We build a general theory of microlocal (homogeneous) Fourier Integral Operators in real-analytic
regularity, following the general construction in the smooth case by
Hörmander and Duistermaat. In particular, we prove that the Boutet-Sjöstrand parametrix for the Szegő
projector at the boundary of a strongly pseudo-convex real-analytic domain
can be realised by an analytic Fourier Integral Operator. We then study some applications,
such as FBI-type transforms on compact, real-analytic Riemannian
manifolds and propagators of one-homogeneous (pseudo)differential operators.

§ INTRODUCTION

This project began as an investigation of the following problem: let
$\Omega\subset \C^n$ be an open set with real-analytic boundary; can we
characterise the analytic singularities of the functions on the boundary $\partial \Omega$ which are
boundary values of holomorphic functions inside $\Omega$?

The problem is well-posed when $\Omega$ is strongly
pseudoconvex (see Definition <ref>) and a precise
description modulo smooth functions of the Szegő projector $S$ (from $L^2(\partial \Omega)$
onto the subspace of boundary values of holomorphic functions) is
well-known in the case where $\partial \Omega$ is smooth
[7, 14]. It is then
natural to expect that, if $\partial \Omega$ is analytic, then an even more
precise description of $S$ can be reached. This microlocal problem admits a
semiclassical (small parameter) counterpart, about the description of
spaces of holomorphic sections of high powers of ample line bundles
over Kähler manifolds. In the latter case, a description of the
Szegő kernel in analytic regularity was recently obtained
[30, 11, 18, 24, 9, 12].

It quickly became obvious to the author that a natural course of action
to study the analytic singularities of $S$ was to adapt the method of
[7], where $S$ is described as a Fourier
Integral Operator with complex-valued phase function, and where the
proof uses a microlocal normal form procedure (and in particular, a
conjugation by a Fourier Integral Operator). Building a theory of
analytic Fourier Integral Operators that lies as close as possible to the smooth
constructions [22] became our next goal and
constitutes the bulk of this article.

In the microlocal as well as semiclassical case, the study of the
Szegő projector has important applications in geometric
quantization and “general purpose” microlocal/semiclassical
analysis. One particular instance of strongly pseudoconvex open set
with real-analytic boundary is the Grauert tube around a real-analytic
Riemannian manifold; the range of $S$ is then the target space of a
natural FBI-type transformation. Thus the analytic counterpart of the
results of [7] serves as a tool for other
problems related to the spectrum and dynamics of differential and
pseudo-differential operators. These techniques could be
particularly relevant to the study of non-self-adjoint analytic
(pseudo)differential operators.

§.§ What this article contains

Section <ref> reviews some basic properties of
the boundary Cauchy-Riemann problem for pseudoconvex open sets. The
impatient reader only interested in “off-the-shelf” properties of analytic
Fourier Integral Operators may skip it, however the pseudoconvex open
set $\R^n\times S^{n-1}$, on which the Szegő projector admits
an exact formula, is useful in the construction of the general theory.

Section <ref> contains the “formal analytic”
content of our construction. We describe spaces of analytic amplitudes
(formal or not) in Section <ref>, then we study the geometric properties of
homogeneous phase functions and their associated conical
Lagrangians in Section <ref>. We also prove a
stationary phase type theorem in Section <ref>. Most of
the contents of this section will look familiar to experts, but many
details (in particular, the sense in which we perform Borel summations
of formal analytic amplitudes in the microlocal setting) turn out to
be non-trivial and crucial.

The core of the construction of analytic Fourier Integral Operators is
Section <ref>. We properly define analytic
Fourier Integral Operators in Section <ref>, in
an ascending level of geometric generality, and study the composition
rule and the action of stationary phase; our results are gathered in Theorem <ref>. The algebra of analytic pseudodifferential operators, which of
course is of utmost importance, is described in Section
<ref>; in our framework, to microlocalize the action
of these operators we introduce a FBI transform, which conjugates the
problem to operators on $\R^n\times S^{n-1}$ whose microlocal
structure resembles that of the Szegő projector. As in
[1], we call them “Toeplitz
operators”, even though they are not best described as $SQS$ for $Q$
a pseudodifferential operators, but rather by a Fourier Integral
Operator formula, akin to the “covariant Toeplitz operators” of
[8]. These Toeplitz operators turn
out to be a very efficient microlocal replacement of pseudodifferential
operators, allowing us to describe refined properties of Fourier Integral Operators
(notably inversion) microlocally in Section <ref>. Besides
pseudodifferential operators and Toeplitz operators, important
examples of Fourier Integral Operators are “quantized contact
transformations”: one can quantize the action of a one-homogeneous
symplectic change of variables into a unitary operator, and we do so
in Proposition <ref>.

We then turn our attention to the specific problem of the Szegő
projector for a general pseudoconvex open set in Section
<ref>. We prove a normal form theorem for the
$\overline{\partial}_b$ operator in analytic regularity, and use it to
describe the Szegő projector as an analytic Fourier Integral
Operator in Theorem <ref>. We
then generalise the properties of Toeplitz operators obtained in
Section <ref> to this more general geometric setting.

Some illustrations of our techniques are presented in Section
<ref>. We discuss in particular the case of
Grauert tubes, and how a natural FBI-type transformation on compact
analytic Riemannian manifolds helps to translate “usual” questions
concerning differential or pseudodifferential operators to the
Toeplitz framework, where they are more easily studied. We prove in
Proposition <ref> that propagators of one-homogeneous, self-adjoint analytic
pseudodifferential operators are analytic Fourier Integral
Operators. We also discuss briefly how to obtain semiclassical results from
our microlocal toolbox. To our knowledge, there is no general theory of analytic
Fourier Integral Operators available in the semiclassical case (and
certainly not one which preserves analytic function spaces), and we hope that
our results can be of use in this direction.

The Appendix presents some well-known facts about analytic
semiclassical analysis, which we gather for the comfort of the
reader. We discuss in particular the properties of spaces of analytic
functions, their dual spaces (called analytic functionals), and
a convenient quotient space of analytic functionals called

§.§ Comparison with earlier work

Microlocal analysis in real-analytic regularity has a long history,
but its early
achievements have been shadowed by the success and popularity of the
$C^{\infty}$ techniques.

Building on an observation that in some
formal sense, there are classes of pseudodifferential operators
adapted to real-analytic regularity
[2], a general framework was
proposed [35] but ultimately, this
theory never reached the same foundational status as the $C^{\infty}$
counterpart. In particular, a method of proof of Theorem
using the tools of [35] appears in
[26], but the status of this proof and of the
statement itself is disputed; we mention
[25] for a more detailed, but incomplete,
discussion of Kashiwara's approach. One of the initial goals of this
work was the hope to settle once and for all the status of Theorem <ref>.

In contrast, the theory of semiclassical analysis in
real-analytic regularity is now well-developed, after the seminal work
[32]. In this setting, one is
interested in improving remainders from $O(\hbar^{\infty})$ to
$O(e^{-c\hbar^{-1}})$ for some $c>0$, rather than improving remainders
from smooth to real-analytic. The semiclassical theory makes full use
of complex-valued phase functions, whose manipulation is perhaps even
easier than in the $C^{\infty}$ case because of the naturalness of
extensions into the complex.

It is often said that microlocal analysis is a particular case of
semiclassical analysis. We found out, however, that the usual semiclassical
constructions of pseudodifferential operators, other
Fourier Integral operators, or even Borel summations of analytic
symbols, was not applicable here. This is chiefly due to the fact that,
when presented with an oscillating integral with small parameter, of
the form $\int e^{i\frac{\phi(x)}{\hbar}}a(x)\dd x$, we may remove the
areas where $\im(\phi)>0$ by a simple (smooth) cut-off argument,
leading to $O(e^{-c\hbar^{-1}})$ errors . As a consequence, at its present
stage of development, analytic semiclassical analysis deals with
errors of size $O(e^{-c\hbar^{-1}})$ in $C^{k}$ topology for fixed
$k$. In the microlocal setting, we cannot afford to introduce cut-offs
even in areas of phase space which we presume are away from the
analytic wave front set. Even apart from stationary phase, cut-and-paste arguments, used to pass from local
definitions of objects in coordinate charts to global descriptions,
must be replaced by slightly more refined arguments of cohomological
nature: broadly speaking, we are able to define how objects act
locally and then we can use somewhat abstract gluing tools. This is a
standard approach in the study of holomorphic or real-analytic
functions, but it turns out (see the Appendix) that these arguments
already work in the $C^{\infty}$ setting.

Conversely, at least the elementary results of analytic semiclassical
analysis can be obtained from our microlocal setting by considering
Fourier modes in an auxiliary variable; we describe this briefly in Section

The book [38] presents an
overview of the existing literature on general-purpose analytic
microlocal analysis and some of its applications. Sections <ref> and
<ref> have been
somewhat motivated by this book, as we found out that adapting the
homogeneous FIO framework to the real-analytic case is not as
straightforward as it seems and many statements and constructions of
this book
deserved a more thorough approach.

§.§ Acknowledgements

We are most grateful to Michael Hitrik and Johannes Sjöstrand for
their encouragement and useful remarks throughout the realisation of
this project.

This work is dedicated to the memory of Steve Zelditch, who motivated
us to give a complete proof of the analytic parametrix of the
Szegő kernel in the first place.

§ THE BOUNDARY CAUCHY-RIEMANN PROBLEM

This section is a primer on the theory of boundary Cauchy-Riemann
problems; we present some facts which, besides being the basis for
Theorem <ref>, will be useful for the general construction of Fourier
Integral Operators in real-analytic regularity.

§.§ Strongly pseudoconvex domains and CR geometry

Let $Y$ be a (paracompact, boundaryless) complex manifold of
dimension $n\geq 2$.

* Let $U\subset Y$ open. A function $\rho\in
C^2(U,\R)$ is strongly plurisubharmonic when at every point of
$U$, in local coordinates, the Hermitian matrix
$[\frac{\partial^2\rho}{\partial z_j\partial
\overline{z_k}}]_{j,k}$ is positive definite. (This does not
depend on the coordinates).

* A compact real hypersurface $X\subset Y$ is strongly
pseudoconvex when there exists an open neighbourhood $U$ of
$X$ and a strongly plurisubharmonic function $\rho:U\to R$, such that $\dd \rho$ never
vanishes and $X=\{\rho=0\}$.

The situation of interest is the case where $X$ separates $Y$ into two
connected components; the “inside” component $\Omega$ (the one which
$\{\rho<0\}$) is then called a strongly pseudoconvex domain,
independently on the behaviour of $\rho$ far inside $\Omega$.

Given a general open set $\Omega\subset Y$ with regular boundary, one
may ask how holomorphic functions on $\Omega$ behave near $\partial
\Omega$: whether they can be extended (using, for instance, the
Hartogs theorem) or whether their restrictions to $\partial \Omega$
admit a simple description. As it turns out, these two questions are
related and strongly pseudoconvex domains are exactly those for
which restrictions of holomorphic functions are solutions of a
hypoelliptic linear PDE, the boundary Cauchy-Riemann operator $\overline{\partial}_b$
[27]. The Szegő
projector $S:L^2(X)\to \ker_{L^2}(\overline{\partial}_b)$ allows to describe
the space of solutions, as well as (after application of the Poisson
kernel) the Bergman space of holomorphic functions on $\Omega$. The
description of $S$ modulo smooth functions
[14, 7] when $X$ is
smooth has numerous crucial applications in complex and Kähler
geometry as well as geometric quantization. For example, the local
definition of a strongly pseudoconvex hypersurface above, or even an
intrinsic definition mimicking the properties of $X$ gathered in
the next proposition, are equivalent to the realisation of $X$ as the
boundary of a strongly pseudoconvex open set in $\C^n$, with a globally
defined and strongly plurisubharmonic $\rho$, as soon as $n\geq 3$

A strongly pseudoconvex hypersurface $X\subset Y$ inherits a
CR-structure as follows: $TX\otimes \C$ decomposes as a direct sum of
three involutive components $T^{(0,1)}X\oplus T^{(1,0)}X\oplus V$, where

* $T^{(1,0)}X=T^{(1,0)}Y\cap (TX\otimes \C)$ is the
$n-1$-dimensional bundle of holomorphic tangent vectors.

* $T^{(0,1)}X=T^{(0,1)}Y\cap (TX\otimes \C)$ is the
$n-1$-dimensional bundle of anti-holomorphic tangent vectors.

* $V$ is a line bundle over $X$.

The line bundle $\Sigma=(T^{(0,1)}X\oplus T^{(1,0)}X)^*\subset T^*X\otimes
\C$ is spanned by the real one-form $\alpha:(x,\xi)\mapsto \dd
\rho(x)(J\xi)$ whenever $\rho$ is a pseudoconvex defining function
for $X$. Moreover $\alpha$ is a contact form on $X$, and $\Sigma$ is
a symplectic submanifold of $T^*X$.

Let us prove that $\alpha$ is a contact form; this follows from an explicit computation in local coordinates

\[\alpha\wedge (\dd\alpha)^{n-1}=\|\nabla
\rho\|
\det([\partial_{z_j}\partial_{\overline{z_k}}\rho]_{j,k})\dd {\rm

where $\dd {\rm vol}$ is the standard volume form in coordinates and
the Hessian of $\rho$ is restricted to $T^{(0,1)}X\otimes
T^{(1,0)}X$, where it is still non-degenerate.

The other claims follow directly from there.

Given $u\in C^1(X,\C)$, one naturally forms $\overline{\partial}_bu$
as the section of $(T^{(0,1)}X)^*$ obtained by restricting $\dd u$ to
$T^{(0,1)}X$. Restrictions to $X$ of holomorphic functions naturally
satisfy $\overline{\partial}_bu=0$, and reciprocally elements of $\ker_{L^2}
\overline{\partial}_b$ extend into $H^{\frac 12}$ holomorphic
functions on $\Omega$ in the strongly pseudoconvex case [27].

In this article we will only be interested in the situation where $X$
(that is, $\rho$) is real-analytic. The symbol of the differential
operator $\overline{\partial}_b$ can then be extended in the complex,
which allows to define several manifolds of interest.

Let $Z=\{(x,\xi)\in \widetilde{T^*X},
\widetilde{\sigma(\overline{\partial}_b)}(x,\xi)=0\}$. Then $Z$ is
a complex submanifold of $\widetilde{T^*X}$ of complex codimension

$Z$ is coisotropic, and defining $\overline{Z}=\{(x,\xi)\in \widetilde{T^*X},
\widetilde{\sigma(\partial_b)}(x,\xi)=0\}$, then $Z$ and
$\overline{Z}$ intersect cleanly and

\[
Z\cap \overline{Z}=\widetilde{\Sigma}.
\]

Moreover $T\Sigma$ is transverse to the
null-distribution $(TZ)^{\perp}=\{v\in T(\widetilde{T^*X}), \forall
v'\in TZ, \omega(v,v')=0\}$.

Let us consider holomorphic coordinates mapping a neighbourhood in $Y$ of a
point of $X$ to a neighbourhood of $0$ in $\C^n$, with
$\partial_{\overline{z_n}}\rho\neq 0$. Then as a subset of
$\widetilde{T^*\C^n}=\{(x,y,\xi,\eta)\in \C^{4n}\}$,
$Z$ is the joint zero set of the functions

\begin{align*}
\xi_j+i\eta_j-\frac{(\partial_{x_j}+i\partial_{y_j})\widetilde{\rho}(x,y)}{(\partial_{x_n}+i\partial_{y_n})\widetilde{\rho}(x,y)}(\xi_n+i\eta_n)
\qquad \qquad 1\leq j \leq n-1\\
\widetilde{\rho}:(x,y,\xi,\eta)&\mapsto \widetilde{\rho}(x,y)\\
\widetilde{\dd \rho}:(x,y,\xi,\eta)&\mapsto \sum_{j=1}^n\xi_j\partial_{x_j}\widetilde{\rho}(x,y)+\eta_j\partial_{y_j}\widetilde{\rho}(x,y)
\end{align*}

The differentials of these functions are linearly independent, and
one can check that the functions $p_j$ Poisson-commute with each
other. Therefore $Z$ is a coisotropic submanifold of
$\widetilde{T^*X}$ whose codimension is the number of functions
$p_j$ (i.e. $n-1$).

To define $\overline{Z}$, we replace $p_j$ with the functions

\[
\xi_j-i\eta_j-\frac{(\partial_{x_j}-i\partial_{y_j})\widetilde{\rho}(x,y)}{(\partial_{x_n}-i\partial_{y_n})\widetilde{\rho}(x,y)}(\xi_n-i\eta_n)
\qquad \qquad 1\leq j \leq n-1.
\]

These functions again Poisson-commute with each other.

Now, clearly $\Sigma=Z_{\R}=(Z\cap \overline{Z})_{\R}$. The symplectic
gradients of the functions $p_1,\ldots,p_{n-1},p_1^*,\ldots,p_{n-1}^*$ are
linearly independent on $\Sigma$, and in fact the matrix of Poisson brackets
$\frac{1}{i}[\{p_j,p_k^*\}]_{j,k}$ is everywhere either positive definite (on
the half-line bundle $\Sigma_+$ of positive multiples of $\alpha$)
or negative definite (on the half-line bundle $\Sigma_-$ of negative
multiples of $\alpha$), see the computations in
[7]. This concludes the proof; in
particular, the Hamiltonian vector fields of $p_1,\ldots,p_{n-1}$, which span $(TZ)^{\perp}$, are transverse to $\Sigma$.

From now on we decompose $\Sigma=\Sigma_+\cup\Sigma_-\cup\{0\}$ where
$\Sigma_\pm$ is the half-line bundle of positive multiples of $\pm
\alpha$, on which $\pm\frac{1}{i}[\{p_j,p_k^*\}]_{j,k}$ is positive definite.

An important consequence of the geometry of $\Sigma$ and $Z$ is the
following CR-extension result for $\Sigma$-Lagrangians.

Let $\lambda\subset \Sigma_+\times \Sigma_+$ be a real-analytic conical
Lagrangian for the twisted symplectic form $\omega_1-\omega_2$.

There exists a unique $\Lambda\subset Z\times \overline{Z}$,
Lagrangian in $T^*X\times T^*X$ for the twisted symplectic form,
which contains $\lambda$. Moreover $\Lambda$ is positive (in the
sense of [29], Definition 3.3).

We proceed as in [7]. Let $\Lambda\subset Z\times \overline{Z}$ be a Lagrangian which
contains $\widetilde{\lambda}$. Then the null-distribution $(TZ^{\perp}\times
\{0\})\oplus(\{0\}\times
T\overline{Z}^{\perp})$ belongs to $T\Lambda$. By
Proposition <ref>, this
distribution is involutive (because $Z$ is coisotropic); it is
transverse to $T\tilde{\lambda}$; its complex dimension is
$2(n-1)=\dim(\Lambda)-\dim(\widetilde{\lambda})$. Therefore
$\Lambda$ is exactly the union of the leaves of this distribution
which pass through $\widetilde{\lambda}$. The positivity of $\Lambda$ is then a
direct consequence of the positivity of the matrix
$\tfrac 1i[\{p_j,p_k^*\}]_{j,k}$ on $\Sigma_+$.

§.§ Classical normal form

At the level of the operator $\overline{\partial}_b$, the
strong pseudoconvexity of $X$ is reflected in the following facts:

* the real
characteristic $\Sigma=\{(x,\xi)\in
T^*X,\sigma(\overline{\partial}_b)=0\}$ is symplectic

* denoting
$(p_1,\ldots,p_{n-1})$ the components of $\sigma(\overline{\partial}_b)$,
one has $\{p_j,p_k\}=0$ for all $j,k$, while the matrix
$\frac{1}{i}[\{p_j,\overline{p_k}\}]_{j,k}$ is everywhere either positive
definite or negative definite.

