# The weak (1,1) boundedness of Fourier integral Operators with complex phases

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Michael Ruzhansky
Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete
Mathematics Ghent University, Belgium and School of Mathematical Sciences
Queen Mary University of London United Kingdom E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $T$ be a Fourier integral operator of order $-(n-1)/2$ associated with a
canonical relation locally parametrised by a real-phase function. A
fundamental result due to Seeger, Sogge, and Stein proved in the 90’s, gives
the boundedness of $T$ from the Hardy space $H^{1}$ into $L^{1}.$
Additionally, it was shown by T. Tao the weak (1,1) type of $T$. In this work,
we establish the weak (1,1) boundedness of a Fourier integral operator $T$ of
order $-(n-1)/2$ when it has associated a canonical relation parametrised by a
complex phase function.

###### Key words and phrases:

Fourier Integral Operator, Complex Phase, Weak (1,1) continuity

###### 2020 Mathematics Subject Classification:

35S30, 42B20; Secondary 42B37, 42B35

The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis
and Partial Differential Equations and by the Methusalem programme of the
Ghent University Special Research Fund (BOF) (Grant number 01M01021). Duván
Cardona was supported by the Research Foundation-Flanders (FWO) under the
postdoctoral grant No 1204824N. Michael Ruzhansky is also supported by EPSRC
grant EP/R003025/2.

###### Contents

  1. 1 Overview
    1. 1.1 Outline
    2. 1.2 Fourier integral operators with real-phase functions
    3. 1.3 Fourier integral operators with complex-phase functions
    4. 1.4 Main result
  2. 2 Preliminaries
    1. 2.1 Basics on symplectic geometry
    2. 2.2 Almost analytic continuation of real manifolds
    3. 2.3 Fourier integral operators with real-valued phases
    4. 2.4 Fourier integral operators with complex-valued phases
    5. 2.5 Asymptotics for oscillatory integrals
  3. 3 Reduction to phases of strictly positive type
  4. 4 Proof of Theorem 3.1
    1. 4.1 Degenerate and non-degenerate components
      1. 4.1.1 Symbol classes in projective co-ordinates
    2. 4.2 Boundedness of the degenerate component
    3. 4.3 Boundedness of the non-degenerate component
      1. 4.3.1 Construction of the operators $A$ and $S$
    4. 4.4 Control of the error operator
  5. 5 Final remarks
    1. 5.1 Applications
    2. 5.2 Conclusions and bibliographical discussion
    3. 5.3 Acknowledgement

##  1\. Overview

In this paper we establish the weak (1,1) boundedness of Fourier integral
operators with complex phases and of order $-(n-1)/2.$ Our approach combines
two methods. First, we adapt to our setting the factorisation procedure
developed by Tao [45] in the proof of the weak (1,1) inequality of these
operators in the setting of real-valued phase functions, and the other one
came from the global parametrisation of the canonical relation associated to
the operator with an appropriate complex phase as observed by the second
author in [38], when extending to Fourier integral operators with complex
phases the $H^{1}$-$L^{1}$ boundedness result established by Seeger, Sogge and
Stein in [43]. To present our main Theorem 1.7 we are going to introduce some
preliminaries. We consider $\mathbb{R}^{n},$ with $n\geq 2,$ and we denote by

$f\mapsto\widehat{f},\,\widehat{f}(\xi):=\smallint_{\mathbb{R}^{n}}e^{-2\pi
ix\cdot\xi}f(x)dx,\,\,\xi\in\mathbb{R}^{n},$

the Fourier transform of an integrable function $f\in L^{1}(\mathbb{R}^{n}).$

###  1.1. Outline

When working with the real methods of the harmonic analysis one can find three
substantial categories related to the boundedness properties of operators:
maximal averages, singular integrals, and oscillatory integrals. In the latter
category, oscillatory integrals provide a necessary tool for exploiting the
geometric properties related to the notions of curvature and of orthogonality.
As it was pointed out by Stein in [42], this family of objects is not easily
classified and appears in multiple forms: Bochner-Riesz operators, variants of
pseudo-differential operators and of Fourier multipliers, and in the main
object treated here, Fourier integral operators.

Fourier integral operators appear as solutions to hyperbolic differential
equations and then their boundedness properties have a direct relation with
the validity of a priori estimates for these equations. As it was shown e.g.
in [38, 43, 45], the geometry and the singularities of the canonical relation
of an operator are reflected in the boundedness properties of operators in
$L^{p}$\- spaces. Due to the compatibility of the space $H^{1}$ with the
Littlewood-Paley theory, see Fefferman and Stein [17], and with the complex
interpolation of operators, a main step in the proof of the
$L^{p}$-boundedness properties of Fourier integral operators (and for other
operators arising in harmonic analysis), is the proof of a suitable criterion
for its $H^{1}$-$L^{1}$-boundedness which is usually interpolated with its
corresponding $L^{2}$-theory. However, as in the case of pseudo-differential
operators, once proved the $H^{1}$-$L^{1}$-continuity criterion one expects
that the corresponding class of $H^{1}$-$L^{1}$-bounded Fourier integral
operators must satisfy the $L^{1}$-$L^{1,\infty}$ boundedness property. This
last property is called the weak (1,1) inequality/boundedness for these
operators.

The purpose of this paper is to prove the weak (1,1) boundedness for the class
of Fourier integral operators associated to symbols of order $m=-(n-1)/2$ and
of complex phases (our main result is Theorem 1.7 below). This order is
expected from the weak (1,1) inequality for Fourier integral operators with
real-valued phase functions proved by Tao in [45], where it was applied a
‘second decomposition’ on the symbol of the operator adapted to a suitable
family of ellipsoids with a smooth variation of eccentricity and orientation
that avoid Kakeya-type covering lemmas, see Theorem 1.2. This approach was
combined with a factorisation strategy. The order $m=-(n-1)/2$ also provides
bounded Fourier integral operators with real-valued phase functions from the
Hardy space $H^{1}$ into $L^{1}$, see Segger, Sogge, and Stein [43], as well
as in the case of complex phase functions as proved by the second author in
[38], see Theorems 1.3 and 1.6, respectively.

The extension of the theory of Fourier integral operators due to Hörmander and
Duistermaat to the case of complex-valued phase functions was systematically
developed by Melin and Sjöstrand in [29] motivated by the problem of the
construction of parametrices, or fundamental solutions for operators of
principal type with non-real principal symbols, see for instance [30].

The algebraic properties for the theory of Fourier integral operators with
complex phases are well developed but their regularity has not been much
studied, with the exception, e.g. of [38]. In some sense, the use of complex
phases provides a more natural approach to Fourier integral operators. Indeed,
when working with the complex phase approach, the geometric obstructions of
the global theory with real phase functions can be avoided and it is a
remarkable fact that every Fourier integral operator with a real phase can be
globally parametrized by a single complex phase, see e.g. [38]. We also refer
to Laptev, Safarov, and Vassiliev [28] for this construction.

###  1.2. Fourier integral operators with real-phase functions

In order to present our main result, we record the definition of Fourier
integral operators based on the real-phase function approach. So, we present
the following simplified version which is suitable for our purposes.

######  Definition 1.1.

A Fourier integral operator $T$ of order $m\in\mathbb{R}$ and with real phase
function $\Phi,$ is a continuous linear operator
$T:C^{\infty}_{0}(\mathbb{R}^{n})\rightarrow C^{\infty}(\mathbb{R}^{n})$ of
the form

$Tf(x)=\smallint\limits_{\mathbb{R}^{n}}e^{i\Phi(x,\xi)}a(x,\xi)\widehat{f}(\xi)d\xi,$
(1.1)

where $\widehat{f}$ is the Fourier transform of $f.$ The symbol $a:=a(x,\xi)$
is of order $m\in\mathbb{R}^{n},$ that is $a$ satisfies the symbol
inequalities

$\forall\alpha\in\mathbb{N}_{0}\,\forall\beta\in\mathbb{N}_{0},\,\,|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leq
C_{\alpha,\beta}(1+|\xi|)^{m-|\alpha|}.$ (1.2)

The symbol $a$ has compact support in $x.$ We assume that in an open
neighbourhood $V$ of
$\pi_{1}(\textnormal{supp}(a))=\\{x:(x,\xi)\in\textnormal{supp}(a)\\},$ (with
$\overline{V}$ being a compact set) the real-valued phase function $\Phi$ is
homogeneous of degree $1$ in $\xi\neq 0,$ smooth in
$(x,\xi)\in\overline{V}\times(\mathbb{R}^{n}\setminus\\{0\\}),$ and satisfies
the non-degeneracy condition

$\forall(x,\xi)\in\overline{V}\times(\mathbb{R}^{n}\setminus\\{0\\}),\,\textnormal{det}\left(\frac{\partial^{2}\Phi(x,\xi)}{\partial{x_{i}}\partial{\xi_{j}}}\right)_{1\leq
i,j\leq n}\neq 0.$ (1.3)

The following result provides the weak(1,1) inequality for a Fourier integral
operator of order $m=-(n-1)/2,$ or equivalently its boundedness from $L^{1}$
into (the weak $L^{1}$-space) $L^{1,\infty}.$

######  Theorem 1.2 (Tao [45], 2004).

Let $T$ be a Fourier integral operator of order $m=-(n-1)/2$ with a real phase
function. Then $T$ is of weak $(1,1)$ type.

The corresponding $L^{p}$-theory for these operators as developed in [43], can
be summarised as follows.

######  Theorem 1.3 (Seeger, Sogge and Stein [43], 1991).

Let $1<p<\infty.$ Let $T$ be a Fourier integral operator of order
$m=-(n-1)|\frac{1}{p}-\frac{1}{2}|,$ with a real phase function. Then $T$ is
bounded from $L^{p}$ to $L^{p}.$ When $p=1,$ then $T$ is bounded from $H^{1}$
to $L^{1}.$

The case $p=2$ in Theorem 1.3 was proved by Duistermaat and Hörmander
[15]111See also Èskin [16]. using the $T^{*}T$-method and extended
independently to the setting of Fourier integral operators with complex phases
by Hörmander in [24] and by Melin and Sjöstrand in [30]. One of the
fundamental differences when working on the boundedness of operators in the
setting of complex-valued phase functions, compared with the real-valued
setting is that one has to deal with the boundedness properties of pseudo-
differential operators in the class $S^{0}_{1/2,1/2}.$ One reason for this is
the analysis of the oscillatory term $e^{\pm(1+i\tau)\textnormal{Im}(\Phi)},$
see Melin and Sjöstrand [30, Page 394].

###  1.3. Fourier integral operators with complex-phase functions

Let $X$ and $Y$ be smooth manifolds of dimension $n$ and let us denote by
$\widetilde{T^{*}(X\times Y)\setminus 0}$ the almost analytic continuation of
$T^{*}(X\times Y)\setminus 0.$ For this notion see Hörmander [22] and
Nirenberg [33], see also Subsection 2.2. Let

$C\equiv
C_{\Phi}=\\{(x,d_{x}\Phi,y,d_{y}\Phi):d_{\theta}\Phi=0\\}\subset\widetilde{T^{*}(X\times
Y)\setminus 0}$

be a smooth positive homogeneous canonical relation parametrised by a phase
function of positive type, namely, it satisfies $\textnormal{Im}(\Phi)\geq 0,$
see Subsections 2.2 and 2.4 for details.

The class $I^{\mu}_{\rho}(X,Y;C)$ of Fourier integral operators is determined
modulo $C^{\infty}$ by those integral operators $T$ that locally have an
integral kernel of the form

$\displaystyle
A(x,y)=\smallint\limits_{\mathbb{R}^{N}}e^{i\Phi(x,y,\theta)}a(x,y,\theta)d\theta,$
(1.4)

where $a(x,y,\theta)$ is a symbol of order $\mu+\frac{N-n}{2}.$ By the
equivalence-of-phase-function theorem, we can always microlocally assume that
$N=n,$ and that the symbol $a$ satisfies the Kohn-Nirenberg type estimates

$|\partial_{x,y}^{\beta}\partial_{\theta}^{\alpha}a(x,y,\theta)|\leq
C_{\alpha,\beta}(1+|\theta|)^{\mu-|\alpha|},$ (1.5)

locally uniformly in $(x,y).$ According to Lemma 2.1 of [30], one can always
re-write the kernel in (1.4) as

$\displaystyle A(x,y)=\smallint\limits_{\mathbb{R}^{N}}e^{2\pi
i(\Phi(x,\theta)-y\cdot\theta)}a(x,y,\theta)d\theta,$ (1.6)

where the function $\Phi$ is of positive type. Note that the amplitude $a$ in
(1.6) is probably different from the one in (1.4), however, we keep the same
notation for simplicity. To be able to use the Fourier inversion formula we
will assume that $a(x,y,\theta)$ is independent of $y\in\mathbb{R}^{n},$ and
compactly supported in $x.$ So, we will work with symbols instead of
amplitudes, in order to recover the simplified form

$Tf(x)=\smallint\limits_{\mathbb{R}^{n}}e^{i\Phi(x,\xi)}a(x,\xi)\widehat{f}(\xi)d\xi.$
(1.7)

To analyse the boundedness of a Fourier integral operator associated to the
phase $\Phi$ we require the following local graph condition.

######  Assumption 1.4.

There exists $\tau\in\mathbb{C},$ such that

$\Phi_{\tau}:=\textnormal{Re}(\Phi)+\tau\textnormal{Im}(\Phi)\textnormal{
defines }\textnormal{ a }\textnormal{ local }\textnormal{ graph }\textnormal{
in }T^{*}(\widetilde{X\times Y})\setminus 0.$ (1.8)

Locally, this means that

$\textnormal{det}\partial_{x}\partial_{\theta}(\textnormal{Re}(\Phi(x,\xi))+\tau\textnormal{Im}(\Phi(x,\xi)))\neq
0,$ (1.9)

on the support of the symbol $a,$ for all $\xi\neq 0.$

######  Remark 1.5.

According to Lemma 1.8.1 of [38], the local graph condition (1.8) in
Assumption 1.4 is equivalent to the same assumption for almost every
$\tau\in\mathbb{C}.$ Moreover, the determinant in (1.9) is a polynomial
function in $\tau$ which means that if (1.8) holds for some
$\tau\in\mathbb{C},$ then it holds for all but finitely many
$\tau\in\mathbb{C}.$ To summarise this discussion, Assumption 1.4 is
equivalent to the existence of the same condition for some, and then for
almost every $\tau\in\mathbb{C}.$

Before presenting our main result we record the $L^{p}$-theory of Fourier
integral operators with complex phases as developed by the second author in
[38, Theorems 1.7.1 and 1.7.2] including the endpoint case $p=1.$

######  Theorem 1.6.

Let $T$ be a Fourier integral operator of order
$\mu=-(n-1)|\frac{1}{p}-\frac{1}{2}|$ associated with a canonical relation
parametrised by a complex phase $\Phi$ of positive type and satisfying the
local graph condition (1.8). If $p=1$ then $T$ is bounded from $H^{1}_{comp}$
to $L^{1}_{loc},$ and for $1<p<\infty,$ $T$ is bounded from $L^{p}_{comp}$ to
$L^{p}_{loc}.$

###  1.4. Main result

The following is the main result of this paper. We will assume that the
complex phase function is of positive type, in such a way that
$\textnormal{Im}(\Phi(x,\xi))>0,$ when $|\xi|\neq 0.$

######  Theorem 1.7.

Let $T$ be a Fourier integral operator of order $\mu=-(n-1)/2$ associated with
a canonical relation parametrised by a complex phase $\Phi$ of positive type
in such a way that $\textnormal{Im}[\Phi(x,\xi)]>0$ when $|\xi|\neq 0,$ and
satisfying the local graph condition (1.8). Then $T$ is of weak $(1,1)$ type.

Below we briefly discuss some aspects of the main differences between the
regularity properties of Fourier integral operators in the setting of complex-
valued phase functions in comparison with the setting of real-valued phases.

######  Remark 1.8.

There is a simple argument to show that our main Theorem 1.7 does not follow
from its real counterpart in Theorem 1.2. Indeed, any Fourier integral
operator $T\in I^{\mu}(X,Y;C)$ in the form

$Tf(x)=\smallint\limits_{\mathbb{R}^{n}}e^{i(\textnormal{Re}(\Phi(x,\xi))+\tau\textnormal{Im}(\Phi(x,\xi))}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\widehat{f}(\xi)d\xi,$
(1.10)

with real $\tau\neq 0$ and with real phase function

$\Phi_{\tau}(x,\xi):=\textnormal{Re}(\Phi(x,\xi))+\tau\textnormal{Im}(\Phi(x,\xi))$

has the symbol

$a_{\tau}(x,\xi)=a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}$

in the class $S^{\mu}_{1/2,1/2}$ if $a\in S^{\mu}.$ $T$ is bounded on
$L^{p}(\mathbb{R}^{n}),$ with $1<p<\infty$ if, for example,

$\mu\leq-\left(n-\frac{1}{2}\right)|1/p-1/2|.$

However when $p\rightarrow 1^{+}$ this order is not good enough as the one in
Theorem 1.2, or compared with the order $-(n-1)/2$ for the
$H^{1}$-$L^{1}$-boundedness proved by Seeger, Sogge and Stein in [43] for real
phase functions or even as proved in [38] for complex phase functions. In
order to improve this order in Theorem 1.7 we need to avoid the reduction to a
real phase function, making some strategic modifications to the argument given
in Tao [45], in particular taking into account that we have to deal with
dyadic components of the symbol
$a_{\tau}(x,\xi)=a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}.$

######  Remark 1.9.

We remark that in Tao [45] it has been introduced a factorisation approach for
the proof of the weak (1,1) inequality of Fourier integral operators, see
Section 4 for details. When $T$ has a real-valued phase function, according to
such an approach a portion of the operator, the non-degenerate part
$T_{\textnormal{non-deg}},$ is factorised as $T_{\textnormal{non-deg}}=SA$
modulo an error operator $E=T_{\textnormal{non-deg}}-SA.$ Here, $S$ is a
pseudo-differential operator of order zero and its symbol is essentially
$|\xi_{n}|^{(n-1)/2}a(x,\xi),$ when $|\xi_{n}|\gg 1,$ where $a$ is the symbol
of $T.$ Note that the weak (1,1) boundedness of $S$ is a consequence of the
Calderón-Zygmund theory. However, compared with the case of Fourier integral
operators with complex phases, one has to redefine the operator $S,$ because
in this setting $|\xi_{n}|^{(n-1)/2}a_{\tau}(x,\xi)\in S^{0}_{1/2,1/2},$ for
$|\xi_{n}|\gg 1.$ Nevertheless, according to the $L^{1}$-theory for the
Hörmander classes $S^{m}_{1/2,1/2},$ $m\in\mathbb{R},$ see Álvarez and Hounie
[1], a pseudo-differential operator with a symbol in these classes is of weak
(1,1) type when $m\leq-n/4.$ In order to avoid the loss of regularity caused
by the order $-n/4$, we will keep the symbol of the operator $a$ in the
pseudo-differential part $S,$ and we will absorb the oscillating term
$e_{\Phi}=e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}$ into the average
operator $A.$ Essentially, this term will modify the argument in [45], where
the $L^{1}$-boundedness of the averaging operator $A$ depends on the uniformly
$L^{1}$-boundedness of the Littlewood-Paley operators, but even, perturbating
this part with the oscillating symbol $e_{\Phi}$ we are able to preserve the
$L^{1}$-boundedness of $A.$ In order to follow this strategy, we will make a
fundamental reduction of our Theorem 1.7 to the case of phase functions of
strictly positive type in Section 3, see Theorem 3.1. We present the proof of
Theorem 1.7 in Section 3. The authors thank Terence Tao for suggesting the
factorisation approach used in this manuscript.

