# An optimal regularity result on the quasi-invariant Gaussian measures for
the cubic fourth order nonlinear Schrödinger equation

Tadahiro Oh, Philippe Sosoe, and Nikolay Tzvetkov  Tadahiro Oh
School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Philippe Sosoe
Department of Mathematics
Cornell University
584 Malott Hall
Ithaca
New York 14853
USA<EMAIL_ADDRESS>Nikolay Tzvetkov
Université de Cergy-Pontoise
2, av. Adolphe Chauvin
95302 Cergy-Pontoise Cedex
France<EMAIL_ADDRESS>
###### Abstract.

We study the transport properties of the Gaussian measures on Sobolev spaces
under the dynamics of the cubic fourth order nonlinear Schrödinger equation on
the circle. In particular, we establish an optimal regularity result for
quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The
main new ingredient is an improved energy estimate established by performing
an infinite iteration of normal form reductions on the energy functional.
Furthermore, we show that the dispersion is essential for such a quasi-
invariance result by proving non quasi-invariance of the Gaussian measures
under the dynamics of the dispersionless model.

###### Key words and phrases:

fourth order nonlinear Schrödinger equation; biharmonic nonlinear Schrödinger
equation; quasi-invariant measure; normal form method

###### 2010 Mathematics Subject Classification:

35Q55

###### Contents

  1. 1 Introduction
    1. 1.1 The equation
    2. 1.2 Quasi-invariance of $\mu_{s}$
    3. 1.3 Non quasi-invariance under the dispersionless model
    4. 1.4 Organization of the paper
  2. 2 Notations
  3. 3 Proof of Theorem 1.2: Quasi-invariance of $\mu_{s}$ under the cubic 4NLS
    1. 3.1 Basic reduction of the problem
    2. 3.2 Truncated dynamics
    3. 3.3 Energy estimate
    4. 3.4 Weighted Gaussian measures
    5. 3.5 A change-of-variable formula
    6. 3.6 On the measure evolution property and the proof of Theorem 1.2
  4. 4 Proof of Proposition 3.4: Normal form reductions
    1. 4.1 First few steps of normal form reductions
    2. 4.2 Notations: index by ordered bi-trees
    3. 4.3 Arithmetic lemma
    4. 4.4 Normal form reductions
    5. 4.5 On the error term $\mathcal{N}_{2}^{(J+1)}$
    6. 4.6 Improved energy bound
    7. 4.7 On the proof of Lemma 3.5
  5. 5 Proof of Theorem 1.6: Non quasi-invariance under the dispersionless model
    1. 5.1 Brownian/Ornstein-Uhlenbeck loop
    2. 5.2 Fractional Brownian motion
    3. 5.3 Critical case: $s\in\frac{1}{2}+\mathbb{N}$

##  1\. Introduction

In this paper, we complete the study of the transport properties of Gaussian
measures on Sobolev spaces for the cubic nonlinear Schrödinger equation (NLS)
with quartic dispersion, initiated by the first and third authors in [32].

The question addressed in this work is motivated by a number of perspectives.
In probability theory, absolute continuity properties for the pushforward of
Gaussian measures under linear and nonlinear transformations have been studied
extensively, starting with the classical work of Cameron-Martin; see [7, 25,
40]. More generally, questions of absolute continuity of the distribution of
solutions to differential and stochastic differential equations with respect
to a given initial distribution or some chosen reference measure are also
central to stochastic analysis. For example, close to the topic of the current
paper, see the work of Cruzeiro [11, 12]. We also note a recent work [30]
establishing absolute continuity of the Gaussian measure associated to the
complex Brownian bridge on the circle under certain gauge transformations.

On the other hand, in the analysis of partial differential equations (PDEs),
Hamiltonian PDE dynamics with initial data distributed according to measures
of Gibbs type have been studied intensively over the last two decades,
starting with the work of Bourgain [4, 5]. See [32] for the references
therein. These Gibbs-type measures are constructed as weighted Gaussian
measures and are usually supported on Sobolev spaces of low regularity with
the exception of completely integrable Hamiltonian PDEs such as the cubic NLS
on the circle. In the approach initiated by Bourgain and successfully applied
to many equations since then, invariance of such Gibbs-type measures under the
flow of the equation has been established by combining the Hamiltonian
structure of suitable finite dimensional approximations, in particular
invariance of the finite dimensional Gibbs-type measures, with PDE
approximation arguments. Invariance of such weighted Gaussian measures implies
absolute continuity of the pushforward of the base Gaussian measures. If we
substitute the underlying measure with a different Gaussian measure, however,
the question of absolute continuity becomes non-trivial. See also [6] for a
related question by Gel’fand on building a direct method to prove absolute
continuity properties without relying on invariant measures.

In [43], the third author initiated the study of transport properties of
Gaussian measures under the flow of a Hamiltonian PDE, combining probabilistic
and PDE techniques. The result proved there for a specific Hamiltonian
equation (the generalized BBM equation) went beyond general results on the
pushforwards of Gaussian measures by nonlinear transformations such as Ramer’s
[40]. It was shown in [43] that a key step to showing absolute continuity is
to establish a smoothing effect on the nonlinear part. In [32], the first and
third authors studied the transport of Gaussian measures for the cubic NLS
with quartic dispersion. An additional difficulty compared to [43] is the
absence of explicit smoothing coming from the nonlinearity, thus requiring the
use of dispersion in an explicit manner. In [32], such dispersion was
manifested through the normal form method. In this paper, we improve the
result in [32] to the optimal range of Sobolev exponents by pushing the normal
form method to the limit. Furthermore, we present a result showing that, in
the absence of dispersion, the distribution of the solution of the resulting
dispersionless equation is not absolutely continuous with respect to the
Gaussian initial data for any non-zero time. This in particular establishes
the necessity of dispersion for an absolute continuity property. Since the
linear equation is easily seen to leave the distribution of the Gaussian
initial data invariant, this highlights that the question of transport
properties for a Hamiltonian PDE is a probabilistic manifestation of the
competition between the dispersion and the nonlinear part, familiar for the
study of nonlinear dispersive equations.

###  1.1. The equation

We consider the cubic fourth order nonlinear Schrödinger equation (4NLS) on
the circle $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$:

$\begin{cases}i\partial_{t}u=\partial_{x}^{4}u\pm|u|^{2}u\\\
u|_{t=0}=u_{0},\end{cases}\qquad(x,t)\in\mathbb{T}\times\mathbb{R},$ (1.1)

where $u$ is a complex-valued function on $\mathbb{T}\times\mathbb{R}$. The
equation (1.1) is also called the biharmonic NLS and it was studied in [21,
42] in the context of stability of solitons in magnetic materials. See also
[23, 24, 3, 13] for a more general class of fourth order NLS:

$\displaystyle
i\partial_{t}u=\lambda\partial_{x}^{2}u+\mu\partial_{x}^{4}u\pm|u|^{2}u.$
(1.2)

The equation (1.1) is a Hamiltonian PDE with the conserved Hamiltonian:

$H(u)=\frac{1}{2}\int_{\mathbb{T}}|\partial_{x}^{2}u|^{2}dx\pm\frac{1}{4}\int_{\mathbb{T}}|u|^{4}dx.$

In addition to the Hamiltonian, the flow of the equation (1.1) preserves the
$L^{2}$-norm, or the so-called “mass”:

$M(u)=\int_{\mathbb{T}}|u|^{2}dx.$

This mass conservation law was used in [32] to prove the following sharp
global well-posedness result.