There is a universal local model (depending only on the dimension) for
the principal symbols of such operators, whose construction is presented in Appendix II of
[5] in the smooth case. We
review this proof in the analytic setting. This local model is
an important tool in the description of the Szegő projector in the
smooth case, and we will also use it in our setting. The local model
we will use is the vector-valued differential operator $D_0$ acting on
$\R^{n-1}_x\times \R^n_y$:

\[
(D_0)_j=-i\frac{\partial}{\partial x_j}-x_j\frac{\partial}{\partial
y_1}\qquad \qquad 1\leq j \leq n-1.
\]

This differs from the local model considered in
[7]. Our choice of $D_0$, with a polynomial total
symbol, will facilitate
handling subprincipal terms in the quantum normal form in Section

[See also
[5], Proposition 10.10, for
the $C^{\infty}$ statement] Let $1\leq r\leq m$. Let $\Sigma\subset T^*\R^m$ be an open cone. Let $p_1,\ldots,p_r$ be
$\frac 12$-homogeneous, real-analytic, complex-valued functions near $\Sigma$, such

* All the Poisson brackets $\{p_j,p_k\}$ and $\{\overline{p_j},\overline{p_k}\}$, for
$1\leq j,k\leq r$, vanish on $\Sigma$.

* All the Poisson brackets $\frac{1}{i}\{p_j,\overline{p_k}\}$, for $1\leq j,k\leq r$,
are equal to $\delta_{jk}$ on $\Sigma$.

Then, in a conical neighbourhood of $\Sigma$, there exist
$p_1',\ldots,p_r'$, homogeneous, real-analytic,
equal respectively to $p_1,\ldots,p_r$ along with
their first derivatives on $\Sigma$, and such that
$\{p'_j,p'_k\}=\{p'_j,\overline{p'_k}\}=\frac{1}{i}\{p'_j,\overline{p'_k}\}-\delta_{jk}$ is equal to
$0$ everywhere, and the ideal generated by $p_1,\ldots,p_r$
coincides with the ideal generated by $p_1',\ldots,p_r'$.

We proceed by induction on $r$. The claim follows immediately from the following
two properties:

* Let $V\subset T^*\R^n$ be a conical open set. Let $p:V\to \C$
be $\tfrac 12$-homogeneous, real-analytic; suppose that
$\{p,\overline{p}\}$ bounded away from $0$ on $V$. (In particular,
$\dd p$ is bounded away from $0$ on $V$.)

Then there exists a small neighbourhood $W$ of $\{p=0\}$, and $p':W\to \C$, with the same properties, such
that $\{\widetilde{p}=0\}=\{\widetilde{p'}=0\}$, and
$\frac{1}{i}\{p',\overline{p'}\}=1$. Moreover, for every function
$q:V\to \C$ such that $\{p,q\}=\{\overline{p},q\}=0$, one has also

* Let $V\subset T^*\R^n$ be a conical open set. Let $p_1,\ldots,p_r,q:V\to \C$
be $\tfrac 12$-homogeneous and real-analytic; suppose that
$\{\widetilde{p}_1=\ldots=\widetilde{p}_r=0\}$ is a regular energy
level near $V$, and that
$\{p_j,p_k\}=\{\overline{p_j},\overline{p_k}\}=\frac{1}{i}\{p_j,\overline{p_k}\}-\delta_{jk}=0$. Suppose
also that $\{p_j,q\},\{\overline{p_j},q\}$ vanish on
$\{p_j=\overline{p_j}=0\}$, on which $\{q,\overline{q}\}$ is
bounded away from $0$.

Then there exists a small neighbourhood $W$ of $\{p_j=\overline{p_j}=0\}$ and
$q':W\to \C$ a $\frac{1}{2}$-homogeneous, real-analytic function,
which is equal to $q$ along with its first total differential on
$\{p_j=\overline{p_j}=0\}$ (in particular $\{q',\overline{q'}\}$
is bounded away from $0$ and the ideal generated by
$p_1,\ldots,p_r,q$ coincides with the ideal generated by
$p_1,\ldots,p_r,q'$), and such that
$\{p_j,q\}=\{\overline{p_j},q\}=0$ everywhere.

Let us prove property 1 first. We want to correct
$p$ into $p'=ap$, where $a$ is real-valued. After this modification, the symplectic bracket
with the complex conjugate reads

\[
\{p',\overline{p'}\}=|a|^2\{p,\overline{p}\}+a\overline{p}\{p,a\}+ap\{a,\overline{p}\}.
\]

Letting $A=a^2$, this simplifies into

\[
\{p',\overline{p'}\}=A\{p,\overline{p}\}+\frac 12
p\{A,\overline{p}\}+\frac 12 \overline{p}\{p,A\}.
\]

Therefore we want to solve the transport equation

\[
Y\cdot A + \tfrac{1}{i}\{p,\overline{p}\}A=1
\]

where $Y$ is the (real-valued) vector field
$\frac{1}{i}(-p\Xi_{\overline{p}}+\overline{p}\Xi_p)$ (where $\Xi_f$ denotes the
symplectic gradient of $f$).

This vector field is singular along the codimension 2 set
$\{Y=0\}=\{p=\overline{p}=0\}$. Moreover it enjoys an absorbing property:
all trajectories of $\pm Y$ converge exponentially fast to the
singularity in positive time. Indeed,
${\rm div}(Y)=\frac{1}{i}\{p,\overline{p}\}$ is bounded away from
$0$ (and therefore is either positive everywhere or negative

Therefore, by the method of characteristics, there exists a unique
$A$ in a neighbourhood of $\{Y=0\}$, real-analytic, and such that
$Y\cdot A + \frac{1}{i}\{p,\overline{p}\}A=1$, and moreover it is real-valued.

If $q$ is such that $\{p,q\}=\{\overline{p},q\}=0$, then both  $Y$
and $\{p,\overline{p}\}$ commute with $q$, so that $A$ does as well.

To prove property 2, we simply observe that the complex submanifold
$\{\widetilde{p_j}=\widetildeoverline{p_j}=0\}$ is transverse to
the flows of $\widetilde{p_j}$ and $\widetilde{\overline{p_j}}$. We
can therefore set $q'=\widetilde{q}$ on this set, then extend it to
be constant along the flows of $\widetilde{p_j}$ and

Let $P_0\in \Sigma_{\pm}$. There exists a real-analytic contact
transformation $\kappa$ from a conical neighbourhood of $P_0$ in
$T^*X\setminus \{0\}$ to a conical neighbourhood of
$(0,0, 0,(\pm 1,0))$ in $T^*(\R^{n-1}\times \R^{n})$, mapping
$P_0$ to $(0,0,0,(\pm 1, 0))$, and a real-analytic, $0$-homogenous
map $C$ from a conical neighbourhood of $x_0$ in
$T^*X\setminus \{0\}$ to $GL(\C^{n-1},\Omega^{(1,0)}(X))$, such that

\[
\sigma(\overline{\partial}_b)(z,\zeta)=C\sigma(D_0)(\kappa(z,\zeta)).
\]

(of Proposition <ref>)
Let us apply Lemma <ref> to the
$\overline{\partial}_b$ operator: the components
$Z_1,\ldots,Z_{n-1}$ of its symbol satisfy, near any point of the
characteristic set $\Sigma^{\pm}$,

\[
\{Z_j,Z_k\}=0 \text{ on }\Sigma \qquad \qquad
\pm\tfrac{1}{i}\{Z_j,\overline{Z_k}\}>>0.
\]

Let $M:\Sigma^+\to GL(n-1)$ be the positive square root of the
matrix $\pm\tfrac{1}{i}\{Z_j,\overline{Z_k}\}$; extend $M$ arbitrarily into a
$\frac 12$-homogeneous, real-analytic function from a neighbourhood
of $\Sigma^+$ into $GL(n-1)$. Let $z=M^{-1}Z$. The ideal generated
by $z$ coincides with that generated by $Z$, and on particular $z$
vanishes on $\Sigma^+$, where

\[
\{z_j,z_k\}=\sum_{l,m}(M^{-1})_{jl}(M^{-1})_{mk}\{Z_l,Z_m\}=0,
\]

\[
\tfrac{1}{i}\{z_j,\overline{z_k}\}=\sum_{l,m}(M^{-1})_{jl}\overline{(M^{-1})_{km}}\tfrac{1}{i}\{Z_l,\overline{Z_m}\}=\pm
\delta_{jk}.
\]

Then, we let $u_j=\re(z_j)$ and $v_j=\pm \im(z_j)$ and directly
obtain the situation of Lemma <ref>:

\[\{u_j,u_k\}=\{v_j,v_k\}=0 \qquad \qquad \{v_j,v_k\}=\delta_{jk}.\]

Letting $z'_j=u'_j+iv'_j$, then the ideal generated by $z'$
coincides with that generated by $Z$.

We are almost ready for the normal form. We let $w$ be any
non-vanishing, $\frac 12$-homogeneous, real-valued function on a
neighbourhood of $\Sigma$ which commutes with $z'_1,\ldots,z'_{d}$,
and we consider the following functions:

\[
x_j=u'_jw^{-1} \qquad \xi_j=v_j'w \qquad \eta_1=\pm w^2.
\]

By construction, the $x_j$ are $0$-homogeneous while $\eta_1$ and
the $\xi_j$ are $1$-homogeneous, and
$\{x_j,x_k\}=\{x_j,\eta_1\}=\{\xi_j,\eta_1\}=\{x_j,\xi_k\}-\delta_{jk}=0$. The
ideal of real-analytic functions generated by the one-homogeneous
family $(d_0)_j=x_j\eta_1+i\xi_j=wz'_j$ coincides with the ideal of
real-analytic functions generated by $z'$, which is the same as the
ideal generated by $Z$. All in all, we obtain that, for some
invertible matrix $C$, one has $Z=Cd_0$.

Completing the coordinates
$(x_1,\ldots,x_{d},\xi_1,\ldots,\xi_{d},\eta_1)$ into a canonical
transformation of a neighbourhood of $\Sigma^+$ into a neighbourhood
of $x=\xi=0,y=0,\pm \eta_1>0,(\eta_2,\ldots,\eta_n)=0$ in $T^*\R^{n-1}\times T^*\R^n$, we can
exhibit a canonical transform $\kappa$ as requested.

§.§ Results from the smooth theory

<ref>, <ref>,
<ref> and <ref>
are also valid in the $C^{\infty}$ category, after a suitable
replacement of holomorphic extensions by almost holomorphic
extensions (in fact, our method of proof for the classical normal form applies to this situation as
well, and is much simpler than the one presented in Appendix 2 of [5]). The main result of [7] is
then that, with $\lambda={\rm diag}(\Sigma_+)$, and $\Lambda$ as in
Proposition <ref>, then the Szegő
projector $S$ on $X$ is a Fourier Integral operator associated with
$\Lambda$, modulo an operator which sends $\mathcal{D}'(X)$ into
$\mathcal{D}(X)$. Since $\Lambda$ is a half-line bundle over $X\times
X$, one can represent this Fourier Integral operator, globally, using
a phase function with one phase variable, as follows:

\[
S(x,y)=\int_0^{+\infty}e^{it\psi(x,y)}s(x,y;t)\dd t.
\]

One can construct $\psi$ using the almost holomorphic extension of a
defining function $\rho$.

The Lagrangian $\Lambda$ is idempotent (and indeed $S$ is a
projector), and a study of Fourier Integral Operators with
Lagrangian $\Lambda$ is  performed in
[1]. This book avoids the notion of
Fourier Integral operators with complex phases and only considers
operators of the form $SPS$ where $P$ is a pseudodifferential operator
on $X$, but it turns out that general
Fourier Integral Operators with Lagrangian $\Lambda$, in the smooth
category, are of this form. We will use the following precise version
of this result.

Let $A$ be a (smooth) Fourier Integral operator with Lagrangian
$\Lambda$. Then there exists a
pseudodifferential operator $Q$ on $X$ such that $[Q,S]$ and $A-SQD$ are
continuous from $\mathcal{D}'$ to $\mathcal{D}$.

By induction we only need to prove the result at principal order, and by
Proposition 2.13 in [1] we may
drop the condition that $Q$ and $S$ commute. Therefore we are only
searching for the principal symbol $q_0$ of a pseudodifferential
operator $Q$, such that the principal symbol of $SQS$ is that of
$A$. An application of stationary phase shows that $SQS$ has an
distributional kernel of the form

\[
SQS(x,y)=\int_0^{+\infty}e^{it\psi(x,y)}a(x,y;t)\dd t,
\]

with principal symbol $a:\Sigma_+\to \R$ equal to $q_0$. Hence,
given $A$, we can extend arbitrarily its principal symbol to the
whole of $T^*X$ and obtain an approximation at first order.

§.§ The case of the cylinder

A particular case of a pseudoconvex hypersurface (albeit not compact)
is $\{z\in \C^n, |\im(z)|=1\}$; indeed the function $\rho:z\mapsto
|\im(z)|^2$ is strongly plurisubharmonic everywhere.

There is an explicit description of the Szegő kernel in this case,
and one can directly establish many properties which we will
generalise only in Section <ref>. This
particular case will be useful in our construction and
characterisations of general Fourier Integral Operators in
real-analytic regularity in Section <ref>.

The Szegő kernel on $\R^n\times S^{n-1}$ is

\[
\xi}}{\int_{S^{n-1}}e^{2y\cdot \xi}\dd y}\dd \xi,
\]

which defines a holomorphic function of $(z,\overline{w})$ on $(\R^n\times \overline{B_{\R^n}(0,1)})^2\setminus {\rm
diag}(\R^n\times S^{n-1})$.

In particular, the Szegő projector $S$ has a distributional
kernel analytic away from the

A related formula concerning the Bergman kernel appears on p. 100 of the manuscript [13].

The first step is the study of the function

\[
m_n:\xi\mapsto \int_{S^{n-1}}e^{y\cdot \xi}\dd y.
\]

This defines an holomorphic function on $\C^n$.
When restricted to $\R^n$, it is a radial function and, by the
Laplace method, its asymptotics for large argument are

\begin{equation}\label{eq:mn_first_term}
\end{equation}

where $|S^{n-2}|$ is the area of the unit $n-2$-sphere.
Let $(z,w)\in (\R^n\times \overline{B_{\R^n}(0,1)})^2\setminus {\rm
diag}(\R^n\times S^{n-1})$. Then $|\im(z-\overline{w})|\leq 2$ and
furthermore $|\im(z-\overline{w})|=2\Rightarrow
\re(z-\overline{w})\neq 0$. Thus, by formula (<ref>), the integral

\[
\int_{\R^n}e^{i(z-\overline{w})\cdot \xi}(m_n(2\xi))^{-1}\dd \xi
\]

is well-defined, real-analytic on its domain, and holomorphic with
respect to $(z,\overline{w})$.

One can directly check that $K(z,w)$ is

It remains to prove that, for every $f\in L^2(\R^n\times
S^{n-1})$ which extends into a holomorphic function inside
$\R^n\times B(0,1)$, one has $f(z)=\int K(z,w)f(w)\dd w$. First,
by the Fourier inversion formula,
for every $x+iy\in \R^n+iS^{n-1}$,

\[
\R^n}e^{i\xi\cdot(x-x')}f(x'+iy)\dd x'\dd \xi.
\]

we deform the contour of integration over $x'$ by translating by $i(w-y)$ where $w\in S^{n-1}$; we obtain, for every $w\in S^{n-1}$,

\[
\R^n}e^{i\xi\cdot(x+iy-(x'+iw))}f(x'+iw)\dd x'\dd \xi.
\]

We now average this over $w\in S^{n-1}$ with respect to the probability density

\[
w\mapsto \frac{e^{-2w\cdot \xi}}{m_n(2\xi)},
\]

and obtain exactly

\[
f(x+iy)=\frac{1}{(2\pi)^n}\int_{\R^n\times \R^n\times
B(0,1)}e^{i\xi\cdot(x+iy-(x'-iw))}(m_n(2\xi))^{-1}f(x'+iw)\dd \xi\dd
x' \dd w.
\]

This concludes the proof.

$\overline{\partial}_b$ is locally analytic
hypoelliptic on $\R^n\times S^{n-1}$, by which we mean the following
property: if $u\in L^2(\R^n\times
S^{n-1})$ and $\overline{\partial}_bu$ is real-analytic on
an open set $V$, then $(1-S)u$ is real-analytic on $V$.

It suffices to prove the claim in the case where $V$ is bounded. Let $V_1\Subset V$ be a smaller open set and let
$(\chi_N)_{N\in \N}$ be a family of smooth functions supported on
$V$, such that $\chi_N=1$ on $V_1$, and

\[
\exists \rho, \forall j\leq N, |\nabla^j\chi_N|\leq (\rho N)^j.
\]

Such cutoffs will be used systematically in Section
<ref>, and we postpone until Proposition
<ref> the proof of their existence. Let us
recall from the Leibniz formula that

\[
u\in C^{\omega}(V)\Rightarrow \exists C,R, \forall n\in \N, \|\nabla^n(\chi_nu)\|_{L^2}\leq
C(Rn)^n\Rightarrow u\in C^{\omega}(V_1) .
\]

Since $V$ is relatively compact, we have more generally the
following characterisation: let $(Z_1,\ldots,Z_k)$ be a finite family of
real-analytic vector fields on a neighbourhood of $V$, which span
$TV$ at each point. Then

\[
u\in C^{\omega}(V)\Rightarrow \exists C,R,\forall n\in \N, \forall \mu\in
\llbracket 1,k\rrbracket^n,\|Z_{\mu_1}Z_{\mu_2}\cdots
Z_{\mu_n}(\chi_nu)\|_{L^2}\leq C(Rn)^n\Rightarrow u\in C^{\omega}(V_1).
\]

The following vector fields are particularly suited to our problem:

\begin{align*}
X_j&=\frac{\partial}{\partial x_j}\qquad  & 1\leq j\leq n\\
\omega_j}-\omega_j\frac{\partial}{\partial
\omega_k}-x_k\frac{\partial}{\partial
x_j}+x_j\frac{\partial}{\partial x_k} \qquad &
\end{align*}

Indeed, these vector fields span $T(\R^n\times S^{n-1})$, and they commute
with $\overline{\partial}$; hence they also commute with the
Szegő projector $S$. Now, given $u\in L^2$ such that
$\overline{\partial}u$ is real-analytic on $V$, one has, for every
$n\in \N$ and every multi-index $\mu\in \llbracket

\begin{align*}
\|Z_{\mu_1}\cdots
&\leq \|\overline{\partial}Z_{\mu_1}\cdots
\end{align*}

where we used Kohn's hypoellipticity result $\|(1-S)u\|_{L^2}\leq
\|\overline{\partial}w\|_{L^2}$ [27].

To conclude, since $\chi_n=1$ on $V_1$ and the kernel of $S$ is analytic away
from the diagonal by Proposition <ref>, then for every $V_2\Subset
V_1$, the sequence $((1-S)(1-\chi_n)u)_{n\in \N}$ is bounded in some fixed
space of analytic functions on $V_2$. Therefore we can conclude that

\[
\exists C,R, \forall \mu\in \llbracket 1,k\rrbracket^n,\|Z_{\mu_1}\cdots
Z_{\mu_n}\overline{\partial}(1-S)u\|_{L^2(V_2)}\leq C(Rn)^n,
\]

and the proof is complete: $(1-S)u$ is real-analytic on $V_2$.

§ AMPLITUDES AND PHASES

This section serves as a technical preliminary for the definition and
study of Fourier Integral Operators in Section
<ref>. We study in detail spaces of amplitudes
adapted to the real-analytic regularity, and we state and prove a stationary
phase result for these spaces of amplitudes.

