# Internal mode-induced growth
in $3$d nonlinear Klein-Gordon equations

Tristan Léger  Tristan Léger, Princeton University, Department of Mathematics,
Fine Hall, Washington Road, Princeton, NJ, 08544, United States.
<EMAIL_ADDRESS>and  Fabio Pusateri  Fabio Pusateri, University of
Toronto, Department of Mathematics, 40 St George Street, Toronto, ON, M5S 2E4,
Canada<EMAIL_ADDRESS>
###### Abstract.

This note complements the paper [20] by proving a scattering statement for
solutions of nonlinear Klein-Gordon equations with an internal mode in $3$d.
We show that small solutions exhibit growth around a one-dimensional set in
frequency space and become of order one in $L^{\infty}$ after a short
transient time. The dynamics are driven by the feedback of the internal mode
into the equation for the field (continuous spectral) component.

The main part of the proof consists of showing suitable smallness for a “good”
component of the radiation field. This is done in two steps: first, using the
machinery developed in [20], we reduce the problem to bounding a certain
quadratic normal form correction. Then we control this latter by establishing
some refined estimates for certain bilinear operators with singular kernels.

F.P. was supported in part by a start-up grant from the University of Toronto,
and NSERC grant RGPIN-2018-06487.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 The main result of [20]
    3. 1.3 New result: Scattering in $L^{\infty}_{\xi}$
    4. 1.4 Estimates from [20]
  2. 2 Preliminaries
  3. 3 Bilinear estimates
  4. 4 Asymptotic behavior of $h$
  5. 5 Estimating the correction

##  1\. Introduction

###  1.1. Background

The study of the small data regime for dispersive PDEs with quadratic
nonlinearities has attracted a lot of attention in recent years, and is
closely related to several open questions about the asymptotic stability of
coherent structures, such as (topological) solitons. This has turned out to be
a challenging problem, since quadratic nonlinearities are strong enough to
influence the long-time behavior of solutions and may lead to blow-up even in
a weakly nonlinear regime.

In recent years, several general methods have been developed to tackle
problems about global existence and long-time asymptotics of nonlinear (and
quasilinear) dispersive equations with low power nonlinearities. Without
trying to be exhaustive, we mention works by Germain-Masmoudi-Shatah [9] and
[10] and the development of the ‘space-time resonance’ method, parallel work
by Gustafson-Nakanishi-Tsai [13] on Gross-Pitaevski, Guo-Ionescu-Pausader [12]
and Deng-Ionescu-Pausader [6] on the Euler-Maxwell system, Ionescu and the
second author on $2$d water waves [14, 15], and Deng-Ionescu-Pausader and the
second author [7] on $3$d gravity-capillary waves. All of these results can be
interpreted as proving asymptotic stability of the zero or ‘trivial’ solution.

To prove similar results for non-trivial solutions, it is natural to consider
corresponding linearized problems. This generically introduces a potential
term in the equation. A typical model example is
$i\partial_{t}u+L(\sqrt{-\Delta+V+m^{2}})=u^{2},$ for some real-valued
dispersion relation $L$ and potential $V=V(x)$. Consequently, a new line of
active research has emerged seeking to understand the global behavior of small
solutions to such equations.

A first result was obtained for the quadratic NLS
$i\partial_{t}u+(-\Delta+V)u=\bar{u}^{2}$ in $3$d by Germain-Hani-Walsh in
[8]. The first author of this note dealt with the case of a $u^{2}$
nonlinearity with small, time-dependent $V$ [18]. Note that while a
$\bar{u}^{2}$ nonlinearity is non-resonant (for any type of evolution), a
$u^{2}$ nonlinearity is space-time resonant at the origin even in the case
$V=0$. See also [19] for the case of electromagnetic perturbations. More
recently, Soffer and the second author [23] treated a $u^{2}$ nonlinearity
with a large potential, and developed a general approach based on the
distorted Fourier transform, that is, the transform adapted to the Schrödinger
operator $H:=-\Delta+V(x)$. The starting point of this is a refined study of
the so-called ‘nonlinear spectral distribution’ (NSD)

$\displaystyle\mu(\xi,\eta,\sigma):=(2\pi)^{-9/2}\int_{{\mathbb{R}}^{3}}\overline{\psi(x,\xi)}\psi(x,\eta)\psi(x,\sigma)dx,$
(1.1)

where $\psi=\psi(x,\xi)$, $\xi\in{\mathbb{R}}^{3}$ are the generalized
eigenfunctions of the Schrödinger operator,
$(-\Delta+V(x))\psi=|\xi|^{2}\psi$. In particular, a precise knowledge of the
structure and singularities of $\mu$ allows the exploitation of oscillations
in (distorted) frequency space, which is a key step to extend the techniques
from the works on the case $V=0$ cited above.

We also mention that there have been several recent works in $1$d on equations
with potentials and low power nonlinearities. Since this is not the main focus
of this note, we refer the reader to the introductions of [11] and [22] for
extensive discussions and references.

Up until now, global dispersive solutions for low powers (in particular
quadratic) in111In $1$d we mention important recent works [17] and [5] on the
stability under odd perturbations of the kink for the $\phi^{4}$ model. The
linearization at the kink gives exactly a $1$d quadratic Klein-Gordon equation
with an internal mode like (1.2)-(1.3) satisfying the Fermi Golden rule (1.6).
$d\leqslant 3$ had only been constructed in the case of ‘trivial’ spectrum,
$\sigma(-\Delta+V)=\sigma(-\Delta)$. In [20] we dealt with a problem where an
internal mode is present. More precisely, we considered the following initial
value problem:

$\begin{cases}\partial_{t}^{2}u+(-\Delta+V(x)+1)u=u^{2},\\\
u(0,x)=u_{0},\quad\partial_{t}u(0,x)=u_{1},\end{cases}$ (1.2)

where $u:{\mathbb{R}}_{t}\times{\mathbb{R}}^{3}_{x}\rightarrow{\mathbb{R}}$,
with a sufficiently regular and decaying generic external potential
$V:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$, and smooth, localized initial
data $u_{0}$ and $u_{1}$. We will make the assumptions on $V$ and the initial
data more precise below.

In this setting, the spectrum of the Schrödinger operator $H:=-\Delta+V$
consists of the purely absolutely continuous part $[0,\infty)$ and a finite
number of negative eigenvalues, with corresponding smooth and fast decaying
eigenfunctions [26]. We further assume that the operator $L^{2}:=-\Delta+V+1$
has a unique strictly positive eigenvalue with a corresponding normalized
eigenfunction $\phi$:

$\displaystyle(-\Delta+V+1)\phi=\lambda^{2}\phi,\qquad
1/2<\lambda<1,\qquad{\|\phi\|}_{L^{2}}=1.$ (1.3)

The eigenvalue $\lambda$ is usually called the “internal frequency of
oscillation”, and $\phi$ is referred to as an “internal mode” of the dynamics.

(1.3) gives rise to a two parameter family of solutions to the linear equation
$\partial_{t}^{2}u+(-\Delta+V+1)u=0$ of the form:

$\displaystyle\phi_{A,\theta}(t,x)=A\cos(\lambda t+\theta)\phi(x),\qquad
A,\theta\in{\mathbb{R}}.$ (1.4)

These solutions - referred to as “bound states”, or “internal modes” with a
slight abuse - are time-periodic, oscillating and spatially localized. In this
weakly nonlinear regime, we expect nonlinear solutions to retain features of
the linear system. Therefore, a natural question is whether such periodic
solutions persist under the nonlinear flow. Our main result in [20] answered
in the negative: in a neighborhood of zero, the bound states (1.4) are
destroyed by the quadratic nonlinearity, and do not continue to quasi-periodic
or other non-decaying solutions. Moreover, solutions exhibit weak dispersive
features and obey decay estimates, although with a rate that is much slower
compared to solutions of $(\partial_{t}^{2}-\Delta+1+V)u=0.$

The first result of this kind was obtained by Sigal [25], who showed
instability of bound states for very general classes of equations, introducing
the “nonlinear Fermi Golden Rule” (FGR). Soffer and Weinstein [27] then
derived decay and asymptotics in the case of cubic Klein-Gordon equations,
that is, (1.2) with a $u^{3}$ nonlinearity. See also the recent work [21] on
the case $\lambda\in(1/3,1/2)$. The problem of meta-stability in the presence
of multiple bound states in $3$d was treated by Bambusi-Cuccagna [2]. For the
nonlinear Schrödinger equation (NLS) the problem of meta-stability (for
excited states) was studied by Tsai-Yau [28]; see also the recent advances by
Cuccagna-Maeda [3], their survey [4] and references therein.

Our main result in this note complements [20] with precise scattering
statements and asymptotics. In particular, we will show scattering in
$L^{\infty}_{\xi}$, i.e., pointwise in (distorted) frequency space, for the
‘good’ component of the radiation identified in [20]. As a consequence we will
obtain that solutions of (1.2) become of $O(1)$ in $L^{\infty}_{\xi}$ after a
short time, and exhibit a certain oscillatory behavior over time. This appears
to be the first rigorous proof for this type of asymptotic dynamics.

###  1.2. The main result of [20]

Set-up and assumptions. Let $\bf{P_{c}}$ denote the projection onto the
continuous spectral subspace, namely, for every $\psi\in
L^{2}({\mathbb{R}}^{3};{\mathbb{R}})$

$\displaystyle{\bf
P_{c}}\psi:=\psi-(\phi,\psi)\phi,\qquad(\psi_{1},\psi_{2}):=\int_{{\mathbb{R}}^{3}}\psi_{1}\psi_{2}\,dx.$
(1.5)

We make the following assumptions:

  * •

Regularity and decay of $V$: We assume $V\in\mathcal{S}$, but a finite amount
of regularity and decay (measured in weighted Sobolev spaces) would suffice.

  * •

Coupling to continuous spectrum: $\lambda\in(1/2,1)$ so that
$\lambda\notin\sigma_{ac}(-\Delta+V+1)$ and
$2\lambda\in\sigma_{ac}(-\Delta+V+1).$

  * •

Fermi Golden Rule: The “Fermi Golden Rule” resonance condition holds:

$\displaystyle\Gamma:=\frac{\pi}{2\lambda}\Big{(}{\bf{P_{c}}}\phi^{2},\delta(L-2\lambda){\bf
P_{c}}\phi^{2}\Big{)}>0,$ (1.6)

where $L:=\sqrt{-\Delta+V+1}$.

  * •

Genericity of $V$: the $0$ energy level is regular for $H:=-\Delta+V$, that
is, $0$ is not an eigenvalue, nor a resonance, i.e., there is no
$\psi\in\langle x\rangle^{1/2+}L^{2}({\mathbb{R}}^{3})$ such that $H\psi=0$.
Such a potential $V$ is said to be ‘generic’.

Distorted Fourier transform. For a regular and decaying $V$ (as in our
assumptions above) we can decompose $L^{2}$ into absolutely continuous and
pure point subspaces:
$L^{2}({\mathbb{R}}^{d})=L^{2}_{ac}({\mathbb{R}}^{d})\oplus
L^{2}_{pp}({\mathbb{R}}^{d})$ where
$L^{2}_{pp}(\mathbb{R}^{d})=\mathrm{span}(\phi)$. Let $\psi(x,\xi)$ denote the
generalized eigenfunctions solving for $\xi,x\in{\mathbb{R}}^{3}$

$\displaystyle\begin{split}&(-\Delta+V)\psi(x,\xi)=|\xi|^{2}\psi(x,\xi),\quad\mbox{with}\quad\big{|}\psi(x,\xi)-e^{ix\cdot\xi}\big{|}\longrightarrow
0,\quad\mbox{as}\quad|x|\rightarrow\infty,\end{split}$ (1.7)

with the Sommerfeld radiation condition
$r(\partial_{r}-i|k|)(\psi(x,k)-e^{ix\cdot k})\rightarrow 0$, for
$r=|x|\rightarrow\infty$. We can then define a unitary operator
$\widetilde{\mathcal{F}}$, the distorted Fourier Transform (dFT), as

$\displaystyle\widetilde{\mathcal{F}}f(\xi):=\widetilde{f}(\xi)=\frac{1}{(2\pi)^{3/2}}\lim_{R\to+\infty}\int_{|x|\leqslant
R}f(x)\overline{\psi(x,\xi)}dx,$ (1.8)

with inverse

$\displaystyle\big{(}\widetilde{\mathcal{F}}^{-1}f\big{)}(x)=\frac{1}{(2\pi)^{3/2}}\lim_{R\to+\infty}\int_{|\xi|\leqslant
R}f(\xi)\psi(x,\xi)d\xi.$ (1.9)

In particular, for any $g\in L^{2}$ we can write
$g(x)=\widetilde{\mathcal{F}}^{-1}\big{(}\widetilde{{\bf
P}_{c}g}\big{)}+(g,\phi)\phi$. Moreover $\widetilde{\mathcal{F}}$ diagonalizes
the Schrödinger operator (on the continuous spectrum),
$\widetilde{\mathcal{F}}{\bf{P_{c}}}H=|\xi|^{2}\widetilde{\mathcal{F}}$.

Continuous-Discrete decomposition. Next, we decompose a solution $u$ into a
discrete and a continuous component:

$\displaystyle u(t)=a(t)\phi+v(t),\quad\mbox{with \, $(v(t),\phi)=0$ \, for
all $t.$}$ (1.10)

Then (1.2) with (1.3) reads

$\begin{cases}\ddot{a}+\lambda^{2}a=\big{(}(a\phi+v)^{2},\phi\big{)}\\\
\partial_{t}^{2}v+L^{2}v={\bf{P_{c}}}\big{(}(a\phi+v)^{2}\big{)},\qquad
L:=\sqrt{-\Delta+V+1}.\end{cases}$ (1.11)

We call $v$ the radiation or field component of the solution $u$, and will
call $a$ the (amplitude of the) discrete component or internal mode of the
solution.

Main result on Radiation Damping.