######  Proposition 1.1.

The cubic 4NLS (1.1) is globally well-posed in $H^{\sigma}(\mathbb{T})$ for
$\sigma\geq 0$.

This global well-posedness result in $L^{2}(\mathbb{T})$ is sharp in the sense
that the cubic 4NLS (1.1) is ill-posed in negative Sobolev spaces in the sense
of non-existence of solutions. See [19, 32, 37].

The defocusing/focusing nature of the equation (1.1) does not play any role in
the following. Hence, we assume that it is defocusing, i.e. with the $+$ sign
in (1.1).

###  1.2. Quasi-invariance of $\mu_{s}$

Given $s>\frac{1}{2}$, we consider the mean-zero Gaussian measures $\mu_{s}$
on $L^{2}(\mathbb{T})$ with covariance operator
$2(\text{Id}-\partial_{x}^{2})^{-s}$, formally written as

$\displaystyle
d\mu_{s}=Z_{s}^{-1}e^{-\frac{1}{2}\|u\|_{H^{s}}^{2}}du=\prod_{n\in\mathbb{Z}}Z_{s,n}^{-1}e^{-\frac{1}{2}\langle
n\rangle^{2s}|\widehat{u}_{n}|^{2}}d\widehat{u}_{n}.$ (1.3)

As we see below, the Gaussian measure $\mu_{s}$ is not supported on
$H^{s}(\mathbb{T})$, i.e. $\mu_{s}(H^{s}(\mathbb{T}))=0$, and we need to work
in a larger space. See (1.5). This is due to the infinite dimensionality of
the problem.

The covariance operator is diagonalized by the Fourier basis on $\mathbb{T}$
and the Gaussian measure $\mu_{s}$ defined above is in fact the induced
probability measure under the map111Henceforth, we drop the harmless factor of
$2\pi$, if it does not play an important role.

$\displaystyle\omega\in\Omega\longmapsto
u^{\omega}(x)=u(x;\omega)=\sum_{n\in\mathbb{Z}}\frac{g_{n}(\omega)}{\langle
n\rangle^{s}}e^{inx},$ (1.4)

where $\langle\,\cdot\,\rangle=(1+|\cdot|^{2})^{\frac{1}{2}}$ and
$\\{g_{n}\\}_{n\in\mathbb{Z}}$ is a sequence of independent standard complex-
valued Gaussian random variables on a probability space
$(\Omega,\mathcal{F},P)$, i.e. $\text{Var}(g_{n})=2$. From this random Fourier
series representation, it is easy to see that $u^{\omega}$ in (1.4) lies in
$H^{\sigma}(\mathbb{T})$ almost surely if and only if

$\displaystyle\sigma<s-\frac{1}{2}.$ (1.5)

Lastly, note that, for the same range of $\sigma$, the triplet
$(H^{s},H^{\sigma},\mu_{s})$ forms an abstract Wiener space. See [17, 26].

In the following, we continue to study the transport property of the Gaussian
measure $\mu_{s}$ under the dynamics of the cubic 4NLS (1.1). Before
proceeding further, recall the following definition of quasi-invariant
measures; given a measure space $(X,\mu)$, we say that the measure is quasi-
invariant under a measurable transformation $T:X\to X$ if $\mu$ and the
pushforward of $\mu$ under $T$, defined by $T_{*}\mu=\mu\circ T^{-1}$, are
equivalent, i.e. mutually absolutely continuous with respect to each other.

Our first result improves the quasi-invariance result in [32] to the optimal
range of Sobolev exponents.

######  Theorem 1.2.

Let $s>\frac{1}{2}$. Then, the Gaussian measure $\mu_{s}$ is quasi-invariant
under the flow of the cubic 4NLS (1.1).

Theorem 1.2 improves the main result in [32], where the first and the third
authors proved quasi-invariance of $\mu_{s}$ under (1.1) for $s>\frac{3}{4}$.
Moreover, the regularity $s>\frac{1}{2}$ is optimal since when
$s=\frac{1}{2}$, the Gaussian measure $\mu_{s}$ is supported in
$H^{-\varepsilon}(\mathbb{T})\setminus L^{2}(\mathbb{T})$, while the cubic
4NLS (1.1) is ill-posed in negative Sobolev spaces in the sense of non-
existence of solutions.

As shown in [43], to prove quasi-invariance of $\mu_{s}$, it is essential to
exhibit a smoothing of the nonlinear part of the equation. This can be
understood at an intuitive level by an analogy to the Cameron-Martin theorem:
the Gaussian measures $\mu_{s}$ are quasi-invariant under translations by
fixed vectors in their respective Cameron Martin spaces $H^{s}(\mathbb{T})$.
Since a typical element under $\mu_{s}$ lies in $H^{\sigma}(\mathbb{T})$,
$\sigma<s-\frac{1}{2}$, one needs to show that the nonlinear part represents a
perturbation which is smoother in the Sobolev regularity. The Cameron-Martin
theorem applies only to translation by fixed vectors, but Ramer’s quasi-
invariance result [40] applies to a more general nonlinear transformation on
an abstract Wiener space, although it requires the translations to be more
regular. This was applied in [43] and [32], where it was noted that a direct
application of Ramer’s result yields a suboptimal range on $s$. In [32], we
applied the normal form reduction to the equation and exhibited
$(1+\varepsilon)$-smoothing on the nonlinearity when $s>1$. We then proved
quasi-invariance of $\mu_{s}$ by invoking Ramer’s result. When
$\frac{3}{4}<s\leq 1$, we followed the general approach introduced by the
third author in the context of the (generalized) BBM equation [43]. This
strategy combines an energy estimate with the analysis of the evolution of
truncated measures. As in [32], showing a smoothing of the nonlinear part for
(1.1) requires normal form reductions. The main improvements over [32] here
comes from a more refined implementation of the normal form reductions,
inspired by [18].

