# Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear
Schrödinger equation

Chenmin Sun  CNRS, Université Paris-Est Créteil, Laboratoire d’Analyse et de
Mathématiques appliquées, UMR 8050 du CNRS, 94010 Créteil cedex, France
<EMAIL_ADDRESS>and  Nikolay Tzvetkov  Ecole Normale Supérieure de Lyon,
UMPA, UMR CNRS-ENSL 5669, 46, allée d’Italie, 69364-Lyon Cedex 07, France
<EMAIL_ADDRESS>Dedicated to Professor Carlos Kenig for his 70’s
birthday with admiration

###### Abstract.

We consider the $3d$ energy critical nonlinear Schrödinger equation with data
distributed according to the Gaussian measure with covariance operator
$(1-\Delta)^{-s}$, where $\Delta$ is the Laplace operator and $s$ is
sufficiently large. We prove that the flow sends full measure sets to full
measure sets. We also discuss some simple applications. This extends a
previous result by Planchon-Visciglia and the second author from $1d$ to
higher dimensions.

##  1\. Introduction

###  1.1. Motivation

The seminal paper [17] initiated the study of Hamiltonian PDE’s with initial
data distributed according to the Gibbs measure which is constructed from the
Hamiltonian functional. The Gibbs measure construction is strongly inspired by
earlier developments in quantum field theory (see e.g. [20, 37]). These Gibbs
measures are absolutely continuous with respect to suitable Gaussian measures
(or shifts of such Gaussian measures). They are at least formally invariant
under the corresponding Hamiltonian flow and therefore the underlying Gaussian
meaure (or its shift) are quasi-invariant under the flow.

In dimensions greater than or equal to two, in order to consider initial data
distributed according to the Gibbs measure a renormalization of the equation
under consideration is required, see e.g. [3, 6, 10, 12, 30, 33]. Such
renormalizations have strong motivations from Physics but they also make the
results not so natural from a classical PDE perspective. A notable exception
is the cubic nonlinear Schrödinger equation for which a gauge transform links
the (truncated) equation and its renormalized version.

One may also observe that full Gibbs measure sets cover a very tiny part of
the phase space of a Hamiltonian PDE and also that the Gibbs measure plays no
role in the dynamics of most of the initial distributions of the initial data.
Observe that this is in sharp contrast with Langevin type dynamics where the
(same) Gibbs measure plays a truly distinguished role because it attracts all
initial distributions.

Motivated by the above observations, in recent years there has been an
activity aiming to show that a more general set of gaussian measures are
quasi-invariant under Hamiltonian PDE’s, see [7, 11, 14, 15, 16, 18, 19, 21,
25, 26, 31, 32, 35, 34, 36, 38, 39]. Such results allow to give a statistical
description of the Hamiltonian flow for a larger class of initial
distributions of the initial data (out of equilibrium dynamics). In
particular, one obtains results for data of arbitrary Sobolev regularity while
the Gibbs measures live in low regularity Sobolev spaces. Moreover, no
renormalization of the equation is required (even if renormalized energies may
be used in the proof, see [36, 21, 39]). It is also worth to observe that it
looks that the question of quasi-invariance of gaussian measures for
Hamiltonian PDE’s does not seem to have an analogue in the context of
dissipative PDE’s.

Most of the results quoted in the previous paragraph are dealing with $1d$
models. The only results in dimensions $\geq 2$ are [36, 21, 39]. They are
dealing with non linear wave equations. The approach used in these works,
based on renormalized energies, does not apply to the nonlinear Schrödinger
equation (NLS) because of the lack of explicit smoothing in the equation. Our
goal here is to resolve this issue and prove the quasi-invariance of Gaussian
measures supported by sufficiently regular functions under the NLS flow in
higher dimensions. Our approach is based on normal form reductions as in [31,
32, 35] combined with a soft analysis initiated in [25]. The main new idea in
this paper is the identification of a remarkable cancellation of the worse
pairing when estimating the divergence of the Hamiltonian vector field with
respect to a weighted Gaussian measure (see Section 7 below). The weight is
naturally produced by the normal form reduction and therefore is related to
the nature of the resonant set (while is the wave equation case the weight is
related to the potential energy). This remarkable cancellation is certainly
related to the Hamiltonian structure and hopefully may be used in other
contexts. Our result is only giving qualitative quasi-invariance for
sufficiently regular initial distributions. Therefore several challenging
issues remain open (see the remarks after the statement of the main result).

###  1.2. Main result

In this work we will study the most challenging model to which we succeeded to
make work our approach. Therefore, we consider the defocusing energy-critical
NLS

$\displaystyle i\partial_{t}u+\Delta
u=|u|^{4}u,\quad(t,x)\in\mathbb{R}\times\mathbb{T}^{3},$ (1.1)

where $\mathbb{T}^{3}:=\mathbb{R}^{3}/(2\pi\mathbb{Z})^{3}$. Equation (1.1) is
a Hamiltonian system with the conserved mass and energy:

$M[u]:=\int_{\mathbb{T}^{3}}|u|^{2}\mathrm{d}x,\quad
H[u]:=\frac{1}{2}\int_{\mathbb{T}^{3}}|\nabla
u|^{2}\mathrm{d}x+\frac{1}{6}\int_{\mathbb{T}^{3}}|u|^{6}\mathrm{d}x.$

These conservation laws allow to construct relatively easily global weak
solutions of (1.1) in the Sobolev spaces $H^{1}(\mathbb{T}^{3})$ via basic
compactness arguments. Unfortunately such techniques are not suitable to prove
uniqueness and propagation of higher Sobolev regularities. Thanks to the
remarkable work by Ionescu-Pausader [23] (based on the previous contributions
[2, 4, 9, 22, 24]) we know that (1.1) is globally well posed in
$H^{s}(\mathbb{T}^{3})$, $s\geq 1$. Namely, for every $u_{0}\in
H^{s}(\mathbb{T}^{3})$, $s\geq 1$ there exists a unique global solution of
(1.1) in $C(\mathbb{R};H^{s}(\mathbb{T}^{3}))$ such that $u(0,x)=u_{0}(x).$
Let us denote by $\Phi(t)$ the Ionescu-Pausader flow of (1.1).

When studying the statistical properties of (1.1), we assume that the initial
data are distributed according to the Gaussian probability measure $\mu_{s}$,
formally defined as
“$\frac{1}{\mathcal{Z}}\mathrm{e}^{-\frac{1}{2}\|u\|_{H^{s}}^{2}}du$”, induced
by the random Fourier series

$\displaystyle\phi^{\omega}(x):=\sum_{k\in\mathbb{Z}^{3}}\frac{g_{k}(\omega)}{\sqrt{1+|k|^{2s}}}\mathrm{e}^{ik\cdot
x}\,,$ (1.2)

where $(g_{k})_{k\in\mathbb{Z}^{3}}$ are independent, identically distributed
standard complex Gaussian random variables. Thanks to the Kakutani theorem we
know that for at least $s\geq 10$ the measure $\mu_{s}$ is absolutely
continuous with respect to the Gaussian measure with covariance operator
$(1-\Delta)^{-s}$ (see e.g. [36] for such an application of the Kakutani
theorem). It is well-known that

$\mathrm{supp}(\mu_{s})=H^{(s-\frac{3}{2})-}(\mathbb{T}^{3}):=\bigcap_{\sigma<s-\frac{3}{2}}H^{\sigma}(\mathbb{T}^{3})$

and $\mu_{s}(H^{s-\frac{3}{2}}(\mathbb{T}^{3}))=0$. Therefore bigger $s$ is
more regular are typical functions with respects to $\mu_{s}$. Thanks to [23],
when $s>\frac{5}{2}$, the flow $\Phi(t)$ of (1.1) exists globally on
$H^{\sigma}(\mathbb{T}^{3})$ for any $1\leq\sigma<s-\frac{3}{2}$. In
particular, a unique global solution exists for any initial data on
supp$(\mu_{s})$, $s>\frac{5}{2}$. Our main result reads as follows.

######  Theorem 1.1.

Assume that $s\geq 10$. Then $\mu_{s}$ is quasi-invariant under $\Phi(t)$.
More precisely, for every $t\in\mathbb{R}$,
$(\Phi(t))_{*}\mu_{s}\ll\mu_{s}\ll(\Phi(t))_{*}\mu_{s}$, where
$(\Phi(t))_{*}\mu_{s}$ is the push forward of $\mu_{s}$ by $\Phi(t)$.

In the statement above, the notation $\mu\ll\nu$ for two measures $\mu,\nu$
means that $\mu$ is absolutely continuous with respect to $\nu$.

In the proof of Theorem 1.1 below, we retain us of using arithmetic arguments
such as the divisor bound. Therefore the result of Theorem 1.1 remains valid
for irrational tori with essentially the same proof.

In view of [1], it seems hopeless to construct a Gibbs measure for (1.1) (and
any other energy critical problem). This gives a further motivation for
studying quasi-invariant Gaussian measures for (1.1) or any other model for
which the Gibbs measure construction fails.

The result of Theorem 1.1 remains true (with a simpler proof) for the cubic
$3d$ NLS

$i\partial_{t}u+\Delta
u=|u|^{2}u,\quad(t,x)\in\mathbb{R}\times\mathbb{T}^{3},$

and also for the $2d$ NLS with an arbitrary polynomial defocusing
nonlinearity.

As already mentioned, Theorem 1.1 only gives qualitative quasi-invariance. It
would be interesting to obtain quantitative bounds on the resulting Radon-
Nikodym derivatives. Such quantitative bounds were obtained in some previous
works on the subject, the most notable being the paper by Forlano-Tolomeo [16]
where such quantitative informations on the Radon-Nikodym derivative are used
in order to perform the Bourgain globalization argument, i.e. quasi-invariance
is used in order to construct the flow. The Forlano-Tolomeo argument is
performed for a $1d$ model and it would be very interesting to extend it to
higher dimensions. In particular, it would be interesting to decide whether
Theorem 1.1 holds in the supercritical regime, i.e. for some $s<\frac{5}{2}$
(in this regime the existence of the flow should rely on a probabilistic well-
posedness in the spirit of [28]). But at this stage it is even not clear to us
how to prove Theorem 1.1 in the natural subcritical range $s>\frac{5}{2}$. By
using the dispersive effects we can relax slightly the assumption $s\geq 10$
but it would still be away for the natural subcritical assumption
$s>\frac{5}{2}$. In summary, much more remains to be understood concerning the
transport of $\mu_{s}$ under the NLS flow and its connection with the
probabilistic well-posedness theory.

###  1.3. Applications

In this section we present two simple corollaries of Theorem 1.1. Recall that
the random field (1.2) is a stationary Gaussian process on $\mathbb{T}^{2}$.
In particular, for each fixed $x\in\mathbb{T}^{2}$, $\phi^{\omega}(x)$ is a
complex Gaussian random variable with law
$\mathcal{N}_{\mathbb{C}}(0,\sigma^{2})$, where
$\sigma^{2}=\sum_{k\in\mathbb{Z}^{3}}\frac{1}{1+|k|^{2s}}.$ Consequently, the
probability density of $\phi^{\omega}(x)$ is
$\frac{1}{2\pi\sigma}{\mathrm{e}}^{-\frac{|y|^{2}}{2\sigma}}dy$ on
$\mathbb{C}=\mathbb{R}^{2}$. In particular, the law of $\phi^{\omega}(x)$ is
absolutely continuous with respect to the Lebesgue measure. A natural question
is to study the regularity of the law for the random variable $u(t,x)$,
evolved by (1.1) with the initial data $\phi^{\omega}$. This type of problems
has been intensively studied in the field of Stochastic analysis. For many
classes of stochastic (partial) differential equations, the regularity of laws
of solutions can be obtained via the Malliavin Calculus (see the book of
Nualart [29] and references therein). The Malliavin Calculus was originally
developed by P. Malliavin [27] to bring a new proof of Hörmander’s theorem for
hypoelliptic operators. We do not intend to include any element of the
Malliavin Calculus in this article, but to give a simple application of the
quasi-invariance property to obtain the absolutely continuity for the law of
solutions of NLS with random initial data, which can be viewed as a pointwise
version of the quasi-invariance property of the NLS equation displayed by
Theorem 1.1.

######  Corollary 1.2.

Assume that $s\geq 10$ and fix
$(t_{0},x_{0})\in\mathbb{R}\times\mathbb{T}^{3}$. Let $u(t,x,\omega)$ be the
solution of (1.1) with data (1.2). Then the law of the complex random variable
$\omega\mapsto u(t_{0},x_{0},\omega)$ has a density with respect to the
Lebesgue measure on $\mathbb{C}$.

In order to prove Corollary 1.2, we observe that we need to study the
composition of $\Phi(t)$ and the evaluation map $u\mapsto u(t_{0},x_{0})$.
Then it suffices to apply Theorem 1.1 for $\Phi(t)$ and the observation before
the statement of Corollary 1.2 for the evaluation map. It is likely that the
Malliavin Calculus can be useful to get regularity properties of the densities
appearing in the statement of Corollary 1.2. In Corollary 1.2, one may replace
the evaluation map by other finite dimensional projections. For instance, one
may show that for every $k\in\mathbb{Z}^{3}$, the law of the Fourier
coefficient $\widehat{u}(t,k,\omega)$ has a density with respect to the
Lebesgue measure on $\mathbb{C}$.

Let us also emphasize that the Malliavin Calculus methods can be applied to
prove quasi-invariance for maps from infinite dimensional gaussian spaces to
finite dimensional spaces, while in Theorem 1.1 we deal with the more complex
situation of a map from an infinite dimensional gaussian space to itself.

Another simple consequence of Theorem 1.1 is the following $L^{1}$-stability
result.

######  Corollary 1.3.

Assume that $s\geq 10$. Let $f_{1},f_{2}\in L^{1}(d\mu_{s})$ and $\Phi(t)$ the
flow of (1.1). Then for any $t\in\mathbb{R}$, the transports of measures
$f_{1}(u)d\mu_{s}(u)$, $f_{2}(u)d\mu_{s}(u)$ by $\Phi(t)$ are given by
$F_{1}(t,u)d\mu_{s}(u)$ and $F_{2}(t,u)d\mu_{s}(u)$ respectively, for suitable
$F_{1}(t,\cdot),F_{2}(t,\cdot)\in L^{1}(d\mu_{s})$. Moreover,

$\|F_{1}(t,\cdot)-F_{2}(t,\cdot)\|_{L^{1}(d\mu_{s})}=\|f_{1}-f_{2}\|_{L^{1}(d\mu_{s})}.$

One may prove Corollary 1.3 by performing the computations from [40]. A more
direct proof can be given by observing that $\Phi(t)$ is a measurable map and
therefore the total variation distance between $F_{1}(t,u)d\mu_{s}(u)$ and
$F_{2}(t,u)d\mu_{s}(u)$ is smaller than the total variation distance between
$f_{1}(u)d\mu_{s}(u)$ and $f_{2}(u)d\mu_{s}(u)$. This implies that

$\|F_{1}(t,\cdot)-F_{2}(t,\cdot)\|_{L^{1}(d\mu_{s})}\leq\|f_{1}-f_{2}\|_{L^{1}(d\mu_{s})}.$

Using the reversibility of the NLS flow we get the inverse inequality.

The remaining part of this paper is devoted to the proof of Theorem 1.1. In
Section 2 we perform the normal form reduction, we define accordingly suitable
weighted Gaussian measures and we state the key energy estimates. In Section 3
we perform the soft analysis leading from the energy estimates to the quasi-
invariance result stated in Theorem 1.1. In Section 4 we introduce our basic
counting tool and the Wiener chaos estimate useful for our purposes. In
Section 5 we decompose the divergence of the Hamiltonian vector field with
respect to the weighted Gaussian measures in several pieces according to the
possible pairings. In Section 6 we estimates the contributions of the first
generation. Section 7 deals with the most singular contribution resulting from
pairings between different generations. This is the most delicate part of our
analysis containing the remarkable algebraic cancellations mentioned above. In
Section 8 we treat the remainder terms in which the singular pairings are not
presented. Finally in an Appendix we prove some approximation results for
(1.1), crucially exploited in Section 3. Let us emphasize that because of the
critical nature of the Cauchy problem for (1.1), the approximation argument is
much more delicate compared to the previous literature on quasi-invariant
Gaussian measures for Hamiltonian PDE’s.

### Acknowledgments

This work is partially supported by the ANR project Smooth ANR-22-CE40-0017.
The authors would like to thank Alexis Knezevitch for pointing out an error in
a previous version of the manuscript.

##  2\. Modified energy and the weighted Gaussian measure

###  2.1. An approximated system

Fix a radial cutoff function $\chi\in C_{c}^{\infty}(\mathbb{R}^{3})$ such
that $\chi\equiv 1$ on $[-\frac{1}{2},\frac{1}{2}]^{3}$ and
supp$(\chi)\subset\\{|x|<1\\}$. For $N\in\mathbb{N}$, set
$\chi_{N}(\cdot):=\chi(N^{-1}\cdot)$ and $S_{N}=\chi_{N}(\sqrt{-\Delta})$ the
smooth frequency truncation and $\Pi_{N}=\mathbf{1}_{\sqrt{-\Delta}\leq N}$
the sharp frequency truncation. By definition,

$S_{N}\Pi_{N}=\Pi_{N}S_{N}=S_{N},\quad S_{N}^{*}=S_{N}.$

The advantage of using the operator $S_{N}$ is that $S_{N}$ is uniformly
bounded on $L^{p}(\mathbb{T}^{3})$ for $1<p<\infty$, which is crucial when
taking the limit of the approximated system in the energy critical case.
Similar to the situation in [8], we consider the following smoothly
approximated NLS equation

$\displaystyle\begin{cases}&i\partial_{t}u_{N}+\Delta
u_{N}=S_{N}(|S_{N}u_{N}|^{4}S_{N}u_{N}),\\\ &u_{N}|_{t=0}=u_{0}\in
H^{\sigma}(\mathbb{T}^{3}).\end{cases}$ (2.1)

As in [8], the solution of (2.1) can be decomposed as two components on
$\mathcal{E}_{N}:=\Pi_{N}L^{2}(\mathbb{T}^{3})$ and
$\mathcal{E}_{N}^{\perp}:=(\mathrm{Id}-\Pi_{N})L^{2}(\mathbb{T}^{3})$. This
naturally leads to a splitting of $\mu_{s}$ as $d\mu_{s}=d\mu_{s,N}\otimes
d\mu_{s,N}^{\perp}$ for every $N\in\mathbb{N}$, where $\mu_{s,N}$ is a measure
on $\mathcal{E}_{N}$ while $\mu_{s,N}^{\perp}$ is a measure on
$\mathcal{E}_{N}^{\perp}$. The finite-dimensional part of (2.1) on
$\mathcal{E}_{N}$ is a Hamiltonian system (see [8, Lemma 8.1]), while the
infinite-dimensional part is the linear evolution ${\mathrm{e}}^{it\Delta}$.
Thanks to the Cauchy-Lipchitz theorem and the defocusing nature, the solution
of (2.1) is global and we denote by $\Phi_{N}(t)$ its flow, which can be
factorized as $(\widetilde{\Phi}_{N}(t),{\mathrm{e}}^{it\Delta})$ on
$\mathcal{E}_{N}\times\mathcal{E}_{N}^{\perp}$, where
$\widetilde{\Phi}_{N}(t)$ is the restriction of $\Phi_{N}(t)$ on the finite-
dimensional space $\mathcal{E}_{N}$, which is a Hamiltonian flow on
$\mathcal{E}_{N}$. By convention, we denote $\Phi(t)$ by $\Phi_{\infty}(t)$.

###  2.2. Poincaré-Dulac normal form and the modified Energy

To construct suitable weighted measures for our study, we must identify a
modified energy functional. Consider a smooth solution $u_{N}(t)$ of (2.1). We
introduce a new unknown:

$v(t)={\mathrm{e}}^{-it\Delta}u_{N}(t).$

Expanding $v(t)$ in the Fourier series, we have:

$v(t,x)=\sum_{k\in\mathbb{Z}^{3}}v_{k}(t)\mathrm{e}^{ik\cdot x},$

from which it follows that $v_{k}(t)$ satisfies the equation:

$\displaystyle
i\partial_{t}v_{k}(t)=\chi_{N}(k)\sum_{k_{1}-k_{2}+k_{3}-k_{4}+k_{5}=k}{\mathrm{e}}^{-it\Omega(\vec{k})}\cdot\left(\prod_{j=1}^{5}\chi_{N}(k_{j})\right)\cdot
v_{k_{1}}(t)\overline{v}_{k_{2}}(t)\cdots v_{k_{5}}(t),$ (2.2)

where

$\Omega(\vec{k})=\sum_{j=1}^{5}(-1)^{j-1}|k_{j}|^{2}-|k|^{2}$

is the resonant function.

