#  Global solution of 3D anisotropic wave equations with the same speed in one
direction

Xuecheng Wang  YMSC, Tsinghua University & BIMSA
Beijing, China 100084<EMAIL_ADDRESS>
###### Abstract.

Motivated by the study of uniaxial crystal optics, we consider a coupled
system of $3D$ anisotropic wave equations, in which they have the same speed
only in one direction but distinct speeds in the other directions. Moreover,
this system has a $1D$-type null structure along the common direction.

Unlike the isotropic case, in which the celebrated Klainerman vector fields
method is very successful, the main difficulty of the anisotropic case lies in
the absence of enough joint commuting vector fields. After carefully analyzing
the differences between group velocities of two waves, which are delicate near
the same speed direction, the role of null structure, and a fine structure of
the phase of oscillation in time, we prove global stability and scattering for
small localized initial data.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation and the problem set-up
    2. 1.2 Previous results
    3. 1.3 The main result
    4. 1.4 Main ideas of proof and the plan of this paper
  2. 2 A bootstrap argument with a divided plan
    1. 2.1 Notation
    2. 2.2 The bootstrap assumption
    3. 2.3  $L^{\infty}_{x}$-decay estimate under the bootstrap assumption
    4. 2.4 Proof of the main theorem 1.1
  3. 3 Energy estimate and the $L^{\infty}_{\xi}$-estimate of profiles
    1. 3.1 A fixed time $Z$-norm estimate of nonlinearities
    2. 3.2 The $L^{\infty}_{\xi}$-estimate of profiles
  4. 4  The $Z$-norm estimate of profiles
    1. 4.1 The High $\times$ High type interaction
    2. 4.2 The High $\times$ Low type and the Low $\times$ High type interactions
    3. 4.3 The frequencies-comparable case

##  1\. Introduction

###  1.1. Motivation and the problem set-up

According to Courant-Hilbert [9][Chapter VI: section 3a] and Born-Wolf [6],
the system of crystal optics is Maxwell’s system in homogeneous media.
Moreover, from Born-Wolf[6][page 841], based on the optical properties, we can
classify transparent crystals into three distinct groups. Both group (II) and
group (III) are anisotropic crystals. More precisely,

  1. (I)

Cubic systems. Cubic system is optically isotropic. Dielectric constants are
all same, e.g., $\epsilon_{x}=\epsilon_{y}=\epsilon_{z}$. An example of cubic
systems is glass.

  2. (II)

Biaxial crystals. Dielectric constants are all different, e.g.,
$\epsilon_{x}\neq\epsilon_{y}\neq\epsilon_{z}$. An example of biaxial crystals
is biotite.

  3. (III)

Uniaxial crystals. Only two out of three dielectric constants are same, e.g.,
$\epsilon_{x}=\epsilon_{y}\neq\epsilon_{z}$. An example of uniaxial crystals
is quartz.

In the seminal work of Klainerman [25], Klainerman makes use of the conformal
structure of the Minkowski spacetime, more precisely, that is the existence of
a collection of vector fields commute with the wave operator, which allows him
to prove decay estimate of the nonlinear solution. This celebrated Klainerman
vector fields method has been very influential in the study of NLW for the
past 30 years. We will discuss more about history in the next subsection. Due
to the isotropic property of cubic systems, Klainerman vector field method is
very successful for the study of crystal optics in group (I).

Comparing with the rich literature in the study of the NLW with the same
speed, the study of anisotropic system of nonlinear waves is rather at the
preliminary stage. A general picture of thinking questions related to the
anisotropic system of NLW started with Fritz John, which in particular
includes questions on anisotropic elastic dynamics and the crystal optics. A
fascinating folklore program, which initiated by Sergiu Klainerman in 80s, is
to improve our understanding of anisotropic nonlinear wave system, in
particular the strength of vector field method in the case of without enough
symmetry. This folklore program includes at least the following questions,

  1. $\bullet$

The role of joint symmetries, which is much less than the isotropic case. In
particular, how to make use of joint symmetries even at the linear level? Due
to the absence of enough joint commuting vector fields, one don’t expect to
have an analogue of the Klainerman-Sobolev embedding for the anisotropic case,
but is there a good linear estimate that makes the most of joint symmetries,
e.g., the scaling vector field?

  2. $\bullet$

Global existence (Blow up) of anisotropic nonlinear wave system for small
data? It would be interesting to know that the linear estimate obtained in the
above question plays a role in the nonlinear problem.

Although the above questions are very general and far from complete, moreover,
the level of difficulty varies from case to case, there are many interesting
questions can be explored. To the best knowledge of the author, the first
result in this line of research is due to Otto Liess [31, 32], which initially
suggested to him by Klainerman. Comparing with the $t^{-1}$ decay rate in the
$3D$ isotropic case, the linear estimate obtained in [32][Theorem 1.3] for the
anisotropic case is only $t^{-1/2}.$ We refer readers to [32][page 2] right
after “Two difficulties then appear right away$\cdots$” for more detailed
explanation about this drastic difference.

Recently, there is a beautiful paper by John Anderson [3], in which he studies
the small data global stability problem for the 2D cubic transversal
anisotropic system of quasilinear wave equations. This system considered by
Anderson [3] is related to the crystal optics in group (II), i.e., biaxial
crystals.

The goal of this paper is to add an important missing piece to the picture,
which is the tangential case. Moreover, this is also related to the crystal
optics in group (III), i.e., uniaxial crystals. As mentioned in [40],
“optically uniaxial crystals are important materials used in fabricating
various optical elements and optical devices$\cdots$”.

Comparing this paper with [3], there are two main differences. Firstly, we
study 3D quadratic systems. Lastly and most importantly, we study the
tangential case instead of the transversal case.

The difference between the tangential case and the transversal is drastic,
subtle and technical. In terms of the space-time resonance language, there is
no space-time resonance set for the transversal case. However, the space-time
resonance set for the tangential case is not empty. In fact, the space-time
resonance set lies exactly in the direction of same speed or the tangent
hypersurface if there are more than one direction of same speed.

When light propagates in uniaxial crystals, there are two types of wave,
ordinary wave (o-wave) and extraordinary wave (e-wave). The normal surfaces of
these two waves are explained in Figure 1, see also Born-Wolf [6][section
15.3] for more physical backgrounds.

Figure 1. Normal surfaces of uniaxial crystals

We are interested in the question of nonlinear interaction between these two
types of wave. Motivated from the above discussion, in this paper, we consider
the following system of nonlinear wave (NLW) equations,

$\left\\{\begin{array}[]{l}\partial_{t}^{2}u_{a}-\Delta_{a}u_{a}=\mathcal{N}_{a}(\nabla_{t,x}u_{1},\nabla_{t,x}u_{2}),\quad
a\in\\{1,2\\},\\\
\Delta_{1}:=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}+\partial_{x_{3}}^{2},\quad\Delta_{2}:=\partial_{x_{1}}^{2}+c_{1}^{2}\partial_{x_{2}}^{2}+c_{2}^{2}\partial_{x_{3}}^{2},\\\
u_{a}=f_{a},\quad\partial_{t}u_{a}=g_{a},\quad a=1,2,\end{array}\right.$ (1.1)

where either $c_{1},c_{2}>1$ or $0<c_{1},c_{2}<1$. That’s to say, $u_{1}$ and
$u_{2}$ have the same speed, which is $1$, in $x_{1}$ direction and have
different speeds in the other two directions, i.e., $x_{2},x_{3}$ directions.

Since the goal of this paper is to study the interaction of nonlinear waves
with different speeds, for concreteness of discussion, we focus on the
$\mathcal{N}_{i}(\nabla_{t,x}u_{1},\nabla_{t,x}u_{2})$-type interaction and
rule out the self-interactions of $u_{1}$ and $u_{2}$ in the nonlinearities.

From the well-known finite time blow up result of John [20], we know that
certain null structure needs to be imposed to ensure the global existence of
the nonlinear solution. Since $u_{1}$ and $u_{2}$ have the same speed in
$x_{1}$ direction, to prevent the blow up mechanism due to lack of null
structure in $x_{1}$ direction, we impose a $1D$-type null structure in
$x_{1}$ direction.

More precisely, let

$\tilde{u}_{i}(t,x_{1}):=\int_{\mathbb{R}}\int_{\mathbb{R}}{u}_{i}(t,x_{1},x_{2},x_{3})dx_{2}dx_{3},\qquad
i\in\\{1,2\\},\quad\tilde{u}_{i}:\mathbb{R}_{t}\times\mathbb{R}\longrightarrow\mathbb{R}.$

It’s expected that, which will also be clear in later $L^{\infty}_{\xi}$-norm
analysis, $\nabla_{x_{2},x_{3}}$ plays the role of null structure. Hence, the
asymptotic system of (1.1) is determined the system satisfied by
$\tilde{u}_{i}(t,x_{1}),i\in\\{1,2\\}$ as follows,

$\partial_{t}^{2}\tilde{u}_{i}(t,x_{1})-\partial_{x_{1}}^{2}\tilde{u}_{i}(t,x_{1})=\mathcal{N}_{i}(\nabla_{t,x_{1}}\tilde{u}_{1},\nabla_{t,x_{1}}\tilde{u}_{2}),\quad
i\in\\{1,2\\}.$ (1.2)

To prevent the blow up mechanism of the asymptotic system (1.2) because of
lacking null structure, we impose $1D$-type null structure for the
nonlinearities
$N_{i}(\nabla_{t,x_{1}}\tilde{u}_{1},\nabla_{t,x_{1}}\tilde{u}_{2}),i=1,2.$
For simplicity, we consider $Q_{0,0}(\cdot,\cdot)$-type null structure for
$N_{i}(\nabla_{t,x_{1}}u_{1},\nabla_{t,x_{1}}u_{2})$ in this paper. For $1D$
NLW with $Q_{0,0}(\cdot,\cdot)$ type null structure, the global existence is
known, see Luli-Yang-Yu [37].

With the above discussion, we study the following system of equations, which
aims to reveal the subtle interaction of waves with different speeds in
general but with the same speed in only one direction,

$\left\\{\begin{array}[]{l}(\partial_{t}^{2}-\Delta_{1})u_{1}=\partial_{t}u_{1}\partial_{t}u_{2}-\partial_{x_{1}}u_{1}\partial_{x_{1}}u_{2}+\sum_{i,j\in\\{2,3\\}}\mathcal{Q}^{1}_{i,j}(\nabla_{x_{i}}u_{1},\nabla_{x_{j}}u_{2}),\\\
(\partial_{t}^{2}-\Delta_{2})u_{2}=\partial_{t}u_{1}\partial_{t}u_{2}-\partial_{x_{1}}u_{1}\partial_{x_{1}}u_{2}+\sum_{i,j\in\\{2,3\\}}\mathcal{Q}^{2}_{i,j}(\nabla_{x_{i}}u_{1},\nabla_{x_{j}}u_{2}),\\\
\Delta_{1}:=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}+\partial_{x_{3}}^{2},\quad\Delta_{2}:=\partial_{x_{1}}^{2}+c_{1}^{2}\partial_{x_{2}}^{2}+c_{2}^{2}\partial_{x_{3}}^{2},\\\
u_{a}=f_{a},\quad\partial_{t}u_{a}=g_{a},\quad a=1,2.\end{array}\right.$ (1.3)

where $\mathcal{Q}^{a}_{i,j},a\in\\{1,2\\},i,j\in\\{2,3\\},$ are bilinear
operators with zero order symbols $\mathfrak{q}^{a}_{i,j}(\cdot,\cdot),$ s.t.,
the following estimate holds for any $k,k_{1},k_{2}\in\mathbb{Z}$,

$\sup_{|\alpha|,|\beta|\leq
100}\sum_{a\in\\{1,2\\},i,j\in\\{2,3\\}}2^{|\alpha|k_{1}+|\beta|k_{2}}\|\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}\mathfrak{q}^{a}_{i,j}(\xi-\eta,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\|_{L^{\infty}_{\xi,\eta}}\lesssim
1.$ (1.4)

Despite we mainly work with the system (1.3) in this paper, we remark that the
Fourier method used in this paper is applicable for a larger class of
nonlinearities.

###  1.2. Previous results

For concreteness, nonlinear wave equations that we discuss in this subsection
usually have nonlinearities depending on the derivatives of solution. To
better understand the state of the art, we provide a short history on the
global stability problem of nonlinear waves equations and related results for
readers’ convenience. Also, we refer readers to the review by Anderson in [3]
for more discussion.

The study of the long time behavior of NLW with small initial data started a
long time ago, e.g., classic works of F. John. We refer readers to the review
by Klainerman in the proceeding of ICM 1983 [23] for more details. Despite
many previous results on long time behavior, a natural question is that
whether smooth solutions exist for all time for small initial data. The first
work that answers such question is due to Klainerman [22], in which he showed
that this is true at least for space dimensions greater than or equal to six.
Later a simplified proof was provided by Klainerman-Ponce in [28]. In the
seminal work [25], the celebrated Klainerman vector field method (as well as
the Klainerman-Sobolev embedding) was introduced, in which the symmetries of
the wave operator are exploited to prove decay estimate of the nonlinear
solution. As corollaries, small data global regularity result is improved to
four and five space dimensions. Moreover, it provides another proof of the
almost global result for three space dimensions case obtained in Klainerman
[24], and John-Klainerman [21].

Due to the critical decay rate of $3D$ wave, from the work of John[20], it’s
not expected to have global solution in general. In [23], null condition was
introduced by Klainerman, which eliminates the parallel interaction of
nonlinearity. As shown by Klainerman [26] and Christodoulou [7], for NLW with
null structure, global stability for small data indeed holds. A great
achievement of the vector field method lies in the monumental work of
Christodoulou-Klainerman [8], in which they proved global stability for the
Einstein-Vacuum equation around the Minkowski spacetime. A crucial step in
[26] is to identify certain kind of null condition inside the coupled system,
which is different from the classic null structures.

It turns out that the classic null condition is sufficient but not always
necessary to ensure the global existence of 3D NLW for small data. As first
shown by Lindblad [33] for the radial case and later by Alinac [2] for the
general case, the equation $\square\phi=\phi\Delta\phi$ has global solution
for small initial data. Very interestingly, this equation doesn’t have classic
null structure. In nowadays language, this equation has the so-called weak
null structure. For the Einstein-vacuum equation, Lindblad-Rodnianski [34, 35]
identified that it also has the weak null structure in the wave coordinates
system which allows them to give an alternative and conceptually simpler proof
of the stability of Minkowski spacetime in the wave coordinates system in
[35], see also Klainerman-Nicolò [27], Bieri[4]. This observation plays a key
role in later study of Einstein-scalar field systems, e.g., the study of
Einstein-Maxwell system in [43], the study of Einstein-Klein-Gordon system in
[18, 30, 41], the study of Einstein-Vlasov system in [5, 11, 36, 42].

However, for the anisotropic system of NLW, due to different speeds, the main
mathematical challenge is that we don’t enough symmetry. The first work that
only uses the scaling vector field and rotational vector fields was due to
Klainerman-Sideris [29]. A refinement was introduced by Sideris [39] to study
the prestressed nonlinear elastic waves. For the $2D$ transversal anisotropic
system of NLW, Anderson [3] performed a careful analysis of the spacetime
geometry together with the vector field method. One should expect that the
geometry would be more difficult in the $3D$ case and even more delicate for
the tangential case. For the system (1.1), as a result of computations, we
know that $S=t\partial_{t}+x\cdot\nabla_{x}$ and
$L_{1}:=x_{1}\partial_{t}+t\partial_{x_{1}}$ commute with the system (1.1) but
$\Omega_{12}$ and $\Omega_{13}$
($\Omega_{ij}:=x_{i}\partial_{x_{j}}-x_{j}\partial_{x_{i}}$) don’t commute
with the system (1.1). Generally speaking, the set of joint symmetry varies
from case to case for the study of crystal optics. However, due to the fact
that wave operators of different waves are all second order operators, we do
always have the scaling vector field. How to make use of the scaling vector
field appears to be a key.

Besides the vector field method, there is another approach for the small data
global regularity type problems. It mainly studies nonlinear solutions on the
Fourier side. Conceptually, the goal is same, which is to control the pull
back of the nonlinear solution along the linear flow over time. At the
technical level, it varies dramatically from case to case. For the past ten
years, this approach plays an important role in the study of nonlinear
dispersive PDEs and NLW.

Without trying to be exhaustive, we briefly mention two methods on the Fourier
approach here, which are the spacetime resonance method and the $Z$-norm
method. The spacetime resonance method was introduced by Germain-Masmoudi-
Shatah [12] in the study of NLS. Now, it has very wide applications in the
study of nonlinear dispersive equations, see e.g., [13, 14, 15, 16, 17, 18]
and the nonlinear wave equations, see [10, 38]. The designer $Z$-norm method
was firstly introduced by Ionescu-Pausader in [16]. This method is often used
together with the spacetime resonance method. It depends essentially on
identifying the “correct” $Z$-norm (often atomic space type norm), depending
on the problem, to prove sharp or almost sharp decay estimates for the
nonlinear solution.

Due to the internal structure of the system (1.3), it turns out that we are
able to consider the small data problem for the system (1.3) purely on the
Fourier side in this paper. But we believe that the combination of the Fourier
approach and the vector field method by exploiting the benefit of the joint
commuting vector fields has wider application in the study of crystal optics.
This is also our interest in the future. Using the strength of both methods is
not new, see e.g., [18, 42].