In the following, we first describe a rough idea behind this method introduced
in [43]. Let $\Phi(t)$ denote the solution map of (1.1) sending initial data
$u_{0}$ to the solution $u(t)$ at time $t\in\mathbb{R}$. Suppose that we have
a measurable set $A\subset L^{2}(\mathbb{T})$ with $\mu_{s}(A)=0$. Fix non-
zero $t\in\mathbb{R}$. In order to prove quasi-invariance of $\mu_{s}$, we
would like to prove $\mu_{s}(\Phi(t)(A))=0$.222By time reversibility, this
would also yield $\Phi(t)_{*}\mu_{s}(A)=\mu_{s}(\Phi(-t)(A))=0$. The main idea
is to establish the following two properties:

  * (i)

Energy estimate (with smoothing):

$\displaystyle\frac{d}{dt}\|\Phi(t)(u)\|_{H^{s}}^{2}\,\text{``}\leq\text{''}\,C(\|u\|_{L^{2}})\|\Phi(t)u\|_{H^{s-\frac{1}{2}-\varepsilon}}^{2-\theta}$
(1.6)

for some $\theta>0$,333In [34], the first and third authors recently proved
quasi-invariance of $\mu_{s}\otimes\mu_{s-1}$ on $(u,\partial_{t}u)$ under the
dynamics of the two-dimensional cubic nonlinear wave equation (NLW), where
they showed that even when $\theta=0$, we can still apply Yudovich’s argument
in the limiting case and establish a desired estimate of the form (1.7). This
was crucial in proving quasi-invariance of $\mu_{s}\otimes\mu_{s-1}$ under the
cubic NLW on $\mathbb{T}^{2}$.

  * (ii)

A change-of-variable formula:

$\displaystyle\mu_{s}(\Phi(t)(A))$
$\displaystyle=Z_{s}^{-1}\int_{\Phi(t)(A)}e^{-\frac{1}{2}\|u\|_{H^{s}}^{2}}du\
\text{``}=\text{''}\
Z_{s}^{-1}\int_{A}e^{-\frac{1}{2}\|\Phi(t)(u)\|_{H^{s}}^{2}}du.$

Step (i) is an example of local analysis, studying a trajectory of a single
solution, while Step (ii) is an example of global analysis on the phase space.
Combining (i) and (ii), we can study the evolution of $\mu_{s}(\Phi(t)A)$ by
estimating $\frac{d}{dt}\mu_{s}(\Phi(t)(A))$. In particular, by applying
Yudovich’s argument [46], we obtain444Compare (1.7) with a much stronger
estimate in Lemma 3.9.

$\displaystyle\mu_{s}(\Phi(t)(A))\leq
C(t,\delta)\big{(}\mu_{s}(A)\big{)}^{1-\delta}$ (1.7)

for any $\delta>0$. In particular, if $\mu_{s}(A)=0$, then we would have
$\mu_{s}(\Phi(t)(A))=0$. See [43, 32, 33] for details.

As the quotation marks indicate, both (i) and (ii) are not quite true as they
are stated above. In [32], we first performed two transformations to (1.1) and
transformed the equation into the following renormalized equation:

$\displaystyle\partial_{t}\widehat{v}_{n}$
$\displaystyle=-i\sum_{\begin{subarray}{c}n=n_{1}-n_{2}+n_{3}\\\ n\neq
n_{1},n_{3}\end{subarray}}e^{-i\phi(\bar{n})t}\widehat{v}_{n_{1}}\overline{\widehat{v}_{n_{2}}}\widehat{v}_{n_{3}}+i|\widehat{v}_{n}|^{2}\widehat{v}_{n},$
(1.8)

where the phase function $\phi(\bar{n})$ is given by

$\displaystyle\phi(\bar{n})=\phi(n_{1},n_{2},n_{3},n)=n_{1}^{4}-n_{2}^{4}+n_{3}^{4}-n^{4}.$
(1.9)

Note that this reduction of (1.1) to (1.8) via two transformations on the
phase space is another instance of global analysis. See Subsection 3.1. This
reformulation exhibits resonant and non-resonant structure of the nonlinearity
in an explicit manner and moreover it removes certain resonant interactions,
which was crucial in establishing an effective energy estimate in Step (i). By
applying a normal form reduction, we introduced a modified energy
$E_{t}=\|u(t)\|_{H^{s}}^{2}+R_{t}$ for some appropriate correction term
$R_{t}$. See (1.10) - (1.12) below. We then established an energy estimate on
the modified energy $E_{t}$, provided $s>\frac{3}{4}$. In Step (ii), in order
to justify such a change-of-variable formula, we considered a truncated
dynamics. Moreover, we needed to introduce and consider a change-of-variable
formula for a modified measure associated with the modified energy $E_{t}$
introduced in Step (i).

The regularity restriction $s>\frac{3}{4}$ in the previous paper [32] comes
from the energy estimate in Step (i), where we applied the normal form
reduction (namely integration by parts in time) once to the equation:
$\partial_{t}\|u\|_{H^{s}}^{2}=\cdots$ satisfied by the $H^{s}$-energy
functional $\|u\|_{H^{s}}^{2}$. In the following, we prove Theorem 1.2 by
performing normal form reductions infinitely many times. Our normal form
approach is analogous to the approach employed in [1, 27, 18]. In particular,
in [18], the first author (with Guo and Kwon) implemented an infinite
iteration scheme of normal form reductions to prove unconditional well-
posedness of the cubic NLS on $\mathbb{T}$ in low regularity. In [18], we
performed integration by parts in a successive manner, introducing nonlinear
terms of higher and higher degrees. While the nonlinear terms thus introduced
are of higher degrees, they satisfy better estimates. In order to keep track
of all possible ways to perform integration by parts, we introduced the notion
of ordered trees. See also [8] for another example of an infinite iteration of
normal form reductions to prove unconditional well-posedness.

In establishing an improved energy estimate (Proposition 3.4), we perform an
infinite iteration of normal form reductions. It is worthwhile to note that,
unlike [18], we do not work at the level of the equation (1.1). Instead, we
work at the level of the evolution equation
$\partial_{t}\|u\|_{H^{s}}^{2}=\cdots$ satisfied by the $H^{s}$-energy
functional. Let us first go over the computation performed in [32] to show a
flavor of this method. Using (1.8), we have

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|u(t)\|_{H^{s}}^{2}\bigg{)}$
$\displaystyle=\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}$
$\displaystyle=-\operatorname*{Re}i\sum_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}n=n_{1}-n_{2}+n_{3}\\\
n\neq n_{1},n_{3}\end{subarray}}e^{-i\phi(\bar{n})t}\langle
n\rangle^{2s}\widehat{v}_{n_{1}}\overline{\widehat{v}_{n_{2}}}\widehat{v}_{n_{3}}\overline{\widehat{v}_{n}},$
(1.10)

where $v$ is the renormalized variable as in (1.8). Then, differentiating by
parts, i.e. integrating by parts without an integral symbol,555This is indeed
a Poincaré-Dulac normal form reduction applied to the evolution equation
(1.10) for $\frac{1}{2}\|v(t)\|_{H^{s}}^{2}$. See Section 1 in [18] for a
discussion on the relation between differentiation by parts and normal form
reductions. we obtain