To construct the modified energy, it is more convenient to use an equivalent
of the Sobolev norm for $s\geq 0$:

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{H^{s}(\mathbb{T}^{3})}^{2}:=\sum_{k\in\mathbb{Z}^{3}}(1+|k|^{2s})|\widehat{f}(k)|^{2}.$

A simple computation using symmetry of indices yields

$\displaystyle\frac{1}{2}\frac{d}{dt}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v(t)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{H^{s}}^{2}=$
$\displaystyle-\frac{1}{6}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\psi_{2s}(\vec{k}){\mathrm{e}}^{-it\Omega(\vec{k})}\Big{(}\prod_{j=1}^{6}\chi_{N}(k_{j})\Big{)}v_{k_{1}}\overline{v}_{k_{2}}\cdots\overline{v}_{k_{6}},$
(2.3)

where in the above expression, we abuse the notation slightly and denote

$\psi_{2s}(\vec{k})=\sum_{j=1}^{6}(-1)^{j-1}|k_{j}|^{2s},\quad\Omega(\vec{k})=\sum_{j=1}^{6}(-1)^{j-1}|k_{j}|^{2}.$

The basic estimate for $\psi_{2s}(\vec{k})$ is

$|\psi_{2s}(\vec{k})|\lesssim|k_{(1)}|^{2s-2}(|k_{(3)}|^{2}+|\Omega(\vec{k})|),$

where $|k_{(1)}|\geq|k_{(2)}|\geq\cdots\geq|k_{(6)}|$ is rearrangement of
$k_{1},\cdots,k_{6}$ and $k_{1}-k_{2}+\cdots-k_{6}=0$ (see Lemma 4.1 below).
Note that each $v_{k_{j}}$ will be accompanied with $\chi_{N}(k_{j})$, and the
capital $N$ plays no role in our analysis, we will simply write
$w_{k_{j}}:=\chi_{N}(k_{j})v_{k_{j}}$ in the sequel. Note that

$\displaystyle
i\partial_{t}w_{k}=\chi_{N}(k)^{2}\sum_{k_{1}-k_{2}+k_{3}-k_{4}+k_{5}=k}{\mathrm{e}}^{-it\Omega(\vec{k})}\cdot
w_{k_{1}}\overline{w}_{k_{2}}\cdots w_{k_{5}}.$ (2.4)

In order to truncate the level set of the resonant function, we further
introduce the symmetric factor

$\lambda(\vec{k})=\Big{(}\sum_{j=1}^{6}|k_{j}|^{2}\Big{)}^{\frac{1}{2}}.$

As the resonant function $\Omega(\vec{k})$ takes integer values111This fact is
not essential for our result and the proof. Though we keep to work on the
rational torus for convenience. , we will decompose the set of indices
$k_{1},\cdots,k_{6}$ according to the level set of $\Omega(\vec{k})$. In order
to perform the differentiations by parts in time, we further write

$\displaystyle\frac{1}{2}\frac{d}{dt}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v(t)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{H^{s}}^{2}=$
$\displaystyle-\frac{1}{6}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k}){\mathrm{e}}^{-it\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$
$\displaystyle-\frac{1}{6}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{-i\Omega(\vec{k})}\partial_{t}\Big{(}{\mathrm{e}}^{-it\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}\Big{)}$
$\displaystyle+\frac{1}{6}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{-i\Omega(\vec{k})}{\mathrm{e}}^{-it\Omega(\vec{k})}\partial_{t}(w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}),$
(2.5)

where $0<\delta_{0}<\frac{2}{3}$ is close to $\frac{2}{3}$ (here we denote by
$\chi$ a standard bump function from $\mathbb{R}$ to $\mathbb{R}$). Motivated
by the above formula, we define the modified energy (with
$w=\chi_{N}(\sqrt{-\Delta})v$)

$\mathcal{E}_{s,t}(v):=\frac{1}{2}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|v\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{H^{s}(\mathbb{T}^{2})}^{2}+\mathcal{R}_{s,t}(w),$
(2.6)

where

$\mathcal{R}_{s,t}(w):=\frac{1}{6}\mathop{\rm
Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{-i\Omega(\vec{k})}{\mathrm{e}}^{-it\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}.$

Changing back to the variable $u$, the modified energy is

$\mathcal{E}_{s,t}(v)=E_{s,N}(u):=\frac{1}{2}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|u\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{H^{s}(\mathbb{T}^{2})}^{2}+R_{s,N}(u),$
(2.7)

where

$R_{s,N}(u):=\frac{1}{6}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{-i\Omega(\vec{k})}\cdot\Big{(}\prod_{j=1}^{6}\chi_{N}(k_{j})\Big{)}\cdot\widehat{u}_{k_{1}}\overline{\widehat{u}}_{k_{2}}\cdots\overline{\widehat{u}}_{k_{6}}\;.$
(2.8)

We define $R_{s}(u)$ as $R_{s,N}(u)$ without $\prod_{j=1}^{6}\chi_{N}(k_{j})$.
Sometimes $R_{s}(u)$ will be denoted by $R_{s,\infty}(u)$. We similarly define
$E_{s}(u)$ which may also be denoted by $E_{s,\infty}(u)$.

The modified energy (2.7) will play a crucial role in our analysis. We refer
to [42] for a survey on the use of modified energies in the analysis of
dispersive PDE’s.

Then from (2.5) and the equation (2.2) of $u_{N}(t)$, and symmetry of indices,
we have (with $w_{k}=\chi_{N}(k)v_{k}$)

$\displaystyle\frac{d}{dt}E_{s,N}(u_{N}(t))=\frac{d}{dt}\mathcal{E}_{s,t}(v)$
$\displaystyle=-\frac{1}{6}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k})e^{-it\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$
$\displaystyle+\frac{1}{2}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}$
$\displaystyle\qquad\times\sum_{\begin{subarray}{c}k_{1}=p_{1}-p_{2}+\cdots+p_{5}\end{subarray}}e^{-it\big{(}\Omega(\vec{k})+\Omega(\vec{p})\big{)}}\chi_{N}(k_{1})^{2}w_{p_{1}}\overline{w}_{p_{2}}\cdots
w_{p_{5}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$
$\displaystyle-\frac{1}{2}\mathop{\rm Im}\nolimits\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}$
$\displaystyle\qquad\times\sum_{k_{2}=q_{1}-q_{2}+\cdots+q_{5}}e^{-it\big{(}\Omega(\vec{k})-\Omega(\vec{q})\big{)}}\chi_{N}(k_{2})^{2}w_{k_{1}}\overline{w}_{q_{1}}\cdots\overline{w}_{q_{5}}w_{k_{3}}\cdots\overline{w}_{k_{6}},$
(2.9)

where

$\displaystyle\Omega(\vec{p})=\sum_{j=1}^{5}(-1)^{j-1}|p_{j}|^{2}-|k_{1}|^{2},\quad\Omega(\vec{q})=\sum_{j=1}^{5}(-1)^{j-1}|q_{j}|^{2}-|k_{2}|^{2}.$

###  2.3. The weighted measure

Using the modified energy, we define the weighted Gaussian measure for given
$R>1$

$\displaystyle
d\rho_{s,R,N}(u)=\chi_{R}(\|u\|_{H^{\sigma}})\cdot{\mathrm{e}}^{-R_{s,N}(u)}d\mu_{s,N}(u),\quad
d\overline{\rho}_{s,R,N}(u):=d\rho_{s,R,N}\otimes d\mu_{s,N}^{\perp},$ (2.10)

where the functional $R_{s,N}(u)$ is defined by (2.8) and
$\chi_{R}(\cdot)=\chi(R^{-1}\cdot)$ is a cutoff.

######  Proposition 2.1 (Local existence of the weighted measure).

Let $s\geq 10,R\geq 1$, $\sigma<s-\frac{3}{2}$, close to $s-\frac{3}{2}$ and
$N\in\mathbb{N}$. Then for any $p\in[1,\infty)$, there exists a uniform in $N$
constant $C(p,s,R)>0$, such that

$\big{\|}\chi_{R}(\|u\|_{H^{\sigma}})\cdot{\mathrm{e}}^{|R_{s,N}(u)|}\big{\|}_{L^{p}(d\mu_{s})}\leq
C(p,s,R).$

Moreover, for fixed $R>0$,

$\lim_{N\rightarrow\infty}\big{\|}\chi_{R}(\|u\|_{H^{\sigma}}){\mathrm{e}}^{-R_{s,N}(u)}-\chi_{R}(\|u\|_{H^{\sigma}}){\mathrm{e}}^{-R_{s}(u)}\big{\|}_{L^{p}(d\mu_{s})}=0.$

Recall that $\Phi_{N}(t)$ is the flow of (2.1) while
$\Phi_{\infty}(t)=\Phi(t)$ is the flow of (1.1). Another key proposition is
the following weighted energy estimate:

######  Proposition 2.2 (Weighted energy estimate).

Let $s\geq 10,R\geq 1$, $\sigma<s-\frac{3}{2}$, close to $s-\frac{3}{2}$ and
$N\in\mathbb{N}\cup\\{\infty\\}$. Set

$Q_{s,N}(u)=\frac{d}{dt}E_{s,N}(\Phi_{N}(t)u)|_{t=0}$

and denote by $B_{R}^{H^{\sigma}}$ the centered ball in
$H^{\sigma}(\mathbb{T}^{3})$ of radius $R$. Then there exist constants
$C(s,R)>0$ and $\beta\in(0,1)$, such that for all $p\in[2,\infty)$ and
$N\in\mathbb{N}\cup\\{\infty\\}$,

$\|\mathbf{1}_{B_{R}^{H^{\sigma}}}(u)\cdot Q_{s,N}(u)\|_{L^{p}(d\mu_{s})}\leq
C(s,R)p^{\beta}.$

Thanks to Proposition 2.1, we have also for all
$N\in\mathbb{N}\cup\\{\infty\\},p\in[1,\infty)$,

$\|\mathbf{1}_{B_{R}^{H^{\sigma}}}(u)\cdot
Q_{s,N}(u)\|_{L^{p}(\overline{\rho}_{s,R,N})}\leq C(s,R)p^{\beta}.$

The proof of above two propositions will occupy the main part of the article.
To prove the quasi-invariance of the full system, we need to pass to the limit
$N\rightarrow\infty$ in the approximated equation (2.1). This will be done in
the next section.

##  3\. Proof of the quasi-invariance assuming energy estimates

In this section we prove Theorem 1.1, assuming Proposition 2.1 and Proposition
2.2.

###  3.1. Approximation theory for the energy-critical NLS

######  Proposition 3.1.

Assume that $\sigma\geq 1$. There exists a constant $\Lambda(R,T)>0$,
depending only on $T>0$, $R>0$ and $\sigma\geq 1$, such that for any $u\in
B_{R}^{H^{\sigma}}$,

$\sup_{|t|\leq T}\|\Phi(t)u\|_{H^{\sigma}}+\sup_{|t|\leq
T}\|\Phi_{N}(t)u\|_{H^{\sigma}}\leq\Lambda(R,T),\quad\forall\,N\in\mathbb{N}.$

######  Proposition 3.2.

Assume that $\sigma\geq 1$. Let $K$ be a compact subset of
$H^{\sigma}(\mathbb{T}^{3})$ and $T>0$. Then uniformly in $|t|\leq T$ and
$u\in K$,

$\lim_{N\rightarrow\infty}\|\Phi_{N}(t)u-\Phi(t)u\|_{H^{\sigma}}=0.$

Observe that since $\Phi_{N}(t)$ and $\Phi(t)$ are continuous we have that for
any $|t|\leq T$ and $N\in\mathbb{N}$, $\Phi_{N}(t)(K),\Phi(t)(K)$ are also
compacts in $H^{\sigma}(\mathbb{T}^{3})$. The proof of Proposition 3.1,
Proposition 3.2 will be given in the Appendix.

###  3.2. Proof of quasi-invariance

First, we prove:

######  Lemma 3.3.

Let $T\geq 1$. Let $A\subset B_{R}^{H^{\sigma}}$ be a Borel measurable set.
Then there exist $\epsilon_{0}>0$ and $C_{s,R,T}>0$, such that for all
$N\in\mathbb{N}$ and $|t|\leq T$, $\mu_{s}(\Phi_{N}(t)(A))\leq
C_{s,R,T}\cdot\mu_{s}(A)^{\frac{1-\epsilon_{0}}{4}}$.

###### Proof.

Let $\Lambda(R,T)>0$ be the constant in Proposition 3.1, such that for all
$R>0,N\in\mathbb{N}\cup\\{\infty\\}$,

$\Phi_{N}(t)(B_{R}^{H^{\sigma}})\subset
B_{\Lambda(R,T)}^{H^{\sigma}},\quad|t|\leq T.$

Denote $R_{1}:=\Lambda(\Lambda(R,T),T)$, and we consider the weighted measure

$\displaystyle d\overline{\rho}_{s,R_{1},N}(u)=$
$\displaystyle\rho_{s,R_{1},N}(u)\otimes d\mu_{s,N}^{\perp}$ $\displaystyle=$
$\displaystyle\chi_{R_{1}}(\|u\|_{H^{\sigma}})\frac{1}{\mathcal{Z}_{N}}{\mathrm{e}}^{-E_{s,N}(u)}\Big{(}\prod_{|k|\leq
N}d\widehat{u}_{k}\Big{)}\otimes d\mu_{s,N}^{\perp},$

where $\mathcal{Z}_{N}>0$ is the normalizing constant appearing in the finite-
dimensional truncation of the Gaussian measure

$d\mu_{s,N}(u)=\frac{1}{\mathcal{Z}_{N}}{\mathrm{e}}^{-\frac{1}{2}\sum_{|k|\leq
N}(1+|k|^{2s})|\widehat{u}_{k}|^{2}}\Big{(}\prod_{|k|\leq
N}d\widehat{u}_{k}\Big{)}.$

For $A\subset B_{R}^{H^{\sigma}}$, from Proposition 3.1, for any
$|t_{1}|,|t_{2}|\leq T$ and $N\in\mathbb{N}$,

$\Phi_{N}(t_{2})\circ\Phi_{N}(t_{1})(A)\subset B_{R_{1}}^{H^{\sigma}}.$

In particular, for any $u\in A$, $|t|\leq 2T$,
$\|\Phi_{N}(t)u\|_{H^{\sigma}}\leq R_{1}$. Now for $|t_{0}|\leq T,|t|\leq 1$,
using that for $\chi_{R_{1}}(\|\Phi_{N}(t)u\|_{H^{\sigma}})\equiv 1$ for $u\in
A$, as in [35, 36, 38], we can obtain the following change of variable formula

$\displaystyle\overline{\rho}_{s,R_{1},N}(\Phi_{N}(t_{0}+t)(A))=\int_{A}\frac{1}{\mathcal{Z}_{N}}{\mathrm{e}}^{-E_{s,N}(\Pi_{N}\Phi_{N}(t_{0}+t)u)}\prod_{|k|\leq
N}d\widehat{u}_{k}\,d\mu_{s,N}^{\perp}(u).$

Observe that

$Q_{s,N}(u)=\frac{d}{dt}E_{s,N}(\Phi_{N}(t)u)|_{t=0}=\frac{d}{dt}E_{s,N}(\Pi_{N}\Phi_{N}(t)u)|_{t=0}\,.$

Taking the time derivative of the above equality and evaluate it at $t=0$, we
obtain the identity

$\displaystyle\frac{d}{dt}\overline{\rho}_{s,R_{1},N}(\Phi_{N}(t_{0}+t)(A))|_{t=0}=$
$\displaystyle-\int_{A}\frac{1}{\mathcal{Z}_{N}}Q_{s,N}(\Phi_{N}(t_{0})u)\,{\mathrm{e}}^{-E_{s,N}(\Pi_{N}\Phi_{N}(t_{0})u)}\prod_{|k|\leq
N}d\widehat{u}_{k}\,d\mu_{s,N}^{\perp}(u)$ $\displaystyle=$
$\displaystyle-\int_{\Phi_{N}(t_{0})(A)}\frac{1}{\mathcal{Z}_{N}}Q_{s,N}(u)\,{\mathrm{e}}^{-E_{s,N}(\Pi_{N}u)}\prod_{|k|\leq
N}d\widehat{u}_{k}\,d\mu_{s,N}^{\perp}(u),$

where we again used the change of variable formula. As for
$u\in\Phi_{N}(t_{0})(A)$, $1\leq\chi_{R_{1}}(\|u\|_{H^{\sigma}})$, we obtain
the inequality

$\displaystyle\Big{|}\frac{d}{dt}\overline{\rho}_{s,R_{1},N}$
$\displaystyle(\Phi_{N}(t)(A))|_{t=t_{0}}\Big{|}$ $\displaystyle\leq$
$\displaystyle\int_{\Phi_{N}(t_{0})(A)}\chi_{R_{1}}(\|u\|_{H^{\sigma}})\frac{1}{\mathcal{Z}_{N}}|Q_{s,N}(u)|\,{\mathrm{e}}^{-E_{s,N}(\Pi_{N}u)}\prod_{|k|\leq
N}d\widehat{u}_{k}\,d\mu_{s,N}^{\perp}(u)$ $\displaystyle=$
$\displaystyle\int_{\Phi_{N}(t_{0})(A)}|Q_{s,N}(u)|d\overline{\rho}_{s,R_{1},N}(u)$
$\displaystyle\leq$
$\displaystyle\|Q_{s,N}(u)\|_{L^{p}(d\overline{\rho}_{s,R_{1},N})}\cdot\overline{\rho}_{s,R_{1},N}(\Phi_{N}(t_{0})(A))^{1-\frac{1}{p}}.$

Thanks to the last assertion of Proposition 2.2, the function

$F(t):=\overline{\rho}_{s,R_{1},N}(\Phi_{N}(t)(A)),$

satisfies the inequality

$F^{\prime}(t)\leq C_{s,R}\cdot
p^{\beta}F(t)^{1-\frac{1}{p}},\quad\forall|t|\leq T,\;\quad p<\infty$

Integrating the differential inequality above, we obtain that

$F(t)\leq\big{(}F(0)^{\frac{1}{p}}+C_{s,R}\cdot p^{-(1-\beta)}t\big{)}^{p}\leq
F(0){\mathrm{e}}^{C_{R,s}tp^{\beta}F(0)^{-\frac{1}{p}}}.$

Without loss of generality, we assume that $F(0)=\mu_{s}(A)>0$. By optimizing
the choice

$p=2+\log\Big{(}\frac{1}{F(0)}\Big{)},$

we conclude that there exists $\epsilon_{0}\in(0,1)$, such that

$F(t)\leq C_{R,s,T}F(0)^{1-\epsilon_{0}},\quad\forall|t|\leq T,$

namely

$\overline{\rho}_{s,R_{1},N}(\Phi_{N}(t)(A))\leq
C_{R_{1},s,T}\overline{\rho}_{s,R_{1},N}(A)^{1-\epsilon_{0}},\quad\forall|t|\leq
T.$

Finally, as $\Phi_{N}(t)(A)\subset B_{R_{1}/2}^{H^{\sigma}}$,

$\mu_{s}(\Phi_{N}(t)(A))=\int_{\Phi_{N}(t)(A)}\chi_{R_{1}}(\|u\|_{H^{\sigma}})\cdot{\mathrm{e}}^{R_{s,N}(u)}d\overline{\rho}_{s,R,N}(u).$

By Cauchy-Schwarz and the $L^{2}$-integrability of
$\chi_{R_{1}}(\|u\|_{H^{\sigma}})\cdot{\mathrm{e}}^{R_{s,N}(u)}$ with respect
to $d\mu_{s}$ (Proposition 2.1),

$\displaystyle\mu_{s}(\Phi_{N}(t)(A))\leq$
$\displaystyle\|\chi_{R_{1}}(\|u\|_{H^{\sigma}}){\mathrm{e}}^{R_{s,N}(u)}\|_{L^{2}(d\overline{\rho}_{s,R,N})}\overline{\rho}_{s,R,N}(\Phi(t)(A))^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\|\chi_{R_{1}}(\|u\|_{H^{\sigma}}){\mathrm{e}}^{|R_{s,N}(u)|}\|_{L^{1}(d\mu_{s})}^{\frac{1}{2}}\cdot\sqrt{C_{R_{1},s,T}}\overline{\rho}_{s,R_{1},N}(A)^{\frac{1-\epsilon_{0}}{2}}$
$\displaystyle\leq
C_{R_{1},s,T}^{\prime}\overline{\rho}_{s,R_{1},N}(A)^{\frac{1-\epsilon_{0}}{2}},$
(3.1)

Again, since $A\subset B_{R}^{H^{\sigma}}\subset B_{R_{1}}^{H^{\sigma}}$,

$\overline{\rho}_{s,R_{1},N}(A)\leq\|\chi_{R_{1}}(\|u\|_{H^{\sigma}}){\mathrm{e}}^{R_{s,N}(u)}\|_{L^{2}(d\mu_{s})}\mu_{s}(A)^{\frac{1}{2}}\leq
C_{R_{1},s}\mu_{s}(A)^{\frac{1}{2}}.$

Plugging into (3.2), we complete the proof of Lemma 3.3. ∎

###### Proof of Theorem 1.1.

Let $T>0$, we first show that for any compact set $K\subset
B_{R}^{H^{\sigma}}$, $|t|\leq T$,

$\mu_{s}(\Phi(t)(K))\leq
C_{s,2R,T}\cdot\mu_{s}(K)^{\frac{1-\epsilon_{0}}{4}}.$

Indeed, applying the approximation theory (Proposition 3.2) to the set
$\Phi(t)(K)$, which is compact, for any small $\epsilon>0$, there exists
$N_{0}\in\mathbb{N}$, such that for all $N\geq N_{0}$,

$\Phi_{N}(-t)(\Phi(t)(K))\subset\Phi(-t)(\Phi(t)(K))+B_{\epsilon}^{H^{\sigma}}=K+B_{\epsilon}^{H^{\sigma}},$

thus $\Phi(t)(K)\subset\Phi_{N}(t)(K+B_{\epsilon}^{H^{\sigma}})$,
consequently,

$\displaystyle\mu_{s}(\Phi(t)(K))\leq\mu_{s}(\Phi_{N}(t)(K+B_{\epsilon}^{H^{\sigma}})).$

Since for small $\epsilon>0$, $K+B_{\epsilon}^{H^{\sigma}}\subset
B_{2R}^{H^{\sigma}}$, Lemma 3.3 implies that

$\displaystyle\mu_{s}(\Phi(t)(K))\leq\mu_{s}(\Phi_{N}(t)(K+B_{\epsilon}^{H^{\sigma}}))\leq
C_{s,2R,T}\cdot\mu_{s}(K+B_{\epsilon}^{H^{\sigma}})^{\frac{1-\epsilon_{0}}{4}}.$
(3.2)

We are going to take the limit $\epsilon\rightarrow 0$ in the inequality
above, using the fact that $\mu_{s}$ is regular. Before doing that, we have to
show that for any open set $G\supset K$, there exists $\epsilon>0$, such that

$G\supset K+B_{\epsilon}^{H^{\sigma}}.$

Since $K$ is compact, for any open set $G\supset K$, there exist finitely many
balls $B_{1},\cdots B_{m}$ of $H^{\sigma}$ such that

$K\subset\bigcup_{j=1}^{m}B_{j}\subset\bigcup_{j=1}^{m}2B_{j}\subset G,$

where $2B_{j}$ is the ball with the same center as $B_{j}$ and with radius
twice of $B_{j}$. In particular, there exists $\epsilon_{1}>0$, such that for
all $0<\epsilon<\epsilon_{1}$,

$K+B_{\epsilon}^{H^{\sigma}}\subset G.$

To see this, we take
$\epsilon_{1}<\frac{1}{4}\min\\{\mathrm{radius}(B_{j}):j=1,\cdots,m\\}$. Then
for any $u\in K+B_{\epsilon_{1}}^{\sigma}$, there exists $u_{0}\in K$, such
that $\|u-u_{0}\|_{H^{\sigma}}<\epsilon_{1}$. As $K$ is covered by $B_{j}$,
there is a ball, say $B_{1}$ with center $u_{1}$, such that
$\|u_{0}-u_{1}\|_{H^{\sigma}}<\mathrm{radius}(B_{1})$. Hence $u\in
2B_{1}\subset G$.