######  Theorem 1.1 ([20]).

Consider (1.2), with initial data $u(0,x)=u_{0}(x)$, $u_{t}(0,x)=u_{1}(x)$
such that:

$\displaystyle\begin{split}&|(u_{0},\phi)|+|(u_{1},\phi)|\leqslant{\varepsilon}_{0},\\\
&{\|(\langle\nabla\rangle u_{0},u_{1})\|}_{W^{3,1}}+{\|\langle
x\rangle(\langle\nabla\rangle u_{0},u_{1})\|}_{H^{3}}+{\|(\langle\nabla\rangle
u_{0},u_{1})\|}_{H^{N}}\leqslant{\varepsilon}_{0}\end{split}$ (1.12)

for $N\gg 1$. Then, there exists $\overline{{\varepsilon}}\in(0,1)$ such that,
for all ${\varepsilon}_{0}\leq\bar{{\varepsilon}}$,

  * •

The equation (1.2) under the assumptions stated above has a unique global
solution $u\in C({\mathbb{R}};H^{N+1}({\mathbb{R}}^{3}))$ such that the
following hold: $u=a(t)\phi+v(t)$ with

$\displaystyle|a(t)|\approx{\varepsilon}_{0}(1+{\varepsilon}_{0}^{2}t)^{-1/2},$
(1.13)

and

$\displaystyle\langle
t\rangle{\big{\|}\partial_{t}v+iLv\big{\|}}_{L^{\infty}_{x}}\approx 1,\qquad
t\gtrsim{\varepsilon}_{0}^{-2},$ (1.14)
$\displaystyle{\big{\|}\partial_{t}v+iLv\big{\|}}_{H^{N}_{x}}\lesssim{\varepsilon}_{0}^{1-\delta},$
(1.15)

for arbitrarily small $\delta>0$.

  * •

Furthermore, we have the following asymptotic behavior: Let
$f:=e^{-itL}(\partial_{t}+iL)v.$ Then, there exists
$f_{\infty}\in{\varepsilon}_{0}^{1-\delta}H^{N}_{x}$, with $\delta>0$
arbitrarily small as above, such that

$\displaystyle{\|f(t)-f_{\infty}\|}_{H^{N}_{x}}\lesssim{\varepsilon}_{0}^{1-\delta}\langle
t\rangle^{-\delta^{\prime}}$ (1.16)

for some small $\delta^{\prime}>0$. Finally, we have the asymptotic growth

$\displaystyle{\big{\|}\partial_{\xi}\widetilde{f}(t)\big{\|}}_{L^{2}}\gtrsim\langle
t\rangle^{1/2},\qquad t\gtrsim{\varepsilon}_{0}^{-2}.$ (1.17)

Set-up and notation. In this note we work on the solutions constructed in
Theorem 1.1, and will use notation which is consistent with [20]. The starting
point for our analysis is the natural definition of a ‘good’ component, $h$,
and a ‘bad’ component, $g$, of the profile $f=e^{-itL}(\partial_{t}+iL)v$:

$\displaystyle\begin{split}&f=h-g,\qquad\widetilde{g}(t,\xi):=-\chi_{C}(\xi)\int_{0}^{t}B^{2}(s)e^{-is(\langle\xi\rangle-2\lambda)}ds\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi),\end{split}$ (1.18)

where $\chi_{C}(\xi):=\varphi_{\leqslant-C}(\langle\xi\rangle-2\lambda)$, with
$\varphi_{\leqslant-C}(x)$ a bump function supported on $|x|\leqslant
2^{-C+1}$ for $C=C(\lambda)$ large enough, and where $B=B(s)$ can be defined
through the profile of the amplitude of the internal oscillations given by

$\displaystyle A(t):=\frac{1}{2i\lambda}e^{-i\lambda t}(\dot{a}+i\lambda a).$
(1.19)

We refer the reader to Lemma 3.5 in [20] for the details of the definition of
$B$ which are not necessary here, and only recall the fact that (see (1.21)
below for the definition of $\rho$)

$\displaystyle\begin{split}&|A(s)-B(s)|=O(|A(s)|^{2})\approx\rho(t),\\\
&\dot{B}(t)=\rho(t)^{3/2}+\rho(t)^{3/2-a}\langle t\rangle^{-1},\quad
a\in(1/2,1).\end{split}$ (1.20)

We also define two useful quantities

$\displaystyle\rho(t):=\varepsilon_{0}\big{(}1+\varepsilon_{0}^{2}(\Gamma/\lambda)t\big{)}^{-1},\qquad
Z:=Z_{\beta}({\varepsilon},m):=\big{(}2^{m}\rho(2^{m})\big{)}^{1-\beta}\big{(}2^{m}\varepsilon\big{)}^{\beta}$
(1.21)

for some absolute constant $0<\beta\ll 1$, and
$\varepsilon:=C_{0}\varepsilon_{0}$ for some large constant $C_{0}>0.$

###  1.3. New result: Scattering in $L^{\infty}_{\xi}$

As already mentioned, our goal is to provide additional and more precise
information on the asymptotic behavior of solution of (1.2). In particular, we
identify a growth phenomenon due to the presence of the internal mode, and
describe the asymptotic behavior of the solution in $L^{\infty}_{\xi}.$ This
allows us to see that after a short time the solution becomes of size $O(1)$
around a certain frequency, despite arising from small initial data. Our main
result can be stated as follows:

######  Proposition 1.2.

There exist $F_{\infty}\in\varepsilon^{\delta}L^{\infty}_{t}L^{\infty}_{\xi}$
and a real-valued $\Psi_{\infty}=O(\varepsilon^{\delta})$, with some small
$\delta>0$, such that

$\displaystyle\begin{split}\widetilde{f}(t,\xi)=\int_{0}^{t}\exp\Big{(}is(\langle\xi\rangle-2\lambda)+i\tfrac{2c_{2}\lambda}{\Gamma}\log\big{(}1+\tfrac{\Gamma}{\lambda}Y_{0}^{2}s\big{)}\Big{)}y(s)^{2}ds\,e^{i2\Psi_{\infty}}\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi)\\\
+F_{\infty}(t,\xi)+\,O_{L^{\infty}_{\xi}}\big{(}\varepsilon^{\delta}\langle
t\rangle^{-\delta}\big{)},\end{split}$ (1.22)

where $c_{2}$ denotes a non-zero numerical constant, and
$y(s):=Y_{0}\,(1+(\Gamma/\lambda)sY^{2}_{0})^{-1/2}$ with
$Y_{0}=O({\varepsilon}_{0})$ a function of the initial data
$(a(0),\dot{a}(0))$.

In particular, for $t\gtrsim{\varepsilon}_{0}^{-2}$ and
$|\langle\xi\rangle-2\lambda|\leqslant\delta_{0}t^{-1}$ where $\delta_{0}$
denotes a sufficiently small constant, we have

$\displaystyle\widetilde{f}(t,\xi)$
$\displaystyle=\frac{1}{2ic_{2}}\Big{[}e^{2ic_{2}\frac{\lambda}{\Gamma}\log\big{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}t\big{)}}-1\Big{]}\,e^{i2\Psi_{\infty}}\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi)+O_{L^{\infty}_{\xi}}(\delta_{0}).$ (1.23)

######  Remark 1.3.

The leading term in (1.23) exhibits an oscillatory behavior, and is $O(1)$ for
most times. Note that this does not contradict (1.16) since this only happens
around a one-dimensional set.

######  Remark 1.4.

Although the equation we study is quadratic, we expect the same behavior to
take place in many other settings, typically when an internal mode is present.

We can reduce the proof of the main Proposition 1.2 to a scattering statement
for the field $h$ plus a quadratic (normal form) correction through the
following:

######  Lemma 1.5.

Assume there exists a correction
$\mathcal{N}(\widetilde{f})\in\varepsilon^{\delta}L^{\infty}_{t}L^{\infty}_{\xi}$
and $h_{\infty}\in{\varepsilon}^{\delta}L^{\infty}_{\xi}$ such that

$\displaystyle{\big{\|}\widetilde{h}(t)-\mathcal{N}(\widetilde{f})(t)-h_{\infty}\big{\|}}_{L^{\infty}_{\xi}}\lesssim{\varepsilon}^{\delta}\langle
t\rangle^{-\delta}$ (1.24)

for some $\delta>0$. Then (1.22) holds true.

###### Proof of Proposition 1.2 using Lemma 1.5.

Recall $(3.64)$ in [20]:

$\displaystyle\begin{split}\widetilde{g}(t,\xi)=-\int_{0}^{t}\exp\Big{(}is(2\lambda-\langle\xi\rangle)+i\frac{2c_{2}\lambda}{\Gamma}\log\Big{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\Big{)}\Big{)}y(s)^{2}ds\,e^{2i\Psi_{\infty}}\widetilde{{\bf
P_{c}}\phi^{2}}(\xi)\\\
+g_{\infty}(\xi)+O_{L^{\infty}_{\xi}}({\varepsilon}^{\delta})\langle
t\rangle^{-\delta},\end{split}$ (1.25)

for some $g_{\infty}\in\varepsilon^{\delta}L^{\infty}_{\xi}.$ Therefore,
assuming that there exist $\mathcal{N}(\widetilde{f})$ as in Lemma 1.5 the
result (1.22) follows by letting
$F_{\infty}(t):=\mathcal{N}(\widetilde{f})(t)+h_{\infty}-g_{\infty}\in{\varepsilon}^{\delta}L^{\infty}_{t}L^{\infty}_{\xi}$
since $f=h-g.$

To verify the last statement of Proposition 1.2, we look at the integral in
(1.22): when $|\langle\xi\rangle-2\lambda|\leqslant\delta_{0}t^{-1}$ and
$t\gtrsim Y_{0}^{-2}\approx{\varepsilon}_{0}^{-2}$ we have

$\displaystyle\begin{split}&\int_{0}^{t}\exp\Big{(}is(\langle\xi\rangle-2\lambda)+i2c_{2}\tfrac{\lambda}{\Gamma}\log\big{(}1+\tfrac{\Gamma}{\lambda}Y_{0}^{2}s\big{)}\Big{)}y(s)^{2}ds\,e^{2i\Psi_{\infty}}\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi)\\\
&=\frac{1}{2ic_{2}}\int_{0}^{t}e^{is(\langle\xi\rangle-2\lambda)}\partial_{s}\bigg{(}e^{2ic_{2}\frac{\lambda}{\Gamma}\log\big{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\big{)}}\bigg{)}\,ds\,e^{i2\Psi_{\infty}}\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi)\\\ &=\frac{\,e^{i2\Psi_{\infty}}\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi)}{2ic_{2}}\Bigg{(}\Big{[}e^{it(\langle\xi\rangle-2\lambda)}e^{2ic_{2}\frac{\lambda}{\Gamma}\log\big{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}t\big{)}}-1\Big{]}-i\big{(}\langle\xi\rangle-2\lambda\big{)}\int_{0}^{t}e^{is(\langle\xi\rangle-2\lambda)}e^{2ic_{2}\frac{\lambda}{\Gamma}\log\big{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\big{)}}ds\Bigg{)}\\\
&=\frac{1}{2ic_{2}}\Big{[}e^{2ic_{2}\frac{\lambda}{\Gamma}\log\big{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}t\big{)}}-1\Big{]}\,e^{i2\Psi_{\infty}}\,\widetilde{{\bf
P}_{c}\phi^{2}}(\xi)+O(\delta_{0}).\end{split}$

∎

###  1.4. Estimates from [20]

Before embarking on the proof, we collect some facts from [20] that will be
used below. We will use notation from §1.6 of [20], in particular concerning
cutoffs and corresponding Littlewood-Paley projections, such as projection to
(distorted) frequencies of size $\approx 2^{k}$, $k\in{\mathbb{Z}}$, i.e.,
$P_{k}f=\widetilde{\mathcal{F}}^{-1}(\varphi_{k}\widetilde{f})$, or size
$\lesssim 2^{k}$, i.e. $P_{\leqslant
k}f=\widetilde{\mathcal{F}}^{-1}(\varphi_{\leqslant k}\widetilde{f})$. We
begin with some bounds satisfied by $h$ and $g$:

######  Lemma 1.6.

For all $t\geqslant\varepsilon^{-1},t\approx 2^{m},$

$\displaystyle\|\partial_{\xi}\widetilde{h}(t)\|_{L^{2}}$
$\displaystyle\lesssim Z,$ (1.26a) $\displaystyle\|G(t)\|_{H^{N}}$
$\displaystyle\lesssim\varepsilon^{1-},\ \ G\in\\{g,h\\},$ (1.26b)
$\displaystyle\|e^{itL}G(t)\|_{L^{q}}$ $\displaystyle\lesssim
2^{\big{(}3\big{(}\frac{1}{q}-\frac{1}{2}\big{)}+10\delta_{N}\big{)}m}Z,\ \
G\in\\{g,h\\},\ \ q\in[2,6],$ (1.26c)
$\displaystyle\|\varphi_{k_{1}}\widetilde{G}(t)\|_{L^{1}_{\xi}}$
$\displaystyle\lesssim 2^{(5/2-)k_{1}}\cdot Z,\ \ G\in\\{g,h\\},$ (1.26d)

where $\delta_{N}:=5/N.$

For the component $g$ we also have, for $t\approx 2^{m},$

$\displaystyle{\|\widetilde{g}(t,\xi)\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim 2^{m}\rho(2^{m})m,$ (1.27a)
$\displaystyle{\|\partial_{\xi}\widetilde{g}(t,\xi)\|}_{L^{1}_{\xi}}$
$\displaystyle\lesssim 2^{m}\rho(2^{m})m^{2},$ (1.27b)
$\displaystyle\|\partial_{\xi}\widetilde{g}(t,\xi)\|_{L^{2}}$
$\displaystyle\lesssim 2^{3m/2}\rho(2^{m})m^{2}.$ (1.27c)

Finally we have estimates for the time-derivative of the two components: with
$M_{0}:=\delta_{N}m$

$\displaystyle\|e^{itL}\partial_{t}g(t)\|_{L^{p}}$
$\displaystyle\lesssim\rho(t),\quad p\geq 1,$ (1.28a)
$\displaystyle\|e^{itL}\partial_{t}h(t)\|_{L^{q}}$ $\displaystyle\lesssim
2^{-m+30M_{0}}Z^{2},\quad q>(3/2)-.$ (1.28b)

###### Proof.

(1.26a) is one of the bootstrap bounds established on $h$ as part of the proof
of Theorem 1.1; see $(1.50)$ in [20]. (1.26b) is a consequence of the second a
priori assumption in $(1.50)$ of [20] and a direct estimation of the $H^{N}$
norm of $g,$ see for example $(1.57)$ in [20]. (1.26c) is a consequence of
dispersive estimates for the free Klein-Gordon semi-group and our bootstrap
estimates, see Lemma $4.5$ in [20]. (1.26d) is a consequence of Bernstein’s
and Hardy’s inequality, the bounds on $h$ and the $L^{\infty}_{\xi}$ bound on
$g,$ see (1.27a).