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}$
$\displaystyle=\operatorname*{Re}\frac{d}{dt}\bigg{[}\sum_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}n=n_{1}-n_{2}+n_{3}\\\
n\neq
n_{1},n_{3}\end{subarray}}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}\widehat{v}_{n_{1}}\overline{\widehat{v}_{n_{2}}}\widehat{v}_{n_{3}}\overline{\widehat{v}_{n}}\bigg{]}$
$\displaystyle\hphantom{X}-\operatorname*{Re}\sum_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}n=n_{1}-n_{2}+n_{3}\\\
n\neq
n_{1},n_{3}\end{subarray}}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}\partial_{t}\big{(}\widehat{v}_{n_{1}}\overline{\widehat{v}_{n_{2}}}\widehat{v}_{n_{3}}\overline{\widehat{v}_{n}}\big{)}.$
(1.11)

This motivates us to define the first modified energy $E_{t}^{(1)}(v)$ with
the correction term $R_{t}^{(1)}(v)$ by

$\displaystyle E^{(1)}_{t}(v)=$
$\displaystyle\,\frac{1}{2}\|v\|_{H^{s}}^{2}+R_{t}^{(1)}(v)$ $\displaystyle:=$
$\displaystyle\,\frac{1}{2}\|v\|_{H^{s}}^{2}-\operatorname*{Re}\sum_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}n=n_{1}-n_{2}+n_{3}\\\
n\neq
n_{1},n_{3}\end{subarray}}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}\widehat{v}_{n_{1}}\overline{\widehat{v}_{n_{2}}}\widehat{v}_{n_{3}}\overline{\widehat{v}_{n}}.$
(1.12)

This is the modified energy used in the previous work [32] (up to a constant
factor). Note that the time derivative of $E_{t}^{(1)}(v)$ is given by the
second term on the right-hand side of (1.11).

In the second step, we divide the the second term on the right-hand side of
(1.11) into nearly resonant and non-resonant parts and apply differentiation
by parts only to the non-resonant part. When we apply differentiation by parts
as in (1.11) in an iterative manner, the time derivative may fall on any of
the factors $\widehat{v}_{n_{j}}$ and $\widehat{v}_{n}$, generating higher
order nonlinear terms. In general, the structure of such terms can be very
complicated, depending on where the time derivative falls. In [18], ordered
(ternary) trees played an important role for indexing such terms. In our case,
we work on the evolution equation satisfied by the $H^{s}$-energy functional
and we need to consider tree-like structures that grow in two directions. In
Section 4, we introduce the notion of bi-trees and ordered bi-trees for this
purpose.

After $J$ steps of the normal form reductions, we arrive at

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}=\frac{d}{dt}\bigg{(}\sum_{j=2}^{J+1}\mathcal{N}^{(j)}_{0}(v)$
$\displaystyle(t)\bigg{)}+\sum_{j=2}^{J+1}\mathcal{N}^{(j)}_{1}(v)(t)$
$\displaystyle+\sum_{j=2}^{J+1}\mathcal{R}^{(j)}(v)(t)+\mathcal{N}^{(J+1)}_{2}(v)(t).$
(1.13)

Here, $\mathcal{N}^{(j)}_{0}(v)$ consists of certain $2j$-linear terms, while
$\mathcal{N}^{(j)}_{1}(v)$ and $\mathcal{R}^{(j)}(v)$ consist of
$(2j+2)$-linear terms. In practice, we obtain (1.13) for smooth functions with
a truncation parameter $N\in\mathbb{N}$. Here, we can only show that the
remainder term $\mathcal{N}^{(J+1)}_{2}$ satisfies the bound of the form:

$\big{|}\mathcal{N}^{(J+1)}_{2}\big{|}\leq F(N,J),$

with the upper bound $F(N,J)$ satisfying

$\lim_{N\to\infty}F(N,J)=\infty$

for each fixed $J\in\mathbb{N}$. This, however, does not cause an issue since
we also show that

$\lim_{J\to\infty}F(N,J)=0$

for each fixed $N\in\mathbb{N}$. Therefore, by first taking the limit
$J\to\infty$ and then $N\to\infty$, we conclude that the error term
$\mathcal{N}^{(J+1)}_{2}$ vanishes in the limit. See Subsection 4.5. While it
is simple, this observation is crucial in an infinite iteration of the normal
form reductions.

At the end of an infinite iteration of the normal form reductions, we can
rewrite (1.10) as

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}=\frac{d}{dt}\bigg{(}\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0}(v)(t)\bigg{)}+\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{1}(v)(t)+\sum_{j=2}^{\infty}\mathcal{R}^{(j)}(v)(t),$
(1.14)

involving infinite series. The main point of this normal form approach is
that, while the degrees of the nonlinear terms appearing in (1.14) can be
arbitrarily large, we can show that they are all bounded in
$L^{2}(\mathbb{T})$ (in a summable manner over $j$). In particular, by
defining the modified energy $\mathcal{E}_{t}(v)$ by

$\displaystyle\mathcal{E}_{t}(v):=\frac{1}{2}\|v(t)\|_{H^{s}}^{2}-\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0}(v)(t),$
(1.15)

we see that its time derivative is bounded:

$\displaystyle\bigg{|}\frac{d}{dt}\mathcal{E}_{t}(v)\bigg{|}\leq
C_{s}(\|u\|_{L^{2}}),$

satisfying the energy estimate (1.6) in Step (i) with $\theta=2$. See
Proposition 3.4 below. This is the main new ingredient for proving Theorem
1.2. See also the recent work [37] by the first author (with Y. Wang) on an
infinite iteration of normal form reductions for establishing a crucial energy
estimate on the difference of two solutions in proving enhanced uniqueness for
the renormalized cubic 4NLS (see (1.17) below) in negative Sobolev spaces.

######  Remark 1.3.

(i) Heuristically speaking, this infinite iteration of normal form reductions
allows us to exchange analytical difficulty with algebraic/combinatorial
difficulty.

(ii) The “correction term” $R_{t}^{(1)}$ in (1.12) is nothing but the
correction term in the spirit of the $I$-method [9, 10]. In fact, at each step
of normal form reductions, we obtain a correction term
$\mathcal{N}^{(j)}_{0}(v)$. Hence, our improved energy estimate (Proposition
3.4) via an infinite iteration of normal form reductions can be basically
viewed as an implementation of the $I$-method with an infinite sequence
$\big{\\{}\mathcal{N}^{(j)}_{0}(v)\big{\\}}_{j=2}^{\infty}$ of correction
terms. Namely, the modified energy $\mathcal{E}_{t}(v)$ defined in (1.15) is a
modified energy of an infinite order in the $I$-method terminology.666The
highest order of modified energies used in the literature is three in the
application of the $I$-method to the KdV equation [10], corresponding to two
iterations of normal form reductions.

(iii) We point out that a finite iteration of normal form reductions is not
sufficient to go below $s>\frac{3}{4}$. See (6.14) in [32], showing the
restriction $s-\frac{1}{2}>\frac{1}{4}$.