Recall that Gaussian measures are regular, namely, for any Borel set $A$

$\displaystyle\mu_{s}(A)=$ $\displaystyle\inf\\{\mu_{s}(G):\;G\supset
A,\;G\text{ open in }H^{\sigma}\\}$ $\displaystyle=$
$\displaystyle\sup\\{\mu_{s}(F):\;F\subset A,\;F\text{ compact and Borel in
}H^{\sigma}\\}$

we can take $\epsilon\rightarrow 0$ on the right hand side of (3.2) to obtain
the estimate

$\displaystyle\mu_{s}(\Phi(t)(K))\leq
C_{s,2R,T}\cdot\mu_{s}(K)^{\frac{1-\epsilon_{0}}{4}},$ (3.3)

as desired.

Finally we assume that $A\subset B_{R}^{H^{\sigma}}$ is an arbitrary Borel
set. Since $\Phi(t)$ is a continuous bijection on
$H^{\sigma}(\mathbb{T}^{3})$, $\Phi(t)(A)$ is also a Borel set (view
$\Phi(t)(A)=(\Phi(-t))^{-1}(A)$). Thus there exists a sequence of compact sets
$K_{n}\subset\Phi(t)(A)$, such that

$\mu_{s}(\Phi(t)(A))=\lim_{n\rightarrow\infty}\mu_{s}(K_{n}).$

For fixed $|t|\leq T$, set $F_{n}=\Phi(-t)(K_{n})$, by the bijectivity of
$\Phi(t)$, $K_{n}=\Phi(t)(F_{n})$. Since $F_{n}$ are also compact (Proposition
3.2), we deduce that

$\mu_{s}(K_{n})=\mu_{s}(\Phi(t)(F_{n}))\leq
C_{R_{1},s,T}\cdot\mu_{s}(F_{n})^{\frac{1-\epsilon_{0}}{4}}.$

Observe that $K_{n}=\Phi(t)(F_{n})\subset\Phi(t)(A)$, again from the
bijectivity, $F_{n}\subset A$, thus

$\mu_{s}(K_{n})\leq C_{R_{1},s,T}\cdot\mu_{s}(A)^{\frac{1-\epsilon_{0}}{4}}.$

Letting $n\rightarrow\infty$, we deduce that

$\mu_{s}(\Phi(t)(A))\leq
C_{R_{1},s,T}\cdot\mu_{s}(A)^{\frac{1-\epsilon_{0}}{4}}.$

In particular, if $\mu_{s}(A)=0$, we must have $\mu_{s}(\Phi(t)(A))=0$. This
proves the quasi-invariance property of $\mu_{s}$ along the flow $\Phi(t)$. ∎

##  4\. Preliminaries for the energy estimates

In this section, we summarize several frequently used preliminary results as
well as some notations.

###  4.1. Deterministic tools

For a given set of frequencies $k_{1},k_{2},\cdots,k_{m}$, we denote
$k_{(1)},k_{(2)},\cdots,k_{(m)}$ a non-increasing rearrangement such that

$|k_{(1)}|\geq|k_{(2)}|\geq\cdots\geq|k_{(m)}|.$

Similarly, for a given set of dyadic integers $N_{1},N_{2},\cdots,N_{m}$, we
denote $N_{(1)},N_{(2)},\cdots,N_{(m)}$ a non-increasing rearrangement such
that

$N_{(1)}\geq N_{(2)}\geq\cdots\geq N_{(m)}.$

We have the following estimate on the the function $\psi_{s}$ which measures
the lack of conservation of $H^{s}$ based quantities.

######  Lemma 4.1.

Set

$\psi_{2s}(\vec{k})=\sum_{j=1}^{6}(-1)^{j-1}|k_{j}|^{2s},\quad\Omega(\vec{k})=\sum_{j=1}^{6}(-1)^{j-1}|k_{j}|^{2}.$

Then for $k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0$,

$|\psi_{2s}(\vec{k})|\lesssim|k_{(1)}|^{2s-2}[|\Omega(\vec{k})|+|k_{(3)}|^{2}].$

###### Proof.

We can suppose that $|k_{(3)}|\ll|k_{(2)}|$, otherwise the estimate reduces to
the straightforward bound $|\psi_{2s}(\vec{k})|\lesssim|k_{(1)}|^{2s}.$
Essentially, there are two different cases : $k_{(1)}=k_{1}$, $k_{(2)}=k_{2}$
and $k_{(1)}=k_{1}$, $k_{(2)}=k_{3}$. In the second case we can again use the
bound $|\psi_{2s}(\vec{k})|\lesssim|k_{(1)}|^{2s}.$ Let is now suppose that
$k_{(1)}=k_{1}$, $k_{(2)}=k_{2}$. By the mean-value theorem,

$\big{|}|k_{1}|^{2s}-|k_{2}|^{2s}\big{|}\lesssim|k_{(1)}|^{2(s-1)}\big{|}|k_{1}|^{2}-|k_{2}|^{2}\big{|}\lesssim|k_{(1)}|^{2s-2}[|\Omega(\vec{k})|+|k_{(3)}|^{2}].$

This completes the proof of Lemma 4.1. ∎

For linear constraints, we denote

$\mathfrak{h}_{k_{1}^{\iota_{1}}k_{2}^{\iota_{2}}\cdots
k_{m}^{\iota_{m}}}:=\mathbf{1}_{\iota_{1}k_{1}+\iota_{2}k_{2}+\cdots+\iota_{m}k_{m}=0},$

where $\iota_{j}\in\\{+1,-1\\}$, identified also as $\\{+,-\\}$, the signature
of frequencies $k_{1},\cdots,k_{m}$. For example,

$\mathfrak{h}_{k_{1}^{+}k_{2}^{-}k_{3}^{+}k_{4}^{-}k_{5}^{+}k_{6}^{-}}=\mathbf{1}_{k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0}.$

We will frequently use the following elementary counting bound:

######  Lemma 4.2.

Assume that $n\geq 2$ and given dyadic numbers $N_{1},N_{2},\cdots N_{n}$.
Then uniformly in $K\in\mathbb{Z}^{3}$, $\kappa\in\mathbb{R}$ and
$\iota_{j}\in\\{+1,-1\\}$, we have

$\sum_{\begin{subarray}{c}k_{1},k_{2},\cdots,k_{n}\\\
\iota_{i}k_{i}+\iota_{j}k_{j}\neq 0,\forall i\neq
j\end{subarray}}\mathbf{1}_{\iota_{1}k_{1}+\iota_{2}k_{2}+\cdots+\iota_{n}k_{n}=K}\cdot\mathbf{1}_{\iota_{\mathfrak{l}_{1}}|k_{1}|^{2}+\cdots+\iota_{\mathfrak{l}_{n}}|k_{n}|^{2}=\kappa}\Big{(}\prod_{j=1}^{n}\mathbf{1}_{|k_{j}|\sim
N_{j}}\Big{)}\lesssim N_{(2)}^{2}\prod_{j=3}^{n}N_{(j)}^{3},$

where we adopt the convention that when $n=2$, the bound on the right hand
side is $N_{(2)}^{2}$.

######  Remark 4.3.

The counting bound stated here is very rough but it already fits our need. By
using some arithmetic, one can improve it when $n\geq 3$ or $n=2$ and
$\iota_{1}=\iota_{2}$. We refer to Lemma 4.5 of [13] for such an improvement .
The estimate of Lemma 4.2 has the advantage to hold with the same (trivial)
proof on a general torus.

Next we recall the following conditional Wiener chaos estimate for multi-
linear expression of complex Gaussian random variables. In the sequel we adopt
the notation $z^{+}=z$ and $z^{-}=\overline{z}$ for a complex number
$z\in\mathbb{C}$.

######  Lemma 4.4 (Wiener chaos estimate).

Consider the multi-linear expression of Gaussian:

$F(\omega^{\prime},\omega)=\sum_{k_{1},\cdots,k_{n}}c_{k_{1},\cdots,k_{n}}(\omega^{\prime})\cdot\prod_{j=1}^{n}g_{k_{j}}^{\iota_{j}}(\omega),$

where the random variables $c_{k_{1},\cdots,k_{n}}(\omega^{\prime})$ are
independent of complex standard i.i.d. Gaussians $g_{k_{j}}(\omega)$. Then for
any $p\geq 2$, we have

$\|F(\omega^{\prime},\omega)\|_{L_{\omega}^{p}}\leq
Cp^{\frac{n}{2}}\|F(\omega^{\prime},\omega)\|_{L_{\omega}^{2}}.$

We state the Wiener chaos estimate in above form since later on we will use
Lemma 4.4 for $L^{p}$ estimates for high-frequency Gaussians conditioning to
some $\sigma$-algebra generated by low-frequency Gaussians (see also [41] for
a statement involving conditional expectation). Starting from [5], in recent
years such conditioned Wiener chaos estimates were extensively use in the
field of random dispersive PDE’s.

##  5\. Decomposion of the differential of the modified energy

Recall from (2.2) that

$Q_{s,N}(w)=\mathop{\rm
Im}\nolimits\big{(}-\frac{1}{6}\mathcal{R}_{0}(w)+\frac{1}{2}\mathcal{R}_{1}(w)-\frac{1}{2}\mathcal{R}_{2}(w)\big{)},$

where

$\mathcal{R}_{0}(w):=\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k})w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}},$
(5.1)
$\displaystyle\mathcal{R}_{1}(w):=\sum_{\begin{subarray}{c}k_{1}-k_{2}+\cdots-
k_{6}=0\\\ k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}\end{subarray}}$
$\displaystyle\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\chi_{N}(k_{1})^{2}w_{p_{1}}\overline{w}_{p_{2}}\cdots
w_{p_{5}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$ (5.2)

and

$\displaystyle\mathcal{R}_{2}(w):=\sum_{\begin{subarray}{c}k_{1}-k_{2}+\cdots-
k_{6}=0\\\
k_{2}=q_{1}-q_{2}+q_{3}-q_{4}+q_{5}\end{subarray}}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\chi_{N}(k_{2})^{2}w_{k_{1}}\overline{w}_{q_{1}}\cdots\overline{w}_{q_{5}}w_{k_{3}}\cdots\overline{w}_{k_{6}}.$
(5.3)

Comparing to the estimate for $\mathcal{R}_{0}(w)$, the major difficulty in
estimating $\mathcal{R}_{1}(w),\mathcal{R}_{2}(w)$ is the existence of pairing
contributions between different generations ($w_{k_{j}}$ and $w_{p_{j}}$ (or
$w_{k_{j}}$ and $w_{q_{j}}$)). Roughly speaking, the singular pairing
contributions in $\mathcal{R}_{1}(v)$ are (up to symmetry)

  * •

$|k_{1}|\sim|k_{2}|\gg|k_{3}|+|k_{4}|+|k_{5}|+|k_{6}|$,
$|k_{1}|\sim|k_{2}|\gg|p_{2}|+|p_{3}|+|p_{4}|+|p_{5}|$ and $p_{1}=k_{2}$;

  * •

$|k_{1}|\sim|k_{3}|\gg|k_{2}|+|k_{4}|+|k_{5}|+|k_{6}|,|k_{1}|\sim|k_{3}|\gg|p_{1}|+|p_{3}|+|p_{4}|+|p_{5}|$
and $p_{2}=k_{3}$.

Now we identify these pairing contributions precisely:

$\displaystyle\Lambda_{1,1}:=\Big{\\{}(p_{1},\cdots,p_{5},k_{2},\cdots,k_{6}):$
$\displaystyle\sum_{j=1}^{5}(-1)^{j-1}p_{j}+\sum_{i=2}^{6}(-1)^{i-1}k_{i}=0,$
$\displaystyle k_{2}=p_{1},\
\sum_{i\in\\{3,4,5,6\\}}|k_{i}|\leq|k_{1}|^{\theta}+|k_{2}|^{\theta},\;\sum_{j\in\\{2,3,4,5\\}}|p_{j}|\leq|k_{1}|^{\theta}+|k_{2}|^{\theta}\Big{\\}}$
(5.4)

and

$\displaystyle\Lambda_{1,2}:=\Big{\\{}(p_{1},\cdots,p_{5},k_{2},\cdots,k_{6}):$
$\displaystyle\sum_{j=1}^{5}(-1)^{j-1}p_{j}+\sum_{i=2}^{6}(-1)^{i-1}k_{i}=0,$
$\displaystyle
k_{3}=p_{2},\;\sum_{i\in\\{2,4,5,6\\}}|k_{i}|\leq|k_{1}|^{\theta}+|k_{3}|^{\theta},\;\sum_{j\in\\{1,3,4,5\\}}|p_{j}|\leq|k_{1}|^{\theta}+|k_{3}|^{\theta}\Big{\\}},$
(5.5)

where $0<\theta<\frac{\delta_{0}}{2}<\frac{1}{3}$ is close to $\frac{1}{3}$.
We define correspondingly

$\displaystyle\mathcal{S}_{1,1}(w):=\sum_{\Lambda_{1,1}}\chi_{N}(k_{1})^{2}|w_{k_{2}}|^{2}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{p_{2}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}$
(5.6)

and

$\displaystyle\mathcal{S}_{1,2}(w):=\sum_{\Lambda_{1,2}}\chi_{N}(k_{1})^{2}|w_{k_{3}}|^{2}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\overline{w}_{k_{2}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot
w_{p_{1}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}.$ (5.7)

Similarly, the pairing contributions in $\mathcal{R}_{2}$ are (up to symmetry)

$\displaystyle\mathcal{S}_{2,1}(w):=\sum_{\Lambda_{2,1}}\chi_{N}(k_{2})^{2}|w_{k_{1}}|^{2}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{q_{3}}w_{q_{2}}\overline{w}_{q_{5}}w_{q_{4}}$
(5.8)

and

$\displaystyle\mathcal{S}_{2,2}(w):=\sum_{\Lambda_{2,2}}\chi_{N}(k_{2})^{2}|w_{k_{4}}|^{2}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}w_{k_{1}}w_{k_{3}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{q_{1}}\overline{w}_{q_{3}}\overline{w}_{q_{5}}w_{q_{4}}.$
(5.9)

where

$\displaystyle\Lambda_{2,1}:=\Big{\\{}(k_{1},q_{1},\cdots,q_{5},k_{3},\cdots,k_{6}):$
$\displaystyle\sum_{j=1}^{5}(-1)^{j}q_{j}+\sum_{i\in\\{1,3,4,5,6\\}}(-1)^{i-1}k_{i}=0,$
$\displaystyle
k_{1}=q_{1},\;\sum_{i\in\\{3,4,5,6\\}}|k_{i}|\leq|k_{1}|^{\theta}+|k_{2}|^{\theta},\;\sum_{j\in\\{2,3,4,5\\}}|q_{j}|\leq|k_{1}|^{\theta}+|k_{2}|^{\theta}\Big{\\}},$
(5.10)

and

$\displaystyle\Lambda_{2,2}:=\Big{\\{}(k_{1},q_{1},\cdots,q_{5},k_{3},\cdots,k_{6}):$
$\displaystyle\sum_{j=1}^{5}(-1)^{j}q_{j}+\sum_{i\in\\{1,3,4,5,6\\}}(-1)^{i-1}k_{i}=0,$
$\displaystyle
k_{4}=q_{2},\;\sum_{i\in\\{1,3,5,6\\}}|k_{i}|\leq|k_{2}|^{\theta}+|k_{4}|^{\theta},\;\sum_{j\in\\{1,3,4,5\\}}|q_{j}|\leq|k_{2}|^{\theta}+|k_{4}|^{\theta}\Big{\\}}.$
(5.11)

$k_{6}$$p_{1}$$k_{1}$$k_{2}$$\mathcal{S}_{1,1}$: $p_{1}$ is paired with
$k_{2}$$k_{6}$$p_{2}$$k_{1}$$k_{3}$$\mathcal{S}_{1,2}$: $p_{2}$ is paired with
$k_{3}$

$k_{6}$$k_{1}$$q_{1}$$k_{2}$$\mathcal{S}_{2,1}$: $q_{1}$ is paired with
$k_{1}$$k_{6}$$q_{2}$$k_{4}$$k_{2}$$\mathcal{S}_{2,2}$: $q_{2}$ is paired with
$k_{4}$

Figure 1: Configurations of trees with two generations. The filled nodes have
a $+$ sign, indicating the absence of a conjugate bar of the Fourier mode,
while the unfilled nodes have a $-$ sign, signifying the presence of a
conjugate bar of the Fourier mode. Each parent node generates five children
nodes, displaying alternating signs. Up to symmetry, there are four distinct
pairings across the two generations, giving the contributions
$\mathcal{S}_{i,j},i,j=1,2$.

By symmetry, we have

$\displaystyle\mathcal{R}_{1}(w)=9\mathcal{S}_{1,1}(w)+4\mathcal{S}_{1,2}(w)+\mathcal{R}_{1,3}(w),$
(5.12)

and

$\displaystyle\mathcal{R}_{2}(w)=9\mathcal{S}_{2,1}(w)+4\mathcal{S}_{2,2}(w)+\mathcal{R}_{2,3}(w),$
(5.13)

where in the expression of remainders $\mathcal{R}_{1,3}(w)$ we have either
$|k_{(3)}|\gtrsim|k_{(1)}|^{\theta}$ or $|k_{(3)}|\lesssim|k_{(1)}|^{\theta}$
and the dominating frequencies are either non-paired or paired within the same
generation. Here $k_{(1)}\cdots,k_{(10)}$ is a rearrangement of leaves
$p_{1},p_{2},p_{3},p_{4},p_{5},k_{2},k_{3},k_{4},k_{5},k_{6}$ such that
$|k_{(1)}|\geq|k_{(2)}|\geq\cdots\geq|k_{(10)}|$. We define similarly the
remainder $\mathcal{R}_{2,3}(w)$. More precisely, we distinguish three
different types in $\mathcal{R}_{1,3}(w)$ (and in $\mathcal{R}_{2,3}(w)$) with
the corresponding constraints in the sum

$\sum_{\begin{subarray}{c}k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0\\\
k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}\\\ \end{subarray}}(\cdots):$

  * •

Type A: $\sum_{j=3}^{10}|k_{(j)}|>|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$.

  * •

Type B: $\sum_{j=3}^{10}|k_{(j)}|\leq|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$
and $\\{k_{(1)},k_{(2)}\\}\subset\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\}$ or
$\\{k_{(1)},k_{(2)}\\}\subset\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}$.

  * •

Type C: $\sum_{j=3}^{10}|k_{(j)}|\leq|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$,
$k_{(1)}\neq k_{(2)}$ and

$k_{(1)}\in\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\},\quad
k_{(2)}\in\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}$

or

$k_{(2)}\in\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\},\quad
k_{(1)}\in\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}.$

Recall that we have to estimate the $L^{p}(d\mu_{s})$ norm of

$Q_{s,N}(w)=-\frac{1}{6}\mathop{\rm
Im}\nolimits\mathcal{R}_{0}(w)+\frac{1}{2}\mathop{\rm
Im}\nolimits\mathcal{R}_{1}(w)-\frac{1}{2}\mathop{\rm
Im}\nolimits\mathcal{R}_{2}(w).$

In Section 6, we estimate the first generation contribution
$\mathcal{R}_{0}(w)$, and in Section 7, we estimate the pairing contributions

$\mathop{\rm Im}\nolimits(\mathcal{S}_{1,j}-\mathcal{S}_{2,j}),\quad j=1,2.$

Finally in Section 8, we finish the estimate for remainders in the second
generation $\mathcal{R}_{1,3}(w),\mathcal{R}_{2,3}(w)$.

##  6\. Energy estimate I: the first generation

Denote

$\displaystyle\mathcal{R}(w):=\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}},$
(6.1) $\displaystyle\mathcal{R}_{0}(w):=\sum_{k_{1}-k_{2}+\cdots-
k_{6}=0}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k})w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}.$
(6.2)

######  Proposition 6.1.

Assume that $\delta_{0}<\frac{2}{3}$. There exists $\beta\in(0,1)$, such that
for any $R>0$ and $p\in[2,\infty)$ we have

$\displaystyle\|\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\mathcal{R}(w)\|_{L^{p}(d\mu_{s})}+\|\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\mathcal{R}_{0}(w)\|_{L^{p}(d\mu_{s})}\leq
C(R)p^{\beta}.$

###### Proof.

Before proceeding to the estimates, we first observe that without loss of
generality, we may assume that there is no pairing between frequencies with
different signatures in the sum defining $\mathcal{R}_{0}(w)$ or
$\mathcal{R}(w)$. Indeed, if this is the case, say $k_{1}=k_{2}$ are paired,
then the resonant function will degenerate to
$|k_{3}|^{2}-|k_{4}|^{2}+|k_{5}|^{2}-|k_{6}|^{2}$ and the energy weight
$\psi_{2s}(\vec{k})$ will degenerate to
$|k_{3}|^{2s}-|k_{4}|^{2s}+|k_{5}|^{2s}-|k_{6}|^{2s}$. Therefore, this pairing
contribution in $\mathcal{R}_{0}(w)$ or $\mathcal{R}(w)$ reduces to some power
of $\|w\|_{L^{2}}^{2}$ times a similar term with two less degrees of
homogeneity222For over pairing contributions, the over-paired part can be
controlled by a power of $\|w\|_{L^{2}}^{2}$., and the treatment of such a
reduced term is similar (simpler) than $\mathcal{R}_{0}(w)$ or
$\mathcal{R}(w)$. So in the sequel of the proof, we implicitly assume that
there is no pairing between frequencies with different signatures in all the
sums.

