# Modified scattering for the cubic Schrödinger equation on Diophantine
waveguides

Nicolas Camps  Nantes Université, CNRS, Laboratoire de Mathématiques Jean
Leray, LMJL, F-44000 Nantes, France<EMAIL_ADDRESS>and
Gigliola Staffilani  Department of Mathematics, Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA 02139-4307 USA.
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We consider the cubic Schrödinger equation posed on a product space subject to
a generic Diophantine condition. Our analysis shows that the small-amplitude
solutions undergo modified scattering to an effective dynamics governed by
some interactions that do not amplify the Sobolev norms. This is in sharp
contrast with the infinite energy cascade scenario observed by
Hani–Pausader–Tzvetkov–Visciglia in the absence of Diophantine conditions.

###### 2020 Mathematics Subject Classification:

35B40 primary

###### Contents

  1. 1 Introduction
  2. 2 Preparations
  3. 3 Nonlinear interactions
  4. 4 A priori bounds and asymptotic behavior
  5. A Some elementary Lemmas

##  1\. Introduction

###  1.1. Motivations

The present work aims at testing the persistence of turbulent mechanisms
observed in the long-time dynamics of certain solutions of nonlinear
dispersive equations. On compact domains, the interplay between weaker
dispersion and nonlinear resonances can lead to energy transfers from low to
high frequency scales of oscillations as time evolves. These forward cascade
mechanisms manifest themselves through the growth of the high-order Sobolev
norms. Bourgain’s original formulation in [5] addresses the potential
explosion of Sobolev norms in infinite time, known as _the infinite energy
cascade_. Despite considerable efforts to elucidate this conjecture, our
understanding of the infinite-time behavior of nonlinear dispersive equations
on compact domains is still incomplete.

One of the very few results constructing infinite energy cascades is due to
Hani–Pausader–Tzvetkov–Visciglia [24]. They consider the cubic defocusing
Schrödinger equation posed on the product space
$\mathbb{R}\times\mathbb{T}^{d}$, with
$\mathbb{T}^{d}=\mathbb{R}/\mathbb{Z}^{d}$:

$i\partial_{t}U+\partial_{x}^{2}U+\Delta_{y}U=|U|^{2}U\,,\quad(x,y)\in\mathbb{R}\times\mathbb{T}^{d}\,.$
(NLS)

The dispersion of the Schrödinger free evolution along the non-compact
direction $\mathbb{R}$ serves to dampen and scatter the non-resonant
interactions, reducing the effective dynamics to a system governed by the
four-waves resonant set of the cubic Schrödinger equation posed on the compact
direction $\mathbb{T}^{d}$. This enables the construction of small amplitude
initial data that ignite a forward energy cascade ([24, Corollary 1.4]):

For all $d\in\\{2,3,4\\}$ and $s>1$ large enough, there exists $U(0)$ with
small amplitude 111The precise statement actually requires the smallness of a
strong norm, with extra derivatives and space decay in the
$\mathbb{R}$-direction in $H^{s}(\mathbb{R}\times\mathbb{T}^{d})$ such that
the corresponding solution $U$ is global and satisfies

$\limsup_{t\to+\infty}\|U(t)\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}=+\infty\,.$

In this work, we allow for a more general dispersion relation in the compact
direction $\mathbb{T}^{d}$:

$i\partial_{t}U+\partial_{x}^{2}U+\mathrm{div(A}\nabla_{y})U=|U|^{2}U\,,\quad(x,y)\in\mathbb{R}\times\mathbb{T}^{d}\,.$
(A-NLS)

The symmetric positive definite constant matrix $\mathrm{A}$ encodes the
dispersion relation. Changing the dispersion via $\mathrm{A}$ corresponds to
rescaling $\mathbb{T}^{d}$ through a change of variables. For instance when
$\mathrm{A}=\operatorname{Id}$, we retrieve the case of the square torus
$\mathbb{T}^{d}$, while rectangular tori correspond to diagonal matrices
$\mathrm{A}$. We will focus on generic matrices satisfying an explicit
Diophantine condition presented in Definition 1.1. They correspond to generic
non-rectangular tori.

A natural question is whether the turbulent mechanisms observed in the case of
the square torus [24] persist under generic and non-periodic boundary
conditions.

Our main results, stated in Theorem 1.3 and Theorem 1.4, address this
question. When (A-NLS) is posed on a waveguide with a generic Diophantine
property, we establish that there is no amplification of the Sobolev norms
over infinite time for the solutions associated with small amplitude initial
data. Hence, the asymptotic behavior of (A-NLS) on the product space
$\mathbb{R}\times\mathbb{T}^{d}$ with $d\geq 2$ is very different when
$\mathbb{T}^{d}$ is replaced by Diophantine tori. This sharp dichotomy that we
see very well manifested analytically should also be of warning when one uses
periodic boundary conditions to model nonlinear dispersive equations on
bounded domains. What we prove here is that any infinitesimal perturbation of
a periodic domain may result in a dramatically different behavior of the
energy spectrum of the solutions.

###  1.2. Context

In this subsection we recall some previous related works and we stress out
further the main motivations.

####  1.2.1. Growth of Sobolev norms for nonlinear Schrödinger equations on
compact domains and waveguides.

The existence of infinite cascades realized via the growth of Sobolev norms
has drawn considerable interest, and a survey of all the existing results goes
much beyond the scope of this paper, hence here we recall only the works that
are strictly related to it.

A line of research within the study of the infinite energy cascade consists in
proving polynomial upper bounds for the growth of Sobolev norms 222The Cauchy
theory and the conservation of the energy give exponential bounds to quantify
how fast do the transfer of energy occurs. This was done originally by
Bourgain [3], and several other works have appeared in recent years [34, 30,
31, 32, 33]. On rectangular Diophantine tori [13] improved the polynomials
bounds, and [12] obtained polynomial bounds for the energy-critical quintic
Schrödinger equation (for the square tori such a result is not known). These
results suggest that energy transfers, if they exist, take longer to occur
when the boundary conditions are Diophantine rather than periodic.

As mentioned before, much less is known about the lower bounds and the
infinite growth of the Sobolev norms. The infinite energy cascade evidenced in
[24] is a consequence of the analysis of the Toy Model, a subsystem of the
four-waves resonant set, which was introduced in [9] where the equation in
(NLS) was considered in a square torus $\mathbb{T}^{2}$ (see also [8]). In
that case, using a stability lemma, one could only partially transfer the
growth dynamics of the Toy Model and prove a norm-amplification result in
finite time:

For all $s>1$ and $C>0$, there exist $T>0$ and $u(0)\in H^{s}(\mathbb{T}^{2})$
such that the solution $u$ to (NLS) posed on $\mathbb{T}^{2}$ satisfies

$\|u(T)\|_{H^{s}(\mathbb{T}^{2})}\geq C\|u(0)\|_{H^{s}(\mathbb{T}^{2})}\,.$

Along these lines we also recall the work in [20, 21, 25, 22]. Apart from the
case of completely resonant equations (the Szegő equation [17], which is the
resonant system of the Half-wave equation), this strategy gives norm
amplification results in finite time.

####  1.2.2. Infinite energy cascade on waveguides

On the waveguide manifold the dispersion along the $\mathbb{R}$ direction
allowed for the full growth of the Toy Model to be harvested in infinite time.
The strategy developed in [24] (also explained in [23]) consists in
propagating a _weak norm_ based on the energy space which is preserved by the
effective resonant system, while allowing the stronger norms to slightly grow
in time. By doing so, one can both establish modified scattering and construct
wave operators from which one can transfer the turbulent dynamics known for
the Toy Model. The time-oscillations in the non-resonant interactions are
easily 333Indeed, the resonant function for NLS on $\mathbb{T}^{d}$ (periodic
boundary conditions) takes integer values. If it does not vanish then it takes
integer values greater than 1, unlike in the present work where small divisor
problems arise. converted into decay via a Poincaré–Dulac normal form, while
non-stationary phase arguments control space-oscillations, in the spirit of
[29].

Subsequent works, such as [38, 7], adopt the same strategy for weakly
dispersive equations, resulting in infinite energy cascades 444In these two
works, however, the Cauchy theory at low regularity is not well understood and
poses considerable problems. They could achieve the construction of modified
wave operators, which is sufficient to prove infinite energy cascade, without
proving the modified scattering result.. Another notable example is [16],
which extracts interesting dynamics with a beating effect from the resonant
system of a coupled system of nonlinear Schrödinger equations.

In the spirit of the present work, [19] consider the cubic Schrödinger
equation perturbed by a generic convolution potential and prove that there is
no norm amplification. For this model, the exact resonances are trivial, and
the quasi-resonances are controlled by the third largest frequency so that one
can reproduce the strategy of [24]. However, our admissibility condition (1.1)
on Diophantine tori without potential provides control by the highest
frequency, necessitating new approaches that will be discussed in Section 1.4
below. To conclude this subsection, we stress out that on
$\mathbb{R}^{n}\times\mathbb{T}^{d}$ with $n\geq 2$ there is scattering [36,
37]. Indeed, all the interactions are attenuated by the stronger dispersion on
$\mathbb{R}^{n}$.

####  1.2.3. Nonlinear Schrödinger equations posed on Diophantine tori: longer
time-scales?

By considering irrational tori instead of the square torus, dispersion is
enhanced and the refocusing time needed for the waves to explore the geometry
is also longer. This was quantified by [14, 15] who obtained longer time
versions of the Strichartz estimates on Diophantine rectangular and non-
rectangular tori than in [6].

Moreover, for the Schrödinger equation in (NLS) on irrational tori some exact
resonant interactions are eliminated, whereas quasi-resonant interactions
emerge: the resonant function is not valued in $\mathbb{Z}$ and can take
arbitrarily small values without vanishing. In [18], it was proved that the
quasi-resonant interactions can also lead to turbulent mechanisms by extending
the norm inflation result of [9] for small amplitude initial data to the case
of some rectangular irrational tori in dimension $d=2$, after exponential
time-scale (the Diophantine approximation condition for the length of the
torus is detailed in [18, paragraph 2.1]). This result indicates that on
irrational tori quasi-resonant interactions have a role to play in the
dynamics.

The first author with J. Bernier recently developed in [1] a normal form
approach to show stability over polynomial time-scales for the cubic
Schrödinger equation posed on Diophantine tori satisfying the Diophantine
condition (1.1). In [35], the second author with B. Wilson proved that on
rectangular irrational tori the resonant system is much smaller than on
$\mathbb{T}^{2}$ and showed that the energy transfers observed in [8] out of
the Toy Model when finitely many modes are excited do not longer occur.
Moreover in [28] the second author with A. Hrabski, Y. Pan and B. Wilson also
gave some numerical evidence that the high Sobolev norms should not grow as
rapidly as the ones on a square torus. For more numerical results one should
also consult [10, 27]. Finally, let us point out the result [11] in the
context of wave kinetic theory, which goes in the direction of considering
generic Diophantine tori in order to bypass the time-scales dictated by the
resonant interactions.

###  1.3. Set up and main results

We consider the equation (A-NLS) on the waveguide manifold with the following
explicit Diophantine condition on the coefficients of $\mathrm{A}$.

######  Definition 1.1.

The matrix $\mathrm{A}$ is admissible with a parameter $\tau>0$ if there
exists $c>0$ such that for all $(a,b)\in(\mathbb{Z}^{d}\setminus\\{0\\})^{2}$,

$|a^{\intercal}\mathrm{A}b|\geq\frac{c}{|a|^{\tau}|b|^{\tau}}\,,$ (1.1)

where for $n\in\mathbb{Z}^{d}$ we denote by $|n|$ the $\ell^{2}$-norm.

######  Remark 1.2.

According to a result from Khinchin, for each $\tau>\frac{d(d+1)}{2}$ the set
of admissible matrices with parameter $\tau$ is of full Lebesgue measure.

The operator $-\operatorname{div}(\mathrm{A}\nabla)$ has discrete spectrum,
with eigenvalues indexed by the points $n\in\mathbb{Z}^{d}$:

$-\operatorname{div}(\mathrm{A}\nabla)\operatorname{e}^{iny}=\lambda_{n}^{2}\operatorname{e}^{iny}\,,\quad\text{with}\quad\lambda_{n}^{2}:=n^{\intercal}\mathrm{A}n=\sum_{i,j=1}^{d}n_{i}n_{j}\mathrm{A}_{i,j}\,.$

Since $\mathrm{A}$ is symmetric and positive definite, there exist $c,C>0$
such that for all $n\in\mathbb{Z}^{d}$,

$c|n|^{2}\leq\lambda_{n}^{2}\leq C|n|^{2}\,,\quad\text{where we recall
that}\quad|n|^{2}:=\sum_{i=1}^{d}n_{i}^{2}\,.$ (1.2)

The four-waves resonant function is

$\displaystyle\Omega(\vec{n})$
$\displaystyle=\lambda_{n_{1}}^{2}-\lambda_{n_{2}}^{2}+\lambda_{n_{3}}^{2}-\lambda_{n}^{2}\,.$
Under the zero momentum condition $n=n_{1}-n_{2}+n_{3}$ it factorizes to
$\displaystyle\Omega(\vec{n})$
$\displaystyle=(n_{1}-n_{2})^{\intercal}\mathrm{A}(n_{2}-n_{3})\,.$

This factorization motivates Definition 1.1 of the Diophantine admissibility
condition for the matrix $\mathrm{A}$. Firstly, it eliminates the exact
resonances, as proved in Lemma 2.6 below. Secondly, it introduces quasi-
resonant interactions but provides a quantitative polynomial decay of the
resonant function. This gives hope the exploit the time-oscillations in
$H^{s}(\mathbb{T}^{d})$ despite the presence of small divisors, which
appearing in the normal forms method used to convert time-oscillations into
decay in time.

Note that rectangular tori are not admissible. If the vector
$(\mathrm{A}_{i,j})_{i,j}$ is diophantine then $\mathrm{A}$ is admissible, but
this is not an equivalence. We refer to the introduction of [1] for more
details on the admissibility condition.

We give the precise definitions of the norms in Section 2.2 below, but at this
stage we stress that the $S$-norm, that we will use below in the statement of
one of the main theorems, contains $s$ derivatives in the compact direction
$\mathbb{T}^{d}$ and $s+\sigma$ in the $\mathbb{R}$ direction, for $\sigma>0$
arbitrarily small (which is an improvement compared to [24]). Other spaces
that we use in the statements are the Sobolev space
$H^{s}(\mathbb{R}\times\mathbb{T}^{d})$ and the space $h^{s}(\mathbb{T}^{d})$,
which is equivalent to the homogeneous Sobolev space
$\dot{H}^{s}(\mathbb{T}^{d})$ as will be proved also in Section 2.2. We also
anticipate that for $\tau>0$ and $c(d)\in(0,2]$ given in Lemma 2.1, the
regularity threshold needed for our results is

$s_{0}(\tau,d):=\frac{d}{2}+\frac{4\tau}{c(d)}\,.$ (1.3)

We are now ready to state the main theorems.