(1.27) and (1.28a) are directly borrowed from Lemmas 4.7 and 4.6 in [20]. Only
(1.28b) requires a proof. It follows from the identity

$\displaystyle\partial_{s}h(s)=e^{-isL}\big{(}L^{-1}\Im
w\big{)}^{2}+e^{-isL}O(\rho(s))\theta+e^{-isL}\big{(}a(s)\phi\cdot L^{-1}\Im
w\big{)},$

and we can conclude using (1.26c) and interpolation with (1.26b). ∎

##  2\. Preliminaries

We start by recording a consequence of (1.25) which improves (1.27a):

######  Lemma 2.1.

We have $\|\widetilde{g}(t)\|_{L^{\infty}_{\xi}}\lesssim 1.$

###### Proof.

It suffices to show that the time integral in (1.25) is $O(1)$. We first
assume $|\langle\xi\rangle-2\lambda|^{-1}\leqslant t$ and split the time
integral into an integral from $0$ to $|\langle\xi\rangle-2\lambda|^{-1}$ and
its complement. For the first piece, recalling the definition of
$y(s)=Y_{0}\,(1+(\Gamma/\lambda)sY^{2}_{0})^{-1/2}$, we write

$\displaystyle\bigg{|}\int_{0}^{|\langle\xi\rangle-2\lambda|^{-1}}\exp\bigg{(}is(\langle\xi\rangle-2\lambda)+i\frac{2c_{2}\lambda}{\Gamma}\log\bigg{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\bigg{)}\bigg{)}y^{2}(s)\,ds\bigg{|}$
$\displaystyle=\frac{1}{2c_{2}}\bigg{|}\int_{0}^{|\langle\xi\rangle-2\lambda|^{-1}}\exp\big{(}is(\langle\xi\rangle-2\lambda)\big{)}\partial_{s}\exp\bigg{(}i\frac{2c_{2}\lambda}{\Gamma}\log\bigg{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\bigg{)}\bigg{)}\,ds\bigg{|}\lesssim
1,$

where the last inequality can be seen integrating by parts in $s$. For the
other piece we instead write

$\displaystyle\bigg{|}\int_{|\langle\xi\rangle-2\lambda|^{-1}}^{t}\exp\bigg{(}is(\langle\xi\rangle-2\lambda)+i\frac{2c_{2}\lambda}{\Gamma}\log\bigg{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\bigg{)}\bigg{)}y^{2}(s)\,ds\bigg{|}$
$\displaystyle=|\langle\xi\rangle-2\lambda|^{-1}\bigg{|}\int_{|\langle\xi\rangle-2\lambda|^{-1}}^{t}\Big{(}\partial_{s}\exp\big{(}is(\langle\xi\rangle-2\lambda)\big{)}\Big{)}\,\exp\bigg{(}i\frac{2c_{2}\lambda}{\Gamma}\log\bigg{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\bigg{)}\bigg{)}y^{2}(s)\,ds\bigg{|}.$

Integrating by parts gives the boundary term

$\displaystyle|\langle\xi\rangle-2\lambda|^{-1}\bigg{|}\bigg{(}\exp\big{(}is(\langle\xi\rangle-2\lambda)\big{)}\bigg{)}\exp\bigg{(}i\frac{2c_{2}\lambda}{\Gamma}\log\bigg{(}1+\frac{\Gamma}{\lambda}Y_{0}^{2}s\bigg{)}\bigg{)}y^{2}(s)\,ds\bigg{|}_{s=|\langle\xi\rangle-2\lambda|^{-1}}^{s=t}\bigg{|}\lesssim
1,$

and a bulk term which is also $O(1)$ since $|(d/ds)y^{2}(s)|\lesssim
y^{4}(s)$. Finally, if $|\langle\xi\rangle-2\lambda|^{-1}\geqslant t$ one can
estimate as we did for the first integral above. ∎

The control of the $L^{\infty}_{\xi}$ norm and the proof of the main
scattering statement for $\widetilde{h}$, see (1.24), is based on a
decomposition obtained from Duhamel’s formula. This is done in two steps
below, Lemma 2.2 and Lemma 2.6 (see also (2.7)).

We introduce a partition of unity of the interval $[0,t]$ to decompose our
time integrals. Let $\tau_{0},\tau_{1},\cdots,\tau_{L+1}:{\mathbb{R}}\to[0,1]$
denote cut-off functions, where the integer $L$ is chosen so that
$|L-\log_{2}(t+2)|<2$, with the following properties:

$\displaystyle\begin{split}&\sum_{n=0}^{L+1}\tau_{n}(s)=\mathbf{1}_{[0,t]}(s),\qquad\mathrm{supp}(\tau_{0})\subset[0,2],\quad\mathrm{supp}(\tau_{L+1})\subset[t/4,t],\\\
&\mbox{and}\quad\mathrm{supp}(\tau_{n})\subseteq[2^{n-1},2^{n+1}],\quad|\tau_{n}^{\prime}(t)|\lesssim
2^{-n},\quad\mbox{for}\quad n=1,\dots,L.\end{split}$ (2.1)

Recall the definitions (1.18) and (1.19)-(1.20). Lemma 4.1 in [20] gives us
the following:

######  Lemma 2.2.

We have

$\displaystyle\widetilde{h}=\widetilde{f_{0}}+F+\sum_{m=0}^{L+1}(S_{m}+M_{m}),$
(2.2)

where the ‘source terms’ are (recall the definition of $\chi_{C}$ below
(1.18))

$\displaystyle S_{m}$ $\displaystyle=S_{1,m}+S_{2,m}+\\{\mbox{similar and
better terms}\\},$ $\displaystyle
S_{1,m}:=\int_{0}^{t}e^{-is(\langle\xi\rangle-2\lambda)}(A^{2}(s)-B^{2}(s))\,\widetilde{\theta}(\xi)\,\tau_{m}(s)ds,$
(2.3a) $\displaystyle
S_{2,m}:=(1-\chi_{C}(\xi))\int_{0}^{t}e^{-is(\langle\xi\rangle-2\lambda)}B^{2}(s)\,\widetilde{\theta}(\xi)\,\tau_{m}(s)ds,$
(2.3b)

the ‘mixed terms’ are

$\displaystyle M_{m}$ $\displaystyle=-\frac{i}{2}M_{1,m}+\\{\mbox{similar and
better terms}\\},$ $\displaystyle M_{1,m}$
$\displaystyle:=\int_{0}^{t}B(s)\int_{{\mathbb{R}}^{3}}e^{-is(\langle\xi\rangle-\langle\eta\rangle-\lambda)}\langle\eta\rangle^{-1}\widetilde{f}(s,\eta)\nu(\xi,\eta)d\eta\,\tau_{m}(s)ds,$
(2.4) $\displaystyle\nu(\xi,\eta)$
$\displaystyle:=\frac{1}{(2\pi)^{3}}\int_{{\mathbb{R}}^{3}}\overline{\psi(x,\xi)}\psi(x,\eta)\phi(x)dx,$

and the ‘continuous/field’ self interactions are

$\displaystyle\begin{split}F&:=-\frac{1}{4}\sum_{{\epsilon_{1},\epsilon_{2}}\in\\{+,-\\}}{\epsilon}_{1}{\epsilon}_{2}F_{{\epsilon_{1},\epsilon_{2}},m}\big{(}f_{{\epsilon}_{1}},f_{{\epsilon}_{2}}\big{)},\qquad
f_{+}=f,\,\,\,f_{-}=\bar{f},\\\
F_{{\epsilon_{1},\epsilon_{2}}}(a,b)&:=\int_{0}^{t}\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{a}(s,\eta)\widetilde{b}(s,\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\,ds,\\\
\mu(\xi,\eta,\sigma)&:=\frac{1}{(2\pi)^{9/2}}\int_{{\mathbb{R}}^{3}}\overline{\psi(x,\xi)}\psi(x,\eta)\psi(x,\sigma)dx,\end{split}$
(2.5)

where the phases are defined by

$\displaystyle\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta,\sigma)=\langle\xi\rangle-{\epsilon}_{1}\langle\eta\rangle-{\epsilon}_{2}\langle\sigma\rangle.$
(2.6)

We split the $F$ terms from the above lemma into low and high frequencies by
letting

$\displaystyle\begin{split}F:=F_{L}+F_{H},\qquad
F_{L}(t,\xi):=\int_{0}^{t}\varphi_{\leqslant
M_{0}(s)}(\xi)\,e^{-is\langle\xi\rangle}\widetilde{\mathcal{F}}\Big{(}\frac{P_{\leqslant
M_{0}(s)}\Im w}{L}\Big{)}^{2}\,ds,\\\ M_{0}(s):=\delta_{N}\log\langle
s\rangle,\quad\delta_{N}:=5/N.\end{split}$ (2.7)

We will also let $F_{H,m}$ denote the term $F_{H}$ where the integral in time
is localized to $s\approx 2^{m}$ using the cutoffs $\tau_{m}$ (as in (2.4) for
example).

We need to further decompose $F_{L}$ using some more refined information on
the measure $\mu$ contained in Proposition 5.6 of [20], which we reproduce
below.

######  Lemma 2.3.

Let $M_{0}$ be a fixed parameter, and let222The parameter $N_{2}$ can be fixed
large enough as a fraction of $N$. $N_{2}$ be a sufficiently large integer. We
can decompose $\mu$ into a singular part (apex $S$), a regular part (apex $R$)
and a remainder part (index $Re$)

$\displaystyle\mu=\mu^{S}+\mu^{R}+\mu^{Re}.$ (2.8)

These three components satisfy the following structural properties (up to
irrelevant constants):

$\displaystyle\begin{split}&\mu^{S}(\xi,\eta,\sigma):=\delta_{0}(\xi-\eta-\sigma)+\mu_{1}^{S}(\xi,\eta,\sigma)+\mu_{2}^{S}(\xi,\eta,\sigma)+\mu_{3}^{S}(\xi,\eta,\sigma),\\\
&\mu^{R}(\xi,\eta,\sigma):=\mu_{1}^{R}(\xi,\eta,\sigma),\end{split}$ (2.9)

where the right-hand sides are defined as follows:

  * •

$\mu_{1}^{S}$ and $\mu_{1}^{R}$ are given by the formulas

$\displaystyle\mu_{1}^{\ast}(\xi,\eta,\sigma)$
$\displaystyle:=\nu_{1}^{\ast}(-\xi+\eta,\sigma)+\nu_{1}^{\ast}(-\xi+\sigma,\eta)+\overline{\nu_{1}^{\ast}(-\eta-\sigma,\xi)},\qquad\ast\in\\{S,R\\}$
(2.10)

with

$\displaystyle\begin{split}\nu_{1}^{S}(p,q)&:=\varphi_{\leqslant-
M_{0}-5}(|p|-|q|)\Big{[}\nu_{0}(p,q)\\\
&+\frac{1}{|p|}\sum_{a=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}b_{a,J}(p,q)\cdot
2^{J}K_{a}\big{(}2^{J}(|p|-|q|)\big{)}\Big{]}\end{split}$ (2.11)

where $K_{a}$ are Schwartz functions, $b_{a,J}$ are symbols satisfying the
bounds

$\displaystyle\sum_{J\in{\mathbb{Z}}}\big{|}\varphi_{P}(p)\varphi_{Q}(q)\nabla_{p}^{\alpha}\nabla_{q}^{\beta}b_{a,J}(p,q)\big{|}\lesssim
2^{-|\alpha|P}\big{(}2^{|\alpha|Q}+2^{(1-|\beta|)Q_{-}}\big{)}\cdot\mathbf{1}_{\\{|P-Q|<5\\}},$
(2.12)

for all $P,Q\leq M_{0}$, $|\alpha|+|\beta|\leqslant N_{2},$ and333We are using
the notation $b_{0}(p/|p|,q)$ here which slightly differs from the one in
Proposition 5.1 of [23] (or Proposition 5.6 in [20]); in this latter the
symbol appearing in the analogous formulas (5.11)-(5.12) was denoted
$b_{0}(p,q)$ instead. The symbol in (2.13) actually denotes (up to a constant)
the symbol $g(-p/|p|,q)$ which appears in Lemma 5.2 of [23]. The relation
between the two symbols in [23] is (up to a constant)
$b_{0}(p,q)=g(-p/|p|,q)$, consistently with (2.13)-(2.14).

$\displaystyle\nu_{0}(p,q):=\frac{b_{0}(p/|p|,q)}{|p|}\Big{[}i\pi\,\delta(|p|-|q|)+\mathrm{p.v.}\frac{1}{|p|-|q|}\Big{]}$
(2.13)

with

$\displaystyle\begin{split}\big{|}\varphi_{P}(p)\varphi_{Q}(q)\nabla_{q}^{\beta}b_{0}(p/|p|,q)\big{|}&\lesssim
2^{(1-|\beta|)Q_{-}}\cdot\textbf{1}_{\\{|P-Q|<5\\}},\ \ Q_{-}:=\min(Q,0),\ \
1\leqslant|\beta|\leqslant N_{2}.\end{split}$ (2.14)

  * •

$\mu_{2}^{S}$ is given by the formulas

$\displaystyle\mu_{2}^{S}(\xi,\eta,\sigma)$
$\displaystyle:=\nu_{2}^{S,1}(\xi,\eta,\sigma)+\nu_{2}^{S,2}(\xi,\eta,\sigma)+\nu_{2}^{S,2}(\xi,\sigma,\eta),$
(2.15)

with

$\displaystyle\begin{split}\nu_{2}^{S,1}(\xi,\eta,\sigma):=\frac{1}{|\xi|}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{\leqslant-
M_{0}-5}(|\xi|-|\eta|-|\sigma|)\\\ \times b_{i,J}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|-|\eta|-|\sigma|)\big{)},\end{split}$ (2.16)

and

$\displaystyle\begin{split}\nu_{2}^{S,2}(\xi,\eta,\sigma)=\frac{1}{|\eta|}\sum_{\epsilon\in\\{1,-1\\}}\sum_{i=1}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{\leqslant-
M_{0}-5}(|\xi|+\epsilon|\eta|-|\sigma|)\\\ \times
b_{i,J}^{\epsilon}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|+\epsilon|\eta|-|\sigma|)\big{)},\end{split}$ (2.17)

where $K_{i}$ are Schwartz functions and $b_{i,J}$ and $b_{i,J}^{\epsilon}$
are symbols satisfying the bounds

$\displaystyle\begin{split}&\big{|}\varphi_{k}(\xi)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}b_{i,J}(\xi,\eta,\sigma)\big{|}\\\
&\lesssim
2^{-|a|k}\cdot\big{(}2^{|a|\max(k_{1},k_{2})}+2^{(1-|\alpha|)k_{1}}2^{(1-|\beta|)k_{2}}\big{)}\mathbf{1}_{\\{|k-\max(k_{1},k_{2})|<5\\}},\end{split}$
(2.18)

and

$\displaystyle\begin{split}&\big{|}\varphi_{k}(\xi)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}b_{i,J}^{\pm}(\xi,\eta,\sigma)\big{|}\\\
&\lesssim
2^{-|\alpha|k_{1}}\cdot\big{(}2^{|\alpha|\max(k,k_{2})}+2^{(1-|a|)k}2^{(1-|\beta|)k_{2}}\big{)}\mathbf{1}_{\\{\max(k,k_{1},k_{2})-\mbox{\tiny$\mathrm{med}$}(k,k_{1},k_{2})<5\\}},\end{split}$
(2.19)

for all $k,k_{1},k_{2}\leq M_{0}$, and $|a|+|\alpha|+|\beta|\leq N_{2},$
respectively.