In the next Section 2, we present the preliminaries and the notation used in
this work. We will continue the discussion about the regularity properties of
Fourier integral operator in Section 3, its main result is Theorem 3.1 and its
proof is presented in Section 4. We finish this manuscript with Section 5
where we present an application of our main theorem and also we provide a
bibliographic discussion.

##  2\. Preliminaries

Let $M,$ $X$ and $Y$ be paracompact smooth real manifolds. We assume that $X$
and $Y$ are manifolds of dimension $n.$ Here, we can take $M=X,Y$ or
$M=X\times Y.$ When $Y\subset\mathbb{R}^{n},$ $L^{1}(Y)$ is defined in the
usual form, namely, $f\in
L^{1}(Y)\Longleftrightarrow\|f\|_{L^{1}(Y)}:=\smallint_{Y}|f(x)|dx<\infty,$
where $dx$ denotes the Lebesgue measure on $\mathbb{R}^{n}.$ The measure of a
Borel set $A\subset\mathbb{R}^{n}$ is denoted by $|A|.$ We also consider the
weak $L^{1}$-space $L^{1,\infty}(Y)$ defined by the seminorm
$\|f\|_{L^{1,\infty}(Y)}:=\sup_{t>0}t|\\{y\in Y:|f(y)|>t\\}|.$

If $X$ is a compact manifold without boundary we define the spaces $L^{1}(X)$
and $L^{1,\infty}(X)$ by using partitions of unity (subordinated to local
coordinate systems). We say that an operator $T:L^{1}(Y)\rightarrow
L^{1,\infty}(Y)$ is of weak (1,1) type if there exists a constant $C>0$ such
that for all $f\in L^{1}(Y),$ the inequality $\|Tf\|_{L^{1,\infty}}(Y)\leq
C\|f\|_{L^{1}(Y)}$ holds. A prototype of operators of weak (1,1) type is the
family of Calderón-Zygmund singular integral operators, see Calderón and
Zygmund [8].

Below we precise the notation and the fundamental notions related to the
theory of Fourier integral operators of importance for this work. We start
with the properies of conic Lagrangian manifolds in the next subsection.

###  2.1. Basics on symplectic geometry

Now, let us follow [38, Chapter I]. We recall that a $2$-form $\omega$ is
called symplectic on $M$ if $d\omega=0,$ and for all $x\in M,$ the bilinear
form $\omega_{x}$ is antisymmetric and non-degenerate on $T_{x}M$. The
canonical symplectic form $\sigma_{M}$ on $M$ is defined as follows. Let
$\pi:=\pi_{M}:T^{*}M\rightarrow M,$ be the canonical projection. For any
$(x,\xi)\in T^{*}M,$ let us consider the linear mappings

$d\pi_{(x,\xi)}:T_{(x,\xi)}(T^{*}M)\rightarrow T_{x}M\textnormal{ and,
}\xi:T_{x}M\rightarrow\mathbb{R}.$

The composition $\alpha_{(x,\xi)}:=\xi\circ d\pi_{(x,\xi)}\in
T^{*}_{(x,\xi)}(T^{*}M),$ that is $\xi\circ
d\pi_{(x,\xi)}:T_{(x,\xi)}(T^{*}M)\rightarrow\mathbb{R},$ defines a 1-form
$\alpha$ on $T^{*}M.$ With the notation above, the canonical symplectic form
$\sigma_{M}$ on $M$ is defined by

$\sigma_{M}:=d\alpha.$ (2.1)

Since $\sigma_{M}$ is an exact form, it follows that $d\sigma_{M}=0$ and then
that $\sigma_{M}$ is symplectic. If $M=X\times Y,$ it follows that
$\sigma_{X\times Y}=\sigma_{X}\oplus-\sigma_{Y}.$ Now we record the family of
submanifolds that are necessary when one defines the canonical relations.

  * •

Let $M$ be of dimension $n.$ A submanifold $\Lambda\subset T^{*}M$ of
dimension $n$ is called Lagrangian if

$T_{(x,\xi)}\Lambda=(T_{(x,\xi)}\Lambda)^{\sigma}:=\\{v\in
T_{(x,\xi)}(T^{*}M):\sigma_{M}(v,v^{\prime})=0,\,\forall v^{\prime}\in
T_{(x,\xi)}\Lambda\\}.$

  * •

We say that $\Lambda\subset T^{*}M\setminus 0$ is conic if $(x,\xi)\in\Gamma,$
implies that $(x,t\xi)\in\Gamma,$ for all $t>0.$

  * •

Let $\Sigma\subset X$ be a smooth submanifold of $X$ of dimension $k.$ Its
conormal bundle in $T^{*}X$ is defined by

$N^{*}\Sigma:=\\{(x,\xi)\in
T^{*}X:\,x\in\Sigma,\,\,\xi(\delta)=0,\,\forall\delta\in T_{x}\Sigma\\}.$
(2.2)

The following facts characterise the conic Lagrangian submanifolds of
$T^{*}M.$

  * •

Let $\Lambda\subset T^{*}M\setminus 0,$ be a closed sub-manifold of dimension
$n.$ Then $\Lambda$ is a conic Lagrangian manifold if and only if the 1-form
$\alpha$ in (2.1) vanishes on $\Lambda.$

  * •

Let $\Sigma\subset X,$ be a submanifold of dimension $k.$ Then its conormal
bundle $N^{*}\Sigma$ is a conic Lagrangian manifold.

The Lagrangian manifolds have the following property.

  * •

Let $\Lambda\subset T^{*}M\setminus 0,$ be a conic Lagrangian manifold and let

$d\pi_{(x,\xi)}:T_{(x,\xi)}\Lambda\rightarrow T_{x}M,$ (2.3)

have constant rank equal to $k,$ for all $(x,\xi)\in\Lambda.$ Then, each
$(x,\xi)\in\Gamma$ has a conic neighborhood $\Gamma$ such that

    * 1.

$\Sigma=\pi(\Gamma\cap\Lambda)$ is a smooth manifold of dimension $k.$

    * 2.

$\Gamma\cap\Lambda$ is an open subset of $N^{*}\Sigma.$

The Lagrangian manifolds have a local representation defined in terms of phase
functions that can be defined as follows. For this, let us consider a local
trivialisation $M\times(\mathbb{R}^{n}\setminus 0),$ where we can assume that
$M$ is an open subset of $\mathbb{R}^{n}.$

######  Definition 2.1 (Real-valued phase functions).

Let $\Gamma$ be a cone in $M\times(\mathbb{R}^{N}\setminus 0).$ A smooth
function $\phi:M\times(\mathbb{R}^{N}\setminus 0)\rightarrow\mathbb{R},$
$(x,\theta)\mapsto\phi(x,\theta),$ is a real phase function if, it is
homogeneous of degree one in $\theta$ and has no critical points as a function
of $(x,\theta),$ that is

$\forall t>0,\,\phi(x,t\theta)=t\phi(x,\theta),\textnormal{ and
}d_{(x,\theta)}\phi(x,\theta)\neq 0,\forall(x,\theta)\in
M\times(\mathbb{R}^{N}\setminus 0).$ (2.4)

Additionally, we say that $\phi$ is a non-degenerate phase function in
$\Gamma,$ if for any $(x,\theta)\in\Gamma$ such that
$d_{\theta}\phi(x,\theta)=0,$ one has that

$d_{(x,\theta)}\frac{\partial\phi}{\partial\theta_{j}}(x,\theta),\,\,1\leq
j\leq N,$ (2.5)

is a system of linearly independent vectors on $\mathbb{R}$.

The following facts describe locally a Lagrangian manifold in terms of a phase
function.

  * •

Let $\Gamma$ be a cone in $M\times(\mathbb{R}^{N}\setminus 0),$ and let $\phi$
be a non-degenerate phase function in $\Gamma.$ Then, there exists an open
cone $\tilde{\Gamma}$ containing $\Gamma$ such that the set

$U_{\phi}=\\{(x,\theta)\in\tilde{\Gamma}:d_{\theta}\phi(x,\theta)=0\\},$ (2.6)

is a smooth conic sub-manifold of $M\times(\mathbb{R}^{N}\setminus 0)$ of
dimension $n.$ The mapping

$L_{\phi}:U_{\phi}\rightarrow T^{*}M\setminus
0,\,\,L_{\phi}(x,\theta)=(x,d_{x}\phi(x,\theta)),$ (2.7)

is an inmersion. Let us denote $\Lambda_{\phi}=L_{\phi}(U_{\phi}).$

  * •

Let $\Lambda\subset T^{*}M\setminus 0$ be a sub-manifold of dimension $n.$
Then, $\Lambda$ is a conical Lagrangian manifold if and only if every
$(x,\xi)\in\Lambda$ has a conic neighborhood $\Gamma$ such that
$\Gamma\cap\Lambda=\Lambda_{\phi},$ for some non-degenerate phase function
$\phi.$

######  Remark 2.2.

The cone condition on $\Lambda$ corresponds to the homogeneity of the phase
function.

######  Remark 2.3.

Although we have presented the previous Definition 2.1 of non-degenerate real
phase function in the case of a real function of $(x,\theta),$ the same can be
defined if one considers functions of $(x,y,\theta).$ Indeed, a real-valued
phase function $\phi(x,y,\theta)$ homogeneous of order 1 at $\theta\neq 0$
that satisfies the following two conditions

$\textnormal{det}\partial_{x}\partial_{\theta}(\phi(x,y,\theta))\neq
0,\,\,\textnormal{det}\partial_{y}\partial_{\theta}(\phi(x,y,\theta))\neq
0,\,\,\theta\neq 0,$ (2.8)

is called non-degenerate.

######  Remark 2.4.

For a symplectic manifold $M$ of dimension $2n,$ we will denote by
$\widetilde{M}$ its almost analytic continuation in $\mathbb{C}^{2n}$ (see
[38, Page 10] for details about the construction of $\widetilde{M},$ as well
as many details in [29]). For completeness, we present such a notion in the
following subsection.

###  2.2. Almost analytic continuation of real manifolds

To define the almost analytic continuation of a manifold we require some
preliminaries. We record that a function $f:U\to\mathbb{C}$ defined on an open
subset $U\subset\mathbb{C}^{n}$ is called almost analytic in
$U_{\mathbb{R}}:=U\cap\mathbb{R}^{n},$ if $f$ satisfies the Cauchy-Riemann
equations on $U_{\mathbb{R}},$ that is, if $\overline{\partial}f$ and all its
derivatives vanish in $U_{\mathbb{R}}.$ Here, as usual,
$\overline{\partial}=1/2(\partial_{x}+i\partial_{y}).$

One defines an almost analytic extension of a manifold $M$ requiring that the
corresponding coordinate functions are almost analytic in $M.$ Here, we are
going to present this notion carefully.

Let $\Omega\subset\mathbb{R}^{n}$ and let $\rho:\Omega\rightarrow\mathbb{R}$
be a non-negative and Lipschitz function. A function
$f:\Omega\rightarrow\mathbb{R}$ is called $\rho$-flat in $\Omega$ if for every
compact set $K\subset\Omega,$ and for every integer $N\geq 0,$ one has the
growth estimate $|f(x)|\lesssim_{K,N}\rho(x)^{N},$ for all $x\in K.$ This
notion defines an equivalence relation on the space of mapping on $\Omega:$

$f\textnormal{ and }g\textnormal{ are }\rho\textnormal{-equivalent if
}f-g\textnormal{ is }\rho\textnormal{-flat on }\Omega.$

If $K_{0}\subset\Omega$ is a compact set, we will say that $f$ is flat  on
$K_{0},$ if it is $\rho$-flat with $\rho(x):=\textnormal{dist}(x,K_{0}).$ One
has the following property:

  * •

let $f\in C^{\infty}(\Omega)$ be $\rho$-flat. Then all its derivatives are
$\rho$-flat and $f$ is flat on $K_{0}$ if and only if $D^{\alpha}f=0$ for all
$x\in K_{0}$ and all $\alpha.$

Let $G$ be an open subset of $\mathbb{C}^{n}$ and let $K$ be a closed subset
of $G.$ A function $f\in C^{\infty}(G)$ is almost analytic on $K$ if the
functions $\partial_{j}f$ are flat on $K,$ for all $1\leq j\leq n.$ For an
open set $\Omega\subset\mathbb{R}^{n},$ we will denote

$\tilde{\Omega}:=\Omega+i\mathbb{R}^{n}.$ (2.9)

One can make the identification
$\Omega\cong\tilde{\Omega}\cap\\{z:\textnormal{Im}(z)=0\\}.$ Note that every
function $f\in C^{\infty}(G_{\mathbb{R}})$ defines an equivalence class of
almost analytic functions on $G_{\mathbb{R}},$ which consist of functions in
$C^{\infty}(G)$ which are almost analytic in $G_{\mathbb{R}},$ modulo
functions being flat on $G_{\mathbb{R}}.$ Any representative of this class is
called an almost analytic continuation of $f$ in $G.$

Let $O$ be an open subset of $\mathbb{C}^{n}.$ Let $M$ be a smooth submanifold
of codimension $2k$ of $O,$ and let $K$ be a closed subset of $O.$ Then, $M$
is called almost analytic on $K,$ if for any point $z_{0}\in K,$ there exists
an open set $U\subset O,$ and $k$-functions $f_{j},$ such that every $f_{j}$
is almost analytic on $K\cap U,$ and such that in $U,$ $M$ is defined by the
zero-level sets $f_{j}(z)=0,$ with the differentials $df_{j},$ $1\leq j\leq
k,$ being linearly independent over $\mathbb{C}.$

Two almost analytic submanifolds $M_{1}$ and $M_{2}$ of $O$ are equivalent if
they have the same dimension and the same intersection with $\mathbb{R}^{n},$
namely, that

$M_{1,\mathbb{R}}=M_{1}\cap\mathbb{R}^{n}=M_{2}\cap\mathbb{R}^{n}=M_{2,\mathbb{R}}=:M_{\mathbb{R}},$

and locally $f_{j}-g_{j}$ are flat functions on $M_{\mathbb{R}}$ where $f_{j}$
and $g_{j}$ are the functions that define $M_{1}$ and $M_{2},$ respectively.

A real manifold $\Omega$ defines an equivalence class of almost analytic
manifolds in $\tilde{\Omega}.$ A representative of this class is called an
almost analytic continuation  of $\Omega$ in $\mathbb{C}^{n}.$ Now, let us
consider:

  * •

a real symplectic manifold $M$ of dimension $d=2n,$ and let $\tilde{M}$ be
(modulo its equivalence class) its almost analytic continuation in
$\mathbb{C}^{2n}.$

  * •

Let $\Lambda\subset\tilde{M}$ be an almost analytic sub-manifold containing
the real point $\rho_{0}\in M,$ and let $(x,\xi)$ be its real symplectic
coordinates in a neighbourhood $W$ of $\rho_{0}.$

  * •

Let $(\tilde{x},\tilde{\xi})$ be almost analytic continuations of the
coordinates $(x,\xi)$ in $W,$ in such a way that $(\tilde{x},\tilde{\xi})$ map
$\tilde{W}$ diffeomorphically on an open subset of $\mathbb{C}^{2n}.$

  * •

Let $g$ be an almost analytic function such that $\textnormal{Im}(g)\geq 0,$
in $\mathbb{R}^{n},$ and such that $\Lambda$ is defined in a neighbourhood of
$\rho_{0}$ by the equations $\tilde{\xi}=\partial_{\tilde{x}}g(x),$
$\tilde{x}\in\mathbb{C}^{n}.$

Then, an almost analytic manifold $\Lambda$ satisfying this property, in some
real symplectic coordinate system at every point is called a positive
Lagrangian manifold. We conclude this discussion with the following result,
see e.g. Theorem 1.2.1 in [38].

######  Proposition 2.5.

Let $M,$ $\Lambda$ and $W$ be as before. If $(\tilde{y},\tilde{\eta})$ is
another almost analytic continuation of coordinates in $\tilde{W}$ and
$\Lambda$ is defined by the equation $\tilde{\eta}=H(\tilde{y}),$ in a
neighbourhood of $\rho_{0},$ then $\Lambda$ is locally equivalent to the
manifold $\tilde{\eta}=\partial_{\tilde{y}}h,$ $\tilde{y}\in\mathbb{C}^{n},$
where $h$ is an almost analytic function and $\textnormal{Im}(h)\geq 0.$

###  2.3. Fourier integral operators with real-valued phases

We can assume that $X,Y$ are open sets in $\mathbb{R}^{n}$. One defines the
class of Fourier integral operators $T\in I^{\mu}_{\rho}(X,Y;\Lambda)$ by the
(microlocal) formula

$Tf(x)=\smallint\limits_{Y}\smallint\limits_{\mathbb{R}^{N}}e^{i\Psi(x,y,\theta)}a(x,y,\theta)f(y)d\theta\;dy,$
(2.10)

where the symbol $a$ is a smooth function locally in the class
$S^{\mu}_{\rho,1-\rho}(X\times Y\times(\mathbb{R}^{n}\setminus 0)),$ with
$1/2\leq\rho\leq 1.$ This means that $a$ satisfies the symbol inequalities

$|\partial_{x,y}^{\beta}\partial_{\theta}^{\alpha}a(x,y,\theta)|\leq
C_{\alpha\beta}(1+|\theta|)^{\mu-\rho|\alpha|+(1-\rho)|\beta|},$

for $(x,y)$ in any compact subset $K$ of $X\times Y,$ and
$\theta\in\mathbb{R}^{N}\setminus 0,$ while the real-valued phase function
$\Psi$ satisfies the following properties:

  * 1.

$\Psi(x,y,\lambda\theta)=\lambda\Psi(x,y,\theta),$ for all $\lambda>0$;

  * 2.

$d\Psi\not=0$;

  * 3.

$d_{\theta}\Psi=0$ is smooth (e.g. $d_{\theta}\Psi=0$ implies
$d_{(x,y,\theta)}\frac{\partial\Psi}{\partial\theta_{j}}$ are linearly
independent).

Here $\Lambda\subset T^{*}(X\times Y)\setminus 0$ is a Lagrangian manifold
locally parametrised by the phase function $\Psi,$

$\Lambda=\Lambda_{\Psi}=\\{(x,d_{x}\Psi,y,d_{y}\Psi):d_{\theta}\Psi=0\\}.$

######  Remark 2.6.

The canonical relation associated with $T$ is the conic Lagrangian manifold in
$T^{*}(X\times Y)\backslash 0$, defined by
$\Lambda^{\prime}=\\{(x,\xi,y,-\eta):(x,\xi,y,\eta)\in\Lambda\\}.$ In view of
the Hörmander equivalence-of-phase-functions theorem (see e.g. Theorem 1.1.3
in [38, Page 9]), the notion of Fourier integral operator becomes independent
of the choice of a particular phase function associated to a Lagrangian
manifold $\Lambda.$ Because of the diffeomorphism
$\Lambda\cong\Lambda^{\prime},$ we do not distinguish between $\Lambda$ and
$\Lambda^{\prime}$ by saying also that $\Lambda$ is the canonical relation
associated with $T.$

###  2.4. Fourier integral operators with complex-valued phases

Now we record the following result for canonical relations, see e.g. Theorem
1.2.1 in [38].