######  Theorem 1.3.

Let $d\geq 2$ and $\mathrm{A}$ be admissible with parameter $\tau$. Given
$s>s_{0}(\tau,d)$, there exist $C>0$ and $\varepsilon_{\ast}>0$ such that for
all $\varepsilon\in(0,\varepsilon_{\ast})$, if

$\|U(0)\|_{S}\leq\varepsilon\,,$

then the solution $U$ to (A-NLS) with Cauchy data $U(0)$ exists globally in
$C(\mathbb{R};S)$. Moreover, for all $t\in\mathbb{R}$,

$\|U(t)\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}\leq
C\varepsilon\,,\quad\|U(t)\|_{L^{\infty}h^{s}(\mathbb{R}\times\mathbb{T}^{d})}\leq
C(1+|t|)^{-\frac{1}{2}}\varepsilon\,.$ (1.4)

Note in particular that (1.4) shows that the solution preserves the dispersion
time rate of $(1+|t|)^{-\frac{1}{2}}$ inherited from the absence of boundaries
in the $\mathbb{R}$ direction. The proof of Theorem 1.3 is based on the
dynamics of an effective system presented below in (3.4) and (3.9), to which
the small amplitude solutions undergo modified scattering. The effective
system does not only contain the resonant interactions (which are trivial when
$\mathrm{A}$ is admissible), but also some specific quasi-resonant
interactions described in (3.6).

######  Theorem 1.4 (Modified scattering).

Under the same assumptions as in Theorem 1.3, we have that for all $U(0)$ with

$\|U(0)\|_{S}\leq\varepsilon\,,$

there exists a global solution $G\in C(\mathbb{R};S)$ to the system (3.9) such
that $\|G(0)\|_{H^{s}}\lesssim\varepsilon$ and

$\lim_{t\to+\infty}\|\operatorname{e}^{-it\Delta}U(t)-G(t)\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}=0\,.$

We make some remarks.

######  Remark 1.5.

Infinite energy cascades are excluded, in sharp contrast to the case of the
square torus studied in [24]. To the best of the authors’ knowledge, Theorem
1.3 is the first result that shows that the dynamics in infinite time is
completely different on Diophantine tori from that on the square torus. This
indicates that the growth mechanism established in [24] depends on the
periodic boundary conditions and is somehow unstable.

######  Remark 1.6.

The $Z$-norm, that will be defined in Section 2.2 and used for the proofs of
the theorems stated above, is the strongest norm preserved by the effective
system. Unlike the case of the square torus [24] this norm is not based on the
energy space but on $H^{s}(\mathbb{T}^{d})$, for $s>\frac{d}{2}$. As a result,
we no longer have low regularity problems and our results cover all the
dimensions $d\geq 2$.

######  Remark 1.7.

The case when $d=1$, which is also covered by the proof, is less interesting
since there is no quasi-resonant resonant interactions: there is modified
scattering to the (trivial) resonant system (3.5), as in the case with the
convolution potential [19].

######  Remark 1.8.

The regularity threshold $s>s_{0}(\tau,d)=\frac{d}{2}+\frac{4\tau}{c(d)}$
depends on the Diophantine properties of the dispersion relation (through the
parameter $\tau$) and on the dimension $d$ (through the constant $c(d)$
related to the frequency separation property given in Lemma 2.1). Since
$c(d)\leq 2$ we have

$s_{0}(\tau,d)\geq\frac{d}{2}+2\tau\,.$

One can expect to save some of the $\frac{d}{2}$-loss by using the Strichartz
estimates from [6]. However, it is not clear how to avoid the loss $2\tau$ due
to the quasi-resonant interactions. Since this loss is dominant 555In order to
have a generic condition we need to suppose $2\tau>d(d+1)$, we prefer not to
use the machinery of Strichartz estimates. In the unbounded direction
$\mathbb{R}$, we are able to make the loss of regularity arbitrarily small.

###  1.4. Idea of the proof

The proof is inspired by the analysis developed in [24] and also in [29], with
two important differences.

$-$ Firstly, the analysis of the time non-resonant interactions is more
complicated due to small divisor problems. For some quasi-resonant
interactions, we can still perform a Poincaré–Dulac normal form reduction to
obtain an integrable decay in time, thanks to a frequency separation property
based on a cluster decomposition recalled in Lemma 2.1. The remaining quasi-
resonant interactions, which cannot be treated with a normal form method,
contribute to the effective system. These non-resonant contributions are
precisely the  high $\times$ low $\to$ high frequency interactions, where the
incoming high frequency and the outgoing frequency are in the same dyadic
cluster. We give a precise definition of the effective system in (3.6).

$-$ Secondly, we propose a general strategy leveraging dispersive decay
alongside the stability of Sobolev norms to prove modified scattering.
Specifically, we make use of the natural dispersive estimate claimed in
Theorem 1.3 to simplify the analysis of the space non-resonant interactions,
and to propose a concise self-contained proof. This allows us to prove the
result in any dimension, and to have an arbiltrarilly small loss of
derivatives in the $\mathbb{R}$-direction compared to the strong norm.

###  1.5. Organization of the paper

After introducing the Hamiltonian system in the transverse direction and
exploiting the frequency separation property in Section 2, we analyze
nonlinear interactions in Section 3, establishing bounds with decay in time.
Finally, in Section 4, we establish a priori bounds for the time-dependent
norm, as presented in Proposition 4.2. These bounds yield the global existence
result and the dispersive decay (1.4) asserted in Theorem 1.3. Additionally,
we prove Theorem 1.4 and deduce the boundedness of the $H^{s}$-norm, as also
claimed in Theorem 1.3.

###  1.6. Notations

For $F\in L^{2}(\mathbb{R}\times\mathbb{T}^{d})$, we define the (partial)
Fourier transform: for $y\in\mathbb{T}^{d}$,

$\displaystyle\xi\in\mathbb{R}\quad\mapsto\quad\widehat{F}(\xi,y)$
$\displaystyle=(2\pi)^{-1}\int_{\mathbb{R}}\operatorname{e}^{-i\xi
x}F(x,y)dx\,$ and, for $x\in\mathbb{R}$, $\displaystyle
n\in\mathbb{Z}^{d}\quad\mapsto\quad F_{n}(x)$
$\displaystyle=(2\pi)^{-d}\int_{\mathbb{T}^{d}}\operatorname{e}^{-in\cdot
y}F(x,y)dy\,.$

The (full) spatial Fourier transform of $F$ is

$(\xi,n)\in\mathbb{R}\times\mathbb{Z}^{d}\ \mapsto\
(\mathcal{F}F)(\xi,n):=(2\pi)^{-(d+1)}\int_{\mathbb{R}\times\mathbb{T}^{d}}\operatorname{e}^{-i(\xi
x+n\cdot y)}F(x,y)dxdy\,.$ (1.5)

Following [24], we write for $(\xi,n)\in\mathbb{R}\times\mathbb{T}^{d}$

$\widehat{F}_{n}(\xi):=(\mathcal{F}F)(\xi,n)\,,\quad\text{so
that}\quad\widehat{F}(\xi,y)=\sum_{n\in\mathbb{Z}^{d}}\widehat{F}_{n}(\xi)\operatorname{e}^{in\cdot
y}\,.$

Given three incoming waves $(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}$, we
define the decreasing rearrangement of $(|n_{1}|,|n_{2}|,|n_{3}|)$ by

$n_{1}^{\ast}\geq n_{2}^{\ast}\geq n_{3}^{\ast}\,.$

We denote by $n=n_{1}-n_{2}+n_{3}$ the outgoing wave and by
$\vec{n}=(n_{1},n_{2},n_{3},n)\in(\mathbb{Z}^{d})^{4}$ the four interacting
waves.

### Acknowledgments

During the preparation of this work the first author benefited from the
support of the Centre Henri Lebesgue ANR-11-LABX-0020-0, the region Pays de la
Loire through the project MasCan and the ANR project KEN ANR-22-CE40-0016. The
second author was supported by the NSF grant DMS-2052651 and the Simons
Foundation Collaboration Grant on Wave Turbulence.

##  2\. Preparations

###  2.1. Clustering of the eigenvalues

Let us recall the frequency separation property, which will play a key role in
the present work. This property was proved by Bourgain [4] on $\mathbb{T}^{d}$
for the flat Laplacian (when $\mathrm{A}=\operatorname{Id}$), and generalized
by Berti and Maspero [2] for all $\mathrm{A}$ symmetric and positive definite
(or, equivalently, for any flat torus).

######  Lemma 2.1 (Frequency separation, Theorem 2.1 in [2]).

There exist $c(d)\in(0,2]$ and $C(A,d)\geq 2$, and a partition of
$\mathbb{Z}^{d}$ into disjoint clusters $(\mathscr{C}_{\alpha})_{\alpha\geq
1}$ satisfying the following:

  1. (1)

The first cluster contains the origin $0\in\mathscr{C}_{0}$ and is bounded:

$\max_{n\in\mathscr{C}_{0}}|n|\leq C(A,d)\,.$

  2. (2)

The other clusters are dyadic: for all $\alpha\geq 1$,

$\max_{n\in\mathscr{C}_{\alpha}}|n|\leq
2\min_{n\in\mathscr{C}_{\alpha}}|n|\,.$

  3. (3)

For all
$(n_{1},n_{2})\in\mathscr{C}_{\alpha_{1}}\times\mathscr{C}_{\alpha_{2}}$ with
$\alpha_{1}\neq\alpha_{2}$,

$|n_{1}-n_{2}|+|\lambda_{n_{1}}^{2}-\lambda_{n_{2}}^{2}|>(|n_{1}|+|n_{2}|)^{c(d)}\,.$
(2.1)

The last condition is a separation property that plays a key role in the
analysis, to control the quasi-invariant interactions, and we refer to Lemma
2.10.

###  2.2. Norms

We fix the regularity $s>s_{0}$ (see (1.3)) as in the statement of Theorem
1.3. In our analysis, the relevant Sobolev norms in the compact direction are
based on the cluster decomposition. We prove in Lemma 2.9 that they are
preserved by the effective system defined in (3.9).

######  Definition 2.2 (Sobolev norms based on the clusters).

We define

$K_{0}:=1\,,\quad
K_{\alpha}:=\min_{n\in\mathscr{C}_{\alpha}}|n|\quad\text{for}\ \alpha\geq
1\,,$

and $\pi_{\alpha}$ is the orthogonal projector onto $\operatorname{span}\
\\{e^{in\cdot y}\\}_{n\in\mathscr{C}_{\alpha}}$: for a function
$u\in\ell^{2}(\mathbb{T}^{d})$,

$\pi_{\alpha}u(y)=\sum_{n\in\mathscr{C}_{\alpha}}u_{n}\operatorname{e}^{in\cdot
y}\,,\quad
u_{n}=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}\operatorname{e}^{-in\cdot
y}u(y)dy\,.$

For $s\in\mathbb{R}$, we define the Sobolev norms for functions on
$\mathbb{T}^{d}$ and on $\mathbb{R}\times\mathbb{T}^{d}$ by

$\displaystyle\|u\|_{h^{s}(\mathbb{T}^{d})}$
$\displaystyle:=\big{(}\sum_{\alpha\geq
0}K_{\alpha}^{2s}\|\pi_{\alpha}u\|_{\ell^{2}(\mathbb{T}^{d})}^{2}\big{)}^{\frac{1}{2}}\,,$
(2.2) $\displaystyle\|U\|_{H_{x,y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}$
$\displaystyle:=(\|U\|_{L_{x}^{2}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}^{2}+\|U\|_{H_{x}^{s}\ell_{y}^{2}(\mathbb{R}\times\mathbb{T}^{d})}^{2})^{\frac{1}{2}}\,.$
(2.3)

######  Remark 2.3.

Since, for all $n\in\mathbb{Z}^{d}\setminus\\{0\\}$ and $\alpha\geq 1$ with
$n\in\mathscr{C}_{\alpha}$, we have

$K_{\alpha}\leq|n|\leq 2K_{\alpha}\,.$

Therefore, the Sobolev norms (2.2) are equivalent to the standard Sobolev
ones.

The motivation for considering this equivalent Sobolev norm
$h^{s}(\mathbb{T}^{d})$ is that it does not capture the possible transfers of
energy within a same cluster. Hence, we expect this norm to stay bounded under
the effective dynamics.

######  Definition 2.4 (Strong, weak and time-dependent norms).

Let $s>s_{0}$ (see (1.3)) and $\sigma>0$ arbitrarily small. The strong norm is

$\displaystyle\|F\|_{S}$
$\displaystyle:=\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}F\|_{H_{x,y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}+\|xF\|_{L_{x}^{2}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\,,$
(2.4) and the weak norm is $\displaystyle\|F\|_{Z}$
$\displaystyle:=\sup_{\xi\in\mathbb{R}}\|\widehat{F}(\xi)\|_{h_{y}^{s}(\mathbb{T}^{d})}\,.$
(2.5) For $\delta>0$ small enough, the space-time norm is
$\displaystyle\|F\|_{X_{T}}$ $\displaystyle:=\sup_{0\leq t\leq
T}\big{(}\|F(t)\|_{Z}+\langle t\rangle^{-\delta}\|F(t)\|_{S}+\langle
t\rangle^{1-\delta}\|\partial_{t}F(t)\|_{S}\big{)}\,.$ (2.6)

######  Remark 2.5.

Observe that

$\|F\|_{Z}=\sup_{\xi\in\mathbb{R}}\|\widehat{F}(\xi)\|_{h_{y}^{s}}\lesssim\|\langle
x\rangle F\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|F\|_{S}\,.$

###  2.3. The Hamiltonian system in the transverse direction

Let $u\in C^{1}(\mathbb{R};\mathbb{T}^{d})$ be a solution on $\mathbb{T}^{d}$
of

$i\partial_{t}u+\mathrm{div}(\mathrm{A}\nabla)u=|u|^{2}u\,,$

with Fourier coefficients

$u_{n}(t)=(2\pi)^{-d}\int_{\mathbb{T}^{d}}\operatorname{e}^{-in\cdot
y}u(y)dy\,,\quad n\in\mathbb{Z}^{d}\,.$

The vector $a\in C^{1}(\mathbb{R};\mathbb{C}^{\mathbb{Z}^{d}})$ defined by

$a_{n}(t):=\operatorname{e}^{it\lambda_{n}^{2}}u_{n}(t)\,,\quad
n\in\mathbb{Z}^{d}\,,$

is solution to the interaction system

$i\partial_{t}a_{n}=\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}a_{n_{1}}\overline{a_{n_{2}}}a_{n_{3}}\,,\quad
n\in\mathbb{Z}^{d}\,,$ (2.7)

where we recall that for
$\vec{n}=(n_{1},n_{2},n_{3},n)\in(\mathbb{Z}^{d})^{4}$, the resonant function
reads

$\Omega(\vec{n})=\lambda_{n_{1}}^{2}-\lambda_{n_{2}}^{2}+\lambda_{n_{3}}^{2}-\lambda_{n}^{2}\,.$

A subsystem of (2.7) is the resonant system:

$i\partial_{t}a_{n}=\sum_{(n_{1},n_{2},n_{3})\,:\,\vec{n}\in\Gamma_{0}}a_{n_{1}}\overline{a_{n_{2}}}a_{n_{3}}\,,$
(2.8)

where we define the 4 waves resonant set by

$\Gamma_{0}=\big{\\{}\vec{n}\in(\mathbb{Z}^{d})^{4}\quad:\quad
n_{1}-n_{2}+n_{3}-n=\Omega(\vec{n})=0\big{\\}}\,.$ (2.9)

Under the zero momentum condition $n_{1}-n_{2}+n_{3}=n$ the resonant function
factorizes:

$\Omega(\vec{n})=(n_{1}-n)^{\intercal}\mathrm{A}(n_{2}-n_{3})\,.$

######  Lemma 2.6 (Four-waves resonances).

Under the Diophantine admissibility condition (1.1) on $\mathrm{A}$, the
nonlinear term on the four-waves resonant set is trivial:

$\sum_{\vec{n}=(n_{1},n_{2},n_{3},n)\in\Gamma_{0}}a_{n_{1}}\overline{a_{n_{2}}}a_{n_{3}}\overline{a_{n}}=2(\sum_{n\in\mathbb{Z}^{d}}|a_{n}|^{2})^{2}-\sum_{n\in\mathbb{Z}^{d}}|a_{n}|^{4}\,.$
(2.10)

###### Proof.

If we factorize the resonant function as above (using the zero-momentum
condition) we obtain from the Diophantine admissibility condition (1.1) on
$\mathrm{A}$ that

$(n_{1}-n_{2})^{\intercal}\mathrm{A}(n_{2}-n_{3})=0\quad\implies
n_{2}=n_{1}\quad\text{or}\quad n_{2}=n_{3}\,.$

We conclude from the zero momentum condition that

$\Gamma_{0}=\big{\\{}\vec{n}=(n_{1},n_{2},n_{3},n)\in\mathbb{Z}^{4}\ :\
\\{n_{1},n_{3}\\}=\\{n,n_{2}\\}\big{\\}}\,.$

This proves Lemma 2.6. ∎

Let us now define the effective quasi-resonant interactions. They consist in
the interactions of type high $\times$ low $\to$ high, where the two high
frequencies are in a same dyadic cluster, and where the resonant function is
smaller than 1. We denote

$\alpha_{0}:=\min\\{\alpha\geq 1\ :\ K_{\alpha}\geq
10^{10}C(\mathrm{A},d)\\}\,,$ (2.11)

where we recall that $K_{\alpha}:=\min_{n\in\mathscr{C}_{\alpha}}|n|$ when
$\alpha\geq 1$. We refer to high frequency clusters when
$\alpha\geq\alpha_{0}$.

######  Definition 2.7 (Effective quasi-resonant interactions).

For $\alpha\geq\alpha_{0}$ as above and for all $n\in\mathscr{C}_{\alpha}$, we
let

$\Lambda^{(1)}(n):=\big{\\{}(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}\ :\
n_{1}-n_{2}+n_{3}=n\,,\quad|\Omega(\vec{n})|<1\,,\\\
n_{1}\in\mathscr{C}_{\alpha}\,,\quad|n_{2}|,|n_{3}|<10^{-5}K_{\alpha}\big{\\}}$

and

$\Lambda^{(3)}(n):=\big{\\{}(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}\ :\
n_{1}-n_{2}+n_{3}=n\,,\quad|\Omega(\vec{n})|<1\,,\\\
n_{3}\in\mathscr{C}_{\alpha}\,,\quad|n_{2}|,|n_{1}|<10^{-5}K_{\alpha}\big{\\}}\,.$

The effective quasi-resonant interaction equation is as follows:

  * •

If $n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha}$ then

$i\partial_{t}a_{n}=2\sum_{(n_{1},n_{2},n_{3})\in\Lambda^{(1)}(n)}\operatorname{e}^{it\Omega(\vec{n})}a_{n_{1}}\overline{a_{n_{2}}}a_{n_{3}}\,.$

  * •

Otherwise,

$i\partial_{t}a_{n}=0\,.$

######  Remark 2.8.

In the definition of the effective quasi-resonant interaction we implicitly
used that $n_{1}$ and $n_{3}$ play symmetric roles, hence the factor 2
(instead of summing over $\Lambda^{(1)}(n)$ and over $\Lambda^{(3)}(n)$).

We now prove that the effective system preserves the high-frequency super
actions based on the dyadic clusters. Recall that high-frequencies refer to
the clusters $\mathscr{C}_{\alpha}$ with $\alpha\geq\alpha_{0}$ defined in
(2.11). Note that if $(n_{1},n_{2},n_{3})\in\Lambda^{(1)}(n)$ then
$n_{2},n_{3}\notin\mathscr{C}_{\alpha}$.

######  Lemma 2.9.

Suppose that $a\in C(\mathbb{R};\ell^{2}(\mathbb{T}^{d}))$

$a(t,y)=\sum_{n\in\mathbb{Z}^{d}}a_{n}(t)\operatorname{e}^{in\cdot y}\,,\quad
y\in\mathbb{T}^{d}\,,$

where $(a_{n}(t))_{n\in\mathbb{Z}^{d}}$ is solution to the system in
Definition (2.7). Then, for all $\alpha\geq\alpha_{0}$, we have

$\frac{d}{dt}\|\pi_{\alpha}a(t)\|_{\ell^{2}(\mathbb{T}^{d})}^{2}=0\,.$

###### Proof.

Let $\alpha\geq\alpha_{0}$. We have

$\displaystyle\frac{1}{4}\frac{d}{dt}\|\pi_{\alpha}a(t)\|_{\ell^{2}(\mathbb{T}^{d})}^{2}$
$\displaystyle=\operatorname{Im}\big{(}\sum_{n\in\mathscr{C}_{\alpha}}\overline{a_{n}}\sum_{(n_{1},n_{2},n_{3})\in\Lambda^{(1)}(\vec{n})}\mathbf{1}_{|\Omega(\vec{n})|<1}\operatorname{e}^{it\Omega(\vec{n})}a_{n_{1}}\overline{a_{n_{2}}}a_{n_{3}}\big{)}$
$\displaystyle=\operatorname{Im}\big{(}\sum_{n,n_{1}\in\mathscr{C}_{\alpha}}\sum_{|n_{2}|,|n_{3}|\leq
10^{-5}K_{\alpha}}\mathbf{1}_{\begin{subarray}{c}|\Omega(\vec{n})|<1\\\
n_{1}+n_{3}=n+n_{2}\end{subarray}}\operatorname{e}^{it\Omega(\vec{n})}\overline{a_{n}}a_{n_{1}}\overline{a_{n_{2}}}a_{n_{3}}\big{)}$
$\displaystyle=0\,.$

To see this cancellation, one can take the complex conjugate and interchange
the roles of $n$ with $n_{1}$ (resp. $n_{2}$ with $n_{3}$). ∎

In the following Lemma, we use the frequency separation property written in
Lemma 2.1 in order to control the small divisors losses that will appear in
the Poincaré–Dulac normal form reduction that will be performed in Section
3.2.3. We defined the set $\Lambda^{(1)}(n)$ and $\Lambda^{(2)}(n)$ in the
above Definition 2.7, and the high-frequency threshold $\alpha_{0}$ in (2.11).

######  Lemma 2.10.

Let $s\geq 0$. There exists $C>0$ (depending on $d$, $s$ and $\mathrm{A}$)
such that for all $\alpha\geq\alpha_{0}$, $n\in\mathscr{C}_{\alpha}$ and

$(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}\setminus(\Lambda^{(1)}(n)\cup\Lambda^{(3)}(n)),\quad\text{with}\quad
n_{1}-n_{2}+n_{3}=n\,,\quad\Omega(\vec{n})\neq 0\,,$

we have

$\frac{1}{|\Omega(\vec{n})|}K_{\alpha}^{s}\leq
C(n_{1}^{\ast})^{s}(n_{2}^{\ast})^{\frac{4\tau}{c(d)}}\,,$

where we recall that $n_{1}^{\ast}\geq n_{2}^{\ast}\geq n_{3}^{\ast}$ is the
decreasing rearrangement of $|n_{1}|,|n_{2}|,|n_{3}|$.

###### Proof.

Let $\alpha\geq\alpha_{0}$ and $\vec{n}=(n_{1},n_{2},n_{3},n)$, with
$n\in\mathscr{C}_{\alpha}$ and $n_{1}-n_{2}+n_{3}=n$ (zero momentum
condition). In particular,

$K_{\alpha}\leq|n|\leq 2K_{\alpha}\,,\quad\text{and}\quad
K_{\alpha}\leq|n|\leq 3n_{1}^{\ast}\,.$

When $|\Omega(\vec{n})|\geq 1$, the desired bound follows from the above
inequalities between the indices. According to the admissibility condition we
can suppose that

$1<\frac{1}{|\Omega(\vec{n})|}\lesssim(n_{1}^{\ast})^{2\tau}\,.$

Moreover, we suppose without loss of generality that $|n_{1}|\geq|n_{3}|$
(they play symmetric roles). Hence, $|n_{1}|\geq n_{2}^{\ast}$. We separate
three cases.

$\bullet$ Case 1: $|n_{1}|=n_{2}^{\ast}$. In this case, we have
$n_{1}^{\ast}=|n_{2}|$ and the condition $|\Omega(\vec{n})|<1$ together with
the coercive bound on the eigenvalues (1.2) yield

$(n_{1}^{\ast})^{2}\lesssim_{\mathrm{A}}\lambda_{n_{2}}^{2}+\lambda_{n}^{2}<|\Omega(\vec{n})|+\lambda_{n_{1}}^{2}+\lambda_{n_{3}}^{2}\lesssim_{\mathrm{A}}(n_{2}^{\ast})^{2}\,,$

for some implicit constants only depending on $\mathrm{A}$. Therefore,

$\frac{1}{|\Omega(\vec{n})|}K_{\alpha}^{s}\leq(n_{1}^{\ast})^{s}(n_{1}^{\ast})^{2\tau}\lesssim_{\mathrm{A}}(n_{1}^{\ast})^{s}(n_{2}^{\ast})^{2\tau}\,$

which is conclusive (we recall that $c(d)\leq 2$, so that
$2\tau\leq\frac{4\tau}{c(d)}$).

$\bullet$ Case 2: $|n_{1}|=n_{1}^{\ast}$ and
$n_{1}\notin\mathscr{C}_{\alpha}$. The frequency separation property (2.1)
gives

$|\lambda_{n}^{2}-\lambda_{n_{1}}^{2}|+|n_{2}-n_{1}|\geq(n_{1}^{\ast})^{c(d)}\,.$

We deduce from the condition $|\Omega(\vec{n})|<1$ and from the zero momentum
condition that

$|\lambda_{n}^{2}-\lambda_{n_{3}}^{2}|+|n_{2}-n_{3}|\geq(n_{1}^{\ast})^{c(d)}-1\geq\frac{1}{2}(n_{1}^{\ast})^{c(d)}\,,$

which in turn implies that

$n_{2}^{\ast}=\max(|n_{2}|,|n_{3}|)\gtrsim_{\mathrm{A}}(n_{1}^{\ast})^{\frac{c(d)}{2}}\,.$

Hence,

$\frac{1}{|\Omega(\vec{n})|}K_{\alpha}^{s}\lesssim_{\mathrm{A}}(n_{1}^{\ast})^{s}(n_{2}^{\ast})^{\frac{4\tau}{c(d)}}\,,$

which is conclusive.

$\bullet$ Case 3: $|n_{1}|=n_{1}^{\ast}$ and $n_{1}\in\mathscr{C}_{\alpha}$.
In particular, $n_{1}^{\ast}\geq|n_{1}|\geq K_{\alpha}$. Moreover, since
$(n_{1},n_{2},n_{3})\leavevmode\nobreak\ \notin\leavevmode\nobreak\
\Lambda^{(1)}(n)$ we have from the Definition 2.7 that

$10^{-5}K_{\alpha}\leq\max(|n_{2}|,|n_{3}|)\,.$

Therefore, $K_{\alpha}\lesssim n_{2}^{\ast}$ and we conclude that

$\frac{1}{|\Omega(\vec{n})|}K_{\alpha}^{s}\lesssim(n_{1}^{\ast})^{2\tau_{\ast}}(n_{2}^{\ast})^{s}\,.$

This completes the proof of Lemma 2.10. ∎

###  2.4. Dispersion in the unbounded direction

Now that we have analyzed the key interaction of the system in the compact
direction, we turn to the dispersive properties of the one-dimensional
Schrödinger free evolution in the unbounded direction $\mathbb{R}$. In the
following lemma we recall the dispersive estimate and its consequence in our
context.

######  Lemma 2.11.

There exists $C>0$ such that for all $f\in L^{2}(\mathbb{R})\cap
x^{-1}L^{2}(\mathbb{R})$ and $t>0$,

$\|\operatorname{e}^{it\partial_{x}^{2}}f(x)-\frac{\operatorname{e}^{i\frac{x^{2}}{4t}}}{(4i\pi
t)^{\frac{1}{2}}}\widehat{f}(\frac{x}{2t})\|_{L_{x}^{\infty}(\mathbb{R})}\leq
Ct^{-\frac{3}{4}}\|xf\|_{L_{x}^{2}(\mathbb{R})}\,.$ (2.12)

Considering the profile $F(t)=\operatorname{e}^{-it\Delta}U(t)$, we have that
for all $T\geq 1$, $t\in[1,T]$,

$t^{\frac{1}{2}}\|U(t)\|_{L_{x}^{\infty}h_{y}^{s}}\lesssim\|F\|_{Z}+t^{-\frac{1}{4}}\|F\|_{S}\lesssim\|F\|_{X_{T}}\,.$
(2.13)

######  Remark 2.12.

The above estimate is a key ingredient in our analysis, especially to obtain
the trilinear estimate claimed in Lemma 2.13. Note that such an estimate is
not available in the case of the square torus, where the space-time norm
$X_{T}$ provides the dispersive estimate for the
$L_{x}^{\infty}h_{y}^{1}(\mathbb{R}\times\mathbb{T}^{d})$-norm only (see [24,
Theorem 1.1]).