  * •

$\mu_{3}^{S}$ is given by the formula

$\displaystyle\begin{split}\mu_{3}^{S}(\xi,\eta,\sigma):=\sum_{i=0}^{N_{2}}\sum_{J\in{\mathbb{Z}}}\varphi_{\leqslant-
M_{0}-5}(|\xi|-|\eta|-|\sigma|)\\\ \times b_{i,J}(\xi,\eta,\sigma)\cdot
K_{i}\big{(}2^{J}(|\xi|-|\eta|-|\sigma|)\big{)},\end{split}$ (2.20)

where $K_{i}$ are Schwartz functions and $b_{i,J}$ are symbols satisfying the
bounds stated in (2.12).

We list some pointwise bounds satisfied by the regular measures
$\nu,\nu_{1}^{R}$ and $\mu^{Re}$, which can again be found in [20] (see
Proposition 5.6 and estimate (8.6)):

######  Lemma 2.4.

Let $M_{0}$ and $N_{2}$ be as in Lemma 2.3. For the measure $\nu(\xi,\eta)$ we
have

$\displaystyle\big{|}\varphi_{k}(\xi)\varphi_{k_{1}}(\eta)\nabla_{\xi}^{a}\nabla^{b}_{\eta}\nu(\xi,\eta)\big{|}\lesssim
1,\qquad|a|,|b|\geqslant 1.$ (2.21)

For $\mu^{Re}$ we have, for all $|a|+|\alpha|+|b|\leqslant N_{2}$,

$\displaystyle|\varphi_{k}(\xi)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}\mu^{Re}(\xi,\eta,\sigma)|$
$\displaystyle\lesssim 2^{-2\max\\{k_{1},k_{2},k_{3}\\}}\cdot
2^{-(|\alpha_{a}|+|\alpha_{e}|)\max\\{k_{1},k_{2},k_{3}\\}}$ (2.22)
$\displaystyle\cdot\max\big{(}1,2^{(1-|\alpha_{i}|)\min\\{k_{1},k_{2},k_{3}\\}}\big{)}\cdot
2^{5M_{0}(|a|+|\alpha|+|\beta|+1)},$

where $\alpha_{i}$ denotes the order of differentiation of the variable that
is located at the smallest frequency, $\alpha_{e}$ is similarly defined for
the middle frequency and $\alpha_{a}$ for the largest one.

Finally for $\nu_{1}^{R},$ we have for all $|a|+|b|\leq N_{2},$

$\displaystyle\begin{split}&|\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)\nabla_{p}^{a}\nabla_{q}^{b}\nu_{1}^{R}(\eta,\sigma)|\\\
&\lesssim\begin{cases}2^{-2k_{1}}\,2^{-(|a|+|b|)k_{1}}\cdot
2^{(2+|a|+|b|)5M_{0}}&\textrm{if}\quad|k_{1}-k_{2}|<5\\\ 2^{-2(k_{1}\vee
k_{2})}\cdot 2^{-|a|(k_{1}\vee k_{2})}\max\\{1,2^{(1-|b|)k_{2}^{-}}\\}\cdot
2^{(|a|+|b|+2)5M_{0}}&\textrm{if}\quad|k_{1}-k_{2}|\geqslant
5.\end{cases}\end{split}$ (2.23)

We now define bilinear operators associated with the measures above.

######  Definition 2.5.

For a general symbol $b=b(\xi,\eta,\sigma)$ let us denote

$\displaystyle T_{0}[b](G,H)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(\eta)H(\sigma)\,b(\xi,\eta,\sigma)\,\delta_{0}(\xi-\eta-\sigma)\,d\eta
d\sigma,$ (2.24a) $\displaystyle T_{1}^{\ast,1}[b](G,H)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(\xi-\eta)H(\sigma)\,b(\xi,\eta,\sigma)\,\nu_{1}^{\ast}(\eta,\sigma)\,d\eta
d\sigma,\quad\ast\in\\{S,R\\},$ (2.24b) $\displaystyle
T_{1}^{\ast,2}[b](G,H)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(-\eta-\sigma)H(\sigma)\,b(\xi,\eta,\sigma)\,\overline{\nu_{1}^{\ast}(\eta,\xi)}\,d\eta
d\sigma,\quad\ast\in\\{S,R\\},$ (2.24c) $\displaystyle T_{2}^{S}[b](G,H)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(\eta)H(\sigma)\,b(\xi,\eta,\sigma)\,\mu_{2}^{S}(\xi,\eta,\sigma)\,d\eta
d\sigma,$ (2.24d) $\displaystyle T_{3}^{S}[b](G,H)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(\eta)H(\sigma)\,b(\xi,\eta,\sigma)\,\mu_{3}^{S}(\xi,\eta,\sigma)\,d\eta
d\sigma,$ (2.24e) $\displaystyle T^{Re}[b](G,H)(x)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(\eta)H(\sigma)\,b(\xi,\eta,\sigma)\,\mu^{Re}(\xi,\eta,\sigma)\,d\eta
d\sigma.$ (2.24f)

We also adopt the following notation: for a general symbol $\underline{b}$ we
let (omitting the dependence on $k_{1},k_{2}\in{\mathbb{Z}}$)

$\displaystyle\begin{split}&b(\xi,\eta,\sigma):=\underline{b}(\xi,\eta,\sigma)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma),\qquad
b_{1}(\xi,\eta,\sigma):=\underline{b}(\xi,\xi-\eta,\sigma)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma),\\\
&b_{1}^{\prime}(\xi,\eta,\sigma):=\underline{b}(\xi,\sigma,\xi-\eta)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma),\qquad
b_{2}(\xi,\eta,\sigma):=\underline{b}(\xi,-\eta-\sigma,\sigma)\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma).\end{split}$
(2.25)

With the building blocks (2.24) we can further decompose $F_{L}$ into
‘singular’ and ‘regular’ parts. The singular parts (denoted with an apex $S$)
are defined so that the phases (2.6) are lower bounded on their support, and
we can apply a normal form transformation; this will give rise to boundary
terms that are quadratic (see (2.27) below) and cubic bulk terms (see (2.28)
below). The regular parts (see (2.30) and (2.32) below) cannot be transformed
but will satisfy better bilinear estimates. This singular vs. regular
decomposition is the content of the next lemma, and we refer the reader to
§5.4 of [20] for more details.

######  Lemma 2.6.

We have (up to irrelevant constants)

$\displaystyle
F_{L}(t)=\sum_{{\epsilon_{1},\epsilon_{2}}\in\\{+,-\\}}F^{S,(1)}_{{\epsilon_{1},\epsilon_{2}}}(t)-F^{S,(1)}_{{\epsilon_{1},\epsilon_{2}}}(0)+\sum_{m=0}^{L+1}F^{S,(2)}_{{\epsilon_{1},\epsilon_{2}},m}(t)+F^{R}_{{\epsilon_{1},\epsilon_{2}},m}(t)+F^{Re}_{{\epsilon_{1},\epsilon_{2}},m}(t)$
(2.26)

with

$\displaystyle\begin{split}F^{S,(1)}_{{\epsilon_{1},\epsilon_{2}}}(t)&=e^{-it\langle\xi\rangle}\varphi_{\leqslant
M_{0}(t)}(\xi)\sum_{k_{1},k_{2}\leqslant
M_{0}(t)}\sum_{G,H\in\\{g,h\\}}\widehat{\mathcal{F}}_{x\rightarrow\xi}\Bigg{[}T_{0}[b]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\\\
&+T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}+T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}}\big{)}\\\
&+T_{1}^{S,2}[b_{2}]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}+\sum_{i=2,3}T_{i}^{S}[b]\big{(}e^{{\epsilon}_{1}it\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}it\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\Bigg{]},\end{split}$
(2.27)

and (disregarding analogous symmetric terms)

$\displaystyle\begin{split}F^{S,(2)}_{{\epsilon_{1},\epsilon_{2}},m}(t)&=\int_{0}^{t}e^{-is\langle\xi\rangle}\sum_{k_{1},k_{2}\leqslant
M_{0}(s)}\sum_{G,H\in\\{g,h\\}}\widehat{\mathcal{F}}_{x\rightarrow\xi}\Bigg{[}T_{0}[b]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\\\
&+T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}+T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}}\big{)}\\\
&+T_{1}^{S,2}[b_{2}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}+\sum_{i=2,3}T_{i}^{S}[b]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\Bigg{]}\,\tau_{m}(s)ds,\end{split}$
(2.28)

where, using the notation (2.25), we define the symbols through

$\displaystyle\underline{b}(\xi,\eta,\sigma)=\frac{\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)}{\langle\eta\rangle\langle\sigma\rangle\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta,\sigma)}.$
(2.29)

Moreover

$\displaystyle\begin{split}F^{R}_{{\epsilon_{1},\epsilon_{2}},m}&:=F^{R,(1)}_{{\epsilon_{1},\epsilon_{2}},m}+F^{R,(2)}_{{\epsilon_{1},\epsilon_{2}},m}\\\
F^{R,(1)}_{{\epsilon_{1},\epsilon_{2}},m}(t)&:=\int_{0}^{t}e^{-is\langle\xi\rangle}\sum_{k_{1},k_{2}\leqslant
M_{0}(s)}\sum_{G,H\in\\{g,h\\}}\widehat{\mathcal{F}}_{x\rightarrow\xi}\Bigg{[}T_{1}^{R,1}[m_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\\\
&+T_{1}^{R,1}[m_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}}\big{)}\Bigg{]}\,\tau_{m}(s)ds,\\\
F^{R,(2)}_{{\epsilon_{1},\epsilon_{2}},m}(t)&:=\int_{0}^{t}e^{-is\langle\xi\rangle}\sum_{k_{1},k_{2}\leqslant
M_{0}(s)}\sum_{G,H\in\\{g,h\\}}\widehat{\mathcal{F}}_{x\rightarrow\xi}\bigg{[}T_{1}^{R,2}[m_{2}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\bigg{]}\,\tau_{m}(s)ds,\end{split}$
(2.30)

with, using again the notation (2.25),

$\displaystyle\underline{m}(\xi,\eta,\sigma):=\varphi_{\leqslant
M_{0}}(\langle\xi\rangle)\langle\eta\rangle^{-1}\langle\sigma\rangle^{-1}.$
(2.31)

Finally

$\displaystyle F^{Re}_{{\epsilon_{1},\epsilon_{2}},m}(t)$
$\displaystyle:=\int_{0}^{t}\int_{{\mathbb{R}}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\widetilde{f_{\epsilon_{1}}}(s,\eta)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\mu^{Re}(\xi,\eta,\sigma)d\eta
d\sigma\,\tau_{m}(s)ds.$ (2.32)

In what follows, we will often omit the indexes
${\epsilon}_{1},{\epsilon}_{2}$ in the notation for the bilinear terms above,
since these should create no confusion (and they will often play no role).

Next we recall Hölder type bounds satisfied by the operators in (2.24):

######  Theorem 2.7.

Let a symbol $b=b(\xi,\eta,\sigma)$ be given such that

$\displaystyle\textrm{supp}(b)\subseteq\big{\\{}(\xi,\eta,\sigma)\in\mathbb{R}^{9}\,:\,|\xi|+|\eta|+|\sigma|\leq
2^{A}\big{\\}},$

for some $A\geq 1$ and such that, for $|\xi|\approx 2^{K},|\eta|\approx
2^{L},|\sigma|\approx 2^{M}$,

$\displaystyle\big{|}\nabla_{\xi}^{a}\nabla_{\eta}^{\alpha}\nabla_{\sigma}^{\beta}b(\xi,\eta,\sigma)\big{|}\lesssim
2^{-|a|K-|\alpha|L-|\beta|M}\cdot
2^{(|a|+|\alpha|+|\beta|)A},\qquad|a|,|\alpha|,|\beta|\leqslant 4.$ (2.33)

Let $p,q,\in[1,\infty),$ and $r\geqslant 1$ with

$\frac{1}{p}+\frac{1}{q}=\frac{1}{r},$

and assume there is $10A\leqslant D\leqslant 2^{A/10}$ such that

$\displaystyle\mathcal{D}(G,H):=\|G\|_{L^{2}}\|H\|_{L^{2}}+\min\big{(}\|\partial_{\xi}G\|_{L^{2}}\|H\|_{L^{2}},\|G\|_{L^{2}}\|\partial_{\xi}H\|_{L^{2}}\big{)}\leqslant
2^{D}.$ (2.34)

Then, there exists an absolute constant $C_{0}$ (for example, $C_{0}:=65$ is
suitable), such that the following bilinear bounds hold for the operators
defined in (2.24):

$\displaystyle\big{\|}P_{K}T_{1}^{S,i}[b](G,H)\big{\|}_{L^{r}},\lesssim\|\widehat{G}\|_{L^{p}}\|\widehat{H}\|_{L^{q}}\cdot
2^{C_{0}M_{0}}+2^{-D}\mathcal{D}(G,H),\ \ i=1,2,$ (2.35a)
$\displaystyle\big{\|}P_{K}T_{i}^{S}[b](G,H)\big{\|}_{L^{r}}\lesssim\|\widehat{G}\|_{L^{p}}\|\widehat{H}\|_{L^{q}}\cdot
2^{C_{0}M_{0}},\qquad i=2,3,$ (2.35b)
$\displaystyle\big{\|}P_{K}T^{Re}[b](G,H)\big{\|}_{L^{r}},\big{\|}P_{K}T_{1}^{R,i}[b](G,H)\big{\|}_{L^{r}}\lesssim\|\widehat{G}\|_{L^{p}}\|\widehat{H}\|_{L^{q}}\cdot
2^{C_{0}A},\ \ i=1,2.$ (2.35c)

######  Remark 2.8 (Bilinear estimates).

Note that the bilinear estimates in (2.35) differ from those of [20] since we
allow for the endpoint $r=1.$ Note that it is possible at the expense of
adding a projection $P_{K}$ in front, see $(6.35)$ in [23]. This is harmless
since our estimates are done in $L^{\infty}_{\xi}.$

######  Remark 2.9 (Estimating $\mathcal{D}$).

In the bilinear bound (2.35a) the quantity $\mathcal{D}$ appears. This is a
lower order remainder term in all our estimates, and is already adequately
estimated in [20]; therefore we can disregard these terms in our proof.