###  1.3. The main result

For convenience in studying the evolution of nonlinear waves on the Fourier
side, we define the following half-waves and their corresponding profiles as
follows,

$U_{a}:=(\partial_{t}-i\Lambda_{a})u_{a},\quad\Lambda_{a}:=\sqrt{-\Delta_{a}},\quad\forall
a\in\\{1,2\\}.$ (1.5)

Our main result is stated as follows,

######  Theorem 1.1.

Let
$N_{0}=10^{5},{}\delta\in(0,10^{-10}),\alpha=10^{-3},(c_{1},c_{2})\in(0,1)\times(0,1)\cup(1,\infty)\times(1,\infty)$.
There exists a small constant $\epsilon_{0}\in\mathbb{R}_{+}$ such that if
initial data $(f_{a},g_{a})\in H^{N_{0}}(\mathbb{R}^{3})$ satisfies the
following estimate,

$\sum_{a=1,2}\|U_{a}(0)\|_{H^{N_{0}}}+\|\langle
x\rangle^{1+\alpha}U_{a}(0)\|_{L^{2}_{x}}+\|\langle\xi\rangle^{8}\widehat{U}_{a}(0,\xi)\|_{L^{\infty}_{\xi}}\leq\epsilon_{0},\quad
U_{a}(0):=g_{a}-i\Lambda_{a}f_{a}.$ (1.6)

then the system (1.3) admits global solutions. The following estimate holds
for the nonlinear solution,

$\sup_{t\in[0,\infty)}\sum_{a=1,2}\langle
t\rangle^{-\delta}\|U_{a}(t)\|_{H^{N_{0}}}+\|\langle\xi\rangle^{8}U_{a}(t,\xi)\|_{L^{\infty}_{\xi}}+\langle
t\rangle\|U_{a}(t)\|_{W^{4,\infty}}\lesssim\epsilon_{0}.$ (1.7)

Moreover, the nonlinear solution of (1.3) scatters to a linear solution in
$L^{\infty}_{\xi}$-space as time goes to infinity.

A few remarks are in order.

######  Remark 1.1.

For the purpose of studying the anisotropic system of NLW in larger context,
e.g., with variable coefficients or in a curved spacetime, it would be
interesting to identify a good set of first order pseduo-differential
operators and second order pseduo-differential operators from the study of
this paper, which allows us to prove sharp (or almost sharp) decay estimates
for the nonlinear solution. This will be our future project. Moreover, it’s
also our interest to study the $3D$ anisotropic system of waves with the same
speed along two different directions and with different speeds along the other
direction.

######  Remark 1.2.

Since the method we use here only depends on the properties of symbols of
nonlinearities, the above theorem also holds for a class of anisotropic
systems of NLW. Moreover, to run the argument presented in this paper, we
don’t need null structure as strong as the null structure presented in (1.3).

######  Remark 1.3.

For the purpose of simplifying the presentation of theorem, we choose to use a
weighted $L^{2}$-type space for the initial data $U_{a}(0),a\in\\{1,2\\}$,
which is stronger than necessary. From our later bootstrap assumption (2.9),
it would be sufficient if the initial data $U_{a}(0),a\in\\{1,2\\}$ lies in
the atomic $Z$-normed space defined in (2.3).

######  Remark 1.4.

Last and also the least, the plausible goal of optimizing $N_{0},\alpha$,
etc., is not pursued here.

###  1.4. Main ideas of proof and the plan of this paper

Due to the critical $1/t$ decay rate of the $3D$ wave equation, by using the
standard energy estimate, the small data global regularity problem is reduced
to prove that the nonlinear solution indeed decays sharply over time. To this
end, in the following, we discuss main steps and main ideas in each step.

$\bullet$ Step (i) The pointwise linear decay estimate.

Since the small data global existence problem has been a subject under active
investigation for many years, there are many linear decay estimates available
in literature. We don’t try to give a complete list here. Instead, we give two
representative estimates. Firstly, we have the celebrated Klainerman-Sobolev
embedding, which states as follows in $3D$,

$|u(t,x)|\lesssim\sum_{|\alpha|\leq
2}\sum_{\Gamma\in\\{S,\Omega_{ij},L_{i}\\}}\frac{1}{(1+|t|+|x|)(1+||t|-|x||)^{1/2}}\|\Gamma^{\alpha}u(t,\cdot)\|_{L^{2}_{x}}.$

Moreover, we also have the following linear decay estimate by Pusateri-Shatah
[38][Lemma A.1], which doesn’t use any vector field,

$\|e^{it|\nabla|}f\|_{L^{\infty}}\lesssim\frac{1}{t}\|\langle
x\rangle|\nabla|^{2}f\|_{L^{2}}^{1/2}\|\langle
x\rangle^{2}|\nabla|^{2}f\|_{L^{2}}^{1/2},\quad|\nabla|:=\sqrt{-\Delta_{1}}.$

Generally speaking, choosing what kind of decay estimate to work with is the
very part of whole argument. Since one still needs to control the right hand
side over time to show that the nonlinear solution indeed decays sharply over
time. Whether such a goal can be achieved depends on the nonlinear system
itself.

For the anisotropic system (1.3), since we don’t have the full set of vector
fields and it doesn’t look promising to propagate a very high order weight for
the profile due to the presence of the nonempty spacetime resonance set, we
can’t use the above two estimates directly. Instead, roughly speaking, we use
the following linear decay estimate,

$\textup{(Rough version)}\qquad\|e^{\pm
it\Lambda_{a}}f\|_{L^{\infty}}\lesssim\frac{1}{t}\big{[}\||\xi|^{2}\widehat{f}(\xi)\|_{L^{\infty}_{\xi}}+\langle
t\rangle^{-\beta}\|\langle
x\rangle^{1+}|\nabla|^{(3/2-\beta)+}f\|_{L^{2}}\big{]},$ (1.8)

where $a\in\\{1,2\\}$ and $\beta$ is some absolute constant, see Lemma 2.1 for
the precise estimate.

$\bullet$ Step (ii) Reduction to the estimates of profiles.

To obtain decay estimate of the nonlinear solution, we pull back the nonlinear
solution along linear flow, which is called the profile, as follows

$\forall a\in\\{1,2\\},\quad
h_{a}(t)=e^{it\Lambda_{a}}U_{a},\quad\Longrightarrow
U_{a}(t)=e^{-it\Lambda}h_{a}(t).$ (1.9)

From the linear decay estimate (1.8), the goal of sharp decay is reduced as
follows,

$\textup{(Rough
version)}\quad\|U_{a}(t)\|_{L^{\infty}}\lesssim\frac{1}{t}\big{[}\underbrace{\||\xi|^{2}\widehat{h_{a}}(t,\xi)\|_{L^{\infty}_{\xi}}}_{\text{doesn't
grow over time }}+\langle t\rangle^{-\beta}\underbrace{\|\langle
x\rangle^{1+}|\nabla|^{(3/2-\beta)+}f\|_{L^{2}}}_{\text{only grow slightly
over time }}\big{]}.$ (1.10)

The weighted $L^{2}$-type space we actually use is the atomic type ($Z$-norm
space). More precisely, we view the profile as a superposition of different
atoms, which have different scales of localization in both the frequency space
and the physical space, see (2.3) for the detailed definition. This type of
$Z$-norm space was firstly introduced by Ionescu-Pausader [16] in the study of
Klein-Gordon equations.

We remark that, we reluctantly use a fractional weighted space because it
doesn’t promising to propagate $\langle x\rangle^{2}$ for the profiles over
time due to the nonempty spacetime resonance set, which will be discussed in
the next step.

$\bullet$ Step (iii) Exploiting properties of the spacetime resonance set.

As mentioned in the previous results, we refer readers to [12, 13, 14, 15, 16,
17, 18] for more details on the spacetime resonance method. At large scale,
the methodology is robust. However, at technical level, difficulties vary from
case to case.

Since $(c_{1},c_{2})\in(0,1)\times(0,1)\cup(1,\infty)\times(1,\infty)$, we
know that the group velocities of $u_{1}$ and $u_{2}$ are different as long as
one of their frequencies is not parallel to $(1,0,0)$, which is the direction
of two characteristic hypersurfaces tangential to each other. More precisely,
the spacetime resonance set is contained in the following set,

$\\{(\xi,\eta):\xi=\lambda(1,0,0),\eta=\mu(1,0,0),\lambda,\mu\in\mathbb{R}\\}.$

It’s the above structure motivates to consider the asymptotic system (1.2).
Moreover, we know that $(\xi_{2},\xi_{3})$ and $(\eta_{2},\eta_{3})$ are very
small when close to the spacetime resonance set. Therefore, one expects that
$\nabla_{x_{2},x_{3}}$ play the role of null structure.

Moreover, because of this observation, the dyadic decomposition we do for the
frequency is also anisotropic. More precisely, besides the size of $|\xi|$, we
also localize the size of $|\xi_{2}|$ and $|\xi_{3}|$.

$\bullet$ Step (iv) Splitting on the Fourier side.

To control the $L^{\infty}_{\xi}$-norm and the weighted $L^{2}$-type norm, we
use the Duhamel’s formula. The question is reduced to control bilinear forms
in different spaces. Because of the structure of the spacetime resonance set,
we split into the non-resonance case and the resonance case. Thanks to the
null structure presented in the system (1.3), it enables us to choose the
threshold of splitting on the Fourier side not too small. However, due to the
$1+\alpha$ order weight we are propagating for the $Z$-norm, the threshold
appears to be critical.

For the resonance case, we obtain fine expansion formulas for the phases, see
(4.49), which are crucial. The time resonance set is very delicate due to the
possible cancellation because of the opposite sign in the leading term in
(4.49). Fortunately, this happens only in the frequency comparable case. For
this case, we will do more sophisticated localization in subsection 4.3.
Roughly speaking, the key observation for the worst scenario is that the
frequency is localized inside a very thin annulus instead of a normal annulus
in which the difference between two radiuses is comparable to radiuses.

For the non-resonance case, we perform normal form transformation if away from
the time resonance set and exploiting the difference of group velocities if
away from the space resonance set.

The rest of this paper is organized as follows.

  1. $\bullet$

In section 2, we introduce notation and the bootstrap assumption and finish
the bootstrap argument with a divided plan.

  2. $\bullet$

In section 3, we finish the energy estimate part and the $L^{\infty}$-norm
estimate part. Moreover, we prove a fixed time $Z$-norm estimate for
nonlinearities, which play a role in later normal form transformation.

  3. $\bullet$

In section 4, we finish the $Z$-norm estimate part.

Acknowledgment This problem was introduced to me by my colleague Pin Yu, to
him, I am very grateful. I believe this problem was introduced to him by
Sergiu Klainerman. The author thanks Pin Yu for many helpful discussions. The
author acknowledges support from NSFC-12141102 and MOST-2020YFA0713003.

##  2\. A bootstrap argument with a divided plan

The plan of this section is listed as follows. Firstly, we introduce the
notation used throughout this paper. Secondly, we state our main bootstrap
assumptions and then obtain the $L^{\infty}$-decay estimate of nonlinear
solutions under the bootstrap assumption by using the linear decay estimate in
Lemma 2.1. Moreover, a $L^{1}$-estimate of kernel for general Fourier symbol
is also provided in Lemma 2.2. Lastly, we finish the bootstrap argument, hence
finishing the proof of our main theorem 1.1, by assuming the validity of
divided propositions in later sections.

###  2.1. Notation

For $\xi=(\xi_{1},\xi_{2},\xi_{3})\in\mathbb{R}^{3}$, we let
$\not{\xi}:=(\xi_{2},\xi_{3})\in\mathbb{R}^{2}.$ For any
$x\in\cup_{n\in\mathbb{Z}_{+}}\mathbb{R}^{n}$, we use the Japanese bracket
$\langle x\rangle$ to denote $(1+|x|^{2})^{1/2}$.

We fix an even smooth function $\tilde{\psi}:\mathbb{R}\rightarrow[0,1]$,
which is supported in $[-3/2,3/2]$ and equals to “$1$” in $[-5/4,5/4]$. For
any $k\in\mathbb{Z}$, we define the cutoff functions $\psi_{k},\psi_{\leq
k},\psi_{\geq
k}:\cup_{n\in\mathbb{Z}_{+}}\mathbb{R}^{n}\longrightarrow\mathbb{R}$ as
follows,

$\psi_{k}(x):=\tilde{\psi}(|x|/2^{k})-\tilde{\psi}(|x|/2^{k-1}),\quad\psi_{\leq
k}(x):=\tilde{\psi}(|x|/2^{k})=\sum_{l\leq k}\psi_{l}(x),\quad\psi_{\geq
k}(x):=1-\psi_{\leq k-1}(x).$

Let $P_{k}$, $P_{\leq k}$, and $P_{\geq k}$ be the Fourier multiplier
operators with symbols $\psi_{k}(\xi),\psi_{\leq k}(\xi),$ and $\psi_{\geq
k}(\xi)$ respectively. We use $f^{+}$ to denote $f$ and use $f^{-}$ to denote
$\bar{f}$. For $k\in\mathbb{Z}$, we use $f_{k}$ to denote $P_{k}f$.

For an integer $k\in\mathbb{Z}$, we use $k_{+}$ to denote $\max\\{k,0\\}$ and
use $k_{-}$ to denote $\min\\{k,0\\}$. For any unit vectors
$u,v\in\mathbb{S}^{2}$, we use $\angle(u,v)$ to denote the angle between $u$
and $v$ and use the convention that $\angle(u,v)\in[0,\pi]$. For any
$k\in\mathbb{Z}$, we define the following set of integers,

$\begin{split}\chi_{k}^{1}&:=\\{(k_{1},k_{2}):k_{1},k_{2}\in\mathbb{Z},|k_{1}-k_{2}|\leq
10,k\leq k_{1}+10\\},\\\
\chi_{k}^{2}&:=\\{(k_{1},k_{2}):k_{1},k_{2}\in\mathbb{Z},k_{1}\leq
k_{2}-10,|k-k_{2}|\leq 5\\},\\\
\chi_{k}^{3}&:=\\{(k_{1},k_{2}):k_{1},k_{2}\in\mathbb{Z},k_{2}\leq
k_{1}-10,|k-k_{1}|\leq 5\\}.\end{split}$ (2.1)

For any $k\in\mathbb{Z},j\in\mathbb{Z}_{+}$, we define the cutoff function
$\varphi_{j,k}:\cup_{n\in\mathbb{Z}_{+}}\mathbb{R}^{n}\longrightarrow\mathbb{R}$
as follows,

$\varphi_{j,k}=\left\\{\begin{array}[]{ll}\psi_{\leq-k}(x)&j=-k_{-}\\\
\psi_{j}(x)&j>-k_{-}.\end{array}\right.$ (2.2)

Correspondingly, we define the localization operators $Q_{j,k}$, $Q_{\leq
j,k}$ and $Q_{>j,k}$ as follows,

$Q_{j,k}f(x):=P_{k}\big{(}\varphi_{j,k}(\cdot)P_{k}f(\cdot)\big{)}(x),\quad
Q_{\leq
j,k}f(x):=\sum_{j^{\prime}\in[-k_{-},j]\cap\mathbb{Z}}Q_{j^{\prime},k}f(x),\quad
Q_{>j,k}:=P_{k}-Q_{\leq j,k}.$

From (2.2), the following partition of unity holds,

$Id=\sum_{k\in\mathbb{Z}}P_{k},\quad
P_{k}=\sum_{j\in[-k_{-},\infty)\cap\mathbb{Z}}Q_{j,k}$

For convenience, we use $f_{j,k}$, $f_{\leq j,k}$, and $f_{>j,k}$ to
abbreviate $Q_{j,k}f$, $Q_{\leq j,k}f$, and $Q_{>j,k}f$ respectively.

With the above notation, we define the following weighted atomic $Z$-normed
space as follows,

$\|f\|_{Z}:=\sup_{k\in\mathbb{Z}}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|Q_{j,k}f(x)\|_{L^{2}_{x}},\qquad\alpha:=10^{-3}.$
(2.3)

To measure the distance of $|c_{1}|,|c_{2}|$ with respect to $1$, we define

$\begin{split}c&:=\min\big{\\{}\inf\big{\\{}k:k\in\mathbb{Z},2^{k}\geq\min\\{|1-|c_{1}||,|1-|c_{2}||\\}\big{\\}},0\big{\\}},\\\
\tilde{c}&:=\max\\{\inf\big{\\{}k:k\in\mathbb{Z},2^{k}\geq\max\\{|c_{1}|,|c_{2}|\\}\big{\\}},0\\},\qquad
c\leq 0\leq\tilde{c}.\\\ \end{split}$ (2.4)

We remark that constants “$c$” and $\tilde{c}$ will not play much role in the
argument. Since we will measure the difference between different waves in
$x_{2},x_{3}$ directions, it’s convenient to use these constants to measure
the lower bound and the upper bound.