###### Proof.

The dispersive estimate (2.12) on $\mathbb{R}$ is standard (see for instance
Hayachi–Naumkin [26]), and follows from the expression of the semi-group
$\operatorname{e}^{it\partial_{x}^{2}}$. We write a short proof in Appendix A.
Let us now prove (2.13): for all $(x,y)\in\mathbb{R}\times\mathbb{T}^{d}$, we
expand

$U(t,x,y)=\sum_{n\in\mathbb{Z}^{d}}\operatorname{e}^{iy\cdot
n-it\lambda_{n}^{2}}\operatorname{e}^{it\partial_{x}^{2}}F_{n}(x)\,.$

Using the equivalence of the $h^{s}$-norms with the standard Sobolev norms on
$\mathbb{T}^{d}$ we have

$\|U(t)\|_{L^{\infty}h^{s}(\mathbb{R}\times\mathbb{T}^{d})}\lesssim_{s}\big{\|}\big{(}\sum_{n\in\mathbb{Z}^{d}}\langle
n\rangle^{2s}|\operatorname{e}^{it\partial_{x}^{2}}F_{n}(x)|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L^{\infty}(\mathbb{R})}\,.$

According to the dispersive estimate (2.12), we have that for fixed
$x\in\mathbb{R}$, $n\in\mathbb{Z}^{d}$ and $t>0$,

$|\operatorname{e}^{it\partial_{x}^{2}}F_{n}(x)|^{2}\lesssim
t^{-1}|\widehat{F}_{n}(\frac{x}{2t})|^{2}+t^{-\frac{3}{4}}\|xF_{n}\|_{L_{x}^{2}(\mathbb{R})}^{2}\,.$

When summing over $n$, the first contribution is bounded by

$t^{-1}\sum_{n\in\mathbb{Z}^{d}}\langle
n\rangle^{2s}|\widehat{F}_{n}(\frac{x}{2t})|^{2}=t^{-1}\|\widehat{F}(\frac{x}{2t})\|_{h^{s}(\mathbb{T}^{d})}^{2}\leq
t^{-1}\sup_{\xi\in\mathbb{R}}\|\widehat{F}(\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2}\,.$

For the second contribution, we have

$\displaystyle\sum_{n}\langle
n\rangle^{2s}\|xF_{n}\|_{L_{x}^{2}(\mathbb{R})}^{2}$
$\displaystyle=\|xF\|_{L_{x}^{2}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\,.$

This proves that for all $t>0$,

$\|U(t)\|_{L^{\infty}h^{s}(\mathbb{R}\times\mathbb{T}^{d})}\lesssim
t^{-\frac{1}{2}}\sup_{\xi\in\mathbb{R}}\
\|\widehat{F}(\xi)\|_{h^{s}(\mathbb{T}^{d})}+t^{-\frac{1}{2}-\frac{1}{4}}\|xF\|_{L^{2}h^{s}(\mathbb{R}\times\mathbb{T}^{d})}\,,$
(2.14)

which gives (2.13) provided $\delta<\frac{1}{4}$ in the definition of the
$X_{T}$-norm. ∎

###  2.5. Profile equation

We detail the standard preparation in which we pull-back the nonlinear
solution $U$ by the linear flow:

$\displaystyle U(t,x,y)$
$\displaystyle=\operatorname{e}^{it\Delta}F(t,x,y)\,.$

The profile $F(t)$ is solution to the interaction equation

$i\partial_{t}F(t)=\operatorname{e}^{-it\Delta}(\operatorname{e}^{it\Delta}F(t)\operatorname{e}^{-it\Delta}\overline{F(t)}\operatorname{e}^{it\Delta}F(t))\,.$

We write

$i\partial_{t}F(t)=\mathcal{N}^{t}[F(t),F(t),F(t)]\,,$ (2.15)

where the trilinear form $\mathcal{N}^{t}$ on
$(H^{s}(\mathbb{R}\times\mathbb{T}^{d}))^{3}$ is

$\mathcal{N}^{t}[F,G,H]:=\operatorname{e}^{-it\Delta}(\operatorname{e}^{it\Delta}F\operatorname{e}^{-it\Delta}\overline{G}\operatorname{e}^{it\Delta}H)\,.$

The full spatial Fourier transform (as defined in (1.5)) of the trilinear
interaction at $(\xi,n)\in\mathbb{R}\times\mathbb{T}^{d}$ reads

$\mathcal{F}\mathcal{N}^{t}[F,G,H](\xi,n):=\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}\\\
\int_{\mathbb{R}^{2}}\operatorname{e}^{2it(\eta-\xi)(\sigma-\eta)}\widehat{F}_{n_{1}}(\xi-\eta+\sigma)\overline{\widehat{G}_{n_{2}}}(\sigma)\widehat{H}_{n_{3}}(\eta)d\eta
d\sigma\,.$

where the sums run over $(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}$. Once
again we denoted

$\vec{n}=(n_{1},n_{2},n_{3},n)\,,\quad\Omega(\vec{n}):=\lambda_{n_{1}}^{2}-\lambda_{n_{2}}^{2}+\lambda_{n_{3}}^{2}-\lambda_{n}^{2}\,.$

Changing variables in the integrals over $(\eta,\sigma)\in\mathbb{R}^{2}$ we
obtain

$\mathcal{F}\mathcal{N}^{t}[F,G,H](\xi,n):=\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}\\\
\int_{\mathbb{R}^{2}}\operatorname{e}^{it2\eta\sigma}\widehat{F}_{n_{1}}(\xi-\sigma)\overline{\widehat{G}_{n_{2}}}(\xi-\eta-\sigma)\widehat{H}_{n_{3}}(\xi-\eta)d\eta
d\sigma\,.$ (2.16)

###  2.6. A priori estimates for the strong norm

We conclude this preparatory section with the proof of an a priori trilinear
bound for the strong norm $S$, which is a stronger analog of [24, Lemma 2.1].
Indeed, we can readily use the dispersive estimate (2.13), which does not hold
in the case of the square torus, to obtain a control of the
$L_{x,y}^{\infty}(\mathbb{R}\times\mathbb{T}^{d})$-norm by the $X_{T}$-norm
with the optimal decay in $t^{-\frac{1}{2}}$.

######  Lemma 2.13.

Suppose that the profile $F$ is solution to (2.15). Then, for all $T\geq 0$
and $t\in[0,T]$, we have

$\|\mathcal{N}^{t}[F,F,F]\|_{S}\lesssim\langle
t\rangle^{-1+\delta}\|F\|_{X_{T}}^{3}\,.$ (2.17)

###### Proof.

Recall that

$\mathcal{N}^{t}[F]:=\mathcal{N}^{t}[F,F,F]=\operatorname{e}^{it\Delta}(|U(t)|^{2}U(t))\,.$

The proof when $t\in[0,1]$ is easier (we do not need to gain decay in time),
and we only detail the case when $T\geq 1$ and $1\leq t\leq T$. We have from
the product rule of Lemma A.1 and the dispersive bound (2.13) that for
$\sigma\geq 0$,

$\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}\mathcal{N}^{t}[F]\|_{H_{x,y}^{s}}\lesssim\|U(t)\|_{L_{x}^{\infty}h_{y}^{s}}^{2}\|(1-\partial_{x})^{\frac{\sigma}{2}}U(t)\|_{H_{x,y}^{s}}\\\
\lesssim
t^{-1}\|F\|_{X_{T}}^{2}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}F(t)\|_{H_{x,y}^{s}}\,.$

We deduce from the definition (2.6) of the $X_{T}$-norm that

$\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}\mathcal{N}^{t}[F]\|_{H_{x,y}^{s}}\lesssim
t^{-1+\delta}\|F\|_{X_{T}}^{3}\,.$

As for the second component of the $S$-norm we have

$\|x\mathcal{N}^{t}[F]\|_{L_{x}^{2}h_{y}^{s}}=\|\partial_{\xi}\mathcal{F}\mathcal{N}^{t}[F]\|_{L_{x}^{2}h_{y}^{s}}\,.$

For fixed $n\in\mathbb{Z}^{d}$, following [29], we have from the expression
(2.16) that

$\partial_{\xi}\mathcal{F}\mathcal{N}^{t}[F](\xi,n)=\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}\\\
\partial_{\xi}\int_{\mathbb{R}^{2}}\operatorname{e}^{2it\eta\sigma}\widehat{F}_{n_{1}}(\xi-\sigma)\overline{\widehat{F}_{n_{2}}}(\xi-\eta-\sigma)\widehat{F}_{n_{3}}(\xi-\eta)d\eta
d\sigma\,.$

Redistributing the phases, we obtain that

$\|x\mathcal{N}^{t}(F,F,F)\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|\operatorname{e}^{-it\Delta}(|U(t)|^{2}xU(t))\|_{L_{x}^{2}h_{y}^{s}}\,.$

We deduce from the dispersive bound (2.13) that

$\|x\mathcal{N}^{t}(F,F,F)\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|U(t)\|_{L_{x}^{\infty}h_{y}^{s}}^{2}\|xU(t)\|_{L_{x}^{2}h_{y}^{s}}\lesssim
t^{-1+\delta}\|F\|_{X_{T}}^{3}\,,$

which finishes the proof of Lemma 2.13. ∎

##  3\. Nonlinear interactions

Lemma 2.13 gives an a priori estimate on the strong norm $S$ of the solution
by the $X_{T}$-norm of the solution, with the appropriate decay in time. To
close the a priori estimate for the $X_{T}$-norm, it remains to bound the weak
norm $Z$ by an integrable time decay.

To this end, we will analyze the trilinear interaction $\mathcal{N}^{t}$ more
carefully. We first isolate the space-resonant interactions (denoted
$\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}$) and we use space oscillations to
prove that the contribution of the rest
($\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}$) is acceptable.

Then, we further decompose $\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}$ into an
effective Hamiltonian system (denoted $\mathcal{N}_{\mathrm{eff}}^{t}$) in the
transverse direction $\mathbb{T}^{d}$ and a remainder interaction (denoted
$\mathcal{E}^{t}$) oscillating sufficiently fast in time.

###  3.1. Decomposition of the nonlinearity

Following [29], we apply the Plancherel’s identity in (2.16) and use the
explicit expression of the Fourier transform of a Gaussian function to isolate
the space-resonant component of the nonlinearity. We obtain (see also the
proof of [24, Lemma 3.10]) that for fixed $\xi\in\mathbb{R}$, $t\geq 1$ and
$(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}$,

$\displaystyle\int_{\mathbb{R}^{2}}$
$\displaystyle\operatorname{e}^{i2t\eta\sigma}\widehat{F}_{n_{1}}(\xi-\sigma)\overline{\widehat{G}_{n_{2}}}(\xi-\eta-\sigma)\widehat{H}_{n_{3}}(\xi-\eta)d\eta
d\sigma$
$\displaystyle=\frac{\pi}{t}\int_{\mathbb{R}^{3}}F_{n_{1}}(x_{1})\overline{G_{n_{2}}}(x_{2})H_{n_{3}}(x_{3})\operatorname{e}^{-i\xi(x_{1}-x_{2}+x_{3})}\operatorname{e}^{-i[\frac{1}{2t}(x_{1}-x_{2})(x_{2}-x_{3})]}dx_{1}dx_{2}dx_{3}$
$\displaystyle=\frac{\pi}{t}\widehat{F}_{n_{1}}(\xi)\overline{\widehat{G}_{n_{2}}}(\xi)\widehat{H}_{n_{3}}(\xi)+\mathcal{O}^{t}[F_{n_{1}},G_{n_{2}},H_{n_{3}}](\xi)\,,$

where $\mathcal{O}^{t}$ contains the space non-resonant interactions: for
functions $f,g,h\in H_{x}^{s}(\mathbb{R})$,

$\mathcal{O}^{t}[f,g,h](\xi):=\frac{\pi}{t}\int_{\mathbb{R}^{3}}f(x_{1})\overline{g(x_{2})}h(x_{3})\operatorname{e}^{-i\xi(x_{1}-x_{2}+x_{3})}\\\
\big{(}\operatorname{e}^{-i[\frac{1}{2t}(x_{1}-x_{2})(x_{2}-x_{3})]}-1\big{)}dx_{1}dx_{2}dx_{3}\,.$
(3.1)

One can prove that $\mathcal{O}^{t}$ has an improved decay in time under some
assumptions on the space localization of the functions $f,g,h$ (two of them
should be in $\langle x\rangle^{-1}L_{x}^{2}(\mathbb{R})$, say). We detail
this part of the analysis in Section 3.3.

Recalling the expression (2.16), we have that for $t\geq 1$ and
$(\xi,n)\in\mathbb{R}\times\mathbb{Z}^{d}$,

$\mathcal{F}\mathcal{N}^{t}[F,G,H](\xi,n)=\frac{\pi}{t}\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}\widehat{F}_{n_{1}}(\xi)\overline{\widehat{G}_{n_{2}}}(\xi)\widehat{H}_{n_{3}}(\xi)\\\
+\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}\mathcal{O}^{t}[F_{n_{1}},G_{n_{2}},H_{n_{3}}](\xi)\,.$
(3.2)

According to this decomposition we write

$\mathcal{F}\mathcal{N}^{t}[F,G,H](\xi,n)=:\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F,G,H](\xi,n)+\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F,G,H](\xi,n)\,.$
(3.3)

We analyze the term $\mathcal{N}^{t}_{\mathbb{R},\mathrm{res}}$ (which
regroups the space-resonant interactions) in Section 3.2 and the term
$\mathcal{N}^{t}_{\mathbb{R},\mathrm{nr}}$ in Section 3.3, respectively.
First, we further decompose the space-resonant system involving
$\mathcal{N}^{t}_{\mathbb{R},\mathrm{res}}$ into two parts: an effective
system consisting of resonant interactions ($\mathcal{R}^{t}$) and of some
quasi-resonant interactions preserving the cluster of the outgoing wave
($\mathcal{Q}^{t}$), and a remainder ($\mathcal{E}^{t}$):

$\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F,G,H]:=\mathcal{F}\mathcal{R}^{t}[F,G,H]+\mathcal{F}\mathcal{Q}^{t}[F,G,H]+\mathcal{F}\mathcal{E}^{t}[F,G,H]\,.$

The effective system is

$\mathcal{N}_{\mathrm{eff}}^{t}=\mathcal{R}^{t}+\mathcal{Q}^{t}\,.$ (3.4)

Recalling definition 2.7 of $\Lambda^{(1)}(n)$, the different interactions are
defined as follows:

  * •

Resonant interaction:

$\mathcal{F}\mathcal{R}^{t}[F,G,H](\xi,n):=\frac{2\pi}{t}\sum_{n_{1}\in\mathbb{Z}^{d}}\widehat{F}_{n_{1}}(\xi)\overline{\widehat{G}_{n_{1}}}(\xi)\widehat{H}_{n}(\xi)-\frac{\pi}{t}\widehat{F}_{n}(\xi)\overline{\widehat{G}_{n}}(\xi)\widehat{H}_{n}(\xi)\,,$
(3.5)

  * •

Effective quasi-resonant interactions: for
$n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha}$,

$\mathcal{F}\mathcal{Q}^{t}[F,G,H](\xi,n)=\frac{2\pi}{t}\sum_{\begin{subarray}{c}(n_{1},n_{2},n_{3})\in\Lambda^{(1)}(n)\end{subarray}}\operatorname{e}^{it\Omega(\vec{n})}\widehat{F}_{n_{1}}(\xi)\overline{\widehat{G}_{n_{2}}}(\xi)\widehat{H}_{n_{3}}(\xi)\,.$
(3.6)

  * •

When $n\in\bigcup_{\alpha<\alpha_{0}}\mathscr{C}_{\alpha}$ is a low frequency,

$\mathcal{F}\mathcal{Q}^{t}[F,G,H](\xi,n)=0\,.$

  * •

Remainder:

$\mathcal{F}\mathcal{E}^{t}[F,G,H]:=\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nres}}^{t}[F,G,H]-\mathcal{F}\mathcal{R}^{t}[F,G,H]-\mathcal{F}\mathcal{Q}^{t}[F,G,H]\,.$
(3.7)

######  Remark 3.1.

In particular, when $n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha}$,
we have the expression

$\mathcal{F}\mathcal{E}^{t}[F,G,H](\xi,n):=\frac{\pi}{t}\sum_{\begin{subarray}{c}(n_{1},n_{2},n_{3})\notin\Lambda^{(1)}(n)\cup\Lambda^{(3)}(n)\\\
n_{1}-n_{2}+n_{3}=n\end{subarray}}\operatorname{e}^{it\Omega(\vec{n})}\widehat{F}_{n_{1}}(\xi)\overline{\widehat{G}_{n_{2}}}(\xi)\widehat{H}_{n_{3}}(\xi)\,.$

###  3.2. The space resonant interactions

Let us analyze the system involving
$\mathcal{N}^{t}_{\mathbb{R},\mathrm{res}}$, decomposed into two parts. On the
one hand, the effective system defined in (3.9) preserves the super-actions
based on the frequency cluster decomposition. On the other hand, the remainder
$\mathcal{E}^{t}$ defined in (3.7) oscillates fast enough in time to gain
decay.