##  3\. Bilinear estimates

This section contains the new bilinear estimates we need for the proof of
Proposition 1.2.

######  Lemma 3.1.

Let $m=m(\xi,\eta)$, $\xi,\eta\in{\mathbb{R}}^{3}$ be a bounded multiplier
localized at frequencies $|\xi|\approx 2^{K},|\eta|\approx 2^{L}$ such that

$\displaystyle\big{|}\nabla_{\xi}^{\alpha}\nabla_{\eta}^{\beta}m(\xi,\eta)\big{|}\lesssim
2^{-|\alpha|K-|\beta|L},\ \ 0\leqslant|\alpha|+|\beta|\leqslant 4,$ (3.1)

and let $b_{0}=b_{0}(\omega,\xi)$,
$\omega\in\mathbb{S}^{2},\xi\in{\mathbb{R}}^{3}$, be such that

$\displaystyle\big{|}\nabla_{\xi}^{\beta}b_{0}(\omega,\xi)\big{|}\lesssim
1+\big{(}|\xi|/\langle\xi\rangle\big{)}^{1-|\beta|},\ \
0\leqslant|\beta|\leqslant 4.$ (3.2)

We have (note the frequency localization on the profile $h$)

$\displaystyle\begin{split}\Bigg{\|}\int_{\mathbb{R}^{6}}\widetilde{f}(\xi-\eta)(\varphi_{M}\widetilde{h})(\sigma)m(\xi,\eta)b_{0}(\eta/|\eta|,\sigma)\mathrm{p.v.}\frac{1}{|\eta|-|\sigma|}d\eta
d\sigma\Bigg{\|}_{L^{\infty}_{\xi}}\\\ \lesssim 2^{L+M}\langle
2^{M}\rangle^{-2}\|m\|_{L^{\infty}_{\xi,\eta}}\|f\|_{L^{2}}\|h\|_{H^{2}},\end{split}$
(3.3)

and

$\displaystyle\Bigg{\|}\int_{\mathbb{R}^{6}}\widetilde{f}(\eta+\sigma)\widetilde{h}(\sigma)m(\eta,\sigma)\mathrm{p.v.}\frac{1}{|\eta|-|\xi|}b_{0}(\eta/|\eta|,\xi)d\eta
d\sigma\Bigg{\|}_{L^{\infty}_{\xi}}\lesssim 2^{2K}\langle
2^{K}\rangle^{-2}\|f\|_{H^{2}}\|h\|_{H^{2}}.$ (3.4)

###### Proof.

Let $\epsilon\in(0,1).$ We stress that the implicit constants in the argument
below do not depend on $\epsilon.$ Denote $\varphi_{\epsilon}$ a radial cut-
off function supported on the region $\epsilon<r<\epsilon^{-1}.$

We first prove the more challenging inequality (3.4). We start by inserting a
cut-off $\varphi_{k}(\xi)$ and pass to polar coordinates $\xi=\rho\phi$ and
$\eta=r\omega$, $\phi,\omega\in\mathbb{S}^{2}$, and estimate the left-hand
side of (3.4) by

$\displaystyle\sup_{\epsilon>0}\sup_{k\in\mathbb{Z}}\sup_{\phi\in\mathbb{S}^{2},\rho>0}\Big{|}\int_{\mathbb{S}^{2}}\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\widetilde{f}(r\omega+\sigma)m(r\omega,\sigma)\widetilde{h}(\sigma)d\sigma\,\frac{\varphi_{\epsilon}(r-\rho)}{r-\rho}\varphi_{k}(\rho)b_{0}(\omega,\rho\phi)r^{2}drd\omega\Big{|}.$
(3.5)

Let us define $f_{1}:=L^{2}f$ and $h_{1}:=L^{2}h$, and

$\displaystyle
F(r,\omega):=\int_{\mathbb{R}^{3}}\widetilde{f_{1}}(r\omega+\sigma)m_{1}(r\omega,\sigma)\widetilde{h_{1}}(\sigma)d\sigma,\quad
m_{1}(\eta,\sigma):=\frac{m(\eta,\sigma)}{\langle\eta+\sigma\rangle^{2}\langle\sigma\rangle^{2}}.$

We then take an inverse Fourier transform in $\rho$ in (3.5); using the fact
that the Fourier transform of the Hilbert transform kernel is bounded, we can
bound (3.5) by a constant times

$\displaystyle\sup_{\epsilon>0}\sup_{k\in\mathbb{Z}}\sup_{\phi\in\mathbb{S}^{2}}\Bigg{\|}\int_{\mathbb{S}^{2}}\widehat{\mathcal{F}}^{-1}_{\rho\rightarrow
x}\Bigg{\\{}\Bigg{[}\int_{\mathbb{R}}r^{2}F(r,\omega)\textbf{1}_{[0,\infty)}(r)\frac{\varphi_{\epsilon}(r-\rho)}{r-\rho}dr\Bigg{]}\varphi_{k}(\rho)b_{0}(\omega,\rho\phi)\Bigg{\\}}d\omega\Bigg{\|}_{L^{1}_{x}(\mathbb{R})}$
$\displaystyle\lesssim\sup_{\epsilon>0}\sup_{k\in\mathbb{Z}}\sup_{\omega,\phi\in\mathbb{S}^{2}}\bigg{\|}\widehat{\mathcal{F}}^{-1}_{\rho\rightarrow
x}\bigg{(}\frac{\varphi_{\epsilon}(r)}{r}\ast\big{(}r^{2}F(r,\omega)\textbf{1}_{[0,\infty)}(r)\big{)}\bigg{)}\bigg{\|}_{L^{1}_{x}(\mathbb{R})}{\big{\|}\mathcal{F}^{-1}_{\rho\rightarrow
x}\big{(}\varphi_{k}(\cdot)b_{0}(\omega,\cdot\phi)\big{)}\big{\|}}_{L^{1}_{x}(\mathbb{R})}$
$\displaystyle\lesssim\bigg{\|}\widehat{\mathcal{F}}^{-1}_{r\rightarrow
x}\big{(}r^{2}F(r,\omega)\textbf{1}_{[0,\infty)}(r)\big{)}\bigg{\|}_{L^{\infty}_{\omega,\phi}L^{1}_{x}}\cdot\sup_{k\in\mathbb{Z}}\bigg{\|}\widehat{\mathcal{F}}^{-1}_{\rho\rightarrow
x}\big{(}\varphi_{k}(\cdot)b_{0}(\omega,\cdot\phi)\big{)}\bigg{\|}_{L^{\infty}_{\omega,\phi}L^{1}_{x}}.$
(3.6)

To bound the second term in (3.6) we use the pointwise bound

$\displaystyle\big{|}\widehat{\mathcal{F}}^{-1}_{\rho\rightarrow
x}\big{(}\varphi_{k}(\cdot)b_{0}(\omega,\cdot\phi)\big{)}(x)\big{|}\lesssim\frac{2^{k}}{(1+|x|)(1+2^{k}/\langle
2^{k}\rangle|x|)^{2}},$

which follows by integration by parts in $\rho$ and our assumptions (3.2) on
$b_{0}.$ This yields

$\displaystyle\bigg{\|}\widehat{\mathcal{F}}^{-1}_{\rho\rightarrow
x}\big{(}\varphi_{k}(\cdot)b_{0}(\omega,\cdot\phi)\big{)}\bigg{\|}_{L^{\infty}_{\omega,\phi}L^{1}_{x}}\lesssim\langle
2^{k}\rangle.$

We then focus on the first term. We denote the wave operator by
$\mathcal{W}:=\widetilde{\mathcal{F}}^{-1}\widehat{\mathcal{F}},\mathcal{W}^{*}=\widehat{\mathcal{F}}^{-1}\widetilde{\mathcal{F}},$
and recall that it is bounded on Sobolev spaces under our assumptions. We
change coordinates by denoting $R_{\omega}$ the rotation that sends $\omega$
to $e_{1}:=(1,0,0)$ and denoting $g^{\omega}:=g\circ R_{\omega}^{-1}$, so that
we find (omitting irrelevant constants)

$\displaystyle\widehat{\mathcal{F}}^{-1}_{r\rightarrow
x}\big{(}r^{2}F(r,\omega)\textbf{1}_{[0,\infty)}(r)\big{)}=\int_{\mathbb{R}}e^{ixr}\int_{\mathbb{R}^{3}}\widetilde{f_{1}}(\sigma+r\omega)m_{1}(r\omega,\sigma)\widetilde{h_{1}}(\sigma)d\sigma\,r^{2}\textbf{1}_{[0,\infty)}(r)dr$
$\displaystyle=\int_{\mathbb{R}^{6}}\Bigg{[}\int_{\mathbb{R}^{4}}e^{ir(x-y\cdot\omega)}e^{-i\sigma\cdot(y+z)}m_{1}(r\omega,\sigma)r^{2}\textbf{1}_{[0,\infty)}(r)drd\sigma\Bigg{]}\mathcal{W}^{*}f_{1}(y)\mathcal{W}^{*}h_{1}(z)dydz$
$\displaystyle:=\int_{\mathbb{R}^{6}}T[m_{1}](x-y_{1},R_{\omega}^{-1}y+R_{\omega}^{-1}z)(\mathcal{W}^{*}f_{1})^{\omega}(y)(\mathcal{W}^{*}h_{1})^{\omega}(z)dydz.$

Note that the last identity defines $T[m_{1}]$. We then observe that, given
the localization of $m_{1}(\eta,\sigma)$ to $|\eta|\approx 2^{K}$ and
$|\sigma|\approx 2^{L}$, using integration by parts and (3.1) we have

$\displaystyle\big{|}T[m_{1}](y_{1},z)\big{|}=\Bigg{|}\int_{\mathbb{R}^{4}}e^{iy_{1}r}e^{-i\sigma\cdot
z}m_{1}(r\omega,\sigma)r^{2}\textbf{1}_{[0,\infty)}(r)drd\sigma\Bigg{|}\lesssim\frac{2^{3K}\langle
2^{K}\rangle^{-2}}{\big{(}1+2^{K}|y_{1}|\big{)}^{2}}\frac{2^{3L}\langle
2^{L}\rangle^{-2}}{\big{(}1+2^{L}|z|\big{)}^{4}}.$

This yields the bound

$\displaystyle\big{\|}T[m_{1}]\big{\|}_{L^{1}({\mathbb{R}}\times{\mathbb{R}}^{3})}\lesssim
2^{2K}\langle 2^{K}\rangle^{-2}.$

Changing variables $z\mapsto y-z,y\mapsto(x-y_{1},y_{2},y_{3})$ and using the
Cauchy-Schwarz inequality as well as the boundedness of wave operators on
$L^{2}$, we have

$\displaystyle\big{|}\widehat{\mathcal{F}}^{-1}_{r\rightarrow
x}\big{(}r^{2}F(r,\omega)\textbf{1}_{[0,\infty)}(r)\big{)}\big{|}\lesssim\int_{\mathbb{R}^{4}}\big{|}T[m_{1}](y_{1},R_{\omega}^{-1}z)\big{|}dy_{1}dz$
$\displaystyle\times\sup_{(y_{1},z)\in\mathbb{R}^{4}}\int_{\mathbb{R}^{3}}\big{|}(\mathcal{W}^{*}f_{1})^{\omega}(x-y_{1},y_{2},y_{3})\big{|}\cdot\big{|}(\mathcal{W}^{*}h_{1})^{\omega}(x-y_{1}-z_{1},y_{2}-z_{2},y_{3}-z_{3})\big{|}dxdy_{2}dy_{3}$
$\displaystyle\lesssim 2^{2K}\langle
2^{K}\rangle^{-2}{\|f_{1}\|}_{L^{2}}{\|h_{1}\|}_{L^{2}}\lesssim\|f\|_{H^{2}}\|h\|_{H^{2}}.$

We now prove the easier inequality (3.3). Passing to polar coordinates
$\eta=r\omega$ and $\sigma=\rho\phi,$ $\omega,\phi\in\mathbb{S}^{2}$, we
estimate using the boundedness of the Hilbert transform on $L^{2},$ and
recalling that on the support of the integral $|\xi|\approx 2^{K}$,
$|\eta|\approx 2^{L}$ and $|\sigma|\approx 2^{M}$:

$\displaystyle\Bigg{|}\int_{0}^{\infty}\int_{\mathbb{S}^{2}}\widetilde{f}(\xi-r\omega)m(\xi,r\omega)\int_{0}^{\infty}\int_{\mathbb{S}^{2}}b_{0}(\omega,\rho\phi)\frac{\varphi_{\epsilon}(r-\rho)}{r-\rho}\widetilde{h}(\rho\phi)\varphi_{M}(\rho)\rho^{2}d\rho
d\phi\,r^{2}drd\omega\Bigg{|}$
$\displaystyle\lesssim\int_{\mathbb{S}^{2}}{\big{\|}\widetilde{f}(\xi-r\omega)m(\xi,r\omega)\,r\|}_{L^{2}_{r}(r^{2}dr)}d\omega\cdot\sup_{\omega\in\mathbb{S}^{2}}\bigg{\|}\frac{\varphi_{\epsilon}(r)}{r}\ast\bigg{(}\int_{\mathbb{S}^{2}}b_{0}(\omega,r\phi)\widetilde{h}(r\phi)\varphi_{M}(r)r^{2}d\phi\bigg{)}\bigg{\|}_{L^{2}_{r}}$
$\displaystyle\lesssim{\|m\|}_{L^{\infty}_{\xi,\eta}}\cdot
2^{L}{\big{\|}\varphi_{L}(\cdot)\widetilde{f}(\xi-\cdot)\big{\|}}_{L^{2}}\cdot
2^{M}{\big{\|}\varphi_{M}\widetilde{h}\big{\|}}_{L^{2}({\mathbb{R}}^{3})}.$

∎

##  4\. Asymptotic behavior of $h$

In this section we begin the verification of the hypotheses in Lemma 1.5. We
start by reducing these to proving a set of bounds for the bilinear and
trilinear terms appearing in Lemma 2.6.