######  Remark 1.4.

Let us briefly discuss the situation for the more general cubic fourth order
NLS (1.2). For this equation, the following phase function

$\displaystyle\phi_{\lambda,\mu}(\bar{n})=-\lambda(n_{1}^{2}-n_{2}^{2}+n_{3}^{2}-n^{2})+\mu(n_{1}^{4}-n_{2}^{4}+n_{3}^{4}-n^{4})$

plays an important role in the analysis. In view of Lemma 3.1 below, we have

$\displaystyle\phi_{\lambda,\mu}(\bar{n})=(n_{1}-n_{2})(n_{1}-n)\big{\\{}-2\lambda+\mu\big{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n^{2}+2(n_{1}+n_{3})^{2}\big{)}\big{\\}}.$
(1.16)

If the last factor in (1.16) does not vanish for any
$n_{1},n_{2},n_{3},n\in\mathbb{Z}$, then we can establish quasi-invariance of
$\mu_{s}$ under (1.2) for $s>\frac{1}{2}$ with the same proof as in [32] and
this paper. It suffices to note that, while we make use of the divisor
counting argument in the proof, we only apply it to
$\mu(\bar{n})=(n_{1}-n_{2})(n_{1}-n)$ and thus the integer/non-integer
character of the last factor in (1.16) is irrelevant.

For example, when $\lambda\mu<0$, the last factor in (1.16) does not vanish
and thus Theorem 1.2 applies to this case. When $\lambda\mu>0$, the non-
resonant condition $2\lambda\not\in\mu\mathbb{N}$ also guarantees the non-
vanishing of the last factor in (1.16). It seems of interest to investigate
the transport property of the Gaussian measure $\mu_{s}$ in the resonant case
$2\lambda\in\mu\mathbb{N}$. In this case, there are more resonant terms and
thus further analysis is required.

######  Remark 1.5.

On the one hand, the cubic 4NLS (1.1) is ill-posed in negative Sobolev spaces
and hence the quasi-invariance result stated in Theorem 1.2 is sharp. On the
other hand, the first author and Y. Wang [37] considered the following
renormalized cubic 4NLS on $\mathbb{T}$:

$\displaystyle\textstyle
i\partial_{t}u=\partial_{x}^{4}u+\big{(}|u|^{2}-2\fint|u|^{2}dx\big{)}u,$
(1.17)

where $\fint_{\mathbb{T}}f(x)dx:=\frac{1}{2\pi}\int_{\mathbb{T}}f(x)dx$. In
particular, they proved global well-posedness of (1.17) in $H^{s}(\mathbb{T})$
for $s>-\frac{1}{3}$. In a very recent work [35], the first and third authors
with Y. Wang went further and constructed global-in-time dynamics for (1.17)
almost surely with respect to the white noise, i.e. the Gaussian measure
$\mu_{s}$ with $s=0$ supported on $H^{\sigma}(\mathbb{T})$,
$\sigma<-\frac{1}{2}$. As a result, they proved invariance of the white noise
$\mu_{0}$ under the renormalized cubic 4NLS (1.17). Invariance is of course a
stronger property than quasi-invariance and hence the white noise is in
particular quasi-invariant under (1.17). The question of quasi-invariance of
$\mu_{s}$ for $s\in(0,\frac{1}{2}]$ under the dynamics of the renormalized
cubic 4NLS (1.17) is therefore a natural sequel of the analysis of this paper.

###  1.3. Non quasi-invariance under the dispersionless model

To motivate our second result, note that, by invariance of the complex-valued
Gaussian random variable $g_{n}$ in (1.4) under rotations, it is clear that
the Gaussian measure $\mu_{s}$ is invariant under the linear dynamics:

$\begin{cases}i\partial_{t}u=\partial_{x}^{4}u\\\
u|_{t=0}=u_{0},\end{cases}\qquad(x,t)\in\mathbb{T}\times\mathbb{R}.$ (1.18)

See Lemma 3.2 (i) below. In particular, $\mu_{s}$ is quasi-invariant under the
linear dynamics (1.18).

In the proof of the quasi-invariance of $\mu_{s}$ under the cubic 4NLS (1.1)
(Theorem 1.2 above), the dispersion plays an essential role. The strong
dispersion allows us to show that the nonlinear part in (1.1) is a
perturbation to the linear equation (1.18). Our next result shows that the
dispersion is indeed essential for Theorem 1.2 to hold.

Consider the following dispersionless model:

$\begin{cases}i\partial_{t}u=|u|^{2}u\\\
u|_{t=0}=u_{0},\end{cases}\qquad(x,t)\in\mathbb{T}\times\mathbb{R}.$ (1.19)

Recall that there is an explicit solution formula for (1.19) given by:

$u(x,t)=e^{-it|u_{0}(x)|^{2}}u_{0}(x)$ (1.20)

at least for continuous initial data such that the pointwise product makes
sense.

Let $s>\frac{1}{2}$. Then, it is easy to see that the random function
$u^{\omega}$ in (1.4) is continuous almost surely. Indeed, by the equivalence
of Gaussian moments and the mean value theorem, we have

$\displaystyle\mathbb{E}\big{[}|u^{\omega}(x)-u^{\omega}(y)|^{p}\big{]}$
$\displaystyle\leq
C_{p}\Big{(}\mathbb{E}\big{[}|u^{\omega}(x)-u^{\omega}(y)|^{2}\big{]}\Big{)}^{\frac{p}{2}}\sim\bigg{(}\sum_{n\in\mathbb{Z}}\frac{1}{\langle
n\rangle^{2s-2\varepsilon}}\bigg{)}^{\frac{p}{2}}|x-y|^{\varepsilon p}$
$\displaystyle\lesssim|x-y|^{\varepsilon p},$

provided that $\varepsilon>0$ is sufficiently small such that
$2s-2\varepsilon>1$. Now, by choosing $p\gg 1$ such that $\varepsilon p>1$, we
can apply Kolmogorov’s continuity criterion and conclude that $u^{\omega}$ in
(1.4) is almost surely continuous when $s>\frac{1}{2}$. This in particular
implies that the solution formula (1.20) is well defined for initial data
distributed according to $\mu_{s}$, $s>\frac{1}{2}$, and the corresponding
solutions exist globally in time. We denote by $\widetilde{\Phi}(t)$ the
solution map for the dispersionless model (1.19).

We now state our second result.

######  Theorem 1.6.

Let $s>\frac{1}{2}$. Then, given $t\neq 0$, the pushforward measure
$\widetilde{\Phi}(t)_{*}\mu_{s}$ under the dynamics of the dispersionless
model (1.19) is not absolutely continuous with respect to the Gaussian measure
$\mu_{s}$. Namely, the Gaussian measure $\mu_{s}$ is not quasi-invariant under
the dispersionless dynamics (1.19).

This is a sharp contrast with the quasi-invariance result for the cubic 4NLS
in Theorem 1.2 and for the cubic NLS for $s\in\mathbb{N}$ (see Remark 1.4 in
[32]). In particular, Theorem 1.6 shows that dispersion is essential for
establishing quasi-invariance of $\mu_{s}$.