######  Proposition 2.7.

Let $\Psi$ be a phase function of the positive type that is defined in a
conical neighbourhood. Let $\tilde{\Psi}$ be an almost analytic homogeneous
continuation of $\Psi$ in a canonical neighbourhood in
$\mathbb{C}^{n}\times\mathbb{C}^{n}\times({\mathbb{C}^{n}}\setminus\\{0\\}).$
Let

$C_{\tilde{\Psi}}=\\{(\tilde{x},\tilde{y},\tilde{\theta})\in\mathbb{C}^{n}\times\mathbb{C}^{n}\times({\mathbb{C}^{n}}\setminus\\{0\\}):\partial_{\theta}\tilde{\Psi}(\tilde{x},\tilde{y},\tilde{\theta})=0\\}.$
(2.11)

Then, the image of the set $C_{\tilde{\Psi}}$ under the mapping

$C_{\tilde{\Psi}}\ni(\tilde{x},\tilde{y},\tilde{\theta})\mapsto(\tilde{x},\partial_{\tilde{x}}\tilde{\Psi},\partial_{\tilde{y}}\tilde{\Psi})|_{(\tilde{x},\tilde{y},\tilde{\theta})}\in\mathbb{C}^{n}\times({\mathbb{C}^{n}}\setminus\\{0\\})\times({\mathbb{C}^{n}}\setminus\\{0\\})$
(2.12)

is a local positive Lagrangian manifold.

Let $X$ and $Y$ be smooth manifolds of dimension $n$ and let us denote by
$\widetilde{T^{*}(X\times Y)\setminus 0}$ the almost analytic continuation of
$T^{*}(X\times Y)\setminus 0.$ Let

$C\equiv
C_{\Phi}=\\{(x,d_{x}\Phi,y,d_{y}\Phi):d_{\theta}\Phi=0\\}\subset\widetilde{T^{*}(X\times
Y)\setminus 0}$

be a smooth positive homogeneous canonical relation. This means that $C$ is
locally parametrised by a complex phase function $\Phi(x,y,\theta),$
satisfying the following properties:

  * 1.

$\Phi$ is homogeneous of order one in $\theta:$
$\Phi(x,y,t\theta)=t\Phi(x,y,\theta),$ for all $t>0,$

  * 2.

$\Phi$ has no critical points on its domain, i.e. $d\Phi\neq 0;$

  * 3.

$\\{d_{\theta}\Phi(x,y,\theta)=0\\}$ is smooth (e.g. $d_{\theta}\Phi=0$
implies that $d\frac{\partial\Phi}{\partial\theta_{j}}$ are independent over
$\mathbb{C}$),

  * 4.

$\textnormal{Im}(\Phi)\geq 0.$

When the last condition is satisfied one says that the complex phase function
$\Phi$ is of positive type. Then, the class $I^{\mu}_{\rho}(X,Y;C)$ of Fourier
integral operators $T$ is determined modulo $C^{\infty}$ by those integral
operators $T$ that locally have an integral kernel of the form

$\displaystyle
A(x,y)=\smallint\limits_{\mathbb{R}^{N}}e^{i\Phi(x,y,\theta)}a(x,y,\theta)d\theta,$

where $a(x,y,\theta)$ is a symbol of order $\mu+\frac{N-n}{2}.$ By the
equivalence-of-phase-function theorem, we can always assume that $N=n,$ and
that the symbol $a$ satisfies the type $(\rho,1-\rho)$-estimates

$|\partial_{x,y}^{\beta}\partial_{\theta}^{\alpha}a(x,y,\theta)|\leq
C_{\alpha,\beta}(1+|\theta|)^{\mu-\rho|\alpha|+(1-\rho)|\beta|},\,\,\frac{1}{2}\leq\rho\leq
1,$ (2.13)

locally uniformly in $(x,y).$

###  2.5. Asymptotics for oscillatory integrals

Now, we recall two principles used in this work: the principle of non-
stationary phase and the principle of stationary phase, respectively.

######  Proposition 2.8 ([42], Page 342).

Suppose that $a$ is smooth and supported in the unit ball, also let $\phi$ be
real-valued so that for some multi-index $\alpha$ with $k=|\alpha|>0,$ we have
that

$|\partial_{x}^{\alpha}\phi|\geq 1$

throughout the support of $a.$ Then,

$\left|\smallint_{\mathbb{R}^{n}}e^{i\lambda\phi(\xi)}a(\xi)d\xi\right|\leq
C_{k,\phi}\lambda^{-\frac{1}{k}}(\|a\|_{L^{\infty}}+\|\nabla_{\xi}a\|_{L^{1}}).$

The constant $C_{k,\phi}$ is independent of $\lambda>0,$ and of $a,$ and
remains bounded as long as the $C^{k+1}$ norm of $a$ remains bounded.

######  Proposition 2.9 ([47], Proposition 6.4).

Let $\phi$ be a smooth function and assume that $\nabla\phi(p)=0.$ Assume also
that the Hessian $H_{\phi}(p)=(\partial_{ij}^{2}\phi(\xi))|_{\xi=p}$ of $\phi$
at $\xi=p,$ is an invertible matrix. Let $\mu$ be the signature of
$H_{\phi}(p),$ and let $\Delta=2^{-n}\det(H_{\phi}(p)).$ Let $a:=a(\xi)\in
C^{\infty}_{0}(\mathbb{R}^{n})$ be supported in a small neighbourhood of
$\xi=p.$ Then for any $N\in\mathbb{N},$ the integral

$I(\lambda)=\smallint_{\mathbb{R}^{n}}e^{i\pi\lambda\phi(\xi)}a(\xi)d\xi,$

satisfies the asymptotic expansion

$I(\lambda)=e^{\pi i\lambda\phi(p)}e^{-\frac{\mu\pi
i}{4}}\Delta^{-\frac{1}{2}}\lambda^{-\frac{n}{2}}\left(a(p)+\sum_{j=1}^{N}\lambda^{-j}D_{2j}a(p)+O(\lambda^{-(N+1)})\right),$
(2.14)

where $D_{2j}$ are differential operators of order $2j,$ with smooth
coefficients depending on $\phi,$ and bounds for finitely many derivatives of
$a.$

##  3\. Reduction to phases of strictly positive type

We say that a complex phase function is of strictly positive type, if
$\textnormal{Im}(\Phi(x,\theta))>0,$ when $|\theta|\sim 1.$ The aim of this
section is to show that our main Theorem 1.7 follows from the following
reduced version.

######  Theorem 3.1.

Let $T$ be a Fourier integral operator of order $-(n-1)/2$ associated with a
canonical relation parametrised by a complex phase $\Phi$ of strictly positive
type, that is, there exists $\varkappa_{0}>0$ such that for all
$(x,\theta)\in\overline{V}\times\overline{\\{\theta:|\theta|\sim 1\\}},$ one
has the lower bound

$\textnormal{Im}(\Phi(x,\theta))>\varkappa_{0},$

and additionally satisfying the local graph condition (1.8). Then $T$ is of
weak type (1,1).

The proof of Theorem 3.1 will be addressed in Section 4. Assuming this
statement we give a proof of our main theorem.

###### Proof of Theorem 1.7.

Since $\textnormal{Im}(\Phi(x,\xi))>0,\,\,|\xi|\neq 0,$ the compactness of the
closed annulus $\overline{\\{\theta:|\theta|\sim 1\\}},$ and of
$\overline{V},$ imply that there exists $\varkappa_{0}>0$ such that for all
$(x,\theta)\in\overline{V}\times\overline{\\{\theta:|\theta|\sim 1\\}},$ one
has the lower bound

$\textnormal{Im}(\Phi(x,\theta))>\varkappa_{0}.$

In view of Theorem 3.1 we end the proof. ∎

#### Notation

We consider the dimension $n\geq 2$ and the parameter $0<\varepsilon\ll 1.$ We
will write $2^{C\varepsilon k},$ and the constant $C>0$ will be independent of
$\varepsilon,$ and probably it is not the same in different parts of the
manuscript. Always we will keep in mind that the constant $C\varepsilon$ can
be chosen small enough by allowing $\varepsilon$ to be arbitrarily small.

Since $n\geq 2,$ we can define the projective system of co-ordinates
$(\lambda,\omega),$ via

$\lambda:=\xi_{n},\,\omega:=\overline{\xi}/\xi_{n},\,\textnormal{ where
}\,\xi=(\overline{\xi},\xi_{n}),\,\xi_{n}\in\mathbb{R},\,\overline{\xi}\in\mathbb{R}^{n-1}.$
(3.1)

In the same way we will denote

$\lambda^{\prime}=\xi_{n}^{\prime},\,\omega^{\prime}=\overline{\xi^{\prime}}/\xi^{\prime}_{n},\,\,\xi^{\prime}=(\overline{\xi}^{\prime},\xi_{n}^{\prime}),\,\xi^{\prime}_{n}\in\mathbb{R},\,\overline{\xi^{\prime}}\in\mathbb{R}^{n-1},$
(3.2)

when $\xi\neq\xi^{\prime}.$

We will use the standard notation for Littlewood-Paley decompositions. So, we
fix a non-negative radial test function $\phi(\xi)=\phi_{0}(\xi),$ supported
on the ball $B(0,2)=\\{\xi:|\xi|\leq 2\\},$ and for any $k\in\mathbb{R},$ we
define the functions

$\phi_{k}(\xi):=\phi(\xi/2^{k}),\,\,\eta_{k}(\xi)=\phi_{k}(\xi)-\phi_{k-1}(\xi).$
(3.3)

Even, when these functions are defined on $\mathbb{R}^{d},$ $d=1,n-1$ we keep
the same notation.

We also will denote by $[x,y]:=\\{(1-t)x+ty:0\leq t\leq 1\\}$ the segment
joining two arbitrary points $x,y\in\mathbb{R}^{n}.$

##  4\. Proof of Theorem 3.1

The aim of this section is to prove the reduced version of our main Theorem
1.7, namely, the one in Theorem 3.1 for phase functions of strictly positive
type. To do this we will combine the following two approaches:

  * •

The first one, is a strategy traced back to the works of Melin and Sjöstrand
[29, 30], where any Fourier integral operator $T\in I^{\mu}(X,Y;C)$ can be
written in the form

$Tf(x)=\smallint\limits_{\mathbb{R}^{n}}e^{i(\textnormal{Re}(\Phi(x,\xi))+\tau\textnormal{Im}(\Phi(x,\xi))}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\widehat{f}(\xi)d\xi,$
(4.1)

with the real parameter $\tau\neq 0$ as in the local graph condition (1.8),
the real phase function $\Phi_{\tau}$ is given by

$\Phi_{\tau}(x,\xi):=\textnormal{Re}(\Phi(x,\xi))+\tau\textnormal{Im}(\Phi(x,\xi)),$
(4.2)

and the symbol

$a_{\tau}(x,\xi)=a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}$

belongs to the class $S^{-(n-1)/2}_{1/2,1/2}.$ This formula for the operator
has shown to be effective when dealing with its $H^{1}$-$L^{1}$-boundedness,
see [38].

  * •

The second one, introduced by Tao [45] where for the proof of the weak (1,1)
inequality one decomposes the operator $T$ into its degenerate and non-
degenerate components, $T_{\textnormal{deg}}$ and $T_{\textnormal{non-deg}}.$
Indeed, this is done using the curvature notion, which measures the extent to
which the phase function fails to be linear. The portion where the curvature
is small on each dyadic component of the operator defines the degenerate
component $T_{\textnormal{deg}}$ of $T$, and its ‘complement’
$T-T_{\textnormal{deg}},$ defines the non-degenerate component
$T_{\textnormal{non-deg}}.$ Among other things, the following are the key
points in the factorisation approach in [45]: (i) to prove that
$T_{\textnormal{deg}}$ is bounded on $L^{1}$; (ii) to construct a pseudo-
differential operator $S$ of order zero, and then of weak (1,1) type,
according to the Calderón-Zygmund theory and an average operator $A,$ which is
bounded on $L^{1},$ and essentially has the same phase function that
$T_{\textnormal{non-deg}},$ in such a way that the error operator
$E=T_{\textnormal{nondeg}}-SA$ is bounded on $L^{1}.$

The aim of the next subsections is to show that the two previous approaches
are compatible, allowing to factorise the operator $T$ with a complex phase
satisfying the conditions in Theorem 3.1, into more manageable operators.

###  4.1. Degenerate and non-degenerate components

Let us use the notation $\phi_{k}$ in (3.3) for the Littlewood-Paley partition
of unity. Typically, $k$ runs over the set of integers, but also we are going
to consider the functions $\phi_{-\varepsilon k}$ where $k$ runs over the set
of integers and $\varepsilon>0$ is fixed, and then $\phi_{-\varepsilon
k}(\xi):=\phi(2^{\varepsilon k}\xi).$

In terms of the projective co-ordinates $(\lambda,\omega)$ in (3.1), in view
of the homogeneity of the phase function $\Phi_{\tau}$ as defined in (4.2), we
can write $\Phi_{\tau}(x,\xi)=\lambda\Phi_{\tau}(x,(\omega,1)),$ when
$\xi_{n}=\lambda\neq 0.$ In such a case we are going to simplify the notation
by writing $\Phi_{\tau}(x,\omega):=\Phi_{\tau}(x,(\omega,1)).$

Using the Littlewood-Paley decomposition for the Fourier integral operator
$T,$ we have

$Tf(x)=\sum_{k\gg 1}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi(x,\xi)}a(x,\xi)\eta_{k}(\xi)\widehat{f}(\xi)d\xi,$

where the inequality $k\gg 1,$ with $k\in\mathbb{N},$ is justified because the
symbol $a(x,\xi)$ is supported on $|\xi|\gg 1.$ Now, we decompose
$T=T_{\textnormal{deg}}+T_{\textnormal{nondeg}}$ with

$T_{\textnormal{deg}}f(x)=\sum_{k\gg
1}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\phi_{-\varepsilon
k}(J_{\tau}(x,\omega))\eta_{k}(\xi)\widehat{f}(\xi)d\xi,$

where, as above,

$\Phi_{\tau}(x,\xi):=\textnormal{Re}(\Phi(x,\xi))+\tau\textnormal{Im}(\Phi(x,\xi)),$
(4.3)

and

$J_{\tau}(x,\omega)=\textnormal{det}(\nabla_{\omega}^{2}(\Phi_{\tau}(x,\omega))),\,\Phi_{\tau}(x,\omega):=\Phi_{\tau}(x,(\omega,1)),$
(4.4)

where $\Phi_{\tau}(x,\xi)=\lambda\Phi_{\tau}(x,(\omega,1)).$ Here,
$J_{\tau}(x,\omega)$ is the “curvature” associated to the real-valued phase
function $\Phi_{\tau}(x,\omega).$ Since $\tau\neq 0,$ sometimes we also write

$J(x,\omega)=J_{\tau}(x,\omega)$

for simplicity. Note then that $T_{\textnormal{nondeg}}$ is given by

$T_{\textnormal{nondeg}}f(x)=\sum_{k\gg
1}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}(1-\phi_{-\varepsilon
k}(J_{\tau}(x,\omega)))\eta_{k}(\xi)\widehat{f}(\xi)d\xi.$ (4.5)

####  4.1.1. Symbol classes in projective co-ordinates

The aim of this subsection is to analyse the behaviour of symbols in the
frequency variable with respect to the projective co-ordinates
$(\lambda,\omega),$ defined via

$\lambda=\xi_{n},\,\omega=\overline{\xi}/\xi_{n},\,\,\xi:=(\overline{\xi},\xi_{n}),\,\xi_{n}\in\mathbb{R},\,\overline{\xi}\in\mathbb{R}^{n-1}.$
(4.6)

Although $\lambda$ and $\xi_{n}$ are equal in value, the radial derivative
$\partial_{\lambda}$ keeps $\omega$ fixed, and the vertical derivative
$\partial_{\xi_{n}}$ keeps $\overline{\xi}$ fixed. In the next lemma we
analyse the derivatives of a symbol in terms of the radial and of the angular
variables. We note that $\omega\in\mathbb{R}^{n-1},$ and we write
$\omega^{J}=\omega_{1}^{J_{1}}\cdots\omega_{n-1}^{J_{n-1}},$ when
$J\in\mathbb{N}_{0}^{n-1},$ and also
$\omega^{T}=(\omega_{1},\cdots,\omega_{n-1})$ for a row vector.

######  Lemma 4.1.

Let us assume that the domain of $a$ contains an open neighborhood of the
region $\Omega_{M}=\\{(x,\xi)\in T^{*}\mathbb{R}^{n}:|\xi_{n}|>M\\},$ where
$M>0$ is fixed. Assuming $a:=a(x,\xi)$ smooth in $\Omega_{M},$ we have that

$\partial^{\ell}_{\lambda}a=\sum_{|J|\leq\ell}C_{J}\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a\cdot\omega^{J},$
(4.7)

for some family of coefficients $C_{J}$. Also,

$\partial_{\omega}^{\beta}a=\partial_{\overline{\xi}}^{\beta}a\cdot\lambda^{|\beta|},$
(4.8)

on $\Omega_{M}.$

###### Proof.

Let us prove (4.7). When $\ell=1,$ the chain rule gets

$\displaystyle\partial_{\lambda}a=\sum_{j=1}^{n-1}\partial_{\xi_{j}}a\times\partial_{\xi_{j}}/\partial{\lambda}+\partial_{\xi_{n}}a\times\partial{\xi_{n}}/\partial\lambda=\sum_{j=1}^{n-1}\omega_{j}\partial_{\xi_{j}}a+\partial_{\xi_{n}}a=\omega^{T}\cdot\nabla_{\overline{\xi}}a+\partial_{\xi_{n}}a.$
(4.9)

When $\ell=2,$ observe that

$\displaystyle\partial_{\lambda}^{2}a$
$\displaystyle=\partial_{\lambda}(\omega^{T}\cdot\nabla_{\overline{\xi}}a+\partial_{\xi_{n}}a)=\omega^{T}\cdot\partial_{\lambda}\nabla_{\overline{\xi}}a+\omega^{T}\cdot\nabla_{\overline{\xi}}(\partial_{\xi_{n}}a)+\partial_{\xi_{n}}^{2}a$
$\displaystyle=\omega^{T}\nabla^{2}_{\overline{\xi}}a\cdot\partial{\overline{\xi}}/\partial\lambda+\omega^{T}\cdot\nabla_{\overline{\xi}}(\partial_{\xi_{n}}a)+\partial_{\xi_{n}}^{2}a$
$\displaystyle=\omega^{T}\cdot[\nabla^{2}_{\overline{\xi}}a\cdot\omega]+\omega^{T}\cdot\nabla_{\overline{\xi}}(\partial_{\xi_{n}}a)+\partial_{\xi_{n}}^{2}a.$

Mathematical induction on $\ell\in\mathbb{N},$ implies that
$\partial^{\ell}_{\lambda}a$ can be written in terms of some coefficients
$C_{J},$ $J\in\mathbb{N}_{0}^{k}$ as

$\partial^{\ell}_{\lambda}a=\sum_{|J|\leq\ell}C_{J}\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a\cdot\omega^{J}.$
(4.10)

For the proof of (4.10) observe that assuming this identity valid for
$\ell>2,$ we have

$\displaystyle\partial^{\ell+1}_{\lambda}a=\partial_{\lambda}\partial^{\ell}_{\lambda}a=\sum_{|J|\leq\ell}C_{J}\partial_{\lambda}(\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a)\cdot\omega^{J}.$

With $b_{J}=\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a,$ we
apply (4.9) to deduce that