######  Lemma 4.1.

Recall the decomposition for $h$ in (2.2)-(2.7) (see also the notation for
$F_{H,m}$ below (2.7)), and for $F_{L}$ in (2.26)-(2.32). Let (see (2.27))

$\displaystyle\mathcal{N}(\widetilde{f})(t,\xi):=F^{S,(1)}(t,\xi).$ (4.1)

Assume a priori that, for all $t\geqslant 0$,

$\displaystyle{\big{\|}\widetilde{h}(t)\big{\|}}_{L^{\infty}_{\xi}}\leqslant\varepsilon^{\delta},$
(4.2)

and that the following bounds hold for all $m=0,1,\dots,L+1$ (recall (2.1)):

$\displaystyle\|S_{m}\|_{L^{\infty}_{\xi}},\,\|M_{m}\|_{L^{\infty}_{\xi}},\,\|F_{H,m}\|_{L^{\infty}_{\xi}},\,\|F^{S,(2)}_{m}\|_{L^{\infty}_{\xi}},\,\|F^{R}_{m}\|_{L^{\infty}_{\xi}},\,\|F^{Re}_{m}\|_{L^{\infty}_{\xi}}\lesssim
2^{-2\delta m}\varepsilon^{\delta}.$ (4.3)

Then the following hold true:

  * (i)

There exists $h_{\infty}\in{\varepsilon}^{\delta}L^{\infty}_{\xi}$ such that
(1.24) holds true.

  * (ii)

We have

$\displaystyle{\big{\|}\mathcal{N}(\widetilde{f})(t)\big{\|}}_{L^{\infty}_{t}L^{\infty}_{\xi}}\lesssim\varepsilon^{2\delta}.$
(4.4)

  * (iii)

The a priori assumption (4.2) can be improved to

$\displaystyle{\big{\|}\widetilde{h}(t)\big{\|}}_{L^{\infty}_{\xi}}\leqslant\tfrac{1}{2}\varepsilon^{\delta}.$
(4.5)

The main goal of this section is to prove the estimates (4.3); these proofs
rely on the machinery developed in [20]. We will also use the following:

######  Remark 4.2.

Note that obtaining a bound by $2^{-3\delta m}$ for all the quantities on the
left-hand side of (4.3) is sufficient since, as in [20], we may assume that we
are working past the local time of existence of $O(1/\varepsilon).$

Below we show how (i) and (iii) in Lemma 4.1 follow from the assumptions (4.2)
and (4.3), using (ii) as well. The bound (4.4) in (ii) is proved in Section 5
relying on the new bilinear bounds from Section 3 and the a priori assumption
(4.2).

###### Proof of (i) and (iii) in Lemma 4.1.

From the formulas (2.2), (2.7) (see also the notation for $F_{H,m}$ below
that), and (2.26), we see that (4.3) implies, for all $t_{1}<t_{2}$,

$\displaystyle{\big{\|}\widetilde{h}(t_{1})-\mathcal{N}(\widetilde{f})(t_{1})-\big{(}\widetilde{h}(t_{2})-\mathcal{N}(\widetilde{f})(t_{2})\big{)}\big{\|}}_{L^{\infty}_{\xi}}\lesssim{\varepsilon}^{\delta}\langle
t_{1}\rangle^{-2\delta}.$

Then $\widetilde{h}(t)-\mathcal{N}(\widetilde{f})(t)$ is Cauchy in time with
values in $L^{\infty}_{\xi}$ and the first claim follows.

For the bound (4.5) we use again the formula for $h$ given by (2.2) with (2.7)
and (2.26), and the bounds on each of the terms appearing in this formula
provided by (4.3) (under the assumption that $2^{m}\gtrsim{\varepsilon}^{-1}$,
see Remark 4.2) and (4.4) (see the definition (4.1)). ∎

To summarize, in order to prove Lemma 4.1 and obtain our main result in
Proposition 1.2, it suffices to prove all the bounds in (4.3), and the bound
(4.4) under the assumption (4.2).

######  Remark 4.3.

We remark that the uniform bound (4.2) for $\widetilde{h}$ is not proven in
[20]. Moreover, as a consequence of (4.2) and (2.1) we also have, for all
$t\geqslant 0$,

$\displaystyle\|\widetilde{f}(t)\|_{L^{\infty}_{\xi}}\lesssim 1.$ (4.6)

Proof of (4.3). We estimate each of the terms on the right-hand side of (4.3)
by $2^{-3\delta m}$. We will constantly use the bilinear bounds stated in
Theorem 2.7 (see also Remark 2.9) without referring to them at each
application.

Step 1: Bounding $F^{S,(2)}_{m}.$ From (2.28) we have that

$\displaystyle\|F^{S,(2)}_{m}\|_{L^{\infty}_{\xi}}\lesssim
2^{m}\cdot\sup_{k\in\mathbb{Z}}\sum_{G,H\in\\{g,h\\}}\sum_{k_{1},k_{2}\leqslant\delta_{N}m}\sup_{s\approx
2^{m}}\Big{[}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T_{0}[b]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle+{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{\infty}_{\xi}}+{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T_{1}^{S,1}[b_{1}^{\prime}]\big{(}e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}}\big{)}\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle+{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T_{1}^{S,2}[b_{2}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{\infty}_{\xi}}+\sum_{i=2,3}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T_{i}^{S}[b]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{G_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{\infty}_{\xi}}\Big{]}.$

Since the sum over $k_{1},k_{2}$ will only contribute $O(m)$ terms we will
often disregard them.

Case 1: $G=g$. We estimate using (2.35a), the decay estimates (1.26c) and
(1.28a), that

$\displaystyle{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim{\big{\|}P_{k}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{g_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{1}_{x}}$
$\displaystyle\lesssim{\big{\|}e^{isL}\partial_{s}g\big{\|}}_{L^{6/5}_{x}}\cdot{\|e^{isL}P_{k_{2}}H\|}_{L^{6}_{x}}\cdot
2^{C_{0}M_{0}}$ $\displaystyle\lesssim\rho(2^{m})\cdot 2^{-m+10M_{0}}Z\cdot
2^{C_{0}M_{0}}.$

This is an acceptable bound.

Case 2: $G=h$. Using (2.35a), (1.26c) and (1.28b), we estimate similarly that

$\displaystyle\begin{split}{\big{\|}P_{k}T_{1}^{S,1}[b_{1}]\big{(}e^{{\epsilon}_{1}is\langle\xi\rangle}\partial_{s}\widetilde{h_{{\epsilon}_{1}}},e^{{\epsilon}_{2}is\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}}\big{)}\big{\|}}_{L^{1}_{x}}&\lesssim{\big{\|}e^{isL}\partial_{s}h\big{\|}}_{L^{3/2-}_{x}}\cdot{\|e^{isL}P_{k_{2}}H\|}_{L^{3}_{x}}\cdot
2^{C_{0}M_{0}}\\\ &\lesssim 2^{-m+10M_{0}}Z^{2}\cdot 2^{-m/2+10M_{0}}Z\cdot
2^{C_{0}M_{0}},\end{split}$ (4.7)

which yields a sufficient bound.

Step 2: Bounding $F^{R}_{m},F^{Re}_{m}.$ From (2.22) and (2.23) one can see
that $F^{Re}_{m}$ satisfies better estimates than $F^{R}_{m},$ and is
therefore strictly easier to treat. Therefore we only explain how to deal with
$F^{R}_{m}.$ Starting with the first piece we write that, using (2.30),

$\displaystyle\begin{split}\|F^{R,(1)}_{m}\|_{L^{\infty}_{\xi}}&\lesssim
2^{m}\sum_{G,H\in\\{g,h\\}}\sum_{k_{1},k_{2}\leqslant\delta_{N}m}\sup_{s\approx
2^{m}}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}})\big{\|}}_{L^{\infty}_{\xi}}\\\
&+{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}^{\prime}](e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}},e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}})\big{\|}}_{L^{\infty}_{\xi}};\end{split}$

Recall the definition for the symbols (2.31) with the notation (2.25), and
note that these always localize the integration variables $\eta,\sigma$ in the
formulas (2.24) to $|\eta|\approx 2^{k_{1}}$, $|\sigma|\approx 2^{k_{2}}$, but
do not localize the inputs of the operators necessarily. It suffices to
estimate the term $T^{R,1}_{1}[b_{1}]$.

Case 1: $k_{2}\leqslant-m/2+\delta m.$ Then integrating directly we find that,
using (1.26d) and (2.23),

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}})\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim
2^{m}\|\varphi_{k_{1}}(\cdot-\eta)\widetilde{G}(s,\eta)\|_{L^{1}_{\eta}}\cdot\|\varphi_{k_{2}}\widetilde{H}\|_{L^{1}_{\sigma}}\cdot\sup_{|\eta|\approx
2^{k_{1}},|\sigma|\approx 2^{k_{2}}}|\nu^{R}_{1}(\eta,\sigma)|$
$\displaystyle\lesssim 2^{m}\cdot 2^{(5/2-)k_{1}}Z\cdot 2^{(5/2-)k_{2}}Z\cdot
2^{-2(k_{1}\vee k_{2})}2^{10M_{0}}.$

This gives the desired bound.

Case 2: $k_{2}\geqslant-m/2+\delta m$ and $k_{1}\leqslant-m/10.$

Case $H=h.$ Integrating by parts in $\sigma$ and using (1.26d) and (2.23)
(note that the case where the derivative hits the cut-offs or measure are
better and can be disregarded) we obtain the bound

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{{\epsilon}_{2}}})\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\sup_{s\approx
2^{m}}\|\varphi_{k_{1}}(\cdot-\eta)\widetilde{G}(s,\eta)\|_{L^{1}_{\eta}}\cdot\bigg{\|}\varphi_{k_{2}}(\sigma)\big{(}\frac{\sigma}{|\sigma|^{2}}\cdot\nabla_{\sigma}\big{)}\widetilde{h}\bigg{\|}_{L^{1}_{\sigma}}\cdot\bigg{(}\sup_{|\eta|\approx
2^{k_{1}},|\sigma|\approx 2^{k_{2}}}|\nu_{1}^{R}(\eta,\sigma)|\bigg{)}$
$\displaystyle\lesssim 2^{(5/2-)k_{1}}Z\cdot 2^{k_{2}/2}Z\cdot 2^{-2(k_{1}\vee
k_{2})}2^{10M_{0}},$

which is sufficient to conclude.

Case $H=g.$ In this case note first that $k_{2}\approx 0.$ We let
$\ell_{0}:=\lfloor-m+\delta m\rfloor$ and insert cut-offs
$\varphi_{\ell}^{(\ell_{0})}(\langle\sigma\rangle-2\lambda)$ defined for
$\ell\geqslant\ell_{0}$ that are such that
$\varphi_{\ell_{0}}^{(\ell_{0})}=\varphi_{\leqslant\ell_{0}}$ and
$\varphi_{\ell}^{(\ell_{0})}=\varphi_{\ell}$ if $\ell>\ell_{0}$. In the case
where $\ell=\ell_{0}$ we integrate directly and find that, using (1.27a),
(1.26d) and (2.23)

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\varphi_{\ell}^{(\ell_{0})}(\langle\xi\rangle-2\lambda)\widetilde{g_{{\epsilon}_{2}}})\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim 2^{m}\cdot\sup_{s\approx
2^{m}}\|\varphi_{k_{1}}(\cdot-\eta)\widetilde{G}(s,\eta)\|_{L^{1}_{\eta}}\cdot
2^{\ell_{0}}\|\widetilde{g}\|_{L^{\infty}_{\sigma}}\cdot 2^{10M_{0}}\lesssim
2^{m}\cdot 2^{(5/2-)k_{2}}Z\cdot 2^{\ell_{0}}\rho(2^{m})2^{m}m\cdot
2^{10M_{0}},$

which yields an acceptable bound provided $\delta>0$ is chosen small enough.
If $\ell>\ell_{0}$ we can integrate by parts in $\sigma$ and using (1.27b),
(1.26d) and (2.23) we obtain

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\widetilde{G_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{g_{{\epsilon}_{2}}})\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\sum_{\ell>\ell_{0}}\sup_{s\approx
2^{m}}\|\varphi_{k_{1}}(\cdot-\eta)\widetilde{G}(s,\eta)\|_{L^{1}_{\eta}}\cdot\|\varphi_{\ell}^{(\ell_{0})}(\langle\sigma\rangle-2\lambda)\nabla_{\sigma}\widetilde{g}\|_{L^{1}_{\sigma}}\cdot
2^{10M_{0}}\lesssim 2^{(5/2-)k_{1}}Z\cdot 2^{m}\rho(2^{m})m^{2}\cdot
2^{10M_{0}},$

which is an acceptable bound.

Case 3: $k_{1}\geqslant-m/10,k_{2}\geqslant-m/2+\delta m.$ In this case we
have from (2.23) that

$\displaystyle\sup_{|\eta|\approx 2^{k_{1}},|\sigma|\approx
2^{k_{2}}}|\nu_{1}^{R}(\eta,\sigma)|\lesssim 2^{m/10+10M_{0}}.$ (4.8)

Next we insert another cut-off $\varphi_{k_{3}}(\xi-\eta)$ to localize the
frequency of the input $G$.

Subcase 3.1: $k_{3}\leqslant-m/2+\delta m.$ We can integrate directly as in
Case 1 and obtain, using (1.26d) and (4.8),

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}{\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\varphi_{k_{3}}(\xi)\widetilde{G_{{\epsilon}_{1}}}(s,\xi),e^{is\epsilon_{2}\langle\xi\rangle}\varphi_{\sim
k_{2}}(\xi)\widetilde{H_{{\epsilon}_{2}}})\big{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim 2^{m}\cdot 2^{(5/2-)(k_{1}\wedge k_{3})}Z\cdot
2^{(5/2-)k_{2}}Z\cdot 2^{m/10+10M_{0}}.$

Subcase 3.2: $k_{3}>-m/2+\delta m.$ In this case we integrate by parts in
$\eta$ and $\sigma.$ Treating first the case where $(G,H)=(h,h)$ we write,
using (1.26a) and (4.8),

$\displaystyle 2^{-m}\sup_{s\approx
2^{m}}{\bigg{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}]\big{(}e^{is{\epsilon}_{1}\langle\xi\rangle}\varphi_{k_{3}}(\xi)\big{(}\frac{\xi}{|\xi|^{2}}\cdot\nabla_{\xi}\big{)}\widetilde{h_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\varphi_{\sim
k_{2}}(\xi)\big{(}\frac{\xi}{|\xi|^{2}}\cdot\nabla_{\xi}\big{)}\widetilde{h_{{\epsilon}_{2}}}\big{)}\bigg{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim 2^{-m}\cdot 2^{-k_{3}}2^{(3/2)(k_{3}\wedge
k_{1})}Z\cdot 2^{k_{2}/2}Z\cdot 2^{m/10+10M_{0}}.$

The desired bound follows.