####  3.2.1. Trilinear bounds

Before analyzing the individual contributions of $\mathcal{R}^{t}$,
$\mathcal{Q}^{t}$ and $\mathcal{E}^{t}$ we first prove trilinear bounds for
the whole resonant system involving
$\mathcal{N}^{t}_{\mathbb{R},\mathrm{res}}$.

######  Lemma 3.2.

For all $t>0$,

$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}$
$\displaystyle\lesssim t^{-1}\|F^{a}\|_{Z}\|F^{b}\|_{Z}\|F^{c}\|_{Z}\,,$
$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F^{a},F^{b},F^{c}]\|_{S}$
$\displaystyle\lesssim
t^{-1}\sum_{\sigma\in\mathfrak{S}_{3}}\|F^{\sigma(a)}\|_{Z}\|F^{\sigma(b)}\|_{Z}\|F^{\sigma(c)}\|_{S}\,,$
$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}$
$\displaystyle\lesssim
t^{-1}\sum_{\sigma\in\mathfrak{S}_{3}}\|F^{\sigma(a)}\|_{Z}\|F^{\sigma(b)}\|_{Z}\|F^{\sigma(c)}\|_{H_{x,y}^{s}}\,,$
(3.8)

where $\mathfrak{S}_{3}$ is the group of permutations of three elements. The
same estimates hold for the subsystems involving $\mathcal{R}^{t}$,
$\mathcal{Q}^{t}$ and $\mathcal{E}^{t}$.

###### Proof.

For fixed $\xi\in\mathbb{R}$ and $t>0$ we have from the Young’s convolution
inequality that

$\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h_{y}^{s}}\lesssim
t^{-1}\sum_{\sigma\in\mathfrak{S}_{3}}\|\widehat{F^{\sigma(a)}}(\xi)\|_{\ell_{y}^{1}}\|\widehat{F^{\sigma(b)}}(\xi)\|_{\ell_{y}^{1}}\|\widehat{F^{\sigma(c)}}(\xi)\|_{h_{y}^{s}}\,,$

and the bounds follow from the assumption $s>\frac{d}{2}$. ∎

####  3.2.2. The effective system

We now handle the contribution of the effective interactions defined in (3.4).
Here, we use the key property that the effective system preserves the high-
frequency super-actions, proved in Lemma 2.9.

######  Lemma 3.3.

Suppose that $G\in C([-T,T];S)$ is a solution to the effective system

$i\partial_{t}G(t)=\mathcal{N}_{\mathrm{eff}}^{t}[G(t)]\,.$ (3.9)

Then, for all $|t|\leq T$, $\alpha\in\mathbb{N}$ and $\xi\in\mathbb{R}$ we
have

$\|\pi_{\alpha}\widehat{G}(t,\xi)\|_{\ell^{2}(\mathbb{T}^{d})}=\|\pi_{\alpha}\widehat{G}(0,\xi)\|_{\ell^{2}(\mathbb{T}^{d})}\,.$
(3.10)

Morevoer,

$\|G(t)\|_{Z}=\|G(0)\|_{Z}\,,\quad\|G(t)\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}=\|G(0)\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}\,.$
(3.11)

###### Proof.

Fix $\xi\in\mathbb{R}$, and consider

$a(\xi;t,y):=\widehat{G}(t,\xi,y)=\sum_{n\in\mathbb{Z}^{d}}\widehat{G}_{n}(t,\xi)\operatorname{e}^{in\cdot
y}\,,\quad a_{n}(\xi;t):=\widehat{G}_{n}(t,\xi)\,.$

In what follows we forget the variable $\xi$, which is parametric. By the
definition (3.4) of the effective system, $a$ is solution to

$i\partial_{t}a_{n}(t)=\mathcal{F}\mathcal{R}[a(t)](n)+\mathcal{F}\mathcal{Q}^{t}[a(t)](n)\,,\quad
n\in\mathbb{Z}^{d}\,.$

and therefore for $\alpha\geq 0$ we have

$\frac{1}{2}\frac{d}{dt}\|\pi_{\alpha}a\|_{\ell^{2}(\mathbb{T}^{d})}^{2}=\operatorname{Im}\big{(}\sum_{n\in\mathscr{C}_{\alpha}}\mathcal{F}\mathcal{R}[a(t)](n)\overline{a_{n}}(t)+\mathcal{F}\mathcal{Q}^{t}[a(t)](n)\overline{a_{n}}(t)\big{)}\,.$

According to Lemma 2.9, the quasi-resonant interactions $\mathcal{Q}^{t}$ do
not contribute. We easily prove that the resonant interactions do not
contribute either:

$\operatorname{Im}\big{(}\sum_{n\in\mathscr{C}_{\alpha}}\mathcal{F}\mathcal{R}[a(t)](n)\overline{a_{n}}(t)\big{)}=\operatorname{Im}\big{(}2\sum_{n\in\mathscr{C}_{\alpha}}\sum_{n_{1}\in\mathbb{Z}^{d}}|a_{n_{1}}(t)|^{2}|a_{n}(t)|^{2}-\sum_{n\in\mathscr{C}_{\alpha}}|a_{n}(t)|^{4}\big{)}=0\,.$

This proves (3.10). Then, the identities (3.11) follow form Definition 2.2 of
the $h^{s}$-norm based on the cluster decomposition: for all
$\xi\in\mathbb{R}$,

$\|\widehat{G}(t,\xi)\|_{h^{s}(\mathbb{T}^{d})}=\big{(}\sum_{\alpha\in\mathbb{N}}K_{\alpha}^{2s}\|\pi_{\alpha}\widehat{G}(t,\xi)\|_{\ell^{2}(\mathbb{T}^{d})}^{2}\big{)}^{\frac{1}{2}}\\\
=\big{(}\sum_{\alpha\in\mathbb{N}}K_{\alpha}^{2s}\|\pi_{\alpha}\widehat{G}(0,\xi)\|_{\ell^{2}(\mathbb{T}^{d})}^{2}\big{)}^{\frac{1}{2}}=\|\widehat{G}(0,\xi)\|_{h^{s}(\mathbb{T}^{d})}\,,$

This completes the proof of Lemma 3.3. ∎

####  3.2.3. The time oscillations

Let us turn to the contributions of $\mathcal{E}^{t}$ defined in (3.7). First,
we prove a trilinear estimate which will allow us to overcome small divisor
problems that arise when performing the Poincaré–Dulac normal form. This
estimate is a consequence of Lemma 2.10, which was in turn a consequence of
the frequency configuration of the interactions contained in
$\mathcal{E}^{t}$.

######  Lemma 3.4.

For $n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha}$, we let

$\mathcal{F}\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}](\xi,n):=\sum_{\begin{subarray}{c}(n_{1},n_{2},n_{3})\notin\Lambda^{(1)}(n)\cup\Lambda^{(3)}(n)\\\
n_{1}-n_{2}+n_{3}=n\end{subarray}}\frac{\operatorname{e}^{it\Omega(\vec{n})}}{\Omega(\vec{n})}\widehat{F}_{n_{1}}^{a}(\xi)\overline{\widehat{F}_{n_{2}}^{b}}(\xi)\widehat{F}_{n_{3}}^{c}(\xi)\,.$

Then, for all $t>0$ and $\xi\in\mathbb{R}$,

$\displaystyle\|\widehat{\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}]}(\xi)\|_{h^{s}(\mathbb{T}^{d})}$
$\displaystyle\lesssim\|\widehat{F^{a}}(\xi)\|_{h^{s}(\mathbb{T}^{d})}\|\widehat{F^{b}}(\xi)\|_{h^{s}(\mathbb{T}^{d})}\|\widehat{F^{c}}(\xi)\|_{h^{s}(\mathbb{T}^{d})}\,.$
In particular,
$\displaystyle\|\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}$
$\displaystyle\lesssim\|F^{a}\|_{Z}\|F^{b}\|_{Z}\|F^{c}\|_{Z}\,.$
$\displaystyle\|\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}]\|_{S}$
$\displaystyle\lesssim\sum_{\sigma\in\mathfrak{S}_{3}}\|F^{\sigma(a)}\|_{S}\|F^{\sigma(b)}\|_{Z}\|F^{\sigma(c)}\|_{Z}\,.$

###### Proof.

For fixed $\xi\in\mathbb{R}$, we have by duality that

$\|\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}\lesssim\sup_{\|v\|_{\ell^{2}}\leq
1}\sum_{\alpha\geq\alpha_{0}}\sum_{n\in\mathscr{C}_{\alpha}}K_{\alpha}^{s}|\mathcal{F}\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}](\xi,n)||v_{n}|\,.$

According to Lemma 2.10, we have that for all
$n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha}$ and
$(n_{1},n_{2},n_{3})\in(\mathbb{Z}^{d})^{3}\setminus(\Lambda^{(1)}(n)\cup\Lambda^{3}(n))$
with $n=n_{1}-n_{2}+n_{3}$,

$\frac{1}{|\Omega(\vec{n})|}K_{\alpha}^{s}\lesssim(n_{1}^{\ast})^{s}(n_{2}^{\ast})^{\frac{4\tau}{c(d)}}$

We deduce that

$|\mathcal{F}\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}](\xi,n)|\leq\sum_{n_{1}-n_{2}+n_{3}=n}(n_{1}^{\ast})^{s}(n_{2}^{\ast})^{\frac{4\tau}{c(d)}}|\widehat{F}_{n_{1}}^{a}(\xi)\widehat{F}_{n_{2}}^{b}(\xi)\widehat{F}_{n_{3}}^{c}(\xi)|\,.$

We conclude with the Young’s convolution inequality that for all
$\xi\in\mathbb{R}$ and from the Sobolev embedding that

$\|\mathcal{F}\widetilde{\mathcal{E}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}\lesssim\sum_{\sigma\in\mathfrak{S}_{3}}\|\langle
n\rangle^{s}\widehat{F_{n}^{\sigma(a)}}(\xi)\|_{\ell_{n}^{2}}\|\langle
n\rangle^{\frac{4\tau}{c(d)}}\widehat{F_{n}^{\sigma(b)}}(\xi)\|_{\ell_{n}^{1}}\|\widehat{F_{n}^{\sigma(c)}}(\xi)\|_{\ell_{n}^{1}}\,,\\\
\lesssim_{\eta}\sum_{\sigma\in\mathfrak{S}_{3}}\|\langle
n\rangle^{s}\widehat{F_{n}^{\sigma(a)}}(\xi)\|_{\ell_{n}^{2}}\|\langle
n\rangle^{\frac{4\tau}{c(d)}+\frac{d}{2}+\eta}\widehat{F_{n}^{\sigma(b)}}(\xi)\|_{\ell_{n}^{2}}\|\langle
n\rangle^{\frac{d}{2}+\eta}\widehat{F_{n}^{\sigma(c)}}(\xi)\|_{\ell_{n}^{2}}\,,$

for any $\eta>0$, which is conclusive under our assumption that
$s>\frac{d}{2}+\frac{4\tau}{c(d)}$. ∎

######  Lemma 3.5 (Poincaré–Dulac normal form).

Suppose that $T\geq 1$ and $F\in X_{T}$. Then, for all $t\in[1,T]$ and
$\xi\in\mathbb{R}$

$\displaystyle\sum_{\alpha\in\mathbb{N}}N_{\alpha}^{2s}\big{|}\sum_{n\in\mathscr{C}_{\alpha}}\int_{1}^{t}\mathcal{F}\mathcal{E}^{\tau}[F,F,F](\xi,n)\overline{\widehat{F}_{n}}(\tau,\xi)d\tau\big{|}$
$\displaystyle\lesssim\|F\|_{X_{T}}^{4}\,.$ (3.12) Moreover,
$\displaystyle\|\int_{\frac{t}{4}}^{t}\mathcal{E}^{t}[F,F,F]d\tau\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}$
$\displaystyle\lesssim t^{-1+4\delta}\|F\|_{X_{T}}^{3}\,.$ (3.13)

###### Proof.

Recalling the expression (3.7) of $\mathcal{E}^{t}$, we have that for fixed
$\xi\in\mathbb{R}$, for $n\leavevmode\nobreak\ \in\leavevmode\nobreak\
\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha}$

$\displaystyle\mathcal{F}\mathcal{E}^{\tau}[F,F,F](\xi,n)$
$\displaystyle=\frac{\pi}{\tau}\sum_{\begin{subarray}{c}(n_{1},n_{3},n_{3})\notin(\Lambda^{(1)}(n)\cup\Lambda^{(3)}(n))\\\
n_{1}-n_{2}+n_{3}=n\end{subarray}}\operatorname{e}^{i\tau\Omega(\vec{n})}\widehat{F}_{n_{1}}(\tau,\xi)\overline{\widehat{F}_{n_{2}}}(\tau,\xi)\widehat{F}_{n_{3}}(\tau,\xi)\,,$
and for $n\in\bigcup_{\alpha<\alpha_{0}}\mathscr{C}_{\alpha}$,
$\displaystyle\mathcal{F}\mathcal{E}^{\tau}[F,F,F](\xi,n)$
$\displaystyle=\frac{\pi}{\tau}\sum_{\begin{subarray}{c}n_{1}-n_{2}+n_{3}=n\\\
\Omega(\vec{n})\neq
0\end{subarray}}\operatorname{e}^{i\tau\Omega(\vec{n})}\widehat{F}_{n_{1}}(\tau,\xi)\overline{\widehat{F}_{n_{2}}}(\tau,\xi)\widehat{F}_{n_{3}}(\tau,\xi)\,.$

Since the terms contributing to $\mathcal{E}^{t}$ are not resonant,
$\Omega(\vec{n})\neq 0$ and the Leibniz rule gives

$\displaystyle\frac{\pi}{\tau}\operatorname{e}^{i\tau\Omega(\vec{n})}\widehat{F}_{n_{1}}(\tau,\xi)\overline{\widehat{F}_{n_{2}}}(\tau,\xi)\widehat{F}_{n_{3}}(\tau,\xi)=$
$\displaystyle\frac{d}{d\tau}\big{(}\frac{\pi}{\tau}\frac{\operatorname{e}^{i\tau\Omega(\vec{n})}}{i\Omega(\vec{n})}\widehat{F}_{n_{1}}(\tau,\xi)\overline{\widehat{F}_{n_{2}}}(\tau,\xi)\widehat{F}_{n_{3}}(\tau,\xi)\big{)}$
$\displaystyle+\frac{\pi}{\tau^{2}}\frac{\operatorname{e}^{i\tau\Omega(\vec{n})}}{i\Omega(\vec{n})}\widehat{F}_{n_{1}}(\tau,\xi)\overline{\widehat{F}_{n_{2}}}(\tau,\xi)\widehat{F}_{n_{3}}(\tau,\xi)$
$\displaystyle-\frac{\pi}{\tau}\frac{\operatorname{e}^{i\tau\Omega(\vec{n})}}{i\Omega(\vec{n})}\frac{d}{dt}\big{(}\widehat{F}_{n_{1}}(\tau,\xi)\overline{\widehat{F}_{n_{2}}}(\tau,\xi)\widehat{F}_{n_{3}}(\tau,\xi)\big{)}\,.$

Hence, we can gain decay in time at the price of a small divisor
$|\Omega(\vec{n})|^{-1}$. For the contributions to low-frequencies
$n\in\bigcup_{\alpha<\alpha_{0}}\mathscr{C}_{\alpha}$, this loss is easy to
overcome in the energy estimates (3.12) and (3.13): indeed, we can bound

$K_{\alpha}^{s}\leq K_{\alpha(0)}^{s}:=C(s,d,\mathrm{A})\,,$

and the loss $|\Omega(\vec{n})|^{-1}\lesssim(n_{1}^{\ast})^{2\tau}$ can be
absorbed (since $s>2\tau$).