###  2.2. The bootstrap assumption

From (1.5) and (1.9), we can recover $u_{a},a\in\\{1,2\\},$ from the half-
waves $U_{a},a\in\\{1,2\\},$ as follows,

$\partial_{t}u_{a}=\sum_{\mu\in\\{+,-\\}}\frac{1}{2}U_{a}^{\mu},\quad
u_{a}=\sum_{\mu\in\\{+,-\\}}c_{\mu}\Lambda_{a}^{-1}U_{a}^{\mu},\quad
c_{\mu}=\frac{\mu}{2i},\quad U_{a}^{+}:=U_{a},\quad
U_{a}^{-}:=\overline{U_{a}}.$ (2.5)

Recall (1.3) and (1.5). On the Fourier side, after doing dyadic decomposition
for $\xi-\eta$ and $\eta$, we have

$\begin{split}\partial_{t}\widehat{h_{a}}(t,\xi)&=\sum_{\mu,\nu\in\\{+,-\\}}\sum_{k\in\mathbb{Z}}\sum_{(k_{1},k_{2})\in\chi_{k}^{1}\cup\chi_{k}^{2}\cup\chi_{k}^{3}}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi),\quad
a\in\\{1,2\\},\\\
\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1}^{\mu}}(t,\xi-\eta)\widehat{h_{2}^{\nu}}(t,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)d\eta,\end{split}$
(2.6)

where the oscillating phases $\Phi_{\mu,\nu}^{a}(\xi,\eta),a\in\\{1,2\\}$, are
defined as follows,

$\begin{split}\Phi_{\mu,\nu}^{a}(\xi,\eta)&:=\Lambda_{a}(\xi)-\mu\Lambda_{1}(\xi-\eta)-\nu\Lambda_{2}(\eta),\\\
\forall\xi\in\mathbb{R}^{3},\quad\Lambda_{1}(\xi)&:=\sqrt{\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}},\quad\Lambda_{2}(\xi):=\sqrt{\xi_{1}^{2}+c_{1}^{2}\xi_{2}^{2}+c_{2}^{2}\xi_{3}^{2}}.\end{split}$
(2.7)

and symbols
$q^{a}_{\mu,\nu}(\xi-\eta,\eta),\mu,\nu\in\\{+,-\\},a\in\\{+,-\\},$ are given
as follows,

$q^{a}_{\mu,\nu}(\xi-\eta,\eta)=\frac{1}{4}+c_{\mu}c_{\nu}\frac{(\xi_{1}-\eta_{1})}{\Lambda_{1}(\xi-\eta)}\frac{\eta_{1}}{\Lambda_{2}(\eta)}-\sum_{a\in\\{1,2\\},i,j\in\\{2,3\\}}\mathfrak{q}^{a}_{i,j}(\xi-\eta,\eta)\frac{c_{\mu}c_{\nu}(\xi_{i}-\eta_{i})\eta_{j}}{\Lambda_{1}(\xi-\eta)\Lambda_{2}(\eta)}.$
(2.8)

Let $\delta\in(0,10^{-10})$ be a small absolute constant. We make the
following bootstrap assumption,

$\begin{split}\sup_{t\in[0,T]}\sum_{a=1,2}\langle
t\rangle^{-\delta}&\|h_{a}(t)\|_{H^{N_{0}}}+\langle
t\rangle^{-\delta}\|h_{a}(t)\|_{Z}+\sup_{k\in\mathbb{Z}}2^{8k_{+}}\|\widehat{h_{a}}(t,\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}\lesssim\epsilon_{1}:=\epsilon^{5/6}_{0},\end{split}$
(2.9)

where the $Z$-normed space was defined in (2.3).

###  2.3.  $L^{\infty}_{x}$-decay estimate under the bootstrap assumption

Let $t\in[2^{m-1},2^{m}]\subset[0,T]$. From the Sobolev embedding and the
bootstrap assumption (2.9), we can rule out the case
$k\notin[-m/2,m/(N_{0}-12)]$ as follows,

$\begin{split}&\sum_{k\in\mathbb{Z},k\notin[-m/2,m/(N_{0}-12)]}2^{-k+5k_{+}}\|e^{-it\Lambda_{a}}P_{k}(h_{a}(t))\|_{L^{\infty}}\\\
&\lesssim\sum_{k\in\mathbb{Z},\notin[-m/2,m/(N_{0}-12)]}2^{-k+5k_{+}}\min\\{2^{3k}\|\widehat{h_{a}}(t,\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}},2^{3k/2}\|\widehat{h_{a}}(t,\xi)\psi_{k}(\xi)\|_{L^{2}_{\xi}}\\}\\\
&\lesssim 2^{-m}\epsilon_{1}.\end{split}$ (2.10)

It remains to consider the case $k\in[-m/2,m/(N_{0}-12)]\cap\mathbb{Z}$. For
this case, from the linear decay estimates (2.13) in Lemma 2.1, we have

$\|e^{-it\Lambda_{i}}P_{k}(h_{a}(t))\|_{L^{\infty}}\lesssim
2^{-m+k-5k_{+}}\epsilon_{1}.$ (2.11)

To sum up, after combining the obtained estimates (2.10) and (2.11), we have

$\sup_{t\in[0,T]}\sum_{k\in\mathbb{Z},a=1,2}2^{-k+5k_{+}}\langle
t\rangle\|e^{-it\Lambda_{a}}P_{k}(h_{a}(t))\|_{L^{\infty}}\lesssim\epsilon_{1}.$
(2.12)

In later study of the High $\times$ High type interaction, we will localize
$\xi$ inside a thin annulus. Hence, in the following Lemma, we provide a
super-localized version of the linear decay estimate.

######  Lemma 2.1.

Let $\delta\in(0,10^{-10})$ be some fixed absolute constant,
$a\in\\{1,2\\},t\in[2^{m-1},2^{m}]\subset\mathbb{R},x\in\mathbb{R}^{3},\mu\in\\{+,-\\}$,
$k\in\mathbb{Z}$, $n,\tilde{k}\in\mathbb{Z}$, s.t., $|t|\gg 1$, $\tilde{k}\leq
k$, $k,\tilde{k}\geq-m+\delta m$,
$n\in[2^{k-10-\tilde{c}},2^{k+10+\tilde{c}}]\cap\mathbb{Z}$, we have

$\begin{split}&\big{|}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}m(\xi)\widehat{f}(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)d\xi\big{|}\lesssim
2^{-m}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\big{[}2^{k+\tilde{k}+k_{+}}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}\\\
&\quad+2^{-(\alpha-4\delta)m/2+(\alpha+\delta)k/2}2^{(2k+\tilde{k})/2}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}\big{]},\end{split}$
(2.13)

where the Fourier multiplier operator $T^{a}_{\tilde{k},n}$ is defined by the
symbol $\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)$.

###### Proof.

Let $n\in[2^{k-10},2^{k+10}]$ be fixed. Note that, as a result of direct
computations, we have

$\forall a\in\\{1,2\\},\quad N_{a}(\xi):=\nabla_{\xi}\Lambda_{a}(\xi),\quad
N_{1}(\xi)=\frac{(\xi_{1},\xi_{2},\xi_{3})}{|\Lambda_{1}(\xi)|},\quad
N_{2}(\xi)=\frac{(\xi_{1},c_{1}^{2}\xi_{2},c_{2}^{2}\xi_{3})}{|\Lambda_{2}(\xi)|},\quad|N_{a}(\xi)|\sim
1.$

Based on the possible size of $|x|$, we split into two cases as follows.

$\bullet$ If $|x|\leq 2^{m-10}$. Note that, for this case, we have

$\big{|}\nabla_{\xi}\big{[}x\cdot\xi-\mu t\Lambda_{a}(\xi)\big{]}\big{|}\sim
2^{m}.$

Base on the possible size of $j$, we split into two parts as follows.

$\begin{split}\int_{\mathbb{R}^{3}}&e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}m(\xi)\widehat{f}(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)d\xi=H_{1}+H_{2},\\\
H_{1}&=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}m(\xi)\widehat{f_{<m-\delta
m,k}}(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)d\xi,\\\
H_{2}&=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}m(\xi)\widehat{f_{\geq m-\delta
m,k}}(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)d\xi.\end{split}$

Recall that $k,\tilde{k}\geq-m+\delta m$. Note that, by doing integration in
$\xi$ once for $H_{1}$, we gain at least $2^{-\delta m}$. Therefore, after
doing integration by parts in $\xi$ $\delta^{-2}$ times, the following
estimate holds from the volume of support of $\xi$,

$|H_{1}(t,x)|\lesssim 2^{-10m+2k+\tilde{k}}\|\widehat{f_{<m-\delta
m,k}}(\xi)\|_{L^{\infty}_{\xi}}\lesssim
2^{-10m+2k+\tilde{k}}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}.$
(2.14)

From the volume of support of $\xi$ and the Cauchy-Schwarz inequality, we have

$\begin{split}|H_{1}(t,x)|&\lesssim\sum_{j\geq m-\delta
m}2^{(2k+\tilde{k})/2}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\\\ &\lesssim
2^{-(1+\alpha)(1-\delta)m+(2k+\tilde{k})/2}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}.\end{split}$
(2.15)

After combining the obtained estimates (2.14) and (2.15), we have

$\begin{split}&\Big{|}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}m(\xi)\widehat{f}(\xi)\psi_{\tilde{k}}(\Lambda_{i}(\xi)-n)d\xi\big{|}\\\
&\lesssim
2^{-m}\big{[}2^{2k+\tilde{k}}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}+2^{-(\alpha-\delta-\alpha\delta)m+(2k+\tilde{k})/2}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}\big{]}.\end{split}$
(2.16)

$\bullet$ If $|x|\geq 2^{m-10}$. Note that, as a result of direct
computations, we have

$N_{a}(\xi)\times\nabla_{\xi}\big{(}x\cdot\xi-\mu
t\Lambda_{a}(\xi)\big{)}=N_{a}(\xi)\times x.$

Let $\bar{l}$ be the least integer such that $2^{\bar{l}}\geq
2^{-k/2}|x|^{-1/2}$. From the volume of support of $\xi$ and the Sobolev
embedding in angular variables, we have

$\Big{|}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}\widehat{f}(\xi)m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{\leq\bar{l}}(\tilde{x}\times
N_{a}(\xi))d\xi\Big{|}\lesssim
2^{\tilde{k}+2k+2\bar{l}}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}.$
(2.17)

For the case when the angle is localized around $2^{l}$ where $l>\bar{l}$, we
first do integration by parts in $\xi$ along directions perpendicular to
$N_{a}(\xi)$ once. As a result, after splitting $f$ into two parts based on
the size of $j$ when $\nabla_{\xi}$ hits $f$, we have

$\begin{split}&\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}\widehat{f}(\xi)m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))d\xi=I_{l}^{0}+I_{l}^{1}+I_{l}^{2},\\\
&I_{l}^{0}=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}\big{[}(N_{a}(\xi)\times\nabla_{\xi})\cdot\big{(}\frac{ix\times
N_{a}(\xi)}{|x\times
N_{a}(\xi)|^{2}}m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))\big{)}\big{]}\widehat{f}(\xi)d\xi,\\\
&I_{l}^{1}=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}\big{(}\frac{ix\times N_{a}(\xi)}{|x\times
N_{a}(\xi)|^{2}}\cdot(N_{a}(\xi)\times\nabla_{\xi})\widehat{f_{\leq
m+l,k}}(\xi)m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))d\xi,\\\ &I_{l}^{2}=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}\big{(}\frac{ix\times N_{a}(\xi)}{|x\times
N_{a}(\xi)|^{2}}\cdot(N_{a}(\xi)\times\nabla_{\xi})\widehat{f_{>m+l,k}}(\xi)m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))d\xi,\\\ \end{split}$ (2.18)

From the volume of support of $\xi$ and the Cauchy-Schwarz inequality, we have

$\begin{split}\sum_{l>\bar{l}}\big{|}I_{l}^{2}\big{|}&\lesssim\sum_{l>\bar{l}}\sum_{j\geq
m+l}|x|^{-1}2^{-l}2^{(2k+2l+\tilde{k})/2+j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\\\
&\lesssim\sum_{l>\bar{l}}\sum_{j\geq m+l}|x|^{-1}2^{(2k+\tilde{k})/2-\alpha
j}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}\\\
&\lesssim 2^{-(1+\alpha/2)m+\alpha
k/2}2^{(2k+\tilde{k})/2}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}.\end{split}$
(2.19)

For $I_{l}^{0}$ and $I_{l}^{1}$, we do integration by parts in $\xi$ along
directions perpendicular to $N_{a}(\xi)$ one more time. As a result, we have

$\begin{split}I_{l}^{0}=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}&(N_{a}(\xi)\times\nabla_{\xi})\cdot\Big{[}\frac{ix\times
N_{a}(\xi)}{|x\times
N_{a}(\xi)|^{2}}\big{[}(N_{a}(\xi)\times\nabla_{\xi})\cdot\big{(}\frac{ix\times
N_{a}(\xi)}{|x\times N_{a}(\xi)|^{2}}\\\ &\times
m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))\big{)}\widehat{f}(\xi)\big{]}\Big{]}d\xi,\\\
I_{l}^{1}=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi-i\mu
t\Lambda_{a}(\xi)}&(N_{a}(\xi)\times\nabla_{\xi})\cdot\Big{[}\frac{ix\times
N_{a}(\xi)}{|x\times N_{a}(\xi)|^{2}}\big{(}\frac{ix\times
N_{a}(\xi)}{|x\times
N_{a}(\xi)|^{2}}\cdot(N_{a}(\xi)\times\nabla_{\xi})\widehat{f_{\leq
m+l,k}}(\xi)\\\ &\times
m(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))\big{)}\big{]}d\xi.\end{split}$ (2.20)

From the volume of support of $\xi$, the Cauchy-Schwarz inequality, and the
Sobolev embedding on sphere $\mathbb{S}^{2}$, we have

$\begin{split}\sum_{l>\bar{l}}\big{|}I_{l}^{0}\big{|}&\lesssim\sum_{l>\bar{l}}|x|^{-2}2^{-2l}2^{-k-l}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\big{[}2^{-k-l}2^{(2k+2l+\tilde{k})}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}\\\
&+2^{(2k+2l+\tilde{k})/2}\|\nabla_{\xi}\widehat{f}(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\psi_{{l}}(\tilde{x}\times
N_{a}(\xi))\|_{L^{2}_{\xi}}\big{]}\\\ &\lesssim
2^{-m+k+\tilde{k}}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}+\sum_{l>\bar{l}}|x|^{-2}2^{-2l}2^{-k-l}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}\\\
&\times\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}2^{(2k+2l+\tilde{k})/2}\big{[}\sum_{j\in[-k_{-},m+l]\cap\mathbb{Z}}2^{k+(1-\delta)l+(1-\alpha)j}+\sum_{j\in[m+l,\infty)\cap\mathbb{Z}}2^{-\alpha
j}\big{]}\\\ &\lesssim
2^{-m}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\big{[}2^{k+\tilde{k}}\|\widehat{f}(\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}\\\
&+2^{-(\alpha-\delta)m/2+(\alpha+\delta)k/2}2^{(2k+\tilde{k})/2}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}\big{]}.\end{split}$
(2.21)

Similarly, from the volume of support of $\xi$, the Cauchy-Schwarz inequality,
and the Sobolev embedding on sphere $\mathbb{S}^{2}$, we have

$\begin{split}\sum_{l>\bar{l}}\big{|}I_{l}^{1}\big{|}&\lesssim\sum_{l>\bar{l}}|x|^{-2}2^{-2l}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\big{[}2^{-k-l}2^{(2k+2l+\tilde{k})/2}\|\widehat{f_{\leq
m+l,k}}(\xi)\psi_{{l}}(\tilde{x}\times N_{a}(\xi))\|_{L^{2}_{\xi}}\\\
&+2^{(2k+2l+\tilde{k})/2}\|\nabla_{\xi}\widehat{f_{\leq
m+l,k}}(\xi)\psi_{\tilde{k}}(\Lambda_{a}(\xi)-n)\|_{L^{2}_{\xi}}\big{]}\\\
&\lesssim\sum_{l>\bar{l}}\sum_{j\in[-k_{-},m+l]\cap\mathbb{Z}}2^{-2m-2l}2^{(2k+2l+\tilde{k})/2-\delta
l+(1-\alpha)j}\|m(\xi)\|_{\mathcal{S}^{\infty}_{k}}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}\\\
&\lesssim
2^{-m-(\alpha-\delta)m/2+(\alpha+\delta)k/2}2^{(2k+\tilde{k})/2}\big{(}\sup_{j\in[-k_{-},\infty)\cap\mathbb{Z}}2^{(1+\alpha)j}\|T^{a}_{\tilde{k},n}Q_{j,k}f\|_{L^{2}}\big{)}.\end{split}$
(2.22)

To sum up, our desired estimate (2.13) holds after combining the obtained
estimates (2.16), (2.17), (2.21), and (2.22). ∎

######  Lemma 2.2.

Let $\delta\in(0,10^{-10})$ be some fixed absolute constant. For any
$m(\xi,\eta)\in C^{4}(\mathbb{R}^{3}\times\mathbb{R}^{3})$, the following
estimate holds for the kernel of the symbol $m(\xi,\eta)$,