$\displaystyle\sum_{|J|\leq\ell}C_{J}\partial_{\lambda}(\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a)\cdot\omega^{J}$
$\displaystyle=\sum_{|J|\leq\ell}C_{J}\partial_{\lambda}(b_{J})\cdot\omega^{J}=\sum_{|J|\leq\ell}\left(\sum_{j=1}^{n-1}C_{J}\partial_{\xi_{j}}b_{J}\omega_{j}\omega^{J}+\partial_{\xi_{n}}b_{J}\right)$
$\displaystyle=\sum_{|J|\leq\ell}\left(\sum_{j=1}^{n-1}C_{J}\partial_{\xi_{j}}\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a\omega_{j}\omega^{J}+\partial_{\xi_{n}}\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a\right)$
$\displaystyle=\sum_{|J|\leq\ell}\left(\sum_{j=1}^{n-1}C_{J}\partial_{\overline{\xi}}^{J+e_{j}}\partial_{\xi_{n}}^{\ell+1-|J+e_{j}|}a\omega^{J+e_{j}}+\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell+1-|J|}a\right)$
$\displaystyle=\sum_{|\tilde{J}|\leq\ell+1}C_{\tilde{J}}\partial_{\overline{\xi}}^{\tilde{J}}\partial_{\xi_{n}}^{\ell+1-|\tilde{J|}}a\cdot\omega^{\tilde{J}},$

for some family of coefficients $C_{\tilde{J}}.$ The proof of (4.7) is
complete. For the proof of (4.8) note that for any $j=1,2,\cdots,n-1,$

$\displaystyle\partial_{\omega_{j}}a=\partial_{\xi_{j}}a\cdot\partial\xi_{j}/\partial\omega_{j}=\partial_{\xi_{j}}a\cdot\lambda,$

which verifies (4.8) when $|\beta|=1.$ Assuming (4.8) with $\beta$ fixed,
namely, that
$\partial_{\omega}^{\beta}=\partial_{\overline{\xi}}^{\beta}a\cdot\lambda^{|\beta|},$
note that for any $1\leq j\leq n-1,$

$\partial_{\omega}^{\beta+e_{j}}a=\partial_{\omega_{j}}\partial_{\omega}^{\beta}a=\partial_{\xi_{j}}(\partial_{\omega}^{\beta}a)\cdot\lambda=\partial_{\xi_{j}}\partial_{\overline{\xi}}^{\beta}a\cdot\lambda^{|\beta|}\cdot\lambda=\partial_{\overline{\xi}}^{\beta+e_{j}}a\cdot\lambda^{|\beta+e_{j}|},$

from where the mathematical induction proves $\eqref{lemma:2}$ for all
$\beta\in\mathbb{N}_{0}^{n}.$ ∎

For $1/2\leq\rho\leq 1,$ let $S^{m}_{\rho,\mathscr{C}}$ be the class of
symbols that consists of all functions $a:=a(x,\xi)$ which are Borel
measurable in $x\in\mathbb{R}^{n},$ supported in the cone bundle

$\mathscr{C}=\\{(x,\xi)\in T^{*}\mathbb{R}^{n}:\xi_{n}\gg
1,\,|\overline{\xi}|\leq\lambda\\},$ (4.11)

and satisfying the Hörmander type estimates

$|\partial_{\xi}^{\alpha}a(x,\xi)|\leq
C_{\alpha}|\xi|^{m-\rho|\alpha|},\,\,(x,\xi)\in\textnormal{supp}(a).$ (4.12)

######  Lemma 4.2.

Let $k\gg 1,$ $C>0$ and let $\varepsilon>0.$ Let $a\in
S^{m}_{\rho,\mathscr{C}}.$ Consider the symbol

$a_{k}(x,\xi)=a(x,\xi)\phi_{-\varepsilon k}(J_{\tau}(x,\omega))\eta_{k}(\xi),$
(4.13)

supported in the region

$\mathscr{C}_{\lambda}=\\{(x,\xi)\in\textnormal{supp}(a):\lambda\sim
2^{k};\,|J_{\tau}(x,\xi)|\lesssim 2^{-\varepsilon k}\\}.$

Then, we have the following symbol estimates

$|\partial_{\omega}^{\beta}\partial_{\lambda}^{\gamma}a_{k}(x,\xi)|\lesssim_{\beta,\gamma,C}2^{mk}2^{-\rho|\gamma|k+(1-\rho)|\beta|k}2^{C\varepsilon
k|\beta|},$ (4.14)

for $(x,\xi)$ in the support of $a:=a(x,\xi).$

###### Proof.

Using that $a$ satisfies the estimates in (4.12) and the identity (4.7) we
have that

$\displaystyle\forall\ell\geq
1,\,\,|\partial_{\lambda}^{\ell}a|\lesssim\sup_{|J|\leq\ell}|\partial_{\overline{\xi}}^{J}\partial_{\xi_{n}}^{\ell-|J|}a|\lesssim|\xi|^{m-\rho\ell}.$

Observing that, on the region $\mathscr{C}_{\lambda},$ $|\xi|\sim|\lambda|\sim
2^{k}$ we have proved that

$|\partial_{\lambda}^{\ell}a|\lesssim 2^{k(m-\rho\ell)}.$ (4.15)

By the Leibniz rule we have that

$\displaystyle\partial_{\omega}^{\beta}a_{k}=\sum_{\beta_{1}+\beta_{2}+\beta_{3}=\beta}C_{\beta_{1},\beta_{2},\beta_{2}}\partial_{\omega}^{\beta_{1}}a\times\partial_{\omega}^{\beta_{2}}(\phi_{-\varepsilon
k}(J_{\tau}(x,\omega)))\times\partial_{\omega}^{\beta_{3}}\eta_{k}.$ (4.16)

In view of (4.8), we can estimate

$|\partial_{\omega}^{\beta_{1}}a|=|\partial_{\overline{\xi}}^{\beta_{1}}a||\lambda|^{|\beta_{1}|}\lesssim|\xi|^{m-\rho|\beta_{1}|}|\lambda|^{|\beta_{1}|}\sim|\lambda|^{m-\rho|\beta_{1}|}|\lambda|^{|\beta_{1}|}\sim
2^{km}2^{k(1-\rho)|\beta_{1}|}$ (4.17)

Since the sequence $\eta_{k}$ belongs uniformly in $k$ to $S^{0},$ we have
that

$|\partial_{\omega}^{\beta_{3}}\eta_{k}|=|\partial_{\overline{\xi}}^{\beta_{3}}\eta_{k}||\lambda|^{|\beta_{3}|}\lesssim|\xi|^{-|\beta_{3}|}|\lambda|^{|\beta_{3}|}\sim
1.$ (4.18)

Now, let us analyse the behaviour of the cutt-off $\psi_{-\varepsilon k}$ at
the “curvature” term $J_{\tau}(x,\omega).$ Observe that for any
$j=1,\cdots,n-1,$

$\displaystyle\partial_{\omega_{j}}(\phi_{-\varepsilon
k}(J_{\tau}(x,\omega)))$ $\displaystyle=\phi_{-\varepsilon
k}^{\prime}(J_{\tau}(x,\omega)))\times\partial_{\omega_{j}}J_{\tau}(x,\omega).$

Then, we have the equalities

$\displaystyle|\partial_{\omega_{j}}(\phi_{-\varepsilon
k}(J_{\tau}(x,\omega)))|$ $\displaystyle=|\phi^{\prime}_{-\varepsilon
k}(t)|_{t=J_{\tau}(x,\omega)}\times\partial_{\omega_{j}}J_{\tau}(x,\omega)|$
$\displaystyle=|\partial_{t}[\phi(t/2^{-\varepsilon
k})]|_{t=J(x,\omega)}\times\partial_{\omega_{j}}J(x,\omega)|$
$\displaystyle=|(\partial_{t}\phi)(t/2^{-\varepsilon
k})2^{k\varepsilon}|_{t=J_{\tau}(x,\omega)}\times\partial_{\omega_{j}}J_{\tau}(x,\omega)|.$

There is $C_{\phi}>0$ such that

$|\partial_{t}\phi(\lambda)|\leq C_{\phi}.$

Consequently,

$\displaystyle|(\partial_{t}\phi)(t/2^{-\varepsilon
k})|_{t=J_{\tau}(x,\omega)}2^{k\varepsilon}\lesssim 2^{k\varepsilon}.$

Thus, the mathematical induction gives the estimate

$|\partial_{\omega}^{\beta_{2}}(\phi_{-\varepsilon
k}(J_{\tau}(x,\omega)))|\lesssim 2^{k\varepsilon|\beta_{2}|}.$ (4.19)

We have proved the estimates

$|\partial_{\lambda}^{\gamma}a|\lesssim
2^{km}2^{-k\gamma},\,|\partial_{\omega}^{\beta_{1}}a|\lesssim
2^{km}2^{k(1-\rho)|\beta_{1}|},$

and

$|\partial_{\omega}^{\beta_{3}}\eta_{k}|\lesssim
1,\,|\partial_{\omega}^{\beta_{2}}(\phi_{-\varepsilon
k}(J_{\tau}(x,\omega)))|\lesssim 2^{k\varepsilon|\beta|}.$ (4.20)

Note that when $\gamma=\beta_{1}=0,$ we recover the information that $a$ has
order $m.$ Indeed, let $a_{\beta_{1}}:=\partial_{\omega}^{\beta_{1}}a.$ Since
$\lambda\sim 2^{k},$ note that the order of $a_{\beta_{1}}$ with respect to
$\lambda$ is $m_{\beta_{1}}:=m+(1-\rho)|\beta_{1}|.$ If we replace the
analysis above with the symbol $a_{\beta_{1}}$ instead of $a,$ then
$m_{\beta_{1}}$ takes the place of $m,$ and we get the estimate

$|\partial_{\lambda}^{\gamma}a_{\beta_{1}}|\lesssim
2^{km_{\beta_{1}}}2^{-k\gamma}=2^{km+k(1-\rho)|\beta_{1}|-k\gamma}.$

Therefore we have proved that

$|\partial_{\lambda}^{\gamma}\partial_{\omega}^{\beta_{1}}a|\lesssim
2^{km_{\beta_{1}}}2^{-k\gamma}=2^{km+k(1-\rho)|\beta_{1}|-k\gamma}.$

The estimates for the derivatives of $a$ in $\lambda$ and in $\omega$ above,
the inequality in (4.20) and the Leibniz rule (4.16) conclude the proof of
(4.14). ∎

###  4.2. Boundedness of the degenerate component

In this section we prove that the operator

$T_{\textnormal{deg}}f(x)=\sum_{k\gg
1}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\phi_{-\varepsilon
k}(J_{\tau}(x,\omega))\eta_{k}(\xi)\widehat{f}(\xi)d\xi,$ (4.21)

is bounded on $L^{1}(\mathbb{R}^{n}).$ Before continuing with the proof of the
$L^{1}$-boundedness of $T_{\textnormal{deg}},$ we discuss the second dyadic
partition that will be employed in our further analysis.

######  Remark 4.3 (About discrete vs continuous ‘second’ dyadic
decompositions).

In [43], a discrete ‘second dyadic decomposition’ was introduced where the
variable $\omega$ is decomposed in a finite family of disks
$\\{D\\}_{D\in\mathcal{I}},$ see Figure 1. At first glance, for the analysis
of the degenerate component $T_{\textnormal{deg}},$ one could apply this
discrete ‘second dyadic partition’ on the symbol:

$a_{k}(x,\xi)=a(x,\xi)\phi_{-\varepsilon k}(J_{\tau}(x,\omega))\eta_{k}(\xi),$

generating the new family of symbols $a_{k,D},$ in such a way that
$a_{k}=\sum_{D}a_{k,D}.$ However, as it was explained in [45], this discrete
decomposition provides the estimate

$\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a_{k}(x,\xi)e^{-2\pi
iy\cdot\xi}d\xi\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim 1,$ (4.22)

for the $L^{1}$-norm of the kernel associated with each dyadic part $a_{k}.$
This estimate is not enough to deduce the $L^{1}$ boundeness of
$T_{\textnormal{deg}}.$ The reason is that one needs a more accurate decay of
this $L^{1}$-norm. Indeed, this is proved in Lemma 4.6 improving the $O(1)$
term in (4.22), by the smaller one $2^{-\varepsilon k}.$ In order to make this
improvement, we will use the degeneracy condition $|J(x,\omega)|\lesssim
2^{-\varepsilon k}.$ To have the control of lower terms in the Taylor
expansion of the phase $\Phi_{\tau}-y\cdot\xi$ at $\omega_{0}=\omega_{D},$
especially if $\omega-\omega_{D}$ is in the direction where
$\nabla_{\omega}^{2}\Phi_{\tau}$ degenerates, the disks $D$ will be replaced
by a suitable family of ellipsoids, that allows us to decompose the
$(x,\xi)$-variable in a family of tubular regions where we gain the decay
$2^{-\varepsilon k}.$

Figure 1.  Here we illustrate the partition of the $\omega$ variable smoothly
into about $2^{(n-1)k/2}$ disks $D$ is radius $2^{-k/2}$ when
$|\omega|\lesssim 1.$ First, in the system of coordinates
$(\overline{\xi},\xi_{n}),$ the projective coordinates $(\lambda,\omega)$
determine each point $(\overline{\xi},\xi_{n})\in\mathscr{C}.$ The unit ball
$B_{n-1}=\\{|\omega|\leq 1\\}$ on $\mathbb{R}^{n-1}$ is saturated to the
‘segment’ $\overline{AB}.$ Each partition of the ball $B_{n-1}$ in a family of
disks $\\{D\\}_{D\in\mathcal{I}}$ is saturated to a partition of the segment
$\overline{AB}.$ In particular if the radii of the disks are proportional to
$r=2^{-k/2},$ one can estimate $|\mathcal{I}|\sim 2^{(n-1)k/2}.$

As it was pointed out in [45], there is an apparent difficulty in the case
when $n\geq 3,$ where the ellipsoid around $\omega_{0}=\omega_{D},$ depends on
the spectrum of $\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega_{D}).$ Then, the
eccentricity and the orientation of this ellipsoid vary in function of
$\omega_{D}.$ However, by adapting to the setting of the complex phases, the
approach in [45], we will define a continuous ‘second dyadic partition’ in
$\omega,$ using a family of functions $\psi_{x,\omega_{D}}$ with a family of
corresponding supports determined by a family of ellipsoids with a smooth
variation of eccentricity and orientation, centred at the points $\omega_{D}$
that avoid Kakeya-type covering lemmas. According to this partition the part
of the support of each symbol $a_{k,D}$ on the hyperplane $\xi_{n}=\lambda$ is
illustrated in Figure 2.

Figure 2. Note that the dyadic partition of the variable $\xi_{n}=\lambda$
together with the family of ‘ellipsoids’ $\\{D\\}_{D\in\mathcal{I}}$
decomposing the variable $\omega,$ provide a partition of the cone bundle
$\mathscr{C}.$

The following is the main result of this subsection.

######  Proposition 4.4.

Let $S^{-(n-1)/2}_{\rho,\mathscr{C}}$ be the symbol of the Fourier integral
operator $T$ of order $m=-(n-1)/2$ and let $\Phi$ be its complex phase
function. Then, the degenerate component $T_{\textnormal{deg}}$ in (4.21)
extends to a bounded operator from $L^{1}(\mathbb{R}^{n})$ into
$L^{1}(\mathbb{R}^{n}).$

######  Remark 4.5.

For the proof of Proposition 4.4, it is suffices to prove the estimate

$\exists C>0,\forall f\in
C^{\infty}_{0}(\mathbb{R}^{n}),\,\|T_{\textnormal{deg}}f\|_{L^{1}(\mathbb{R}^{n})}\leq
C\|f\|_{L^{1}(\mathbb{R}^{n})}.$ (4.23)

Note that the kernel of $T_{\textnormal{deg}}$ is given by

$K_{\textnormal{deg}}(x,y):=\sum_{k\gg
1}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\phi_{-\varepsilon
k}(J_{\tau}(x,\omega))\eta_{k}(\xi)e^{-2\pi iy\cdot\xi}d\xi.$

By the triangle inequality, for the proof of (4.23), it suffices to prove that

$\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\phi_{-\varepsilon
k}(J_{\tau}(x,\omega))\eta_{k}(\xi)e^{-2\pi
iy\cdot\xi}d\xi\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim 2^{-\varepsilon
k},$ (4.24)

in view of Minkowski integral inequality. Indeed, from (4.24) one deduces that

$C:=\sup_{y\in\mathbb{R}^{n}}\|K_{\textnormal{deg}}(x,y)\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim\sum_{k\gg
1}2^{-\varepsilon k}<\infty.$ (4.25)

Consequently,

$\|T_{\textnormal{deg}}f\|_{L^{1}(\mathbb{R}^{n})}=\|\smallint
K_{\textnormal{deg}}(x,y)f(y)dy\|_{L^{1}(\mathbb{R}^{n}_{x})}=\smallint\left|\smallint
K_{\textnormal{deg}}(x,y)f(y)dy\right|dx$
$\leq\smallint\smallint|K_{\textnormal{deg}}(x,y)|dx|f(y)|dy\leq\sup_{y\in\mathbb{R}^{n}}\smallint|K_{\textnormal{deg}}(x,y)|dx\smallint|f(y)|dy$
$=\sup_{y\in\mathbb{R}^{n}}\smallint|K_{\textnormal{deg}}(x,y)|dx\|f\|_{L^{1}(\mathbb{R}^{n})},$

from where one deduces (4.23) with $C>0$ defined in (4.25).

For the proof of Proposition 4.4 we require some preliminary results. We start
with the following estimates for radial and angular derivatives of symbols in
the class $S^{-(n-1)/2}_{\rho,\mathscr{C}}$ and with an estimate of the
$L^{1}$-norm of the kernel of $T_{\textnormal{deg}}.$ This is our first step
in the proof of (4.24).