For this reason, we only give details for the terms contributing to the high
frequencies $n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha_{0}}$. We
will exploit Lemma 3.4, in which we proved how to overcome the small divisor
loss. Defining the trilinear interaction $\widetilde{\mathcal{E}}^{t}$ by

$\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[F^{a},F^{b},F^{c}](\xi,n):=\sum_{\begin{subarray}{c}(n_{1},n_{3},n_{3})\notin\Lambda(n)\\\
n_{1}-n_{2}+n_{3}=n\end{subarray}}\frac{\pi}{\tau}\frac{\operatorname{e}^{i\tau\Omega(\vec{n})}}{\Omega(\vec{n})}\widehat{F}_{n_{1}}^{a}(\xi)\overline{\widehat{F}_{n_{2}}^{b}}(\xi)\widehat{F}_{n_{3}}^{c}(\xi)\,,$

we need to bound for
$n\in\bigcup_{\alpha\geq\alpha_{0}}\mathscr{C}_{\alpha_{0}}$

$\frac{d}{d\tau}\big{(}\frac{\pi}{\tau}\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[F,F,F](\xi,n)\big{)}+\frac{\pi}{\tau^{2}}\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[F,F,F](\xi,n)\\\
-\frac{\pi}{\tau}\big{(}\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[\partial_{\tau}F,F,F]+\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[F,\partial_{\tau}F,F]+\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[F,F,\partial_{\tau}F]\big{)}(\xi,n)\,.$
(3.14)

Let us first prove the energy estimate (3.12). We integrate by parts, and
control the first contribution in (3.14):

$\int_{1}^{t}\frac{d}{d\tau}\big{(}\frac{\pi}{\tau}\mathcal{F}\widetilde{\mathcal{E}}^{t}[F,F,F](\xi,n)\big{)}\overline{\widehat{F}_{n}}(\xi,\tau)d\tau=-\int_{1}^{t}\frac{\pi}{\tau}\mathcal{F}\widetilde{\mathcal{E}}^{\tau}[F,F,F](\xi,n)\partial_{\tau}\overline{\widehat{F}_{n}}(\xi,\tau)d\tau\\\
+\mathcal{F}\widetilde{\mathcal{E}}^{t}[F,F,F](\xi,n)\overline{\widehat{F}_{n}}(\xi,t)-\mathcal{F}\widetilde{\mathcal{E}}^{1}[F,F,F](\xi,n)\overline{\widehat{F}_{n}}(\xi,1)\,.$

Then, we deduce from the definition of the $Z$-norm that

$\sum_{\alpha\in\mathbb{N}}N_{\alpha}^{2s}\big{|}\sum_{n\in\mathscr{C}_{\alpha}}\int_{1}^{t}\frac{d}{d\tau}\big{(}\frac{\pi}{\tau}\mathcal{F}\widetilde{\mathcal{E}}^{t}[F,F,F](\xi,n)\big{)}\overline{\widehat{F}_{n}}(\xi,\tau)d\tau\big{|}\\\
\lesssim\int_{1}^{t}\|\mathcal{F}\widetilde{\mathcal{E}}^{t}[F,F,F]\|_{Z}\|\partial_{\tau}\widehat{F}\|_{Z}\frac{d\tau}{\tau}+\|\mathcal{F}\widetilde{\mathcal{E}}^{t}[F,F,F]\|_{Z}\|\widehat{F}_{n}(t)\|_{Z}+\|\mathcal{F}\widetilde{\mathcal{E}}^{1}[F,F,F]\|_{Z}\|\widehat{F}_{n}(1)\|_{Z}\\\
\lesssim\int_{1}^{t}\|F(\tau)\|_{Z}^{3}\|\partial_{\tau}F(\tau)\|_{Z}\frac{d\tau}{\tau}+\|F(t)\|_{Z}^{4}+\|F(1)\|_{Z}^{4}\,.$

In the last estimate, we applied the trilinear bound obtained in Lemma 3.4. By
definition (2.6) of the $X_{T}$-norm we obtain that for all $t\in[1,T]$,

$\sum_{\alpha\in\mathbb{N}}N_{\alpha}^{2s}\big{|}\int_{1}^{t}\frac{d}{d\tau}\big{(}\frac{\pi}{\tau}\mathcal{F}\widetilde{\mathcal{E}}^{t}[F,F,F](\xi,n)\big{)}\overline{\widehat{F}_{n}}(\xi,\tau)d\tau\big{|}\lesssim\|F\|_{X_{T}}^{4}\big{(}1+\int_{1}^{t}\frac{d\tau}{\tau^{2(1-\delta)}}\big{)}\lesssim\|F\|_{X_{T}}^{4}\,.$

The control of the other contributions follows in a very similar way. We now
turn to the proof of the bound (3.13). We have

$\|\int_{\frac{t}{4}}^{t}\mathcal{E}^{t}[F,F,F]d\tau\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}^{2}=\int_{\mathbb{R}}\langle\xi\rangle^{2s}\|\int_{\frac{t}{4}}^{t}\widehat{\mathcal{E}^{\tau}[F,F,F]}(\xi)d\tau\|_{h^{s}}^{2}d\xi$

Considering only the contributions to the high frequencies, it remains to
control

$\int_{\mathbb{R}}\langle\xi\rangle^{2s}\big{(}\|\frac{1}{t}\widehat{\widetilde{\mathcal{E}}^{t}[F,F,F](\xi)}-\frac{4}{t}\widehat{\widetilde{\mathcal{E}}^{\frac{t}{4}}[F,F,F]}(\xi)\|_{h^{s}}^{2}\\\
+\|\int_{\frac{t}{4}}^{t}(\frac{1}{\tau}\widehat{\widetilde{\mathcal{E}}^{\tau}[F,F,F]}(\xi)-\widehat{\widetilde{\mathcal{E}}^{\tau}[\partial_{\tau}F,F,F]}(\xi)-\widehat{\widetilde{\mathcal{E}}^{\tau}[F,\partial_{\tau}F,F]}(\xi)-\widehat{\widetilde{\mathcal{E}}^{\tau}[F,F,\partial_{\tau}F]}(\xi))\frac{d\tau}{\tau}\|_{h^{s}}^{2}\big{)}d\xi\,.$

To bound the first contribution we apply Lemma 3.4:

$\int_{\mathbb{R}}\langle\xi\rangle^{2s}(\|\frac{1}{t}\widehat{\widetilde{\mathcal{E}}^{t}[F,F,F](\xi)}-\frac{4}{t}\widehat{\widetilde{\mathcal{E}}^{\frac{t}{4}}[F,F,F]}(\xi)\|_{h^{s}}^{2})d\xi\\\
\lesssim\frac{1}{t}\int_{\mathbb{R}}\langle\xi\rangle^{2s}(\|\widehat{F}(\xi)\|_{h^{s}}^{2}\|\widehat{F}(\xi)\|_{h^{s}}^{2}\|\widehat{F}(\xi)\|_{h^{s}}^{2})d\xi\lesssim\frac{1}{t}\|F\|_{H^{s}}^{3}\lesssim\frac{1}{t^{1-3\delta}}\|F\|_{X_{T}}^{3}\,.$

The bounds for the other contributions follow similarly, using the rapid decay
of $\partial_{t}F$ encoded by the $X_{T}$-norm. This concludes the proof of
Lemma 3.5. ∎

###  3.3. The space oscillations

We now control the contribution of the space non-resonant interactions
$\mathcal{N}^{t}_{\mathbb{R},\mathrm{nr}}$ defined in (3.2). For this purpose,
we adapt the argument from Kato–Pusateri [29] (see also [24, Lemma 3.10]).
Here, we do not use the oscillations in the transverse direction
$\mathbb{T}^{d}$.

######  Lemma 3.6 (Space oscillations).

For all $T\geq 1$, $t\in[1,T]$, and any $\gamma\in(0,\frac{1}{4})$ we have

$\displaystyle\|\mathcal{N}^{t}_{\mathbb{R},\mathrm{nr}}[F^{a},F^{b},F^{c}]\|_{Z}$
$\displaystyle\lesssim_{\gamma}t^{-1-\gamma+3\delta}\|F^{a}\|_{X_{T}}\|F^{b}\|_{X_{T}}\|F^{c}\|_{X_{T}}\,.$
(3.15) Moreover, for $\delta>0$ sufficiently small (depending on $s,\sigma,d$
and $\gamma$), we have
$\displaystyle\|\mathcal{N}^{t}_{\mathbb{R},\mathrm{nr}}[F^{a},F^{b},F^{c}]\|_{H^{s}(\mathbb{R}\times\mathbb{T}^{d})}$
$\displaystyle\lesssim
t^{-1-2\delta}\|F^{a}\|_{X_{T}}\|F^{b}\|_{X_{T}}\|F^{c}\|_{X_{T}}\,.$ (3.16)

In both cases we obtain an integrable decay in time provided that
$3\delta<\gamma$. The second bound (3.16) will be useful to prove the modified
scattering result in the topology of $H^{s}(\mathbb{R}\times\mathbb{T}^{d})$,
from which we deduce the stability in the space
$H^{s}(\mathbb{R}\times\mathbb{T}^{d})$ claimed in the main Theorem 1.3.

###### Proof.

We first prove (3.15). We show that for all $\gamma\in(0,\frac{1}{4})$,

$\|\mathcal{N}^{t}_{\mathbb{R},\mathrm{nr}}[F^{a},F^{b},F^{c}]\|_{Z}\lesssim_{\gamma}t^{-1-\gamma}\|F^{a}\|_{S}\|F^{b}\|_{S}\|F^{c}\|_{S}\,.$
(3.17)

Following [29] we have that for all $\gamma\in(0,\frac{1}{4}]$, $t>0$, and
$f^{a},f^{b},f^{c}$,

$\displaystyle\sup_{\xi\in\mathbb{R}}\,
|\mathcal{O}^{t}[f^{a},f^{b},f^{c}](\xi)|$ $\displaystyle\lesssim
t^{-1-\gamma}\int_{\mathbb{R}^{3}}|\eta|^{\gamma}|\sigma|^{\gamma}|f^{a}(t,x-\eta)||f^{b}(t,x)||f^{c}(t,x-\sigma)|dxd\eta
d\sigma$ $\displaystyle\lesssim
t^{-1-\gamma}\max_{\sigma\in\mathfrak{S}_{3}}\|f^{\sigma(a)}\|_{L_{x}^{1}(\mathbb{R})}\||x|^{2\gamma}f^{\sigma(b)}\|_{L_{x}^{1}(\mathbb{R})}\||x|^{2\gamma}f^{\sigma(c)}\|_{L_{x}^{1}(\mathbb{R})}$
$\displaystyle\lesssim t^{-1-\gamma}\|\langle
x\rangle^{\frac{1}{2}+2\gamma+0}f^{a}\|_{L_{x}^{2}(\mathbb{R})}\|\langle
x\rangle^{\frac{1}{2}+2\gamma+0}f^{b}\|_{L_{x}^{2}(\mathbb{R})}\|\langle
x\rangle^{\frac{1}{2}+2\gamma+0}f^{c}\|_{L_{x}^{2}(\mathbb{R})}\,,$

where we denoted by $\mathfrak{S}_{3}$ the group of permutations of three
elements. We obtain that for fixed $(n_{1},n_{2},n_{3})$ and
$\gamma\in(0,\frac{1}{4})$,

$\sup_{\xi\in\mathbb{R}}|\mathcal{O}^{t}[F_{n_{1}}^{a},F_{n_{2}}^{b},F_{n_{3}}^{c}](\xi)|\lesssim_{\gamma}t^{-1-\gamma}\mathbf{f}_{n_{1}}^{a}\mathbf{f}_{n_{2}}^{b}\mathbf{f}_{n_{3}}^{c}\,,$

where we denote for $n\in\mathbb{Z}^{d}$

$\mathbf{f}_{n}^{a}:=\|\langle x\rangle
F_{n}^{a}\|_{L_{x}^{2}(\mathbb{R})}\,,\quad\mathbf{f}_{n}^{b}:=\|\langle
x\rangle
F_{n}^{b}\|_{L_{x}^{2}(\mathbb{R})}\,,\quad\mathbf{f}_{n}^{c}:=\|\langle
x\rangle F_{n}^{c}\|_{L_{x}^{2}(\mathbb{R})}\,.$

Observe that, since $s>\frac{d}{2}$,

$\|\mathbf{f}_{n}^{a}\|_{\ell_{n}^{1}(\mathbb{Z}^{d})}\lesssim\|\langle
n\rangle^{2s}\,\mathbf{f}_{n}^{a}\|_{\ell_{n}^{2}(\mathbb{Z}^{d})}\lesssim\|\langle
x\rangle
F^{a}\|_{L^{2}h^{s}(\mathbb{R}\times\mathbb{T}^{d})}\lesssim\|F^{a}\|_{S}\,.$
(3.18)

We obtain

$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}$
$\displaystyle=\sup_{\xi\in\mathbb{R}}\|\sum_{n_{1}-n_{2}+n_{3}=n}\operatorname{e}^{it\Omega(\vec{n})}\mathcal{O}^{t}[F_{n_{1}}^{a},F_{n_{2}}^{b},F_{n_{3}}^{c}](\xi)\|_{h^{s}(\mathbb{Z}^{d})}$
$\displaystyle\lesssim_{\gamma}t^{-1-\gamma}\|\sum_{n_{1}-n_{2}+n_{3}=n}\mathbf{f}_{n_{1}}^{a}\mathbf{f}_{n_{2}}^{b}\mathbf{f}_{n_{3}}^{c}\|_{h^{s}(\mathbb{Z}^{d})}$
$\displaystyle\lesssim_{\gamma}t^{-1-\gamma}\sum_{\sigma\in\mathfrak{S}_{3}}\|\mathbf{f}_{n}^{\sigma(a)}\|_{\ell^{1}_{n}(\mathbb{Z}^{d})}\|\mathbf{f}_{n}^{\sigma(b)}\|_{\ell^{1}_{n}(\mathbb{Z}^{d})}\|\langle
n\rangle^{2s}\,\mathbf{f}_{n}^{\sigma(c)}\|_{\ell^{2}(\mathbb{Z}^{d})}\,,$

and the observation (3.18) concludes the proof of estimate (3.17).