We prove this negative result in Theorem 1.6 by establishing that typical
elements under $\mu_{s}$, $s>\frac{1}{2}$, possess an almost surely constant
modulus of continuity at each point. This is the analogue of the classical law
of the iterated logarithm for the Brownian motion. We show that this modulus
of continuity is destroyed with a positive probability by the nonlinear
transformation (1.20) for any non-zero time $t\in\mathbb{R}\setminus\\{0\\}$.

Our proof is based on three basic tools: the Fourier series representation of
the (fractional) Brownian loops, the law of the iterated logarithm, and the
solution formula (1.20) to the dispersionless model (1.19). We will use three
different versions of the law of the iterated logarithm, depending on (i)
$s=1$ corresponding to the Brownian/Ornstein-Uhlenbeck loop, (ii)
$\frac{1}{2}<s<\frac{3}{2}$, corresponding to the fractional Brownian loop,
and (iii) the critical case $s=\frac{3}{2}$. In Cases (i) and (ii), we make
use of the mutual absolute continuity of the function $u$ given in the random
Fourier series (1.4) and the (fractional) Brownian motion on $[0,2\pi)$ to
deduce the law of the iterated logarithm for the random function $u$. In Case
(iii), we directly establish the relevant law of the iterated logarithm for
$u$ in (1.4).777We could apply this argument to directly establish the
relevant law of the iterated logarithm in Cases (i) and (ii) as well. See
Proposition 5.7.

On the one hand, the law of the iterated logarithm yields almost sure
constancy of the modulus of continuity at time $t=0$. On the other hand, we
combine this almost sure constancy of the modulus of continuity at time $t=0$
and the solution formula (1.20) to show that the modulus of continuity at non-
zero time $t\neq 0$ does not satisfy the conclusion of the law of the iterated
logarithm with a positive probability. Lastly, for $s>\frac{3}{2}$, we reduce
the proof to one of Cases (i), (ii), or (iii) by differentiating the random
function.

######  Remark 1.7.

The existence of a quasi-invariant measure shows a delicate persistence
property of the dynamics. In particular, this persistence property due to the
quasi-invariance is stronger than the persistence of regularity888 In the
scaling sub-critical case, by persistence of regularity, we mean the
following; if one proves local well-posedness in $H^{s_{0}}$ for some
$s_{0}\in\mathbb{R}$ and if $u_{0}$ lies in a smoother space $H^{s}$ for some
$s>s_{0}$, then the corresponding solution remains smoother and lies in
$C([-T,T];H^{s})$, where the local existence time $T>0$ depends only on the
$H^{s_{0}}$-norm of the initial condition $u_{0}$.  obtained by the usual
well-posedness theory. While the dispersionless model (1.19) enjoys the
persistence of regularity in $H^{\sigma}(\mathbb{T})$, $\sigma>\frac{1}{2}$,
Theorem 1.6 shows that the Gaussian measure $\mu_{s}$ is not quasi-invariant
under the dynamics of (1.19).

######  Remark 1.8.

(i) For $\varepsilon\in\mathbb{R}$, consider the following 4NLS:

$i\partial_{t}u=\varepsilon\partial_{x}^{4}u+|u|^{2}u.$ (1.21)

For smooth initial data, it is easy to show that, on the unit time interval
$[0,1]$, the corresponding solutions to the small dispersion 4NLS, i.e. (1.21)
with small $\varepsilon\neq 0$, converge to those to the dispersionless model
(1.19) as $\varepsilon\to 0$. See Lemma 4.1 in [36]. In this sense, the small
dispersion 4NLS (1.21) with $|\varepsilon|\ll 1$ is “close” to the
dispersionless model (1.19).

On the other hand, there is a dichotomy in the statistical behavior of
solutions to the small dispersion 4NLS (1.21) and to the dispersionless model
(1.19). When $\varepsilon\neq 0$, one can easily adapt the proof of Theorem
1.2 and prove quasi-invariance of the Gaussian measure $\mu_{s}$. When
$\varepsilon=0$, however, Theorem 1.6 shows that $\mu_{s}$ is not quasi-
invariant under (1.19). This shows a dichotomy between quasi-invariance for
$\varepsilon\neq 0$ and non quasi-invariance for $\varepsilon=0$, while there
is a good approximation property for the deterministic dynamics of the
dispersionless model (1.19) by that of the small dispersion 4NLS (1.21) with
$|\varepsilon|\ll 1$.

(ii) We mention recent work [31, 14] on establishing quasi-invariance of the
Gaussian measure $\mu_{s}$ for Schrödinger-type equations with less
dispersion. In particular, Forlano-Trenberth [14] studied the following
fractional NLS:

$i\partial_{t}u=(-\partial_{x}^{2})^{\frac{\alpha}{2}}u+|u|^{2}u$ (1.22)

and proved quasi-invariance of $\mu_{s}$ (for some non-optimal range of
$s>s_{\alpha}$), provided that $\alpha>1$. When $\alpha=1$, the equation
(1.22) corresponds to the half-wave equation, which does not possess any
dispersion. See [15, 39, 16]. It would be of interest to study the transport
properties of the Gaussian measure $\mu_{s}$ under (1.22) for $0<\alpha<1$. We
also point out that (1.22) for $0<\alpha<1$ also appears as a model in the
study of one-dimensional wave turbulence [29] and hence is a natural model for
statistical study of its solutions.

###  1.4. Organization of the paper

In Section 2, we introduce some notations. In Section 3, we prove Theorem 1.2,
assuming the improved energy estimate (Proposition 3.4). We then present the
proof of Proposition 3.4 in Section 4 by implementing an infinite iteration of
normal form reductions. Lastly, by studying the random Fourier series (1.4)
and the relevant law of the iterated logarithm, we prove non quasi-invariance
of the Gaussian measure $\mu_{s}$ under the dispersionless model (Theorem 1.6)
in Section 5.

##  2\. Notations

Given $N\in\mathbb{N}$, we use $\mathbf{P}_{\leq N}$ to denote the Dirichlet
projection onto the frequencies $\\{|n|\leq N\\}$ and set
$\mathbf{P}_{>N}:=\text{Id}-\mathbf{P}_{\leq N}$. When $N=\infty$, it is
understood that $P_{\leq N}=\text{Id}$. Define $E_{N}$ and $E_{N}^{\perp}$ by

$\displaystyle E_{N}$ $\displaystyle=\mathbf{P}_{\leq
N}L^{2}(\mathbb{T})=\text{span}\\{e^{inx}:|n|\leq N\\},$ $\displaystyle
E_{N}^{\perp}$
$\displaystyle=\mathbf{P}_{>N}L^{2}(\mathbb{T})=\text{span}\\{e^{inx}:|n|>N\\}.$

Given $s>\frac{1}{2}$, let $\mu_{s}$ be the Gaussian measure on
$L^{2}(\mathbb{T})$ defined in (1.3). Then, we can write $\mu_{s}$ as