Moving on to the $(g,g)$ case, we insert cutoffs
$\varphi_{\ell_{1}}^{(\ell_{0})}(\langle\xi-\eta\rangle-2\lambda)$ and
$\varphi_{\ell_{2}}^{(\ell_{0})}(\langle\sigma\rangle-2\lambda)$ for
$\ell_{0}=\lfloor-m+\delta m\rfloor$ as above. Note that $k_{3},k_{2}\approx
0.$ First we treat the case where $\ell_{1},\ell_{2}>\ell_{0}.$ Then we can
integrate by parts in both $\eta$ and $\sigma$ to obtain, using (1.27b), (4.8)

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}{\bigg{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,1}_{1}[b_{1}](e^{is{\epsilon}_{1}\langle\xi\rangle}\varphi_{\sim
0}(\xi)\widetilde{g_{{\epsilon}_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\varphi_{\sim
0}(\xi)\widetilde{g_{{\epsilon}_{2}}})\bigg{\|}}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim
2^{-m}\sum_{\ell_{1},\ell_{2}>\ell_{0}}\|\varphi_{\ell_{1}}^{(\ell_{0})}(\langle\eta\rangle-2\lambda)\nabla_{\eta}\widetilde{g_{{\epsilon}_{1}}}\|_{L^{1}_{\eta}}\cdot\|\varphi_{\ell_{2}}^{(\ell_{0})}(\langle\sigma\rangle-2\lambda)\nabla_{\sigma}\widetilde{g_{{\epsilon}_{2}}})\|_{L^{1}_{\sigma}}\cdot
2^{m/10+10M_{0}}$ $\displaystyle\lesssim
2^{-m}\cdot\big{(}2^{m}\rho(2^{m})m^{2}\big{)}^{2}\cdot 2^{m/10+10M_{0}}.$

In the case where one of the indices $\ell_{i}$ equals $\ell_{0},$ we can
integrate directly instead of integrating by parts. We skip the details.

Similarly for the mixed cases $(G,H)=(h,g)$ or $(g,h)$ we can combine the
approaches above to obtain the desired bound.

Next, we deal with the piece $F^{R,(2)}_{m}.$

Case 1: $k_{2}\leqslant-m/2+m/100$. Note that we can add a cut-off
$\varphi_{\leqslant k_{1}\vee k_{2}+5}(\sigma)$ on the profile $H.$ Then we
can write, using (1.26d) and (2.23), that

$\displaystyle 2^{m}\sup_{s\approx
2^{m}}\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,2}_{1}[b_{2}](e^{is\epsilon_{1}\langle\xi\rangle}\widetilde{G_{\epsilon_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{\epsilon_{2}}})\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim
2^{m}\cdot\|\varphi_{k_{1}}(\cdot+\eta)\widetilde{G}(s,\eta)\|_{L^{1}_{\eta}}\|\varphi_{k_{2}}\widetilde{H_{\epsilon_{2}}}\|_{L^{1}_{\sigma}}\cdot
2^{-2k_{1}}2^{10M_{0}}\lesssim 2^{m}\cdot 2^{(5/2-)k_{1}}Z\cdot
2^{(5/2-)k_{2}}Z\cdot 2^{-2k_{1}}2^{10M_{0}}.$

This is sufficient to conclude.

Case 2: $k_{2}>-m/2+m/100$.

Subcase 2.1: $k_{1}\leqslant-m/2+m/200$. In this case $|\eta+\sigma|\approx
2^{k_{2}},$ and we can add a cutoff $\varphi_{\sim k_{2}}(\eta+\sigma)$ to the
expression.

Subcase 2.1.1: $(\epsilon_{1},\epsilon_{2})\neq(+,-),(-,+)$. In this case we
note that $|\nabla_{\sigma}\Phi_{{\epsilon_{1},\epsilon_{2}}}|\gtrsim
2^{k_{2}}\langle 2^{k_{2}}\rangle^{-1}$ and, therefore, we can integrate by
parts in $\sigma.$ We may assume that $k_{2}<0$, as otherwise the bound is
easier to obtain. Integration by parts gives two types of main terms, one when
the derivative hits $H$ and another when it hits $G.$ In the first case we
integrate directly, in the second we change variables
$\eta\mapsto-\eta-\sigma$ and then integrate. Overall, using (1.26d), (1.26a),
(1.27b), (1.27a), (2.23) we get,

$\displaystyle 2^{m}\sum_{G,H\in\\{g,h\\}}\sup_{s\approx
2^{m}}\big{\|}\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,2}_{1}[b_{2}](e^{is\epsilon_{1}\langle\xi\rangle}\widetilde{G_{\epsilon_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{\epsilon_{2}}})\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\sum_{H\in\\{g,h\\}}2^{m}\cdot\big{\|}\varphi_{k_{1}}(\eta)\widetilde{G}(s,\eta+\sigma)\big{\|}_{L^{\infty}_{\sigma}L^{1}_{\eta}}\bigg{[}2^{-m}2^{-k_{2}}\big{\|}\varphi_{\sim
k_{2}}\nabla_{\sigma}\widetilde{h}\big{\|}_{L^{1}_{\sigma}}+2^{\ell_{0}}\|\widetilde{g}\|_{L^{\infty}_{\sigma}}$
$\displaystyle+\sum_{\ell>\ell_{0}}2^{-m}\,\big{\|}\varphi_{\ell}^{(\ell_{0})}(\langle\sigma\rangle-2\lambda)\nabla_{\sigma}\widetilde{g}\big{\|}_{L^{1}_{\sigma}}\bigg{]}\cdot\sup_{\xi\in\mathbb{R}^{3},|\eta|\approx
2^{k_{1}}}|\nu^{R}_{1}(\xi,\eta)|$
$\displaystyle+\sum_{G\in\\{g,h\\}}2^{m}\cdot\big{\|}\varphi_{k_{1}}(\cdot+\sigma)\widetilde{H}(s,\sigma)\big{\|}_{L^{\infty}_{\eta}L^{1}_{\sigma}}\bigg{[}2^{-m}2^{-k_{2}}\big{\|}\varphi_{\sim
k_{2}}\widetilde{h}\big{\|}_{L^{1}_{\eta}}+2^{\ell_{0}}\|\widetilde{g}\|_{L^{\infty}_{\eta}}$
$\displaystyle+\sum_{\ell>\ell_{0}}2^{-m}\,\big{\|}\varphi_{\ell}^{(\ell_{0})}(\langle\sigma\rangle-2\lambda)\nabla_{\sigma}\widetilde{g}\big{\|}_{L^{1}_{\eta}}\bigg{]}\cdot\sup_{\xi\in\mathbb{R}^{3},|\eta|\approx
2^{k_{1}}}|\nu^{R}_{1}(\xi,\eta)|$ $\displaystyle\lesssim
2^{(5/2-)k_{1}}Z\cdot\big{[}2^{k_{2}/2}Z+2^{\delta
m}2^{m}\rho(2^{m})+2^{m}\rho(2^{m})m^{3}\big{]}\cdot 2^{-2k_{1}}2^{10M_{0}},$

which is an acceptable bound.

Subcase 2.1.2: $(\epsilon_{1},\epsilon_{2})\in\\{(+,-),(-,+)\\}$. In this case
we note that $|\Phi_{{\epsilon_{1},\epsilon_{2}}}|\gtrsim 1.$ Therefore we can
proceed as for $F_{L}$ and integrate by parts in time to produce terms of the
type $F^{S}_{m}$. Since the operator $T^{R,2}_{1}$ satisfies the same bilinear
estimates as $T^{S,2}_{1},$ the bounds are the exact same; we then skip the
details and refer the reader to the bounds for $T^{S,2}_{1}$ in the next
section.

Subcase 2.2: $k_{2}>-m/2+m/100$ and $k_{1}>-m/2+m/200$. We insert an
additional cutoff $\varphi_{k_{3}}(\eta+\sigma)$ to localize the frequency of
the input $G$. We fix $\delta_{0}\in(0,1/200)$.

Subcase 2.2.1: $k_{3}\leqslant-m/2+\delta_{0}m.$ In this case
$|\eta|\approx|\sigma|\approx 2^{k_{1}}\approx 2^{k_{2}}.$ Integrating
directly we get, using (1.26d), (2.23),

$\displaystyle 2^{m}\sum_{G,H\in\\{g,h\\}}\sup_{s\approx
2^{m}}\|\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,2}_{1}[b_{2}](e^{is\epsilon_{1}\langle\xi\rangle}\widetilde{G_{\epsilon_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{\epsilon_{2}}}\|_{L^{\infty}_{\xi}}\lesssim
2^{m}\cdot 2^{(5/2-)k_{3}}Z\cdot 2^{(5/2-)k_{1}}Z\cdot
2^{-2k_{1}}2^{10M_{0}},$

which is acceptable.

Subcase 2.2.2: $k_{3}>-m/2+\delta_{0}m.$ Here we integrate by parts in both
$\sigma$ and $\eta$ after changing variables $\eta\mapsto-\eta-\sigma.$ We can
then bound, using (1.26a), (1.27b), (1.27a), (2.23) (note that the worse terms
correspond to the derivative falling on the profile)

$\displaystyle 2^{m}\sum_{G,H\in\\{g,h\\}}\sup_{s\approx
2^{m}}\|\varphi_{k}\widehat{\mathcal{F}}_{x\rightarrow\xi}T^{R,2}_{1}[b_{2}](e^{is\epsilon_{1}\langle\xi\rangle}\widetilde{G_{\epsilon_{1}}},e^{is\epsilon_{2}\langle\xi\rangle}\widetilde{H_{\epsilon_{2}}}\|_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim 2^{-m}\big{[}2^{k_{3}/2}Z+2^{\delta
m}2^{m}\rho(2^{m})+2^{m}\rho(2^{m})m^{3}\big{]}\cdot\big{[}2^{k_{2}/2}Z+2^{\delta
m}2^{m}\rho(2^{m})+2^{m}\rho(2^{m})m^{3}\big{]}\cdot 2^{-2k_{1}}2^{10M_{0}},$

which is acceptable provided $\delta$, $\delta_{0}$ and $\delta_{N}$ are small
enough.

Step 3: Bounding $M_{1,m}$. Taking an inverse Fourier transform of $M_{1,m}$
and relying on the dispersive estimates (1.26c), we have

$\displaystyle\big{\|}M_{1,m}\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\leqslant\bigg{\|}\int_{0}^{t}B(s)\,\big{(}L^{-1}e^{isL}f\big{)}\phi\,\tau_{m}(s)ds\bigg{\|}_{L^{1}_{x}}\lesssim
2^{m}\sup_{s\approx 2^{m}}|B(s)|\|L^{-1}e^{isL}f\|_{L^{6}_{x}}\lesssim
2^{m}\cdot 2^{-m/2}\cdot 2^{-m+10\delta_{N}}Z,$

which yields the desired result.

Step 4: Bounding $F_{H}$. From the definition of $F_{H}=\sum_{m}F_{H,m}$ given
through (2.7), we see that when we express this term in distorted Fourier
space at least one of the three frequencies (the output $\xi$, and the two
input frequencies $\eta,\sigma$) is of size $\gtrsim
2^{M_{0}(s)}\approx\langle s\rangle^{\delta_{N}}$. Then we can split $F_{H,m}$
into seven terms by inserting frequency projections $P_{+}$ and $P_{-}$
associated to cutoffs $\varphi_{+}:=\varphi_{\leqslant M_{0}}$ and
$\varphi_{-}:=\varphi_{>M_{0}}$ (see (2.7)). We denote these terms by
$F_{H,m}^{(i,j,k)}$ where the signs $i,j,k\in\\{+,-\\}$ correspond to the
presence of a cutoff $\varphi_{+}$ or $\varphi_{-}$, and $(i,j,k)\neq(+,+,+)$.

Case 1: $j=-$ or $k=-$. Without loss of generality we let $j=-$. Then we
write, for $k\in\\{+,-\\}$ and using (1.26b),

$\displaystyle\|F_{H,m}^{(i,-,k)}\|_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\Bigg{\|}\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\Bigg{[}\varphi_{i}(\xi)\int_{0}^{t}\int_{\mathbb{R}^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\varphi_{-}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta)\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\,\tau_{m}(s)ds\Bigg{]}\Bigg{\|}_{L^{1}_{x}}$ $\displaystyle\lesssim
2^{m}\bigg{\|}L^{-1}e^{itL}P_{-}f_{\epsilon_{1}}\cdot
L^{-1}e^{itL}P_{k}f_{\epsilon_{2}}\bigg{\|}_{L^{1}_{x}}\lesssim 2^{m}\cdot
2^{-NM_{0}}\|f_{\epsilon_{1}}\|_{H^{N}}\|f_{\epsilon_{2}}\|_{L^{2}},$

which is more than sufficient since $M_{0}=\delta_{N}m=\frac{5}{N}m.$

Case 2: $(i,j,k)=(-,+,+).$ In this case using the boundedness of wave
operators on $L^{1}$ and distributing derivatives, we can estimate

$\displaystyle\|F_{H,m}^{(-,+,+)}\|_{L^{\infty}_{\xi}}$ $\displaystyle\lesssim
2^{-NM_{0}}\bigg{\|}\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}|\xi|^{N}\varphi_{i}(\xi)\int_{0}^{t}\int_{R^{6}}\frac{e^{-is\Phi_{\epsilon_{1},\epsilon_{2}}(\xi,\eta,\sigma)}}{\langle\eta\rangle\langle\sigma\rangle}\varphi_{-}(\eta)\widetilde{f_{\epsilon_{1}}}(s,\eta)\varphi_{k}(\sigma)\widetilde{f_{\epsilon_{2}}}(s,\sigma)\mu(\xi,\eta,\sigma)d\eta
d\sigma\,\tau_{m}(s)ds\bigg{\|}_{L^{1}_{x}}$ $\displaystyle\lesssim
2^{-NM_{0}}\sum_{N_{1}+N_{2}\leqslant
N}\big{\|}D^{N_{1}}L^{-1}e^{itL}f_{\epsilon_{1}}\big{\|}_{L^{2}_{x}}\big{\|}D^{N_{2}}L^{-1}e^{itL}f_{\epsilon_{2}}\big{\|}_{L^{2}_{x}};$

we then conclude as we did above using (1.26b).