We now derive the bound (3.16). For this purpose, we use the slightly non-
integrable 666Namely with time decay $t^{-1+\delta}$, with $\delta>0$. bound
in the $S$-norm already proved for the whole system involving
$\mathcal{N}^{t}$ and for the space-resonant system with
$\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}$, and somehow interpolate with the
decay estimate (3.17). For all $\alpha\in(0,1)$,

$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}$
$\displaystyle[F^{a},F^{b},F^{c}]\|_{L_{x}^{2}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}=\big{(}\int_{\mathbb{R}}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2}d\xi\big{)}^{\frac{1}{2}}$
$\displaystyle\leq\sup_{\xi\in\mathbb{R}}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{\alpha}\big{(}\int_{\mathbb{R}}\langle\xi\rangle^{-2\alpha}\langle\xi\rangle^{2\alpha}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2(1-\alpha)}d\xi\big{)}^{\frac{1}{2}}\,.$
By Hölder’s inequality, and assuming that $\frac{\alpha}{1-\alpha}<\sigma$, we
obtain
$\displaystyle\lesssim\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}^{\alpha}\big{(}\int_{\mathbb{R}}\langle\xi\rangle^{\frac{2\alpha}{1-\alpha}}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2}d\xi\big{)}^{\frac{1-\alpha}{2}}$
$\displaystyle\lesssim\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}^{\alpha}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x}^{s}\ell_{y}^{2}(\mathbb{R}\times\mathbb{T}^{d})}^{1-\alpha}.$

Similarly, we have that for all $\alpha\in(0,1)$,

$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}$
$\displaystyle[F^{a},F^{b},F^{c}]\|_{H_{x}^{s}\ell_{y}^{2}(\mathbb{R}\times\mathbb{T}^{d})}=\big{(}\int_{\mathbb{R}}\langle\xi\rangle^{2s}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2}d\xi\big{)}^{\frac{1}{2}}$
$\displaystyle\leq\sup_{\xi\in\mathbb{R}}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{\alpha}\big{(}\int_{\mathbb{R}}\langle\xi\rangle^{-2\alpha}\langle\xi\rangle^{2(s+\alpha)}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2(1-\alpha)}d\xi\big{)}^{\frac{1}{2}}$
We deduce from Hölder’s inequality in the integral over $\xi$ that
$\displaystyle\lesssim\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}^{\alpha}\big{(}\int_{\mathbb{R}}\langle\xi\rangle^{\frac{2(s+\alpha)}{1-\alpha}}\|\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}](\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2}d\xi\big{)}^{\frac{1-\alpha}{2}}.$

By choosing $\alpha>0$ sufficiently small (only depending on $s$ and $\sigma$
from the definition of the $S$-norm) to ensure that

$\frac{s+\alpha}{1-\alpha}<s+\sigma\,,$

we conclude that

$\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}\lesssim\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{Z}^{\alpha}\
\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}^{1-\alpha}\,.$

We have from the trilinear bounds (2.17) and (3.8)

$\displaystyle\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}$
$\displaystyle\lesssim\|\mathcal{N}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}+\|\mathcal{N}_{\mathbb{R},\mathrm{res}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}$
$\displaystyle\lesssim\langle
t\rangle^{-1+\delta}\|F^{a}\|_{X_{T}}\|F^{b}\|_{X_{T}}\|F^{c}\|_{X_{T}}\,.$

Hence, we deduce from the first bound (3.17) (in the $Z$-norm) that for all
$t\geq 1$,

$\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{t}[F^{a},F^{b},F^{c}]\|_{H_{x,y}^{s}}\lesssim
t^{-1-\alpha\gamma+(1-\alpha)\delta}\|F^{a}\|_{X_{T}}\|F^{b}\|_{X_{T}}\|F^{c}\|_{X_{T}}\,,$

which is conclusive when

$-\alpha\gamma+(1-\alpha)\delta<-2\delta\,,$

satisfied for $\delta>0$ small enough (for fixed $\alpha$, $\gamma$). This
proves of Lemma 3.6. ∎

##  4\. A priori bounds and asymptotic behavior

In this section we bring together the multilinear bounds collected in the
previous sections and prove the main results, namely Theorem 1.3 and Theorem
1.4.

###  4.1. A priori bounds

We already obtained the slightly divergent bounds for the strong norm $S$ in
Lemma 2.13. It remains to prove the a priori bounds for the weak norm $Z$.

######  Proposition 4.1 (Energy estimate).

Suppose that $T\geq 1$ and that $F$ is a profile solution to the interaction
system (2.15) on $[0,T]$ satisfying

$\|F\|_{X_{T}}\lesssim 1\,.$

Then, for all $t\in[1,T]$, we have

$\|F(t)-F(1)\|_{Z}\leq C\|F\|_{X_{T}}^{2}\,.$

###### Proof.

For fixed $\xi\in\mathbb{R}$, we have

$\|\widehat{F}(t,\xi)-\widehat{F}(1,\xi)\|_{h^{s}(\mathbb{T}^{d})}^{2}\leq\sum_{\alpha\in\mathbb{N}}N_{\alpha}^{2s}\big{|}\int_{1}^{t}\frac{d}{d\tau}\|\pi_{\alpha}\widehat{F}(\xi,\tau)\|_{\ell^{2}(\mathbb{T}^{d})}^{2}d\tau\big{|}\,.$

According to the decomposition (3.3) we have that for all
$\alpha\in\mathbb{N}$,

$\frac{1}{2}\frac{d}{d\tau}\|\pi_{\alpha}\widehat{F}(\tau,\xi)\|_{\ell^{2}(\mathbb{Z})}^{2}\\\
=\mathrm{Im}\Big{(}\sum_{n\in\mathscr{C}_{\alpha}}\overline{\widehat{F}_{n}}(\tau,\xi)\big{(}\mathcal{F}\mathcal{N}_{\mathrm{eff}}^{\tau}[F](\xi,n)+\mathcal{F}\mathcal{E}^{\tau}[F](\xi,n)+\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{\tau}[F](\xi,n)\big{)}\Big{)}\,.$

We proved in Lemma 3.3 that the effective system involving
$\mathcal{N}_{\mathrm{eff}}$ does not contribute to the above energy estimate.
This gives

$\frac{1}{2}\frac{d}{d\tau}\|\pi_{\alpha}\widehat{F}(\xi,\tau)\|_{\ell^{2}(\mathbb{Z})}^{2}=\mathrm{Im}\Big{(}\sum_{n\in\mathscr{C}_{\alpha}}\overline{\widehat{F}_{n}}(\xi,\tau)\big{(}\mathcal{F}\mathcal{E}^{\tau}[F](\xi,n)+\mathcal{F}\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{\tau}[F](\xi,n)\big{)}\Big{)}\,,$

and we obtain

$\|\widehat{F}(t,\xi)-\widehat{F}(1,\xi)\|_{h^{s}}^{2}\leq\sum_{\alpha\in\mathbb{N}}N_{\alpha}^{2s}\big{|}\sum_{n\in\mathscr{C}_{\alpha}}\int_{1}^{t}\mathcal{F}\mathcal{E}^{\tau}[F](\xi,n)\overline{\widehat{F}_{n}}(\tau,\xi)d\tau\big{|}\\\
+\int_{1}^{t}\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{\tau}[F](\xi)\|_{h^{s}}\|\widehat{F}(\tau,\xi)\|_{h^{d}}d\tau\,.$

A Poincaré–Dulac normal form performed in Lemma 3.5 allowed us to control the
contributions of $\mathcal{E}^{\tau}$: we have from (3.12) that

$\|\widehat{F}(t,\xi)-\widehat{F}(1,\xi)\|_{h^{s}}^{2}\lesssim\|F\|_{X_{T}}^{4}+\int_{1}^{t}\|\mathcal{N}_{\mathbb{R},\mathrm{nr}}^{\tau}[F](\xi)\|_{h^{s}}\|\widehat{F}(\tau,\xi)\|_{h^{s}}d\tau\,.$

Then, we apply (3.15) from Lemma 3.6 to control the space non-resonant
interactions:

$\|\widehat{F}(t,\xi)-\widehat{F}(1,\xi)\|_{h^{s}}^{2}\lesssim\|F\|_{X_{T}}^{4}+\|F\|_{X_{T}}^{3}\int_{1}^{t}\tau^{-1-\gamma+3\delta}\|F(\tau)\|_{Z}d\tau\lesssim\|F\|_{X_{T}}^{4}\,.$

provided $3\delta<\gamma$. This concludes the proof of Proposition 4.1. ∎

######  Proposition 4.2 (A priori bounds).

There exists $C>0$ such that for all $\varepsilon>0$ and $U(0)\in S$ with

$\|U(0)\|_{S}\leq\varepsilon\,,$ (4.1)

the solution $U$ to (A-NLS) is global in $C(\mathbb{R};S)$ and satisfies for
all $T\geq 0$,

$\|F\|_{X_{T}}\leq C\|U(0)\|_{S}\,,$ (4.2)

where $F(t):=\operatorname{e}^{-it\Delta}U(t)$.

###### Proof.

Let $T\geq 0$ and $0\leq t\leq T$. It follows from Lemma 2.13 that when $0\leq
t\leq 1$,

$\|F(t)-F(0)\|_{S}\leq\sup_{\tau\in[0,t]}\|\mathcal{N}^{\tau}[F]\|_{S}\lesssim\|F\|_{X_{T}}^{3}\,.$

Recalling that the $S$ norm controls the $Z$ norm (remark 2.5), we deduce from
the definition (2.6) of the $X_{T}$-norm that it remains to show that for all
$T\geq 1$ and $1\leq t\leq T$,

$\|F(t)-F(1)\|_{Z}+t^{-\delta}\|F(t)-F(1)\|_{S}+t^{1-\delta}\|\partial_{t}F(t)\|_{S}\lesssim\|F\|_{X_{T}}^{2}+\|F\|_{X_{T}}^{3}\,.$
(4.3)

We proved in Proposition 4.1 that

$\|F(t)-F(1)\|_{Z}\lesssim\|F\|_{X_{T}}^{2}\,,$

In Lemma 2.13 we proved that for all $t\in[1,T]$,

$\|\mathcal{N}^{t}[F]\|_{S}\lesssim t^{-1+\delta}\|F\|_{X_{T}}^{3}\,.$

This implies that

$\displaystyle\|\partial_{t}F(t)\|_{S}$
$\displaystyle=\|\mathcal{N}^{t}[F]\|_{S}\lesssim
t^{-1+\delta}\|F\|_{X_{T}}^{3}\,,$ and $\displaystyle\|F(t)-F(1)\|_{S}$
$\displaystyle\leq\int_{1}^{t}\|\mathcal{N}^{\tau}[F]\|_{S}d\tau\lesssim
t^{\delta}\|F\|_{X_{T}}^{3}\,.$

Hence,

$t^{-\delta}\|F-F(1)\|_{S}+t^{1-\delta}\|\partial_{t}F\|_{S}\lesssim\|F\|_{X_{T}}^{3}\,.$

This gives (4.3) and completes the proof of Proposition 4.2. ∎

###  4.2. Proof of the main Theorems

We have all the ingredients to prove our main Theorems.

###### Proof that Theorem 1.4 $\implies$ Theorem 1.3..

The global existence and the dispersive estimate (second inequality in (1.4))
follow from the a priori bound (4.2) proved in Proposition 4.2 and from Lemma
2.11. Then, let $G$ be the global solution to the effective system (3.9) given
by Theorem 1.4. The uniform bound on the $H^{s}$-norm of $U$ (first inequality
in (1.4)) is a consequence of the conservation of the $H^{s}$-norm of $G$ from
Lemma 3.3, from the modified scattering result in $H^{s}$ and from the initial
bound $\|G(0)\|_{H^{s}}\lesssim\varepsilon$. ∎

It remains to prove Theorem 1.4. The proof follows the same line as the proof
of [24, Theorem 6.1] but we write it in our context for completeness.

###### Proof of Theorem 1.4.

Given, $U(0)$ satisfying the smallness condition (4.1) in the $S$-norm, we
first construct the initial state $G_{\infty}(0)$ such that the global
solution $U$ to (A-NLS) with data $U(0)$ scatters to $G$, the global solution
to the effective system (3.9) with data $G(0)=G_{\infty}(0)$.