$\displaystyle\mu_{s}=\mu_{s,N}\otimes\mu_{s,N}^{\perp},$

where $\mu_{s,N}$ and $\mu_{s,N}^{\perp}$ are the marginal distributions of
$\mu_{s}$ restricted onto $E_{N}$ and $E_{N}^{\perp}$, respectively. In other
words, $\mu_{s,N}$ and $\mu_{s,N}^{\perp}$ are the induced probability
measures under the following maps:

$\displaystyle u_{N}:\omega\in\Omega\longmapsto u_{N}(x;\omega)=\sum_{|n|\leq
N}\frac{g_{n}(\omega)}{\langle n\rangle^{s}}e^{inx},$ $\displaystyle
u_{N}^{\perp}:\omega\in\Omega\longmapsto
u_{N}^{\perp}(x;\omega)=\sum_{|n|>N}\frac{g_{n}(\omega)}{\langle
n\rangle^{s}}e^{inx},$

respectively. Formally, we can write $\mu_{s,N}$ and $\mu_{s,N}^{\perp}$ as

$\displaystyle d\mu_{s,N}=Z_{s,N}^{-1}e^{-\frac{1}{2}\|\mathbf{P}_{\leq
N}u_{N}\|_{H^{s}}^{2}}du_{N}\quad\text{and}\quad
d\mu_{s,N}^{\perp}=\widehat{Z}_{s,N}^{-1}e^{-\frac{1}{2}\|\mathbf{P}_{>N}u_{N}^{\perp}\|_{H^{s}}^{2}}du_{N}^{\perp}.$
(2.1)

Given $r>0$, we also define a probability measure $\mu_{s,r}$ with an
$L^{2}$-cutoff by

$\displaystyle d\mu_{s,r}=Z_{s,r}^{-1}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}d\mu_{s}.$

Given a function $v\in L^{2}(\mathbb{T})$, we simply use $v_{n}$ to denote the
Fourier coefficient $\widehat{v}_{n}$ of $v$, when there is no confusion. This
shorthand notation is especially useful in Section 4.

We use $a+$ (and $a-$) to denote $a+\varepsilon$ (and $a-\varepsilon$,
respectively) for arbitrarily small $\varepsilon\ll 1$, where an implicit
constant is allowed to depend on $\varepsilon>0$ (and it usually diverges as
$\varepsilon\to 0$). Given $x\in\mathbb{R}$, we use $\lfloor x\rfloor$ to
denote the integer part of $x$.

In view of the time reversibility of the equations (1.1) and (1.19), we only
consider positive times in the following.

##  3\. Proof of Theorem 1.2: Quasi-invariance of $\mu_{s}$ under the cubic
4NLS

In this section, we present the proof of Theorem 1.2. The main new ingredient
is the improved energy estimate (Proposition 3.4) whose proof is postponed to
Section 4. The remaining part of the proof follows closely the presentation in
[32] and thus we keep our discussion concise.

###  3.1. Basic reduction of the problem

We first go over the basic reduction of the problem from [32]. Given
$t\in\mathbb{R}$, we define a gauge transformation $\mathcal{G}_{t}$ on
$L^{2}(\mathbb{T})$ by setting

$\displaystyle\mathcal{G}_{t}[f]:=e^{2it\fint|f|^{2}}f.$

Given a function $u\in C(\mathbb{R};L^{2}(\mathbb{T}))$, we define
$\mathcal{G}$ by setting

$\mathcal{G}[u](t):=\mathcal{G}_{t}[u(t)].$

Note that $\mathcal{G}$ is invertible and its inverse is given by
$\mathcal{G}^{-1}[u](t)=\mathcal{G}_{-t}[u(t)]$.

Given a solution $u\in C(\mathbb{R};L^{2}(\mathbb{T}))$ to (1.1), let
$\widetilde{u}=\mathcal{G}[u]$. Then, it follows from the mass conservation
that $\widetilde{u}$ is a solution to the following renormalized fourth order
NLS:

$\displaystyle\textstyle
i\partial_{t}\widetilde{u}=\partial_{x}^{4}\widetilde{u}+\big{(}|\widetilde{u}|^{2}-2\fint_{\mathbb{T}}|\widetilde{u}|^{2}dx\big{)}\widetilde{u}.$
(3.1)

This is precisely the renormalized cubic 4NLS in (1.17).

Let $S(t)=e^{-it\partial_{x}^{4}}$ be the solution operator for the linear
fourth order Schrödinger equation (1.18). Denoting by $v=S(-t)\widetilde{u}$
the interaction representation of $\widetilde{u}$, we see that $v$ satisfies
the following equation for $\\{v_{n}\\}_{n\in\mathbb{Z}}$:

$\displaystyle\partial_{t}v_{n}$
$\displaystyle=-i\sum_{\Gamma(n)}e^{-i\phi(\bar{n})t}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}+i|v_{n}|^{2}v_{n}$
$\displaystyle=:\mathcal{N}(v)_{n}+\mathcal{R}(v)_{n},$ (3.2)

where the phase function $\phi(\bar{n})$ is as in (1.9) and the plane
$\Gamma(n)$ is given by

$\displaystyle\Gamma(n)=\\{(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}:\,n=n_{1}-n_{2}+n_{3}\text{
and }n_{1},n_{3}\neq n\\}.$ (3.3)

Recall that the phase function $\phi(\bar{n})$ admits the following
factorization. See [32] for the proof.

######  Lemma 3.1.

Let $n=n_{1}-n_{2}+n_{3}$. Then, we have

$\displaystyle\phi(\bar{n})=(n_{1}-n_{2})(n_{1}-n)\big{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n^{2}+2(n_{1}+n_{3})^{2}\big{)}.$

It follows from Lemma 3.1 that $\phi(\bar{n})\neq 0$ on $\Gamma(\bar{n})$.
Namely, $\mathcal{N}(v)$ and $\mathcal{R}(v)$ on the right-hand side of (3.2)
correspond to the non-resonant and resonant parts, respectively. It follows
from Lemma 3.1 that there is a strong smoothing property on the non-resonant
term $\mathcal{N}(v)$ due to the fast oscillation caused by $\phi(\bar{n})$.

Given $t,\tau\in\mathbb{R}$, let $\Phi(t):L^{2}(\mathbb{T})\to
L^{2}(\mathbb{T})$ be the solution map for (1.1) and
$\Psi(t,\tau):L^{2}(\mathbb{T})\to L^{2}(\mathbb{T})$ be the solution map for
(3.2),999Note that (3.2) is non-autonomous. We point out that this non-
autonomy does not play an essential role in the remaining part of the paper,
since all the estimates hold uniformly in $t\in\mathbb{R}$.  sending initial
data at time $\tau$ to solutions at time $t$. When $\tau=0$, we may denote
$\Psi(t,0)$ by $\Psi(t)$ for simplicity. Then, from
$v=S(-t)\circ\mathcal{G}[u]$, we have

$\displaystyle\Phi(t)=\mathcal{G}^{-1}\circ S(t)\circ\Psi(t).$ (3.4)

Recall the following lemma from [32].