Step 5: Bounding $S_{m}.$ Recall the definitions (2.3). To treat $S_{2,m}$ we
integrate by parts in time and then use the estimate for $\dot{B}$ in (1.20)
to obtain

$\displaystyle{\|S_{2,m}(t)\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\bigg{\|}\tau_{m}(t)B^{2}(t)\frac{e^{-it(\langle\xi\rangle-2\lambda)}}{\langle\xi\rangle-2\lambda}(1-\chi_{C}(\xi))\,\widetilde{\theta(\xi)}\bigg{\|}_{L^{\infty}_{\xi}}$
$\displaystyle+\bigg{\|}\frac{1-\chi_{C}(\xi)}{\langle\xi\rangle-2\lambda}\int_{0}^{t}e^{-is(\langle\xi\rangle-2\lambda)}\big{(}2\dot{B}(s)B(s)\tau_{m}(s)+B^{2}(s)2^{-m}\tau^{\prime}_{m}(s)\big{)}ds\,\widetilde{\theta(\xi)}\bigg{\|}_{L^{2}}$
$\displaystyle\lesssim\rho(2^{m})+\int_{s\approx
2^{m}}s^{-1}\rho(s)ds\lesssim\rho(2^{m}).$

For the term $S_{1,m}$ we note that $A^{2}-B^{2}=O(A^{3}),$ see (1.20), and
therefore we can bound directly $\|S_{1,m}\|_{L^{\infty}_{\xi}}\lesssim
2^{m}\rho(2^{m})^{3/2}$, which suffices. The proof of (4.3) is complete.
$\hfill\Box$

##  5\. Estimating the correction

We conclude the proof by estimating the normal form correction defined in the
previous section, namely showing (4.4), under the a priori assumption (4.2).
Recalling (4.1) and (2.27), we have to estimate $T_{1}^{S,1},T_{1}^{S,2}$ and
$T_{i}^{S},i=2,3.$

Step 1: Bounding $T_{1}^{S,1}.$ Using the identities (2.11), (2.13) and (2.14)
we can reduce matters to estimating the following three operators:

$\displaystyle T_{\delta}^{S,1}[b](G,H)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}\widetilde{G}(\xi-\eta)\widetilde{H}(\sigma)\,b(\xi,\eta,\sigma)\,\frac{b_{0}(\hat{\eta},\sigma)}{|\eta|}\delta(|\eta|-|\sigma|)\,d\eta
d\sigma,$ $\displaystyle T_{p.v.}^{S,1}[b](G,H)$
$\displaystyle:=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}\widetilde{G}(\xi-\eta)\widetilde{H}(\sigma)\,b(\xi,\eta,\sigma)\,\frac{b_{0}(\hat{\eta},\sigma)}{|\eta|}\mathrm{p.v.}\frac{\varphi_{\leqslant-
M_{0}-5}(|\eta|-|\sigma|)}{|\eta|-|\sigma|}\,d\eta d\sigma,$

where we denoted $\hat{\eta}:=-\eta/|\eta|$, and, for $a=1,\dots,N_{2}$,

$\displaystyle T_{a}^{S,1}[b](G,H):=\widehat{\mathcal{F}}^{-1}_{\xi\rightarrow
x}\iint_{{\mathbb{R}}^{3}\times{\mathbb{R}}^{3}}G(\xi-\eta)H(\sigma)\,b(\xi,\eta,\sigma)\,\varphi_{\leqslant-
M_{0}-5}(|\eta|-|\sigma|)\frac{1}{|\eta|}$
$\displaystyle\times\sum_{J\in\mathbb{Z}}b_{a,J}(\eta,\sigma)\cdot
2^{J}K_{a}(|\eta|-|\sigma|)\,d\eta d\sigma.$

Recall the definition of $b$ in (2.29). Notice that in the formulas above the
inputs $(G,H)$ play the role of either $e^{\pm itL}h$ or $e^{\pm itL}g$. In
particular the $H^{s}_{x}$ and $\widetilde{\mathcal{F}}^{-1}L^{\infty}_{\xi}$
norms of $(G,H)$ are those of $g$ and $h$.

We can deal with $T_{\delta}^{S,1},T_{a}^{S,1}$ by direct integration, using
the lower bound
$|\Phi(\xi,\eta,\sigma)|\gtrsim\big{(}\langle\xi\rangle+\langle\eta\rangle+\langle\sigma\rangle\big{)}^{-1}$
see (5.47), Lemma 5.10 in [20]. More precisely,

$\displaystyle\big{\|}T_{\delta}^{S,1}[b](G,H)\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\Bigg{\|}\frac{\varphi_{k_{1}}(\eta)}{\Phi(\xi,\xi-\eta,\eta)\langle\xi-\eta\rangle\langle\eta\rangle}\Bigg{\|}_{L^{\infty}_{\xi,\eta}}\|b_{0}(\omega,\xi)\|_{L^{\infty}_{\omega,\xi}}$
$\displaystyle\times\sup_{\eta\in\mathbb{R}^{3}}\Bigg{|}\int_{\mathbb{R}^{3}}\widetilde{H}(\sigma)\varphi_{k_{2}}(\sigma)\frac{\delta(|\eta|-|\sigma|)}{|\eta|}d\sigma\Bigg{|}\cdot\big{\|}G(\xi-\eta)\varphi_{k_{1}}(\eta)\big{\|}_{L^{\infty}_{\xi}L^{1}_{\eta}}$
$\displaystyle\lesssim\langle 2^{k_{1}}\rangle^{-1}\cdot
2^{-k_{1}}{\|\widetilde{H}\|}_{L^{\infty}_{\sigma}}\cdot
2^{3k_{1}/2}{\|G\|}_{L^{2}}\lesssim 2^{k_{1}/2}\langle
2^{k_{1}}\rangle^{-1}{\varepsilon}^{1-}.$

We can then sum over $k_{1}$ and get a bound consistent with (4.4). Note that
we have used (1.26b). Similarly we can bound

$\displaystyle\big{\|}T_{a}^{S,1}[b](G,H)\big{\|}_{L^{\infty}_{\xi}}$
$\displaystyle\lesssim\Bigg{\|}\frac{\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)}{\Phi(\xi,\xi-\eta,\sigma)\langle\xi\rangle\langle\xi-\eta\rangle\langle\sigma\rangle}\Bigg{\|}_{L^{\infty}_{\xi,\eta,\sigma}}\|b_{0}(\omega,\xi)\|_{L^{\infty}_{\omega,\xi}}\cdot\big{\|}\langle\xi\rangle\widetilde{G}(\xi-\eta)\varphi_{k_{1}}(\eta)\big{\|}_{L^{1}_{\eta}}$
$\displaystyle\times\sup_{\xi,\eta\in\mathbb{R}^{3}}\sum_{J\in\mathbb{Z}}\Bigg{|}\varphi_{\sim
k_{1}}(\eta)\int_{\mathbb{R}^{3}}\widetilde{H}(\sigma)\varphi_{\sim
k_{2}}(\sigma)b_{a,J}(\eta,\sigma)\cdot
2^{J}K_{a}\big{(}2^{J}(|\eta|-|\sigma|)\big{)}d\sigma\Bigg{|}$
$\displaystyle\lesssim\langle 2^{k_{1}}\rangle^{-1}\langle
2^{k_{2}}\rangle^{-1}\|\widetilde{H}\|_{L^{\infty}_{\sigma}}\cdot
2^{k_{1}/2}\langle
2^{k_{1}}\rangle\|G\|_{H^{2}}\cdot\sum_{J\in\mathbb{Z}}\big{|}\varphi_{\sim
k_{1}}(\eta)\varphi_{\sim k_{2}}(\sigma)b_{a,J}(\eta,\sigma)\big{|}$
$\displaystyle\lesssim 2^{k_{1}/2}\langle
2^{k_{2}}\rangle^{-1}\varepsilon^{1-}.$

Note that here we have used again (1.26b), and also the a apriori assumption
(4.2) and Lemma 2.1 to bound $\widetilde{H}$ in $L^{\infty}_{\sigma}$. We can
then sum over $k_{1},k_{2}$ using $|k_{1}-k_{2}|<5$, see (2.11).

We then look at the principal value part and bound

$\displaystyle\big{|}T_{p.v.}^{S,1}[b](G,H)(\xi)\big{|}\leqslant|I(\xi)|+|II(\xi)|,$
$\displaystyle
I(\xi):=\int_{\mathbb{R}^{3}}\frac{\widetilde{G}(\xi-\eta)}{\langle\xi-\eta\rangle}\frac{\varphi_{k_{1}}(\eta)}{|\eta|}\int_{\mathbb{R}^{3}}\frac{b_{0}(\hat{\eta},\sigma)}{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\xi-\eta,\eta)\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\xi-\eta,\sigma)}$
$\displaystyle\qquad\qquad\qquad\times\widetilde{H}(\sigma)\frac{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\xi-\eta,\sigma)-\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\xi-\eta,\eta)}{|\eta|-|\sigma|}\frac{\varphi_{k_{2}}(\sigma)}{\langle\sigma\rangle}d\sigma
d\eta,$ $\displaystyle
II(\xi):=\int_{\mathbb{R}^{3}}\frac{\widetilde{G}(\xi-\eta)}{\langle\xi-\eta\rangle}\frac{\varphi_{k_{1}}(\eta)}{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\xi-\eta,\eta)|\eta|}\,\mathrm{p.v.}\int_{\mathbb{R}^{3}}\frac{b_{0}(\hat{\eta},\sigma)}{|\eta|-|\sigma|}\frac{\widetilde{H}(\sigma)}{\langle\sigma\rangle}\varphi_{k_{2}}(\sigma)d\sigma
d\eta.$

The term $I$ can be treated using direct integration as $T_{\delta}^{S,1}$ and
$T_{a}^{S,1}$ since the singularity is canceled by the difference of the
phases. For $II$ we use (3.3) and the fact that
$|\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\xi-\eta,\eta)\langle\xi-\eta\rangle\langle\eta\rangle|\gtrsim
1,$ and find

$\displaystyle\|II\|_{L^{\infty}_{\xi}}\lesssim\langle 2^{k_{1}}\rangle\cdot
2^{k_{2}}\langle 2^{k_{2}}\rangle^{-3}(\varepsilon^{1-})^{2}.$

Step 2: Bounding $T_{1}^{S,2}.$ We can use similar argument and eventually
rely on (3.4) for the hardest term. Defining $T_{\delta}^{S,2},T_{p.v.}^{S,2}$
and $T_{a}^{S,2}$ in an analogous way to what was done above, i.e., replacing
the measure $\mu_{1}^{S}$ in the expression $T_{1}^{S,2}$ (see (2.24c)) by the
three pieces in (2.11), (2.13).

We only focus on the principal value part since the other terms can be bounded
by direct integration as above. We start by writing that

$\displaystyle\big{|}T_{p.v.}^{S,2}[b](G,H)\big{|}$
$\displaystyle\leqslant|III(\xi)|+|IV(\xi)|,$ $\displaystyle III(\xi)$
$\displaystyle:=\iint_{\mathbb{R}^{6}}\frac{\widetilde{G}(\eta+\sigma)}{\langle\eta+\sigma\rangle}\frac{\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)}{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\eta,\eta+\sigma,\sigma)}\frac{\widetilde{H}(\sigma)}{\langle\sigma\rangle}\frac{b_{0}(\hat{\eta},\xi)}{|\eta|}\mathrm{p.v.}\frac{1}{|\eta|-|\xi|}d\eta
d\sigma,$ $\displaystyle IV(\xi)$
$\displaystyle:=\iint_{\mathbb{R}^{6}}\frac{\widetilde{G}(\eta+\sigma)}{\langle\eta+\sigma\rangle}\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)\frac{\widetilde{H}(\sigma)}{\langle\sigma\rangle}\frac{b_{0}(\hat{\eta},\xi)}{|\eta|}$
$\displaystyle\times\frac{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\eta,\eta+\sigma,\sigma)-\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta+\sigma,\sigma)}{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\eta,\eta+\sigma,\sigma)\Phi_{{\epsilon_{1},\epsilon_{2}}}(\xi,\eta+\sigma,\sigma)}\mathrm{p.v.}\frac{1}{|\eta|-|\xi|}d\eta
d\sigma.$

Since the term $IV$ is not singular we can treat it by direct integration as
above; we skip the details. To bound the main term $III$ we apply (3.4) to the
multiplier

$\displaystyle
m(\eta,\sigma):=\frac{\varphi_{k_{1}}(\eta)\varphi_{k_{2}}(\sigma)}{\Phi_{{\epsilon_{1},\epsilon_{2}}}(\eta,\eta+\sigma,\sigma)\langle\eta+\sigma\rangle^{5}\langle\sigma\rangle^{5}}\cdot\frac{2^{k_{1}}}{|\eta|},$

and, using again (1.26b), we obtain (here $K=k_{1}$)

$\displaystyle\|III\|_{L^{\infty}_{\xi}}$ $\displaystyle\lesssim
2^{k_{1}}\langle 2^{k_{1}}\rangle^{-2}{\|G\|}_{H^{6}}\|P_{\sim
k_{2}}H\|_{H^{6}}$ $\displaystyle\lesssim 2^{k_{1}}\langle
2^{k_{1}}\rangle^{-2}{\varepsilon}^{1-}\cdot\min\big{\\{}\langle
2^{k_{2}}\rangle^{-1}\|H\|_{H^{7}},2^{3k_{2}/2}\|H\|_{L^{\infty}_{\sigma}}\big{\\}}.$

We can then use Lemma 2.1 and (4.2) to control the last $L^{\infty}_{\sigma}$
norm and sum over $k_{1},k_{2}$ to conclude.

Step 3: Bounding $T_{i}^{S}$. Using (2.15), (2.16), (2.17) and (2.20), we can
treat these terms by direct integration as we did above for
$T_{\delta}^{S,1}.$ Estimates are easier in this case, therefore we omit the
details.

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