$\begin{split}\sup_{k,k_{1}\in\mathbb{R}^{3}}\|&\mathcal{F}^{-1}_{(\xi,\eta)\rightarrow(x,y)}\big{[}m(\xi,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\eta)\big{]}(x,y)\|_{L^{1}_{x,y}}\\\
&\lesssim\big{(}\sum_{|\alpha|,|\beta|\leq
3}\||\xi|^{\alpha}|\eta|^{\beta}\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}m(\xi,\eta)\psi_{[k-1,k+1]}(\xi)\psi_{[k_{1}-1,k_{1}+1]}(\eta)\|_{L^{\infty}_{\xi,\eta}}\big{)}^{1-\delta}\\\
&\times\big{(}\sum_{|\alpha|,|\beta|\leq
4}\||\xi|^{\alpha}|\eta|^{\beta}\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}m(\xi,\eta)\psi_{[k-1,k+1]}(\xi)\psi_{[k_{1}-1,k_{1}+1]}(\eta)\|_{L^{\infty}_{\xi,\eta}}\big{)}^{\delta}.\end{split}$
(2.23)

###### Proof.

Note that

$\begin{split}K_{k,k_{1}}(x,y)&=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+iy\cdot\eta}m(\xi,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\eta)d\xi
d\eta,\\\
(ix)^{\alpha}(iy)^{\beta}K_{k,k_{1}}(x,y)&=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+iy\cdot\eta}\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}\big{(}m(\xi,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\eta)\big{)}d\xi
d\eta.\end{split}$

From the above equality and the volume of support of $\xi,\eta$, we have

$\begin{split}\langle 2^{k}x\rangle^{3}\langle
2^{k_{1}}y\rangle^{3}\big{|}K_{k,k_{1}}(x,y)\big{|}&\lesssim\sum_{|\alpha|,|\beta|\leq
3}2^{3k+3k_{1}}\||\xi|^{\alpha}|\eta|^{\beta}\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}m(\xi,\eta)\psi_{[k-1,k+1]}(\xi)\psi_{[k_{1}-1,k_{1}+1]}(\eta)\|_{L^{\infty}_{\xi,\eta}},\\\
\langle 2^{k}x\rangle^{4}\langle
2^{k_{1}}y\rangle^{4}\big{|}K_{k,k_{1}}(x,y)\big{|}&\lesssim\sum_{|\alpha|,|\beta|\leq
4}2^{3k+3k_{1}}\||\xi|^{\alpha}|\eta|^{\beta}\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}m(\xi,\eta)\psi_{[k-1,k+1]}(\xi)\psi_{[k_{1}-1,k_{1}+1]}(\eta)\|_{L^{\infty}_{\xi,\eta}}.\\\
\end{split}$

Hence our desired estimate (2.23) holds after interpolating the above two
estimates. ∎

###  2.4. Proof of the main theorem 1.1

From the estimate (3.1) in Proposition 3.1, the estimate (4.1) in Proposition
4.1, and the estimate (3.18) in Proposition 3.2, we have

$\sup_{t\in[0,T]}\sum_{a=1,2}\langle
t\rangle^{-\delta}\|h_{a}(t)\|_{H^{N_{0}}}+\langle
t\rangle^{-\delta}\|h_{a}(t)\|_{Z}+\sup_{k\in\mathbb{Z}}\|\langle\xi\rangle^{8}\widehat{h_{a}}(t,\xi)\psi_{k}(\xi)\|_{L^{\infty}_{\xi}}\lesssim\epsilon_{0}+\epsilon_{1}^{2}\lesssim\epsilon_{0}.$

Hence the bootstrap assumption is improved, i.e., nonlinear solutions exist
globally. Moreover the desired estimate (1.7) holds from the above bootstrap
assumption and the obtained estimate (2.12).

As a byproduct, from the estimate (3.18) in Proposition 3.2, for any
$t\in[0,\infty)$, we have

$\sum_{a=1,2}\sup_{\xi\in\mathbb{R}^{3}}\Big{|}\widehat{U_{a}}(t,\xi)-e^{-it\Lambda_{a}(\xi)}\widehat{U_{\infty}^{a}}(\xi)\Big{|}\lesssim\langle
t\rangle^{-\delta},$
$\widehat{U_{\infty}^{a}}(\xi):=\widehat{U_{a}}(\xi)+\int_{0}^{\infty}e^{is\Lambda_{a}(\xi)}\sum_{\mu,\nu\in\\{+,-\\}}\sum_{k\in\mathbb{Z}}\sum_{(k_{1},k_{2})\in\chi_{k}^{1}\cup\chi_{k}^{2}\cup\chi_{k}^{3}}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(s,\xi)ds,$

where $\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(s,\xi)$ are defined in (2.6).
That’s to say, the nonlinear solution scatters to a linear solution in
$L^{\infty}_{\xi}$ as time goes to infinity.

##  3\. Energy estimate and the $L^{\infty}_{\xi}$-estimate of profiles

In this section, we first close the bootstrap assumption for the energy
estimate and the $L^{\infty}_{\xi}$-estimate, which are relatively easier than
the $Z$-norm estimate part. This allows us to focus on the $Z$-norm estimate
part for the rest of this paper. Moreover, as a preparation for readers for
more complicated analysis in the next section, we give a fixed time $Z$-norm
estimate for nonlinearities, which is not only useful when we do normal form
transformation in the next section but also contains some standard analysis.

Since the nonlinear solution has the sharp $1/t$ decay rate over time, the
energy estimate part is standard. More precisely, we have

######  Proposition 3.1.

Under the bootstrap assumption (2.9), for any
$t_{1},t_{2}\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, we have

$\sum_{a=1,2}\|h_{a}(t_{2})-h_{a}(t_{1})\|_{H^{N_{0}}}\lesssim 2^{\delta
m}\epsilon_{1}^{2}.$ (3.1)

###### Proof.

Recall (1.3) and (1.5). For any $t_{1},t_{2}\in[2^{m-1},2^{m}]\subset[0,T],$
s.t., $t_{1}\leq t_{2}$, we have

$\begin{split}\sum_{a=1,2}\|h_{a}(t_{2})-h_{a}(t_{1})\|_{H^{N_{0}}}\lesssim&\sum_{a=1,2}\sum_{i,j\in\\{2,3\\}}\int_{t_{1}}^{t_{2}}\|\partial_{t}u_{1}(t)\partial_{t}u_{2}(t)-\partial_{x_{1}}u_{1}(t)\partial_{x_{1}}u_{2}(t)\|_{H^{N_{0}}}\\\
&+\|\mathcal{Q}^{a}_{i,j}(\nabla_{x_{i}}u_{1},\nabla_{x_{j}}u_{2})\|_{H^{N_{0}}}dt.\end{split}$
(3.2)

After combining the above estimate with the $L^{\infty}$-decay estimate
(2.12), the estimate (1.4), and the $L^{2}-L^{\infty}$ type bilinear estimate,
we have

$\sum_{a=1,2}\|h_{a}(t_{2})-h_{a}(t_{1})\|_{H^{N_{0}}}\lesssim\int_{t_{1}}^{t_{2}}2^{-m+\delta
m}\epsilon_{1}^{2}dt\lesssim 2^{\delta m}\epsilon_{1}^{2}.$ (3.3)

Hence finishing the proof of our desired estimate (3.1). ∎

###  3.1. A fixed time $Z$-norm estimate of nonlinearities

In this subsection, we first prove a fixed time $Z$-norm estimate of
nonlinearities, which can be used directly later when we do normal form
transformation. Moreover, from the final estimate (3.4), we can see that,
without using the oscillation in time and careful analysis of phases based on
the sign, we are not off much from the goal of $\langle
t\rangle^{-1+\delta}\epsilon_{1}^{2}$, which is sufficient for closing the
bootstrap argument.

######  Lemma 3.1.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, $\mu,\nu\in\\{+,-\\}$,
we have

$\sup_{k\in\mathbb{Z}}\sum_{a=1,2}\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\|\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)\big{]}\|_{Z}\lesssim
2^{-3m/4+(\alpha+10\delta)m}\epsilon_{1}^{2}.$ (3.4)

###### Proof.

Recall (2.6). Note that, after doing space localization for two inputs, we
have

$\begin{split}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)\big{]}&=\sum_{j_{1}\in[-k_{1,-},\infty)\cap\mathbb{Z},j_{2}\in[-k_{2,-},\infty)\cap\mathbb{Z}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]},\\\
\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}&=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)d\eta
d\xi.\end{split}$ (3.5)

Based on the possible size of $x$, i.e., the size of $j$, we split into two
cases as follows.

$\bullet$ If $j\geq\max\\{m,-k_{-}\\}+\delta(m+j).$

Based on the possible sizes of $j_{1},j_{2}$, we split into two sub-cases as
follows.

$\oplus$ If $\min\\{j_{1},j_{2}\\}\geq(1-\delta)j.$

For this case, from the $L^{2}-L^{\infty}$ type bilinear estimate and the
$L^{\infty}\rightarrow L^{2}$-type Sobolev embedding, we have

$\begin{split}&\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\min\\{j_{1},j_{2}\\}\geq(1-\delta)j}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\min\\{j_{1},j_{2}\\}\geq(1-\delta)j}2^{(1+\alpha)j}\min\big{\\{}\|h_{1;j_{1},k_{1}}^{\mu}(t,x)\|_{L^{2}}\\\
&\times\|h_{2;j_{2},k_{2}}^{\mu}(t,x)\|_{L^{\infty}_{x}},\|h_{1;j_{1},k_{1}}^{\mu}(t,x)\|_{L^{\infty}_{x}}\|h_{2;j_{2},k_{2}}^{\mu}(t,x)\|_{L^{2}_{x}}\big{\\}}\\\
&\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\min\\{j_{1},j_{2}\\}\geq(1-\delta)j}2^{(1+\alpha)j}\big{[}2^{-(1+\alpha)j_{1}+\delta
m}2^{3k_{2}/2}\\\ &\times\min\\{2^{-N_{0}k_{2,+}+\delta
m},2^{-(1+\alpha)j_{2}+\delta m}\\}+2^{-(1+\alpha)j_{2}+\delta
m}2^{3k_{1}/2}\min\\{2^{-N_{0}k_{1,+}+\delta m},2^{-(1+\alpha)j_{1}+\delta
m}\\}\big{]}\epsilon_{1}^{2}\\\
&\lesssim\sum_{\min\\{j_{1},j_{2}\\}\geq(1-\delta)j}2^{(1+\alpha)j}2^{-(1+\alpha)j_{2}-(1+\alpha)j_{1}+3\delta
m}\lesssim 2^{-m-10\delta m}\epsilon_{1}^{2}.\\\ \end{split}$ (3.6)

$\oplus$ If $\min\\{j_{1},j_{2}\\}\leq(1-\delta)j.$

Without loss of generality, we assume that $j_{1}=\min\\{j_{1},j_{2}\\}.$
Otherwise, we change coordinates $(\xi,\eta)\longrightarrow(\xi,\xi-\eta)$ to
switch the roles of $\xi-\eta$ and $\eta$. Note that, for the case we are
considering, we have

$|\nabla_{\xi}\big{[}x\cdot\xi+t\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{]}|\sim
2^{j}.$ (3.7)

Therefore, by doing integration by parts in $\xi$ once, we gain $2^{-j}$ and
lose at most $\max\\{2^{j_{1}},2^{-k_{1}},2^{-k}\\}$. Therefore, we gain at
least $2^{-\delta j}$ in this process. After doing integration by parts in
“$\xi$” $\delta^{-2}$ times, the following estimate holds from the
$L^{2}-L^{\infty}$ type bilinear estimate and the $L^{\infty}\rightarrow
L^{2}$-type Sobolev embedding,

$\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\min\\{j_{1},j_{2}\\}\leq(1-\delta)j\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\min\\{j_{1},j_{2}\\}\leq(1-\delta)j\\\
\end{subarray}}2^{-100j}2^{-(1+\alpha)j_{1}+\delta
m}2^{3k_{2}/2}\min\\{2^{-N_{0}k_{2,+}+\delta m},2^{-(1+\alpha)j_{2}+\delta
m}\\}\epsilon_{1}^{2}$
$\lesssim\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\end{subarray}}2^{-10j-j_{1}-j_{2}}\epsilon_{1}^{2}\lesssim
2^{-10m}\epsilon_{1}^{2}.$ (3.8)

$\bullet$ If $j\leq\max\\{m,-k_{-}\\}+\delta(m+j).$

Note that, for this case, we have
$j\leq(1-\delta)^{-1}\big{(}\max\\{m,-k_{-}\\}+\delta m\big{)}$. As a result
of direct computations, we have

$\big{|}\nabla_{\eta}\big{(}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{)}\big{|}=\big{|}-\mu\frac{\eta-\xi}{|\xi-\eta|}-\nu\frac{(\eta_{1},c_{1}^{2}\eta_{2},c_{2}^{2}\eta_{3})}{\Lambda_{2}(\eta)}\big{|}.$
(3.9)

Moreover,

$\big{|}1-\frac{|(\eta_{1},c_{1}^{2}\eta_{2},c_{2}^{2}\eta_{3})|}{\Lambda_{2}(\eta)}\big{|}\sim\frac{|\not{\eta}|^{2}}{|\eta|^{2}},\quad\big{|}\frac{(\eta_{1},0,0)}{\sqrt{\eta_{1}^{2}+c_{1}^{4}\eta_{2}^{2}+c_{2}^{4}\eta_{3}^{2}}}-(sgn(\eta_{1}),0,0)\big{|}\gtrsim\frac{|\not{\eta}|^{2}}{|\eta|^{2}}$

From above estimates, $\forall a\in\\{1,2\\},$ we have

$\big{|}\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{|}\gtrsim\frac{|\not{\eta}|^{2}}{|\eta|^{2}}+\big{|}\mu\frac{\xi_{1}-\eta_{1}}{|\xi_{1}-\eta_{1}|}-\frac{\nu\eta_{1}}{|\eta_{1}|}\big{|}+\frac{|\not{\xi}-\not{\eta}|^{2}}{|\xi-\eta|^{2}}+\big{|}\frac{(\xi_{2}-\eta_{2},\xi_{3}-\eta_{3})}{|\xi_{1}-\eta_{1}|}-\mu\nu\frac{(c_{1}^{2}\eta_{2},c_{2}^{2}\eta_{3})}{|\eta_{1}|}\big{|}:=l_{\mu,\nu}(\xi,\eta).$
(3.10)

After doing dyadic decomposition for the size of $l_{\mu,\nu}(\xi,\eta)$ with
threshold $-m/2+\delta m$, we decompose
$\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$ further as
follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)&=\sum_{l\in[-m/2+\delta
m,\tilde{c}+10]\cap\mathbb{Z}}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi),\\\
\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))d\eta,\end{split}$ (3.11)

where the absolute constant $\tilde{c}$ is defined in (2.4).

$\oplus$ The case
$l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}.$

Recall (3.11). From (3.10), we have

$\forall
a\in\\{1,2\\},\quad\big{|}\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{|}\gtrsim
1.$ (3.12)

From the above estimate, by doing integration by parts in $\eta$ many times,
we can rule out the case $\max\\{j_{1},j_{2}\\}\leq(1-\delta)m$. It would be
sufficient to consider the case $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m$.

From the decay estimate (2.12), the $L^{2}-L^{\infty}$-type bilinear estimate
if $k\geq-m,$ and the volume of support of $\xi$ if $k\leq-m$, we have

$\begin{split}&\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{(1+\alpha)j}2^{-(1+\alpha)\max\\{j_{1},j_{2}\\}}2^{-8(\min\\{k_{1},k_{2}\\})_{+}+3\delta
m}\\\ &\times\min\\{2^{3k/2},2^{-m}\\}\epsilon_{1}^{2}\lesssim 2^{-m+10\delta
m}\epsilon_{1}^{2}.\end{split}$ (3.13)

$\oplus$ The case $l\in[-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}.$

Recall (2.8), (3.9), and the definition of $l_{\mu,\nu}(\xi,\eta)$ in (3.10).
From the estimate (1.4), the following estimate holds for the localized
symbol,

$\big{|}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\big{|}\lesssim
2^{l}.$ (3.14)

We first consider the threshold case $l=-m/2+\delta m$. After using the
$L^{2}-L^{\infty}$ type bilinear estimate and the volume of support of
frequency for the input putted in $L^{2}$, from the estimate (3.14), we have

$\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}2^{(1+\alpha)j}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}$
$\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}2^{l+(1+\alpha)j}2^{\min\\{k_{1},k_{2}\\}/2+l/2}2^{-m+\max\\{k_{1},k_{2}\\}}2^{-8(k_{1,+}+k_{2,+})}\epsilon_{1}^{2}\lesssim
2^{-3m/4+(\alpha+10\delta)m}\epsilon_{1}^{2}.$ (3.15)

Now, we focus on the estimate the non-threshold case, i.e., $l\in(-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$. From (3.10), we have

$\forall
a\in\\{1,2\\},\quad\big{|}\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{|}\gtrsim
2^{l}.$ (3.16)

From the above estimate, by doing integration by parts in “$\eta$” many times,
we can rule out the case $\max\\{j_{1},j_{2}\\}\leq(1-\delta)m+l$. It would be
sufficient to consider the case $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l$. From
the decay estimate (2.12) and the $L^{2}-L^{\infty}$-type bilinear estimate if
$k\geq-m,$ and the volume of support of $\xi$ if $k\leq-m$, we have

$\begin{split}&\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{(1+\alpha)j+l}2^{-(1+\alpha)\max\\{j_{1},j_{2}\\}}2^{-8\min\\{k_{1},k_{2}\\}+3\delta
m}\\\ &\times\min\\{2^{3k/2},2^{-m}\\}\epsilon_{1}^{2}\lesssim 2^{-m+\alpha
m+10\delta m}\epsilon_{1}^{2}.\end{split}$ (3.17)