Let $n\in\mathbb{N}$ and $T_{n}:=\operatorname{e}^{n}$. We let $(G_{n})_{n}$
be the sequence of global solutions in $S$ to (3.9) with Cauchy data

$G_{n}(T_{n}):=F(T_{n})\,,\quad\text{where}\quad
F(t)=\operatorname{e}^{-it\Delta}U(t)\,.$

According to (3.11) proved in Lemma 3.3 we have that for all $t\in\mathbb{R}$,

$\|G_{n}(t)\|_{Z}=\|G_{n}(T_{n})\|_{Z}=\|F(T_{n})\|_{Z}\lesssim\varepsilon\,,$
(4.4)

where we used the global bound (4.2) on the $X_{T}$-norm of $F$ given by
Proposition 4.2. In addition, we have from Lemma 3.2 that for $t\geq T_{n}$,

$\|\partial_{t}G_{n}(t)\|_{H_{x,y}^{s}}\lesssim
t^{-1}\|G_{n}(t)\|_{Z}^{2}\|G_{n}(t)\|_{H_{x,y}^{s}}\lesssim
t^{-1}\varepsilon^{2}\|G_{n}(t)\|_{H_{x,y}^{s}}\,.$ (4.5)

Hence, using also the control on $G_{n}(T_{n})=F(T_{n})$ in $H_{x,y}^{s}$
given by (4.2), we deduce from the Gronwall’s inequality that for all $t\geq
T_{n}$,

$\|G_{n}(t)\|_{H_{x,y}^{s}}\lesssim\varepsilon t^{\delta}\,.$

We now prove that for all $T_{n}\leq t\leq 4T_{n}$,

$\|F(t)-G_{n}(t)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}T_{n}^{-\delta}\,.$
(4.6)

For this, we use the equation

$F(t)-G_{n}(t)=-i\int_{T_{n}}^{t}(\mathcal{E}^{\tau}[F]+\mathcal{N}^{\tau}_{\mathbb{R},\mathrm{nr}}[F])d\tau+\int_{T_{n}}^{t}(\mathcal{N}_{\mathrm{eff}}^{\tau}[F]-\mathcal{N}_{\mathrm{eff}}^{\tau}[G_{n}])d\tau\,.$
(4.7)

To control the source term (namely the first integral term in (4.7)), we use
Lemma 3.5 and Lemma 3.6 together with the a priori bound (4.2) on $F$. This
gives that for $T_{n}\leq t\leq 4T_{n}$,

$\big{\|}\int_{T_{n}}^{t}(\mathcal{E}^{\tau}[F]+\mathcal{N}^{\tau}_{\mathbb{R},\mathrm{nr}}[F])d\tau\big{\|}_{Z}\lesssim
T_{n}^{-1+4\delta}\varepsilon^{3}+\varepsilon^{3}\int_{T_{n}}^{t}\frac{d\tau}{\tau^{-1-2\delta}}\lesssim
T_{n}^{-2\delta}\varepsilon^{3}\,.$

Factorizing

$\mathcal{N}_{\mathrm{eff}}^{t}[F]-\mathcal{N}_{\mathrm{eff}}^{t}[G_{n}]=\mathcal{N}_{\mathrm{eff}}^{t}[F-G_{n},F,F]+\mathcal{N}_{\mathrm{eff}}^{t}[G_{n},F-G_{n},F]+\mathcal{N}_{\mathrm{eff}}^{t}[G_{n},G_{n},F-G_{n}]\,,$

we deduce from the a priori bound (4.4) that

$\|F(t)-G_{n}(t)\|_{Z}\lesssim\varepsilon^{3}T_{n}^{-2\delta}+\int_{T_{n}}^{t}(\|G_{n}(\tau)\|_{Z}^{2}+\|F(\tau)\|_{Z}^{2})\|F(\tau)-G(\tau)\|_{Z}\frac{d\tau}{\tau}\\\
\lesssim\varepsilon^{3}T_{n}^{-2\delta}+\varepsilon^{2}\int_{T_{n}}^{t}\|F(\tau)-G_{n}(\tau)\|_{Z}\frac{d\tau}{\tau}\,.$

Gronwall’s inequality yields that for all $T_{n}\leq t\leq 4T_{n}$,

$\|F(t)-G_{n}(t)\|_{Z}\lesssim\varepsilon^{3}T_{n}^{-2\delta}\,.$

By proceeding similarly, and using also the a-priori bounds (4.5), this in
turn implies (4.6) since

$\|F(t)-G_{n}(t)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}T_{n}^{-2\delta}+\varepsilon^{2}\int_{T_{n}}^{t}\|F(\tau)-G_{n}(\tau)\|_{H_{x,y}^{s}}\frac{d\tau}{\tau}\\\
+\varepsilon\int_{T_{n}}^{t}(\|G_{n}(\tau)\|_{H_{x,y}^{s}}+\|F(\tau)\|_{H_{x,y}^{s}})\|F(\tau)-G_{n}(\tau)\|_{Z}\frac{d\tau}{\tau}\,.$

The a priori bound (4.6) implies that if $\varepsilon$ is small enough
(depending on $\delta^{\frac{1}{2}}$) then for all $n\in\mathbb{N}$

$\|G_{n}(0)-G_{n}(0)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}\operatorname{e}^{-\frac{n\delta}{2}}\,.$

Indeed, since $T_{n+1}\leq 4T_{n}$ we proved that

$\|G_{n}(T_{n+1})-G_{n+1}(T_{n+1})\|_{H_{x,y}^{s}}=\|G_{n}(T_{n+1})-F(T_{n+1})\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}T_{n}^{-\delta}\,,$

and Gronwall’s inequality for the effective system (3.9) (that is solved both
by $G_{n}$ and $G_{n+1}$) gives

$\|G_{n}(0)-G_{n+1}(0)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}T_{n}^{-\delta}\operatorname{e}^{C\varepsilon^{2}\log(T_{n+1})}\lesssim\varepsilon^{3}\operatorname{e}^{-\delta
n+C\varepsilon^{2}(n+1)}\lesssim\varepsilon^{3}\operatorname{e}^{-\frac{n\delta}{2}}$

In particular $(G_{n}(0))$ converges in $H_{x,y}^{s}$ to a limit
$G_{\infty}(0)$, which satisfies

$\|G_{\infty}(0)\|_{Z}\lesssim\varepsilon\,,\quad\|G_{n}(0)-G_{\infty}(0)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}\operatorname{e}^{-\frac{\delta
n}{2}}\,.$

Using that for $n=1$, $\|G_{1}(0)-F(1)\|_{H^{s}}\lesssim\varepsilon$ and
$\|F(1)\|_{H^{s}}\lesssim\varepsilon$ we obtain that

$\|G_{\infty}(0)\|_{H^{s}}\lesssim\varepsilon\,.$

Considering the global solution $G$ with Cauchy data $G(0)=G_{\infty}(0)$,
another Gronwall’s inequality gives

$\sup_{[0,T_{n+2}]}\|G(t)-G_{n}(t)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}\operatorname{e}^{-\frac{\delta
n}{4}}\,,$

and we conclude from (4.6) that

$\sup_{T_{n}\leq t\leq T_{n+1}}\|F(t)-G(t)\|_{H_{x,y}^{s}}\\\
\lesssim\sup_{T_{n}\leq t\leq
T_{n+1}}\|F(t)-G_{n}(t)\|_{H_{x,y}^{s}}+\sup_{T_{n}\leq t\leq
T_{n+1}}\|G(t)-G_{n}(t)\|_{H_{x,y}^{s}}\lesssim\varepsilon^{3}\operatorname{e}^{-\frac{\delta
n}{4}}\,,$

which completes the proof of Theorem 1.4. ∎

##  Appendix A Some elementary Lemmas

In order to make the exposition self contained, we now recall the proof of the
dispersive estimate for the free Schrödinger evolution on $\mathbb{R}$.

###### Proof of (2.12).

We have

$\operatorname{e}^{it\partial_{x}^{2}}f(x)=\frac{1}{(4i\pi
t)^{\frac{1}{2}}}\int_{\mathbb{R}}\operatorname{e}^{\frac{i(x-z)^{2}}{4t}}f(z)dz=\frac{\operatorname{e}^{\frac{ix^{2}}{4t}}}{(4i\pi
t)^{\frac{1}{2}}}\int_{\mathbb{R}}\operatorname{e}^{-\frac{ixz}{2t}}\operatorname{e}^{\frac{iz^{2}}{4t}}f(z)dz\\\
=\frac{\operatorname{e}^{\frac{ix^{2}}{4t}}}{(2it)^{\frac{1}{2}}}\big{[}\widehat{f}(\frac{x}{2t})+r[f](t,x)\big{]}\,,$

where

$r[f](t,x):=\frac{1}{(2\pi)^{\frac{1}{2}}}\int_{\mathbb{R}}\operatorname{e}^{-\frac{ixz}{2t}}(\operatorname{e}^{\frac{iz^{2}}{4t}}-1)f(z)dz\,.$

Proving (2.12) amounts to get the desired bound for $r[f]$. For this, we split
the integral into two different regions: $r_{1}[f](t,x)$ (resp.
$r_{2}[f](t,x)$) corresponds to the region $|z|\leq\sqrt{t}$ (resp.
$|z|>\sqrt{t}$). On the one hand, for $x\in\mathbb{R}$ and $|t|\geq 1$,

$|r_{1}[f](t,x)|\lesssim\frac{1}{t}\int_{|z|\leq\sqrt{t}}z^{2}|f(z)|dz\lesssim\frac{1}{t}\big{(}\int_{|z|\leq\sqrt{t}}z^{2}dz\big{)}^{\frac{1}{2}}\|zf(z)\|_{L^{2}(\mathbb{R})}\\\
\lesssim\frac{1}{t^{\frac{1}{4}}}\|zf(z)\|_{L^{2}(\mathbb{R})}\,.$

On the other hand,

$|r_{2}[f](t,x)|\lesssim\int_{|z|\geq\sqrt{t}}|f(z)|dz\lesssim\big{(}\int_{|z|\geq\sqrt{t}}\frac{1}{|z|^{2}}dz\big{)}^{\frac{1}{2}}\|zf(z)\|_{L^{2}(\mathbb{R})}\\\
\lesssim\frac{1}{t^{\frac{1}{4}}}\|zf(z)\|_{L^{2}(\mathbb{R})}\,.$

This completes the proof of the dispersive estimate (2.12). ∎

The product rule stated in the following Lemma was useful in the trilinear
estimates on the strong norm of Lemma 2.13.

######  Lemma A.1 (Product rule).

For all $s>\frac{d}{2}$ and $\sigma\geq 0$, there exists $C>0$ such that for
every functions
$F,G\in(1-\partial_{x}^{2})^{-\sigma}H_{x,y}^{s}(\mathbb{R}\times\mathbb{T}^{d})\cap
L_{x}^{\infty}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})$,

$\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}(FG)\|_{H_{x,y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\lesssim\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}F\|_{H_{x,y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\|G\|_{L_{x}^{\infty}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\\\
+\|F\|_{L_{x}^{\infty}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}G\|_{H_{x,y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\,.$

###### Proof.

The bound on the norm $H_{x}^{s}\ell_{y}^{2}$ follows from the product rule on
$\mathbb{R}$:

$\displaystyle\|FG\|_{H_{x}^{s}\ell_{y}^{2}}$
$\displaystyle\lesssim\|\|F\|_{H_{x}^{s}}\|G\|_{L_{x}^{\infty}}\|_{\ell_{y}^{2}}+\|\|F\|_{L_{x}^{\infty}}\|G\|_{H_{x}^{s}}\|_{\ell_{y}^{2}}$
$\displaystyle\lesssim\|F\|_{H_{x}^{s}\ell_{y}^{2}}\|G\|_{L_{x}^{\infty}\ell_{y}^{\infty}}+\|F\|_{L_{x}^{\infty}\ell_{y}^{\infty}}\|G\|_{H_{x}^{s}\ell_{y}^{2}}\,.$

It remains to prove the bound for the norm $L_{x}^{2}h_{y}^{s}$.

$\bullet$ Case $\sigma=0$: we have from the assumption $s>\frac{d}{2}$ that

$\|FG\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|\|F\|_{h_{y}^{s}}\|G\|_{h_{y}^{s}}\|_{L_{x}^{2}}\lesssim\|F\|_{L_{x}^{\infty}h_{y}^{s}}\|G\|_{L_{x}^{2}h_{y}^{s}}\,.$

$\bullet$ Case $\sigma>0$: we employ Littlewood–Paley decomposition in the
$x$-variable only, and use that $h^{s}(\mathbb{T}^{d})$ is an algebra when
$s>\frac{d}{2}$. Given $\varphi$ a bump function on $\mathbb{R}$, supported in
$[-2,2]$ and equals to 1 on $[-1,1]$, we set for $N>1$ dyadic,
$\xi\in\mathbb{R}$ and $y\in\mathbb{T}^{d}$

$\widehat{P_{N}F}(\xi,y)=\big{(}\varphi(\frac{\xi}{N})-\varphi(\frac{\xi}{2N})\big{)}\widehat{F}(\xi,y)\,,$

and

$\widehat{P_{\leq 1}F}(\xi,y)=\varphi(\xi)\widehat{F}(\xi,y)\,.$

In particular, for all fixed $y\in\mathbb{T}^{d}$,

$\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}F(y)\|_{L_{x}^{2}(\mathbb{R})}\sim\big{(}\|P_{\leq
1}F(y)\|_{L_{x}^{2}(\mathbb{R})}^{2}+\sum_{N>1}N^{2\sigma}\|P_{N}F(y)\|_{L_{x}^{2}(\mathbb{R})}^{2}\big{)}^{\frac{1}{2}}\,.$

In particular, we obtain from the Fubini–Tonelli’s theorem that

$\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}F\|_{L_{x}^{2}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}\sim\big{(}\|P_{\leq
1}F\|_{L_{x}^{2}h^{s}(\mathbb{R}\times\mathbb{T}^{d})}^{2}+\sum_{N>1}N^{2\sigma}\|P_{N}F\|_{L_{x}^{2}h_{y}^{s}(\mathbb{R}\times\mathbb{T}^{d})}^{2}\big{)}^{\frac{1}{2}}\,.$

For a fixed dyadic $N$ we decompose

$\|P_{N}(FG)\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|P_{<\frac{N}{8}}FP_{\sim
N}G\|_{L_{x}^{2}h_{y}^{s}}+\sum_{M\geq\frac{N}{8}}\|P_{M}FG\|_{L_{x}^{2}h_{y}^{s}}\,,$

where we denoted

$P_{\sim N}=\sum_{\frac{N}{8}<M<8N}P_{M}\,.$

To handle the first contribution, we observe that

$\|P_{<\frac{N}{8}}FP_{\sim
N}G\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|P_{<\frac{N}{8}}F\|_{L_{x}^{\infty}h_{y}^{s}}\|P_{\sim
N}G\|_{L_{x}^{2}h_{y}^{s}}\,.$

Since

$\|P_{<\frac{N}{8}}F\|_{L^{\infty}h^{s}}=\|\frac{1}{2\pi}\int_{\mathbb{R}}\widehat{\varphi}(x^{\prime})F(\cdot+\frac{x^{\prime}}{N})dx^{\prime}\|_{L_{x}^{\infty}h_{y}^{s}}\lesssim\|\widehat{\varphi}\|_{L^{1}(\mathbb{R})}\|F\|_{L_{x}^{\infty}h_{y}^{s}}\,,$

we conclude that

$\|P_{<\frac{N}{8}}FP_{\sim
N}G\|_{L_{x}^{2}h_{y}^{s}}\lesssim\|F\|_{L_{x}^{\infty}h_{y}^{s}}\|P_{\sim
N}G\|_{L_{x}^{2}h_{y}^{s}}\,,$

which gives an acceptable contribution when summing over the frequency scales
$N$. As for the second contribution, we write

$\sum_{M\geq\frac{N}{8}}\|P_{M}FG\|_{L^{2}h^{s}}\lesssim\big{(}\sum_{M\geq\frac{N}{8}}M^{-\sigma}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}P_{M}F\|_{L_{x}^{2}h_{y}^{s}}\big{)}\|G\|_{L_{x}^{\infty}h_{y}^{s}}\,.$

Using Cauchy–Schwarz and the fact that
$\sum_{M\geq\frac{N}{8}}(NM^{-1})^{\sigma}=\mathcal{O}(1)$ when $\sigma>0$ we
obtain

$N^{2\sigma}\big{(}\sum_{M\geq\frac{N}{8}}M^{-\sigma}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}P_{M}F\|_{L_{x}^{2}h_{y}^{s}}\big{)}^{2}\\\
\lesssim\sum_{M\geq\frac{N}{8}}(NM^{-1})^{\sigma}\sum_{M\geq\frac{N}{8}}(NM^{-1})^{\sigma}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}P_{M}F\|_{L_{x}^{2}h_{y}^{s}}^{2}\\\
\lesssim\sum_{M\geq\frac{N}{8}}(NM^{-1})^{\sigma}\|(1-\partial_{x}^{2})^{\frac{\sigma}{2}}P_{M}F\|_{L_{x}^{2}h_{y}^{s}}^{2}\,.$

Since $\sigma>0$ we can then sum over $N$ without loosing derivatives for
fixed $M$, and then sum over $M$ to conclude that this second contribution is
also acceptable. ∎

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