######  Lemma 4.6.

Let $k\gg 1,$ $\frac{1}{2}\leq\rho\leq 1,$ $C>0,$ and let $\varepsilon>0.$ Let
$a\in S^{-(n-1)/2}_{\rho,\mathscr{C}}.$ Consider the symbol

$a_{k}(x,\xi)=a(x,\xi)\phi_{-\varepsilon k}(J_{\tau}(x,\omega))\eta_{k}(\xi).$
(4.26)

Then, we have the following symbol estimates

$|\partial_{\omega}^{\beta}\partial_{\lambda}^{\gamma}a_{k}(x,\xi)|\lesssim_{\beta,\gamma}2^{-\frac{(n-1)k}{2}}2^{-\rho|\gamma|k+(1-\rho)|\beta|k}2^{C\varepsilon
k|\beta|},$ (4.27)

for $(x,\xi)$ in the support of $a.$ Moreover, any function
$a_{k}:=a_{k}(x,\xi)\in S^{-(n-1)/2}_{\rho,\mathscr{C}}$ supported in the
region

$\\{(x,\xi)\in\textnormal{supp}(a):\lambda\sim
2^{k};\,|J_{\tau}(x,\xi)|\lesssim 2^{-\varepsilon k}\\}$ (4.28)

satisfying the symbol estimates in (4.27) defines a Fourier integral operator
whose kernel satisfies the inequality

$\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a_{k}(x,\xi)e^{-2\pi
iy\cdot\xi}d\xi\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim_{\rho}2^{-\varepsilon
k},$ (4.29)

uniformly in $y\in\mathbb{R}^{n}.$

###### Proof.

The estimate in (4.27) for $a_{k}$ as defined in (4.26) is a consequence of
Lemma 4.2 applied to the order $m=-(n-1)/2.$

Now, let us consider a symbol $a_{k}:=a_{k}(x,\xi)$ satisfying the estimates
in (4.27) whose support is contained in (4.28). For the proof of (4.29) we
need a preliminary construction. Indeed, let us define a positive-definite
replacement $Q(x,\omega)$ of $\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega).$ Note
that $\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega)$ is the associated matrix to
the quadratic form

$\zeta\mapsto\zeta^{T}\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega)\zeta=:\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega)(\zeta,\zeta).$

Let

$\lambda_{1}(x,\omega),\cdots,\lambda_{n-1}(x,\omega)$

be the family of eigenvalues of $\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega).$ In
an appropriate bases $B$ of $\mathbb{R}^{n-1}$ we can write

$\displaystyle\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega)(\zeta,\zeta)=\sum_{j=1}^{n-1}\lambda_{j}(x,\omega)|\zeta_{j}|^{2},\,\,\zeta\in\mathbb{R}^{n-1}.$

Define

$Q(x,\omega):=(2^{-2\varepsilon
k}I+(\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega))^{2})^{\frac{1}{2}},$ (4.30)

where $I=I_{n-1}$ is the identity matrix of size $n-1.$ In the same basis $B,$
we can write

$\displaystyle Q(x,\omega)=\textnormal{diag}[2^{-2\varepsilon
k}+\lambda_{j}(x,\omega)^{2}]_{1\leq j\leq n-1}^{\frac{1}{2}}.$

Observe that

$\det(Q(x,\omega))=\prod_{j=1}^{n-1}[2^{-2\varepsilon
k}+\lambda_{j}(x,\omega)^{2}]^{\frac{1}{2}}\geq 2^{-Ck\varepsilon},\,C=n-1.$
(4.31)

On the other hand

$\displaystyle\det(Q(x,\omega))$
$\displaystyle=\prod_{j=1}^{n-1}[2^{-2\varepsilon
k}+\lambda_{j}(x,\omega)^{2}]^{\frac{1}{2}}\leq\prod_{j=1}^{n-1}[2^{-\varepsilon
k}+|\lambda_{j}(x,\omega)|]$ $\displaystyle=2^{-(n-1)\varepsilon
k}+\prod_{j=1}^{n-1}|\lambda_{j}(x,\omega)|+O(2^{-\varepsilon k})$
$\displaystyle\lesssim 2^{-\varepsilon k}+|J_{\tau}(x,\omega)|.$

The matrix $Q(x,\omega)$ has determinant satisfying the estimates

$\displaystyle 2^{-Ck\varepsilon}\leq\det(Q(x,\omega))\lesssim 2^{-\varepsilon
k}+|J_{\tau}(x,\omega)|.$ (4.32)

For any $(x,\omega_{D})\in\mathbb{R}^{n}\times\mathbb{R}^{n-1},$ with the
property that when $\lambda\sim 2^{k},$

$(x,\lambda(\omega_{D},1))=(x,\overline{\xi}_{D},\lambda)\in\mathscr{C},\,\,\overline{\xi}_{D}:=\omega_{D}\lambda,$

define the function $\psi_{x,\omega_{D}}$ by

$\displaystyle\psi_{x,\omega_{D}}(\omega):=\det(Q(x,\omega_{D}))^{\frac{1}{2}}{2^{\frac{(n-1)k}{2}}\psi_{-k}(Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D}))}.$
(4.33)

Since on $\mathscr{C},$ $|\overline{\xi}_{D}|\lesssim\lambda$ we necessarily
have that $|\omega_{D}|\lesssim 1$ and so $\omega_{D}$ is contained in a
compact subset $\Omega$ of $\mathbb{R}^{n-1}.$ Note that

$\displaystyle
Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D})\leq\sup_{x\in\pi_{1}[\textnormal{supp}(a)],\,\omega^{\prime}\in\Omega}\|Q(x,\omega^{\prime})\|_{\textnormal{op}}\|\omega-\omega_{D}\|^{2}\lesssim_{\textnormal{supp}(a),\Omega}\|\omega-\omega_{D}\|^{2}.$

In consequence, if

$\|\omega-\omega_{D}\|\lesssim 2^{-(1-\frac{\varepsilon}{2})k},$

then

$Q(x,\omega)(\omega-\omega_{D},\omega-\omega_{D})\leq
2^{-(2-\varepsilon)k}\lesssim 2^{-k},$

and consequently $\textnormal{supp}(\psi_{x,\omega_{D}})$ contains a small
disk $\\{\omega=\omega_{D}+z:|z|\lesssim 2^{-(1-\frac{\varepsilon}{2})k}\\},$
namely, we have the inclusion

$\displaystyle\\{\omega=\omega_{D}+z:|z|\lesssim
2^{-(1-\frac{\varepsilon}{2})k}\\}\subset\textnormal{supp}(\psi_{x,\omega_{D}}).$
(4.34)

By writing $Q(x,\omega_{D})$ in the basis $B,$ we have

$\displaystyle
Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D})=\sum_{j=1}^{n-1}(2^{-2\varepsilon
k}+\lambda_{j}(x,\omega_{D})^{2})^{\frac{1}{2}}(\omega_{j}-\omega_{D,j})^{2}$

from where one can deduce that $\psi_{x,\omega_{D}}(\omega)$ is supported in
an ellipsoid centred at $\omega=\omega_{D}.$ Observe that, since

$\displaystyle
Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D})\geq\sum_{j=1}^{n-1}2^{-\varepsilon
k}(\omega_{j}-\omega_{D,j})^{2},$ (4.35)

the support of $\psi_{x,\omega_{D}}$ contains the disk
$\\{\omega=\omega_{D}+O(2^{-k/2})\\}.$ Indeed, on the support of
$\psi_{x,\omega_{D}},$
$Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D})\in\textnormal{supp}(\psi_{-k})$
which says that

$Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D})\lesssim 2^{-k}.$

In view of (4.35) the inequality $\sum_{j=1}^{n-1}2^{-\varepsilon
k}(\omega_{j}-\omega_{D,j})^{2}\lesssim 2^{-k}$ implies that

$\sum_{j=1}^{n-1}(\omega_{j}-\omega_{D,j})^{2}\lesssim 2^{-k}2^{\varepsilon
k}=2^{-(1-\varepsilon)k}.$

In consequence,

$\displaystyle\textnormal{supp}(\psi_{x,\omega_{D}})\subset\\{\omega=\omega_{D}+z:|z|\lesssim
2^{-(1-\varepsilon)k}\\}.$ (4.36)

In view of the inclusions (4.34) and (4.36) for any
$\omega\in\textnormal{supp}(\psi_{x,\omega_{D}})$ we have

$\displaystyle
2^{-(1-\frac{\varepsilon}{2})k}\lesssim|\omega-\omega_{D}|\lesssim
2^{-(1-\varepsilon)k}.$ (4.37)

Now, define the averaged function $\psi_{x}(\omega)$ by

$\psi_{x}(\omega)=\smallint_{\omega_{D}\in\mathbb{R}^{n-1}}\psi_{x,\omega_{D}}(\omega)d\omega_{D}.$
(4.38)

We review the properties of the averaged function $\psi_{x}$ in the following
lemma.

######  Lemma 4.7.

The function $\psi_{x}$ is positive, and the following estimates are valid:

  * •

$\psi_{x}(\omega)\sim 1$

  * •

$\forall\beta,\,\,|\partial^{\beta}_{\omega}\psi_{x}(\omega)|\lesssim_{\beta}2^{C\varepsilon
k|\beta|}.$

###### Proof.

In a suitable basis we can write the quadratic form $Q(x,\omega_{D})$ as
follows

$\displaystyle
Q(x,\omega)(\zeta,\zeta)=\sum_{j=1}^{n-1}\lambda_{Q,j}^{\omega}|\zeta_{j}|^{2},\,\lambda_{Q,j}^{\omega}:=(2^{-2\varepsilon
k}+\lambda_{j}(x,\omega)^{2})^{\frac{1}{2}}.$

Observe that

$\displaystyle\psi_{x}(\omega)$
$\displaystyle=\smallint_{\omega_{D}\in\mathbb{R}^{n-1}}\psi_{x,\omega_{D}}(\omega)d\omega_{D}=\smallint_{\textnormal{supp}(\psi_{x,\omega_{D}})}\psi_{x,\omega_{D}}(\omega)d\omega_{D}$
$\displaystyle\sim\smallint_{\omega=\omega_{D}+z:2^{-(1-\frac{\varepsilon}{2})k}\lesssim|z|\lesssim
2^{-(1-\varepsilon)k}}\psi_{x,\omega_{D}}(\omega)d\omega_{D}$
$\displaystyle=\smallint_{\omega=\omega_{D}+z:2^{-(1-\frac{\varepsilon}{2})k}\lesssim|z|\lesssim
2^{-(1-\varepsilon)k}}{2^{\frac{(n-1)k}{2}}\psi_{-k}(Q(x,\omega_{D})(\omega-\omega_{D},\omega-\omega_{D}))}{\det(Q(x,\omega_{D}))^{\frac{1}{2}}}d\omega_{D}$
$\displaystyle=\smallint_{z:\,2^{-(1-\frac{\varepsilon}{2})k}\lesssim|z|\lesssim
2^{-(1-\varepsilon)k}}{2^{\frac{(n-1)k}{2}}\psi_{-k}(Q(x,\omega-z)(z,z))}{\det(Q(x,\omega-z))^{\frac{1}{2}}}dz.$

By making Taylor expansion of the function
$\psi_{-k}(Q(x,\omega-z^{\prime})(z,z))|_{z^{\prime}=z}$ at $z^{\prime}=0$ we
have that

$\displaystyle\psi_{x}(\omega)$
$\displaystyle=\smallint_{z:\,2^{-(1-\frac{\varepsilon}{2})k}\lesssim|z|\lesssim
2^{-(1-\varepsilon)k}}{2^{\frac{(n-1)k}{2}}\psi_{-k}(Q(x,\omega-z)(z,z))}\det(Q(x,\omega-z))^{\frac{1}{2}}dz.$
$\displaystyle=\smallint_{z:\,2^{-(1-\frac{\varepsilon}{2})k}\lesssim|z|\lesssim
2^{-(1-\varepsilon)k}}{2^{\frac{(n-1)k}{2}}\psi_{-k}(Q(x,\omega)(z,z))}{\det(Q(x,\omega))^{\frac{1}{2}}}dz+O(2^{C\varepsilon
k}2^{-\varepsilon k})$
$\displaystyle=\smallint_{z:\,2^{-(1-\frac{\varepsilon}{2})k}\lesssim|z|\lesssim
2^{-(1-\varepsilon)k}}{2^{\frac{(n-1)k}{2}}\psi_{-k}\left(\sum_{j=1}^{n-1}\lambda_{Q,j}^{\omega}|z_{j}|^{2}\right)}{\det(Q(x,\omega_{D}))^{\frac{1}{2}}}dz+O(2^{C\varepsilon
k}2^{-\varepsilon k}).$

The change of variables $z=(\lambda_{j,Q}^{-\frac{1}{2}}\zeta_{j})_{1\leq
j\leq n-1}$ has the new volume element

$dz=\prod_{j=1}^{n-1}(\lambda_{j,Q}^{\omega})^{-\frac{1}{2}}\cdot
d\zeta=\det(Q(x,\omega))^{-\frac{1}{2}}d\zeta$ (4.39)

from where we have that

$\displaystyle\psi_{x}(\omega)$
$\displaystyle\sim\smallint{2^{\frac{(n-1)k}{2}}\psi_{-k}((\zeta,\zeta))}d\zeta=\smallint{2^{\frac{(n-1)k}{2}}\psi(|\zeta|^{2}/2^{-k})}d\zeta=\smallint\psi_{0}(|\overline{\xi}|^{2})d\overline{\xi}\sim
1.$

A similar analysis can be done for the proof of the estimate
$|\partial^{\beta}_{\omega}\psi_{x}(\omega)|\lesssim_{\beta}2^{C\varepsilon
k|\beta|}.$ On the other hand, let $|\beta|\geq 1.$ Note that when applying
derivatives $\partial_{\omega}^{\beta}$ to $\psi_{x}(\omega)$ the main term
$\smallint\psi_{0}(|\overline{\xi}|^{2})d\overline{\xi}$ vanishes and the
error term contribute in $O(2^{C\varepsilon k|\beta|}).$ ∎

In order to prove (4.29) we use the continuous partition of unity taking
averages with respect to ellipsoids centered at $\omega_{D}$ as follows (and
using Fubini’s theorem)

$\displaystyle\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i(\Phi_{\tau}(x,\xi)-y\cdot\xi)}a_{k}(x,\xi)d\xi$
$\displaystyle=\smallint\limits_{\mathbb{R}^{n}}\smallint\limits_{\Omega\subset\mathbb{R}^{n-1}}e^{2\pi
i(\Phi_{\tau}(x,\xi)-y\cdot\xi)}\frac{a_{k}(x,\xi)\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\omega_{D}d\xi$
$\displaystyle=\smallint\limits_{\Omega\subset\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i(\Phi_{\tau}(x,\xi)-y\cdot\xi)}\frac{a_{k}(x,\xi)\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\xi
d\omega_{D}.$

The variable $\omega_{D}$ runs over a compact set $\Omega,$ when
$(x,\xi)\in\mathscr{C}$ as discussed above. Since $\lambda\sim 2^{k},$ on the
support of $a_{k},$ Figure 2 illustrates the points of the support of $a_{k}$
on the hyperplane $\xi_{n}=\lambda.$

Hence, for the proof of (4.29) is enough to show that

$\displaystyle\sup_{\omega_{D}\in\Omega}\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i(\Phi_{\tau}(x,\xi)-y\cdot\xi)}\frac{a_{k}(x,\xi)\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\xi\right\|\lesssim
2^{-\varepsilon k}.$ (4.40)

Indeed, observe that (4.40) implies the estimate

$\displaystyle\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a_{k}(x,\xi)e^{-2\pi
iy\cdot\xi}\right\|_{L^{1}(\mathbb{R}^{n}_{x})}$
$\displaystyle=\left\|\smallint\limits_{\Omega}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i(\Phi_{\tau}(x,\xi)-y\cdot\xi)}\frac{a_{k}(x,\xi)\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\xi
d\omega_{D}\right\|_{L^{1}(\mathbb{R}^{n}_{x})}$
$\displaystyle\leq|\Omega|\sup_{\omega_{D}\in\Omega}\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i(\Phi_{\tau}(x,\xi)-y\cdot\xi)}\frac{a_{k}(x,\xi)\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\xi\right\|$
$\displaystyle\lesssim 2^{-\varepsilon k}.$

For the proof of (4.40) we write the frequency variable $\xi$ in terms of the
co-variables $(\lambda,\omega),$ and of the new volume element

$\displaystyle d\xi=d\overline{\xi}d\xi_{n}=\lambda^{n-1}d\omega d\lambda.$

Then, (4.40) can be written as

$\displaystyle\left\|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,\omega)-y\cdot(\omega,1))}\frac{\lambda^{n-1}a_{k}(x,\lambda(\omega,1))\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\lambda
d\omega\right\|\lesssim 2^{-\varepsilon k}.$ (4.41)

To simplify the previous estimate we will consider the following change of
variables

$\displaystyle
T(\zeta)=\omega=\omega_{D}+2^{-k/2n}Q(x,\omega_{D})^{-1/2}[(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C}\zeta_{j}],$

with a constant $C>1.$ Note that

$\displaystyle|\textnormal{det}T(\zeta)|=2^{-(n-1)k/2}\det(Q(x,\omega_{D}))^{-1/2}(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C(n-1)}.$ (4.42)

In what follows, let us denote

$\alpha:=(2^{k}+2^{(1-\rho)100k}+2^{\rho k})^{-C},$

and

$\chi_{x,\omega_{D}}(\zeta):=\psi(2^{k}Q(x,\omega_{D})^{1/2}(\alpha\zeta,\alpha\zeta))(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C(n-1)}.$

Then, in view of (4.33) and (4.42) we have that

$\displaystyle\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,\omega)-y\cdot(\omega,1))}\frac{\lambda^{n-1}a_{k}(x,\lambda(\omega,1))\psi_{x,\omega_{D}}(\omega)}{\psi_{x}(\omega)}d\lambda
d\omega$
$\displaystyle=\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}\frac{\lambda^{n-1}a_{k}(x,\lambda(T(\zeta),1))\psi_{x,\omega_{D}}(T(\zeta))}{\psi_{x}(T(\zeta))}|\det{T(\zeta)}|d\lambda
d\zeta$
$\displaystyle=\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}\frac{\lambda^{n-1}a_{k}(x,\lambda(T(\zeta),1))2^{(n-1)k/2}}{\psi_{x}(T(\zeta))}\det(Q(x,\omega_{D}))^{\frac{1}{2}}$
$\displaystyle\times\psi(2^{k}Q(x,\omega_{D})^{1/2}(\alpha\zeta,\alpha\zeta))2^{-(n-1)k/2}\det(Q(x,\omega_{D}))^{-1/2}(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C(n-1)}d\lambda d\zeta$
$\displaystyle=\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}\frac{\lambda^{n-1}a_{k}(x,\lambda(T(\zeta),1))}{\psi_{x}(T(\zeta))}$
$\displaystyle\hskip
56.9055pt\times\psi(2^{k}Q(x,\omega_{D})^{1/2}(\alpha\zeta,\alpha\zeta))(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C(n-1)}d\lambda d\zeta$
$\displaystyle=\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}\frac{\lambda^{n-1}a_{k}(x,\lambda(T(\zeta),1))\chi_{x,\omega_{D}}(\zeta)}{\psi_{x}(T(\zeta))}d\lambda
d\zeta.$

Then, by multiplying in both sides of (4.41) by the factor $2^{-(n-1)k/2}$ we
can re-write (4.41) in terms of $\omega=T(\zeta)$ as follows

$\displaystyle\left\|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim
2^{-(n-1)k/2}2^{-\varepsilon k}$ (4.43)

where

$\displaystyle
b_{k,\omega_{D},x}(\lambda,\zeta):=\frac{2^{-k(n-1)/2}\lambda^{n-1}a_{k}(x,\lambda(T(\zeta),1))\chi_{x,\omega_{D}}(T(\zeta))}{\psi_{x}(T(\zeta))}.$

So, to finish the proof we will show the estimate in (4.43). The advantage in
proving (4.43) is that the symbol $b_{k,\omega_{D},x}(\lambda,\zeta)$ allows
the use of the phase stationary theorem since it does satisfy the estimates

$\displaystyle|\partial_{\lambda}^{\beta}\partial_{\zeta}^{\delta}b_{k,\omega_{D},x}(\lambda,\zeta)|\lesssim_{\delta}2^{-k\rho|\beta|},$
(4.44)

Note that, when $|\delta|=1,$ the chain rule gives

$\displaystyle|\partial_{\zeta}^{\delta}\chi_{x,\omega_{D}}(x,\omega_{D})|$
$\displaystyle=|\partial_{\zeta}^{\delta}\left(\psi(2^{k}Q(x,\omega_{D}))^{1/2}(\alpha\zeta,\alpha\zeta)\right)|(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C(n-1)}$
$\displaystyle=|\partial_{\zeta}^{\delta}\left(\psi(2^{k}Q(x,\omega_{D})^{1/2}(\alpha\zeta,\alpha\zeta))\right)|(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C(n-1)}$
$\displaystyle\lesssim\|(\partial^{\delta}\psi)\|_{L^{\infty}}|\partial_{\zeta}^{\delta}(\psi(2^{k}Q(x,\omega_{D})^{1/2}(\alpha\zeta,\alpha\zeta))^{1/2})|$
$\displaystyle\lesssim
2^{k|\delta|}(2^{-k\varepsilon}+|J_{\tau}(x,\omega)|)^{1/2}\alpha^{|\delta|}$
$\displaystyle\lesssim 2^{k|\delta|}2^{-k\varepsilon}\alpha^{|\delta|}$
$\displaystyle\lesssim
2^{k|\delta|}2^{-k\varepsilon}(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C|\delta|}$ $\displaystyle\lesssim_{\delta}1,$

where we have used that on the support of $a_{k},$ $|J_{\tau}(x,\xi)|\lesssim
2^{-\varepsilon k},$ see (4.28). Then, using mathematical induction we can
prove the estimate