After summing up with respect to $l$, which only causes a logarithmic loss,
our desired estimate (3.4) holds from the above obtained estimate, (3.15),
(3.13), and (3.8). ∎

###  3.2. The $L^{\infty}_{\xi}$-estimate of profiles

######  Proposition 3.2.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, $\mu,\nu\in\\{+,-\\}$,
we have

$\sum_{a=1,2}\|\langle\xi\rangle^{8}\partial_{t}\widehat{h_{a}}(t,\xi)\|_{L^{\infty}_{\xi}}\lesssim
2^{-m-\delta m}\epsilon_{1}^{2}.$ (3.18)

###### Proof.

Recall (2.6). Same as what we did in (3.5) and (3.11), we do space
localization for two inputs and decompose
$\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$ into
$\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$ based on the
size of $l_{\mu,\nu}(\xi,\eta)$. Moreover, from the Cauchy-Schwarz inequality,
we have

$\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i},\max\\{k_{1},k_{2}\\}\geq
5m/N_{0}}\|\langle\xi\rangle^{8}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)\|_{L^{\infty}_{\xi}}$
$\lesssim\sum_{(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i},\max\\{k_{1},k_{2}\\}\geq
5m/N_{0}}2^{8k_{+}}2^{3\min\\{k_{1},k_{2}\\}/2-N_{0}\max\\{k_{1},k_{2}\\}}\epsilon_{1}^{2}\lesssim
2^{-2m}\epsilon_{1}^{2}.$ (3.19)

Now, it would be sufficient to consider the case $\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}.$

$\oplus$ The case
$l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}.$

Recall (3.11). As in the proof of Lemma 3.1, by doing integration by parts in
$\eta$ many times, we can rule out the case
$\max\\{j_{1},j_{2}\\}\leq(1-\delta)m$. It would be sufficient to consider the
case $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m$. From the Cauchy-Schwarz
inequality, we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}\|\langle\xi\rangle^{8}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\|_{L^{\infty}_{\xi}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{8k_{+}-(1+\alpha)\max\\{j_{1},j_{2}\\}+3\min\\{k_{1},k_{2}\\}/2+\delta
m}\epsilon_{1}^{2}\lesssim 2^{-m-10\delta m}\epsilon_{1}^{2}.\end{split}$
(3.20)

$\oplus$ The case $l\in[-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}.$

We first consider the threshold case $l=-m/2+\delta m.$ For this case, we use
the volume of support of frequencies variable. Since the definition of
$l_{\mu,\nu}(\xi,\eta)$, see (3.10), is a little tricky, we elaborate how to
estimate the volume of support as follows. From (3.10), for any $(\xi,\eta)\in
supp(\varphi_{l;-m/2+\delta m}(l_{\mu,\nu}(\xi,\eta))$
$\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta))$, we have

$\xi_{2}=(1+\frac{\mu\nu
c_{1}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})\eta_{2}+\mathcal{O}(2^{\max\\{k_{1},k_{2}\\}+l}),\quad\xi_{3}=(1+\frac{\mu\nu
c_{2}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})\eta_{3}+\mathcal{O}(2^{\max\\{k_{1},k_{2}\\}+l}).$
(3.21)

For any fixed $\xi\in\mathbb{R}^{3}$, if $\mu\nu=+$ or
$k\leq\max\\{k_{1},k_{2}\\}-100-2\max\\{\tilde{c},-c\\}$ or
$\min\\{k_{1},k_{2}\\}\leq k-100-2\max\\{\tilde{c},-c\\}$, we have

$\big{|}\\{\eta:|l_{\mu,\nu}(\xi,\eta)|\leq 2^{l},|\eta|\sim
2^{k_{2}},|\xi-\eta|\sim 2^{k_{1}},|\xi|\sim 2^{k}\\}\big{|}\lesssim
2^{\min\\{k_{1},k_{2}\\}+2\max\\{k_{1},k_{2}\\}+2l}.$ (3.22)

It remains to consider the case $\mu\nu=-$, $|\xi|\sim|\eta|\sim|\xi-\eta|$.
For this case, after localizing the size of
$|\eta_{1}|-c_{1}^{2}|\xi_{1}-\eta_{1}|$, from (3.21), we have

$\begin{split}&\big{|}\\{\eta:|l_{\mu,\nu}(\xi,\eta)|\leq 2^{l},|\eta|\sim
2^{k_{2}},|\xi-\eta|\sim 2^{k_{1}},|\xi|\sim
2^{k},||\eta_{1}|-c_{1}^{2}|\xi_{1}-\eta_{1}||\sim 2^{b}\\}\big{|}\\\
&\lesssim\min\\{2^{b}(2^{-b+2k+l})^{2},2^{b}2^{2k+l}\\}.\end{split}$

From the above estimate, we have

$\begin{split}&\big{|}\\{\eta:|l_{\mu,\nu}(\xi,\eta)|\leq 2^{l},|\eta|\sim
2^{k_{2}},|\xi-\eta|\sim 2^{k_{1}},|\xi|\sim 2^{k}\\}\big{|}\lesssim
2^{3k+3l/2}.\end{split}$ (3.23)

To sum up, in whichever case, the following estimate holds for any fixed
$\xi\in\mathbb{R}^{3}$,

$\big{|}\\{\eta:|l_{\mu,\nu}(\xi,\eta)|\leq 2^{l},|\eta|\sim
2^{k_{2}},|\xi-\eta|\sim 2^{k_{1}},|\xi|\sim 2^{k}\\}\big{|}\lesssim
2^{\min\\{k_{1},k_{2}\\}+2\max\\{k_{1},k_{2}\\}+3l/2}.$ (3.24)

From the estimate of the symbol in (3.14), the estimate of the volume of the
support of $\eta$ in (3.24), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}\end{subarray}}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\end{subarray}}\langle\xi\rangle^{8}\mathcal{J}_{k;k_{1},k_{2};l}^{\mu,\nu;j_{1},j_{2}}(t,\xi)\|_{L^{\infty}_{\xi}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}\end{subarray}}2^{8k_{+}+l}2^{\min\\{k_{1},k_{2}\\}+2\max\\{k_{1},k_{2}\\}+3l/2}\epsilon_{1}^{2}\lesssim
2^{-m-10\delta m}\epsilon_{1}^{2}.\end{split}$ (3.25)

Now, we focus on the non-threshold case, i.e, $l\in(-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}.$ As in the proof of Lemma 3.1,
by doing integration by parts in $\eta$ many times, we can rule out the case
$\max\\{j_{1},j_{2}\\}\leq(1-\delta)m+l$. It would be sufficient to consider
the case $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l$.

From the Cauchy-Schwarz inequality, the estimate of kernel in (3.14), the
estimate of the volume of the support of $\eta$ in (3.24), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}\|\langle\xi\rangle^{8}\mathcal{J}_{k;k_{1},k_{2};l}^{\mu,\nu;j_{1},j_{2}}(t,\xi)\|_{L^{\infty}_{\xi}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\cup_{i=1,2,3}\chi_{k}^{i}\\\
\max\\{k_{1},k_{2}\\}\leq
5m/N_{0}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{l}2^{8k_{+}-(1+\alpha)\max\\{j_{1},j_{2}\\}+\delta m}\\\
&\times\big{(}2^{\min\\{k_{1},k_{2}\\}+2\max\\{k_{1},k_{2}\\}+3l/2}\big{)}^{1/2}\epsilon_{1}^{2}\lesssim
2^{-m-10\delta m}\epsilon_{1}^{2}.\\\ \end{split}$ (3.26)

Hence finishing the desired estimate (3.18). ∎

##  4\.  The $Z$-norm estimate of profiles

The rest of this section is devoted to prove the following proposition, which
is the main $Z$-norm estimate part. A divided plan is provided in the proof of
this Proposition.

######  Proposition 4.1.

Under the bootstrap assumption (2.9), for any
$t_{1},t_{2}\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, we have

$\sum_{a=1,2}\|h_{a}(t_{2})-h_{a}(t_{1})\|_{Z}\lesssim 2^{\delta
m}\epsilon_{1}^{2}.$ (4.1)

###### Proof.

Recall (2.6). The desired estimate (4.1) holds after combining (4.2) in Lemma
4.1, (4.11) in Lemma 4.2, (4.33) and (4.34) in Lemma 4.3, (4.40) and 4.41 in
Lemma 4.4, and (4.59) in Lemma 4.5. ∎

###  4.1. The High $\times$ High type interaction

In this subsection, we consider the High $\times$ High type interaction with
the output frequency much smaller than the input frequencies, e.g., $k\leq
k_{1}-5\max\\{\tilde{c},-c\\}-100$ and $|\xi|\ll|\eta|$, where absolute
constants $\tilde{c}$ and $c$ are defined in (2.4).

In the following Lemma, we first consider the non-resonance case, i.e.,
$\mu\nu=+.$

######  Lemma 4.1.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, $\mu,\nu\in\\{+,-\\}$,
s.t., $\mu\nu=+,$ we have

$\sup_{k\in\mathbb{Z}}\sum_{(k_{1},k_{2})\in\chi_{k}^{1},k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}}\|\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)\big{]}dt\|_{Z}\lesssim
2^{-\alpha m/4}\epsilon_{1}^{2}.$ (4.2)

###### Proof.

Recall (2.6). Same as what we did in (3.5), we also do space localization for
two inputs. As in the proof of Lemma 3.1, we split into two cases based on the
possible size of $x$, i.e., the size of $j$.

Since the obtained estimates (3.6) and (3.8) are still valid, it would be
sufficient to consider the case $j\leq\max\\{m,-k_{-}\\}+\delta(m+j).$

Note that, for this case, we have
$j\leq(1-\delta)^{-1}\big{(}\max\\{m,-k_{-}\\}+\delta m\big{)}$. Moreover,
from the fact that $|\xi|\ll|\eta|$, we have
$sgn(\eta_{1}-\xi_{1})=sgn(\eta_{1})$ if
$|\not{\xi}-\not{\eta}|\ll|\xi-\eta|$. As a result, from the estimate (3.10),
we have

$\big{|}\nabla_{\eta}\big{(}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{)}\big{|}\gtrsim\frac{|\not{\eta}|^{2}}{|\eta|^{2}}+|\frac{(\eta_{1}-\xi_{1},\eta_{2}-\xi_{2},\eta_{3}-\xi_{3})}{|\xi-\eta|}+\frac{(\eta_{1},c_{1}^{2}\eta_{2},c_{2}^{2}\eta_{3})}{\sqrt{\eta_{1}^{2}+c_{1}^{2}\eta_{2}^{2}+c_{2}^{2}\eta_{3}^{2}}}|\gtrsim
1.$ (4.3)

Based on the possible size of $\max\\{j_{1},j_{2}\\}$, we split into two sub-
cases as follows.

$\oplus$ If $\max\\{j_{1},j_{2}\\}\leq(1-\delta)m$.

Note that, from the estimate (4.3), by doing integration by parts in $\eta$
once, we gain $2^{-m}$ and lose at most
$\max\\{2^{-k_{1}},2^{\max\\{j_{1},j_{2}\\}}\\}=2^{\max\\{j_{1},j_{2}\\}}.$
Therefore, by doing integration by parts in $\eta$ once, we gain at least
$2^{-\delta m}$. After doing integration by parts in “$\eta$” $\delta^{-2}$
times, the following estimate holds from using either the $L^{2}-L^{\infty}$
type bilinear estimate and the $L^{\infty}\rightarrow L^{2}$-type Sobolev
embedding or using the $L^{2}\rightarrow L^{1}$ type Sobolev embedding first
and then using the $L^{2}-L^{2}$ type bilinear estimate,

$\sum_{(k_{1},k_{2})\in\chi_{k}^{1}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq(1-\delta)m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{(k_{1},k_{2})\in\chi_{k}^{1}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq(1-\delta)m\\\
\end{subarray}}2^{-100m}2^{(1+\alpha)j-(1+\alpha)j_{2}+\delta
m}2^{3\min\\{k,k_{2}\\}/2}$ $\times\min\\{2^{-N_{0}k_{2,+}+\delta
m},2^{-(1+\alpha)j_{1}+\delta m}\\}\epsilon_{1}^{2}\lesssim
2^{-2m}\epsilon_{1}^{2}.$ (4.4)

$\oplus$ If $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m$ and $k_{1}\leq-\alpha m.$

Due to symmetry of inputs in the High $\times$ High type interaction case,
without loss of generality, we assume that $j_{1}=\max\\{j_{1},j_{2}\\}$.
After putting $h_{1;j_{1},k_{1}}$ in $L^{2}$ and putting
$e^{-it\Lambda_{2}}h_{2;j_{2},k_{2}}$ in $L^{\infty}$, from (2.12), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\leq-\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\leq-\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{(1+\alpha)j-8k_{1,+}-(1+\alpha)\max\\{j_{1},j_{2}\\}+2\delta
m}2^{-m+k_{1}}\lesssim 2^{-(1+\alpha/4)m}\epsilon_{1}^{2}\\\ \end{split}$
(4.5)

$\oplus$ If $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m$ and $k_{1}\geq-\alpha m.$

Note that, for this case, we have $j\leq(1+10\delta)m.$ Due to symmetry
between two inputs in the High $\times$ High type interaction, without loss of
generality, we assume that $j_{1}=\max\\{j_{1},j_{2}\\}.$ As
$\mu\nu=+,|\xi|\ll|\eta|$, we have

$|\Phi_{\mu,\nu}^{a}(\xi,\eta)|\sim|\eta|\gtrsim 2^{k_{1}}\gtrsim 2^{-\alpha
m}.$ (4.6)

Recall (3.5). To take advantage of high oscillation in time, we do integration
by parts in time once. As a result, we have

$\begin{split}&\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}dt=\sum_{b=1,2}End_{k;k_{1},k_{2};b}^{a;\mu,\nu;j_{1},j_{2}}(t_{a},\xi)+\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\widetilde{\mathcal{J}}_{k;k_{1},k_{2};b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}dt,\\\
&End_{k;k_{1},k_{2};b}^{a;\mu,\nu;j_{1},j_{2}}(t_{b},\xi):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it_{b}\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{(-1)^{b-1}iq^{a}_{\mu,\nu}(\xi-\eta,\eta)}{\Phi_{\mu,\nu}^{a}(\xi,\eta)}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t_{b},\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t_{b},\eta)\psi_{k}(\xi)d\eta
d\xi,\\\
&\widetilde{\mathcal{J}}_{k;k_{1},k_{2};1}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{iq^{a}_{\mu,\nu}(\xi-\eta,\eta)}{\Phi_{\mu,\nu}^{a}(\xi,\eta)}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\partial_{t}\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)d\eta
d\xi,\\\
&\widetilde{\mathcal{J}}_{k;k_{1},k_{2};2}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi):=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+it\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{iq^{a}_{\mu,\nu}(\xi-\eta,\eta)}{\Phi_{\mu,\nu}^{a}(\xi,\eta)}\partial_{t}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)d\eta
d\xi.\\\ \end{split}$ (4.7)

From the estimate of phases in (4.6), $\forall k_{1}\in[-\alpha
m,\infty)\cap\mathbb{Z},(k_{1},k_{2})\in\chi_{k}^{1},$ the following estimate
holds for the kernel,

$\big{\|}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi+iy\cdot\eta}\frac{iq^{a}_{\mu,\nu}(\xi-\eta,\eta)}{\Phi_{\mu,\nu}^{a}(\xi,\eta)}\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)d\eta
d\xi\big{\|}_{L^{1}_{x,y}}\lesssim 2^{8\alpha m}.$ (4.8)

From the $L^{2}-L^{\infty}$ type bilinear estimate and the above estimate of
kernel, we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq-\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}End_{k;k_{1},k_{2};b}^{a;\mu,\nu;j_{1},j_{2}}(t_{a},\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq-\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}2^{(1+10\alpha)m}\|h_{1;j_{1},k_{1}}(t_{a})\|_{L^{2}}\|e^{-it_{a}\Lambda_{2}}h_{2;j_{2},k_{2}}(t_{a})\|_{L^{\infty}}\\\
&\lesssim 2^{-m/2}\epsilon_{1}^{2}.\end{split}$ (4.9)

Similarly, from the $L^{2}-L^{\infty}$ type bilinear estimate, the estimate of
kernel in (4.8) and the estimate (3.4) in Lemma 3.1, we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq-\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}\sum_{b=1,2}2^{(1+\alpha)j}\|\int_{t_{1}}^{t_{2}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\widetilde{\mathcal{J}}_{k;k_{1},k_{2};b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)dt\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq-\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m\\\
\end{subarray}}\sup_{t\in[2^{m-1},2^{m}]}2^{(2+10\alpha)m}\big{(}\|h_{1;j_{1},k_{1}}(t)\|_{L^{2}}\|e^{-it\Lambda_{2}}\partial_{t}h_{2;j_{2},k_{2}}(t)\|_{L^{\infty}}\\\
&+\|\partial_{t}h_{1;j_{1},k_{1}}(t)\|_{L^{2}}\|e^{-it\Lambda_{2}}h_{2;j_{2},k_{2}}(t)\|_{L^{\infty}}\big{)}\lesssim
2^{-m/2}\epsilon_{1}^{2}.\end{split}$ (4.10)

Hence the desired estimate (4.2) holds after combining the above estimate and
the obtained estimates (4.9), (4.4), and (4.5).