§.§ Amplitudes in analytic microlocal analysis

This section takes inspiration from [32, 11, 38].

Convention: given $U\subset \R^n$ open, $x\in U$, and $u\in
C^j(U,\C)$, we let

\[
\N^n,|\alpha|=j}|\partial^{\alpha}u(x)|^2\right)^{\frac 12}.
\]

Let $\Omega\subset \R^m_x\times \R^n_{\theta}$ be an open cone in
the second variable, and let
$d\in \R$. A (homogeneous classical) formal analytic
amplitude of degree $d$ is a sequence $(a_k)$ of real-analytic functions on
$\Omega$ such that $a_k$ is homogeneous of degree $d-k$, and such
that there exists $C>0, \rho>0,R>0,m>0$ satisfying the following:
for every $j,k,\ell\in \N_0$, for every
$(x,\theta)\in \Omega$,

\[
\]

For every $m,\rho,R$, the best constant $C$ in this inequality is
written $\|a\|_{S^{\rho,R}_m(\Omega)}$.

If $(a_{\ell})_{\ell\in \N}$ is a formal analytic amplitude, then for every
$\ell$, $a_{\ell}$ is a real-analytic function of $(x,\theta)$, which
extends to a fixed size neighbourhood of the real locus, where it
satisfies $\|a_{\ell}(1+|\theta|)^{-d+\ell}\|_{L^{\infty}}\leq
C(R')^{\ell}\ell!$. This turns out to be an equivalent definition of
a formal analytic amplitude. If $\Omega$ is compact with respect to
the first variable,
one-homogeneous analytic changes of
variables in $x,\theta$ preserve
the space of all formal analytic amplitudes, even though they may not
individually preserve the Banach spaces $S^{\rho,R}_m(\Omega)$.

Given a paracompact conical analytic manifold, we may then define
formal analytic amplitudes on $X$ as sequences of real-analytic
functions on $X$ with decreasing homogeneity, which satisfy the
controls of Definition <ref> locally in charts.

Let $\Omega\subset \R^m_x\times \R^n_{\theta}$ be an open cone in
the second variable, and let $d\in \R$. An analytic amplitude of degree $d$ on $\Omega$ is a smooth
function $a$ on $\Omega$ such that  there exists
$R>0,\rho>0,C>0$ satisfying the following:

\[
\forall (x,\theta )\in \Omega,\forall k\leq
|\theta|/R,\forall j,
\]

An analytic amplitude is real-analytic with respect to $x$, and
extends into a holomorphic function of $x$; an
alternative definition of an analytic amplitude is the estimate

\[\forall k \leq |\theta|/R, |\nabla_{\theta}^ka(x,\theta)|\leq C\rho^kk!|\theta|^{d-k}\]

on a whole neighbourhood of the real locus in $x$. Thus, again, we may speak
of analytic amplitudes on $\Omega\subset X\times \R^n_{\theta}$ when
$X$ is a paracompact analytic manifold.

Despite their names, the analytic amplitudes as in Definition <ref> are not
real-analytic functions. The main reason for this
definition is that the Borel summation of a formal analytic amplitude into an
analytic amplitude (Proposition <ref>) will involve a lowest term summation, which requires
cut-off functions in the variable $|\theta|$. Trèves calls such functions
“pseudo-analytic” [38] but unfortunately this
term, like many other variations upon the word “analytic”, already
refer to well-established mathematical objects.

Analytic amplitudes are stable by
“phasification” of the variables: denoting $x=(x_1,x_2)$, where
$x_1\in \R^{m_1}$ and $x_2\in \R^{m_2}$, and similarly
$\R^{m_2+n}\ni \theta=(\theta_1,\theta_2)\ni \R^{m_2}\times \R^n$, if some
amplitude $a$ is analytic on $\Omega\subset \R^{m_1+m_2}\times (\R^{n}\setminus \{0\})$, then so is

\[
(\R^{m_1})\times (\R^{m_2}\times (\R^n\setminus \{0\}))\supset
\Omega \ni (x_1,\theta)\mapsto a((x_1,\theta_1/|\theta_2|),\theta_2).
\]

The converse, however, is not true. We will say that an amplitude $a$
is maximally analytic if, writing $\theta=\lambda\omega$ in
spherical coordinates, the function $((x,\omega),\lambda)\mapsto
a(x,\lambda\omega)$ is an analytic amplitude.

A last immediate consequence of Definition <ref> is that analytic amplitudes belong to the usual
Hörmander symbol classes, even as they are extended holomorphically
with respect to the first variable.

In order to build analytic amplitudes starting from formal analytic amplitudes, the
following construction of cut-off functions will be useful.
(Ehrenpreis cutoffs)
Let $K,L$ be closed sets of $\R^d$ at positive distance from each other. There exists a
sequence of compactly supported functions
$(\chi_N)_{N\geq 1}\in C^{\infty}(\R^d,[0,1])$ such that $\chi_N=0$
on $L$ and $\chi_N=1$ on $K$, and a constant $\rho>0$ such that

\begin{equation}\label{eq:Ehrenpreis}
\forall N\in \N, \forall j\in \N, j\leq N\Rightarrow
|\nabla^j\chi_N|\leq (\rho N)^j.
\end{equation}

Let $K\subset K'$ such that
$K'\cap L=\partial K'\cap K=\emptyset $, and let
$c=\min(\dist(K',L),\dist(K,\partial K'))>0$.

Let $\phi\in C^{\infty}(\R^d,[0,+\infty))$ be supported on $B(0,1)$
and such that $\int_{\R^d}\phi=1$. Given $\alpha>0$, let

\[\phi_{\alpha}:x\mapsto \alpha^{-d}\phi(\alpha^{-1}x).\]

The claim follows from the following choice of cut-off functions:

\[
\chi_N=\underbrace{\phi_{cN^{-1}}*\ldots
*\phi_{cN^{-1}}}_{N\text{ times }}*\mathds{1}_{K'}.
\]

Indeed, letting $\rho=\|\nabla \phi_c\|_{L^1}$, one has $\|\nabla
\phi_{cN^{-1}}\|_{L^1}=\rho N$, while $\|\phi_{cN^{-1}}\|_{L^1}=1$ and
$\|\mathds{1}_{K'}\|_{L^{\infty}}=1$; altogether we find the desired bound.

Let $\Omega\subset \R^m_x\times \R^n_{\theta}$ be an open cone
with respect to the second variable. Let $(a_k)_{k\in \N}$ be a degree $d$ formal analytic amplitude on
$\Omega$. Then there exists a degree $d$ maximally analytic amplitude $a$ on
$\Omega$, and
constants $C,\rho$ such that, uniformly on a complex neighbourhood
of $\Omega$ in the spherical variable,
for every $N\in \N$,

\[
\left|a(x,\theta)-\sum_{k=0}^{N-1}a_k(x,\theta)\right|\leq
\]

Any analytic amplitude whose extension satisfies the bound above
will be called a realisation of the formal amplitude $(a_k)_{k\in
\N}$. Two extensions of analytic amplitudes realising the same formal amplitude
have a difference bounded by $e^{-c|\theta|}$, for some $c>0$, on a
whole conical complex neighbourhood of $\Omega$ with respect to the
first variable.

Let us first prove the last fact: letting $\widetilde{\Omega}$ be a
complex neighbourhood with respect to the first variable, suppose that

\[
\forall (x,\theta)\in \widetilde{\Omega}, \forall N\in \N,
|a(x,\theta)|\leq C\rho^NN!|\theta|^{d-N}.
\]

Writing $N=\alpha|\theta|$ and applying the Stirling formula gives

\[
|(x,\theta)|\leq C|\theta|^d(\alpha\rho/e)^{\alpha|\theta|}
\]

For $\theta$ large enough there is an integer $N$ between $\frac{e|\theta|}{4\rho}$
and $\frac{e|\theta|}{2\rho}$, and we obtain

\[
|a(x,\theta)|\leq C|\theta|^d(1/2)^{\frac{e|\theta|}{4\rho}};
\]

hence the claim.

We now give a direct construction of $a$.
$(\chi_N)_{N\in \N}:[0,+\infty)\to [0,1]$ be a sequence of
Ehrenpreis cutoffs
as in Proposition <ref>, equal to $1$ on $[0,1]$
and to $0$ on
$\R\setminus [0,2]$.

Given $(a_\ell)_\ell\in S^{\rho,R}_0(\Omega)$, the claim follows
from the choice

\[
\sum_{\ell=1}^{\infty}a_{\ell}(x,\theta)(1-\chi_{\ell+1}(\tfrac{c|\theta|}{\ell+1})),
\]
where $c$ is chosen small enough depending on $R$.

From now on we pass to spherical coordinates and write $\theta=\lambda\omega,X=(x,\omega)$.
Let us prove that $a$ is an analytic amplitude. Given
$(X,\lambda)\in \Omega$, $j\in \N$ and $k\leq \lambda/R$, one has

\[
\nabla^j_{X}\partial^k_{\lambda}a(X,\lambda)=\sum_{\ell\leq c\lambda}\nabla^j_{X}\partial^k_{\lambda}(a_{\ell}(X,\lambda))(1-\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1})).
\]

Let us first control the terms where no derivatives hit
$\chi_{\ell+1}$. If $c\leq \frac{1}{4R}$, then

\begin{align*}
\left|\sum_{\ell\leq
c\lambda}(1-\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1}))\nabla^j_{X}\partial^k_{\lambda}a_\ell(X,\lambda)\right| &
\leq \|a\|_{S^{\rho,R}_0}\sum_{\ell\leq
\|a\|_{S^{\rho,R}_0}(j+k)!(2\rho)^{j+k}\lambda^{d-k}\sum_{\ell\leq
&\leq 2\|a\|_{S^{\rho,R}_0}(j+k)!(2\rho)^{j+k}\lambda^{d-k},
\end{align*}

where we used the inequality $(j+k+\ell)!\leq
2^{j+k+\ell}(j+k)!\ell!$ and the fact that, on $0\leq \ell\leq
c\lambda$, the family $(2R)^{\ell}\ell!\lambda^{-\ell}$ decreases
with $\ell$ faster than a geometric sequence of ratio $\frac 12$.

Now we treat the case where one of the derivatives has hit
$\chi_{\ell+1}$. The support of $\lambda\mapsto
\chi_{\ell+1}'(\frac{c\lambda}{\ell+1})$ is included in
$\frac{c\lambda}{2}\leq \ell+1\leq c\lambda$. If moreover $k\leq
\epsilon \lambda$ with $\epsilon \leq \frac{c}{2}$, then for every
$m\leq k$, one has $m\leq \frac{c\lambda}{2}\leq \ell+1$. Therefore

\begin{multline*}
\left|\sum_{\ell\leq
\leq\|a\|_{S^{\rho,R}_0}\sum_{\frac{c\lambda}{2}-1\leq\ell\leq c\lambda-1}\sum_{m=1}^k\binom{k}{m}c^m\rho^{j+k-m}R^\ell(j+k-m+\ell)!\lambda^{d-k-\ell+m}\\
\leq \|a\|_{S^{\rho,R}_0}(4\rho)^{j+k}(j+k)!\lambda^{d-k}\sum_{\frac{c\lambda}{2}-1\leq\ell\leq c\lambda-1}\sum_{m=1}^kc^m(2\rho)^{-m}(2R)^{\ell}(\ell-m)!\lambda^{m-\ell}.
\end{multline*}

Since $m\leq \tfrac{\ell}{2}$, we perform a change of variables and obtain,
provided $c\leq\frac{1}{4R}$,

\begin{multline*}
\left|\sum_{\ell\leq
\leq
\|a\|_{S^{\rho,R}_0}k(8\rho^2R)^{j+k}(j+k)!\lambda^{d-k}\sum_{0\leq
p\leq c\lambda-1}(2R)^p\lambda^{-p}(p+1)!\\
\leq
\end{multline*}

It remains to show that
$a=\sum_{\ell=0}^Na_\ell+O_N(\lambda^{d-N-1})$ with the required growth
in $N\geq
0$. First

\[
\]

has compact support, and is uniformly bounded as follows:

\[\left|\sum_{\ell=0}^Na_\ell(X,\lambda)\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1})\right|
\leq
\|a\|_{S^{\rho,R}_0}\sum_{\ell=0}^NR^{\ell}\ell!\lambda^{d-\ell}\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1});\]

for this sum to be nonzero, we need $c\lambda\leq N+1$, moreover
$\ell\mapsto R^{\ell}\ell!\lambda^{d-\ell}$ is log-convex so that

\[\left|\sum_{\ell=0}^Na_\ell(X,\lambda)\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1})\right|
\leq
\|a\|_{S^{\rho,R}_0}\lambda^d(N+1)\max(1,R^NN!\lambda^{-N})\1_{\lambda\leq
\frac{N+1}{c}}.\]

Moreover if $\lambda\leq \frac{N+1}{c}$ then $N!\lambda^{-N}\geq
\left(\frac{c}{e}\right)^N$, so that, all in all, for some $R_1$ large enough,

\[\left|\sum_{\ell=0}^Na_\ell(X,\lambda)\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1})\right|
\leq
\|a\|_{S^{\rho,R}_0}R_1^{N+1}(N+1)!\lambda^{d-N-1}\]

It remains to bound

\[
\left|\sum_{\ell=N+1}^{+\infty}a_\ell(X,\lambda)(1-\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1})\right|\leq \|a\|_{S^{\rho,R}_0}
\sum_{\ell=N+1}^{c\lambda}R^{\ell}\ell!\lambda^{d-\ell}.
\]

If $c\leq \frac{1}{2R}$, the first term in this sum is
$R^{N+1}(N+1)!\lambda^{d-N-1}$, and the ratio between two consecutive
terms is smaller than $\frac{1}{2}$; thus

\[
\left|\sum_{\ell=N+1}^{+\infty}a_\ell(X,\lambda)\chi_{\ell+1}(\tfrac{c\lambda}{\ell+1})\right|\leq
\]

This concludes the proof.

It is not clear to us which natural conditions one can impose on the
formal amplitude $a$ to be the expansion at large $|\theta|$ of an
analytic function; many estimates and considerations in this section
are much easier in this case. This seems to be an instance of
resurgence. It would be interesting to study if the space of
such formal amplitudes is stable under the natural Fourier Integral
Operator manipulations.

Let $(a_j)_{j\in \N}$, $(b_j)_{j\in \N}$ be two formal analytic
amplitudes on an open cone $\Omega$. Let $a,b$ be respective analytic
realisations of $(a_j)_{j\in \N},(b_j)_{j\in \N}$. Then $ab$ is an
analytic amplitude, which realises the Cauchy product
$((a*b)_j)_{j\in \N}=(\sum_{\ell=0}^ja_{\ell}b_{j-\ell})_{j\in \N}$.

One can directly check that $ab$ is an analytic amplitude and that
$((a*b)_j)_{j\in \N}$ is a formal analytic amplitude. Now (counting some terms twice)

\begin{multline*}
\left|a(x,\theta)b(x,\theta)-\sum_{k=0}^{N-1}(a*b)_k(x,\theta)\right|\\ \leq
\left|a(x,\theta)b(x,\theta)-\sum_{j=0}^{\frac{N-1}{2}}\sum_{\ell=0}^{\frac{N-1}{2}}a_j(x,\theta)b_\ell(x,\theta)\right|+\sum_{j+\ell=\frac{N-1}{2}}^{N-1}|a_j(x,\theta)b_{\ell}(x,\theta)|.
\end{multline*}

The first term of the right-hand side is equal to

\[

so that

\[
\left|a(x,\theta)b(x,\theta)-\sum_{j=0}^{\frac{N-1}{2}}\sum_{\ell=0}^{\frac{N-1}{2}}a_j(x,\theta)b_\ell(x,\theta)\right|\leq
\]

To treat the second term, observe that, given $j+\ell\in

\begin{multline}\label{eq:ajbl}
C_{a,b}|\theta|^{d_a+d_b-j-\ell}\rho^{j+\ell}(j+\ell)!\\ \leq
\end{multline}

where in the last inequality we used the fact that the middle
quantity is a log-convex function of $j+\ell$. Altogether if
$|\theta|^{-\frac{N-1}{2}}\rho^{\frac{N-1}{2}}(\tfrac{N-1}{2})!\leq |\theta|^{-N+1}\rho^{N-1}(N-1)!$, the
proof is complete.

Similarly, one can write

\begin{multline*}
\left|a(x,\theta)b(x,\theta)-\sum_{k=0}^{N-1}(a*b)_k(x,\theta)\right|\\ \leq
\left|a(x,\theta)b(x,\theta)-\sum_{j=0}^{N-1}\sum_{\ell=0}^{N-1}a_j(x,\theta)b_\ell(x,\theta)\right|+\sum_{j+\ell=N-1}^{2N-2}|a_j(x,\theta)b_{\ell}(x,\theta)|,
\end{multline*}

and if $|\theta|^{-2N+2}\rho^{2N-2}(2N-2)!\leq
|\theta|^{-N+1}\rho^{N-1}(N-1)!$, we can conclude using
(<ref>) again.

The sequence $(v_k)_{k\in \N}=(|\theta|^{-k}\rho^kk!)_{k\in \N}$ is decrasing on
$\{k\leq |\theta|/\rho\}$ and increasing on $\{k\geq
|\theta|/\rho\}$. Therefore if both $v_{\frac{N-1}{2}}$ and $v_{2N-2}$
are greater than $v_{N-1}$, then $N-1\in
[\frac{|\theta|}{2\rho},2\frac{|\theta|}{\rho}]$. Under these
circumstances, by the Stirling formula, all of
$v_{\frac{N-1}{2}},v_{N-1},v_{2N-2}$ can be bounded by
$e^{-c|\theta|}$ for some $c>0$, and for $\rho'$ large enough,
$e^{-c|\theta|}\leq (\rho')^{N-1}|\theta|^{-N+1}(N-1)!$; this
concludes the proof.

Let $\Omega_1\subset \R^{n_x}\times \R^{n_{\theta}}\times
\R^{n_y}$ and $\Omega_2\subset \R^{n_y}\times \R^{n_{\tau}}\times
\R^{n_z}$ be open cones with respect to their middle variables. Let
$c>0$ and let

\[
\Omega=\{(x,\theta,y,\tau,z)\in \R^{n_x}\times \R^{n_{\theta}}\times
\R^{n_y}\times \R^{n_{\tau}}\times
\R^{n_z},(x,\theta,y)\in \Omega_1,(y,\tau,z)\in \Omega_2,
\]

Let $a_1:\Omega_1\to \R$ and $a_2:\Omega_2\to \R$ be analytic amplitudes
(real-analytic with respect to their first and third variables). Then

\[
A:\Omega\ni (x,\theta,y,\tau,z)\mapsto
\]

is an analytic amplitude (real-analytic with respect to its first and
last variable). Moreover, if $a_1$ and $a_2$ are respectively
realisations of formal analytic amplitudes $(a_{1;j})_{j\in
\N},(a_{2;j})_{j\in \N}$, then $A$ is a realisation of the Cauchy

\[
\left(\sum_{\ell=0}^ja_{1;\ell}(x,\theta,y)a_{2;j-\ell}(y,\tau,z)\right)_{j\in
\N},
\]

the latter being a formal analytic amplitude on $\Omega$.

The key property is that, on $\Omega$, the quantities
$|\theta|,|\tau|$ and $(|\tau|^2+|\theta|^2)^{\frac 12}$ are
comparable. Therefore $(x,\theta,y,\tau,z)\mapsto a_1(x,\theta,y)$
is an analytic amplitude on $\Omega$, which realises the formal
analytic amplitude $((x,\theta,y,\tau,z)\mapsto
a_{1;j}(x,\theta,y))_{j\in \N}$, and a similar property holds for
$a_2$. By Proposition <ref>, the proof is complete.

