{1}!\dots\nu_{\ell}!}\partial_{y}^{\nu_{1}}a\dots\partial_{y}^{\nu_{\ell}}ah^{-\ell}\partial_{y}^{\mu}g_{h}\wedge\,\mathrm{d}Z_{1}\wedge\dots\wedge\,\mathrm{d}Z_{2n}$

where the sum runs over the set

$\nu+\mu=\lambda,\qquad\nu_{1}+\cdots\nu_{\ell}=\nu,\qquad\ell=1,\cdots,\nu$

and
$g_{h}=\partial_{\overline{Z}}u_{h}+\frac{iu_{h}}{h}\partial_{\overline{Z}}a$.
For $Z\in\bigcup_{\sigma\in[0,1]}$ one has

$\mathop{\rm Im}\nolimits a(y,Z)\geq\frac{1}{C}|\mathop{\rm Im}\nolimits
Z|^{2}.$

Moreover, $\partial_{\overline{Z}}u_{h}$ and $\partial_{\overline{Z}}a$ are
bounded by $C\exp\left(-\frac{1}{C}|\mathop{\rm Im}\nolimits
Z|^{-\frac{1}{s-1}}\right)$ as well as their derivatives by construction. We
deduce estimates for the RHS of (5.6), since

$\frac{|\partial_{y}^{\nu_{1}}a|}{\nu_{1}!}...\frac{|\partial_{y}^{\nu_{l}}a|}{\nu_{l}!}\leq
C^{|\nu|+1},\qquad h|\partial_{y}^{\mu}g_{h}|\leq
C^{1+|\mu|}\mu!^{s}\exp(-\frac{1}{C}|\mathop{\rm Im}\nolimits
Z|^{-\frac{1}{s-1}})$

and

$\left|e^{\frac{i}{h}a(y,Z)}\right|\leq e^{-\frac{1}{Ch}|\mathop{\rm
Im}\nolimits
Z|^{2}}\sum_{\mu+\nu=\lambda}\frac{\lambda!\nu!}{\mu!\nu!\ell!}h^{-\ell}M!N!h^{N}|\mathop{\rm
Im}\nolimits Z|^{-2N}|\mathop{\rm Im}\nolimits Z|^{\frac{M}{s-1}}\mu!^{s},$

for all $N,M>0$. Choosing $N=\ell$ and ·$M=2\ell(s-1)$ one has the bound

$C^{1+|\lambda|}\mu!^{s}\nu!\ell!^{2s-2}\leq
C^{1+|\lambda|}\mu!^{s}\nu!^{2s-1}\leq C^{1+|\lambda|}\lambda!^{2s-1}.$

Hence, $\partial_{y}^{\lambda}R(y)$ is bounded by

$C^{1+|\lambda|}\lambda!^{2s-1}\exp(-\frac{1}{C}h^{\frac{-1}{2s-1}})$

so $R$ is a small ${\mathcal{G}}^{2s-1}$ remainder.

∎

We use next Lemma 5.5 to recover the asymptotic (5.3). We can also check that
on $\Gamma$ $P^{\prime}a$ has the form

$P^{\prime}a=\frac{h}{i}\partial_{\mathop{\rm Re}\nolimits
x_{1}}a+\frac{h}{i}\frac{\partial
p}{\partial\eta}(y,-\varphi_{y}^{\prime})\partial_{y}a+ca+R^{\prime}a$

with $c$ a symbol of order $-1$ and $R^{\prime}$ a linear map
${\mathcal{S}}_{s}^{m^{\prime}}\rightarrow{\mathcal{S}}_{s}^{m^{\prime}-2}$.

We solve inductively the equations on $\Gamma$:

$\frac{h}{i}v^{\prime}a_{0}+ca_{0}=hb,\qquad\frac{h}{i}v^{\prime}a_{j}+ca_{j}+R^{\prime}a_{j-1}=0,\quad
j\geq 1,$

for

$v^{\prime}=\partial_{\mathop{\rm Re}\nolimits x_{1}}+\frac{\partial
p}{\partial\eta}(y,-\varphi_{y}^{\prime})\partial_{y}+\overline{\frac{\partial
p}{\partial\eta}}(y,-\varphi_{y}^{\prime})\partial_{\overline{y}}.$

We notice that $v^{\prime}|_{\Gamma}$ is tangent to $\Gamma$ as a subset of
${\mathbb{C}}^{2n}$.

In order to conclude the proof, we need to check that the remainder

$b^{\prime}(x,y)=\frac{1}{h}e^{\frac{h}{i}\varphi}\left(hD_{\mathop{\rm
Re}\nolimits
x_{1}}-h^{m}P^{t}(y,D_{y})\right)(a^{\prime}e^{\frac{i}{h}\varphi})-b,$

when $a^{\prime}\sim\sum_{j\geq 0}a_{j}^{\prime}$,
$a_{j}^{\prime}\in{\mathcal{S}}_{s}^{-j}({\mathbb{C}}^{n}\times{\mathbb{C}}^{n})$
and $a_{j}^{\prime}|_{\Gamma}=a_{j}$ for $a_{j}$ solving the transport
equations above is small, i.e.: $\left|b^{\prime}(x,y)\right|\leq
c\left\\{\exp(\frac{-1}{c}\left|y-y(x)\right|^{\frac{-1}{s-1}})+\exp(\frac{-1}{c}h^{\frac{-1}{s}})\right\\}$

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