∎

Now, we consider the resonance case $\mu\nu=-$.

######  Lemma 4.2.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, $\mu,\nu\in\\{+,-\\}$,
s.t., $\mu\nu=-,$ we have

$\sup_{k\in\mathbb{Z}}\sum_{a=1,2}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\|\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)dt\big{]}\|_{Z}\lesssim
2^{\delta m}\epsilon_{1}^{2},$ (4.11)

where the absolute constants $c$ and $\tilde{c}$ are defined in (2.4).

###### Proof.

Recall (2.6). Same as what we did in (3.5), we also do space localization for
two inputs. As in the proof of Lemma 3.1, we split into two cases based on the
possible size of $x$, i.e., the size of $j$. Moreover, because the sign of
$\mu\nu$ doesn’t play much role for the case
$j\geq\max\\{m,-k_{-}\\}+\delta(m+j)$ in the proof of Lemma 4.1, it would be
sufficient for us to only consider the case
$j\leq\max\\{m,-k_{-}\\}+\delta(m+j)$.

As in the decomposition in (3.11), based on the size of
$l_{\mu,\nu}(\xi,\eta),$ we do dyadic decomposition for
$\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$. We first
consider the main case $l$ is relatively small. The case
$l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}$ (see (2.4) for
the definitions of $c$ and $\tilde{c}$) can be handled in the same way as one
sub-case of the main case. We will elaborate this point later.

Let $l\in[-m/2+\delta m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$ be fixed.
Recall the detailed formula of $q_{\mu,\nu}(\xi-\eta,\eta)$ in (2.8). From the
estimate (1.4), the following estimate for the localized symbol,

$\big{|}q_{\mu,\nu}(\xi-\eta,\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\big{|}\lesssim
2^{l}.$ (4.12)

$\bullet$ If $k\leq-2m/3-\alpha m$ or $k_{2}\leq-m/4-2\alpha m.$

We first consider the threshold case i.e., $l=-m/2+\delta m$. Recall the
definition of $l_{\mu,\nu}(\xi,\eta)$ in (3.10). From the estimate of the
symbol in (4.12), the estimate of the volume of support of $\eta$ in (3.22),
the $L^{2}-L^{\infty}$-type estimate, the decay estimate (2.12), we have

$\begin{split}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}&2^{(1+\alpha)j}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{l+(1+\alpha)j}\min\\{2^{-m+k_{1}}2^{l/2},2^{3k/2+2l+3k_{1}}\\}\epsilon_{1}^{2}\lesssim
2^{-m}\epsilon_{1}^{2}.\end{split}$ (4.13)

Now we focus on the non-threshold case, $l\in(-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}.$ From the estimate (3.10), by
doing integration by parts in $\eta$ once, we gain at least $2^{-\delta m}$.
After doing integration by parts in $\eta$ $\delta^{-2}$ times, we can rule
out the case $\max\\{j_{1},j_{2}\\}\leq m+l-\delta m$ as follows,

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq m+l-\delta m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq m+2l-\delta m\\\
\end{subarray}}2^{-100m}2^{(1+\alpha)j-(1+\alpha)j_{2}+\delta
m}2^{3\min\\{k,k_{2}\\}/2}\\\ &\quad\times\min\\{2^{-N_{0}k_{2,+}+\delta
m},2^{-(1+\alpha)j_{1}+\delta m}\\}\epsilon_{1}^{2}\lesssim
2^{-2m}\epsilon_{1}^{2}.\end{split}$ (4.14)

It remains to consider the case $\max\\{j_{1},j_{2}\\}\geq m+l-\delta m.$ Due
to symmetry between two inputs in the High $\times$ High type interaction,
without loss of generality, we assume that $j_{1}=\max\\{j_{1},j_{2}\\}.$ From
the estimate (4.12), the $L^{2}-L^{\infty}$ type bilinear estimate and the
decay estimate (2.12), the $L^{2}\rightarrow L^{1}$ type Sobolev embedding and
the $L^{2}-L^{2}$ type bilinear estimate, we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{l}2^{(1+\alpha)j-(1+\alpha)j_{1}+\delta
m}\min\\{2^{3k/2},2^{-m+k_{2}-5k_{2,+}}\\}\epsilon_{1}^{2}\lesssim
2^{-m}\epsilon_{1}^{2}.$ (4.15)

$\bullet$ If $k\geq-2m/3-\alpha m$ and $k_{2}\geq-m/4-2\alpha m.$

We decompose $\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$
further by using the following partition of unity function,

$\begin{split}&\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)=\sum_{i=1,2,3}\mathcal{J}_{k;k_{1},k_{2};l;i}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi),\\\
&\varphi_{1}(\xi,\eta)=\psi_{>-3}(1+\mu\xi_{1}\eta_{1}/|\xi||\eta|)\psi_{[-3+c,3+\tilde{c}]}(|\not{\xi}|/|\not{\eta}|),\\\
&\varphi_{2}(\xi,\eta)=\psi_{\leq-3}(1+\mu\xi_{1}\eta_{1}/|\xi||\eta|)\psi_{[-3+c,3+\tilde{c}]}(|\not{\xi}|/|\not{\eta}|),\quad\varphi_{3}(\xi,\eta)=1-\psi_{[-3+c,3+\tilde{c}]}(|\not{\xi}|/|\not{\eta}|),\\\
&\mathcal{J}_{k;k_{1},k_{2};l;i}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\\\
&\quad\times\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{i}(\xi,\eta)d\eta\end{split}$ (4.16)

where $c,\tilde{c}$ are defined in (2.4).

$\oplus$ The estimate of
$\mathcal{J}_{k;k_{1},k_{2};l;1}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi).$

Recall the definition of the cutoff function $\varphi_{1}(\xi,\eta)$ in
(4.16). As $l\leq-100-2\max\\{\tilde{c},-c\\}$, $k\leq
k_{1}-100-5\max\\{\tilde{c},-c\\}$, and $1+\mu\xi_{1}\eta_{1}/|\xi||\eta|\geq
2^{-4}$, For any $(\xi,\eta)\in
supp\big{(}\varphi_{1}(\xi,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\big{)},$ we have $|\Phi_{\mu,\nu}^{a}(\xi,\eta)|\sim
2^{k}.$ Recall that $k\geq-2m/3-\alpha m$. To take advantage of high
oscillation in time, we do normal form transformation. As a result, we have

$\begin{split}\int_{t_{1}}^{t_{2}}&\mathcal{J}_{k;k_{1},k_{2};l;1}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt=\sum_{n=1,2}(-1)^{n}\int_{\mathbb{R}^{3}}e^{it_{n}\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{q^{a}_{\mu,\nu}(\xi-\eta,\eta)}{i\Phi_{\mu,\nu}^{a}(\xi,\eta)}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t_{n},\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t_{n},\eta)\\\
&\quad\times\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{1}(\xi,\eta)d\eta-\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{i}(\xi,\eta)}\frac{q^{a}_{\mu,\nu}(\xi-\eta,\eta)}{i\Phi_{\mu,\nu}^{a}(\xi,\eta)}\\\
&\quad\times\partial_{t}\big{[}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\big{]}\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{1}(\xi,\eta)d\eta dt.\end{split}$ (4.17)

We first consider the threshold case i.e., $l=-m/2+\delta m$. From (4.17), the
$L^{2}-L^{\infty}$-type estimate, the estimate of symbol in (4.12), the
estimate of the volume of support of $\eta$ in (3.22), and the estimate (3.18)
in Proposition 3.2,

$\begin{split}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}&\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;1}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{-k+l+(1+\alpha)j+2\alpha
m}2^{3k_{1}+2l-8k_{1,+}}\epsilon_{1}^{2}\lesssim\epsilon_{1}^{2}.\end{split}$
(4.18)

For the non-threshold case, $l\in(-m/2+\delta m,-100+c]\cap\mathbb{Z},$ as in
the obtained estimate (4.14), we can rule out the case
$\max\\{j_{1},j_{2}\\}\leq m+l-\delta m$ by doing integration by parts in
“$\eta$” $\delta^{-2}$ times. Now, we focus on the case
$\max\\{j_{1},j_{2}\\}\geq m+l-\delta m$.

Recall (2.6). As $|\Phi_{\mu,\nu}^{a}(\xi,\eta)|\sim 2^{k}$, from the
$L^{2}-L^{\infty}$ type bilinear estimate, the $L^{\infty}$ decay estimate
(2.12), the estimate (2.13) in Lemma 2.1, and the estimate (3.4) in Lemma 3.1,
we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;1}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{m/4+(\alpha+30\delta)m}2^{-k+l}2^{(1+\alpha)j-(1+\alpha)\max\\{j_{1},j_{2}\\}}\\\
&\times 2^{-m+k_{2}-4k_{2,+}}\epsilon_{1}^{2}\lesssim\epsilon_{1}^{2}.\\\
\end{split}$ (4.19)

$\oplus$ The estimate of
$\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi).$

Recall the definition of the cutoff function $\varphi_{2}(\xi,\eta)$ in
(4.16). As $l\leq-100-2\max\\{\tilde{c},-c\\}$, $k\leq
k_{1}-100-5\max\\{\tilde{c},-c\\}$, for any $(\xi,\eta)\in
supp\big{(}\varphi_{2}(\xi,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\big{)},$ we have

$\begin{split}\forall
a\in\\{1,2\\},\quad|\Phi_{\mu,\nu}^{a}(\xi,\eta)|&=\big{|}\Lambda_{a}(\xi)-\mu\Lambda_{2}(\xi-\eta)-\nu\Lambda_{1}(\eta)\big{|}\\\
&=\Big{|}\frac{2\Lambda_{a}(\xi)|\eta|+2\mu\xi\cdot\eta+(c_{1}^{2}-1)\mu(\xi_{2}-\eta_{2})^{2}+(c_{2}^{2}-1)\mu(\xi_{3}-\eta_{3})^{2}}{\mu\Lambda_{1}(\xi)+\Lambda_{2}(\xi-\eta)+\Lambda_{1}(\eta)}\Big{|}\\\
&\gtrsim
2^{-k_{2}}\big{(}\frac{|\not{\xi}|^{2}|\eta_{1}|}{|\xi_{1}|}-2^{3+2\tilde{c}}\big{(}|\not{\eta}|^{2}+|\not{\xi}|^{2}\big{)}\big{)}\gtrsim|\not{\eta}|^{2}|\eta|^{-1}.\\\
\end{split}$ (4.20)

Motivated from the above estimate of phases, we localize further the size of
$|\not{\eta}|/|\eta|$. Recall (3.10), for any $(\xi,\eta)\in
supp\big{(}\varphi_{2}(\xi,\eta)\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\psi_{b}(|\not{\eta}|/|\eta|)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\big{)},b\in\mathbb{Z}$, we have $2b\leq l+10$.

Moreover, recall the detailed formula of $q^{a}_{\mu,\nu}(\xi-\eta,\eta)$ in
(2.8). From the estimate (1.4), the following estimate for the localized
symbol,

$|q^{a}_{\mu,\nu}(\xi-\eta,\eta)\varphi_{2}(\xi,\eta)\psi_{b}(|\not{\eta}|/|\eta|)\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)|\lesssim
2^{2b}.$ (4.21)

Recall that $k_{2}\geq-m/4-2\alpha m.$ Same as the previous case, from the
above estimate of the phase, to take advantage of high oscillation in time, we
do normal form transformation. As a result, we have

$\begin{split}\int_{t_{1}}^{t_{2}}&\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt=\sum_{n=1,2}(-1)^{n}\int_{\mathbb{R}^{3}}e^{it_{n}\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{q^{a}_{\mu,\nu}(\xi-\eta,\eta)}{i\Phi_{\mu,\nu}^{a}(\xi,\eta)}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t_{n},\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t_{n},\eta)\\\
&\quad\times\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{2}(\xi,\eta)d\eta-\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{q^{a}_{\mu,\nu}(\xi-\eta,\eta)}{i\Phi_{\mu,\nu}^{a}(\xi,\eta)}\\\
&\quad\times\partial_{t}\big{[}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\big{]}\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{2}(\xi,\eta)d\eta dt.\end{split}$ (4.22)

We first consider the threshold case i.e., $l=-m/2+\delta m$. From the
estimate of phase (4.20), the estimate of the kernel in (4.21), the estimate
of the volume of support of $\eta$ in (3.22), the decay estimate (2.12), and
the estimate (3.18) in Proposition 3.2, we have

$\begin{split}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}&\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{-k_{2}+(1+\alpha)j+2\delta
m}2^{l/2}2^{3k_{1}+2l-8k_{1,+}}\epsilon_{1}^{2}\lesssim\epsilon_{1}^{2}.\end{split}$
(4.23)

For the non-threshold case, $l\in(-m/2+\delta
m,-100-2\max\\{\tilde{c},-c\\}]\cap\mathbb{Z},$ as in the obtained estimate
(4.14), we can rule out the case $\max\\{j_{1},j_{2}\\}\leq m+l-\delta m$ by
doing integration by parts in “$\eta$” $\delta^{-2}$ times. Now, we focus on
the case $\max\\{j_{1},j_{2}\\}\geq m+l-\delta m$.

From the estimate (4.20), the $L^{2}-L^{\infty}$ type bilinear estimate, the
$L^{\infty}$ decay estimate (2.12), and the estimate (3.4) in Lemma 3.1, we
have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}$
$\begin{split}&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\ \end{subarray}}2^{-k_{2}+m/4+2\alpha
m}2^{(1+\alpha)j-(1+\alpha)\max\\{j_{1},j_{2}\\}}2^{-m+k_{2}-5k_{2,+}}\epsilon_{1}^{2}\\\
&\lesssim 2^{-m/10}\epsilon_{1}^{2}.\end{split}$ (4.24)

$\oplus$ The estimate of
$\mathcal{J}_{k;k_{1},k_{2};l;3}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi).$

Recall the definition of the cutoff function $\varphi_{3}(\xi,\eta)$ in
(4.16). From (3.10), for any $(\xi,\eta)\in supp\big{(}\varphi_{3}(\xi,\eta)$
$\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\psi_{b}(|\not{\eta}|/|\eta|)\big{)},b\in\mathbb{Z}$,
we have

$|\nabla_{\eta}\Phi_{\mu,\nu}^{i}(\xi,\eta)|\gtrsim
2^{2b}+\big{|}\frac{\not{\xi}-\not{\eta}}{|\xi-\eta|}+\frac{(c_{1}^{2}\eta_{2},c_{2}^{2}\eta_{3})}{|\eta|}\big{|}\gtrsim
2^{b}.$ (4.25)

Based on the possible size of $|\not{\eta}|/|\eta|$, we decompose
$\mathcal{J}_{k;k_{1},k_{2};l;3}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$ further as
follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2};l;3}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)&=\sum_{b\in[-m/2+\delta
m,2]\cap\mathbb{Z}}\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\\\
\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{3}(\xi,\eta)\\\
&\quad\times\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{b;-m/2+\delta
m}(|\not{\eta}|/|\eta|)d\eta.\end{split}$ (4.26)

We first consider the threshold case i.e., $b=-m/2+\delta m$. From the
estimate of the symbol in (4.21), and the volume of support of $\xi,\eta$, we
have

$\begin{split}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}&\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};3;b}^{\mu,\nu;j_{1},j_{2};l}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{2b+(1+\alpha)j}2^{b}2^{3k_{1}+2b-8k_{1,+}}\epsilon_{1}^{2}\lesssim
2^{-5m/4}\epsilon_{1}^{2}.\end{split}$ (4.27)

For the non-threshold case, $b\in(-m/2+\delta m,2]\cap\mathbb{Z},$ from the
estimate (4.25), we can rule out the case $\max\\{j_{1},j_{2}\\}\leq
m+b-\delta m$ by doing integration by parts in “$\eta$” $\delta^{-2}$ times.
Moreover, as in obtaining the rough estimate (4.15), from the estimate (4.21),
we rule out further the case $\max\\{j_{1},j_{2}\\}\geq m+b-\delta m$ and
$b\leq-10\delta m.$

Now, it would be sufficient to consider the case $\max\\{j_{1},j_{2}\\}\geq
m+b-\delta m$ and $b\in[-10\delta m,2]\cap\mathbb{Z}.$ Due to symmetry between
two inputs in the High $\times$ High type interaction, without loss of
generality, we assume that $j_{1}=\max\\{j_{1},j_{2}\\}.$ For any fixed
$j\in[0,(1-\delta)^{-1}\big{(}\max\\{m,-k_{-}\\}+\delta
m\big{)}]\cap\mathbb{Z}$, based on the size of $j_{1}$, we split into two sub-
cases as follows.