We conclude this section with a practical example, related to
Proposition <ref>, and which will allow us to prove
that the Szegő kernel takes the form of an analytic Fourier
Integral Operator, in the sense of Section <ref>.

Define the following open cone in $\C^d$:

\[
\Omega=\{z\in \C^d, |\im(z)|^2<|\re(z)|^2\};
\]

recall that the function $r:\Omega\ni z\mapsto
\sqrt{\sum_{j=1}^dz_j^2}$ is well-defined and holomorphic.

\[
a:z\mapsto e^{-r(z)}\int_{S^{n-1}}e^{z\cdot
y}\dd y
\]

is an analytic amplitude which realises the formal analytic

\[
\left(2^{\frac{d-1}{2}}|S^{d-2}|\frac{(d-1)(d-3)\ldots
(d-2j+1)}{2^jj!}\Gamma(\tfrac{d+1}{2}+j)r(z)^{-\frac{d+1}{2}-j}\right)_{j\in \N}.
\]

Let us first study the elementary properties of $a$ on $\Omega$.

$S^{d-1}$ is a real contour in the complex manifold $\widetilde{S^{d-1}}:=\{\omega\in \C^d,
\sum_j \omega_j^2=1\}$, on which $\omega\mapsto e^{z\cdot \omega}$ is
holomorphic. This manifold is preserved by the linear action of

\[
\widetilde{O(d)}:=\{M\in M_d(\C), M^TM=I\},
\]

whose action is transitive; more generally, for every $\lambda\in
\C\setminus \R^-$,  $\widetilde{O(d)}$ acts transitively on
$\{z\in \Omega_1, r(z)^2=\lambda\}$. The function $r$ is itself

To prove that $a$ itself is $\widetilde{O(d)}$-invariant, let $z\in \Omega$. By the previous considerations, there
exists $M\in \widetilde{O(d)}$ such that $Mz$ is of the form $\alpha
x$ where $x\in \R^{d}$ and $\re(\alpha)>0$. Since $\widetilde{O(d)}$
is connected, one can deform the contour of integration into
$\{M^{-1}\omega, \omega\in S^{d-1}\}$, and then $z\cdot
M^{-1}\omega=\alpha z\cdot \omega$. Thus $a(z)=a(Mz)$, so $a$ is $\widetilde{O(d)}$-invariant.

Using this $\widetilde{O(d)}$ invariance, to prove that $a$ is
the realisation of a formal analytic amplitude, it suffices to prove
the required estimate for $a(R\alpha e_1)$, where $R\to +\infty$, $|\alpha|=1$,
$\re(\alpha)>|\im(\alpha)|$, and $e_1$ is the first vector of the base.

One has

\[
a(R\alpha e_1)=|S^{d-2}|\int_{-1}^1 e^{-R\alpha
(x-1)}(1-x^2)^{\frac{d-1}{2}}\dd x,
\]

where $|S^{d-2}|$ is the area of the unit sphere in
$\R^{d-1}$. Letting $x=1-y$, we rewrite this integral into

\[
e^{\alpha R y}y^{\frac{d-1}{2}}(1-\tfrac{y}{2})^{\frac{d-1}{2}}\dd y.
\]

Let us decompose this integral into two parts. If $y\in [1,2]$,
then $|e^{-\alpha Ry}|=e^{-\re(\alpha)Ry}\geq
e^{-\frac{R}{\sqrt{2}}}$ is exponentially small. On $[0,1]$, we will
replace $(1-\tfrac{y}{2})^{\frac{d-1}{2}}$ by its Taylor series at
$0$. First

\[
\frac{\partial^j}{\partial
\]

Let $c$ be a small constant (to be determined later on), and let
$J_R=\lfloor cR\rfloor$. We write, for $y\in [0,1]$,

\[
\]

\[
\sup_{y\in [0,1]}|F_{J_R}(y)|\leq
\frac{(d-1)(d-3)\ldots(d-2J_R+1)}{4^{J_R}J_R!}.
\]

Notice that the sequence

\[
\left(\frac{(d-1)(d-3)\cdots(d-2j+1)}{2^jj!}\right)_{j\in \N}
\]

is bounded; for every $j\geq d/2$, from $j$ to $j+1$ the general
term is multiplied by $(d-2j-1)(j+1)\in [-1,0]$.

For every $k>0$, $y>0$ and $|\alpha|=1$ with

\[
|e^{-\alpha y}y^k|\leq e^{-\re(\alpha)y}y^k\leq
e^{-\frac{y}{\sqrt{2}}}y^k\leq k^ke^{k(\log(\sqrt{2})-1)}\leq k^k.
\]

We apply this inequality at $k=J_R+1+\frac{d-1}{2}$, and obtain

\[
\left|\int_0^1e^{\alpha R y}y^{\frac{d-1}{2}+J_R}F_{J_R}(y)\dd
y\right|\leq (cR)^{cR}R^{-cR}\leq Ce^{-c'R}
\]

for some $c'>0$, if $c$ is small enough. It remains

\[
R y}y^{\frac{d-1}{2}+j}\dd y.
\]

And to this end we use that, for every $k\leq \frac{R}{\sqrt{2}}$,

\[
\sup_{y\in [1,+\infty)}|e^{-\alpha Ry}y^k|\leq e^{-R/\sqrt{2}};
\]

applied at $k=j+\frac{d-1}{2}+2$, we obtain

\[
\left|\sum_0^{J_R}\frac{(d-1)(d-3)\ldots(d-2j+1)}{2^jj!}\int_1^{+\infty}
R y}y^{\frac{d-1}{2}+j}\dd y\right|\leq Ce^{-c'R}.
\]

Now we can conclude:

\[
\left|a(R\alpha
\]

and by $\widetilde{O(d)}$-invariance, for every $z\in \Omega$,

\[
\left|a(z)-\sum_{j=0}^{J_R}2^{\frac{d-1}{2}}|S^{d-2}|\frac{(d-1)(d-3)\ldots
\]

Using the Cauchy formula, we deduce a similar formula for all
derivatives of $a$, which concludes the proof that $a$ satisfies the
derivative bounds of an analytic amplitude.

§.§ Stationary phase

The goal of this section is to obtain a stationary phase theorem
adapted to our notion of analytic amplitudes. We will have to deal
with complex-valued phase functions, and our first step is a study of
the contour deformations necessary to reach the critical points.

Let $\mathcal{K}\Subset \R^n_x\times \R^m_y$. Let $\phi$ be a
real-analytic function from an open neighbourhood of $\mathcal{K}$
to $\{z\in \C,{\rm Im}(z)\geq 0\}$. Suppose that $\nabla^2_y\phi$,
the matrix of second derivatives with respect to the second set of
variables, is invertible everywhere on $\mathcal{K}$ and that ${\rm
Im}(\phi)>0$ on $\partial K$.

Then there exists a constant $c>0$ and a covering of $\mathcal{K}$
by two open sets $U_1$ and $U_2$ of $\R^{n+m}$ such that the
following is true.

* There exists a real-analytic map
$\kappa_1:U_1\to \R^n_x\times \C^m_y$ with the following

* $\kappa_1$ acts as the identity on the first variable.

* $\kappa_1$ is a diffeomorphism between $U_1$ and its image
$\Gamma_1\subset \R^n_x\times \{|{\rm Im}(y)|<c\}$.

* $\Gamma_1$ is a totally real submanifold of
$\C^n\times \C^m$.

* ${\rm Im}(\phi)\geq c$ on $\partial \Gamma_1$.

* $\kappa_1$ is isotopic to the identity among the maps
satisfying the four previous properties. (We will henceforth
call this isotopy a contour deformation, and call
$\Gamma_1$ a contour.)

* For every $x\in \R^n$, in each connected component of
$\Gamma_1(y):=\{y\in \C^m, (x,y)\in \Gamma_1\}$, the holomorphic extension
$\widetilde{\phi}$ of $\phi$ admits exactly one critical point $y^*$ with respect to
$x$ and the real part of $\widetilde{\phi}$ is constant.

* $\nabla^2_y({\rm Im}\widetilde{\phi}\circ \kappa_1)\geq c$, and the
phase, at the critical points, has non-negative imaginary part.

* There exists a contour $\Gamma_2$ deforming $U_2$, on which
${\rm Im}(\widetilde{\phi})\geq c$.

Let $V$ be a neighbourhood of $\mathcal{K}$ on which $\phi$ is
defined and $\det\nabla^2_y\phi$ is bounded away from $0$. Let
$\epsilon>0$ be small enough. We define

\[
U_1=\{(x,y)\in V, |\nabla_y\phi(x,y)|< \epsilon\} \qquad \qquad
U_2=\{(x,y)\in V, |\nabla_y\phi(x,y)|> \epsilon/2\}.
\]

Since $\det\nabla^2_y\phi$ is bounded away from $0$, for every
$(x,y)\in U_1$, there exists a unique critical point $(x,y^*(c))$ of
$\phi$ closeby (at distance of order $\epsilon$). Moreover, again
because of the nondegeneracy, a small neighbourhood of $(x,y^*(x))$
contains the connected component $V$ of
$U_1\cap (\{x\}\times \R^m)$ to which $(x,y)$ belongs.

On this neighbourhood of the critical point, we apply the
holomorphic Morse lemma: there exists a biholomorphism $\sigma$
(with real-analytic dependence on $x$) such that

We are able to deform $V$ into a neighbourhood of $0$ in
$\{\sigma\in i\R^n\}$, first by straightening $\sigma(V)$ into
its tangent space at some point $y_0$, then by translation and
rotation, since the set of the totally real linear subspaces is open
and connected.

On $U_2$, where $|\nabla_y\phi(x,y)|>\epsilon/2$, we deform using
the ascending gradient flow of ${\rm Im}(\widetilde{\phi})$ on $\C^m$. After a
time $\epsilon$, the imaginary part of the phase is incremented by a
quantity bounded below everywhere except in the vicinity of real
points where $\nabla_y{\rm Re}(\widetilde{\phi})$ is small. This can only happen
if $\nabla_y{\rm Im}(\widetilde{\phi})$ is large, which only happens if
${\rm Im}(\phi)$ is large to begin with. Thus, after this gradient
flow, ${\rm Im}(\widetilde{\phi})$ is bounded from below everywhere.

To conclude the proof, it remains to show that, among the critical
points sufficiently close to $U_1$, the imaginary part of the phase
is never negative. We proceed by contradiction and suppose that
there is such a critical point close to a point $P\in U_1$. Up to a
linear transformation by an element of $GL_m(\R)$, near $P$, the
second derivative of the phase with respect to the second variable
$y_1$ is nonzero. We can therefore apply the analytic Morse lemma in
the variable $y_1$, with all other variables as parameters. This
transforms $U_1$ into a piece of real curve in $\C$, parametrised by
the other variables. By hypothesis, the imaginary part of the phase
is strictly smaller on the image of the real locus by the Morse
biholomorphism than at the critical point. Hence, this piece of
curve belongs to either one of the two quarter spaces in
$\C$ where ${\rm Re}(y_1^2)>0$. Since these quarter spaces have a corner at $0$, the image of
$U_1$ cannot get close to $0$ without the second derivative of the
phase along $U_1$ getting large. This is not possible because
$\mathcal{K}$ is compact.

The contour deformation above allows us to prove a
stationary phase result. Mind that there is no assumption of
uniqueness of the critical point, so that the result of the stationary
phase is a sum of different contributions coming from the different
critical points.

Let $\Omega_1$ be an open subset of $\R^{n_{x}}\times
\R^{n_{\theta}}$, conic with respect to the
second variable and relatively compact, and let $\Omega_2$ be an
open, relatively compact subset of
$\R^{n_y}$. Let $\phi:\Omega_1\times \Omega_2\to \C$ be real-analytic and
one-homogeneous with respect to $\theta$, which extends to a small
neighbourhood of $\Omega_1\times \Omega_2$. Suppose the following:

* $\phi$ takes values in $\{{\rm Im(z)}\geq 0\}$.

* $\nabla^2_y\phi$ is nonsingular.

* ${\rm Im}(\phi)>0$ on $\partial(\Omega_1\times \Omega_2)$.

Let $a:\Omega_1\times \Omega_2\to \C$ be such that
$((x,y),\theta)\mapsto a(x,\theta,y)$
is an analytic amplitude.

Then there exists a finite collection of open conic subsets
$V_1,\ldots,V_N$ of $\Omega$, real-analytic zero-homogeneous functions $y^*_j:V_j\to
\widetilde{\Omega_2}$, analytic amplitudes $b_j$ on $V_j$, and $\epsilon>0$, such
that, for every $(x,\theta)$ in a small complex neighbourhood of
$\Omega$ with respect of the first variable,

\[
\int_{\Omega_2}e^{i\phi(x,\theta,y)}a(x,\theta,y)\dd y -
\sum_{j=1}^N\1_{(x,\theta)\in V_i}e^{i\widetilde{\phi}(x,\theta,y^*_j(x,\theta))}b_j(x,\theta)=O(e^{-\epsilon|\theta|}).
\]

Moreover ${\rm Im}(\phi)(x,\theta,y^*_j(x,\theta))\geq 0$ on $V_j$ and
${\rm Im}(\phi)(x,\theta,y^*_j(x,\theta))>\epsilon|\theta|$ on $\partial V_j$.

If $a$ realises a formal analytic amplitude,
then the $b_j$ realise the formal amplitudes obtained by
formal stationary phase (and in particular, said formal amplitudes are analytic).

We first apply the contour deformations of Proposition
<ref>, which allows us to identify the open sets $V_j$
and the functions $y^*_j$. Recalling the open sets $U_1$ and $U_2$
of $\Omega_1\times \Omega_2$, for each $(x,\theta)\in \Omega_1$, each
connected component of $\{y\in \Omega_2,(x,\theta)\in \Omega_1\}$ will
correspond to a single critical point of $\phi$ with respect to
$y$. As $x,\theta$ vary, these connected components never merge or split (as
$|\det(\nabla^2_y\phi)|$ is bounded from below, critical points are
far away from each other) but simply appear or disappear, as the
associated critical point moves too far away from the real locus.

Therefore, if we define the sets $V_j$'s as the projections onto the
$(x,\theta)$ variables of the connected components of $U_1$, and
$y^*_j$ as the corresponding critical points, one has indeed ${\rm
Im}(\phi)(x,\theta,y_j^*(x,\theta))\geq 0$, and positivity on the

Let $(x,\theta)\in \Omega_1$; decompose $\Omega_2=U_2(x,\theta)\cup
\bigcup_{j\in \mathcal{J}(x,\theta)}U_{1;j}(x,\theta)$, where
$U_2(x,\theta)=\{y\in \Omega_2,(x,\theta,y)\in U_2\}$,
$\mathcal{J}(x,\theta)=\{1\leq j\leq n, (x,\theta)\in V_j\}$ and
$U_{1;j}(x,\theta)$ is the connected component of $\{y\in
\Omega_2,(x,\theta,y)\in U_2\}$ whose complex neighbourhood contains
$y_j^*(x,\theta)$. Since there exists $\epsilon>0$ such that ${\rm
Im}(\phi)>\epsilon|\theta|$ on $U_2(x,\theta)$, we are left with

\[
\sum_{j\in
\mathcal{J}(x,\theta)}\int_{U_{1;j}(x,\theta)}e^{i\phi(x,\theta,y)}a(x,\theta,y)\dd
\]

By Proposition <ref>, there exists a homotopy
$\Gamma_t$ of real-analytic
contours deforming $U_{1;j}(x,\theta):=\Gamma_{0;j}(x,\theta)$ into some
$\Gamma_{1;j}(x,\theta)$ containing $y^*_j(x,\theta)$ on which ${\rm Re}(\phi)$ is constant; this homotopy
satisfies ${\rm Im}(\phi)>\epsilon|\theta|$ on $\partial
\Gamma_{t;j}(x,\theta)$.

Extending $a$ and $\phi$ holomorphically on $y$, Stokes' theorem gives

\[
\int_{\Gamma_{0;j}(x,\theta)}e^{i\widetilde{\phi}(x,\theta,y)}a(x,\theta,y)\dd y -
\int_{\Gamma_{1;j}(x,\theta)}e^{i\widetilde{\phi}(x,\theta,y)}\widetilde{a}(x,\theta,y)\dd
\]

here, on $W_1=\bigcup_{t\in [0,1]}\partial \Gamma_{t;j}(x,\theta)$, one
has ${\rm Im}(\widetilde{\phi})>\epsilon |\theta|$. Hence,
the difference between the integrals on the two contours is

Applying the Morse lemma on $\widetilde{\phi}$ on $\Gamma_{1;j}$, and
an analytic change of variables on $y$, it remains to
prove that, when $a$ is an analytic amplitude of $((x,y),\theta)$, then so is

\begin{equation}\label{eq:std_phase}
(x,\theta)\mapsto \int_{B(0,1)}e^{-|\theta|y^2}a(x,\theta,y)\dd y,
\end{equation}

and that this stationary phase commutes with the realisation of
formal analytic amplitudes.

We first show a result “at fixed order”, then re-sum the
obtained amplitude in the spirit of Proposition

Let $N\in \N$ and let $u\in C^N(B_{\R^d}(0,1),\R)$. The Taylor expansion of
$u$ at $0$ is

\[
\]

\[
|R(y)|\leq \frac{\rho_d^N|y|^N}{N!}\|\nabla^Nu\|_{L^{\infty}};
\]

here $\rho_d$ depends only on the dimension $d$.

Since $\frac{|y|^N}{N!}\leq e^{|y|}$, we obtain

\[
\left|    \int_{B(0,1)}e^{-\lambda y^2}R(y)\dd y\right|\leq
C_d\rho_d^N\lambda^{-\frac{N}{2}}(\tfrac N2)!\frac{\|\nabla^Nu\|_{L^{\infty}}}{N!}.
\]

Noting that, on $\R^+$, the function $y\mapsto y^Ne^{-y}$ reaches
its maximum at $y=N$, then as long as $\frac{N+d+1}{2}<\lambda$,

\[
y\notin B(0,1)\rightarrow |y^{\mu}e^{-\lambda y^2}|\leq
\]

\[
\left|\int_{\R^d\setminus B(0,1)}e^{-\lambda
y^2}\sum_{|\mu|<N}\frac{\partial^\mu u(0)y^{\mu}}{\mu!}\dd
\]

It remains to compute the explicit integral

\[
\int_{\R^d}e^{-\lambda y^2}\sum_{|\mu|<N}\frac{\partial^\mu
u(0)}{\mu!}y^{\mu}\dd y=\pi^{\frac d2}\lambda^{-\frac d2}\sum_{j<\frac{N}{2}}\frac{\Delta^ju(0)}{(4\lambda)^jj!}.
\]

Altogether we have proved, for some $\rho_d>0$, for every $N<2\lambda-d-1$, for every $u\in

\[
\left|\int e^{-\lambda y^2}u(y)\dd y-\pi^{\frac d2}\lambda^{-\frac d2}\sum_{j<
\frac{N}{2}}\frac{\Delta^ju(0)}{(4\lambda)^jj!}\right|\leq
N2}(\tfrac N2)!\frac{\|\nabla^Nu\|_{L^{\infty}}}{N!}.
\]

If, on the contrary, $N\geq 2\lambda-d-1$, then

\[
\left|\int e^{-\lambda y^2}u(y)\dd y\right|\leq
C_0\|u\|_{L^{\infty}}\leq C_0\lambda^{-\frac N2}(\tfrac
N2)!\rho_d^{\frac N2}\|u\|_{L^{\infty}}
\]

and, for $j<\frac N2$,

\[
\lambda^{-\frac d2}\frac{\Delta^ju(0)}{(4\lambda)^jj!}\leq
\lambda^{-\frac N2}(\tfrac N2)!\rho_d^{\frac N2}\frac{\|\nabla^{2j}u\|_{L^{\infty}}}{(2j)!}.
\]

Therefore, without condition on $N$ and $\lambda$, for every $u\in C^N$,

\begin{equation}\label{eq:stat_phase_fixed_order}
\left|\int e^{-\lambda y^2}u(y)\dd y-\pi^{\frac d2}\lambda^{-\frac d2}\sum_{j<
\frac{N}{2}}\frac{\Delta^ju(0)}{(4\lambda)^jj!}\right|\leq
N2}(\tfrac N2)!)\sum_{j=0}^N\frac{\|\nabla^ju\|_{L^{\infty}}}{j!}.
\end{equation}

Plugging $u:y\mapsto a(x,\lambda \omega,y)$, we obtain the second part of the result.