$\bullet$ Estimate of $\mathcal{R}_{0}(w)$:

Pick $\alpha\in(0,1)$, close enough to $1$, we split $\mathcal{R}_{0}(w)$ as
$\mathrm{I}+\mathrm{II}$, where

$\mathrm{I}:=\sum_{\begin{subarray}{c}|k_{(3)}|>|k_{(1)}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k})w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$

and

$\mathrm{II}:=\sum_{\begin{subarray}{c}|k_{(3)}|\leq|k_{(1)}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k})w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}.$

To estimate I, we only exploit deterministic analysis. The order of
$w_{k_{j}},\overline{w}_{k_{j}}$ plays no significant role in the analysis.
Therefore, without loss of generality, we assume that in the sum,
$|k_{1}|\geq|k_{2}|\geq|k_{3}|\geq|k_{4}|\geq|k_{5}|\geq|k_{6}|$. Taking the
absolute value in the sum, we have

$\displaystyle\mathrm{I}\lesssim$
$\displaystyle\sum_{\begin{subarray}{c}|k_{3}|>|k_{1}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-
k_{6}=0,\end{subarray}}\mathbf{1}_{|\Omega(\vec{k})|\lesssim|k_{1}|^{\delta_{0}}}\;\mathbf{1}_{|k_{1}|\geq|k_{2}|\geq\cdots\geq|k_{6}|}\cdot|k_{1}|^{2s-2}(|k_{3}|^{2}+|\Omega(\vec{k})|)|w_{k_{1}}\cdots
w_{k_{6}}|$ $\displaystyle\lesssim$
$\displaystyle\sum_{\begin{subarray}{c}N_{1}\geq N_{2}\geq\cdots N_{6}\\\
N_{3}\gtrsim N_{1}^{\alpha}\end{subarray}}\mathrm{I}_{N_{1},\cdots N_{6}},$

where the summations are performed on the dyadic values of $N_{1},\cdots
N_{6}$ and

$\mathrm{I}_{N_{1},\cdots N_{6}}:=\sum_{|\kappa|\lesssim
N_{1}^{\delta_{0}}}\sum_{\begin{subarray}{c}|k_{3}|>|k_{1}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-k_{6}=0\\\
\Omega(\vec{k})=\kappa\end{subarray}}N_{1}^{2s-2}N_{3}^{2}\prod_{j=1}^{6}\mathbf{1}_{|k_{j}|\sim
N_{j}}|w_{k_{j}}|,$

provided that $\alpha>\frac{1}{3}$ thanks to the restriction
$0<\delta_{0}<\frac{2}{3}$. Using the Cauchy-Schwarz inequality in the
$k_{1},k_{2}$ summations, we can write

$\mathrm{I}_{N_{1},\cdots N_{6}}\lesssim
N_{1}^{2s-2}N_{3}^{2}\|P_{N_{1}}w\|_{L^{2}}\|P_{N_{2}}w\|_{L^{2}}\prod_{j=3}^{6}\big{(}\sum_{k_{j}\in\mathbb{Z}}\mathbf{1}_{|k_{j}|\sim
N_{j}}|w_{k_{j}}|\big{)},$

where $P_{N}$ is the frequency projector to $|k|\sim N$. Therefore, for
$N_{3}\gtrsim N_{1}^{\alpha}$ and $w\in B_{R}^{H^{\sigma}}$, we have a crude
estimate

$\mathrm{I}_{N_{1},\cdots N_{6}}\lesssim
R^{6}\,N_{1}^{2s-2}N_{3}^{2}\,N_{1}^{-2\sigma}N_{3}^{\frac{3}{2}-\sigma}\lesssim_{R}N_{1}^{\frac{3}{2}+2s-2\sigma}N_{1}^{-\alpha\sigma}\,,$

which is conclusive as far as $(2+\alpha)\sigma>2s+\frac{3}{2}$. The last
restriction is easily satisfied by taking $\alpha$ close to $1$ as far as
$s>\frac{15}{2}$.

Next, we estimate $\mathrm{II}$. We decompose II dyadically as
$\sum_{N_{1},\cdots,N_{6}}\mathrm{II}_{N_{1},\cdots,N_{6}}$, where

$\displaystyle\mathrm{II}_{N_{1},\cdots,N_{6}}:=\sum_{\begin{subarray}{c}|k_{(3)}|\leq|k_{(1)}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\psi_{2s}(\vec{k})w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}\prod_{j=1}^{6}\mathbf{1}_{|k_{j}|\sim
N_{j}}.$

For this contribution, we mainly rely on the Wiener chaos estimates.

Without loss of generality, we assume that $N_{1}\sim N_{2}\sim
N_{(1)},N_{(3)}=N_{3}$ (since the analysis of cases $N_{1}\sim N_{3}\sim
N_{(1)},N_{2}\sim N_{(3)}$ or $N_{1}\sim N_{3}\sim N_{(1)},N_{5}\sim N_{(3)}$
are similar or simpler).

Denote $\mathcal{B}_{\ll N_{1}}$ the $\sigma$-algebra generated by Gaussians
$(g_{k}(\omega))_{|k|\leq N_{1}/100}$. Note that we have the constraint
$N_{3}\leq N_{1}^{\alpha}$ for some $\alpha<1$, close to 1, and we only need
to consider the contribution where $N_{1}$ is sufficiently large so that
$N_{1}^{\alpha}\ll\frac{N_{1}}{100}$. Consequently,
$\mathbf{1}_{|k_{j}|}w_{k_{j}},j=3,4,5,6$ are all $\mathcal{B}_{\ll N_{1}}$
measurable and the random function

$\sum_{|k|\sim
N_{1}}\frac{g_{k}(\omega)}{\sqrt{1+|k|^{2s}}}{\mathrm{e}}^{ik\cdot x}$

is independent of $\mathcal{B}_{\ll N_{1}}$, we have

$\displaystyle\|\mathrm{II}_{N_{1},\cdots,N_{6}}\cdot\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\|_{L^{p}(d\mu_{s})}\leq$
$\displaystyle\|\mathrm{II}_{N_{1},\cdots,N_{6}}\cdot\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\leq
N_{1}/100}w)\|_{L^{p}(d\mu_{s})}$ $\displaystyle\leq$
$\displaystyle\big{\|}\|\mathrm{II}_{N_{1},\cdots,N_{6}}\|_{L^{p}(d\mu_{s}|\mathcal{B}_{\ll
N_{1}})}\cdot\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\leq
N_{1}/100}w)\big{\|}_{L^{\infty}(d\mu_{s})},$

where $\mathbb{P}_{\leq N_{1}/100}$ is the frequency projection to $|k|\leq
N_{1}/100$ and $L^{p}(d\mu_{s}|\mathcal{B}_{\leq N_{1}/100})$ means the
$L^{p}$ norm conditioned to the $\sigma$-algebra $\mathcal{B}_{\leq
N_{1}/100}$. By the conditional Wiener-chaos estimate (see Lemma 4.4 ), we
have

$\displaystyle\|\mathrm{II}_{N_{1},\cdots,N_{6}}\|_{L^{p}(d\mu_{s}|\mathcal{B}_{\ll
N_{1}})}\lesssim$
$\displaystyle\,\,p\|\mathrm{II}_{N_{1}\cdots,N_{6}}\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
N_{1}})}$ $\displaystyle\lesssim$
$\displaystyle\,\,p(N_{1}N_{2})^{-s}\Big{(}\sum_{\begin{subarray}{c}|k_{1}|\sim
N_{1}\\\ |k_{2}|\sim
N_{2}\end{subarray}}\Big{|}\\!\\!\\!\sum_{\begin{subarray}{c}k_{3},k_{4},k_{5},k_{6}\\\
k_{3}-k_{4}+k_{5}-k_{6}=k_{2}-k_{1}\\\ |\Omega(\vec{k})|\lesssim
N_{1}^{\delta_{0}}\end{subarray}}\\!\\!\\!\\!\\!\\!\psi_{2s}(\vec{k})w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\prod_{j=3}^{6}\mathbf{1}_{|k_{j}|\sim
N_{j}}\Big{|}^{2}\Big{)}^{\frac{1}{2}}.$ (6.3)

By Cauchy-Schwarz,

$\displaystyle\sum_{\begin{subarray}{c}|k_{1}|\sim N_{1}\\\ |k_{2}|\sim
N_{2}\end{subarray}}\Big{|}\\!\sum_{\begin{subarray}{c}k_{3},k_{4},k_{5},k_{6}\\\
k_{3}-k_{4}+k_{5}-k_{6}=k_{2}-k_{1}\\\ |\Omega(\vec{k})|\lesssim
N_{1}^{\delta_{0}}\end{subarray}}\\!\\!\\!\\!\\!\\!\psi_{2s}(\vec{k})w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\prod_{j=3}^{6}\mathbf{1}_{|k_{j}|\sim
N_{j}}\Big{|}^{2}$ $\displaystyle\leq$
$\displaystyle\Big{(}\sum_{\begin{subarray}{c}|k_{1}|\sim N_{1}\\\ |k_{2}|\sim
N_{2}\end{subarray}}\sum_{\begin{subarray}{c}k_{3},k_{4},k_{5},k_{6}\\\
k_{3}-k_{4}+k_{5}-k_{6}=k_{2}-k_{1}\\\ |\Omega(\vec{k})|\lesssim
N_{1}^{\delta_{0}}\end{subarray}}\\!\\!\\!\\!\\!\\!|\psi_{2s}(\vec{k})|^{2}|w_{k_{6}}|^{2}\prod_{j=3}^{6}\mathbf{1}_{|k_{j}|\sim
N_{j}}\Big{)}$ $\displaystyle\hskip 85.35826pt\times\sup_{|k_{1}|,|k_{2}|\sim
N}\sum_{k_{3}-k_{4}+k_{5}-k_{6}=k_{2}-k_{1}}|w_{k_{3}}w_{k_{4}}w_{k_{5}}|^{2}\prod_{j=3}^{6}\mathbf{1}_{|k_{j}|\sim
N_{j}}.$

Since $|\psi_{2s}(\vec{k})|^{2}\lesssim
N_{1}^{4(s-1)}(N_{3}^{4}+|\Omega(\vec{k})|^{2})$, the first sum on the right
hand-side can be bounded by (below we implicitly insert the constraint
$|k_{j}|\sim N_{j}$)

$\displaystyle\sum_{|k_{6}|\sim N_{6}}|w_{k_{6}}|^{2}$
$\displaystyle\sum_{|\kappa|\lesssim
N_{1}^{\delta_{0}}}\,\,\sum_{k_{1},k_{2},k_{3},k_{4},k_{5}}N_{1}^{4(s-1)}(N_{3}^{4}+\kappa^{2})\,\,\mathfrak{h}_{k_{1}k_{2}k_{3}k_{4}k_{5}k_{6}}(\kappa)$
$\displaystyle\lesssim$ $\displaystyle
N_{6}^{-2\sigma}\|w\|_{H^{\sigma}}^{2}N_{1}^{4(s-1)}\big{(}N_{3}^{4}N_{1}^{\delta_{0}}\,N_{2}^{2}\,(N_{3}N_{4}N_{5})^{3}+N_{1}^{3\delta_{0}}\,N_{2}^{2}\,(N_{3}N_{4}N_{5})^{3}\big{)}$
$\displaystyle\lesssim$
$\displaystyle\|w\|_{H^{\sigma}}^{2}N_{6}^{-2\sigma}N_{1}^{4s-4}(N_{3}^{4}N_{1}^{\delta_{0}}+N_{1}^{3\delta_{0}})N_{2}^{2}N_{3}^{3}N_{4}^{3}N_{5}^{3},$

where we used Lemma 4.2 and the notation

$\mathfrak{h}_{k_{1}k_{2}k_{3}k_{4}k_{5}k_{6}}(\kappa):=\mathbf{1}_{k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0}\cdot\mathbf{1}_{\Omega(\vec{k})=\kappa}.$

Plugging into (6), we obtain that

$\displaystyle\|\mathrm{II}_{N_{1},\cdots,N_{6}}\|_{L^{p}(d\mu_{s}|\mathcal{B}_{\ll
N_{1}})}\lesssim$
$\displaystyle\,\,pN_{1}^{-1}(N_{3}^{2}N_{1}^{\frac{\delta_{0}}{2}}+N_{1}^{\frac{3\delta_{0}}{2}})N_{3}^{\frac{3}{2}}N_{3}^{-\sigma}\prod_{j=3}^{6}\|w_{N_{j}}\|_{H^{\sigma}}$
$\displaystyle\lesssim$
$\displaystyle\,\,p\big{(}N_{1}^{\frac{\delta_{0}}{2}-1}N_{3}^{\frac{7}{2}-\sigma}+N_{1}^{\frac{3\delta_{0}}{2}-1}N_{3}^{\frac{3}{2}-\sigma}\big{)}\|w\|_{H^{\sigma}}^{4}.$

Since $\delta_{0}<\frac{2}{3}$ and $\sigma>\frac{7}{2}$, the above quantity
can be controlled by

$pN_{1}^{-(1-\frac{3\delta_{0}}{2})}\|w\|_{H^{\sigma}}^{4}.$

Hence

$\|\mathrm{II}_{N_{1},\cdots,N_{6}}\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\|_{L^{p}(d\mu_{s})}\lesssim
pN_{(1)}^{-(1-\frac{3\delta_{0}}{2})}R^{4}.$

Here since we gain a negative power in $N_{(1)}$, by interpolating with the
crude deterministic estimate

$|\mathrm{II}_{N_{1},\cdots,N_{6}}\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)|\lesssim
N_{(1)}^{2s-2\sigma}\|w\|_{H^{\sigma}}^{6}\leq N_{(1)}^{2(s-\sigma)}R^{6},$

we conclude the estimate for $\mathcal{R}_{0}(w)$.

$\bullet$ Estimate of $\mathcal{R}(w)$: The estimate for $\mathcal{R}(w)$ is
similar (simpler) to the estimate for $\mathcal{R}_{0}(w)$ and we only sketch
the proof. Indeed, comparing to the estimate of $\mathcal{R}_{0}(w)$, the only
difference is that the weight
$\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}$ is
now replaced by

$\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{1}{\Omega(\vec{k})}.$

We similarly split $\mathcal{R}(w)$ similarly as
$\mathrm{I}^{\prime}+\mathrm{II}^{\prime}$, where

$\mathrm{I}^{\prime}:=\sum_{\begin{subarray}{c}|k_{(3)}|>|k_{(1)}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}},$
$\mathrm{II}^{\prime}:=\sum_{\begin{subarray}{c}|k_{(3)}|\leq|k_{(1)}|^{\alpha}\\\
k_{1}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}w_{k_{1}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$

and we invoke the inequalities

$\sum_{N_{(1)}^{\delta_{0}}\lesssim|\kappa|\lesssim
N_{(1)}^{2}}\frac{1}{|\kappa|}\lesssim\log(N_{(1)}),\quad\Big{|}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{|}\lesssim|k_{(1)}|^{2s-2}(|k_{(3)}|^{2}+1).$

The proof of Proposition 6.1 is now complete. ∎

##  7\. Energy estimate II: the pairing contributions in the second generation

In this section, we estimate the singular contributions. Recall the definition
of $\mathcal{S}_{i,j}$ in (5.6)-(5.9).

######  Proposition 7.1.

There exist $C>0$ and $\beta=\beta(\theta,s)\in(0,1)$, such that for
$j\in\\{1,2\\}$, $R\geq 1$ and $p\in[2,\infty)$, we have

$\|\mathop{\rm
Im}\nolimits\big{(}\mathcal{S}_{1,j}(w)-\mathcal{S}_{2,j}(w)\big{)}\mathbf{1}_{B_{R}^{H^{\sigma}}(w)}\|_{L^{p}(d\mu_{s})}\leq
Cp^{\beta}R^{10}.$

######  Remark 7.2.

To explain the difficulty, we remark that the singular contributions
$\mathcal{S}_{i,j}(w)$ ($i,j\in\\{1,2\\}$) prevents us to use Wiener-chaos
estimate to gain the square root cancellation. Nevertheless, it turns out that
there is an extra cancellation when one takes the imaginary part of
$\mathcal{S}_{i,j}(w)$. To understand the hidden cancellation, for
$\mathcal{S}_{1,1}(v)$, one can think about the sum is taken over
$|k_{3}|,\cdots,|k_{6}|,|p_{2}|,\cdots,|p_{5}|=O(1)$, then

$\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\approx\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}},$

and the second sum in the definition of $\mathcal{S}_{1,1}$ is completely
decoupled and we have

$\displaystyle\mathcal{S}_{1,1}(w)=$
$\displaystyle-\sum_{k_{1},k_{2}}\chi_{N}(k_{1})^{2}|w_{k_{2}}|^{2}\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}\Big{|}\sum_{\begin{subarray}{c}|k_{3}|+|k_{4}|+|k_{5}|+|k_{6}|\leq|k_{2}|^{\theta}\\\
k_{3}-k_{4}+k_{5}-k_{6}=k_{2}-k_{1}\end{subarray}}w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\Big{|}^{2}$
$\displaystyle+$ $\displaystyle\text{ error },$

where the main contribution is obviously real.

######  Remark 7.3.

However, it turns out that the cancellation described in Remark 7.2 alone is
not enough to conclude, as the error term in the formula above is not
negligible if we estimate individually $\mathcal{S}_{1,j}(w)$ and
$\mathcal{S}_{1,j}(w)$, $j=1,2$. What saves us is that these expressions there
is some symmetric structure so that we can exploit some extra probabilistic
cancellation and a deterministic smoothing. Let us explain theses points with
more details. With the identification of
$(q_{3},q_{2},q_{5},q_{4})=(p_{2},p_{3},p_{4},p_{5})$ (without changing
$k_{j}$), we observe that $\Lambda_{2,1}=\Lambda_{1,1}$ and

$\displaystyle\mathcal{S}_{2,1}(w)=\sum_{\Lambda_{1,1}}\chi_{N}(k_{2})^{2}|w_{k_{1}}|^{2}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{p_{2}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}.$
(7.1)

Therefore,

$\displaystyle\mathop{\rm Im}\nolimits\mathcal{S}_{1,1}(w)-\mathop{\rm
Im}\nolimits\mathcal{S}_{2,1}(w)=$ $\displaystyle\mathop{\rm
Im}\nolimits\sum_{\Lambda_{1,1}}(\chi_{N}(k_{1})^{2}|w_{k_{2}}|^{2}-\chi_{N}(k_{2})^{2}|w_{k_{1}}|^{2})\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}$
$\displaystyle\hskip 71.13188pt\times
w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{p_{2}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}.$
(7.2)

For $j=2$, due to the special position, there is no such cancellation in
$\mathop{\rm
Im}\nolimits\big{(}\mathcal{S}_{1,2}(w)-\mathcal{S}_{2,2}(w)\big{)}$. Indeed,

$\displaystyle\mathcal{S}_{2,2}(w)=$
$\displaystyle\sum_{k_{2},k_{4}}\chi_{N}(k_{2})^{2}|w_{k_{4}}|^{2}\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{1}+k_{3}+k_{5}-k_{6}=k_{2}+k_{4}\\\
q_{1}+q_{3}+q_{5}-q_{4}=k_{2}+k_{4}\\\
|k_{1}|+|k_{3}|+|k_{5}|+|k_{6}|\leq|k_{2}|^{\theta}+|k_{4}|^{\theta}\\\
|q_{1}|+|q_{3}|+|q_{4}|+|q_{5}|\leq|k_{2}|^{\theta}+|k_{4}|^{\theta}\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}w_{k_{1}}w_{k_{3}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{q_{1}}\overline{w}_{q_{3}}\overline{w}_{q_{5}}w_{q_{4}}$

By switching the indices $(k_{1},k_{3},k_{5})$ with $(k_{2},k_{4},k_{6})$ and
identifying $(q_{1},q_{3},q_{4},q_{5})$ as $(p_{1},p_{3},p_{4},p_{5})$ in
$\Lambda_{2,2}$, we deduce that

$\mathcal{S}_{2,2}(w)=\overline{\mathcal{S}}_{1,2}(w),$

where we used the fact that $\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}$ is
invariant by the switching of indices $(k_{1},k_{3},k_{5})$ and
$(k_{2},k_{4},k_{6})$. Therefore,

$\displaystyle\mathop{\rm Im}\nolimits\mathcal{S}_{1,2}(w)-\mathop{\rm
Im}\nolimits\mathcal{S}_{2,2}(w)=-2\mathop{\rm
Im}\nolimits\mathcal{S}_{2,2}(w).$ (7.3)

The good news is that in the expression $\mathop{\rm
Im}\nolimits\mathcal{S}_{1,2}(v)$, we only need to exploit the first
cancellation explained in Remark 7.2, since the resonant function
$\Omega(\vec{k})\approx|k_{1}|^{2}+|k_{3}|^{2}\sim|k_{(1)}|^{2}$ has a
significantly larger size which provides a smoothing effect.

###### Proof of Proposition 7.1.

We separate the analysis for $j=1$ and $j=2$.

$\bullet$ Estimate for $j=1$:

Set

$\Psi(\vec{k}):=\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}-\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}\Big{(}1-\chi\Big{(}\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}\Big{)}.$

We need an elementary lemma:

######  Lemma 7.4.

On $\Lambda_{1,1}$ defined in (5), for sufficiently large $|k_{(1)}|$, we have

$|\Psi(\vec{k})|\lesssim\frac{|k_{(1)}|^{2s-2}|k_{(3)}|^{2}}{|\Omega(\vec{k})|}\mathbf{1}_{|\Omega(\vec{k})|\gtrsim|k_{(1)}|^{\delta_{0}}},$

where we recall that in the definition of $\Lambda_{1,1}$,
$\theta<\frac{\delta_{0}}{2}$.