$|\partial_{\zeta}^{\delta}\chi_{x,\omega_{D}}(x,\omega_{D})|\lesssim_{C,\delta}1,$

where the previous estimate depends on $C>\delta$ but it is independent of
$\delta$ if $C>0$ is large enough and we take a finite number of derivatives
in $\zeta$. This will be the case because we are going to apply the method of
stationary phase just taking derivatives $\partial_{\zeta}^{\delta}$ in
$\zeta$ with $|\delta|=1.$ Now, using that the derivatives of
$\chi_{x,\omega_{D}}(x,\omega_{D})$ are uniformly bounded, in order to prove
that the derivatives in $\zeta$ satisfy (4.44) we proceed as follows for
$|\delta|=1,$

$\displaystyle|\partial_{\zeta}^{\delta}b_{k,\omega_{D},x}(\lambda,\zeta)|$
$\displaystyle\sim
2^{-k(n-1)/2}2^{k(n-1)}|\partial_{\zeta}^{\delta}\left(\frac{a_{k}(x,\lambda(T(\zeta),1))\psi_{x,\omega_{D}}(T(\zeta))}{\psi_{x}(T(\zeta))}\right)|$
$\displaystyle\lesssim
2^{k(n-1)/2}\sum_{\delta_{1}+\delta_{2}=\delta}|\partial_{\zeta}^{\delta_{1}}(a_{k}(x,\lambda(T(\zeta),1))\psi_{x,\omega_{D}}(T(\zeta)))|\cdot|\partial_{\zeta}^{\delta_{2}}(1/\psi_{x}(T(\zeta)))|$
$\displaystyle\lesssim
2^{k(n-1)/2}\sum_{\delta_{1}+\delta_{2}=\delta}|\partial_{\zeta}^{\delta_{1}}(a_{k}(x,\lambda(T(\zeta),1)))|$
$\displaystyle\hskip
113.81102pt\times\sum_{\delta_{2}^{1}+\cdots+\delta_{2}^{k}=\delta_{2}}\frac{|\partial_{\zeta}^{\delta_{2}^{1}}\cdots\partial_{\zeta}^{\delta_{2}^{k}}\psi_{x}(T(\zeta))|}{|\psi_{x}(T(\zeta)))|^{|\delta_{2}|+1}}$
$\displaystyle\lesssim
2^{k(n-1)/2}\sum_{\delta_{1}+\delta_{2}=\delta}|\partial_{\omega}^{\delta_{1}}(a_{k}(x,\lambda(T(\zeta),1)))\lambda^{|\delta_{1}|}\partial_{\zeta}^{\delta_{1}}T(z)|$
$\displaystyle\hskip
113.81102pt\times\sum_{\delta_{2}^{1}+\cdots+\delta_{2}^{k}=\delta_{2}}|\partial_{\omega}^{(\delta_{2}^{1},\cdots,\delta_{2}^{k})}\psi_{x}(T(\zeta))\partial_{\zeta}^{(\delta_{2}^{1},\cdots,\delta_{2}^{k})}(T(\zeta))|$

since $|\psi_{x}(T(\zeta)))|\sim 1.$ Note that we have the estimate

$\displaystyle|\partial_{\zeta}^{\delta}T(z)|$
$\displaystyle=|\partial_{\zeta}^{\delta}(\omega_{D}+2^{-k/2n}Q(x,\omega_{D})^{-1/2}[(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C}\zeta_{j}])|$
$\displaystyle=|\partial_{\zeta}^{\delta}\left(2^{-k/2n}\sum_{j=1}^{n-1}(2^{-2\varepsilon
k}+\lambda_{j}(x,\omega_{D})^{2})^{-\frac{1}{4}}(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-2C}|\zeta_{j}|^{2}\right)|$ $\displaystyle\lesssim
2^{-k/2n}2^{\varepsilon k/4}(2^{k}+2^{(1-\rho)100k}+2^{\rho k})^{-2C}.$

Moreover, since $T(\zeta)$ is a polynomial of order $2,$ when $|\delta|\geq
3,$ $\partial_{\zeta}^{\delta}T(z)=0,$ and we trivially have the bound

$|\partial_{\zeta}^{\delta}T(z)|\leq C_{\delta}2^{-k/2n}2^{\varepsilon
k/4}(2^{k}+2^{(1-\rho)100k}+2^{\rho k})^{-2C-|\delta|}.$

In consequence, since $\lambda^{|\delta_{1}|}\sim 2^{k|\delta_{1}|},$ we can
estimate

$\displaystyle
2^{k(n-1)/2}\sum_{\delta_{1}+\delta_{2}=\delta}|\partial_{\omega}^{\delta_{1}}(a_{k}(x,\lambda(T(\zeta),1)))\lambda^{|\delta_{1}|}\partial_{\zeta}^{\delta_{1}}T(z)|$
$\displaystyle\hskip
113.81102pt\times\sum_{\delta_{2}^{1}+\cdots+\delta_{2}^{k}=\delta_{2}}|\partial_{\omega}^{(\delta_{2}^{1},\cdots,\delta_{2}^{k})}\psi_{x}(T(\zeta))\partial_{\zeta}^{(\delta_{2}^{1},\cdots,\delta_{2}^{k})}(T(\zeta))|$
$\displaystyle\lesssim
2^{k(n-1)/2}\sum_{\delta_{1}+\delta_{2}=\delta}2^{-\frac{(n-1)k}{2}}2^{(1-\rho)|\delta_{1}|k}2^{\varepsilon
k/4}(2^{k}+2^{(1-\rho)100k}+2^{\rho k})^{-2C-|\delta_{1}|}2^{k|\delta_{1}|}$
$\displaystyle\hskip
56.9055pt\times\sum_{\delta_{2}^{1}+\cdots+\delta_{2}^{k}=\delta_{2}}2^{-k/2n}2^{C\varepsilon
k|(\delta_{2}^{1},\cdots,\delta_{2}^{k})|}2^{\varepsilon
k/4}(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-2C-|(\delta_{2}^{1},\cdots,\delta_{2}^{k})|}$
$\displaystyle\lesssim\sum_{\delta_{1}+\delta_{2}=\delta}\sum_{\delta_{2}^{1}+\cdots+\delta_{2}^{k}=\delta_{2}}2^{-k/2n}2^{(1-\rho)|\delta_{1}|k}2^{\varepsilon
k/4}2^{-Ck}2^{-50(1-\rho)|\delta_{1}|)}2^{C\varepsilon
k|\delta_{2}|}2^{\varepsilon k/4}2^{-(C+|\delta_{2}|/2)k}$
$\displaystyle\lesssim 1.$

On the other hand, to complete the proof of (4.44) we just have to use the
estimate

$|\partial_{\lambda}^{\gamma}a_{k}(x,\xi)|\lesssim_{\beta,\gamma}2^{-\frac{(n-1)k}{2}}2^{-\rho|\gamma|k}.$

Since $\lambda\sim 2^{k},$ we can estimate

$\displaystyle|\partial_{\lambda}^{\beta}b_{k,\omega_{D},x}(\lambda,\zeta)|$
$\displaystyle=|\partial_{\lambda}^{\beta}\left(\frac{2^{-k(n-1)/2}\lambda^{n-1}a_{k}(x,\lambda(T(\zeta),1))\chi_{x,\omega_{D}}(T(\zeta))}{\psi_{x}(T(\zeta))}\right)|$
$\displaystyle\sim\left|\left(\frac{2^{-k(n-1)/2}2^{k(n-1)}\partial_{\lambda}^{\beta}(a_{k}(x,\lambda(T(\zeta),1)))\chi_{x,\omega_{D}}(T(\zeta))}{\psi_{x}(T(\zeta))}\right)\right|$
$\displaystyle\sim\left|{2^{k(n-1)/2}\partial_{\lambda}^{\beta}(a_{k}(x,\lambda(T(\zeta),1)))\chi_{x,\omega_{D}}(T(\zeta))}\right|$
$\displaystyle\lesssim 2^{k(n-1)/2}2^{-\frac{(n-1)k}{2}}2^{-\rho|\beta|k}$
$\displaystyle=2^{-k\rho|\beta|},$

as claimed. Indeed, note that the derivatives
$\delta_{\lambda}^{\beta_{i}}(\lambda^{n-1})$ with $|\beta_{i}|\leq\beta,$ are
bounded from above by $\lesssim 2^{k(n-1)},$ when $\lambda\sim 2^{k}.$ On the
other hand, let us analyse the phase using its Taylor expansion as follows
taking into account that $\omega=T(\zeta)$

$\displaystyle\Phi_{\tau}(x,\omega)-y(\omega,1)$
$\displaystyle=\Phi_{\tau}(x,\omega_{D})-y\cdot(\omega_{D},1)$
$\displaystyle+\nabla_{\omega}(\Phi_{\tau}(x,\omega)-y(\omega,1))|_{\omega=\omega_{D}}\cdot(\omega-\omega_{D})+e_{k,x,y,\omega_{D}}(\zeta)$
$\displaystyle=\Phi_{\tau}(x,\omega_{D})-y\cdot(\omega_{D},1)$
$\displaystyle+(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})\cdot(2^{-k/2n}Q(x,\omega_{D})^{-1/2}(\alpha\zeta))+e_{k,x,y,\omega_{D}}(\zeta)$
$\displaystyle=\Phi_{\tau}(x,\omega_{D})-y\cdot(\omega_{D},1)$
$\displaystyle+(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C}2^{-k/2n}(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})\cdot
Q(x,\omega_{D})^{-1/2}(\zeta)$ $\displaystyle+e_{k,x,y,\omega_{D}}(\zeta)$
$\displaystyle=\Phi_{\tau}(x,\omega_{D})-y\cdot(\omega_{D},1)$
$\displaystyle+(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C}2^{-k/2n}Q(x,\omega_{D})^{-1/2}(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})\cdot\zeta$
$\displaystyle+e_{k,x,y,\omega_{D}}(\zeta).$

Since the phase function

$\displaystyle F(\zeta)$ $\displaystyle=\Phi_{\tau}(x,\omega)-y(\omega,1)$
$\displaystyle=\Phi_{\tau}(x,\omega_{D})-y\cdot(\omega_{D},1)$
$\displaystyle+(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-C}2^{-k/2n}Q(x,\omega_{D})^{-1/2}(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})\cdot\zeta$
$\displaystyle+e_{k,x,y,\omega_{D}}(\zeta),$

has gradient

$\displaystyle\nabla_{\zeta}F(\zeta)=\alpha
2^{-k/2n}Q(x,\omega_{D})^{-1/2}(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})+O(2^{-k}),$

we have that

$\displaystyle|\nabla_{\zeta}F(\zeta)|$ $\displaystyle=\alpha
2^{-k/2n}|\textnormal{diag}[(\lambda_{Q,i}^{\omega})^{-1/2}]_{1\leq i\leq
n-1}(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})|+O(2^{-k})$
$\displaystyle=\alpha 2^{-k/2n}|[(2^{-\varepsilon
k}I+\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega)^{2}]^{-1/4}(\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})-\overline{y})|+O(2^{-k})$
$\displaystyle=O(\alpha 2^{-k/2n}2^{\varepsilon k/4})+O(2^{-k}).$

Then, with $\beta=\alpha 2^{-k/2n}2^{\varepsilon k/4},$ one has the estimate

$|\nabla_{\zeta}\beta^{-1}F(\zeta)|=O(1).$ (4.45)

For our further analysis note that

$\displaystyle\Phi_{\tau}(x,\omega)-y(\omega,1)=\Phi_{\tau}(x,\omega_{D})-y\cdot(\omega_{D},1)+Z(\zeta),\,|Z(\zeta)|=O(\beta).$

So, in order to apply the non-stationary phase principle222See Proposition
2.8. We are going to apply this proposition to the phase
$F_{\beta}:=\beta^{-1}F(\zeta),$ making use of the estimate (4.45). to the
integral

$\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta$

we re-write it as follows

$\displaystyle\left|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta\right|$
$\displaystyle\leq\smallint\limits_{\mathbb{R}}\left|\smallint\limits_{\mathbb{R}^{n-1}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\zeta\right|d\lambda$
$\displaystyle\sim\smallint\limits_{\lambda\sim
2^{k}}\left|\smallint\limits_{\mathbb{R}^{n-1}}e^{2\pi i\lambda
F(\zeta)}b_{k,\omega_{D},x}(\lambda,\zeta)d\zeta\right|d\lambda$
$\displaystyle\lesssim 2^{k}\sup_{\lambda\sim
2^{k}}\left|\smallint\limits_{\mathbb{R}^{n-1}}e^{2\pi i\lambda
F(\zeta)}b_{k,\omega_{D},x}(\lambda,\zeta)d\zeta\right|d\lambda$
$\displaystyle=2^{k}\sup_{\lambda\sim
2^{k}}\left|\smallint\limits_{\mathbb{R}^{n-1}}e^{2\pi
i\lambda\beta\beta^{-1}F(\zeta)}b_{k,\omega_{D},x}(\lambda,\zeta)d\zeta\right|d\lambda$
$\displaystyle\lesssim 2^{k}\sup_{\lambda\sim
2^{k}}(\lambda\beta)^{-1}\sim\beta^{-1}.$

Consequently,

$\displaystyle\left\|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta\right\|_{L^{1}(\mathbb{R}^{n}_{x})}$
$\displaystyle=\left\|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta\right\|_{L^{1}(D)}$ $\displaystyle\leq\beta^{-1}\textnormal{Vol}(D),$

where the eccentric disc $D$ is given by

$D=\left\\{x:\Phi_{\tau}(x,\omega_{D})=y(\omega_{D},1)+O(\beta);\nabla_{\omega}\Phi_{\tau}(x,\omega_{D})=\overline{y}+Q(x,\omega_{D})^{1/2}Z\right\\},$

with $|Z|=O(\beta).$ Taking into account that
$\textnormal{det}(Q(x,\omega_{D}))\lesssim 2^{-\varepsilon k}$ one can
estimate the volume of $D$ as follows, see [45, Page 8]

$\textnormal{Vol}(D)\lesssim 2^{-\varepsilon k}\beta^{n+1}.$

So, we can estimate

$\displaystyle\left\|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\leq\beta^{-1}\textnormal{Vol}(D)$
$\displaystyle\lesssim 2^{-\varepsilon k}\beta^{n}$ $\displaystyle\lesssim
2^{-\varepsilon k}(2^{k}+2^{(1-\rho)100k}+2^{\rho
k})^{-Cn}2^{-k/2}2^{\varepsilon nk/4}$ $\displaystyle\leq 2^{-\varepsilon
k}2^{-k/2}(2^{-Cnk}2^{\varepsilon nk/4}).$

Taking $\varepsilon>0$ small enough and $C>0$ large enough we have the
estimate $2^{-k/2}(2^{-Cnk}2^{\varepsilon nk/4})\leq 2^{-(n-1)k/2}.$
Therefore, we have that

$\left\|\smallint\limits_{\mathbb{R}^{n-1}}\smallint\limits_{\mathbb{R}}e^{2\pi
i\lambda(\Phi_{\tau}(x,T(\zeta))-y\cdot(T(\zeta),1))}b_{k,\omega_{D},x}(\lambda,\zeta)d\lambda
d\zeta\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim 2^{-\varepsilon
k}2^{-(n-1)k/2}$

as desired. We have proved (4.43). In consequence, the proof of Lemma 4.6 is
complete. ∎

###### Proof of Proposition 4.4.

Let us consider the kernel

$K_{\textnormal{deg},k}(x,y):=\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\phi_{-\varepsilon
k}(J_{\tau}(x,\omega))\eta_{k}(\xi)e^{-2\pi iy\cdot\xi}d\xi.$

Since $a(x,\xi)\in S^{-(n-1)/2}_{1,0},$ and
$e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\in S^{0}_{1/2,0},$ the Hörmander
calculus implies that

$\tilde{a}(x,\xi):=a(x,\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}\in
S^{-(n-1)/2}_{1/2,0}.$

By applying Lemma 4.6 to $\tilde{a}(x,\xi)\in S^{-(n-1)/2}_{\rho,0},$ with
$\rho=1/2,$ the symbol

$\tilde{a}_{k}(x,\xi)=\tilde{a}(x,\xi)\phi_{-\varepsilon
k}(J_{\tau}(x,\omega))\eta_{k}(\xi),$ (4.46)

satisfies the inequalities

$|\partial_{\omega}^{\beta}\partial_{\lambda}^{\gamma}a_{k}(x,\xi)|\lesssim_{\beta,\gamma}2^{-\frac{(n-1)k}{2}}2^{-\frac{1}{2}|\gamma|k+\frac{1}{2}|\beta|k}2^{C\varepsilon
k|\beta|},$ (4.47)

and the kernel $K_{\textnormal{deg},k}$ satisfies the estimate

$\left\|\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i\Phi_{\tau}(x,\xi)}a_{k}(x,\xi)e^{-2\pi
iy\cdot\xi}d\xi\right\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim 2^{-\varepsilon
k},$ (4.48)

uniformly in $y\in\mathbb{R}^{n},$ and in $\varepsilon>0$ small enough. In
view of Remark 4.5 the proof of Proposition 4.4 is complete. ∎

###  4.3. Boundedness of the non-degenerate component

Now, we are going to prove the weak (1,1) boundedness of the non-degenerate
component $T_{\textnormal{nondeg}}$ in (4.5). To do this, we follow the
approach in [45], by making the factorisation $T_{\textnormal{nondeg}}=SA+E$
modulo an error operator $E=T_{\textnormal{nondeg}}-SA$ bounded on $L^{1}.$
Here, we are going to construct a pseudo-differential operator $S$ of order
zero and an averaging operator $A,$ which is bounded on $L^{1}.$

####  4.3.1. Construction of the operators $A$ and $S$

Let us consider the projection
$\pi_{1}:\textnormal{supp}(a)\rightarrow\mathbb{R}^{n}$ in the first component
$(x,\xi)\mapsto x,$ and let $V$ be an open subset, such that $\overline{V}$ is
compact with $\pi_{1}(\textnormal{supp}(a))\subset V.$ Let $1_{\overline{V}}$
be the characteristic function of $\overline{V}.$ The operator $A$ will be
defined as follows

$Af(x)$ $\displaystyle:=\sum_{k\gg 1}\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}$ $\displaystyle(1-\phi_{-\varepsilon
k}(J(x,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\times
1_{\overline{V}}(x)|J(x,\omega)|^{1/2}\widehat{f}(\xi^{\prime})d\xi^{\prime}d\omega,$

where $\nabla_{\xi}\Phi_{\tau}(x,\omega)$ denotes the value of
$\nabla_{\xi}\Phi_{\tau}(x,\xi)$ at $\xi=(\omega,1),$ and
$\psi:=\psi(x,\omega)$ is a bump function to be chosen later. The integer
quantities $\mu(x,\omega)$ are such that they satisfy, according to the
principle of stationary phase, the asymptotic expansion (see Proposition 2.9)

$\displaystyle\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\psi(x,\omega)d\omega=e^{2\pi
i\Phi_{\tau}(x,\xi^{\prime})}\psi(x,\omega^{\prime})e^{\mu(x,\omega^{\prime})\pi
i/4}(\lambda^{\prime})^{-(n-1)/2}|J(x,\omega^{\prime})|^{-1/2}+\cdots,$

when $|\xi^{\prime}|\gg 1.$ Since $\mu(x,\omega)$ are smooth functions in the
region of integration they are constant on each connected component of this
region.