It remains to prove that (<ref>) defines an analytic amplitude. The derivatives
on $x$ and on the spherical variables of $\theta$ play no role, and we are left with proving that, for some
$\epsilon>0$ and $\rho>0$,

\[
\forall k\leq \epsilon \lambda, \left|\frac{\partial^k}{\partial
\lambda^k}\int_{B(0,1)}e^{-\lambda y^2}a(x,y,\lambda)\dd y
\right|\leq \rho^{k}k!\lambda^{d-\frac{n_y}{2}-k}.
\]

One term of the binomial sum looks like

\[
\binom{k}{j}\int_{B(0,1)}e^{-\lambda
y^2}y^{2j}\partial_\lambda^{k-j}a(x,y,\lambda)\dd y
\]

and, applying (<ref>) at order $N=2j+n_y$, we obtain

\begin{multline*}
\left|\int_{B(0,1)}e^{-\lambda
y^2}y^{2j}\partial_\lambda^{k-j}a(x,y,\lambda)\dd y\right| \leq
+\left.\lambda^{-\frac{n_y}{2}}\sum_{\ell=j}^{j+\frac{n_y}{2}}\frac{\Delta^\ell_y(y^{2j}\partial_{\lambda}^{k-j}a)(0)}{(4\lambda)^\ell \ell!}+\rho_d^{2j+n_y}\lambda^{-\frac{n_y}{2}-j}\frac{\|\nabla^{2j+n_y}_yy^{2j}\partial^{k-j}_{\lambda}a\|_{L^{\infty}}}{(2j+n_y)!}\right).
\end{multline*}

As long as $\frac{k}{\lambda}$ is small enough, if $a$ is an analytic
amplitude, then this satisfies the required claim.

§.§ Positive nondegenerate phases and their Lagrangians

In this short subsection, we avail ourselves of a few geometrical
results concerning what will be the oscillating phases of the Fourier
Integral Operators considered in this article. We will directly deal
with complex-valued phase functions; real-analytic regularity
makes their treatment quite transparent, and every object will be
defined in a complex-geometric setting. For instance, “Lagrangians”
will always be complex submanifolds (of a larger complex symplectic
space) on which the symplectic form

We will only use the relatively standard notion of non-degenerate
phase functions.

A (strictly) positive non-degenerate phase function
$\phi:U_x\times V_{\theta}\times W_y\to \C$, where
$U\subset \R^{n_x}$, $W\subset \R^{n_y}$ are relatively compact
open sets and $V\subset \R^{n_{\theta}}$ is an open cone,
is a one-homogeneous function of $\theta$ such that, with
$\Sigma_{\phi}=\{(x,\theta,y)\in \widetilde{U}\times
\widetilde{V}\times \widetilde{W},
\nabla_{\theta}\widetilde{\phi}=0\}$, the following conditions

* The gradients $\nabla_x\widetilde{\phi}$ and $\nabla_y\widetilde{\phi}$, restricted
to $\Sigma_{\phi}$, do not vanish.

* $\nabla_{\theta}\widetilde{\phi}$ is a defining function for
$\Sigma_{\phi}$ (that is to say,
$\nabla_{x,\theta,y}\nabla_{\theta}\widetilde{\phi}$ has rank
$n_{\theta}$ and its columns span the conormal bundle of

* There exists $c>0$ such that, on the real set $U\times V\times W$, one has
${\rm Im}(\phi)\geq c|\theta||\nabla_{\theta}\phi|^2$.

Let $\phi$ be a positive non-degenerate phase function. Then
$\Sigma_{\phi}$ is a real-analytic conic submanifold diffeomorphic
to the conical Lagrangian

\[\Lambda_{\phi}=\{(x,\dd_x\widetilde\phi(x,\theta,y),y,-\dd_y\widetilde\phi(x,\theta,y)),(x,\theta,y)\in
\Sigma_{\phi}\}\subset (T^*\C^n\setminus \{0\})^2,\]
called the
canonical relation of $\phi$.

This is an elementary and well-known fact, whose proof is the same
whenever $\phi$ is a real-valued phase function, a $C^{\infty}$
complex-valued phase function with almost analytic extensions, or a
real-analytic complex-valued phase function. See for instance
[22], Proposition 25.4.4.

Definition <ref> is independent of a real-analytic
change of variables in $\theta$, as long as it has a real-analytic
dependence in $x$ and $y$. This motivates the next definition.

Given a real-analytic function $\phi:U\times V\times W\to \C$
satisfying the first two conditions of Definition
<ref>, a totally real, conical submanifold of
$\widetilde{V}$ with analytic dependence in $x$ and $y$ will be
called a good contour if $\widetilde{\phi}$, restricted to
this contour, satisfies the third condition.

Two Lagrangians $\Lambda_1\subset (T^*X\setminus \{0\})\times (T^*Y\setminus \{0\})$ and
$\Lambda_2\subset (T^*Y\setminus \{0\})\times (T^*Z\setminus \{0\})$
are said to have transverse composition at a point of
$\Lambda_1\times \Lambda_2\cap [(T^*X\setminus \{0\})\times
\Delta(T^*Y\setminus \{0\})\times (T^*Z\setminus \{0\})]$ if the
intersection of these two submanifolds is transverse at this

Let $\phi_1:\Omega_1\to \C$ and $\phi_2:\Omega_2$ be positive nondegenerate phase functions
whose Lagrangians have transverse composition. Then for every
$\Phi: (x,z,\theta,y,\tau)\mapsto \phi_1(x,\theta,y)+\phi_2(y,\tau,z)$ is a positive
nondegenerate phase function on

\[
\{(x,\theta,y)\in \Omega_1,(y,\tau,z)\in
\Omega_2,\epsilon|\theta|<|\tau|<\epsilon^{-1}|\theta|\},
\]
up to a small contour homotopy in $y$. Its Lagrangian is

\[
\Lambda_1\circ \Lambda_2:=\{(x,\xi,z,\zeta)\in T^*X\setminus
\{0\}\times T^*Z\setminus \{0\},\exists (y,\eta)\in T^*Y\setminus
\{0\}, (x,\xi,y,\eta)\in \Lambda_1,(y,\eta,z,\zeta)\in
\Lambda_2\},\]

and its real locus is exactly

\begin{multline*}
(\Lambda_1\circ \Lambda_2)_\R\\=\{(x,\xi,z,\zeta)\in (T^*X)_{\R}\setminus
\{0\}\times (T^*Z)_{\R}\setminus \{0\},\exists (y,\eta)\in (T^*Y)_{\R}\setminus
\{0\}, (x,\xi,y,\eta)\in (\Lambda_1)_{\R},(y,\eta,z,\zeta)\in
\end{multline*}

See [22], Proposition 25.5.4. Note that
the notion of positivity used there is more general than ours and we
have to prove positivity by hand; a
priori we only have

\[
\im(\Phi)\geq
\]

We now apply a contour deformation on $y$, consisting in the
positive gradient flow of $(|\theta|^2+|\tau|^2)^{-\frac{1}{2}}\im(\Phi)$. Along this flow, $\im(\Phi)$
increases at a rate
$(|\theta|^2+|\tau|^2)^{-\frac{1}{2}}|\nabla_y\Phi|^2$. Therefore, after a small
time of evolution, we obtain

\[
\im(\Phi)\geq
\]

This is the desired result since $|\theta|, |\tau|,$ and
$(|\theta|^2+|\tau|^2)^{\frac 12}$, are comparable.

The condition on the real locus is an easy consequence of the fact
that the imaginary part of the phases are always nonpositive, and
vanish exactly on the real locus of their Lagrangians.

Phase functions are seldom more than continuous at the
origin (unless they are linear), so that it is necessary to remove
conical neighbourhoods of $\theta=0$ and $\tau=0$ above. When actually
composing Fourier Integral operators in the next section, it will be necessary to deal
with these neighbourhoods in a different way. This will be facilitated
by the fact that $\im(\Phi)+(|\theta|^2+|\tau|^2)^{\frac
12}|\nabla_y\Phi|^2$ is bounded away from $0$
on this set.

In the $C^{\infty}$ theory, Fourier Integral Operators are
completely described by the Lagrangians of their phase functions;
the usual proof consists in a term-by-term identification or
construction (see for instance [22],
Proposition 25.1.5). The analytic theory is a bit subtler, and we
first describe how to remove “spurious” phase variables; we will
prove in effect that Fourier Integral Operators associated with
“complicated” phase functions can also be written using simpler
phases, which will be enough for our needs.

Let $\phi$ be a positive
nondegenerate phase. Then on every point of $\Sigma_{\phi}$,

\[
n_{\theta}-{\rm rank}\nabla^2_{\theta}\phi=n_x+n_y-{\rm

The proof is the same as in the real-valued case, see
[38], Proposition 18.4.1.

This proposition interacts very well with the condition of
positivity in our definition of phase functions. Indeed, on
$\Sigma_{\phi}^{\R}$, in the directions where
$\nabla^2_{\theta}\phi$ is non-degenerate, it has positive
imaginary part; we will therefore be able to apply the stationary
phase without difficulty.

§ FOURIER INTEGRAL OPERATORS

§.§ Basic properties

The main goal of this subsection is to give a suitable definition of
Fourier Integral Operators in a real-analytic setting; they will be
associated with positive phase functions in the sense of Definition
<ref> and analytic amplitudes in the sense of
Definition <ref>. Among other properties, we wish
Fourier Integral Operators to preserve real-analytic functions and to
behave well under composition.

The main obstacle to a direct and general definition is the fact that
most of our objects are only defined locally, in small neighbourhoods
of the subset where the imaginary part of the phase vanishes. The
zones where the phase has positive imaginary part should not
contribute to the integrals modulo real-analytic functions, but
without the possibility of introducing cut-offs, it is difficult to
prove a priori that it is the case.

We first describe a particular case of oscillatory integrals, in
coordinates, then we proceed by successive generalisations and
patching arguments.

Let $U\subset \R^{n_x}$ be an open set, and let
$\Gamma\subset \R^{n_{\theta}}$ be a conical open set.  Let $\phi$
be a one-homogeneous function on $U\times \Gamma$, real-analytic
with respect to $x$, and
such that $\im(\phi)\geq 0$ everywhere.

Then, given an analytic amplitude $a$ on $U\times \Gamma$, we define the
following distribution on $U$:

\[
K_{\Gamma,\phi}(a):x\mapsto \int_{\Gamma}
e^{i\phi(x,\theta)}a(x,\theta)\dd \theta.
\]

Let $U$ and $\Gamma$ be two open sets as above, let $\phi$ be a one-homogeneous function on a neighbourhood of
$U\times \Gamma$, real-analytic with respect to $x$ and such that $\im(\phi)\geq 0$ everywhere. Let
$a$ be an analytic amplitude on a neighbourhood of $U\times \Gamma$. Then $K_{\Gamma,\phi}(a)$
is real-analytic on

\[
\{x\in U, \forall \theta\in \Gamma,
\im(\phi)(x,\theta)>c|\theta|\}\]
as well as on

\[
\{x_0\in U, \exists C,c>0,\forall \theta\in \Gamma, \forall x\in
{\rm Neigh}(x_0,\widetilde{U}),

Under either of the conditions above, the integrand
$(x,\theta)\mapsto e^{i\phi(x,\theta)}a(x,\theta)$ extends holomorphically
into an integrable function of $\theta$ for $x$ close to the real

A particular case of oscillatory integral above consists in
singular integral kernels of operators acting as follows:

\begin{equation}\label{eq:local_FIO}
x\mapsto \int_{\Gamma} e^{i\phi(x,\theta,y)}a(x,\theta,y)v(y)\dd \theta \dd y,
\end{equation}

that is, the singular integral kernel of a Fourier Integral Operator (with phase
$\phi$ and amplitude $a$). If $\phi$ is a positive
phase function and $a$ is also real-analytic with respect to $y$, then
these integral operators preserve real-analytic functions, as long as
$\im(\phi)>0$ on the boundary of $\Gamma$.

Let $U\subset \R^{n_x}$, $\Gamma\subset
\R^{n_{\theta}+n_y}$ be open sets, with $\Gamma$ conical in its
first variable and relatively compact in its second variable. Let $\phi$ be
a positive phase function on a neighbourhood of $U\times \Gamma$ such that

\[
\exists c>0, \forall (x,\theta,y)\in U\times \partial \Gamma, \im(\phi)(x,\theta,y)\geq c|\theta|,
\]

and let
$a$ be an analytic amplitude on a neighbourhood of $U\times \Gamma$. Then
for every $v\in C^{\omega}(\R^{n_y})$, the function

\[
x\mapsto \int_{\Gamma} e^{i\phi(x,\theta,y)}a(x,\theta,y)v(y)\dd \theta \dd y
\]

is real-analytic on $\overline{U}$.

Let $v\in C^{\omega}(\R^{n_y})$ and consider

\[
u:x\mapsto \int_{\Gamma}e^{i\phi(x,\theta,y)}a(x,\theta,y)v(y)\dd \theta \dd y.
\]

We will change the contour of integration in the variable
$y$. Since $\phi$ is a positive phase function, one has

\[
\exists c>0,\forall (x,\theta,y)\in \overline{U}\times \Gamma,
\im(\phi)+|\nabla_y\phi|^2\geq c|\theta|.
\]

We now deform contours by following the flow of
$\nabla_y\im(\widetilde{\phi})$. After a small time, we obtain a
contour $\Gamma_1$ on which $\im(\widetilde{\phi})>c|\theta|$. Now,
by the Stokes formula,

\[
\theta \dd y + \int_{\Gamma_2}e^{i\widetilde{\phi}(x,\theta,y)}\widetilde{a}(x,\theta,y)\widetilde{u}(y)\dd
\theta \dd y,
\]

where $V_2$ lies within a small neighbourhood of $\partial \Gamma$ in
$\R^{n_{\theta}}\times \C^{n_y}$, so that $\im(\widetilde{\phi})>c|\theta|$ on
$V_2$. Thus, by Proposition <ref>, $v$ is real-analytic.

Over the course of this article, the open sets on which phases and
amplitudes are well-defined will seldom be of a product form $U\times
\Gamma$, unless restricted to small neighbourhoods of a point. We have
to patch together expressions of the form (<ref>), which
is possible thanks to Proposition <ref>. Before
doing so we introduce a relevant function space.

If $K$ is a compact set of a real-analytic paracompact manifold, we denote by $\mathcal{E}(K)$ the
space of smooth functions on $K$ in the sense of Whitney
[39], and by $\mathcal{O}(K)$ the space of
functions which extend to real-analytic functions on a neighbourhood
of $K$.

Let $U$ be a relatively compact open set of a real-analytic
paracompact manifold. We denote

\[
\mathcal{F}(U)=\mathcal{E}'(\overline{U})/(\mathcal{E}'(\partial
U)\cup \mathcal{O}(\overline{U})).
\]

We refer to Appendix <ref> for basic properties
of these spaces.

Let $X$ and $Y$ be two paracompact analytic manifolds. Let $U\subset
X$ and $V\subset Y$ be open and relatively compact. Let
$\Omega\subset X\times \R^{n_{\theta}}\times V$ be open and conical
in its second variable.

Let $\phi$ be a positive phase function on a neighbourhood of
$\Omega$, such that $\im(\phi)\geq 0$ everywhere and

\begin{equation}\label{eq:boundary_positivity}
\exists c>0, \forall x\in U, \forall (\theta,y)\in
\partial(\{(\theta',y')\in \R^{n_{\theta}}\times V,(x,\theta,y)\in \Omega\}), \im(\phi)(x,\theta,y)\geq c|\theta|.
\end{equation}

Let $a$ be an analytic amplitude on a neighbourhood of
$\Omega$, and let $v\in \mathcal{F}(V)$.
We define $I_{\phi,\Omega,U}(a)v$ as the element of
$\mathcal{F}(U)$ defined by the following recipe.

* For every $x_0\in U$, for some small neighbourhood
$\Gamma_0\times V_0$ of

\[\{(\theta,y)\in \R^{n_{\theta}}\times V,(x_0,\theta,y)\in
\Omega,\im(\phi)(x,\theta,y)=0\},\]
consider $\chi\in
C^{\infty}(X,\R)$ be equal to $1$ on a small neighbourhood $U_0$ of $x_0$ and supported on a
small neighbourhood of $U_0$. Then

\[
\int_{\Gamma}e^{i\phi(x,\theta,y)}a(x,\theta,y)\chi(x)v(y)\dd \theta
\dd y
\]

is the action of a “usual” Fourier Integral Operator on $v$, and in
particular it belongs to $\mathcal{E}'(X)$ (in fact it is supported
near $U_0$). This element of $\mathcal{E}'(X)$ can then be restricted
to an element of
$\mathcal{E}'(U_0)/\mathcal{E}'(\partial U_0)$, see Proposition
<ref>. By Proposition <ref>, if $v\in
C^{\omega}(\overline{V_0})$ then this produces an element of $C^{\omega}(\overline{U_0})$.

* Perform the last item on a finite family of open sets covering

* By Proposition <ref>, any two such
distributions agree on the intersection of their defining supports
modulo a real-analytic function, and by Proposition
<ref>. Therefore, by Proposition <ref>, we can patch these distributions together into
a uniquely, well-defined element of
$\mathcal{F}(U)$, which is $0$ if $v\in C^{\omega}(\overline{V})$. In conclusion, we've built

* One has then, by Proposition <ref>,

\[
SS_a(I_{\Omega,U,\phi}(a)v)\subset \{x\in U, \exists (\theta,y)\in
\R^{n_{\theta}}\times SS_a(v), (x,\theta,y)\in \Omega,
\im(\phi)(x,\theta,y)=0\},
\]

and in particular, if $v\in \mathcal{E}'(\partial V)$, then
$SS_a(I_{\Omega,U,\phi}(a)v)=0$. Therefore, we can conclude:

\[
I_{\phi,\Omega,U}(a):\mathcal{F}(V)\to \mathcal{F}(U).
\]

It is natural to ask whether this construction works when $v$ is a
singularity hyperfunction, that is, an element of
V)+\mathcal{O}(V))$. The answer is yes, but to do so we need to study the
analytic wave front set of the kernel $K_{\Gamma,\phi}$; we will do
so in Section <ref>, after having defined
the analytic wave front set as the set of analytic singularities
after conjugation by a Fourier Integral Operator; for the moment we
only invoke the fact that the smooth wave front set of
$K_{\Gamma,\phi}$ is known and therefore analytic Fourier Integral
Operators act nicely on distributions. After making sure that
Definition <ref> makes sense as acting on
singularity hyperfunctions, the rest of Section <ref> will extend to this
case without any difficulty.

As in the smooth case, these operators behave well under composition
and stationary phase
under natural geometric hypotheses, and particularly so when the
analytic amplitudes in question are realisations of formal analytic amplitudes.