###### Proof.

Note that on $\Lambda_{1,1}$, $\\{k_{1},k_{2}\\}=\\{k_{(1)},k_{(2)}\\}$ and
$|k_{(3)}|^{2}\lesssim|k_{(1)}|^{2\theta}\ll\lambda(\vec{k})^{\delta_{0}}$.
Thanks to the support property of $\chi$, if
$|\Omega(\vec{k})|\ll|k_{(1)}|^{\delta_{0}}\sim\lambda(\vec{k})^{\delta_{0}}$,
we must have

$\Psi(\vec{k})=-\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}\Big{(}1-\chi\Big{(}\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}\Big{)},$

and
$||k_{1}|^{2}-|k_{2}|^{2}|\gtrsim(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}\sim\lambda(\vec{k})^{\delta_{0}}$,
otherwise $\Psi(\vec{k})=0$. Thus

$|k_{(1)}|^{\delta_{0}}\lesssim||k_{1}|^{2}-|k_{2}|^{2}|=|\Omega(\vec{k})|-O(|k_{(3)}|^{2}),$

which contradicts to the fact that $|k_{(3)}|^{2}\ll|k_{(1)}|^{\delta_{0}}$.
Therefore, we assume that $|\Omega(\vec{k})|\gtrsim|k_{(1)}|^{\delta_{0}}$ and
consequently $|k_{1}|\neq|k_{2}|$ in the sequel.

Set

$G=|k_{3}|^{2}-|k_{4}|^{2}+|k_{5}|^{2}-|k_{6}|^{2},\quad
F=|k_{3}|^{2s}-|k_{4}|^{2s}+|k_{5}|^{2s}-|k_{6}|^{2s}$

and write

$\displaystyle\psi_{2s}(\vec{k})=\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}(\Omega(\vec{k})-G)+F.$

Hence

$\displaystyle\Psi(\vec{k})=$
$\displaystyle\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}\Big{[}\Big{(}1-\frac{G}{\Omega(\vec{k})}\Big{)}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}-\Big{(}1-\chi\Big{(}\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}\Big{)}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{F}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}.$

Since $|F|\lesssim|k_{(3)}|^{2s},|G|\lesssim|k_{(3)}|^{2}$ and

$\Big{|}\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}\Big{|}\lesssim|k_{(1)}|^{2s-2},$

we deduce that

$\displaystyle|\Psi(\vec{k})|\lesssim\frac{|k_{(1)}|^{2s-2}|k_{(3)}|^{2}}{|\Omega(\vec{k})|}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}+|k_{(1)}|^{2s-2}\Big{|}\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}-\chi\Big{(}\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}\Big{|}.$
(7.4)

The first term on the right hand side of (7.4) satisfies the claimed bound. It
remains to evaluate the second one. By the mean value theorem, there exists
$\alpha\in[0,1]$ such that

$\displaystyle\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}-\chi\Big{(}\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}=\chi^{\prime}(\xi_{\alpha})\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)},$

where

$\xi_{\alpha}=\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\alpha\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}.$

Thanks to the support properties of $\chi^{\prime}$, when the second term on
the right hand side of (7.4) is non zero, we must have $|\xi_{\alpha}|\sim 1$.
In this case, a direct computation yields

$\displaystyle\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}=\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\cdot\frac{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}-\lambda(\vec{k})^{\delta_{0}}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}+\frac{G}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}.$

Note that
$|(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}-\lambda(\vec{k})^{\delta_{0}}|\lesssim|k_{(3)}|^{\delta_{0}}$,
we deduce that

$\Big{|}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\xi_{\alpha}\Big{|}\lesssim\alpha\frac{|k_{(3)}|^{\delta_{0}}}{\lambda(\vec{k})^{\delta_{0}}}\Big{|}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\xi_{\alpha}\Big{|}+\alpha\frac{|k_{(3)}|^{\delta_{0}}}{\lambda(\vec{k})^{\delta_{0}}}|\xi_{\alpha}|+\frac{|G|}{\lambda(\vec{k})^{\delta_{0}}}.$

As
$|G|\lesssim|k_{(3)}|^{2}\lesssim\lambda_{1}(\vec{k})^{2\theta}\ll\lambda(\vec{k})^{\delta_{0}}$
for large enough $|k_{(1)}|$, we deduce that

$\Big{|}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}-\xi_{\alpha}\Big{|}\lesssim|k_{(3)}|^{2}\lambda(\vec{k})^{-\delta_{0}}\lesssim\lambda(\vec{k})^{-\delta_{0}+2\theta}\ll
1.$

Since $|\Omega(\vec{k})|\gtrsim|k_{(1)}|^{\delta_{0}}$, the second term on the
right hand side of (7.4) is bounded by

$\mathbf{1}_{|\Omega(\vec{k})|\sim|k_{(1)}|^{\delta_{0}}}\cdot\frac{|k_{(1)}|^{2s-2}|k_{(3)}|^{2}}{\lambda(\vec{k})^{\delta_{0}}}\sim\mathbf{1}_{|\Omega(\vec{k})|\sim|k_{(1)}|^{\delta_{0}}}\cdot\frac{|k_{(1)}|^{2s-2}|k_{(3)}|^{2}}{|\Omega(\vec{k})|}.$

This completes the proof of Lemma 7.4. ∎

The key observation is that

$\displaystyle\sum_{\Lambda_{1,1}}(\chi_{N}(k_{1})^{2}|w_{k_{2}}|^{2}-\chi_{N}(k_{2})^{2}|w_{k_{1}}|^{2})\Big{[}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}-\Psi(\vec{k})\Big{]}$
$\displaystyle\hskip 56.9055pt\times
w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{p_{2}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}$
$\displaystyle=$
$\displaystyle\sum_{k_{1},k_{2}}(\chi_{N}(k_{1})^{2}|w_{k_{2}}|^{2}-\chi_{N}(k_{2})^{2}|w_{k_{1}}|^{2})\frac{|k_{1}|^{2s}-|k_{2}|^{2s}}{|k_{1}|^{2}-|k_{2}|^{2}}\Big{(}1-\chi\Big{(}\frac{|k_{1}|^{2}-|k_{2}|^{2}}{(|k_{1}|^{2}+|k_{2}|^{2})^{\delta_{0}/2}}\Big{)}\Big{)}$
$\displaystyle\hskip
28.45274pt\times\Big{|}\sum_{\begin{subarray}{c}k_{3}-k_{4}+k_{5}-k_{6}=k_{2}-k_{1}\\\
|k_{3}|+|k_{4}|+|k_{5}|+|k_{6}|\leq|k_{1}|^{\theta}+|k_{2}|^{\theta}\end{subarray}}w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\Big{|}^{2}$

is real-valued and it disappears when taking the imaginary part.

Therefore it suffices to show that there exists
$\beta=\beta(s,\theta,\delta_{0})\in(0,1)$, such that

$\displaystyle\|J(w)\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\|_{L^{p}(d\mu_{s})}\lesssim p^{\beta}R^{10}.$ (7.5)

where

$\displaystyle
J(w):=\sum_{\Lambda_{1,1}}\Psi(\vec{k})(\chi_{N}(k_{1})^{2}|w_{k_{2}}|^{2}-\chi_{N}(k_{2})^{2}|w_{k_{1}}|^{2})w_{k_{3}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot\overline{w}_{p_{2}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}.$
(7.6)

Since in the above expression, the contribution of $k_{1}=k_{2}$ is zero,
below we always implicitly assume that $k_{1}\neq k_{2}$.

For dyadic numbers
$N_{1},N_{2},N_{3},N_{4},N_{5},N_{6},M_{2},M_{3},M_{4},M_{5}$, we decompose
accordingly $w_{k_{j}}^{N_{j}}=w_{k_{j}}\mathbf{1}_{|k_{j}|\sim N_{j}}$ and
$w_{p_{i}}^{M_{i}}=v_{p_{i}}\mathbf{1}_{|p_{i}|\sim M_{i}}$. It suffices to
show that

$\displaystyle\|J_{N_{1},\cdots,N_{6};M_{2},\cdots,M_{5}}(w)\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\|_{L^{p}(d\mu_{s})}\lesssim p^{\beta}N_{(1)}^{-\gamma}R^{10}$ (7.7)

for some $\beta\in(0,1)$ and $\gamma>0$, where
$J_{N_{1},\cdots,N_{6};M_{2},\cdots,M_{5}}$ is the same expression as $J(w)$
by replacing the inputs $w_{k_{j}},w_{p_{i}}$ to
$w_{k_{j}}^{N_{j}},w_{p_{i}}^{M_{i}}$. By definition of $\Lambda_{1,1}$, we
have $N_{1}\sim N_{2}$ and

$N_{3}+\cdots+N_{6}+M_{2}+\cdots+M_{5}\lesssim N_{1}^{\theta}.$

By Lemma 7.4 and the fact that $|\Omega(\vec{k})|\gtrsim
N_{(1)}^{\delta_{0}}>N_{(1)}^{2\theta}\gtrsim N_{(3)}^{2}$, a crude
deterministic estimate leads to

$\displaystyle|J_{N_{2},\cdots,M_{5}}(w)|\lesssim$
$\displaystyle\sum_{\Lambda_{1,1}}\frac{N_{1}^{2(s-1)}N_{(3)}^{2}}{|\Omega(\vec{k})|}\mathbf{1}_{k_{1}\neq
k_{2}}(|w^{N_{2}}_{k_{2}}|^{2}+|w^{N_{1}}_{k_{1}}|^{2})|w^{N_{3}}_{k_{3}}\cdots
w^{N_{6}}_{k_{6}}|\cdot|w^{M_{2}}_{p_{2}}\cdots w^{M_{5}}_{p_{5}}|$
$\displaystyle\lesssim$
$\displaystyle\sum_{\Lambda_{1,1}}N_{1}^{2(s-1)}\mathbf{1}_{k_{1}\neq
k_{2}}(|w^{N_{2}}_{k_{2}}|^{2}+|w^{N_{1}}_{k_{1}}|^{2})|w^{N_{3}}_{k_{3}}\cdots
w^{N_{6}}_{k_{6}}|\cdot|w^{M_{2}}_{p_{2}}\cdots w^{M_{5}}_{p_{5}}|,$

and the right hand side ca be bounded by

$\displaystyle
N_{1}^{2(s-1)}(\|w_{k_{1}}^{N_{1}}\|_{l^{2}}^{2}+\|w_{k_{2}}^{N_{2}}\|_{l^{2}}^{2})\|w_{k_{3}}^{N_{3}}\|_{l^{1}}\cdots\|w_{k_{6}}^{N_{6}}\|_{l^{1}}\|w_{p_{2}}^{M_{2}}\|_{l^{1}}\cdots\|w_{p_{5}}^{M_{5}}\|_{l^{1}}$
$\displaystyle\lesssim$ $\displaystyle
N_{1}^{2(s-1)}N_{1}^{-2\sigma}(\|w_{k_{1}}^{N_{1}}\|_{h^{\sigma}}^{2}+\|w_{k_{2}}^{N_{2}}\|_{h^{\sigma}}^{2})\|w_{k_{3}}^{N_{3}}\|_{h^{\sigma}}\cdots\|w_{p_{5}}^{M_{5}}\|_{h^{\sigma}}\cdot(N_{3}\cdots
M_{5})^{-\sigma+\frac{3}{2}}$ $\displaystyle\lesssim$ $\displaystyle
N_{1}^{2(s-1-\sigma)}(N_{3}\cdots
M_{5})^{-\sigma+\frac{3}{2}}\|w\|_{H^{\sigma}}^{10}.$

Therefore, we obtain the first bound

$\displaystyle|J_{N_{1},\cdots,M_{5}}(w)|\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\lesssim N_{1}^{2(s-1-\sigma)}(N_{3}\cdots
M_{5})^{-(\sigma-\frac{3}{2})}R^{10}.$ (7.8)

As $\sigma<s-\frac{3}{2}$, we need to improve the above bound using Wiener
chaos estimates. We further split

$J_{N_{1},\cdots,M_{5}}(w):=\widetilde{J}_{N_{1},\cdots,M_{5}}(w)+R_{N_{1},\cdots,M_{5}}(w),$

where

$\displaystyle\widetilde{J}_{N_{1},\cdots,M_{5}}(w):=\sum_{\Lambda_{1,1}}\Psi(\vec{k})$
$\displaystyle\Big{[}\chi_{N}(k_{1})^{2}\big{(}|w^{N_{2}}_{k_{2}}|^{2}-\frac{1}{\sqrt{1+|k_{1}|^{2s}}}\big{)}-\chi_{N}(k_{2})^{2}\big{(}|w^{N_{1}}_{k_{1}}|^{2}-\frac{1}{\sqrt{1+|k_{2}|^{2s}}}\big{)}\Big{]}$
$\displaystyle\times$ $\displaystyle
w^{N_{3}}_{k_{3}}\overline{w}^{N_{4}}_{k_{4}}w^{N_{5}}_{k_{5}}\overline{w}^{N_{6}}_{k_{6}}\cdot\overline{w}^{M_{2}}_{p_{2}}w^{M_{3}}_{p_{3}}\overline{w}^{M_{4}}_{p_{4}}w^{M_{5}}_{p_{5}},$
(7.9)

and

$\displaystyle
R_{N_{1},\cdots,M_{5}}:=\sum_{\Lambda_{1,1}}\Psi(\vec{k})\Big{(}\frac{\chi_{N}(k_{1})^{2}}{\sqrt{1+|k_{2}|^{2s}}}-\frac{\chi_{N}(k_{2})^{2}}{\sqrt{1+|k_{1}|^{2s}}}\Big{)}w^{N_{3}}_{k_{3}}\overline{w}^{N_{4}}_{k_{4}}w^{N_{5}}_{k_{5}}\overline{w}^{N_{6}}_{k_{6}}\cdot\overline{w}^{M_{2}}_{p_{2}}w^{M_{3}}_{p_{3}}\overline{w}^{M_{4}}_{p_{4}}w^{M_{5}}_{p_{5}}.$

For $|k_{1}|\sim|k_{2}|\sim N_{1}\sim N_{2}$, by the mean-value theorem and
the fact that $\chi_{N}(k_{1})-\chi_{N}(k_{2})$ takes the form
$\widetilde{\chi}(|k_{1}|^{2}/N^{2})-\widetilde{\chi}(|k_{2}|^{2}/N^{2})$, we
have

$\Big{|}\frac{\chi_{N}(k_{1})^{2}}{\sqrt{1+|k_{2}|^{2s}}}-\frac{\chi_{N}(k_{2})^{2}}{\sqrt{1+|k_{1}|^{2s}}}\Big{|}\lesssim\frac{|\Omega(\vec{k})|+|k_{(3)}|^{2}}{|k_{(1)}|^{2(s+1)}}.$

Thus the remainder term $R_{N_{1},\cdots,M_{5}}$ can be estimated by the
previous deterministic manipulation (recall that $|\Omega(\vec{k})|\gtrsim
N_{(1)}^{\delta_{0}}>N_{(1)}^{2\theta}$ in the sum)

$\displaystyle
N_{1}^{2(s-1)+2\theta-2(s+1)}\sum_{\Lambda_{1,1}}\mathbf{1}_{k_{1}\neq
k_{2}}\frac{|\Omega(\vec{k})|+N_{1}^{2\theta}}{|\Omega(\vec{k})|}|w_{k_{3}}^{N_{3}}\cdots
w_{p_{5}}^{M_{5}}|$ $\displaystyle\lesssim$ $\displaystyle
N_{1}^{-4+2\theta}\cdot
N_{1}^{3}\|w_{k_{3}}^{N_{3}}\|_{l^{1}}\cdots\|w_{p_{5}}^{M_{5}}\|_{l^{1}}$
$\displaystyle\lesssim$ $\displaystyle
N_{1}^{-1+2\theta}\|w_{k_{3}}^{N_{3}}\|_{h^{\sigma}}\cdots\|w_{p_{5}}^{M_{5}}\|_{h^{\sigma}}(N_{3}\dots
M_{5})^{-\sigma+\frac{3}{2}}$ $\displaystyle\lesssim$ $\displaystyle
N_{1}^{-1+2\theta}(N_{3}\cdots
M_{5})^{-(\sigma-\frac{3}{2})}\|w\|_{H^{\sigma}}^{8}\lesssim
N_{(1)}^{-1+2\theta}R^{8},$ (7.10)

which is conclusive as far as $\theta<\frac{1}{2}$ and $\sigma>\frac{3}{2}$.

We next estimate $\widetilde{J}_{N_{1},\cdots,M_{5}}$. We will not make use of
the cancellation in the difference

$\frac{\chi_{N}(k_{1})^{2}}{\sqrt{1+|k_{2}|^{2s}}}-\frac{\chi_{N}(k_{2})^{2}}{\sqrt{1+|k_{1}|^{2s}}}$

so we treat separately and in the same manner the contribution of each term. .

Let $\mathcal{B}_{\ll N_{1}}$ be the $\sigma$-algebra generated by Gaussians
$g_{k_{j}},|k_{j}|\ll N_{1}$ and $\mathbf{P}_{\ll N_{1}}$ the frequency
projector to $|k|\ll N_{1}$. In particular, $g_{k_{1}},g_{k_{2}}$ for
$|k_{1}|\sim N_{1},|k_{2}|\sim N_{2}$ are independent of the $\sigma$-algebra
$\mathcal{B}_{\ll N_{1}}$. We have

$\displaystyle\|\widetilde{J}_{N_{1},\cdots,M_{5}}(w)\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\|_{L^{p}(d\mu_{s})}^{p}\leq\mathbb{E}^{\mu_{s}}[\mathbb{E}^{\mu_{s}}[|\widetilde{J}_{N_{1},\cdots,M_{5}}(w)\mathbf{1}_{\|\mathbf{P}_{\ll
N_{1}}w\|_{H^{\sigma}}\leq R}|^{p}|\mathcal{B}_{\ll N_{1}}]].$

As $\mathbf{P}_{\ll N_{1}}w,w_{k_{3}}^{N_{3}},\cdots,w_{p_{5}}^{M_{5}}$ are
$\mathcal{B}_{\ll N_{1}}$-measurable, by the Wiener chaos estimate conditional
to $\mathcal{B}_{\ll N_{1}}$,

$\displaystyle\Big{(}\mathbb{E}^{\mu_{s}}[|\widetilde{J}_{N_{1},\cdots,M_{5}}(w)\mathbf{1}_{\|\mathbf{P}_{\ll
N_{1}}w\|_{H^{\sigma}}\leq R}|^{p}|\mathcal{B}_{\ll
N_{1}}]\Big{)}^{\frac{1}{p}}$ $\displaystyle\leq$ $\displaystyle
Cp\Big{(}\sum_{|k_{2}|\sim N_{2}}\frac{1}{\langle
k_{2}\rangle^{4s}}\Big{|}\sum_{k_{3}-k_{4}+k_{5}-k_{6}=p_{2}-p_{3}+p_{4}-p_{5}}\\!\\!\\!\\!\\!\\!\Psi(\vec{k})w_{k_{3}}^{N_{3}}\cdots
w_{p_{5}}^{M_{5}}\Big{|}^{2}\Big{)}^{\frac{1}{2}}\cdot\mathbf{1}_{\|\mathbf{P}_{\ll
N_{1}}w\|_{H^{\sigma}}\leq R}.$

By Lemma 7.4 and Cauchy-Schwarz, we bound the above expression by

$\displaystyle CpN_{2}^{-2+2\theta}\Big{(}\sum_{\Lambda_{1,1},k_{1}\neq
k_{2}}\frac{1}{|\Omega(\vec{k})|^{2}}\Big{)}^{\frac{1}{2}}\|w_{k_{3}}^{N_{3}}\|_{l^{2}}\cdots\|w_{p_{5}}^{M_{5}}\|_{l^{2}}\cdot\mathbf{1}_{\|\mathbf{P}_{\ll
N_{1}}w\|_{H^{\sigma}}\leq R}$ $\displaystyle\lesssim$ $\displaystyle
CpN_{2}^{-2+2\theta}\cdot
N_{2}\|w_{k_{3}}^{N_{3}}\|_{h^{\sigma}}\cdots\|w_{p_{5}}^{M_{5}}\|_{h^{\sigma}}(N_{3}\cdots
M_{5})^{-\sigma+\frac{3}{2}}\cdot\mathbf{1}_{\|\mathbf{P}_{\ll
N_{1}}w\|_{H^{\sigma}}\leq R}$ $\displaystyle\lesssim$ $\displaystyle
CpN_{1}^{-1+2\theta}R^{8},$

provided that $\sigma>\frac{3}{2}$. Therefore,

$\|\widetilde{J}_{N_{1},\cdots,M_{5}}(w)\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\|_{L^{p}(d\mu_{s})}\lesssim pN_{1}^{-1+2\theta}R^{8}.$

Combining with (7.10) and interpolating with (7.8), we deduce that there exist
constants $C>0$, $\beta=\delta(s,\theta)\in(0,1)$, such that for any $p\geq 2$
and $R\geq 1$,

$\displaystyle\|J_{N_{1},\cdots,M_{5}}(w)\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\|_{L^{p}(d\mu_{s})}\leq Cp^{\beta}N_{(1)}^{-1+2\theta}R^{10},$ (7.11)

which is conclusive since $\theta<\frac{1}{3}$.