######  Remark 4.8.

When $\textnormal{Im}(\Phi(x,\xi))\equiv 0,$ that is if $\Phi$ is real-valued,
the definition of $A$ purely involves the phase function $\Phi_{\tau}$ and the
curvature term  $J(x,\omega)=\det(\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega)),$
the $L^{1}$-boundedness of $A$ follows from the argument for real-valued phase
functions in Tao [45, Pages 14-15].

######  Lemma 4.9.

$A$ extends to a bounded operator on $L^{1}(\mathbb{R}^{n})$ provided that
there exists $\varkappa_{0}>0,$ such that for all
$(x,\theta)\in\overline{V}\times\overline{\\{\theta:|\theta|\sim 1\\}},$ the
lower bound

$\textnormal{Im}(\Phi(x,\theta))>\varkappa_{0},$

holds.

###### Proof.

First, assume that $\widehat{f}(\xi)$ is smooth and compactly supported and
that $f$ vanishes when $|\xi|\leq 1.$ By applying summation by parts, because
of the properties of the support of $f,$ boundary terms vanish and we can
write

$\displaystyle Af(x)$ $\displaystyle:=\sum_{k\gg 1}\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}(1-\phi_{-\varepsilon
k}(J(x,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times|J(x,\omega)|^{1/2}\widehat{f}(\xi^{\prime})d\xi^{\prime}d\omega$
$\displaystyle=-\sum_{k\gg 1}\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}(\phi_{-\varepsilon(k+1)}(J(x,\omega))-\phi_{-\varepsilon
k}(J(x,\omega)))$
$\displaystyle\times\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}1_{\overline{V}}(x)|J(x,\omega)|^{1/2}\widehat{f}(\xi^{\prime})d\xi^{\prime}d\omega.$

In view of the triangle inequality, it suffices to prove that the estimate

$\|A_{k}f\|_{L^{1}(\mathbb{R}^{n})}\lesssim 2^{-\varepsilon
k}\|f\|_{L^{1}(\mathbb{R}^{n})},$ (4.49)

where

$\displaystyle A_{k}f(x)$ $\displaystyle=-\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4})(\phi_{-\varepsilon(k+1)}(J(x,\omega))-\phi_{-\varepsilon
k}(J(x,\omega)))$
$\displaystyle\times\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}1_{\overline{V}}(x)|J(x,\omega)|^{1/2}\widehat{f}(\xi^{\prime})d\xi^{\prime}d\omega.$

Using the Fourier inversion formula, we have that

$\displaystyle\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}1_{\overline{V}}(x)\widehat{f}(\xi^{\prime})d\xi^{\prime}$
$\displaystyle=1_{\overline{V}}(x)[\eta_{k}(D)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,D))}f](\nabla_{\xi}\Phi_{\tau}(x,\omega)).$

Since $\textnormal{supp}[(\phi_{-\varepsilon(k+1)}(J(x,\omega))]$ is contained
in the set where $J(x,\omega)\lesssim 2^{-\varepsilon k},$ the difference
$\phi_{-\varepsilon(k+1)}(J(x,\omega))-\phi_{-\varepsilon k}(J(x,\omega)),$ is
supported on the region $J(x,\omega)\sim 2^{-k\varepsilon}.$ Since
$|\phi_{k}(t)|\lesssim\|\phi\|_{L^{\infty}},$ $\forall t>0,$ we have that

$|(\phi_{-\varepsilon(k+1)}(J(x,\omega))-\phi_{-\varepsilon
k}(J(x,\omega))||J(x,\omega)|^{1/2}\lesssim 2^{-\varepsilon k/2},$

and consequently

$\displaystyle\|A_{k}f\|_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle=\smallint|A_{k}f(x)|dx$
$\displaystyle\lesssim\smallint\smallint|1_{\overline{V}}(x)[\eta_{k}(D)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,D))}f](\nabla_{\xi}\Phi_{\tau}(x,\omega))|2^{-\varepsilon
k/2}|\phi(x,\omega)|dxd\omega.$

Note that the operator

$\Omega(x,D)=1_{\overline{V}}(x)\eta_{k}(D)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,D))}$

is uniformly bounded in $k$ on $L^{1}(\mathbb{R}^{n}).$ Indeed, its kernel
$\Omega(x,D)\delta(y),$ given by

$\Omega(x,D)\delta(y)=\smallint e^{2\pi
i(x-y)\xi}1_{\overline{V}}(x)[\eta_{k}(\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}d\xi$

belongs to $L^{1}(\mathbb{R}_{x}^{n})$ uniformly in $y\in\mathbb{R}^{n}.$
Since the phase function $\Phi$ is of strictly positive type, that is,
$\textnormal{Im}(\phi)>0,$ and $|\eta_{k}(\xi)|\lesssim 1,$ note that

$\displaystyle\smallint|\smallint e^{2\pi i(x-y)\xi}1_{\overline{V}}(x)$
$\displaystyle\eta_{k}(\xi)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi))}d\xi|dx$
$\displaystyle\leq\smallint 1_{\overline{V}}(x)\sup_{|\xi^{\prime}|\sim
2^{k}}e^{-\textnormal{Im}(\Phi(x,\xi^{\prime}))}dx\smallint_{|\xi|\sim
2^{k}}|\eta_{k}(\xi)|d\xi$
$\displaystyle\lesssim|\overline{V}|\sup_{x\in\overline{V}}\sup_{|\xi^{\prime}|\sim
2^{k}}e^{-\textnormal{Im}(\Phi(x,\xi^{\prime}))}\smallint_{|\xi|\sim
2^{k}}d\xi$
$\displaystyle\lesssim|\overline{V}|\sup_{x\in\overline{V}}\sup_{|\xi^{\prime}|\sim
2^{k}}e^{-2^{k}\textnormal{Im}(\Phi(x,2^{-k}\xi^{\prime}))}2^{kn}$
$\displaystyle\lesssim|\overline{V}|\sup_{x\in\overline{V}}\sup_{|\theta|\sim
1}e^{-2^{k}\textnormal{Im}(\Phi(x,\theta))}2^{kn}<\infty.$

Indeed, since $\textnormal{Im}(\phi)>0,$ when $|\xi|\neq 0,$ the compactness
of the closed annulus $\overline{\\{\theta:|\theta|\sim 1\\}},$ and of
$\overline{V},$ imply that there exists $\varkappa_{0}>0$ such that for all
$(x,\omega)\in\overline{V}\times\overline{\\{\theta:|\theta|\sim 1\\}},$ one
has the lower bound

$\textnormal{Im}(\Phi(x,\theta))>\varkappa_{0}.$

Therefore,

$\mathscr{K}:=\sup_{k>0}\sup_{x\in\overline{V}}\sup_{|\theta|\sim
1}e^{-2^{k}\textnormal{Im}(\Phi(x,\theta))}2^{kn}\leq\sup_{k>0}e^{-2^{k}\varkappa_{0}}2^{kn}<\infty.$

Thus, we have proved that
$\sup_{y\in\mathbb{R}^{n}}\|\Omega(x,D)\delta(y)\|_{L^{1}(\mathbb{R}^{n}_{x})}<\infty,$
and Schur’s lemma gives the boundedness of $\Omega(x,D)$ on
$L^{1}(\mathbb{R}^{n}).$ Note also, that if $z=F(x)$ is an smooth change of
coordinates on $\overline{V}$, the operator

$\Omega(F^{-1}(z),D)=1_{\overline{V}}(F^{-1}(z))\eta_{k}(D)e^{-(1+i\tau)\textnormal{Im}(\Phi(F^{-1}(z),D))}$

is also bounded on $L^{1}.$ Moreover, since $1_{\overline{V}}\circ
F^{-1}=1_{F(\overline{V})},$ by following the previous argument, we have that

$\displaystyle\sup_{y}\|\Omega(F^{-1}(z),D)\delta(y)\|_{L^{1}(\mathbb{R}^{n}_{z})}\lesssim|F(\overline{V})|\sup_{z\in
F(\overline{V})}\sup_{|\theta|\sim
1}e^{-2^{k}\textnormal{Im}(\Phi(F^{-1}(z),\theta))}2^{kn}<\infty.$

Now, since for any $\omega,$
$z=F_{\omega}(x)=\nabla_{\xi}\Phi_{\tau}(x,\omega)$ is a diffeomorphism on
$\overline{V},$ and taking $R>0$ large enough such that
$\\{\omega:(x,\omega)\in\textnormal{supp}(\phi(x,\omega))\\}\subset\\{|\omega|\leq
R\\},$ we have that

$\displaystyle\|A_{k}f\|_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\smallint_{|\omega|\leq
R}\smallint|1_{\overline{V}}(x)[\eta_{k}(D)e^{-(1+i\tau)\textnormal{Im}(\Phi(x,D))}f](\nabla_{\xi}\Phi_{\tau}(x,\omega))|2^{-\varepsilon
k/2}|\phi(x,\omega)|dxd\omega$
$\displaystyle\lesssim\|\phi(x,\omega)\|_{L^{\infty}}\smallint_{|\omega|\leq
R}\smallint|\Omega(x,D)f(F_{\omega}(x))|2^{-\varepsilon k/2}dxd\omega$
$\displaystyle=\|\phi(x,\omega)\|_{L^{\infty}}\smallint_{|\omega|\leq
R}\smallint|\Omega(F^{-1}_{\omega}(z),D)f(z)|2^{-\varepsilon
k/2}|\det(DF_{\omega}^{-1}(z))|dzd\omega$ $\displaystyle\lesssim
2^{-\varepsilon k/2}\smallint_{|\omega|\leq
R}\|\Omega(F^{-1}_{\omega}(z),D)f\|_{L^{1}(\mathbb{R}^{n}_{z})}d\omega$
$\displaystyle\lesssim 2^{-\varepsilon k/2}\sup_{|\omega^{\prime}|\leq
R}\|\Omega(F^{-1}_{\omega^{\prime}}(z),D)f\|_{L^{1}(\mathbb{R}^{n}_{z})}\smallint_{|\omega|\leq
R}d\omega$ $\displaystyle\lesssim 2^{-\varepsilon
k/2}\sup_{|\omega^{\prime}|\leq
R}|F_{\omega^{\prime}}(\overline{V})|\sup_{z\in
F_{\omega^{\prime}}(\overline{V})}\sup_{|\theta|\sim
1}e^{-2^{k}\textnormal{Im}(\Phi(F_{\omega^{\prime}}^{-1}(z),\theta))}2^{kn}\|f\|_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle=2^{-\varepsilon k/2}\sup_{|\omega^{\prime}|\leq
R}|F_{\omega^{\prime}}(\overline{V})|\sup_{x\in\overline{V}}\sup_{|\theta|\sim
1}e^{-2^{k}\textnormal{Im}(\Phi(x,\theta))}2^{kn}\|f\|_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle\lesssim_{\mathscr{K}}2^{-\varepsilon
k/2}\sup_{|\omega^{\prime}|\leq
R}|F_{\omega^{\prime}}(\overline{V})|\|f\|_{L^{1}(\mathbb{R}^{n})}.$

Since $F_{\omega}$ is $C^{1}$ in $\omega,$ and $\overline{V}$ is compact,
$\sup_{|\omega^{\prime}|\leq R}|F_{\omega^{\prime}}(\overline{V})|<\infty.$ So
we have the estimate

$\|A_{k}f\|_{L^{1}(\mathbb{R}^{n})}\lesssim
2^{-k\varepsilon/2}\|f\|_{L^{1}(\mathbb{R}^{n})}$ (4.50)

as desired proving the $L^{1}$-boundedness of $A.$ The proof of Lemma 4.9 is
complete. ∎

######  Definition 4.10.

Using that the mapping

$(x,\xi)\mapsto(x,\theta):=(x,\nabla_{x}\Phi_{\tau}(x,\xi))$

is a diffeomorphism on the support of the symbol $a:=a(x,\xi),$ we define the
pseudo-differential operator $S$ associated to the symbol

$s:=s(x,\theta)=\frac{\lambda^{\frac{n-1}{2}}a(x,\xi)}{\psi(x,\omega)}1_{\overline{V}},$
(4.51)

where, $\forall(x,\omega),$ $\psi(x,\omega)\neq 0,$ is a bounded function away
from zero and smooth for all $x$ and for all $\omega$.

######  Remark 4.11.

Since $V$ is an open neighbourhood of the set
$\\{x\in\mathbb{R}^{n}:(x,\xi)\in\textnormal{supp}[a]\\},$ we have that
$a(x,\xi)=0$ on $\overline{V}\setminus V.$ In consequence
$s(x,\theta)=\frac{\lambda^{\frac{n-1}{2}}a(x,\xi)}{\psi(x,\omega)}1_{\overline{V}}=\frac{\lambda^{\frac{n-1}{2}}a(x,\xi)}{\psi(x,\omega)},$
is smooth in $(x,\omega)$ and
$\\{x\in\mathbb{R}^{n}:(x,\theta)\in\textnormal{supp}[s]\\}\subset V.$

Now we prove the following fundamental property of the operator $S.$

######  Lemma 4.12.

The pseudo-differential operator $S$ defined by

$Sf(x)=\smallint e^{2\pi ix\cdot\xi}s(x,\theta(x,\xi))\widehat{f}(\xi)d\xi,$
(4.52)

is of weak (1,1) type.

###### Proof.

Since $\lambda^{\frac{n-1}{2}}=\xi_{n}^{\frac{n-1}{2}}$ and
$a_{\tau}:=a_{\tau}(x,\xi)\in S^{-\frac{n-1}{2}}_{1,0}$ we have that
$s:=s(x,\theta(x,\xi))$ is symbol of order zero. Then, the pseudo-differential
operator $S=s(x,\nabla_{x}\Phi_{\tau}(x,D))$ is essentially a Calderón-Zygund
operator and then it satisfies the weak (1,1) inequality of $S.$ ∎

###  4.4. Control of the error operator

In this subsection we prove that the error operator $E$ is bounded on $L^{1}.$
The idea behind the proof of this fact can be understood from the calculus of
Fourier integral operators. The operator $A$ is essentially a Fourier integral
operator with the same phase function that $T,$ in the sense that its phase
$2\pi i\Phi_{\tau}(x,\omega)$ evaluated at the stationary point
$\omega=\omega^{\prime}$ is $2\pi i\Phi(x,\xi^{\prime}),$ see [45, Page 19,
Eq. (16)]. So, intuitively, if $A$ would have had the form in (1.1), namely,
$Af(x)=\smallint\limits_{\mathbb{R}^{n}}e^{i\phi_{A}(x,\xi)}\sigma_{A}(x,\xi)\widehat{f}(\xi)d\xi,$
one could compute an asymptotic expansion for the symbol $\sigma(x,\xi)$ of
$SA$ as follows

$\sigma(x,\xi)=\smallint\smallint\sigma_{A}(x,\xi)s(x,\zeta)e^{2\pi
i(\phi_{A}(y,\xi)-\phi_{A}(x,\xi)+(x-y)\cdot\zeta)}d\zeta dy,$ (4.53)

and satisfying the asymptotic expansion, see e.g. [26, Page 17],

$\sigma(x,\xi)=s(x,\nabla\phi_{A}(x,\xi))\sigma_{A}(x,\xi)+\sum_{0<|\alpha|<N}\frac{1}{\alpha!}\sigma_{\alpha}(x,\xi)+r_{N}(x,\xi),$
(4.54)

and the symbols $\sigma_{\alpha}$ satisfying the estimates (under suitable
conditions on the phase $\phi_{A}$)

$|\partial_{\xi}^{\beta}\partial_{x}^{\gamma}\sigma_{\alpha}(x,\xi)|\lesssim_{\varepsilon^{\prime}}(1+|\xi|)^{-\frac{n-1}{2}-(\frac{1}{2}-\varepsilon^{\prime})|\alpha|-\frac{|\beta|}{2}+\frac{1}{2}|\gamma|},$

for any $0<\varepsilon^{\prime}<1/2.$ Below we will prove that essentially
$T_{\textnormal{nondeg}}$ and $SA$ have the same principal symbol
$s(x,\nabla\phi_{A}(x,\xi))\sigma_{A}(x,\xi),$ and then the order
$\tilde{\mu}$ of the error operator $E=T_{\textnormal{nondeg}}-SA$ should be,
at least heuristically, the order of the $\sigma_{\alpha}$’s when
$|\alpha|=1,$ that is
$\tilde{\mu}\leq-\frac{n-1}{2}-\frac{1}{2}+\varepsilon^{\prime}.$ This order
$\tilde{\mu}$ is good enough to deduce the $L^{1}$-boundedness of $E.$
However, since the definition of $A$ also involves integration with respect to
$\omega,$ the structure of this operator is not as the one in (1.1), and one
has to use the stationary phase method to prove that each dyadic component of
the error $E,$ has a kernel with a $L^{1}_{x}$-norm satisfying uniformly in
$y,$ a decay proportional to $2^{C\varepsilon k}2^{-k/2}.$ Then, if proved
this we are allowed to use Proposition 6.1 in [45, Page 17] to deduce the
boundedness of $E,$ even under the presence of the oscillating term
$e_{\Phi}(x,\xi)=e^{-(1+i\tau)\Phi(x,\xi)}.$

First, let us calculate the kernel
$E\delta_{z}(x)=T_{\textnormal{nondeg}}\delta_{z}(x)-SA\delta_{z}(x)$ of the
error operator $E.$ Note that

$\displaystyle A\delta_{z}(x)$ $\displaystyle=\sum_{k\gg 1}\smallint\smallint
e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4})(1-\phi_{-\varepsilon
k}(J(x,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times|J(x,\omega)|^{1/2}\widehat{\delta_{z}}(\xi^{\prime})d\xi^{\prime}d\omega$
$\displaystyle=\sum_{k\gg 1}\smallint\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)-2\pi
iy\cdot\xi^{\prime}}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4})(1-\phi_{-\varepsilon k}(J(x,\omega)))$
$\displaystyle\times\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}|J(x,\omega)|^{1/2}\delta_{z}(y)d\xi^{\prime}d\omega
dy$ $\displaystyle=\sum_{k\gg 1}\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)-2\pi
iz\cdot\xi^{\prime}}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}(1-\phi_{-\varepsilon k}(J(x,\omega)))$
$\displaystyle\times\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}|J(x,\omega)|^{1/2}d\xi^{\prime}d\omega.$

Consequently,

$\displaystyle SA\delta_{z}(x)$ $\displaystyle=\smallint e^{2\pi
ix\cdot\zeta}s(x,\zeta)\widehat{A\delta_{z}}(\zeta)d\zeta$
$\displaystyle=\smallint\smallint e^{2\pi
i(x-y)\cdot\zeta}s(x,\zeta)A\delta_{z}(y)dyd\zeta$ $\displaystyle=\sum_{k\gg
1}\smallint\smallint\smallint\smallint e^{2\pi
i\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(y,\omega)-2\pi
iz\cdot\xi^{\prime}+2\pi
i(x-y)\cdot\zeta}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi i/4}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\xi^{\prime}d\omega
dyd\zeta$ $\displaystyle=\sum_{k\gg 1}\smallint\smallint\smallint\smallint
e^{2\pi
i[\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-z)+(x-y)\cdot\zeta]}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\xi^{\prime}d\omega
dyd\zeta.$

On the other hand, the kernel of $T_{\textnormal{nondeg}}$ is given by

$\displaystyle T_{\textnormal{nondeg}}\delta_{z}(x)$ $\displaystyle=\sum_{k\gg
1}\smallint e^{2\pi i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot
z]}a(x,\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))\eta_{k}(\xi^{\prime})d\xi^{\prime}$
$\displaystyle=\sum_{k\gg 1}\smallint e^{2\pi
i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot
z]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))(\lambda^{\prime})^{-\frac{n-1}{2}}e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))\eta_{k}(\xi^{\prime})d\xi^{\prime}.$