Let $X,Y,Z$ be paracompact
real-analytic manifolds, and let $U\subset X,V\subset Y,W\subset Z$ be relatively compact open
Let $\Omega_1\subset X\times \R^{n_{\theta}}\times V$
and $\Omega_2\subset Y\times \R^{n_{\tau}}\times W$ be
open and conic with respect to their middle variables; let
$\phi_1$ and $\phi_2$ positive
non-degenerate phase functions on open neighbourhoods of,
respectively, $\Omega_1$ and $\Omega_2$. Suppose that $(\phi_1,\Omega_1,U)$ and
$(\phi_2,\Omega_2,V)$ both satisfy
(<ref>). Let
$a_1:\Omega_1\to \C$ and $a_2:\Omega_2\to \C$ be analytic

Let $\Phi:(x,\theta,y,\tau,z)\mapsto
\phi_1(x,\theta,y)+\phi_2(y,\tau,z)$ and
$A:(x,\theta,y,\tau,z)\mapsto a_1(x,\theta,y)a_2(y,\tau,z)$.

There exists a contour deformation $\Gamma\in \widetilde{Y}$, with
real-analytic dependence on $x,\theta,\tau,z$, and a
neighbourhood $\Omega$ of

\[
\{(x,\theta,\tau,z)\in
X\times \R^{n_{\theta}+n_{\tau}}\times Z,y\in
\Gamma(x,\theta,\tau,z),\nabla_{y,\theta,\tau}\Phi=0\},
\]

contained in $\{\epsilon|\theta|<|\tau|<\epsilon^{-1}|\theta|\}$ for
some $\epsilon>0$,
such that $(\phi_1+\phi_2,\Omega,U_1)$ satisfies

\[
\]

(as maps from $\mathcal{F}(W)$ to $\mathcal{F}(U)$).
Moreover, if $a_1$ and $a_2$ respectively realise formal analytic
amplitudes, then there exists a maximally analytic amplitude $A'$ on
$\Omega$ such that

\[
\]

The distributional kernel of $I_{\phi_1,\Omega_1,U_1}(a_1)\circ
I_{\phi_2,\Omega_2,U_2}(a_2)$ is

\[
(x,z)\mapsto \int
e^{i\Phi(x,\theta,y,\tau,z)}A(x,\theta,y,\tau,z)\dd y \dd \theta
\dd \tau.
\]

We begin with a contour deformation on $y$, which follows the
positive gradient flow of $(|\theta|^2+|\tau|^2)^{-\frac
12}\nabla_y\im(\Phi)$. After a small time, along this contour
$\Gamma_1$ one

\[
\im(\Phi)\geq c|\theta||\nabla_{\theta}\Phi|^2+c|\tau||\nabla_{\tau}\Phi|^2+c(|\theta|^2+|\tau|^2)^{-\frac
\]

Compared to the integral on the original contour, the difference is,
by Stokes' theorem, an integral of the form

\[
\int_{V}  e^{i\widetilde{\Phi}(x,\theta,y,\tau,z)}\widetilde{A}(x,\theta,y,\tau,z)\dd y \dd \theta
\dd \tau
\]

where on $V= \bigcup_{t\in [0,1]}\partial
\Gamma_t(x,\theta,\tau,z)$, the phase $\widetilde{\Phi}$ has
positive imaginary part, due to the fact that
$(\phi_1,\Omega_1,U_1)$ satisfies
(<ref>). Hence, by Definition
<ref>, the difference maps $\mathcal{E}'(\overline{W})$
into $C^{\omega}(\overline{U})$ (note that, in this application of Definition <ref>, $\Phi$ is not a positive
phase function, because it is singular at $\theta=0$ and $\tau=0$;
for the same reasons $A$ is not an analytic amplitude).

We now use Proposition <ref> again to remove
the complement set of a conical neighbourhood $\Omega$ of

Let us prove that, on $\Omega$, $\theta$ and $\tau$ are bounded away
from $0$. Indeed, if $\theta=0$, then $\Phi=\phi_2$, and
$|\tau||\nabla_\tau\phi_2|^2$ is bounded away from $0$ except in the
vicinity of $\Sigma_2$, where $|\tau|^{-1}|\nabla_y\phi|^2$ is
bounded away from $0$. Therefore $\im(\Phi)$ is bounded away from
$0$ on $\{\theta=0\}$, and on $\{\tau=0\}$ as well, by a symmetrical
argument. This concludes the proof.

To conclude, if $a_1$ and $a_2$ realise formal analytic amplitudes,
then so does $A$, by Proposition <ref>;
letting $A'$ be a maximally analytic realisation of the same
analytic amplitude, one has $|A'-A|\leq Ce^{-c(|\theta|+|\tau|)}$ so
that, by Proposition <ref>, the difference
between the associated Fourier Integral Operators maps
$\mathcal{E}'(\overline{W})$ into
real-analytic functions.

Let $X$ and $Y$ be paracompact real-analytic manifolds and let $U,V$
be relatively compact sets of $X$ and $Y$. Let $\Omega\subset X\times \R^{n_{\theta'}}\times
\R^{n_{\omega}}\times V$ be open and a relatively compact cone with respect
to the second variable; let
$\phi$ be a positive phase function on a neighbourhood of $\Omega$ such that
$(\phi,\Omega,U)$ satisfies condition
(<ref>). Suppose further that
$\nabla_\omega^2\phi$ is nonsingular everywhere.

Let $a:\Omega\to \C$ be such that $((x,y,\omega),\xi)\mapsto
a(x,\xi,\omega,y)$ is an analytic amplitude.

Then there exists $J\in \N$ and, for every $1\leq j\leq J$, a conic
open set $\Omega_j\subset X\times
\R^{n_{\theta'}}\times V$, a non-degenerate phase
function $\phi_j$ on a neighbourhood of $\Omega_j$ such that
$(\phi_j,\Omega_j,U)$ satisifes condition
(<ref>), and an analytic amplitude
$b_j:\Omega_j\to \C$, such that

\[
\]

maps $\mathcal{E}'(\overline{V})$ into $C^{\omega}(\overline{U})$. Moreover, the union of the
Lagrangians of $\phi_j$ is the Lagrangian of $\phi$, and if $a$
realises some formal analytic amplitude $(a_k)_{k\in \N}$, then each
$b_j$ realises a formal analytic amplitude $(b_{j;k})_{k\in \N}$,
which can be obtained from $(a_k)_{k\in \N}$ by the formal
stationary phase.

Choosing local expressions for $I_{\phi,\Omega,U}(a)$ as in
Definition <ref>, this is a restatement of
Proposition <ref>.

natural Fourier Integral Operators make sense of the fibre variable
$\theta$ as belonging to the fibre of a bundle over the base $X\times
Y$ rather than a fixed space $\R^{n_{\theta}}$. Similarly, in
practical applications of Proposition <ref>,
the decomposition of the phase variables into the directions where
$\nabla^2\phi$ is non-degenerate and a complement set is seldom
independent on $x,y$.

Both issues are formally taken care of by a one-homogeneous real-analytic change of
variables $(x,\theta,y)\rightsquigarrow
(x,\xi(x,\theta,y),y)$; however, analytic amplitudes do not have
real-analytic dependence on $\theta$, and therefore, these change of
variables destroy real-analyticity with respect to $x$ and
$y$. We can remediate to this problem in the case of realisations of formal analytic amplitudes.

Let $X$ and $Y$ be
paracompact analytic manifolds, let $U\subset X$ and $V\subset Y$ be
relatively compact. Let $\Omega\subset
X\times \R^{n_{\theta}}_{\theta}\times V$ suppose that $\Omega$ is conical
in its second variable.

Let $\xi_t:\Omega\to X\times \C^{n_{\theta}}\times V$ be a family of contours
(they extend into complex-analytic diffeomorphisms, are
one-homogeneous in the second variable, act as identity on the first and third
variables, depending smoothly on the parameter $t$) with
$\xi_0=id$. Let $\phi$ be a phase function on a neighbourhood of $\Omega$ and suppose that $(\phi_t\circ
\xi_t,\Omega,U)$ satisfies (<ref>) for all
$t\in [0,1]$. Let $a$ be the realisation of a formal analytic
amplitude $(a_k)_{k\in \N}$ on a neighbourhood of $\Omega$.

Then, any realisation $b$ of $(a_k\circ \xi_1)_{k\in \N}$ is such
that (as maps from $\mathcal{F}(V)$ to $\mathcal{F}(U)$)

\[
I_{\phi\circ \xi_1,\Omega,U}(b\det(D_{\theta}\xi_1))=I_{\phi,\Omega',U}(a).
\]

Let $\xi_t$ be a homotopy of complex-analytic diffeomorphisms that are
one-homogeneous and act as identity on the first and third
variables, with $\xi_0=id$. Then $(a_k\circ \xi_t)_{k\in \N}$ is, uniformly for $t\in
[0,1]$, a formal analytic amplitude; as in the proof of Proposition
<ref>, one can realise it into

\[
b(t,x,y,\theta)=\sum_{k\in \N}a_k\circ
\xi_t(x,\theta,y)(1-\chi_k(\tfrac{c\theta}{k}))
\]

for some $c>0$ small, where $\chi_k$ is a radial function, supported
on $B(0,2)$ and equal to $1$ on $B(0,1)$, and such that
$|\nabla^j\chi_k|\leq (Rk)^j$ whenever $j\leq k$.

Let now $x_0\in U$. Following Definition
<ref>, let $U_0$ be a small neighbourhood of
$x_0$ in $U$, let $\Gamma_0,V_0$ an open neighbourhood of
$\{\im(\phi)(x_0,y,\theta)=0\}$, and let $v\in \mathcal{E}'(V_0)$,
then for $t$ small, we are considering

\[
J(t):x\mapsto \int_{\Gamma_0\times V_0 }
\theta\dd y,
\]

For some $t_0>0$,
one has still
$\im(\phi)\circ \xi_t>0$ on $U_0\times \partial \Gamma_0\times V_0$ for all $t\in [0,t_0]$.

For every holomorphic function $f$,

\[
\frac{\dd}{\dd
t}(f(x,\xi_t(x,\theta,y),y)\det(D_{\theta}\xi_t))={\rm div}_{\theta}\left(f(x,\xi_t(x,\theta,y))\partial_t\xi_t\right),
\]

and therefore, by Stokes' formula,

\begin{multline}\label{eq:Jprime}
\frac{\dd J(t)}{\dd t}(x)=\int_{\Gamma_0\times
\N}a_k\circ\xi_t(x,\theta,y) \partial_t\xi_t\cdot
\nabla_{\theta}\chi_k(\tfrac{c\theta}{k})\dd
\theta \dd y\\+\int_{\partial \Gamma_0\times V_0}e^{i\phi(x,\xi_t(x,\theta,y),y)}\det(D_{\theta}\xi_t)v(y)\sum_{k\in
\N}b(t,x,y,\theta)\dd
\theta \dd y.
\end{multline}

We can apply Proposition <ref> to the
second term of the right-hand side, since $b$ is an analytic
amplitude and $\phi\circ\xi_t$ has positive imaginary part on
$\partial \Gamma_0\times V_0$; therefore this term is an analytic
function of $x$. As for the first term, let us prove that there
exists $C>0, \epsilon>0$ such that,
uniformly for $(x,y)$ in a complex neighbourhood,

\[
\left|      \sum_{k\in
\N}a_k\circ\xi_t(x,\theta,y) \partial_t\xi_t\cdot
\nabla_{\theta}\chi_k(\tfrac{c\theta}{k})\right|\leq

Since $(a_k)_{k\in \N}$ is a formal analytic amplitude, one has

\[
|a_k\circ\xi_t(x,\theta,y)|\leq CR^kk!|\theta|^{-k}.
\]

Moreover $\nabla_{\theta}\chi_k(\tfrac{c\theta}{k})$ is only nonzero
whenever $\frac{c|\theta|}{2}\leq k \leq c\theta$. In particular,

\[

Moreover for fixed $\theta$ the number of nonzero terms is of order
$|\theta|$, so that we finally obtain, whenever $c<\frac{1}{K}$, for
some $\epsilon>0$, the required estimate.

In particular, we can apply Proposition <ref>
to the first term of the right-hand-side of (<ref>) as
well; to conclude, for $t$ small, $\frac{\dd J(t)}{\dd t}$
continuously sends $\mathcal{E}'(Y)$ into $\mathcal{O}(U)$.
Integrating this fact
on $[0,t_0]$ for $t_0$ small, we obtain the desired identity with $\xi_1$ replaced
with $\xi_{t_0}$. However the statement is clearly of a local nature, and we can cover
$[0,1]$ with a finite number of open sets on which we can apply the
argument above. This concludes the proof.

From now on we will only consider analytic Fourier Integral Operators
whose amplitudes are realisations of formal analytic amplitudes; by
Proposition <ref>, they do not depend on the
realisation, and we will simply pass formal amplitudes as arguments of $I_{\phi,\Omega,U}$.

Assembling the previous facts, we obtain the following general result.

Let $X,Y$ be paracompact real-analytic manifolds. Let $E$ be a real-analytic
cone bundle over $X\times Y$, and denote the base projection by $\pi=(\pi_X,\pi_Y)$. Let $U\subset X$ and $V\subset Y$ be
open and relatively compact. Let $\Omega\subset \pi^{-1}(X\times V)$ be open and
conical, and let $\phi$ be a positive phase function on a
neighbourhood of $\Omega$. Let $a$ be a formal analytic amplitude on
a neighbourhood of $\Omega$.

* Given $U\subset X$ such that $(\phi,\Omega,U)$ satisifes
(<ref>), any choice of local trivialisations
of $E$ leads (via Definition <ref>) to the same, well-defined
operator $I_{\phi,\Omega,U}(a)$ mapping $\mathcal{F}(V)$ to
$\mathcal{F}(U)$. When further quotiented by smooth
functions, $I_{\phi,\Omega,U}(a)$ coincides with the
usual formal construction of the Fourier Integral operator with
phase $\phi$ and formal amplitude $a$. Furthermore, for all $v\in
\mathcal{E}'(Y)$,

\[
SS_a(I_{\phi,\Omega,U}(a)v)\subset \{x\in U, \exists \Theta \in \Omega,
\pi_X (\Theta)=x, \pi_Y(\Theta)\in SS_a(v), \im(\phi(\Theta))=0\}.
\]

We call analytic Fourier Integral operator such an

* If $U$ is relatively compact, $\Omega$ is a relatively compact
open cone, and ${\rm rank} \nabla^2_{\theta}\phi\geq 1$ on a
neighbourhood of $\Omega$, then
there exists a finite number of real-analytic vector bundles
$(E_j)_{1\leq j\leq J}$, such that ${\rm rank}(E_j)={\rm rank}(E)-1$, open conical sets $\Omega_j\subset E_j$,
positive phase functions $\phi_j$  and formal analytic amplitudes $b_j$
on neighbourhoods of $\Omega_j$, such that

\[I_{\phi,\Omega,U}(a)=\sum_{j=1}^JI_{\phi_j,\Omega_j,U}(b_j).\]
a formal amplitude, $b_j$ is obtained from $a$ by
stationary phase.

* Let $Z$ be a paracompact real-analytic manifold, $F$ a
real-analytic cone bundle over $Y\times Z$, $W\subset Z$ open and
conical, $\Omega'\subset \pi_F^{-1}(Y\times W)$
open and conical, $\phi'$ a positive phase function on a
neighbourhood of $\Omega'$  such
that $(\phi',F,V)$ satisfies (<ref>), $a'$ a formal analytic amplitude on a
neighbourhood of $\Omega'$. Then the composition
$I_{\phi,\Omega,U}(a)I_{\phi',\Omega',V}(a')$ is an analytic Fourier Integral
operator (constructed as in Proposition <ref>).

To some extent, we have reproduced the $C^{\infty}$ theory in the
analytic case. One thing still missing, and a crucial property in the
$C^{\infty}$ case, is the fact that Fourier Integral operators only
depend on the positive Lagrangian associated with their phases: given
two phases $\phi_1$ and $\phi_2$ with same Lagrangian, we expect any analytic
Fourier Integral operator associated with $\phi_1$ to be equal to some
analytic Fourier Integral operator associated with $\phi_2$. The proof
in the $C^{\infty}$ case (see [22],
Proposition 25.1.5) relies on a construction of the amplitude
order by order, which is difficult to translate to the analytic case
(essentially, one would need to prove by hand that the constructed
formal amplitude is analytic). We can, however, prove that some
Lagrangians of interest possess a more or less canonical phase, so
that all Fourier Integral operators associated with said Lagrangians
can be rewritten with this specific case. The first of these
Lagrangians is the diagonal in $T^*\C^n$; we will prove in
Proposition <ref> the Kuranishi trick: all
such Fourier Integral operators are pseudodifferential operators in
the sense that they can be rewritten using the usual phase and any good

The second family of Lagrangians contains the Szegő kernel
parametrix on the boundary of any strictly pseudoconvex domain (the
Lagrangian, of course, depends on the domain). These Lagrangians are
idempotent, and Fourier Integral Operators with same Lagrangian are
called “covariant Toeplitz operators” and form an algebra. To
describe this algebra in practice, one writes all these Fourier
Integral Operators with the same phase. We can do so, more generally, for Lagrangians whose base projection has corank 1.

Let $X,Y$ be paracompact analytic manifolds. Let $\Lambda\subset
(T^*\widetilde{X}\setminus \{0\})\times (T^*\widetilde{Y}\setminus
\{0\})$ be a conical, real-analytic, positive
Lagrangian and suppose that the projection onto the base
$\pi:\Lambda\to \widetilde{X}\times \widetilde{Y}$ satisfies $\dim
\ker \dd\pi=1$ everywhere (so that $\Lambda$ is a half-line bundle
over its projection $\pi(\Lambda)=Z$).

There exists a positive phase function on a half-line bundle over
a neighbourhood of $Z$ such that $\Lambda_{\phi}=\Lambda$. For any such
phase function, and any analytic Fourier Integral Operator with
phase $\phi$, there exists a formal analytic amplitude $a$ on a neighbourhood
of $Z$ such that $I_{\phi}(a)$ coincides with the analytic Fourier
Integral Operator.

By Theorem
<ref> and Propositions <ref> and <ref>,
possibly after application of stationary phase, an analytic Fourier
Integral Operator with Lagrangian $\Lambda$ can be written with a
number of phase variables equal to $1$, which means that the phase
function is defined on a half-line bundle over a neighbourhood of $Z$. This proves the existence of such a phase.

Any such phase $\phi$ satisfies $Z=\{\phi=0\}$ and near $Z$ one has
$\nabla_x\phi\neq 0,\nabla_y\phi\neq 0$. Since $Z$ has codimension
$1$, this means that $\phi$ is a defining function for $Z$. In particular, given another
function $\phi_2$ satisfying the same properties, locally after
identification of the half-line bundles over a neighbourhood of $Z$ on
which the phases are defined, one has $\phi_2=f\phi$ for some
non-vanishing function $f$. Thus, locally, Fourier Integral
Operators with phase $\phi_2$ have integral kernels of the form

\[
(x,y)\mapsto \int_0^{+\infty}e^{iuf(x,y)\phi(x,y,1)}b(x,y,u)\dd
\]

where we trivialised the half-line bundle and $b$ is a realisation
of a formal analytic amplitude.

Now $u\mapsto f(x,y)u$ is a well-defined change of variables, so that by
Proposition <ref>, there exists a
formal analytic amplitude $a$ such that $I_{\phi}(a)$ is the same
Fourier Integral Operator. This concludes the proof.