$\bullet$ Estimate for $j=2$: It suffices to estimate $\mathop{\rm
Im}\nolimits\mathcal{S}_{1,2}(w)$. Recall that

$\mathcal{S}_{1,2}(w):=\sum_{\Lambda_{1,2}}\chi_{N}(k_{1})^{2}|w_{k_{3}}|^{2}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\overline{w}_{k_{2}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot
w_{p_{1}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}},$

and on $\Lambda_{1,2}$,
$|\Omega(\vec{k})|\sim|k_{(1)}|^{2}\gg\lambda(\vec{k})^{\delta_{0}}$, thus

$\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}=\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}.$

Set

$\widetilde{\Psi}(\vec{k}):=\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}-\frac{|k_{1}|^{2s}+|k_{3}|^{2s}}{|k_{1}|^{2}+|k_{3}|^{2}}.$

By mean-value theorem, we easily deduce that:

######  Lemma 7.5.

On $\Lambda_{1,2}$ defined in (5), for sufficiently large $|k_{(1)}|$, we have

$|\widetilde{\Psi}(\vec{k})|\lesssim\frac{|k_{(1)}|^{2s-2}|k_{(3)}|^{2}}{|\Omega(\vec{k})|}\mathbf{1}_{|\Omega(\vec{k})|\sim|k_{(1)}|^{2}}.$

Since

$\sum_{k_{1},k_{3}}\chi_{N}(k_{1})^{2}|w_{k_{3}}|^{2}\frac{|k_{1}|^{2s}+|k_{3}|^{2s}}{|k_{1}|^{2}+|k_{3}|^{2}}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{2}+k_{4}-k_{5}+k_{6}=k_{1}+k_{3}\\\
p_{1}+p_{3}-p_{4}+p_{5}=k_{1}+k_{3}\\\
|k_{2}|+|k_{4}|+|k_{5}|+|k_{6}|\leq|k_{1}|^{\theta}+|k_{3}|^{\theta}\\\
|p_{1}|+|p_{3}|+|p_{4}|+|p_{5}|\leq|k_{1}|^{\theta}+|k_{3}|^{\theta}\end{subarray}}\\!\\!\\!\\!\overline{w}_{k_{2}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot
w_{p_{1}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}$

equals to

$\sum_{k_{1},k_{3}}\chi_{N}(k_{1})^{2}|w_{k_{3}}|^{2}\frac{|k_{1}|^{2s}+|k_{3}|^{2s}}{|k_{1}|^{2}+|k_{3}|^{2}}\Big{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{2}+k_{4}-k_{5}+k_{6}=k_{1}+k_{3}\\\
|k_{2}|+|k_{4}|+|k_{5}|+|k_{6}|\leq|k_{1}|^{\theta}+|k_{3}|^{\theta}\end{subarray}}\\!\\!\\!\\!\\!\\!\
\overline{w}_{k_{2}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\Big{|}^{2},$

which is real-valued, we deduce that

$\displaystyle\mathrm{Im}(\mathcal{S}_{1,2}(w))=I(w):=$
$\displaystyle\sum_{\Lambda_{1,2}}\chi_{N}(k_{1})^{2}|w_{k_{3}}|^{2}\widetilde{\Psi}(\vec{k})\overline{w}_{k_{2}}\overline{w}_{k_{4}}w_{k_{5}}\overline{w}_{k_{6}}\cdot
w_{p_{1}}w_{p_{3}}\overline{w}_{p_{4}}w_{p_{5}}.$ (7.12)

As for the case $j=1$, we will split the sum into dyadic pieces. For dyadic
numbers $N_{j},M_{j}$, we decompose accordingly
$w_{k_{j}}^{N_{j}}=w_{k_{j}}\mathbf{1}_{|k_{j}|\sim N_{j}}$ and
$w_{p_{i}}^{M_{i}}=w_{p_{i}}\mathbf{1}_{|p_{i}|\sim M_{i}}$. It suffices to
show that for some $\beta\in(0,1)$.

$\displaystyle\|I_{N_{1},\cdots,N_{6};M_{2},\cdots,M_{5}}(w)\mathbf{1}_{\|w\|_{H^{\sigma}}\leq
R}\|_{L^{p}(d\mu_{s})}\lesssim p^{\beta}N_{(1)}^{-\frac{1}{100}}R^{10},$
(7.13)

where $I_{N_{1},\cdots,N_{6};M_{2},\cdots,M_{5}}$ is the same expression as
$I(w)$ by replacing the inputs $w_{k_{j}},w_{p_{i}}$ to
$w_{k_{j}}^{N_{j}},w_{p_{i}}^{M_{i}}$.

It turns out that only deterministic estimates suffice. Indeed, from the fact
that $|\Omega(\vec{k})|\sim N_{(1)}^{2}$, by Lemma 7.5, we have

$|I_{N_{1},\cdots,M_{5}}(w)|\lesssim\frac{N_{(1)}^{2s-2}N_{(3)}^{2}}{N_{(1)}^{2}}\,\sum_{k_{3}}|w^{N_{3}}_{k_{3}}|^{2}\sum_{k_{2},k_{4},k_{5},k_{6}}\sum_{p_{1},p_{3},p_{4},p_{5}}|w^{N_{2}}_{k_{2}}w^{N_{4}}_{k_{4}}w^{N_{5}}_{k_{5}}w^{N_{6}}_{k_{6}}w^{M_{1}}_{p_{1}}w^{M_{3}}_{p_{3}}w^{M_{4}}_{p_{4}}w^{M_{5}}_{p_{5}}|,$

which is bounded by

$\displaystyle
N_{(1)}^{2(s-2)}N_{(3)}^{2}\|w_{k_{3}}^{N_{3}}\|_{l^{2}}^{2}\|w_{k_{2}}^{N_{2}}\|_{l^{1}}\cdots\|w_{k_{6}}^{N_{6}}\|_{l^{1}}\|w_{p_{1}}^{M_{1}}\|_{l^{1}}\cdots\|w_{p_{5}}^{M_{5}}\|_{l^{1}}$
$\displaystyle\lesssim$ $\displaystyle
N_{(1)}^{2(s-2-\sigma)+2\theta}(N_{2}N_{4}N_{5}N_{6}M_{1}M_{3}M_{4}M_{5})^{-(\sigma-\frac{3}{2})}\|w\|_{H^{\sigma}}^{10}$
$\displaystyle\leq$ $\displaystyle
N_{(1)}^{2(s-2-\sigma)+2\theta}\|w\|_{H^{\sigma}}^{10},$

provided that $\sigma>\frac{3}{2}$, which is conclusive when
$\theta<\frac{1}{2}$ and $\sigma$ close enough to $s-\frac{3}{2}$. This
completes the proof of Proposition 7.1. ∎

##  8\. Energy estimate III: Remainders in the second generation

In this section, we will estimate $\mathcal{R}_{1,3}(w),\mathcal{R}_{2,3}(w)$.
More precisely, we have the following statement:

######  Proposition 8.1.

Let $\theta<\frac{1}{3}$, close enough to $\frac{1}{3}$ and
$\delta_{0}\in(2\theta,\frac{2}{3})$, close enough to $\frac{2}{3}$. There
exist $C>0$ and $\beta=\beta(\theta,s)\in(0,1)$, such that for
$j\in\\{1,2\\}$, $R\geq 1$ and $p\in[2,\infty)$, we have

$\|\mathcal{R}_{1,3}(w)\mathbf{1}_{B_{R}^{H^{\sigma}}(w)}\|_{L^{p}(d\mu_{s})}+\|\mathcal{R}_{2,3}(w)\mathbf{1}_{B_{R}^{H^{\sigma}}(w)}\|_{L^{p}(d\mu_{s})}\leq
Cp^{\beta}R^{10}.$

Since the estimate for $\mathcal{R}_{2,3}(w)$ is similar, we only do it for
$\mathcal{R}_{1,3}(w)$. Recall in the expression of $\mathcal{R}_{1,3}(w)$, we
distinguish three types of contributions in the decomposition of the sum

$\sum_{\begin{subarray}{c}k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0\\\
k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}\\\ \end{subarray}}(\cdots).$

Recall that $k_{(1)}\cdots,k_{(10)}$ is a rearrangement of leaves
$p_{1},p_{2},p_{3},p_{4},p_{5},k_{2},k_{3},k_{4},k_{5},k_{6}$ such that
$|k_{(1)}|\geq|k_{(2)}|\geq\cdots\geq|k_{(10)}|$.

  * •

Type A: $\sum_{j=3}^{10}|k_{(j)}|>|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$.

  * •

Type B: $\sum_{j=3}^{10}|k_{(j)}|\leq|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$
and $\\{k_{(1)},k_{(2)}\\}\subset\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\}$ or
$\\{k_{(1)},k_{(2)}\\}\subset\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}$.

  * •

Type C: $\sum_{j=3}^{10}|k_{(j)}|\leq|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$,
$k_{(1)}\neq k_{(2)}$ and

$k_{(1)}\in\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\},\quad
k_{(2)}\in\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}$

or

$k_{(2)}\in\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\},\quad
k_{(1)}\in\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\},$

and $k_{(1)},k_{(2)}$ have different signatures.

Let us denote by $\Lambda_{A}$, $\Lambda_{B}$, $\Lambda_{C}$ the sets of
indices $k_{1},\ldots,k_{6},p_{1},\ldots,p_{5}$ that satisfy the linear
constraints

$k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0,\quad
k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}$

and the conditions for Type A, B, C respectively. Furthermore, we denote
$\mathcal{R}_{1,3}^{(A)},\mathcal{R}_{1,3}^{(B)},\mathcal{R}_{1,3}^{(C)}$ the
corresponding contributions to $\mathcal{R}_{1,3}(w)$. We will need the
following elementary lemma.

######  Lemma 8.2.

Assume that $f^{(j)}$ satisfies $f_{k_{j}}^{(j)}\mathbf{1}_{|k_{j}|\sim
M_{j}}=f_{k_{j}}^{(j)}$ for $j=1,2,3,4,5,6$ with $M_{j}\in 2^{\mathbb{N}}$.
Then

$\sum_{k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0}\frac{\psi_{2s}(\vec{k})}{|\Omega(\vec{k})|}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\prod_{j=1}^{6}|f_{k_{j}}^{(j)}|\\\
\lesssim
M_{(1)}^{2s-2}\,M_{(3)}^{2}(M_{(3)}M_{(4)}M_{(5)}M_{(6)})^{\frac{3}{2}}\prod_{j=1}^{6}\|f^{(j)}\|_{l^{2}},$

where $M_{(1)}\geq M_{(2)}\geq\cdots\geq M_{(6)}$ is non-increasing
rearrangement of dyadic integers $M_{1},M_{2},\cdots,M_{6}$.

###### Proof.

Since signature of $k_{j}$ plays no significant role in the proof, without
loss of generality, we assume that $M_{1}\geq M_{2}\geq\cdots\geq M_{6}$.
Since $|\psi_{2s}(\vec{k})|\lesssim
M_{1}^{2s-2}(M_{3}^{2}+|\Omega(\vec{k})|)$, using the Cauchy-Schwarz
inequality in the $k_{1}$ and $k_{2}$ summation, we obtain the bound

$M_{(1)}^{2s-2}\,M_{(3)}^{2}\prod_{j=1}^{2}\|f^{(j)}\|_{l^{2}}\prod_{j=3}^{6}\|f^{(j)}\|_{l^{1}}.$

It remains to use the Cauchy-Schwarz inequality to pass from $l^{1}$ to
$l^{2}$. This completes the proof of Lemma 8.2. ∎

######  Remark 8.3.

Since the crude bound is enough for our need, we do not make use of the
denominator $\frac{1}{|\Omega(\vec{k})|}$ in the estimate above.

###### Proof of Proposition 8.1.

Since the proof follows from tedious estimates, we split it into three
different parts, according to Type A,B,C. We will split the function $w$ into
dyadic pieces, and we denote $w^{K}=\mathbf{P}_{K}w$ in the sequel which means
that $w_{k}^{K}=\mathbf{1}_{|k|\sim K}w_{k}$.

$\bullet$ Estimate of Type A contribution:

We decompose the expression

$\mathcal{R}_{1,3}^{(A)}:=\sum_{\Lambda_{A}}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\chi_{N}(k_{1})^{2}w_{p_{1}}\cdots
w_{p_{5}}\overline{w}_{k_{2}}\cdots\overline{w}_{k_{6}}$

dyadically by

$\displaystyle\sum_{M_{1},\cdots,M_{6},P_{1},\cdots,P_{5}}\mathcal{R}_{1,3}^{(A)}(M_{1},\cdots,P_{5}),$

where $\mathcal{R}_{1,3}^{(A)}(M_{1},\cdots,P_{5})$ is

$\sum_{\Lambda_{A}}\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\chi_{N}(k_{1})^{2}w^{P_{1}}_{p_{1}}\cdots
w^{P_{5}}_{p_{5}}\overline{w}^{M_{2}}_{k_{2}}\cdots\overline{w}^{M_{6}}_{k_{6}}\cdot\mathbf{1}_{|k_{1}|\sim
M_{1}}.$

We denote by $N_{(1)}\geq N_{(2)}\geq\cdots N_{(10)}$ the non-increasing
rearrangement of dyadic integers

$P_{1},P_{2},P_{3},P_{4},P_{5},M_{2},M_{3},M_{4},M_{5},M_{6}.$

Note that the constraint
$\sum_{j=3}^{10}|k_{(j)}|>|k_{(1)}|^{\theta}+|k_{(2)}|^{\theta}$ implies that
$N_{(3)}\gtrsim N_{(1)}^{\theta}$ for non-zero terms
$\mathcal{R}_{1,3}^{(A)}(M_{1},\cdots,P_{5})$. Write

$|\mathcal{R}_{1,3}^{(A)}(M_{1},\cdots,P_{5})|\leq\sum_{k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0}\frac{\psi_{2s}(\vec{k})}{|\Omega(\vec{k})|}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\prod_{j=1}^{6}|f_{k_{j}}^{(j)}|,$

where $f_{k_{j}}^{(j)}=w_{k_{j}}^{M_{j}}$ for $j=2,3,4,5,6$ and

$f_{k_{1}}^{(1)}=\sum_{p_{1}-p_{2}+p_{3}-p_{4}+p_{5}=k_{1}}\mathbf{1}_{|k_{1}|\sim
M_{1}}w_{p_{1}}^{P_{1}}\cdots w_{p_{5}}^{P_{5}}.$

Applying Lemma 8.2, we have

$\displaystyle|\mathcal{R}_{1,3}^{(A)}(M_{1},\cdots,P_{5})|\lesssim$
$\displaystyle
M_{(1)}^{2s-2}\,M_{(3)}^{2}(M_{(3)}M_{(4)}M_{(5)}M_{(6)})^{\frac{3}{2}}\prod_{j=1}^{6}\|f^{(j)}\|_{l^{2}}$
$\displaystyle\lesssim$ $\displaystyle
M_{(1)}^{2s-2}\,M_{(3)}^{2}(M_{(3)}M_{(4)}M_{(5)}M_{(6)})^{\frac{3}{2}}\|f^{(1)}\|_{l^{2}}\prod_{j=2}^{6}(M_{j}^{-\sigma}\|w^{M_{j}}\|_{H^{\sigma}}).$

where $M_{(1)}\geq M_{(2)}\geq\cdots\geq M_{(6)}$ is a non-increasing
rearrangement of $M_{1},M_{2},\cdots,M_{6}$. By Cauchy-Schwarz, we have

$\|f^{(1)}\|_{l^{2}}\lesssim(P_{(2)}P_{(3)}P_{(4)}P_{(5)})^{\frac{3}{2}}\prod_{j=1}^{5}(P_{j}^{-\sigma}\|w^{P_{j}}\|_{H^{\sigma}}),$

where $P_{(1)}\geq P_{(2)}\geq P_{(3)}\geq P_{(4)}\geq P_{(5)}$ is a non-
increasing rearrangement of $P_{1},P_{2},P_{3},P_{4},P_{5}$. Thus we obtain
that

$|\mathcal{R}_{1,3}^{(A)}(M_{1},\cdots,P_{5})|\\\ \lesssim
M_{(1)}^{2s-2}\,M_{(3)}^{2}(M_{(3)}M_{(4)}M_{(5)}M_{(6)})^{\frac{3}{2}}(M_{2}\cdots
M_{6})^{-\sigma}\,P_{(1)}^{-\sigma}(P_{(2)}\cdots
P_{(5)})^{\frac{3}{2}-\sigma}\|w\|_{H^{\sigma}}^{10}.$

Since we are in the regime $s\geq 10$ and $\sigma$ is close to
$s-\frac{3}{2}$, we control the right hand side by

$N_{(1)}^{2(s-1-\sigma)}\,N_{(3)}^{\frac{7}{2}-\sigma}\|w\|_{H^{\sigma}}^{10}\lesssim
N_{(1)}^{2(s-1-\sigma)}\,N_{(1)}^{-\theta(\sigma-\frac{7}{2})}\|w\|_{H^{\sigma}}^{10}.$

For $s\geq 10$, $\sigma$ close to $s-\frac{3}{2}$ and $\theta$ is close to
$\frac{1}{3}$ the last expression can be estimated by
$N_{(1)}^{-\epsilon_{0}(\theta,\sigma)}\|w\|_{H^{\sigma}}^{10}$ for some
$\epsilon_{0}(\theta,\sigma)>0$ which is conclusive.

$\bullet$ Estimate of Type B contribution:

Denote $\Lambda_{B1}$ the set of $(k_{1},\cdots,p_{5})\in\Lambda_{B}$ such
that $k_{(1)},k_{(2)}\in\\{k_{2},k_{3},k_{4},k_{5},k_{6}\\}$ and
$\Lambda_{B2}$ the set of $(k_{1},\cdots,p_{5})\in\Lambda_{B}$ such that
$k_{(1)},k_{(2)}\in\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}$, and denote by
$\mathcal{R}_{1,3}^{(B1)},\mathcal{R}_{1,3}^{(B2)}$ the corresponding
multilinear expressions.

$\bullet$ Subcase: Contribution $\mathcal{R}_{1,3}^{(B1)}$:  We first estimate
$\mathcal{R}_{1,3}^{(B1)}$. By symmetry of indices, we may assume that
$k_{(1)}=k_{3},k_{(2)}=k_{2}$. Then other frequencies satisfy the constraint

$\sum_{j=1}^{5}|p_{j}|+\sum_{j=4}^{6}|k_{j}|<|k_{2}|^{\theta}+|k_{3}|^{\theta}$

on $\Lambda_{B1}$. We decompose $\mathcal{R}_{1,3}^{(B1)}$ by the dyadic sum

$\sum_{M_{1},\cdots,M_{6},P_{1},\cdots,P_{5}}\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})$

as in the estimate for Type (A) terms. Under the constraint of $\Lambda_{B}$
and our convention that $\\{k_{(1)},k_{(2)}\\}=\\{k_{3},k_{2}\\}$, we must
have $M_{2}\sim M_{3}\sim N_{(1)}$ and $\max\\{M_{1},M_{4},M_{5},M_{6}\\}\leq
N_{(3)}$.

Note that for the pairing part $k_{2}=k_{3}$ in
$\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})$, we have
$|\psi_{2s}(\vec{k})|\lesssim|k_{(3)}|^{2s}$, thus we can control it simply by

$\displaystyle\|w^{M_{2}}\|_{l^{2}}^{2}\cdot\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{1}-k_{4}+k_{5}-k_{6}=0\\\
k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}\end{subarray}}|k_{(3)}|^{2s}\prod_{j=1}^{5}|w_{p_{j}}^{P_{j}}|\prod_{j=4}^{6}|w_{k_{j}}^{M_{j}}|$
$\displaystyle\lesssim$ $\displaystyle
N_{(1)}^{-2\sigma}N_{(3)}^{2s}\|w\|_{H^{\sigma}}^{10}\lesssim
N_{(1)}^{2s\theta-2\sigma}\|w\|_{H^{\sigma}}^{10},$ (8.1)

thanks to $\sigma>\frac{3}{2}$. As $\theta<\frac{1}{3}$, $s\geq 10$ and
$\sigma$ is close enough to $s-\frac{3}{2}$, the right hand side is bounded by
a negative power of $N_{(1)}$ times $\|w\|_{H^{\sigma}}^{10}$, which is
conclusive.

It remains to consider the non-pairing contribution in
$\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})$. Recall that

$\displaystyle\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}k_{2}\neq k_{3}\\\ |k_{2}|\sim
M_{2},|k_{3}|\sim
M_{3}\end{subarray}}\\!\\!\\!\\!\overline{w}_{k_{2}}^{M_{2}}w_{k_{3}}^{M_{3}}\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{4},k_{5},k_{6},p_{1},\cdots,p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\\!\\!\\!\\!\\!\\!\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\overline{w}_{k_{4}}^{M_{4}}w_{k_{5}}^{M_{5}}\overline{w}_{k_{6}}^{M_{6}}w_{p_{1}}^{P_{1}}\cdots
w_{p_{5}}^{P_{5}}.$

Denote $\mathcal{B}_{\ll M_{2}}$ the $\sigma$-algebra generated by
$(g_{k})_{|k|\leq M_{2}/100}$. Without loss of generality, we assume that
$M_{2}\sim N_{(1)}$ is large enough such that $N_{(3)}\lesssim
N_{(1)}^{\theta}\ll\frac{M_{2}}{100}$. Consequently, with respect to
$\mu_{s}$, $w^{M_{4}},w^{M_{5}},w^{M_{6}},w^{P_{1}},\cdots,w^{P_{5}}$ are
independent of $w^{M_{2}},w^{M_{3}}$.

By conditional Wiener chaos estimate, we have

$\displaystyle\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\|_{L^{p}(d\mu_{s})}\leq$
$\displaystyle\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
M_{2}}w)\|_{L^{p}(d\mu_{s})}$ $\displaystyle=$
$\displaystyle\big{\|}\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
M_{2}}w)\|_{L^{p}(d\mu_{s}|\mathcal{B}_{\ll
M_{2}})}\big{\|}_{L^{p}(d\mu_{s})}$ $\displaystyle\lesssim$
$\displaystyle\,p\big{\|}\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
M_{2}})}\cdot\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
M_{2}}w)\big{\|}_{L^{p}(d\mu_{s})}.$

It suffices to show that

$\displaystyle\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
M_{2}})}\lesssim$ $\displaystyle
N_{(1)}^{-\frac{1}{2}}\|w\|_{H^{\sigma}}^{8}.$ (8.2)

Indeed, the above estimate yields

$\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\|_{L^{p}(d\mu_{s})}\lesssim
pN_{(1)}^{-\frac{1}{2}}R^{8}.$

Since we have left a negative power of $N_{(1)}$, by interpolating with the
crude deterministic bound which is of the form $N_{(1)}^{O(1)}$, we obtain the
desired estimate.