In order to prove the $L^{1}$ boundedness of $E$ it suffices to prove the
following estimate on each dyadic component

$\displaystyle\|$ $\displaystyle\smallint e^{2\pi
i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot
z]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))(\lambda^{\prime})^{-\frac{n-1}{2}}e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))\eta_{k}(\xi^{\prime})d\xi^{\prime}$
$\displaystyle-\smallint\smallint\smallint\smallint e^{2\pi
i[\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-z)+(x-y)\cdot\zeta]}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta d\xi^{\prime}\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim 2^{C\varepsilon
k}2^{-k/2}.$

However, the integral inside of this $L^{1}$-norm, can be written as follows

$\displaystyle\|$ $\displaystyle\smallint e^{2\pi
i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot
z]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))(\lambda^{\prime})^{-\frac{n-1}{2}}e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))\eta_{k}(\xi^{\prime})d\xi^{\prime}$
$\displaystyle-\smallint\smallint\smallint\smallint e^{2\pi
i[\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-z)+(x-y)\cdot\zeta]}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta d\xi^{\prime}\|_{L^{1}(\mathbb{R}^{n}_{x})}$ $\displaystyle=\|$
$\displaystyle\smallint e^{2\pi
i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot
z]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))(\lambda^{\prime})^{-\frac{n-1}{2}}e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))\eta_{k}(\xi^{\prime})d\xi^{\prime}$
$\displaystyle-\smallint\smallint\smallint\smallint e^{2\pi
i[\Phi_{\tau}(x,\xi^{\prime})-z\cdot\xi^{\prime}+[\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(y,\omega)-\Phi_{\tau}(x,\xi^{\prime})]+(x-y)\cdot\zeta]}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))\eta_{k}(\xi^{\prime})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta d\xi^{\prime}\|_{L^{1}(\mathbb{R}^{n}_{x})}$ $\displaystyle=\|$
$\displaystyle\smallint e^{2\pi
i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot z]}$
$\displaystyle\times\\{s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))(\lambda^{\prime})^{-\frac{n-1}{2}}e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))$ $\displaystyle-\smallint\smallint\smallint e^{2\pi
i[\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(y,\omega)-\Phi_{\tau}(x,\xi^{\prime})]+(x-y)\cdot\zeta]}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta\\}\eta_{k}(\xi^{\prime})d\xi^{\prime}\|_{L^{1}(\mathbb{R}^{n}_{x})}.$

So, let us simplify the notation above by defining

$\displaystyle
W^{0}_{x,k,z}(\xi^{\prime}):=s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))(\lambda^{\prime})^{-\frac{n-1}{2}}e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime}))),$

and

$\displaystyle W_{x,k,z}(\xi^{\prime})$
$\displaystyle:=\smallint\smallint\smallint e^{2\pi
i[\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(y,\omega)-\Phi_{\tau}(x,\xi^{\prime})+(x-y)\cdot\zeta]}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta.$

Observe that in view of the Euler homogeneity identity

$\Phi_{\tau}(x,\xi^{\prime})=\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}),$

we can re-write the phase of the integral $W_{x,k,z}(\xi^{\prime})$ as follows

$\displaystyle\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(y,\omega)-\Phi_{\tau}(x,\xi^{\prime})+(x-y)\cdot\zeta$
$\displaystyle=\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))+(x-y)\cdot\zeta$
$\displaystyle=:\Psi_{x,k,z}(\omega,\zeta,y),$

and then

$\displaystyle W_{x,k,z}(\xi^{\prime})$
$\displaystyle:=\smallint\smallint\smallint e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta.$

The analysis above shows that in order to prove the $L^{1}$ boundedness of $E$
it suffices to prove the following estimate

$\|\smallint e^{2\pi i[\Phi_{\tau}(x,\xi^{\prime})-\xi^{\prime}\cdot
z]}[W_{x,k,z}(\xi^{\prime})-W^{0}_{x,k,z}(\xi^{\prime})]\eta_{k}(\xi^{\prime})d\xi^{\prime}\|_{L^{1}(\mathbb{R}^{n}_{x})}\lesssim
2^{C\varepsilon k}2^{-k/2}.$ (4.55)

To do so, we will estimate the integral $W_{x,k,z}(\xi^{\prime})$ on the set
of points $(\omega,y,\zeta)$ that contribute more information to the
integrand. This set of points is determined by the region:

$\mathscr{R}=\\{(\omega,y,\zeta):\omega-\omega^{\prime}=y-x=O(2^{\varepsilon
k}2^{-k/2})\\}.$

By following the argument in [45, Page 19] one can make the approximation

$s(x,\zeta)=s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))+O(2^{C\varepsilon
k}2^{-k/2})$

where the error term $O(2^{C\varepsilon
k}2^{-k/2})=O_{x,\xi^{\prime},\zeta}(2^{C\varepsilon k}2^{-k/2})$ has
derivatives satisfying the estimate

$|\partial_{\xi^{\prime}}^{\beta}O_{x,\xi^{\prime},\zeta}(2^{C\varepsilon
k}2^{-k/2})|\lesssim O(2^{C|\beta|\varepsilon
k}2^{-k/2}2^{-(n-1)k/2}2^{-k|\beta|}).$ (4.56)

Denoting $\mathscr{R}^{c}:=\mathbb{R}^{n}\setminus\mathscr{R},$ we have that

$\displaystyle W_{x,k,z}(\xi^{\prime})$
$\displaystyle:=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$ $\displaystyle+\smallint\smallint\smallint_{\mathscr{R}^{c}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$
$\displaystyle=:W_{x,k,z}(\xi^{\prime})^{\mathscr{R}}+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}^{c}}.$

Here, the main integral to study is $W_{x,k,z}(\xi^{\prime})^{\mathscr{R}}.$
Indeed, arguing as in [45, Page 17] one can demonstrate that
$W_{x,k,z}(\xi^{\prime})^{\mathscr{R}^{c}}$ is a smoothing symbol. Indeed, one
can estimate this integral over the region $\mathscr{R}^{c}$ by inserting the
cut-off $1-\phi_{-(1/2-\varepsilon)k}(\omega-\omega^{\prime})$ in the
integrand of $W_{x,k,z}(\xi^{\prime})^{\mathscr{R}^{c}}.$ Since on
$\mathscr{R}^{c},$ $|J(y,\omega)|\gg 2^{-\varepsilon k,}$ and in view of the
identity

$\nabla_{\omega}\Phi_{x,k,z}(\omega,\zeta,y)=\lambda^{\prime}(\omega-\omega^{\prime})\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega),$

one has that

$|\nabla_{\omega}\Phi_{x,k,z}(\omega,\zeta,y)|\gg
2^{k}|\omega-\omega^{\prime}|\gg 2^{\varepsilon k}2^{k/2}.$

Repeated integration by parts in the $\omega$-variable provides the factor
$2^{-\varepsilon k}2^{-k/2}$ but the differentiation of the term
$\phi_{-(1/2-\varepsilon)}$ gives the factor $2^{-\varepsilon k}2^{k/2}$ which
shows that this portion of the integral is $O(2^{-Ck})$ for any $k.$

Note that

$\displaystyle W_{x,k,z}(\xi^{\prime})^{\mathscr{R}}$
$\displaystyle=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\zeta)\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$ $\displaystyle=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$ $\displaystyle+$
$\displaystyle\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}O(2^{C\varepsilon
k}2^{-k/2})\psi(y,\omega)e^{-\mu(y,\omega)\pi i/4}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$
$\displaystyle=:W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}$

where

$\displaystyle W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}$
$\displaystyle:=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta,$

and

$\displaystyle W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}$
$\displaystyle:=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}O(2^{C\varepsilon
k}2^{-k/2})\psi(y,\omega)e^{-\mu(y,\omega)\pi i/4}$
$\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta.$

To evaluate $W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}$ we compute the
integral with respect to $\zeta$ using the distributional identity

$\smallint e^{2\pi i(x-y)\cdot\zeta}d\zeta=\delta(x-y),$

as follows

$\displaystyle W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}$
$\displaystyle=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$ $\displaystyle=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i[\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))+(x-y)\cdot\zeta]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta;$ $\displaystyle=\smallint\smallint_{\mathscr{R}}e^{2\pi
i[\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}\left(\smallint
e^{2\pi i(x-y)\cdot\zeta}d\zeta\right)d\omega dy$
$\displaystyle=\smallint\smallint_{\mathscr{R}}e^{2\pi
i[\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(y,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))]}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}\delta(x-y)dyd\omega$
$\displaystyle=\smallint_{\mathscr{R}}e^{2\pi
i\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(x,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(x,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}|J(x,\omega)|^{1/2}d\omega$
$\displaystyle=s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times\smallint_{\omega-\omega^{\prime}=O(2^{\varepsilon
k}2^{-k/2})}e^{2\pi
i\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(x,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}$ $\displaystyle\hskip 56.9055pt(1-\phi_{-\varepsilon
k}(J(x,\omega)))|J(x,\omega)|^{1/2}d\omega.$

Note that the integration with respect to $\omega$ is over the region
$\\{\omega-\omega^{\prime}=O(2^{\varepsilon k}2^{-k/2})\\}$ and then we can
estimate this integral by introducing a cutt-off
$\phi_{-(\frac{1}{2}-\varepsilon)k}(\omega-\omega^{\prime}).$ In accordance
with the principle of stationary phase, we now look at where the phase is
stationary in $\omega.$ From the homogeneity of the phase function we have
that

$\displaystyle\nabla_{\xi}\Phi_{\tau}(x,\omega)$
$\displaystyle=(\nabla_{\overline{\xi}}\Phi_{\tau}(x,(\omega,1)),\partial_{\xi_{n}}\Phi_{\tau}(x,(\omega,1)))$
$\displaystyle=(\nabla_{\omega}\Phi_{\tau}(x,\omega),\Phi_{\tau}(x,\omega)-\omega\cdot\nabla_{\omega}\Phi(x,\omega)).$

Hence

$\displaystyle\nabla_{\omega}(\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega))$
$\displaystyle=\nabla_{\omega}(\lambda^{\prime}\omega^{\prime}\cdot\nabla_{\omega}\Phi_{\tau}(x,\omega)+\lambda^{\prime}(\Phi_{\tau}(x,\omega)-\omega\cdot\nabla_{\omega}\Phi_{\tau}(x,\omega)))$
$\displaystyle=\lambda^{\prime}(\omega^{\prime}-\omega)\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega).$

So, the only stationary points occurs when $\omega=\omega^{\prime},$ and the
Hessian at this stationary point is given by

$\det(\nabla_{\omega}^{2}(\xi^{\prime}\cdot\nabla_{\xi}\Phi_{\tau}(x,\omega)))|_{\omega=\omega^{\prime}}=\det(-\lambda^{\prime}\nabla_{\omega}^{2}\Phi_{\tau}(x,\omega))|_{\omega=\omega^{\prime}}=(-\lambda^{\prime})^{n-1}J(x,\omega^{\prime}).$
(4.57)

Consequently, on the support of
$\phi_{-(\frac{1}{2}-\varepsilon)k}(\omega-\omega^{\prime}),$ we have the
Taylor series expansion

$\displaystyle 2\pi
i\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(x,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))=\frac{1}{2}\lambda^{\prime}\nabla_{\omega}^{2}\Phi(x,\omega^{\prime})(\omega-\omega^{\prime},\omega-\omega^{\prime})+O(2^{C\varepsilon}2^{-k/2}).$

Similarly, in view of the Taylor expansion

$\displaystyle\psi(x,\omega)e^{-\mu(x,\omega)\pi i/4}(1-\phi_{-\varepsilon
k}(J(x,\omega)))|J(x,\omega)|^{\frac{1}{2}}$
$\displaystyle=\psi(x,\omega^{\prime})e^{-\mu(x,\omega^{\prime})\pi
i/4}(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))|J(x,\omega^{\prime})|^{\frac{1}{2}}+O(2^{C\varepsilon}2^{-k/2}),$

we can write

$\displaystyle
s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}$
$\displaystyle\times\smallint_{\omega-\omega^{\prime}=O(2^{\varepsilon
k}2^{-k/2})}e^{2\pi
i\xi^{\prime}\cdot(\nabla_{\xi}\Phi_{\tau}(x,\omega)-\nabla_{\xi}\Phi_{\tau}(x,\xi^{\prime}))}\psi(x,\omega)e^{-\mu(x,\omega)\pi
i/4}$ $\displaystyle\hskip 56.9055pt(1-\phi_{-\varepsilon
k}(J(x,\omega)))|J(x,\omega)|^{\frac{1}{2}}d\omega$
$\displaystyle=s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}\psi(x,\omega^{\prime})e^{-\mu(x,\omega^{\prime})\pi
i/4}(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))|J(x,\omega^{\prime})|^{\frac{1}{2}}$
$\displaystyle\times\smallint_{\omega-\omega^{\prime}=O(2^{\varepsilon
k}2^{-k/2})}e^{\pi
i\lambda^{\prime}\nabla_{\omega}^{2}\Phi(x,\omega^{\prime})(\omega-\omega^{\prime},\omega-\omega^{\prime})}\phi_{-(\frac{1}{2}-\varepsilon)k}(\omega-\omega^{\prime})d\omega.$

In view of the stationary phase principle

$\displaystyle\smallint_{\omega-\omega^{\prime}=O(2^{\varepsilon
k}2^{-k/2})}e^{\pi
i\lambda^{\prime}\nabla_{\omega}^{2}\Phi(x,\omega^{\prime})(\omega-\omega^{\prime},\omega-\omega^{\prime})}\phi_{-(\frac{1}{2}-\varepsilon)k}(\omega-\omega^{\prime})d\omega$
$\displaystyle=e^{\mu(x,\omega^{\prime})\pi
i/4}|J(x,\omega^{\prime})|^{-\frac{1}{2}}(\lambda^{\prime})^{-(n-1)/2}+O(2^{-Ck}).$

Consequently,

$\displaystyle W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}$
$\displaystyle=\smallint\smallint\smallint_{\mathscr{R}}e^{2\pi
i\Psi_{x,k,z}(\omega,\zeta,y)}s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))\psi(y,\omega)e^{-\mu(y,\omega)\pi
i/4}$ $\displaystyle\times(1-\phi_{-\varepsilon
k}(J(y,\omega)))e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}|J(y,\omega)|^{1/2}d\omega
dyd\zeta$
$\displaystyle=s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}\psi(x,\omega^{\prime})e^{-\mu(x,\omega^{\prime})\pi
i/4}(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime})))|J(x,\omega^{\prime})|^{\frac{1}{2}}$
$\displaystyle+O(2^{C\varepsilon}2^{-k/2})$
$\displaystyle\times(e^{\mu(x,\omega^{\prime})\pi
i/4}|J(x,\omega^{\prime})|^{-\frac{1}{2}}(\lambda^{\prime})^{-(n-1)/2}+O(2^{-Ck}))$
$\displaystyle=s(x,\nabla_{x}\Phi_{\tau}(x,\xi^{\prime}))e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}(\lambda^{\prime})^{-(n-1)/2}\psi(x,\omega^{\prime})(1-\phi_{-\varepsilon
k}(J(x,\omega^{\prime}))$ $\displaystyle+O(2^{C\varepsilon}2^{-k/2}))$
$\displaystyle=W^{0}_{x,k,z}(\xi^{\prime})+O(2^{C\varepsilon}2^{-k/2})).$

Because $W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}$ captures the first term of
the Taylor expansion of the symbol $s(x,\zeta)$ around
$\zeta_{0}=\nabla_{z}\Phi_{\tau}(x,\xi^{\prime}),$ we have proved that
$W_{x,k,z}(\xi^{\prime})$ is essentially a symbol of order $-(n-1)/2.$
Moreover, the factor $e^{-(1+i\tau)\textnormal{Im}(\Phi(x,\xi^{\prime}))}\in
S^{0}_{1/2,1/2}$ in $W^{0}_{x,k,z}(\xi^{\prime})$ shows that

$W^{0}_{x,k,z}(\xi^{\prime}),\,W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}\in
S^{-\frac{n-1}{2}}_{1/2,1/2}.$

In consequence, the difference

$W_{x,k,z}(\xi^{\prime})-W^{0}_{x,k,z}(\xi^{\prime})\in S^{\mu}_{1/2,1/2}$

has order $\mu,$ where $\mu\leq-\frac{n-1}{2}.$ Now, what we want to prove is
that $\mu$ satisfies the inequality order
$\mu\leq-\frac{n-1}{2}-\frac{1}{2}+2\varepsilon.$ Since

$W_{x,k,z}(\xi^{\prime})=W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{1}}+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}^{c}}$
$=W^{0}_{x,k,z}(\xi^{\prime})+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}^{c}}\textnormal{
mod }S^{-\infty}$
$=W^{0}_{x,k,z}(\xi^{\prime})+W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}\textnormal{
mod }S^{-\infty}$

since $W_{x,k,z}(\xi^{\prime})^{\mathscr{R}^{c}}\in S^{-\infty}$ is a
smoothing symbol the order $\mu$ corresponds to the order of the symbol
$W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}.$ However, in view of (4.56), one
has the estimates

$\displaystyle|\partial_{\xi^{\prime}}^{\beta}O_{x,\xi^{\prime},\zeta}(2^{C\varepsilon
k}2^{-k/2})|\lesssim O(2^{C|\beta|\varepsilon
k}2^{-k/2}2^{-(n-1)k/2}2^{-k|\beta|}).$

Since $W_{x,k,z}(\xi^{\prime})^{\mathscr{R}_{2}}$ also contains the term
$e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}\in S^{0}_{1/2,1/2},$ the
amplitude $a(x,y,\xi^{\prime},\zeta):=O_{x,\xi^{\prime},\zeta}(2^{C\varepsilon
k}2^{-k/2})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}$ satisfies the
estimates

$|\partial_{\xi^{\prime}}^{\beta}[O_{x,\xi^{\prime},\zeta}(2^{C\varepsilon
k}2^{-k/2})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}]|\lesssim
O(2^{C|\beta|\varepsilon k}2^{-k/2}2^{-(n-1)k/2}2^{-k|\beta|/2}).$

The previous inequality can be written as

$|\partial_{\xi^{\prime}}^{\beta}[O_{x,\xi^{\prime},\zeta}(2^{C\varepsilon
k}2^{-k/2})e^{-(1+i\tau)\textnormal{Im}(\Phi(y,\xi^{\prime}))}]|\lesssim
O(2^{C_{\beta}\varepsilon k}2^{-k/2}2^{-(n-1)k/2}2^{-k|\beta|}),$

where

$C_{\beta}=|\beta|+|\beta|/2.$

In this way, the previous estimate says that, omitting the errors
$2^{C_{\beta}\varepsilon k},$
$W^{0}_{x,k,z}(\xi^{\prime})-W_{x,k,z}(\xi^{\prime})$ is essentially a symbol
of order $-\frac{(n-1)}{2}-\frac{1}{2}.$

Now, we have prepared the way in order to use Proposition 6.1 of [45] for
deducing the estimate (4.55). Indeed, our estimate in (4.55) is the analog of
(19) in [45, Page 19] which is a consequence of [45, Proposition 6.1]. For
completeness, we present such a proposition as follows.

######  Lemma 4.13.

If $V_{x,k,z}:=V_{x,k,z}(\xi^{\prime})$ is a symbol satisfying estimates of