§.§ Pseudodifferential operators, FBI transforms, and Toeplitz

Pseudodifferential operators are a particular case of Fourier Integral
operators, with singular kernels of the form

\begin{equation}\label{eq:kernel_pseudos}
U\times V\ni (x,y)\mapsto \int_{\Gamma(x,y)} e^{i(x-y)\cdot (\xi+i\eta(x,\xi,y))}a(x,\xi,y)\dd \xi,
\end{equation}

where $U,V$ are open sets of $\R^n$, $\Gamma(x,y)$ is
a conical contour, with real-analytic dependence in $(x,y)$, and which
is positive, meaning that

\begin{equation}\label{eq:good_ctr_pseudo}
\exists c, \forall (x,y)\in U\times V, \forall \xi\in \Gamma(x,y),\im((x-y)\cdot \xi)>c|\xi||x-y|^2;
\end{equation}

moreover $a$ realises a formal analytic amplitude on a neighbourhood of
$\{(x,\xi,x),x\in U\}$ in $U\times \R^n\times V$.

By the previous results, operators defined by (<ref>) do not depend on the
realisation, nor do they depend on the choice of $\Gamma$. However, there is an important caveat at this stage: in order for condition
(<ref>) to hold, we need
$\Gamma(x,x)=\R^n$. Indeed, otherwise the phase vanishes
when $x=y\in U,\xi\in \partial \Gamma(x,x)$. This is, of course, a
severe hindrance because we would like to study the action of
real-analytic amplitudes defined near a point of phase space. This issue be
addressed later on through the FBI transform. For the moment, we make note of this limitation
and define pseudodifferential operators as follows.

Let $U\Subset V\subset \R^n$, and let $a$ be a formal analytic amplitude on a
neighbourhood of $\{(x,\xi,x),(x,\xi)\in U\times \R^n\}$ in
$U\times \R^n\times V$. The pseudodifferential operator
$Op(a)$ is defined as the operator with distributional kernel

\[
U\times V\ni (x,y)\mapsto
\int_{\Gamma(x,y)} e^{i(x-y)\cdot\xi}a^{\Gamma}(x,\xi,y)\dd \xi,
\]

where  $\Gamma(x,y)$ is any
contour satisfying (<ref>) and $a^{\Gamma}$ is such
that $(x,\xi,y)\mapsto a^{\Gamma}(x,\theta(x,\xi,y),y)$ is a realisation of
$(a_k(x,\theta(x,\xi,y),y))_{k\in \N}$, for
a contour deformation $\theta$ which sends $\R^n$ to $\Gamma(x,y)$.

These operators
preserve analytic functions, and they are uniquely defined as
operators from $\mathcal{F}(V)$ to $\mathcal{F}(U)$.

The Lagrangian associated with the phase of a pseudodifferential
operator is always the diagonal of $(T^*U\setminus \{0\})^2$. Our first
result is the Kuranishi trick, which allows us to prove a
reciprocal statement.

Let $U\subset V\Subset \R^n$ and let $\pi:E\to \R^n\times \R^n$ be a
cone bundle and let $\Omega\subset \pi^{-1}(E)$ be an open compact cone.. Let $\phi$
be a positive phase function on a neighbourhood of  $\Omega$ such that $(\phi,\Omega,U)$
satisfies (<ref>) and $\Lambda_{\phi}$ is the
diagonal of $(T^*U\setminus \{0\})^2$. Let $a$ realise
a formal analytic amplitude on a neighbourhood of $\Omega$

There exists a formal analytic amplitude $b$
on a neighbourhood of $\{(x,\xi,x),(x,\xi)\in U\times \R^n\}$ in $U\times \R^n\times V$, such that

\[
\]

(as always, in the sense of operators from
$\mathcal{F}(V)$ to $\mathcal{F}(U)$).

We first apply Propositions <ref> and
<ref> to reduce ourselves to the case where
$N=n$, since the right-hand side of the formula in Proposition
<ref> is equal to $n$. Note that when doing so, we obtain a finite sum of
Fourier Integral Operators.

$\Lambda_{\phi}$ is the diagonal of $(T^*U\setminus \{0\})^2$, then
$\phi(x,\theta,y)$ vanishes when $x=y$, and $\theta\mapsto
\partial_{\theta}\phi(x,\theta,x)\in \R^n$ is a surjective local
diffeomorphism. Therefore it is a global diffeomorphism, and it
extends to a neighbourhood of $x=y$ into a map $(x,\theta,y)\mapsto
\zeta(x,\theta,y)$ which is a global diffeomorphism between the real domain
of its second variable and a totally real contour of $\C^n$, and
such that

\[
\phi(x,\theta,y)=(x-y)\cdot \zeta(x,\theta,y).
\]

contour is $\R^n$ when $x=y$, and it satisfies
(<ref>), because $\phi$ is positive. Removing a
neighbourhood of $x=y$ (away from which the phase has positive
imaginary part) and applying
Proposition <ref>, we obtain a
pseudodifferential operator.

Given a compact real-analytic manifold $M$, one can equivalently
define analytic pseudodifferential operators on $M$ as

* gluings of Definition <ref> in coordinate charts;

* Fourier Integral Operators on $M$ (or on open subsets of $M$)
with Lagrangian ${\rm diag}(T^*M)$.

* Fourier Integral Operators defined with a suitable global phase
function. An example is given by the function $\phi:T^*M\times M\to
\C$ defined as follows: $\iota:M\to \R^N$ is an embedding of $M$ in
Euclidian space of sufficiently large dimension, $\pi_x:\R^n\to
T_xM$ is the pull-back of the orthogonal projection from $\R^n$ to
$\iota(T_xM)$, and

\[
\phi(x,\xi,y)=\xi(\pi_x(\iota(y)))+i|\xi|\dist(\iota(x),\iota(y))^2.
\]

Pseudodifferential operators are not uniquely determined by their
amplitudes; there is “one variable too many” among
$(x,\xi,y)$. The notion of symbol is better suited to their analysis.

Let $U\subset \R^n$ open and relatively compact and let $a$ be a formal analytic symbol on a
neighbourhood $\Omega$ of $\{(x,\xi,x)\in \R^{3n},x\in U\}$. Let $V$ be a
small neighbourhood of $U$.

Given $j\in \N$ define

\[
b_j:(x,\xi)=\sum_{|\mu|\leq j}\frac{i^{|\mu|}}{\mu!}(\partial_y^{\mu}\cdot
\partial_{\xi}^{\mu}a_{j-|\mu|})(x,\xi,x).
\]

Then $(b_j)$ is a formal analytic amplitude and
$Op((a_j))=Op((b_j))$. $(b_j)$ is called the (left)
total symbol of $(a_j)$.

It follows from a direct computation that $b_j$ is a formal analytic

Let $V$ be a small neighbourhood of $U$ such that $(b_j)_{j\in \N}$ is
well-defined on a complex neighbourhood of $U\times \R^n\times
V$. Let $\theta:U\times \R^n\times V\to \C^n$ be
such that $\kappa:(x,\xi,y)\mapsto (x,\theta(x,\xi,y),y)$ extends to a
biholomorphism with $\kappa(x,\xi,x)=\xi$ and $\phi:(x,\theta,y)\mapsto \phi\circ \kappa^{-1}$ is a
positive phase function.

Let $a$ and $b$ be respective realisations of $a_j\circ \kappa^{-1}$
and $b_j\circ \kappa^{-1}$ as built in Proposition
<ref>. Namely, with $\chi_j\in
C^{\infty}(\R^n,\R)$ radial and such that

\begin{align*}
&\1_{\R^n\setminus B(0,1)}\leq \chi_j\leq \1_{\R^n\setminus
B(0,\frac 12)}\\
&\exists \rho_0,\forall N\leq j,
\|\nabla^N\chi_j\|_{L^{\infty}}\leq (\rho_0j)^N,
\end{align*}

we have defined, for $c>0$ small,

\begin{align*}
\N}a_j(x,\xi(x,\theta,y),y)(1-\chi_{j+1}(\tfrac{c\theta}{j+1}))\\
\N}b_j(x,\xi(x,\theta,y),y)(1-\chi_{j+1}(\tfrac{c\theta}{j+1})),
\end{align*}

and then

\begin{align*}
Op((a_j))(x,y)&=\int e^{i\phi(x,\theta,y)}a(x,\theta,y)\dd \theta\\
Op((b_j))(x,y)&=\int e^{i\phi(x,\theta,y)}b(x,\theta,y)\dd \theta
\end{align*}

modulo a regularising operator. Define the contour
$\Gamma(x,y)=\{\xi\in \C^n, \theta(x,\xi,y)\in \R^n\}$, so that

\begin{align*}
Op((a_j))(x,y)&=\int_{\Gamma(x,y)} e^{i(x-y)\cdot \xi}a\circ \kappa(x,\xi,y)\dd \xi\\
Op((b_j))(x,y)&=\int_{\Gamma(x,y)} e^{i(x-y)\cdot \xi}b\circ \kappa (x,\xi,y)\dd \xi.
\end{align*}

Now we use the standard manipulation: for every fixed $N\in \N$ and
smooth function $f$ on $V$

\[
\forall x\in U, \forall y \in B(x,\epsilon), f(y)=\sum_{|\mu|<
\]

\[
|R_\mu(x,y)|\leq C\frac{\|\partial^{\mu}f\|_{L^{\infty}}}{N!}.
\]

Let $\psi_j=1-\chi_j$, and let $c_1\ll c$. To apply the above
formula at an order $N$ which grows like $c'|\xi|$, we write

\begin{multline*}
a\circ \kappa(x,\xi,y)=a\circ \kappa(x,\xi,y)\chi_1(c'\theta(x,\xi,y))\\+\sum_{N\in
\N}\left(\sum_{|\mu|<N}\frac{\partial^{\mu}_y}{\mu!}a\circ
\kappa(x,\xi,y)|_{y=x}(y-x)^{\mu}+\sum_{|\mu|=N}R_\mu(a\circ \kappa)(x,\xi,y)(y-x)^{\mu}\right)(\psi_{N+1}(\tfrac{c'\theta(x,\xi,y)}{N+1})-\psi_{N}(\tfrac{c'\theta(x,\xi,y)}{N})).
\end{multline*}

This amounts to

\begin{multline*}
a\circ \kappa(x,\xi,y)=\sum_{\mu\in
\N^d}\frac{\partial_y^{\mu}}{\mu!}(a\circ
\kappa(x,\xi,y))|_{y=x}(y-x)^{\mu}\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})\\+\sum_{\mu\in
\N^d}R_{\mu}(a\circ \kappa)(x,\xi,y)(y-x)^{\mu}(\psi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})-\psi_{|\mu|-1}(\tfrac{c'\theta(x,\xi,y)}{|\mu|-1})).
\end{multline*}

Before proceeding further, we integrate by parts in $Op(a)$ to
trade the powers of $x-y$ for derivatives in $\xi$:

\begin{multline*}
\N^d}\partial_{\xi}^{\mu}\left[\frac{i^{|\mu|}\partial_y^{\mu}}{\mu!}(a\circ
\kappa(x,\xi,y))|_{y=x}\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})\right]\dd
\xi\\
\N^d}|i|^{|\mu|}\partial_{\xi}^{\mu}\left[R_{\mu}(a\circ
\kappa)(x,\xi,y)(\psi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})-\psi_{|\mu|-1}(\tfrac{c'\theta(x,\xi,y)}{|\mu|-1}))\right]\dd \xi.
\end{multline*}

Each term in this expansion is real-analytic with respect to $(x,y)$
in the variables $(x,\theta,y)$. Indeed, given $1\leq k\leq d$,

\begin{align*}
\partial_{y_j}(a\circ \kappa)=&\partial_{y_j}\kappa\cdot (\nabla
a)\circ \kappa=(\partial_{y_j}a)\circ \kappa+\partial_{y_j}\theta\cdot
(\nabla_{\xi}a)\circ \kappa\\
\partial_{\xi_j}(a\circ \kappa)=&\partial_{\xi_j}\theta\cdot (\nabla_{\xi}
a)\circ \kappa
\end{align*}

and both $(\nabla_{\xi,y}) \theta)\circ \kappa^{-1}$ and $a$ are
real-analytic with respect to $(x,y)$. Therefore, by induction, for
every $\mu\in \N^{d}$
$[\partial_y^{\mu}(a\circ \kappa)|_{y=x}]\circ \kappa^{-1}$ is
real-analytic with respect to $(x,y)$. The term $R_{\mu}$ only
depends on these kind of derivatives, as evidenced by the general remainder
formula for Taylor series

\[
\]

Let us now prove that the remainders are small:

\[
\left|\sum_{\mu\in
\N^d}\partial_{\xi}^{\mu}\left[R_{\mu}(a\circ
\kappa)(x,\xi,y)(\psi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})-\psi_{|\mu|-1}(\tfrac{c'\theta(x,\xi,y)}{|\mu|-1}))\right]\right|\leq
\]

thereby obtaining (by Proposition <ref>) that
this term in $a\circ \kappa$ contributes to $Op(a)$ by a
regularising operator. Given $\mu\in \N^d$, one has

\[
\psi_{|\mu|}(\tfrac{c'\theta}{|\mu|})-\psi_{|\mu|-1}(\tfrac{c'\theta}{|\mu|-1})\neq
0 \Rightarrow \frac 12 (|\mu|-1)\leq c'|\theta|\leq |\mu|.
\]

Now, for some $\rho>0,R>0$,

\[
\forall j+k\leq \xi/R, |\nabla^j_y\nabla^k_{\xi}(a\circ \kappa)(x,\xi,y)|\leq
\]

so that, given $\mu\in \N^d$ and $\theta$ such that $ \frac 12
(|\mu|-1)\leq c'|\theta|\leq |\mu|$, given $\N^{d}\ni \nu\leq
\mu$,

\begin{multline*}
\frac{\mu!}{\nu!(\mu-\nu)!}\partial_{\xi}^\nu R_{\mu}(a\circ
\kappa)(x,\xi,y)\partial_{\xi}^{\mu-\nu}(\psi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})-\psi_{|\mu|-1}(\tfrac{c'\theta(x,\xi,y)}{|\mu|-1}))\\
\leq C(a)|\mu|\frac{|\mu|!}{(|\mu|-|\nu|)!}\rho^{2|\mu|}(c')^{|\mu|-|\nu|}|\xi|^{d-|\nu|}.
\end{multline*}

With respect to $|\nu|$, the right-hand side is a log-convex
function; if $c'$ is small enough, at both endpoints $|\nu|=0$ and
$|\nu|=|\mu|$, it is smaller than $Ce^{c_1|\theta|}$. Therefore

\[
\left|\partial_{\xi}^{\mu}\left[R_{\mu}(a\circ
\kappa)(x,\xi,y)(\psi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})-\psi_{|\mu|-1}(\tfrac{c'\theta(x,\xi,y)}{|\mu|-1}))\right]\right|\leq
\]

and for $\theta$ fixed the number of non-zero terms in the sum over
$\mu$ is polynomial in $|\theta|$.

Similarly, within

\[
\sum_{\mu\in
\N^d}\partial_{\xi}^{\mu}\left[\frac{i^{|\mu|}\partial_y^{\mu}}{\mu!}(a\circ
\kappa(x,\xi,y))|_{y=x}\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})\right],
\]

the terms where at least one derivative with respect to $\xi$ hits
$\chi_{|\mu|}$ are exponentially small: indeed, given $\mu\in \N^d$, the support of $\nabla
\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})$ lies within $\{\frac
12 |\mu|\leq c'|\theta|\leq |\mu|\}$, and the estimates on the
derivatives are the same as previously.

To conclude the proof, we inject the expression of $a\circ \kappa$

\[
\sum_{\mu\in
\N^d}\partial_{\xi}^{\mu}\frac{i^{|\mu|}\partial_y^{\mu}}{\mu!}(a\circ
\kappa(x,\xi,y))|_{y=x}\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})
\]

to prove that this is close to $b\circ \kappa$. We obtain

\[
\sum_{\mu\in \N^d}\sum_{j\in
\N}\frac{i^{|\mu|}\partial_{\xi}^{\mu}\partial_y^{\mu}}{\mu!}[a_j(x,\xi,y)\chi_j(\tfrac{c\theta(x,\xi,y)}{j})]|_{y=x}\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|}),
\]

and again one can prove that this is exponentially small unless no
derivative hits $\chi_j$: the support of
$\nabla\chi_j(\tfrac{c\theta(x,\xi,y)}{j})$ lies in $\{\frac 12
j\leq c|\theta|\leq j\}$, and one must also have
$|\mu|<2c'|\theta|$; as long as $c'<\frac{c}{4}$, this ensures $2|\mu|\leq j$, hence for all
$\nu_y\leq \mu$ and $\nu_{\xi}\leq \mu$,

\begin{multline*}
\left|\frac{\mu!}{\nu_y!(\mu-\nu_y)!\nu_{\xi}!(\mu-\nu_{\xi})!}\partial^{\nu_y}_y\partial^{\nu_{\xi}}_\xi
\leq \rho^{j+2|\mu|}\frac{j!|\mu|!}{(|\mu|-|\nu_y|)!(|\mu|-|\nu_\xi|)!}c^{2|\mu|-|\nu_y|-|\nu_{\xi}|}|\xi|^{d-j-|\nu_{\xi}|};
\end{multline*}

for $c$ small this log-convex function of $|\nu_y|$ and $|\nu_{\xi}|$ is smaller
than $e^{-c|\theta|}$ at all four endpoints
$(0,0)$, $(0,|\mu|)$, $(|\mu|,0)$, $(|\mu|,|\mu|)$.

It remains

\[
\sum_{\mu\in \N^d}\sum_{j\in
\N}\frac{i^{|\mu|}\partial_{\xi}^{\mu}\partial_y^{\mu}a_j}{\mu!}(x,\xi,x)\chi_j(\tfrac{c\theta(x,\xi,x)}{j})\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|}),
\]

where we recall that

\[
b(x,\xi,y)=\sum_{\mu\in \N^d}\sum_{j\in
\N}\frac{i^{|\mu|}\partial_{\xi}^{\mu}\partial_y^{\mu}a_j}{\mu!}(x,\xi,x)\chi_{j+|\mu|}(\tfrac{c\theta(x,\xi,x)}{j+|\mu|});
\]

therefore we want to prove that the following is exponentially small

\[
\sum_{\mu\in \N^d}\sum_{j\in
\N}\frac{i^{|\mu|}\partial_{\xi}^{\mu}\partial_y^{\mu}a_j}{\mu!}(x,\xi,x)[\chi_j(\tfrac{c\theta(x,\xi,x)}{j})\chi_{|\mu|}(\tfrac{c'\theta(x,\xi,y)}{|\mu|})-\chi_{j+|\mu|}(\tfrac{c\theta(x,\xi,x)}{j+|\mu|})].
\]

We first observe that this is zero whenever $j\geq 2c|\theta|$, and
also whenever $j+|\mu|\leq c|\theta|$ and $|\mu|\leq
2c'|\theta|$. Therefore we are only interested in situations where
$j\leq 2c|\theta|$ and either $|\mu|\leq 2c'|\theta|$, $j\geq
(c-2c')|\theta|$, or $|\mu|\geq 2c'|\theta|$; since $c'\leq \frac
c4$, this means that we are only considering cases where either $j$
or $|\mu|$ is larger than $2c'|\theta|$. In this setting,

\[
\left|\frac{\partial_{\xi}^{\mu}\partial_y^{\mu}a_j}{\mu!}(x,\xi,x)\right|\leq
C(a)\mu!j!|\xi|^{d-j-|\mu|}\leq Ce^{-c_1|\theta|}.
\]

This concludes the proof.

Define the Moyal product as follows: given two formal analytic symbols
$a(x,\xi),b(x,\xi)$, let

\[
\]

* The symbol product of two formal analytic symbols is a formal
analytic symbol; in fact the topology of formal analytic symbols is
induced by a countable family of Banach spaces of analytic
symbols that are Banach algebras for the Moyal product.