Now we prove (8.2). Thanks to the fact that $k_{2}\neq k_{3}$, we deduce
that333In the summation below, we implicitly assume that $|k_{2}|\sim
M_{2},|k_{3}|\sim M_{3}.$

$\displaystyle\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
M_{2}})}$ $\displaystyle\lesssim$
$\displaystyle(M_{2}M_{3})^{-s}\Big{(}\sum_{k_{2}\neq
k_{3}}\Big{(}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{4},k_{5},k_{6},p_{1},\cdots,p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\\!\\!\\!\\!\\!\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\overline{w}_{k_{4}}^{M_{4}}w_{k_{5}}^{M_{5}}\overline{w}_{k_{6}}^{M_{6}}w_{p_{1}}^{P_{1}}\cdots
w_{p_{5}}^{P_{5}}\Big{)}^{2}\Big{)}^{\frac{1}{2}}$ $\displaystyle\lesssim$
$\displaystyle N_{(1)}^{-2s}\Big{(}\sum_{k_{2}\neq
k_{3}}\sum_{\begin{subarray}{c}k_{4},k_{5},k_{6},p_{1},\cdots,p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\frac{|\psi_{2s}(\vec{k})|^{2}}{1+|\Omega(\vec{k})|^{2}}|w_{k_{(3)}}^{N_{(3)}}|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sup_{k_{2}\neq
k_{3}}\sum_{\begin{subarray}{c}k_{4},k_{5},k_{6},p_{1},\cdots,p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}|w_{k_{(4)}}^{N_{(4)}}\cdots
w_{k_{(10)}}^{N_{(10)}}|^{2}\Big{)}^{\frac{1}{2}}$ $\displaystyle\lesssim$
$\displaystyle
N_{(1)}^{-2s}\Big{(}\prod_{j=4}^{10}\|w^{N_{(j)}}\|_{L^{2}}\Big{)}\cdot\Big{(}\sum_{\begin{subarray}{c}k_{2}\neq
k_{3},k_{4},k_{5},k_{6},p_{1}\cdots,p_{5}\\\ p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\frac{M_{(1)}^{4(s-1)}(M_{(3)}^{4}+|\Omega(\vec{k})|^{2})}{1+|\Omega(\vec{k})|^{2}}|w_{k_{(3)}}^{N_{(3)}}|^{2}\Big{)}^{\frac{1}{2}}.$

By Lemma 4.2, the last sum on the right hand side can be estimated as

$\displaystyle\sum_{\begin{subarray}{c}k_{4},k_{5},k_{6},p_{1},\cdots,p_{5}\end{subarray}}|w_{k_{(3)}}^{N_{(3)}}|^{2}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{2}\neq
k_{3}\\\
k_{2}-k_{3}=p_{1}-\cdots+p_{5}-k_{4}+k_{5}-k_{6}\end{subarray}}\\!\\!\\!\\!\\!\frac{N_{(1)}^{4(s-1)}(N_{(3)}^{4}+|\Omega(\vec{k})|^{2})}{1+|\Omega(\vec{k})|^{2}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{k_{4},k_{5},k_{6},p_{1},\cdots,p_{5}}|w_{k_{(3)}}^{N_{(3)}}|^{2}N_{(1)}^{4(s-1)}(N_{(3)}^{4}N_{(1)}^{2}+N_{(1)}^{3})$
$\displaystyle\lesssim$
$\displaystyle\|w^{N_{(3)}}\|_{L^{2}}^{2}N_{(1)}^{4(s-1)}(N_{(3)}^{4}N_{(1)}^{2}+N_{(1)}^{3})\prod_{j=4}^{10}N_{(j)}^{3},$

thus

$\displaystyle\|\mathcal{R}_{1,3}^{(B1)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
M_{2}})}\lesssim$ $\displaystyle
N_{(1)}^{-2}(N_{(1)}^{\frac{3}{2}}+N_{(3)}^{2}N_{(1)})\Big{(}\prod_{j=3}^{10}\|w^{N_{(j)}}\|_{L^{2}}\Big{)}\Big{(}\prod_{j=4}^{10}N_{(j)}^{\frac{3}{2}}\Big{)}$
$\displaystyle\lesssim$ $\displaystyle
N_{(1)}^{-\frac{1}{2}}N_{(3)}^{-\sigma}\prod_{j=4}^{10}N_{(j)}^{-\sigma+\frac{3}{2}}\cdot\|w\|_{H^{\sigma}}^{8}+N_{(1)}^{-1}N_{(3)}^{-\sigma+2}\prod_{j=4}^{10}N_{(3)}^{-\sigma+\frac{3}{2}}\cdot\|w\|_{H^{\sigma}}^{8}$
$\displaystyle\lesssim$ $\displaystyle
N_{(1)}^{-\frac{1}{2}}\|w\|_{H^{\sigma}}^{8},$

which is conclusive, thanks to the fact that $s\geq 10$ and that $\sigma$ is
close to $s-\frac{3}{2}$.

$\bullet$ Subcase: Contribution $\mathcal{R}_{1,3}^{(B2)}$:  Next we estimate
$\mathcal{R}_{1,3}^{(B2)}$ for which
$k_{(1)},k_{(2)}\in\\{p_{1},p_{2},p_{3},p_{4},p_{5}\\}$. By symmetry of
indices, we assume that $k_{(1)}=p_{1},k_{(2)}=p_{2}$, then

$\sum_{j=3}^{5}|p_{j}|+\sum_{j=2}^{6}|k_{j}|\leq|p_{1}|^{\theta}+|p_{2}|^{\theta}$

on $\Lambda_{B2}$. Similarly, we decompose $\mathcal{R}_{1,3}^{(B2)}$ by
dyadic sum

$\sum_{M_{1},\cdots,M_{6},P_{1},\cdots,P_{5}}\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5}),$

and this time, $P_{1}\sim P_{2}\sim N_{(1)}$ and
$\max\\{M_{1},M_{2},\cdots,M_{6}\\}\lesssim N_{(3)}$. In particular, the
energy weight $\psi_{2s}(\vec{k})$ satisfies $|\psi_{2s}(\vec{k})|\lesssim
N_{(3)}^{2s}$ which is much smaller than in the previous case.

The pairing contribution $p_{1}=p_{2}$ in
$\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})$ can be controlled in the same
way by the same bound as (8). We omit the detail.

For the non-pairing contribution in $\mathcal{R}_{1,3}^{(B2)}$, again we apply
the Wiener chaos estimate. Denote $\mathcal{B}_{\ll P_{1}}$ the
$\sigma$-algebra generated by $(g_{k})_{|k|\leq P_{1}/100}$. Without loss of
generality, we assume that $P_{1}\sim N_{(1)}$ is large enough such that
$N_{(3)}\lesssim N_{(1)}^{\theta}\ll\frac{P_{1}}{100}$. Consequently, with
respect to $\mu_{s}$,
$w^{M_{2}},\cdots,w^{M_{6}},w^{P_{3}},w^{P_{4}},w^{P_{5}}$ are independent of
$w^{P_{1}},w^{P_{2}}$. Recall that

$\displaystyle\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}p_{1}\neq p_{2}\\\ |p_{1}|\sim
P_{1},|p_{2}|\sim
P_{2}\end{subarray}}\\!\\!\\!\\!w_{p_{1}}^{P_{1}}\overline{w}_{p_{2}}^{P_{2}}\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k_{2},\cdots,k_{6},p_{3},p_{4},p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\\!\\!\\!\\!\\!\\!\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}w_{p_{3}}^{P_{3}}\overline{w}_{p_{4}}^{P_{4}}w_{p_{5}}^{P_{5}}\overline{w}_{k_{2}}^{M_{2}}\cdots\overline{w}_{k_{6}}^{M_{6}}.$

By conditional Wiener chaos estimate, we have

$\displaystyle\|\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\|_{L^{p}(d\mu_{s})}\leq$
$\displaystyle\|\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
P_{1}}w)\|_{L^{p}(d\mu_{s})}$ $\displaystyle=$
$\displaystyle\big{\|}\|\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
P_{1}}w)\|_{L^{p}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}\big{\|}_{L^{p}(d\mu_{s})}$ $\displaystyle\lesssim$
$\displaystyle\,p\big{\|}\|\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}\cdot\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
P_{1}}w)\big{\|}_{L^{p}(d\mu_{s})}.$

As in the estimate of $\mathcal{R}_{1,3}^{(B1)}$, here it suffices to show
that

$\displaystyle\|\mathcal{R}_{1,3}^{(B2)}\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}\lesssim N_{(1)}^{-\frac{1}{2}}\|w\|_{H^{\sigma}}^{8}.$ (8.3)

Thanks to the non-pairing condition $p_{1}\neq p_{2}$, we deduce that444In the
summation below, we implicitly assume that $|p_{1}|\sim P_{1},|p_{2}|\sim
P_{2}.$

$\displaystyle\|\mathcal{R}_{1,3}^{(B2)}(M_{1},\cdots,P_{5})$
$\displaystyle\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll P_{1}})}$
$\displaystyle\lesssim$ $\displaystyle(P_{1}P_{2})^{-s}\Big{(}\sum_{p_{1}\neq
p_{2}}\Big{(}\sum_{\begin{subarray}{c}k_{2},\cdots,k_{6},p_{3},p_{4},p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}N_{(3)}^{2s}|w^{P_{3}}_{p_{3}}w^{P_{4}}_{p_{4}}w^{P_{5}}_{p_{5}}w_{k_{2}}^{M_{2}}\cdots
w_{k_{6}}^{M_{6}}|\Big{)}^{2}\Big{)}^{\frac{1}{2}}$ $\displaystyle\lesssim$
$\displaystyle
N_{(1)}^{-2s}N_{(3)}^{2s}\Big{(}\sum_{p_{1}-p_{2}+\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0}|w^{P_{3}}_{p_{3}}w^{P_{4}}_{p_{4}}w^{P_{5}}_{p_{5}}w_{k_{2}}^{M_{2}}\cdots
w_{k_{6}}^{M_{6}}|^{2}\Big{)}^{\frac{1}{2}}$ $\displaystyle\hskip
42.67912pt\times\Big{(}\sup_{p_{1}\neq
p_{2}}\sum_{\begin{subarray}{c}k_{2},\cdots,k_{6},p_{3},p_{4},p_{5}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\\!\\!\\!\\!\\!1\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle
N_{(1)}^{-2s+\frac{3}{2}}N_{(3)}^{2s}\|w\|_{H^{\sigma}}^{8}\lesssim
N_{(1)}^{-2s+\frac{3}{2}+2s\theta}\|w\|_{H^{\sigma}}^{8}\lesssim
N_{(1)}^{-\frac{1}{2}}\|w\|_{H^{\sigma}}^{8},$

thanks to the fact that $s\geq 10$ that $\theta$ is close to $\frac{1}{3}$ and
the restriction $N_{(3)}\lesssim N_{(1)}^{\theta}$.

$\bullet$ Estimate of Type C contribution:

Without loss of generality, we assume that $k_{(1)}=p_{1}$ and
$k_{(2)}=k_{2}$, since the other cases can be treated in the same way. In
particular, $P_{1}\sim M_{2}\sim N_{(1)}$. We write

$\displaystyle\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}p_{1}\neq k_{2}\\\
|p_{1}|\sim P_{1},|k_{2}|\sim
M_{2}\end{subarray}}\\!\\!\\!\\!w_{p_{1}}^{P_{1}}\overline{w}_{p_{2}}^{P_{2}}\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}p_{2}\cdots,p_{5},k_{3},\cdots,k_{6}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\\!\\!\\!\\!\\!\\!\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\overline{w}_{p_{2}}^{P_{2}}\cdots
w_{p_{5}}^{P_{5}}w_{k_{3}}^{M_{3}}\cdots\overline{w}_{k_{6}}^{M_{6}}.$

Denote $\mathcal{B}_{\ll P_{1}}$ the $\sigma$-algebra generated by
$(g_{k})_{|k|\leq P_{1}/100}$. Without loss of generality, we assume that
$P_{1}\sim N_{(1)}$ is large enough such that $N_{(3)}\lesssim
N_{(1)}^{\theta}\ll\frac{P_{1}}{100}$. Consequently, with respect to
$\mu_{s}$, $w^{M_{3}},\cdots w^{M_{6}},w^{P_{2}},\cdots,w^{P_{5}}$ are
independent of $w^{P_{1}},w^{M_{2}}$.

By conditional Wiener chaos estimate, we have

$\displaystyle\|\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(w)\|_{L^{p}(d\mu_{s})}\leq$
$\displaystyle\|\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
P_{1}}w)\|_{L^{p}(d\mu_{s})}$ $\displaystyle=$
$\displaystyle\big{\|}\|\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
P_{1}}w)\|_{L^{p}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}\big{\|}_{L^{p}(d\mu_{s})}$ $\displaystyle\lesssim$
$\displaystyle\,p\big{\|}\|\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}\cdot\mathbf{1}_{B_{R}^{H^{\sigma}}}(\mathbf{P}_{\ll
P_{1}}w)\big{\|}_{L^{p}(d\mu_{s})}.$

As in the estimate for Type (B) terms, it suffices to show that

$\displaystyle\|\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}\lesssim$ $\displaystyle
N_{(1)}^{-\frac{1}{2}}\|w\|_{H^{\sigma}}^{8}.$

Since $p_{1}\neq k_{2}$(recall that the contribution where $p_{1}=k_{2}$ is
contained in $\mathcal{S}_{1,1}$), we estimate555In the summation below, we
implicitly assume that the sum is taken in the range $|p_{1}|\sim
P_{1},|k_{2}|\sim M_{2}$.

$\displaystyle\|\mathcal{R}_{1,3}^{(C)}(M_{1},\cdots,P_{5})\|_{L^{2}(d\mu_{s}|\mathcal{B}_{\ll
P_{1}})}$ $\displaystyle\lesssim$
$\displaystyle(P_{1}M_{2})^{-s}\Big{(}\sum_{p_{1}\neq
k_{2}}\Big{(}\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}p_{2},\cdots,p_{5},k_{3},\cdots,k_{6}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\\!\\!\\!\\!\frac{\psi_{2s}(\vec{k})}{\Omega(\vec{k})}\Big{(}1-\chi\Big{(}\frac{\Omega(\vec{k})}{\lambda(\vec{k})^{\delta_{0}}}\Big{)}\Big{)}\overline{w}_{p_{2}}^{P_{2}}\cdots
w_{p_{5}}^{P_{5}}w_{k_{3}}^{M_{3}}\cdots\overline{w}_{k_{6}}^{M_{6}}\Big{)}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle N_{(1)}^{-2s}\Big{(}\sum_{p_{1}\neq
k_{2}}\sum_{\begin{subarray}{c}p_{2}\cdots,p_{5},k_{3},\cdots,k_{6}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\frac{|\psi_{2s}(\vec{k})|^{2}}{1+|\Omega(\vec{k})|^{2}}|w_{k_{(3)}}^{N_{(3)}}|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sup_{p_{1}\neq
k_{2}}\sum_{\begin{subarray}{c}p_{2}\cdots,p_{5},k_{3},\cdots,k_{6}\\\
p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\prod_{j=4}^{10}|w_{k_{(j)}}^{N_{(j)}}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle
N_{(1)}^{-2s}\Big{(}\prod_{j=4}^{10}\|w^{N_{(j)}}\|_{L^{2}}\Big{)}\cdot\Big{(}\sum_{\begin{subarray}{c}p_{1}\neq
k_{2},k_{3},\cdots,k_{6},p_{2}\cdots,p_{5}\\\ p_{1}-\cdots+p_{5}-k_{2}+\cdots-
k_{6}=0\end{subarray}}\frac{|\psi_{2s}(\vec{k})|^{2}}{1+|\Omega(\vec{k})|^{2}}|w_{k_{(3)}}^{N_{(3)}}|^{2}\Big{)}^{\frac{1}{2}}.$
(8.4)

The estimate the last sum on the right hand side is very similar as for the
estimate of $\mathcal{R}_{1,3}^{(B1)}$. The only difference here is that we
might have the pairing of $k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}$ and $k_{2}$,
although $p_{1}\neq k_{2}$. Note that in the case of pairing

$k_{1}=p_{1}-(p_{2}-p_{3}+p_{4}-p_{5})=k_{2}$

we have $|\psi_{2s}(\vec{k})|\lesssim N_{(3)}^{2s}$, thus we control the
paired contribution crudely by

$\displaystyle\sum_{p_{1}\neq
k_{2}}\sum_{\begin{subarray}{c}p_{2}\cdots,p_{5},k_{3},\cdots,k_{6}\\\
p_{1}-\cdots+p_{5}=k_{2}\\\
k_{3}-k_{4}+k_{5}-k_{6}=0\end{subarray}}N_{(3)}^{4s}|w_{k_{(3)}}^{N_{(3)}}|^{2}\lesssim
N_{(1)}^{3}N_{(3)}^{4s}\|w^{N_{(3)}}\|_{L^{2}}^{2}\prod_{j=4}^{10}N_{(j)}^{3}$

For the non-pairing contribution, we can argue exactly as the last part of the
estimate of $\mathcal{R}_{1,3}^{(B1)}$ by using Lemma 4.2:

$\displaystyle\sum_{k_{3}\cdots,k_{6},p_{2}\cdots,p_{5}}|w_{k_{(3)}}^{N_{(3)}}|^{2}\sum_{\begin{subarray}{c}k_{1}\neq
k_{2}\\\ k_{1}=p_{1}-p_{2}+p_{3}-p_{4}+p_{5}\\\
k_{1}-k_{2}+k_{3}-k_{4}+k_{5}-k_{6}=0\end{subarray}}\frac{M_{(1)}^{4(s-1)}(M_{(3)}^{4}+|\Omega(\vec{k})|^{2})}{1+|\Omega(\vec{k})|^{2}}.$
(8.5)

For fixed $p_{2},\cdots,p_{5},k_{3},\cdots,k_{6}$,

$|\Omega(\vec{k})|=|p_{1}-\mathbf{p}|^{2}-|k_{2}|^{2}+\mathbf{c}$

with $\mathbf{p}=p_{2}-p_{3}+p_{4}-p_{5}$ and
$\mathbf{c}=|k_{3}|^{2}-|k_{4}|^{2}+|k_{5}|^{2}-|k_{6}|^{2}.$ When
$\mathbf{p}\neq p_{1}-k_{2}$, by Lemma 4.2, the choices of $p_{1},k_{2}$ such
that $p_{1}-\mathbf{p}-k_{2}=\mathbf{k}=k_{3}-k_{4}+k_{5}-k_{6}$ are bounded
by $M_{(1)}^{2}$, hence (8.5) is bounded by

$N_{(1)}^{4(s-1)}(N_{(3)}^{4}N_{(1)}^{2}+N_{(1)}^{3})\|w^{N_{(3)}}\|_{L^{2}}^{2}\prod_{j=4}^{10}N_{(j)}^{3}.$

Therefore, the right hand side of (8) can be bounded by

$\big{(}N_{(1)}^{-2s+\frac{3}{2}}N_{(3)}^{2s}+N_{(1)}^{-2}(N_{(1)}^{\frac{3}{2}}+N_{(3)}^{2}N_{(1)})\big{)}\prod_{j=3}^{10}\|w^{N_{(j)}}\|_{L^{2}}\cdot\Big{(}\prod_{j=4}^{10}N_{(j)}^{\frac{3}{2}}\Big{)}\lesssim
N_{(1)}^{-\frac{1}{2}}\|w\|_{H^{\sigma}}^{8},$

thanks to the fact that $s\geq 10$, that $\theta<\frac{1}{3}$, $\theta$ close
to $\frac{1}{3}$ and that $N_{(3)}\lesssim N_{(1)}^{\theta}$. This completes
the proof of Proposition 8.1. ∎

##  Appendix A Long time approximations

In this appendix, we prove the approximation results used in Section 3. The
proof is a consequence of the global regularity theory of [23].

###  A.1. Ingredients in the global regularity theory for the energy critical
NLS on $\mathbb{T}^{3}$

In the sequel, we follow the notations in [23] and [22] about basic
definitions and properties for function spaces
$U^{p},V^{p},U_{\Delta}^{p},V_{\Delta}^{p}$ related to critical problems. Let
us briefly recall some related function spaces as well as multilinear
estimates from [22] and [23] that we will use. For $s\in\mathbb{\mathbb{R}}$,

$\displaystyle\|u\|_{\widetilde{X}^{s}(\mathbb{R})}:=\Big{(}\sum_{k\in\mathbb{Z}^{3}}\langle
k\rangle^{2s}\|{\mathrm{e}}^{it|k|^{2}}(\mathcal{F}u)(t,k)\|_{U_{t}^{2}}^{2}\Big{)}^{\frac{1}{2}},$
(A.1)
$\displaystyle\|u\|_{\widetilde{Y}^{s}(\mathbb{R})}:=\Big{(}\sum_{k\in\mathbb{Z}^{3}}\langle
k\rangle^{2s}\|{\mathrm{e}}^{it|k|^{2}}(\mathcal{F}u)(t,k)\|_{V_{t}^{2}}^{2}\Big{)}^{\frac{1}{2}}.$
(A.2)

We have the continuous embedding property:

$\widetilde{X}^{s}(\mathbb{R})\hookrightarrow\widetilde{Y}^{s}(\mathbb{R})\hookrightarrow
L^{\infty}(\mathbb{R};H^{s}(\mathbb{T}^{3})).$

For intervals $I\subset\mathbb{R}$, the space $X^{s}(I)$ is defined via the
restriction norms:

$\|u\|_{X^{s}(I)}:=\sup_{J\subset I,|J|\leq
1}\inf_{v\mathbf{1}_{J}(t)=u\mathbf{1}_{J}(t)}\|v\|_{\widetilde{X}^{s}}.$

Similarly for the space $Y^{s}(I)$. Note that by definition, for linear
solution $u(t)={\mathrm{e}}^{it\Delta}\phi$,

$\displaystyle\|u(t)\|_{X^{s}(I)}\leq\|\phi\|_{H^{s}(\mathbb{T}^{3})}.$ (A.3)

The critical Strichartz type norm is defined via the norm

$\|u\|_{Z(I)}:=\sum_{p\in\\{p_{0},p_{1}\\}}\sup_{J\subset I,|J|\leq
1}\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{5-\frac{p}{2}}\|\mathbf{P}_{N}u(t)\|_{L_{t,x}^{p}(J\times\mathbb{T}^{3})}^{p}\Big{)}^{\frac{1}{p}},\quad
p_{0}=4+1/10,\,\,p_{1}=100,$

where $\mathbf{P}_{N}=\mathbf{P}_{\leq N}-\mathbf{P}_{\leq N/2}$, and
$\mathbf{P}_{\leq N}$ are square Littlewood-Paley projectors defined in
Section 2 of [23].