$\dagger$ If $j_{1}\geq j+2b/(1+2\alpha).$

From the estimate (4.21), the $L^{2}-L^{\infty}$ type bilinear estimate, and
the $L^{\infty}$ decay estimate (2.12), we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\geq j+2b/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}\tilde{j}\geq
j+2b/(1+2\alpha)\\\
\end{subarray}}2^{m+2a}2^{(1+\alpha)j-(1+\alpha)\tilde{j}+\delta
m}2^{-m+k_{2}-8k_{2,+}}\epsilon_{1}^{2}\lesssim 2^{\delta m+\delta
a}\epsilon_{1}^{2}.$ (4.28)

$\dagger$ If $j_{1}\leq j+2b/(1+2\alpha).$

For this sub-case, we have $m-100\delta m\leq j\leq m+10\delta m$. Recall
(4.26). Note that, for any $\gamma\in\mathbb{Z}_{+}^{3}$, we have

$\begin{split}\nabla_{\xi}^{\gamma}\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)&=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)\\\
\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)&=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}\big{(}it\nabla_{\xi}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{)}^{\gamma_{1}}\nabla_{\xi}^{\gamma_{2}}\big{[}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\\\
&\quad\times\psi_{k}(\xi)\varphi_{3}(\xi,\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\varphi_{b;-m/2+\delta
m}(|\not{\eta}|/|\eta|)\big{]}\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)d\eta.\end{split}$
(4.29)

For $\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)$, we do integration
by parts in “$\eta$” $|\gamma_{1}|$ times. As a result, we have

$\begin{split}\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)&=\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{i}(\xi,\eta)}\nabla_{\eta}\cdot\Big{[}\frac{i\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)|^{2}}\circ\cdots\nabla_{\eta}\cdot\Big{[}\frac{i\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)|^{2}}\big{(}i\nabla_{\xi}\Phi_{\mu,\nu}^{a}(\xi,\eta)\big{)}^{\gamma_{1}}\\\
&\qquad\times\nabla_{\xi}^{\gamma_{2}}\big{[}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\varphi_{3}(\xi,\eta)\psi_{k}(\xi)\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\\\ &\qquad\times\varphi_{b;-m/2+\delta
m}(|\not{\eta}|/|\eta|)\big{]}\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\Big{]}\circ\cdots\Big{]}d\eta.\end{split}$
(4.30)

Recall the range of $k,j,j_{1},k_{1},$ and $l$. The worst scenario happens
when $\nabla_{\eta}^{\kappa}\nabla_{\xi}^{\gamma_{1}}$ hits
$\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)$, where
$\gamma\in\mathbb{Z}_{+}^{3},$ s.t., $|\kappa|=|\gamma_{2}|$. Moreover, from
the estimate (4.25) and (4.21), for any $n\in\mathbb{Z}_{+}$, from the
estimate (2.23) in Lemma 2.2, the following estimate holds for the kernel of
typical symbols appeared in (4.30),

$\begin{split}\|\mathcal{F}^{-1}_{(\xi,\eta)\rightarrow(x,y)}\big{[}&\underbrace{\frac{i\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)|^{2}}\otimes\circ\cdots\circ\otimes\frac{i\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{\mu,\nu}^{a}(\xi,\eta)|^{2}}}_{\text{$n$-times
}}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\\\
&\times\psi_{k}(\xi)\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\varphi_{b;-m/2+\delta
m}(|\not{\eta}|/|\eta|)\big{]}\big{]}\|_{L^{1}_{x,y}}\lesssim
2^{-(6+\delta)b-(n-2)b}.\end{split}$

From the above estimate, the $L^{2}-L^{\infty}$ type bilinear estimate, and
the $L^{\infty}$ decay estimate (2.12), for any $\gamma\in\mathbb{Z}_{+}^{3}$,
we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\leq j+2b/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\sum_{\begin{subarray}{c}\tilde{j}\leq
j+2b/(1+2\alpha)\\\
\end{subarray}}2^{-(6+\delta)b-(|\gamma|-2)b+|\gamma|j_{1}}2^{(1+\alpha)j-(1+\alpha)\tilde{j}+\delta
m}2^{-m+k_{2}-5k_{2,+}}\epsilon_{1}^{2}$ $\lesssim
2^{|\gamma|j-(6+\delta)b-|\gamma|b+2|\gamma|b/(1+2\alpha)}2^{-m+\delta
m}\epsilon_{1}^{2}.$ (4.31)

Now, we let $|\gamma|=10$, which is sufficient for our purpose. Recall (4.29).
From the estimate (4.31), we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\ k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\leq j+2b/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{\begin{subarray}{c}\gamma,\gamma_{1},\gamma_{2}\in\mathbb{Z}_{+}^{3}\\\
|\gamma|=10,\gamma_{1}+\gamma_{2}=\gamma\\\
\end{subarray}}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{1}\\\
k_{1}\geq
k+100+5\max\\{\tilde{c},-c\\}\end{subarray}}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\leq j+2b/(1+2\alpha)\\\
\end{subarray}}2^{(1+\alpha)j-|\gamma|j}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim 2^{-(6+\delta)b+10(1-2\alpha)b/(1+2\alpha)}2^{-m+\delta
m}\epsilon_{1}^{2}\lesssim 2^{-m+\delta m+b}\epsilon_{1}^{2}.$ (4.32)

After summing up the case $b\in[-10\delta m,2]\cap\mathbb{Z}.$ ,
$l\in[2b,-100-2\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$ in the obtained
estimates (4.28) and (4.32), we finish the proof for the case
$l\in[-m/2+\delta m,-100-2\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$.

It remains to consider the case
$l\in[-100-5\max\\{\tilde{c},-c\\},2]\cap\mathbb{Z}$. Note that, from the
rough estimate (3.10), the lower bound $2^{l}$ plays the same role of the
lower bound $2^{b}$ we use in (4.25) for the estimate of
$\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$. Hence,
after rerun the argument used for the estimate
$\mathcal{J}_{k;k_{1},k_{2};l;3;b}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$, the
estimate of $\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu}(t,\xi)$ follows similarly
for the case $l\in[-100-2\max\\{\tilde{c},-c\\},2]\cap\mathbb{Z}$. Hence
finishing the proof of our desired estimate (4.11). ∎

###  4.2. The High $\times$ Low type and the Low $\times$ High type
interactions

In this subsection, we consider the High $\times$ Low type and the Low
$\times$ High type interactions with one of the input frequencies much smaller
than the out frequency, e.g., $\min\\{k_{1},k_{2}\\}\leq
k-5\max\\{\tilde{c},-c\\}-100.$

In the following Lemma, we first consider the non-resonance case.

######  Lemma 4.3.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, we have

$\sup_{k\in\mathbb{Z}}\sum_{a=1,2}\sum_{\mu\in\\{+,-\\}}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\
k_{1}\leq k-5\max\\{\tilde{c},-c\\}-100\\\
\end{subarray}}\|\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,-}(t,\xi)\big{]}dt\|_{Z}\lesssim\epsilon_{1}^{2},$
(4.33)
$\sup_{k\in\mathbb{Z}}\sum_{a=1,2}\sum_{\nu\in\\{+,-\\}}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\leq k-5\max\\{\tilde{c},-c\\}-100\\\
\end{subarray}}\|\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;-,\nu}(t,\xi)\big{]}dt\|_{Z}\lesssim\epsilon_{1}^{2}.$
(4.34)

###### Proof.

Since only rough estimates of the phase $\Phi_{\mu,\nu}^{a}(\xi,\eta)$ will be
used in the non-resonance case, the desired estimates (4.33) and (4.34) can be
proved similarly, we only focus on the proof of the estimate (4.33).

Recall (2.6). Same as what we did in (3.5), we also do space localization for
two inputs. As in the proof of Lemma 3.1, we first rule out the case
$j\geq\max\\{m,-k_{-}\\}+\delta(m+j).$ It would be sufficient to consider the
case $j\leq\max\\{m,-k_{-}\\}+\delta(m+j).$ As in the decomposition in (3.11),
based on the size of $l_{\mu,\nu}(\xi,\eta),$ we do dyadic decomposition for
$\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$.

Moreover, as the estimates (3.12) and (3.16) are still valid, it would be
sufficient to consider the non-threshold case $l\in(-m/2+\delta
m,\tilde{c}+10]\cap\mathbb{Z}$, $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l$ and
the threshold case $l=-m/2+\delta m$.

Based on the size of $k$, we split into two cases as follows.

$\bullet$ If $k\leq-m/4-2\alpha m.$

We first consider the threshold case $l=-m/2+\delta m$. From the
$L^{2}-L^{\infty}$ type bilinear estimate, the volume of support of the input
putted in $L^{2}$, and the decay estimate (2.12), since $k\leq-m/4-2\alpha m$,
we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\
k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}2^{(1+\alpha)j}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\ k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}2^{l+(1+\alpha)j}2^{l/2}2^{-m+k_{1}}\epsilon_{1}^{2}\lesssim
2^{-(1+10\delta)m}\epsilon_{1}^{2}.\\\ \end{split}$ (4.35)

For the non-threshold case $l\in(-m/2+\delta m,\tilde{c}+10]\cap\mathbb{Z}$,
by using the strategies used in obtaining (3.17), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\
k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l}^{\mu,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\ k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{(1+\alpha)j+l}2^{-(1+\alpha)\max\\{j_{1},j_{2}\\}+\delta
m}\min\big{\\{}2^{3k_{1}/2},2^{-m+k}\big{\\}}\epsilon_{1}^{2}\\\ &\lesssim
2^{-m-\delta m}\epsilon_{1}^{2}.\end{split}$ (4.36)

Hence finishing the proof of the case $k\leq-m/4-2\alpha m$.

$\bullet$ If $k\geq-m/4-2\alpha m$.

Since $k_{1}\leq k-5\max\\{\tilde{c},-c\\}-100$ and $\nu=-$, for this case, we
have

$\forall a\in\\{1,2\\},\qquad|\Phi_{\mu,\nu}^{a}(\xi,\eta)|\sim 2^{k}\gtrsim
2^{-m/4-2\alpha m}.$

To take advantage of high oscillation in time, we do normal form
transformation once. As a result, we have

$\begin{split}&\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt=\sum_{n=1,2}\int_{\mathbb{R}^{3}}e^{it_{n}\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{(-1)^{n}q^{a}_{\mu,\nu}(\xi-\eta,\eta)}{i\Phi_{\mu,\nu}^{a}(\xi,\eta)}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t_{n},\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t_{n},\eta)\\\
&\times\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\psi_{k}(\xi)d\eta-\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}\frac{q^{a}_{\mu,\nu}(\xi-\eta,\eta)\psi_{k}(\xi)}{i\Phi_{\mu,\nu}^{a}(\xi,\eta)}\\\
&\times\partial_{t}\big{(}\widehat{h_{1;j_{1},k_{1}}^{\mu}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\big{)}\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))d\eta dt.\end{split}$ (4.37)

By using the same strategy used in obtaining (3.15), from the estimate in
Lemma and the volume of support of frequencies, for the threshold case
$l=-m/2+\delta m$, we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\
k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}2^{(1+\alpha)j}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\ k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}2^{-k+l+(1+\alpha)j}2^{3k/2+l/2}2^{-m+k_{1}-4k_{+}}\epsilon_{1}^{2}\lesssim
2^{-\delta m}\epsilon_{1}^{2}.\\\ \end{split}$ (4.38)

For the non-threshold case $l\in(-m/2+\delta m,2]\cap\mathbb{Z}$, by using the
strategies used in obtaining (3.17), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\
k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{2}\\\ k_{1}\leq
k-5\max\\{\tilde{c},-c\\}-100\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}2^{(1+\alpha)j+l}2^{-(1+\alpha)\max\\{j_{1},j_{2}\\}}2^{-3m/4+(\alpha+10\delta)m-k}\\\
&\times\min\\{2^{3k_{1}/2+m},1\\}\epsilon_{1}^{2}\lesssim
2^{-m/4}\epsilon_{1}^{2}.\end{split}$ (4.39)

Hence finishing the proof of the non-resonance case. ∎

Now, we consider the time-resonance case.

######  Lemma 4.4.

Under the bootstrap assumption (2.9), for any
$t\in[2^{m-1},2^{m}]\subset[0,T],m\in\mathbb{Z}_{+}$, we have

$\sup_{k\in\mathbb{Z}}\sum_{a=1,2}\sum_{\mu\in\\{+,-\\}}\sum_{(k_{1},k_{2})\in\chi_{k}^{2},k_{1}\leq
k-100-5\max\\{\tilde{c},-c\\}}\|\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,+}(t,\xi)\big{]}dt\|_{Z}\lesssim
2^{\delta m}\epsilon_{1}^{2},$ (4.40)
$\sup_{k\in\mathbb{Z}}\sum_{a=1,2}\sum_{\nu\in\\{+,-\\}}\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}}\|\int_{t_{1}}^{t_{2}}\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2}}^{a;+,\nu}(t,\xi)\big{]}dt\|_{Z}\lesssim
2^{\delta m}\epsilon_{1}^{2}.$ (4.41)

###### Proof.

Since after changing coordinates $(\xi,\eta)\rightarrow(\xi,\xi-\eta),$ the
desired estimates (4.40) and (4.41) can be proved similarly, we only focus on
the proof of the estimate (4.41). Recall (2.6). Same as what we did in (3.5),
we also do space localization for two inputs. As in the proof of Lemma 3.1, we
first rule out the case $j\geq\max\\{m,-k_{-}\\}+\delta(m+j).$ It would be
sufficient to consider the case $j\leq\max\\{m,-k_{-}\\}+\delta(m+j).$

As in (3.11), we do dyadic decomposition for
$\mathcal{J}_{k;k_{1},k_{2}}^{a;\mu,\nu;j_{1},j_{2}}(t,\xi)$. Note that the
obtained estimates (3.12) and (3.16) are still valid. Moreover, similar to
what we did in the proof of Lemma 4.2, from the $L^{2}-L^{\infty}$ type
bilinear estimate, the estimate (2.12), and the volume of support, we can rule
out the case $k\leq-m/4-2\alpha m$ and the case $k_{2}\leq-2m/3-\alpha m.$

To sum up, it would be sufficient to consider $k\geq-m/4-2\alpha m$ and
$k_{2}\geq-2m/3-\alpha m$ for the non-threshold case $l\in(-m/2+\delta
m,\tilde{c}+10]\cap\mathbb{Z}$, $\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l$, the
threshold case $l=-m/2+\delta m$.

We first consider the main case $l\in[-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]$ . The case
$l\in[-100-5\max\\{\tilde{c},-c\\},\tilde{c}+10]\cap\mathbb{Z}$ can be handled
in the same way as one sub-case of the main case. We will elaborate this point
later.

To see different behavior of phases, we use the following partition of unity,

$\begin{split}1&=\phi_{\nu,1}(\xi,\eta)+\phi_{\nu,2}(\xi,\eta)+\phi_{\nu,3}(\xi,\eta),\\\
\phi_{\nu,1}(\xi,\eta)&=\psi_{\geq-10}(1-\nu\xi_{1}\eta_{1}/(|\xi||\eta|))\psi_{[-10,10]}(|\not{\xi}-\not{\eta}|\Lambda_{2}(\eta)/(|\xi-\eta|\sqrt{c_{1}^{2}\eta_{2}^{4}+c_{2}^{4}\eta_{3}^{2}})),\\\
\phi_{\nu,2}(\xi,\eta)&=\psi_{<-10}(1-\nu\xi_{1}\eta_{1}/(|\xi||\eta|))\psi_{[-10,10]}(|\not{\xi}-\not{\eta}|\Lambda_{2}(\eta)/(|\xi-\eta|\sqrt{c_{1}^{2}\eta_{2}^{4}+c_{2}^{4}\eta_{3}^{2}})),\\\
\phi_{\nu,3}(\xi,\eta)&=1-\psi_{[-10,10]}(|\not{\xi}-\not{\eta}|\Lambda_{2}(\eta)/(|\xi-\eta|\sqrt{c_{1}^{2}\eta_{2}^{4}+c_{2}^{4}\eta_{3}^{2}})).\end{split}$
(4.42)

Correspondingly, we split
$\mathcal{J}_{k;k_{1},k_{2};l}^{a;+,\nu;j_{1},j_{2}}(t,\xi)$ into three pieces
as follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2};l}^{a;+,\nu;j_{1},j_{2}}(t,\xi)&=\sum_{b\in\\{1,2,3\\}}\mathcal{J}_{k;k_{1},k_{2};l;b}^{a;+,\nu;j_{1},j_{2}}(t,\xi)\\\
\forall
b\in\\{1,2,3\\},\quad\mathcal{J}_{k;k_{1},k_{2};l;b}^{a;+,\nu;j_{1},j_{2}}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{+,\nu}^{a}(\xi,\eta)}q^{a}_{+,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\\\
&\quad\times\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{+,\nu}(\xi,\eta))\phi_{\nu,b}(\xi,\eta)d\eta\\\ \end{split}$ (4.43)

$\bullet$ If $b=1.$

Note that, for any $(\xi,\eta)\in
supp\big{(}\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\phi_{\nu,1}(\eta)\big{)}$, we have

$\big{|}\Phi_{+,\nu}^{a}(\xi,\xi-\eta)\big{|}\sim|\eta|\sim 2^{k_{2}}.$

Recall that $k_{2}\geq-m/4-2\alpha m$. To take advantage of high oscillation
in time, as in (4.17), we do integration by parts in time once. As a result,
similar to the obtained estimate (4.18), for the threshold case $l=-m/2+\delta
m,$ we have

$\begin{split}\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\geq-m/4-2\alpha m\\\
\end{subarray}}2^{(1+\alpha)j}&\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};-m/2+\delta
m;1}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\big{\|}_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\geq-m/4-2\alpha m\\\ \end{subarray}}2^{-k_{2}+l+(1+\alpha)j+2\alpha
m}2^{-m+l/2+k_{1}-4k_{1,+}}\epsilon_{1}^{2}\lesssim\epsilon_{1}^{2}.\end{split}$
(4.44)

Similar to the obtained estimate (4.19), for the non-threshold case
$l\in(-m/2+\delta m,\tilde{c}+10]\cap\mathbb{Z}$, we have

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\geq-m/4-2\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{(1+\alpha)j}\|P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;1}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\geq-m/4-2\alpha m\\\
\end{subarray}}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq m+l-\delta m\\\
\end{subarray}}2^{m/4+(\alpha+30\delta)m}2^{-k_{2}+l}2^{(1+\alpha)j-(1+\alpha)\max\\{j_{1},j_{2}\\}}2^{-m-4k_{2,+}}\epsilon_{1}^{2}\lesssim\epsilon_{1}^{2}.$

$\bullet$ If $b=2.$

For this case, we have $|\xi_{1}|-|\xi_{1}-\eta_{1}|-\nu|\eta_{1}|=0.$ Recall
(3.10). Note that, for any $(\xi,\eta)\in supp(\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\phi_{\nu,2}(\eta)\psi_{k}(\xi)\psi_{k_{2}}(\eta))$,
we have

$\begin{split}\big{|}\nabla_{\xi}\big{(}\Phi_{+,\nu}^{a}(\xi,\eta)\big{)}\big{|}=\big{|}\nabla_{\xi}\Lambda_{a}(\xi)-\frac{\xi-\eta}{|\xi-\eta|}\big{|}\lesssim
2^{l/2},\\\ \end{split}$ (4.45)

Moreover, if we change coordinates $(\xi,\eta)\longrightarrow(\xi,\xi-\eta)$,
for any $(\xi,\eta)\in supp(\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\xi-\eta))\phi_{\nu,2}(\xi-\eta)\psi_{k}(\xi)\psi_{k_{2}}(\xi-\eta))$,
we still have

$\begin{split}\big{|}\nabla_{\xi}\big{(}\Phi_{+,\nu}^{a}(\xi,\xi-\eta)\big{)}\big{|}=\big{|}\nabla_{\xi}\Lambda_{a}(\xi)-\nu\frac{\xi-\eta}{\Lambda_{2}(\xi-\eta)}\big{|}\lesssim
2^{l/2}.\\\ \end{split}$ (4.46)

Thanks to the above improved estimates, we can improve the upper bound of $j$.
As in the proof of Lemma 3.1, by doing integration by parts in “$\xi$”
$\delta^{-2}$ times, we rule out further the case
$j\geq\max\\{m,-k_{-}\\}+l/2+\delta m.$ It would be sufficient to consider the
case $j\leq\max\\{m,-k_{-}\\}+l/2+\delta m.$

$\dagger$ The threshold case, i.e., $l=-m/2+\delta m$.