* If both $a$ and $b$ are defined on
$U\times \R^n\times V$ where $U\Subset V$, then

\[
Op(a)Op(b)-Op(a\sharp b)
\]

is regularising.

* Moreover, for every formal analytic amplitude $a$ such that $a_0$ is bounded
away from $0$, there exists a formal analytic amplitude $b$ such that
$(a\sharp b)=1$, and there exists a formal analytic amplitude $c$ such that
$c\sharp c=a$.

* Define the adjoint $a^*$ of a formal analytic amplitude $a$ as

\[
(a^*)_k:(x,\xi)=\sum_{|\mu|\leq k}\frac{i^{|\mu|}}{\mu!}\partial_x^{\mu}\partial_{\xi}^{\mu}\overline{a}_{k-|\mu|}(x,\xi).
\]

Then, given a formal analytic amplitude $a$ with $a^*=a$ and $a_0>0$, there
exists a formal analytic amplitude $b$ such that
$b^*\sharp b=a$.

Claims 1 and 3 only depend on the formal calculus and
are well-known; see [32]. A related
fact, proved in Appendix <ref>, is the following:
with the symbol norms introduced in Definition <ref>,
for every $m$ large enough (depending on the dimension) and every
$R\geq 2^{d+1}\rho^2$,

\[
\|a\sharp b\|_{S_m^{\rho,R}}\leq
\]

Claim 2 is a particular case of
Proposition <ref>.

To prove claim 4, fix a Banach space $S^{\rho,R}_m$ of symbols as in Definition <ref>, containing
$a,\sqrt{a_0},(\sqrt{a_0})^{-1}$ (where we invert with respect to
the Moyal product) and satisfying the requirements of Proposition <ref>. Letting $b_0=\sqrt{a_0}$, by claim 3,

\[
(b_0^*)^{-1}\sharp a\sharp (b_0)^{-1}=1+r
\]

where the inverse is taken with respect to the Moyal product and
where $r$ is a formal analytic symbol of degree $-1$.

Up to increasing $R$, we can assume
$\|r\|_{S^{\frac{\rho}{2},\frac{R}{4}_m}}<\frac{1}{12}$. Then the power

\[
\sqrt{1+r}:=1+\frac{r}{2}-\frac{r\sharp r}{8}+\frac{r\sharp r
\sharp r}{16}+\ldots
\]

converges to an element $c$ of $S^{\rho,R}_m$ such that $c^*=c$ and
$c\sharp c = 1+r$. Setting $b=c\sharp b_0$ then concludes the proof.

Our next move is to translate the pseudodifferential algebra into one
that is more suited to our microlocal needs, as a workaround to Remark

The FBI transform from $L^2(\R^n)$ to $L^2(\R^n\times S^{n-1})$ is the operator with singular kernel

\[
T:\R^n\times S^{n-1}\times \R^n\ni (x,\omega,y)\mapsto \int_0^{+\infty} e^{it((x-y)\cdot
\omega+i\frac{|x-y|^2}{2})}t^{\frac{n+1}{4}}\dd t.
\]

The power of $t$ is chosen such that $T$ is continuous from
$L^2(\R^n)$ to $L^2(\R^n\times S^{n-1})$ and $T^*T$ is a degree $0$
pseudodifferential operator.

The FBI transform of a compactly supported distribution extends into an holomorphic
function of $x-i\omega$ on $\R^n\times B(0,1)$.

It suffices to remark that, writing the phase as

\[
t((x-y)\cdot \omega+i\tfrac{|x-y|^2}{2}+i\tfrac{(1-|\omega|^2)}{2}),
\]

we obtain a holomorphic function of $x-i\omega$ on $\R^n\times
B(0,1)$, whose imaginary part is non-negative.

Before proceeding further, we introduce a helpful notation: given
$z\in \C^n$ we set $z^2=\sum_{j=1}^nz_j^2$; this is the holomorphic
extension of $\R^n\ni x\mapsto |x|^2$. We will reserve $|z|^2$ for the
Hermitian norm of $z$.

* The Lagrangian associated with $T$ is

\[
\Lambda_T=\{(x,t(\omega+i(x-y)),\omega,t(x-y),y,t(\omega+i(x-y))),t>0,(x-y)\cdot \omega=-i\tfrac{(x-y)^2}{2}.\}
\]

* $\Lambda_{T^*}\circ
\Lambda_T={\rm diag}(T^*\C^n\setminus \{0\})$, so that
$T^*T$ is an elliptic pseudodifferential operator.

* The
projection of
$\Lambda_{T}\circ\Lambda_{T^*}$ onto the
base has corank $1$ everywhere. Moreover the intersection of
$\Lambda_{T}\circ\Lambda_{T^*}$ and the real locus is a half-line
bundle over $\diag(\R^{n}\times S^{n-1})$, so that the integral
kernel of any operator of the form $T^*Op(a)T$,
for a formal analytic amplitude $(a_j)_{j\in \N}$, or more generally of
any Fourier Integral Operator with the same Lagrangian as $T^*T$, is
real-analytic away from the diagonal, and near the diagonal takes the form,

\[
\int_0^{+\infty}e^{it\psi(x_1,\omega_1,x_2,\omega_2)}b(x_1,\omega_1,x_2,\omega_2,t)\dd
\]

In this formula,

\begin{equation}\label{eq:phase_Toep}
\psi(x_1,\omega_1,x_2,\omega_2):=i(1+(\tfrac{x_1-i\omega_1-x_2-i\omega_2}{2})^2),
\end{equation}

and in particular $\psi$ extends into a holomorphic function of
$x_1-i\omega_1$ and $x_2+i\omega_2$ on $\R^n\times B(0,1)\times
\R^n\times B(0,1)$, and $\im(\psi)$ is positive away from
$\diag(\R^n\times S^{n-1})$.

* $b$ is the realisation of a formal analytic amplitude.

* $r$ and $b$ extend into holomorphic functions of
$x_1-i\omega_1$ and $x_2+i\omega_2$ on a neighbourhood of
$\diag(\R^n\times S^{n-1})$ in $\R^{4n}$.

* $b$  realises a formal analytic amplitude obtained from $a$ by usual
stationary phase. The first term $b_0$ is the only function
such that $b_0(x,\omega,x,\omega)=a_0(x,\omega)$ and which extends
holomorphic function of $\omega_1+ix_1$ and $\omega_2-ix_2$. More
generally, for any $j,\ell\in \N$, $b_j(x,\omega,x,\omega)$ only depends on $a_l$ and its
derivatives up to degree $2j-\ell$ at $x,\omega$.

The description of $\Lambda_T$ follows by an immediate computation. Now

\begin{align*}
\Lambda_{T^*}\circ \Lambda_T=\{(y_1,\eta_1,y_2,\eta_2),\exists
&(x-y_2)\cdot \omega=-i\tfrac{(x-y_2)^2}{2}\\
&(y_1-x)\cdot \omega=-i\tfrac{(x-y_1)^2}{2}\}.
\end{align*}

The three first conditions ensure $\eta_1=\eta_2$. In particular,
$\eta_1^2=\eta_2^2$, so that (using the last two conditions)
$t_1=t_2$; finally the fourth condition gives
$y_1=y_2$. Reciprocally, given $(y,\eta)\in T^*\R^d\setminus \{0\}$,
setting $x=y,t_1=t_2=|\eta|,\omega=\eta/|\eta|$ yields
$(y,\eta,y,\eta)\in \Lambda_{T^*}\circ \Lambda_T$.

We now consider

\begin{align*}
\Lambda_{T}\circ \Lambda_{T^*}=\{(x_1,\xi_1,\omega_1,v_1,x_2,\xi_2,\omega_2,v_2),\exists
&(x_1-y)\cdot \omega_1=-i\tfrac{(x_1-y)^2}{2}\\
&(y-x_2)\cdot \omega_2=-i\tfrac{(y-x_2)^2}{2}\}.
\end{align*}

Given $(x_1,\omega_1,x_2,\omega_2)$ fixed, conditions 1, 3, and 5 give $\xi_1=\xi_2$; together with the
two last conditions, $\xi_1^2=\xi_2^2$ yields $t_1=t_2$. Using
again condition 5, we can write
$y=\frac{x_1-i\omega_1+x_2+i\omega_2}{2}$, and then either the
last two conditions amounts to

\[
\]

on this codimension 1 manifold, whose intersection with
the real locus is exactly the diagonal, the space of solutions is a
half-line described by $t_1$.

The rest of the claim follows from there by applying
Propositions <ref>,
<ref>, and <ref>. The property of holomorphic extension
of $b$ comes from that of the kernel of $T$ itself; the fact that
holomorphic extension of real-analytic functions prescribed on the diagonal is unique comes from the fact that $\R^n\times
B(0,1)$ is strongly pseudoconvex.

The microlocal structure of operators of the form $TOp(a)T^*$ is
exactly that of the candidate for the Szegő projector at the
boundary of $\R^n\times B(0,1)$, which is strongly pseudoconvex
since its natural defining function $|\omega|^2-1$ is strongly
p.s.h. Indeed,

\begin{align*}
\psi(x_1,\omega_1,x_2,\omega_2)&=\frac{i}{4}\left((x_1-x_2)^2+(\omega_1-\omega_2)^2+2i(x_1-x_2)\cdot(\omega_1+\omega_2)\right)
\\ &=-i\left(\frac
14[(x_1+i\omega_1)^2+(x_2-i\omega_2)^2-2(x_1+i\omega_1)\cdot (x_2-i\omega_2)]-1\right)
\end{align*}

is the holomorphic extension of the function equal to
$i(1-|\omega|^2)$ on the diagonal $(x_1,\omega_1)=(x_2,\omega_2)$.

The Szegő kernel on $\R^n\times S^{n-1}$ is a Fourier Integral Operator with phase $\psi$.

Recall from Proposition <ref> that $a:\xi\mapsto
e^{-|\xi|}\int_{S^{n-1}}e^{\xi\cdot y}\dd y$ is, up to an exponential factor,
the realisation of an elliptic formal analytic amplitude. Thus

\[
b:\R^n\ni \xi \mapsto \frac{e^{-|\xi|}}{\int_{S^{n-1}}e^{\xi\cdot
y}\dd y}
\]

is a radial function and the realisation of an elliptic formal analytic
amplitude. Passing to spherical coordinates, the Szegő kernel
computed in Proposition <ref> now reads

\[
\nu}\dd \nu\right]r^{n-1}\dd r,
\]

which we simplify into

\[
\]

If $(z,w)$ belongs to a neighbourhood to $\{z=w\in \R^n\times
S^{n-1}\}$, then $i(z-\overline{w})$ lies in a neighbourhood of
$2S^{n-1}$ in $\C^n$, so that $ir(z-\overline{w})\in \Omega$.

To conclude, near the diagonal, the Szegő kernel
takes the form of a Fourier integral operator whose phase is the
holomorphic extension of

\[
\psi_2(z,\overline{z})=-2+2|\im(z)|
\]

which is a defining function for $\R^n\times S^{n-1}$. By
Proposition <ref>, one can rewrite it as a
Fourier Integral Operator with phase $\psi$. Away from the
diagonal we already know by Proposition <ref>
that the Szegő kernel is real-analytic.

Given $U\subset \R^n\times
S^{n-1}$, open and relatively compact, we let

\[\mathcal{H}(U)=\ker_{\mathcal{E}'(\overline{U})}(\partial_b)/(C^{\omega}(\overline{U})+\mathcal{E}'(\partial

and $S(U)$ the space of formal analytic amplitudes on
$\overline{U}\times \R_+$.

* Given $V\subset \R^n\times S^{n-1}$ and $a\in S(V)$, there exists a
neighbourhood $W$ of $\overline{V\times V}$ and a unique formal analytic amplitude $\widetilde{a}$
on $W\times \R_+$, holomorphic in the second variable and
anti-holomorphic in the first variable.

* Given $U\Subset V$ open and
$\epsilon>0$ small enough, letting $\Omega=\{(x,t,y)\in U\times \R^+\times V,
\dist(x,y)<\epsilon\}$, then $(\phi,\Omega,U)$ satisfies
(<ref>) and
$I_{\psi,\Omega,U}(\widetilde{a})$ maps
$\mathcal{F}(V)$ into $\mathcal{H}(U)$.

* There exists a sequence $(B_j)_{j\in \N}$ of bi-differential operators
on $\R^n\times S^{n-1}$ such that for every $j$, $B_j$ has total
degree $2j$, and for every relatively compact $V\subset \R^n\times
S^{n-1}$, $S(V)$ is a unit algebra for the product

\[
(S(V))^2\ni (a,b)\mapsto (a\sharp
\]

* Invertible elements for the product above are exactly the formal amplitudes
$a=(a_j)_{j\in \N}$ such that $a_0$ never vanishes.

* For every $U_1\Subset U_2\Subset V$ and $a,b\in S(V)$,
letting $\epsilon$ small enough and

\begin{align*}
\Omega_1&=\{(x,t,y)\in U_1\times \R^+\times U_2,
\dist(x,y)<\epsilon\}\\
\Omega_2&=\{(x,t,y)\in U_2\times \R^+\times V,
\dist(x,y)<\epsilon\},
\end{align*}

one has

\[
\]

* The unit for $\sharp$ acts as the identity on

Since $\R^n\times S^{n-1}$ is strongly pseudoconvex, the diagonal of
$(\R^n\times S^{n-1})^2$ is a totally real submanifold, where we
reverse the CR structure on the left factor. Therefore any
real-analytic function on an open set $V\subset \R^n\times S^{n-1}$
can be extended CR-holomorphically to a neighbourhood of $\diag(V)$ in
$(\R^n\times S^{n-1})^2$.

Given a formal analytic amplitude $\widetilde{a}$ near the diagonal such
that $\partial_{b;1}\widetilde{a}=0$ (by which we mean: it is anti-CR-holomorphic in
the first variable), since $\partial_{b;1}\psi=0$ as well, local
realisations of $I_{\psi,\Omega,U}(\widetilde{a})$ as in Definition
<ref> will indeed map $\mathcal{E}'(\overline{V})$ into
functions $u$ such that $\partial_{b;1}u$ is real-analytic on a
neighbourhood of $\overline{U}$. Therefore, by Proposition
<ref>, if $S$ denotes the
(antiholomorphic) Szegő
projector one has indeed $u=(1-S)u+Su$ where
$\partial_{b}Su=0$ and $(1-S)u$ is real-analytic on
$\overline{U}$. In particular, $I_{\psi,\Omega,U}(\widetilde{a})$ indeed maps
$\mathcal{F}(V)$ into $\mathcal{H}(U)$.

Recalling from Proposition <ref> that
$\Lambda_{\psi}=\Lambda_{T}\circ \Lambda_{T^*}$ and that
$\Lambda_{T^*}\circ \Lambda_T$ is the diagonal of $(T^*\C^n)^*$, we
obtain that $\Lambda_{\psi}\circ \Lambda_{\psi}=\Lambda_{\psi}$. We
also proved in Proposition <ref> that any analytic Fourier Integral
Operator with Lagrangian $\Lambda_{\psi}$ can be written as a
Fourier Integral Operator with phase $\psi$. Therefore, given
$V,a,b, U_1,U_2,\Omega_1,\Omega_2$ as in the claim, there exists a formal analytic
amplitude $c$ such that
$I_{\psi,\Omega_1,U_1}(\widetilde{a})I_{\psi,\Omega_2,U_2}(\widetilde{b})=I_{\psi,\Omega_1,U_1}(c)$. The
amplitude $c$ is obtained from $a$, $b$ and $\psi$ by stationary phase and
change of variables in the middle variables, so that $c$ is
CR-holomorphic on $\Omega_1$. By the usual $C^{\infty}$ calculus of
these operators (see
for instance [7]), as a formal amplitude, the restriction of
$c$ to the diagonal is indeed of the form $a\sharp b$ as in the

We know by Proposition <ref> that the Szegő
projector has a kernel real-analytic from the diagonal and of the
form $I_{\psi,\Omega,U}(a)$, where $U=\R^n\times S^{n-1}$,
$\Omega=(\R^n\times S^{n-1})^2$ and $a$ depends only on the fibre
variable. By definition, $S$ acts as identity on $\mathcal{H}$; its
amplitude is a unit for the algebra above.

It remains to prove that elliptic amplitudes can be inverted, which is
a claim of formal nature (by the product formula, it suffices to
check it at the level of expansion of amplitudes, and not
a priori as operators making sense from some function space to
another). To this
end, we first suppose that amplitudes are globally defined, so that for every $a\in
S(\R^n\times S^{n-1})$, by Proposition <ref>,
the operator $T^*I_{\psi,\Omega,V_1}(a)T$ is a pseudodifferential
operator on $\R^n$. Moreover if $a$ is elliptic then so
is this pseudodifferential operator, so that by Proposition
<ref> it admits an inverse with amplitude

Now, $M=I_{\psi}(a)TOp(b)T^*$ satisfies, for every $c\in
S_{\mathcal{H}}(\R^n\times S^{n-1})$,

\[
\]

and since $T$ and $T^*$ are invertible, we conclude that $M$ acts as
identity on $\mathcal{H}$. Moreover, by the previous considerations,
$M$ can be written in the form $I_{\psi}(d)$ for some $d\in
S_{\mathcal{H}}(\R^n\times S^{n-1})$.

As a formal amplitude, $d$ is obtained from $a$ by a formula of the
form $d=\sum_{k=0}^{+\infty}C_k(a,a_0^{-1})$, for some
sequence of bidifferential operators $C_k$ (this is a consequence of
Propositions <ref> and <ref>;
one can also directly prove it using the $C^{\infty}$ calculus). Applying this formula to
a locally defined amplitude $a$ leads necessarily its inverse for
the formal product.

In the last proof and from now on, given $V_0\subset \R^n\times S^{n-1}$ and given a
formal analytic amplitude $a$ on $V_0\times (0,+\infty)$, we will
sometimes drop the domains and consider $I_{\psi}(a)$ as the collection of
operators of the form
$I_{\psi,\Omega,U}(\widetilde{a}):\mathcal{F}(V)\to \mathcal{H}(U)$ where $U\Subset
V\Subset V_0$ and $\Omega$ is a small neighbourhood of $U\times
V\times (0,+\infty)$ on which $a$ extends into $\widetilde{a}$,
holomorphic in the first variable and anti-holomorphic in the second
These operators will always strictly decrease the definition set of
the function on which they act.

The operator $T^*$ sends $\mathcal{O}(\R^n\times S^{n-1})$ to
$\mathcal{O}(\R^n)$, and therefore $T$ is well-defined from
$\mathcal{O}'(\R^n)$ to $\mathcal{O}'(\R^n\times S^{n-1})$. Thus we
can define the analytic wave front set of an analytic
functional as follows.

The analytic wave front set of $u\in \mathcal{O}'(\R^n)$ is
defined as the analytic singular support of $Tu$.

This is equivalent to the usual definition of analytic singular
support by means of the FBI transform, which asks for exponential
decay of a related quantity. In particular, it is equivalent to all
of the classical notions of analytic wave front set

We conclude this section with a few comments. Our algebra of operators
defined in Proposition <ref> is inspired from
“covariant Toeplitz operators”, originally defined in the semiclassical
setting in [8] as an alternative
description of operators of the “contravariant Toeplitz”
form $SaS$, where $S$ is the Szegő projector and $a$ acts by
multiplication; these operators also form an algebra in analytic
regularity [11, 30].

It it a bit awkward that
the natural range of $T$, and therefore the space on which our algebra
acts, extends into a space of antiholomorphic functions on
$\R^n\times B(0,1)$. Alternative conventions, however, are not more
satisfactory; one can set the FBI phase as $(y-x)\cdot
\omega+i\frac{|x-y|^2}{2}$ and obtain holomorphic functions, but at
the end we
obtain a characterisation of the analytic wave front set of $u$ as the