By definition we remark that if $T\geq 1$ and $I_{T}=[-T,T]$,

$\|u\|_{Z(I_{T})}\sim_{T}\sum_{p\in\\{p_{0},p_{1}\\}}\|u(t)\|_{L_{t}^{p}([-T,T];\widetilde{B}_{p,p}^{\frac{5}{p}-\frac{1}{2}}(\mathbb{T}^{3}))},$

while if $T<\frac{1}{2}$,

$\|u\|_{Z(I_{T})}=\sum_{p\in\\{p_{0},p_{1}\\}}\|u(t)\|_{L_{t}^{p}([-T,T];\widetilde{B}_{p,p}^{\frac{5}{p}-\frac{1}{2}}(\mathbb{T}^{3}))}$

where $\widetilde{B}_{p,q}^{s}$ are Besov spaces related to the Littlewood-
Paley projectors $\mathbf{P}_{N}$.

The inhomogeneous term on an interval $I=(a,b)$ will be controlled by the
$N^{s}(I)$ norm:

$\|F\|_{N^{s}(I)}:=\Big{\|}\int_{a}^{t}{\mathrm{e}}^{i(t-t^{\prime})\Delta}F(t^{\prime})dt^{\prime}\Big{\|}_{X^{s}(I)}.$

It turns out that ([22], Proposition 2.11)

$\displaystyle\|F\|_{N^{s}(I)}\leq\sup_{\begin{subarray}{c}G\in Y^{-s}(I)\\\
\|G\|_{Y^{-s}(I)}\leq
1\end{subarray}}\Big{|}\int_{I}\int_{\mathbb{T}^{3}}F(t,x)\overline{G}(t,x)dxdt\Big{|}.$

Recall the key Strichartz estimate:

######  Lemma A.1 ([23], Corollary 2.2).

Let $p\in(4,\infty)$ and $\mathbf{P}_{C}$ the frequency projector to some cube
$C$ of size $N$. For any interval $I\subset\mathbb{R}$, $|I|\leq 1$,

$\|\mathbf{P}_{C}u\|_{L_{t,x}^{p}(I\times\mathbb{T}^{3})}\lesssim
N^{\frac{3}{2}-\frac{5}{p}}\|u\|_{U_{\Delta}^{p}(I;L_{x}^{2})},$

where the implicit constant is independent of intervals $I$.

As a consequence of Lemma A.1 and the embedding $X^{0}(I)\hookrightarrow
U_{\Delta}^{p}(I;L_{x}^{2})$ (basically since $U^{2}\hookrightarrow U^{p}$)
for $p>2$, we have

$\displaystyle\|u\|_{Z(I)}\lesssim\|u\|_{X^{1}(I)}$ (A.4)

for any interval $I$, where the implicit constant is uniform. The key
multilinear estimate we will use reads:

######  Lemma A.2 ([23], Lemma 3.2).

Let $\sigma\geq 1$. For $u_{j}\in X^{1}(I)$, $j=1,2,3,4,5$, $|I|\leq 1$. the
estimate

$\displaystyle\Big{\|}\prod_{j=1}^{5}u_{j}^{\pm}\Big{\|}_{N^{\sigma}(I)}\lesssim\sum_{\sigma\in\mathfrak{S}_{5}}\|u_{\sigma(1)}\|_{X^{\sigma}(I)}\prod_{j\geq
2}\|u_{\sigma(j)}\|_{Z(I)}^{\frac{1}{2}}\|u_{\sigma(j)}\|_{X^{1}(I)}^{\frac{1}{2}}$
(A.5)

holds true, where $\mathfrak{S}_{5}$ is the permutation group of $5$ elements
and $u_{j}^{\pm}\in\\{u_{j},\overline{u}_{j}\\}$, and the implicit constant in
the inequality is independent of intervals $I$ such that $|I|\leq 1$.

We remark that [23] treats the case $\sigma=1$. For $\sigma>1$, the proof
follows in the similar way by the more precise estimate

$\displaystyle\int_{I\times\mathbb{T}^{3}}\Big{|}\sum_{N_{0},N_{1}\geq
N_{2}\geq N_{3}\geq N_{4}\geq
N_{5}}\prod_{j=0}^{5}\mathbf{P}_{N_{j}}u_{j}^{\pm}\Big{|}dxdt$
$\displaystyle\lesssim$
$\displaystyle\Big{(}\frac{N_{0}}{N_{1}}\Big{)}^{\sigma}\Big{(}\frac{N_{5}}{N_{1}}+\frac{1}{N_{3}}\Big{)}^{\delta}\Big{(}\frac{N_{4}}{N_{0}}+\frac{1}{N_{2}}\Big{)}^{\delta}\|\mathbf{P}_{N_{1}}u_{1}\|_{Y^{\sigma}(I)}\|\mathbf{P}_{N_{0}}u_{0}\|_{Y^{-\sigma}(I)}\prod_{j=2}^{5}\|\mathbf{P}_{N_{j}}u_{j}\|_{Z(I)}^{\frac{1}{2}}\|\mathbf{P}_{N_{j}}u_{j}\|_{X^{1}(I)}^{\frac{1}{2}}$

for some $\delta>0$. We omit the details of the proof.

Finally, we recall the global regularity theory of Ionescu-Pausader. Following
Section 6 of [23], given $R>0$ and $\tau\geq 0$, consider the non-negative
function (possibly $\infty$)

$\Sigma(R,\tau):=\sup\big{\\{}\|u\|_{Z(I)}^{2}:\;H(u)\leq
R,|I|\leq\tau\big{\\}},$

where $H(u)$ is the energy of $u$ and the supremum is taken over all strong
solutions of (1.1) of energy less than or equal to $R$ and all intervals $I$
of length $|I|\leq\tau$. As an increasing function in $\tau$, the limit
(possibly $\infty$ in a priori)

$\Sigma_{*}(R):=\lim_{\tau\rightarrow 0^{+}}\Sigma(R,\tau)$

exists. Moreover, $\Sigma(R,\tau)$ is quasi subadditive in $\tau$:

$\Sigma(R,\tau_{1}+\tau_{2})\lesssim\Sigma(R,\tau_{1})+\Sigma(R,\tau_{2})$

for any $\tau_{1},\tau_{2}>0$.

The global regularity result of Ionescu-Pausader can be stated as:

######  Theorem A.1 ([23]).

For any $R>0$, $\Sigma_{*}(R)<\infty$. Consequently, for any $\tau>0$,
$\Sigma(R,\tau)<\infty$ and moreover

$\Sigma(R,\tau)\leq\Sigma(R,1){\mathrm{e}}^{C_{0}(1+\tau)},$

where $C_{0}>0$ is an absolute constant. In particular, for any $\phi\in
H^{1}(\mathbb{T}^{3})$ of energy smaller than or equal to $R$, the strong
solution $u(t)$ of (1.1) with initial data $\phi$ is global and

$\|u(t)\|_{Z([0,\tau])}\leq\Sigma(R,1){\mathrm{e}}^{C_{0}(1+\tau)}.$

Finally,

$\|u\|_{X^{1}([0,\tau])}\leq C(R,\|u\|_{Z([0,\tau])}).$

We denote $\Phi(t)$ the global flow of (1.1) in $H^{1}(\mathbb{T}^{3})$. The
following corollary shows that the $H^{\sigma}(\mathbb{T}^{3})$, $\sigma\geq
1$ regularity is propagated by $\Phi(t)$.

######  Corollary A.2.

Let $\sigma\geq 1$. Then
$\Phi(t)(H^{\sigma}(\mathbb{T}^{3}))=H^{\sigma}(\mathbb{T}^{3})$ for every
$t\in\mathbb{R}$.

###### Proof.

Assume that $\phi\in H^{\sigma}(\mathbb{T}^{3})$ such that $H(\phi)\leq R$.
Denote $u(t)=\Phi(t)\phi$. From Theorem A.1,

$\|u\|_{Z([-T,T])}\leq\Sigma(R,T),\quad\|u\|_{X^{1}([-T,T])}\leq C(R,T),$

where $C(R,T)$ depends only on $R$ and $T$. By the Duhamel formula and (A.5)
of Lemma A.2, for any $I=(a,b)\subset[-T,T]$,

$\displaystyle\|u\|_{X^{\sigma}(I)}\leq$
$\displaystyle\|e^{i(t-a)\Delta}u(a)\|_{X^{\sigma}(I)}+\||u|^{4}u\|_{N^{\sigma}(I)}$
$\displaystyle\leq$
$\displaystyle\|u(a)\|_{H_{x}^{\sigma}}+C\|u\|_{X^{\sigma}(I)}\|u\|_{X^{1}(I)}^{2}\|u\|_{Z(I)}^{2}$
$\displaystyle\leq$
$\displaystyle\|u(a)\|_{H_{x}^{\sigma}}+C(R,T)\|u\|_{Z(I)}^{2}\|u\|_{X^{\sigma}(I)},$

where $C(R,T)$ is a constant depending only on $R$ and $T$ that can change
from line to line.

Next, we partition $[0,T]=\bigcup_{j=1}^{\kappa}[a_{j-1},a_{j}]$ such that
$\|u\|_{Z([a_{j-1},a_{j}])}<\frac{1}{\sqrt{2C(R,T)}},$ hence for all
$j=1,2,\cdots,k$,

$\|u\|_{X^{\sigma}([a_{j-1},a_{j}])}\leq 2\|u(a_{j-1})\|_{H_{x}^{\sigma}}.$

By the embedding property, for all $j\geq 1$,

$\|u\|_{X^{\sigma}([a_{j},a_{j+1}])}\leq
C\|u\|_{X^{\sigma}([a_{j},a_{j-1}])}.$

This shows that for all $t\in[-T,T]$,

$\|u(t)\|_{H^{\sigma}}\leq C^{\kappa}\|\phi\|_{H_{x}^{\sigma}}.$

Therefore $\Phi(t)(H^{\sigma}(\mathbb{T}^{3}))\subset
H^{\sigma}(\mathbb{T}^{3})$. Using the time reversibility completes the proof.
∎

######  Remark A.3.

When $\sigma>1$, the above argument does not give a uniform control of the
$H^{\sigma}$ norm for the solution, since the index $\kappa$ depends on the
profile of each individual global $H^{1}$-solution $u(t)$. Later we shall
strengthen the $H^{\sigma}$-estimate uniformly on any bounded ball of
$H^{\sigma}$ by choosing a uniform partition of $[0,T]$.

###  A.2. Local convergence and stability

From now on, denote $\Phi(t)$ the flow of the energy critical NLS. Denote
$\Phi_{N}(t)$ the flow of the truncated NLS:

$\displaystyle i\partial_{t}u_{N}+\Delta
u_{N}=S_{N}(|S_{N}u_{N}|^{4}S_{N}u_{N}),$ (A.6)

where $S_{N}$ is the smooth Fourier truncation at size $N$ defined at the
beginning of Section 2.

######  Proposition A.4 (Local convergence).

Assume that $\sigma\geq 1$. Let $\phi,\widetilde{\phi}\in
H^{\sigma}(\mathbb{T}^{3})$ and $I\subset\mathbb{R}$ be an interval and
$t_{0}\in I$. Suppose $\|\phi\|_{H_{x}^{\sigma}}\leq
A,\|\widetilde{\phi}\|_{H_{x}^{\sigma}}\leq A$. Then for any $\epsilon>0$,
there exist $\delta=\delta(A,\epsilon)>0$ such that if

$\|e^{i(t-t_{0})\Delta}\phi\|_{Z(I)}<\delta,\;\|\phi-\widetilde{\phi}\|_{H_{x}^{\sigma}}<\delta,$

there exist unique solutions $u=\Phi(t-t_{0})\phi$ and
$u_{N}=\Phi_{N}(t-t_{0})\widetilde{\phi}$ in
$C(\overline{I};H_{x}^{\sigma})\cap X^{\sigma}(I)$ satisfying

$\|u_{N}\|_{X^{\sigma}(I)}+\|u\|_{X^{\sigma}(I)}\leq
C_{0}A,\;\|u_{N}\|_{Z(I)}+\|u\|_{Z(I)}<\epsilon,$

where $C_{0}>0$ is an absolute constant. Moreover,

$\displaystyle\|\Phi_{N}(t)\widetilde{\phi}-\Phi(t)\phi\|_{C(\overline{I};H_{x}^{\sigma})}\leq
C_{0}\|\phi-\widetilde{\phi}\|_{H_{x}^{\sigma}}+C_{0}\delta_{N}(A,\epsilon,\phi),$
(A.7)

where $\delta_{N}(A,\epsilon,\phi)\rightarrow 0$ as $N\rightarrow\infty$,
uniformly in $\phi$ on a compact set $K$ of $H^{\sigma}(\mathbb{T}^{3})$ such
that $\|\phi\|_{H_{x}^{\sigma}}\leq A$.

######  Remark A.5.

Consequently, taking $\widetilde{\phi}=\phi$, under the hypothesis of
Proposition A.4, we have

$\|\Phi_{N}(t)\phi-\Phi(t)\phi\|_{X^{\sigma}(I)}\rightarrow 0,$

uniformly on any compact set of $H^{1}(\mathbb{T}^{3})$. Taking the limit
$N\rightarrow\infty$ in (A.7), we obtain also that under the hypothesis of
Proposition A.4,

$\|\Phi(t)\widetilde{\phi}-\Phi(t)\phi\|_{X^{\sigma}(I)}\leq
C_{0}\|\phi-\widetilde{\phi}\|_{H_{x}^{\sigma}}.$

###### Proof.

First we prove the case when $\sigma=1$. Note that the existence of solutions
on $\overline{I}$ is a direct consequence of Proposition 3.3 of [23]. So we
will only concentrate on the bounds for solutions $u_{N}(t)$ and $u(t)$. We
argue by several steps.

Step 1: Uniform bound:

Without loss of generality, we assume that $t_{0}=0$. By the Strichartz
estimate (A.4) and Lemma A.2, we have

$\displaystyle\|u\|_{Z(I)}\leq$
$\displaystyle\|{\mathrm{e}}^{it\Delta}\phi\|_{Z(I)}+C\||u|^{4}u\|_{N^{1}(I)}$
$\displaystyle\leq$
$\displaystyle\|{\mathrm{e}}^{it\Delta}\phi\|_{Z(I)}+C\|u\|_{Z(I)}^{2}\|u\|_{X^{1}(I)}^{3},$
(A.8)

and

$\displaystyle\|u\|_{X^{1}(I)}\leq$
$\displaystyle\|{\mathrm{e}}^{it\Delta}\phi\|_{Z(I)}+C\||u|^{4}u\|_{N^{1}(I)}$
$\displaystyle\leq$
$\displaystyle\|\phi\|_{H_{x}^{1}}+C\|u\|_{Z(I)}^{2}\|u\|_{X^{1}(I)}^{3},$
(A.9)

where $C>0$ is independent of $I$. Note that the same inequalities holds for
$u_{N}$, as the smooth spectral projector $S_{N}$ is bounded from $L_{x}^{r}$
to $L_{x}^{r}$ for any $1<r<\infty$. Then the desired control for
$\|u\|_{Z(I)}$ and $\|u\|_{X^{1}(I)}$, as well as
$\|u_{N}\|_{Z(I)},\|u_{N}\|_{X^{1}(I)}$ follow from the following elementary
lemma:

######  Lemma A.6.

Let $A>0,C_{0}>0$ and $f\in C([0,\tau_{0}];[0,\infty))$ be an increasing
continuous function such that $f(0)=0$ and
$h:[0,\tau_{0}]\rightarrow[0,\infty)$ is increasing. Then for any small
$\epsilon>0$, there exists $\delta=\delta(A,\epsilon)>0$, such that if

$f(t)\leq\delta+C_{0}f(t)^{4}h(t),\quad h(t)\leq
C_{0}A+C_{0}f(t)^{4}h(t),\quad\forall t\in[0,\tau_{0}],$

then $f(t)\leq\epsilon$ and $h(t)\leq 2C_{0}A$.

###### Proof.

This follows from a standard continuity argument. Let $\tau\leq\tau_{0}$ be
the largest time such that $f(t)\leq\epsilon$ ($\tau$ exists since $f(0)=0$).
We claim that if $\epsilon\ll 1$ is such that
$16C_{0}\epsilon^{4}<\frac{1}{2}$ and $32C_{0}^{2}A\epsilon^{3}<\frac{1}{2}$,
then $\tau=\tau_{0}$. By contradiction, if $\tau<\tau_{0}$, by continuity of
$f$, there exists $\tau_{1}\in(\tau,\tau_{0})$ such that
$f(\tau_{1})<2\epsilon$. Then for all $0\leq t\leq\tau_{1}$, $h(t)\leq
C_{0}A+16C_{0}\epsilon^{4}h(t)<2C_{0}A$, thanks to the smallness of
$\epsilon$. Plugging into the inequality of $f(t)$, we deduce that for all
$0\leq t\leq\tau_{1}$

$f(t)\leq\delta+32C_{0}^{2}A\epsilon^{4}.$

Choosing $\delta<\frac{\epsilon}{2}$, hence,
$\delta+32C_{0}^{2}A\epsilon^{4}<\epsilon$, we obtain a contradiction. ∎

Step 2: Quantitative convergence:

By the same application of the Strichartz inequality, we get

$\displaystyle\|u_{N}-u\|_{X^{1}(I)}\leq\|\phi_{N}-\phi\|_{H_{x}^{1}}+\|S_{N}^{\perp}(|S_{N}u_{N}|^{4}S_{N}u_{N})\|_{N^{1}(I)}+\||S_{N}u_{N}|^{4}S_{N}u_{N}-|u|^{4}u\|_{N^{1}(I)}.$

By splitting $S_{N}u_{N}=S_{M}u_{N}+S_{M}^{\perp}S_{N}u_{N},$ we observe that
for $M=\frac{N}{16}$, $S_{N}^{\perp}(|S_{M}u_{N}|^{4}S_{M}u_{N})=0$.
Therefore, by the uniform boundedness of $S_{N},S_{M}$ on $L_{x}^{r}$
($1<r<\infty$) and on $X^{1}$,

$\displaystyle\|S_{N}^{\perp}(|S_{N}u_{N}|^{4}S_{N}u_{N})\|_{N^{1}(I)}\leq$
$\displaystyle
C\|S_{M}^{\perp}u_{N}\|_{Z(I)}\|u_{N}\|_{Z(I)}^{3}\|u_{N}\|_{X^{1}(I)}^{3}$
$\displaystyle+$ $\displaystyle
C\|u_{N}\|_{Z(I)}^{2}\|S_{M}^{\perp}u_{N}\|_{X^{1}(I)}\|u_{N}\|_{X^{1}(I)}^{2}$
$\displaystyle\leq$ $\displaystyle
C\|S_{M}^{\perp}u_{N}\|_{X^{1}(I)}\|u_{N}\|_{Z(I)}^{2}\|u_{N}\|_{X^{1}(I)}^{2}$
(A.10) $\displaystyle\leq$ $\displaystyle
C(\|S_{M}^{\perp}u\|_{X^{1}(I)}+\|u_{N}-u\|_{X^{1}(I)})\|u_{N}\|_{Z(I)}^{2}\|u_{N}\|_{X^{1}(I)}^{2}$
$\displaystyle\leq$ $\displaystyle
CA^{2}\epsilon^{2}(\|S_{M}^{\perp}u\|_{X^{1}(I)}+\|u_{N}-u\|_{X^{1}(I)}).$

For the other term, algebraic manipulation yields

$\displaystyle\||S_{N}u_{N}|^{4}S_{N}u_{N}-|u|^{4}u\|_{N^{1}(I)}\leq$
$\displaystyle
C\|S_{N}u_{N}-u\|_{X^{1}(I)}(\|S_{N}u_{N}\|_{Z(I)}^{2}+\|u\|_{Z(I)}^{2})(\|S_{N}u_{N}\|_{X^{1}(I)}^{2}+\|u\|_{X^{1}(I)}^{2})$
$\displaystyle\leq$ $\displaystyle
CA^{2}\epsilon^{2}(\|S_{N}^{\perp}u\|_{X^{1}(I)}+\|u_{N}-u\|_{X^{1}(I)})$
$\displaystyle\leq$ $\displaystyle
CA^{2}\epsilon^{2}(\|S_{M}^{\perp}u\|_{X^{1}(I)}+\|u_{N}-u\|_{X^{1}(I)}).$

In summary, we have

$\displaystyle\|u_{N}-u\|_{X^{1}(I)}\leq$
$\displaystyle\|\phi-\widetilde{\phi}\|_{H_{x}^{1}}+CA^{2}\epsilon^{2}(\|S_{N/16}^{\perp}u\|_{X^{1}(I)}+\|u_{N}-u\|_{X^{1}(I)}).$
(A.11)

Furthermore, from the Duhamel formula of $u$ and the similar argument as
(A.2), we have the recursive inequality

$\displaystyle\|S_{N}^{\perp}u\|_{X^{1}(I)}\leq$
$\displaystyle\|S_{N}^{\perp}\phi\|_{H_{x}^{1}}+CA^{2}\epsilon^{2}\|S_{N/16}^{\perp}u\|_{X^{1}(I)}.$
(A.12)

To conclude, we invoke the following elementary result:

######  Lemma A.7.

Let $\\{a_{j}\\},\\{b_{j}\\}$ be two positive sequences and $A_{0}>0$ is an
absolute constant. Assume that $0<\theta<1$ and for $1\leq j\leq m$,

$a_{j}\leq A_{0}b_{j}+\theta a_{j-1}.$

Then we have

$\displaystyle a_{m}\leq
A_{0}\sum_{j=0}^{m-1}\theta^{j}b_{m-j}+\theta^{m-1}a_{1}.$

In particular, if $b_{m}\rightarrow 0$, then $a_{m}\rightarrow 0$.

The proof of this elementary lemma is straightforward hence we omit the
detail.

Consequently, we have

$\displaystyle\|S_{N/16}^{\perp}u\|_{X^{1}(I)}\leq
A_{0}\sum_{j=1}^{\log_{16}N-1}(CA\epsilon)^{j}\|S_{N/16^{j}}^{\perp}\phi\|_{H_{x}^{1}}+(CA\epsilon)^{\log_{16}N-1}\|\phi\|_{H_{x}^{1}},$

where $C,A>0$ are absolute constants and $\epsilon\ll 1$ such that
$CA\epsilon<1$. We denote