Similar to what we did in (4.26), we localize further based on the size of
$\not{\eta}/|\eta|$ as follows,

$\begin{split}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)&=\sum_{b\in[-m/2+\delta
m,l/2+2]\cap\mathbb{Z}}\mathcal{J}_{k;k_{1},k_{2};l;2;b}^{a;+,\nu;j_{1},j_{2}}(t,\xi)\\\
\mathcal{J}_{k;k_{1},k_{2};l;2;b}^{a;+,\nu;j_{1},j_{2}}(t,\xi)&:=\int_{\mathbb{R}^{3}}e^{it\Phi_{+,\nu}^{a}(\xi,\eta)}q^{a}_{+,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\\\
&\quad\times\varphi_{l;-m/2+\delta
m}(l_{+,\nu}(\xi,\eta))\varphi_{b;-m/2+\delta
m}(|\not{\eta}|/|\eta|)\phi_{\nu,2}(\xi,\eta)d\eta.\end{split}$ (4.47)

From the volume of support of $\xi,\eta$, the estimate of symbol in (3.14),
and the estimate of the volume of support of frequency variable in (3.22), we
have

$\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}}2^{(1+\alpha)j}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathcal{J}_{k;k_{1},k_{2};l;2;b}^{a;+,\nu;j_{1},j_{2}}(t,\xi)\big{]}\big{]}(x)\big{\|}_{L^{2}}$
$\lesssim\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}}2^{2b+(1+\alpha)j}2^{3k/2+l/2}2^{k_{2}+2k+2l}2^{-5(k_{1,+}+k_{2,+})}\epsilon_{1}^{2}.$
(4.48)

The above estimate is sufficient to rule out the case $b\leq-m/4-2\alpha m$ or
$k_{1}\notin[-2\alpha m,2\alpha m]$.

It remains to consider the case $b\in[-m/4-2\alpha m,-m/4+\delta
m]\cap\mathbb{Z}$ and $k_{1}\in[-2\alpha m,2\alpha m]\cap\mathbb{Z}.$ Note
that, from Taylor expansion and the approximations in (3.10) and (3.21), the
following approximation holds for the resonance case,

$\begin{split}\Phi_{\mu,\nu}^{1}(\xi,\eta)&=\frac{(1-c_{1}^{2})\eta_{2}^{2}}{2|\xi_{1}|}(1+\frac{\mu\nu
c_{1}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})+\frac{(1-c_{2}^{2})\eta_{3}^{2}}{2|\xi_{1}|}(1+\frac{\mu\nu
c_{2}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})+\mathcal{O}(2^{-3m/4+10\alpha m})\\\
&=(1+\frac{\mu\nu
c_{1}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})^{-1}\frac{(1-c_{1}^{2})\xi_{2}^{2}}{2|\xi_{1}|}+(1+\frac{\mu\nu
c_{2}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})^{-1}\frac{(1-c_{2}^{2})\xi_{3}^{2}}{2|\xi_{1}|}+\mathcal{O}(2^{-3m/4+10\alpha
m}),\\\
\Phi_{\mu,\nu}^{2}(\xi,\eta)&=\frac{-\mu\nu|\xi_{1}-\eta_{1}|}{2|\eta_{1}||\xi_{1}|}\big{[}(1-c_{1}^{2})\eta_{2}^{2}(1+\frac{\mu\nu
c_{1}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})+(1-c_{2}^{2})\eta_{3}^{2}(1+\frac{\mu\nu
c_{2}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})\big{]}\\\
&+\mathcal{O}(2^{-3m/4+10\alpha m})=\mathcal{O}(2^{-3m/4+10\alpha m})\\\
&+\frac{-\mu\nu|\xi_{1}-\eta_{1}|}{2|\eta_{1}||\xi_{1}|}\big{[}(1-c_{1}^{2})\xi_{2}^{2}(1+\frac{\mu\nu
c_{1}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})^{-1}+(1-c_{2}^{2})\xi_{3}^{2}(1+\frac{\mu\nu
c_{2}^{2}|\xi_{1}-\eta_{1}|}{|\eta_{1}|})^{-1}\big{]},\\\ \end{split}$ (4.49)

We remark that, the above expansion only depends on the fact that we have
$|\xi|-\mu|\xi_{1}-\eta_{1}|-\nu|\eta_{1}|=0$ in the resonance case and the
fact that $l<-m/2+\alpha m,k\leq 2\alpha m.$

From (4.49), $\forall a\in\\{1,2\\}$, we have a lower bound for the size of
phases for the case we are considering,

$|\Phi_{+,\nu}^{a}(\xi,\eta)|\varphi_{l;-m/2+\delta
m}(l_{+,+}(\xi,\eta))\varphi_{b;-m/2+\delta m}(|\not{\eta}|/|\eta|)\gtrsim
2^{2b+2(k-k_{2})}\gtrsim 2^{-m/2-10\alpha m}.$

To utilize high oscillation in time, we do integration by parts in time once.
As a result, from the volume of support of $\xi,\eta$, the estimate (3.22),
and the estimate (3.18) in Proposition 3.2, we have

$\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}}2^{(1+\alpha)j}\big{\|}\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2;b}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\big{\|}_{L^{2}}$
$\lesssim\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}}2^{10\alpha
m+(1+\alpha)j}2^{3k/2+l/2}2^{k_{2}+2k+2l}2^{-5(k_{1,+}+k_{2,+})}\epsilon_{1}^{2}\lesssim\epsilon_{1}^{2}.$
(4.50)

Hence finishing the proof of the threshold case.

$\dagger$ The non-threshold case, i.e., $l\in(-m/2+\delta
m,-100-5\max\\{\tilde{c},-c\\}]\cap\mathbb{Z}$.

Based on the possible size of $\max\\{j_{1},j_{2}\\}$, we split into two cases
as follows.

$\oplus$ If $\max\\{j_{1},j_{2}\\}\geq j+l/(1+2\alpha).$

If $j_{1}=\max\\{j_{1},j_{2}\\}$, then we put
$e^{it\Lambda_{2}}h_{2;j_{2},k_{2}}^{\mu}$ in $L^{\infty}$ and put
$h_{1;j_{1},k_{1}}^{\mu}$ in $L^{2}$. As a result, from the estimate (3.14),
the $L^{2}-L^{\infty}$ type bilinear estimate, we have

$\begin{split}&\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{1}=\max\\{j_{1},j_{2}\\}\geq j+l/(1+2\alpha)\\\ \end{subarray}}\sum
P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim 2^{l}2^{-m+k_{2}-8k_{1,+}}2^{-(1+\alpha)(j+l/(1+2\alpha))+\delta
m}\epsilon_{1}^{2}.\end{split}$ (4.51)

If $j_{2}=\max\\{j_{1},j_{2}\\}$, from the estimate in (3.14), the
orthogonality in $L^{2}$, and the super-localized decay estimate (2.13) in
Lemma 2.1, the following estimate holds after super-localizing the frequency
of the input $h_{1;j_{1},k_{1}}$ with the step size $2^{k_{2}}$,

$\begin{split}&\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
j_{2}=\max\\{j_{1},j_{2}\\}\geq j+l/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{j_{2}\geq
j+l/(1+2\alpha)}\big{(}\sum_{a\in[2^{k-k_{2}-10},2^{k-k_{2}+10}]\cap\mathbb{Z}}\|\int_{\mathbb{R}^{3}}e^{it\Phi_{\mu,\nu}^{a}(\xi,\eta)}q^{a}_{\mu,\nu}(\xi-\eta,\eta)\widehat{h_{1;\leq
j_{2},k_{1}}}(t,\xi-\eta)\\\
&\times\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\psi_{k_{1}}(|\xi-\eta|-a2^{k_{2}})d\eta\|_{L^{2}_{\xi}}^{2}\big{)}^{1/2}\\\
&\lesssim\sum_{j_{2}\geq
j+l/(1+2\alpha)}2^{l-4k_{+}}2^{-m+k_{2}/3}\epsilon_{1}\|h_{2;j_{2},k_{2}}^{\mu}\|_{L^{2}}\lesssim
2^{l-4k_{+}}2^{-m+k_{2}/3}2^{-(1+\alpha)(j+l/(1+2\alpha))+\delta
m}\epsilon_{1}^{2}.\\\ \end{split}$ (4.52)

After combining the obtained estimates (4.51) and (4.52), we have

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq j+l/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\ k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}\end{subarray}}2^{m+l}2^{(1+\alpha)j-(1+\alpha)(j+l/(1+2\alpha))+\delta
m}2^{-m+k_{1}/3-4k_{+}}\epsilon_{1}^{2}\lesssim 2^{\delta m+\delta
l}\epsilon_{1}^{2}.\\\ \end{split}$ (4.53)

$\oplus$ If $\max\\{j_{1},j_{2}\\}\leq j+l/(1+2\alpha).$

For this case, we will use same strategy used in the proof of Lemma 4.2 for
the estimate of
$\mathcal{J}_{k;k_{1},k_{2};3}^{\mu,\nu;j_{1},j_{2};l}(t,\xi)$, see (4.16).

We first rule out some trivial cases. By using the same strategy used in
obtaining the estimate (4.53), the following estimate holds if
$k_{2}\leq-3\alpha m$ or $l\leq-\alpha m$,

$\begin{split}&\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\
k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\geq(1-\delta)m+l\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\int_{t_{1}}^{t_{2}}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)dt\big{]}\big{]}(x)\|_{L^{2}}\\\
&\lesssim\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}}2^{m+l}2^{(1+\alpha)j-(1+\alpha)((1-\delta)m+l)+\delta
m}2^{-m+k_{2}/3}\epsilon_{1}^{2}\lesssim 2^{-10\delta m}\epsilon_{1}^{2}.\\\
\end{split}$ (4.54)

Now, we focus on the case $k_{1}\geq-3\alpha m$ and $l\geq-\alpha m.$ Similar
to (4.29), for any $\gamma\in\mathbb{Z}_{+}^{3}$, we have

$\begin{split}\nabla_{\xi}^{\gamma}\mathcal{J}_{k;k_{1},k_{2};l;2}^{a;+,\nu;j_{1},j_{2}}(t,\xi)&=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\mathfrak{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)\\\
\mathfrak{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)&=\int_{\mathbb{R}^{3}}e^{it\Phi_{+,\nu}^{a}(\xi,\eta)}\big{(}it\nabla_{\xi}\Phi_{+,\nu}^{a}(\xi,\eta)\big{)}^{\gamma_{1}}\nabla_{\xi}^{\gamma_{2}}\big{[}q^{a}_{+,\nu}(\xi-\eta,\eta)\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\psi_{k}(\xi)\\\
&\quad\times\varphi_{l;-m/2+\delta
m}(l_{+,\nu}(\xi,\eta))\phi_{\nu,2}(\xi,\eta)\big{]}\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)d\eta.\end{split}$
(4.55)

For $\widetilde{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)$, we do integration
by parts in “$\eta$” $|\gamma_{1}|$ times. As a result, we have

$\begin{split}\mathfrak{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)&=\int_{\mathbb{R}^{3}}e^{it\Phi_{+,\nu}^{a}(\xi,\eta)}\\\
&\times\nabla_{\eta}\cdot\Big{[}\frac{i\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)|^{2}}\circ\cdots\nabla_{\eta}\cdot\Big{[}\frac{i\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)|^{2}}\big{(}i\nabla_{\xi}\Phi_{+,\nu}^{a}(\xi,\eta)\big{)}^{\gamma_{1}}\nabla_{\xi}^{\gamma_{2}}\big{[}q^{a}_{+,\nu}(\xi-\eta,\eta)\\\
&\qquad\times\widehat{h_{1;j_{1},k_{1}}}(t,\xi-\eta)\psi_{k}(\xi)\varphi_{l;-m/2+\delta
m}(l_{+,\nu}(\xi,\eta))\phi_{\nu,2}(\xi,\eta)\big{]}\widehat{h_{2;j_{2},k_{2}}^{\nu}}(t,\eta)\Big{]}\circ\cdots\Big{]}d\eta.\end{split}$
(4.56)

Recall the range of $k,j,j_{1},k_{1},$ and $l$. The worst scenario happens
when $\nabla_{\eta}^{\kappa}\nabla_{\xi}^{\gamma_{1}}$ hits
$\widehat{h_{1;j_{1},k_{1}}}(t,\eta)$ if $j_{1}=\max\\{j_{1},j_{2}\\}$
($\widehat{h_{2;j_{2},k_{2}}^{\mu}}(t,\xi-\eta)$ if
$j_{2}=\max\\{j_{1},j_{2}\\}$), where $\gamma\in\mathbb{Z}_{+}^{3},$ s.t.,
$|\kappa|=|\gamma_{2}|$. Moreover, from the estimates (4.45) and (3.16), for
any $n\in\mathbb{Z}_{+}$, the following estimate holds for the kernel of
typical symbols appeared in (4.30),

$\begin{split}\|\mathcal{F}^{-1}_{(\xi,\eta)\rightarrow(x,y)}&\big{[}\underbrace{\frac{i\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)|^{2}}\otimes\circ\cdots\circ\otimes\frac{i\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)}{|\nabla_{\eta}\Phi_{+,\nu}^{a}(\xi,\eta)|^{2}}}_{\text{$n$-times
}}\varphi_{l;-m/2+\delta
m}(l_{\mu,\nu}(\xi,\eta))\big{(}i\nabla_{\xi}\Phi_{+,\nu}^{a}(\xi,\eta)\big{)}^{\gamma_{1}}\\\
&\times\psi_{k_{1}}(\xi-\eta)\psi_{k_{2}}(\eta)\psi_{k}(\xi)q^{a}_{+,\nu}(\xi-\eta,\eta)\big{]}\|_{L^{1}_{x,y}}\lesssim
2^{-(6+\delta)l-(n-1)l+|\gamma_{1}|l/2}.\end{split}$

From the above estimate of kernels, by using the same strategy used in
obtaining the estimate (4.53), the following estimate holds for any
$\gamma\in\mathbb{Z}_{+}^{3}$,

$\sum_{\begin{subarray}{c}(k_{1},k_{2})\in\chi_{k}^{3}\\\ k_{2}\leq
k-100-5\max\\{\tilde{c},-c\\}\end{subarray}}2^{(1+\alpha)j}\|\sum_{\begin{subarray}{c}j_{i}\in[-k_{i,-},\infty)\cap\mathbb{Z},i=1,2\\\
\max\\{j_{1},j_{2}\\}\leq j+l/(1+2\alpha)\\\
\end{subarray}}P_{k}\big{[}\varphi_{j,k}(\cdot)\mathcal{F}^{-1}\big{[}\mathfrak{J}_{\gamma_{1},\gamma_{2}}^{\gamma}(t,\xi)\big{]}\big{]}(x)\|_{L^{2}}$
$\lesssim\sum_{(k_{1},k_{2})\in\chi_{k}^{3},k_{2}\leq