######  Lemma 3.2.

(i) Let $s>\frac{1}{2}$ and $t\in\mathbb{R}$. Then, the Gaussian measure
$\mu_{s}$ defined in (1.3) is invariant under the linear map $S(t)$ and the
gauge transformation $\mathcal{G}_{t}$.

(ii) Let $(X,\mu)$ be a measure space. Suppose that $T_{1}$ and $T_{2}$ are
measurable maps on $X$ into itself such that $\mu$ is quasi-invariant under
$T_{j}$ for each $j=1,2$. Then, $\mu$ is quasi-invariant under $T=T_{1}\circ
T_{2}$.

When $s=1$, Lemma 3.2 (i) basically follows from Theorem 3.1 in [30] which
exploits the properties of the Brownian loop under conformal mappings. For
general $s>\frac{1}{2}$, such approach does not seem to be appropriate. See
Section 4 in [32] for the proof of Lemma 3.2.

In view of this lemma, Theorem 1.2 follows once we prove quasi-invariance of
$\mu_{s}$ under $\Psi(t)$. Therefore, we focus our attention to (3.2) in the
following.

###  3.2. Truncated dynamics

Let us first introduce the following truncated approximation to (3.2):

$\displaystyle\partial_{t}v_{n}$ $\displaystyle=\mathbf{1}_{|n|\leq
N}\bigg{\\{}-i\sum_{\Gamma_{N}(n)}e^{-i\phi(\bar{n})t}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}+i|v_{n}|^{2}v_{n}\bigg{\\}},$
(3.5)

where $\Gamma_{N}(n)$ is defined by

$\displaystyle\Gamma_{N}(n)$
$\displaystyle=\Gamma(n)\cap\\{(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}:|n_{j}|\leq
N\\}$
$\displaystyle=\\{(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}:\,n=n_{1}-n_{2}+n_{3},\,n_{1},n_{3},\neq
n,\text{ and }|n_{j}|\leq N\\}.$

Note that (3.5) is an infinite dimensional system of ODEs for the Fourier
coefficients $\\{v_{n}\\}_{n\in\mathbb{Z}}$, where the flow is constant on the
high frequencies $\\{|n|>N\\}$. We also consider the following finite
dimensional system of ODEs:

$\displaystyle\partial_{t}v_{n}=-i\sum_{\Gamma_{N}(n)}e^{-i\phi(\bar{n})t}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}+i|v_{n}|^{2}v_{n},\qquad|n|\leq
N.$ (3.6)

with $v|_{t=0}=\mathbf{P}_{\leq N}v|_{t=0}$, i.e. $v_{n}|_{t=0}=0$ for
$|n|>N$.

Given $t,\tau\in\mathbb{R}$, denote by $\Psi_{N}(t,\tau)$ and
$\widetilde{\Psi}_{N}(t,\tau)$ the solution maps of (3.5) and (3.6), sending
initial data at time $\tau$ to solutions at time $t$, respectively. For
simplicity, we set

$\displaystyle\Psi_{N}(t)=\Psi_{N}(t,0)\qquad\text{and}\qquad\widetilde{\Psi}_{N}(t)=\widetilde{\Psi}_{N}(t,0)$
(3.7)

when $\tau=0$. Then, we have the following relations:

$\displaystyle\Psi_{N}(t,\tau)=\widetilde{\Psi}_{N}(t,\tau)\mathbf{P}_{\leq
N}+\mathbf{P}_{>N}\qquad\text{and}\qquad\mathbf{P}_{\leq
N}\Psi_{N}(t,\tau)=\widetilde{\Psi}_{N}(t,\tau)\mathbf{P}_{\leq N}.$

We now recall the following approximation property of the truncated dynamics
(3.5).

######  Lemma 3.3 (Proposition 6.21/B.3 in [32]).

Given $R>0$, let $A\subset B_{R}$ be a compact set in $L^{2}(\mathbb{T})$. Fix
$t\in\mathbb{R}$. Then, for any $\varepsilon>0$, there exists
$N_{0}=N_{0}(t,R,\varepsilon)\in\mathbb{N}$ such that we have

$\displaystyle\Psi(t)(A)\subset\Psi_{N}(t)(A+B_{\varepsilon})$

for all $N\geq N_{0}$. Here, $B_{r}$ denotes the ball in $L^{2}(\mathbb{T})$
of radius $r$ centered at the origin.

###  3.3. Energy estimate

In this subsection, we state a crucial energy estimate. The main goal is to
establish an energy estimate of the form (1.6) by introducing a suitable
modified energy functional. We achieve this goal by performing normal form
reductions infinitely many times and thus constructing an infinite sequence of
correction terms.

Let $N\in\mathbb{N}\cup\\{\infty\\}$. In the following, we simply say that $v$
is a solution to (3.6) if $v$ is a solution to (3.6) when $N\in\mathbb{N}$ and
to (3.2) when $N=\infty$.

######  Proposition 3.4.

Let $\frac{1}{2}<s<1$. Then, given $N\in\mathbb{N}\cup\\{\infty\\}$, there
exist multilinear forms
$\big{\\{}\mathcal{N}_{0,N}^{(j)}\big{\\}}_{j=2}^{\infty}$,
$\big{\\{}\mathcal{N}_{1,N}^{(j)}\big{\\}}_{j=2}^{\infty}$, and
$\big{\\{}\mathcal{R}_{N}^{(j)}\big{\\}}_{j=2}^{\infty}$ such that

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}=\frac{d}{dt}\bigg{(}\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0,N}(v)(t)\bigg{)}+\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{1,N}(v)(t)+\sum_{j=2}^{\infty}\mathcal{R}_{N}^{(j)}(v)(t),$

for any solution $v\in C(\mathbb{R};H^{s}(\mathbb{T}))$ to (3.6). Here,
$\mathcal{N}_{0,N}^{(j)}$ are $2j$-linear forms, while
$\mathcal{N}_{1,N}^{(j)}$ and $\mathcal{R}_{N}^{(j)}$ are $(2j+2)$-linear
forms, satisfying the following bounds on $L^{2}(\mathbb{T})$; there exist
positive constants $C_{0}(j)$, $C_{1}(j)$, and $C_{2}(j)$ decaying faster than
any exponential rate101010In fact, by slightly modifying the proof, we can
make $C_{0}(j)$, $C_{1}(j)$, and $C_{2}(j)$ decay as fast as we want as
$j\to\infty$. as $j\to\infty$ such that

$\displaystyle|\mathcal{N}^{(j)}_{0,N}(v)(t)|$ $\displaystyle\lesssim
C_{0}(j)\|v\|_{L^{2}}^{2j},$ $\displaystyle|\mathcal{N}^{(j)}_{1,N}(v)(t)|$
$\displaystyle\lesssim C_{1}(j)\|v\|_{L^{2}}^{2j+2},$
$\displaystyle|\mathcal{R}^{(j)}_{N}(v)(t)|$ $\displaystyle\lesssim
C_{2}(j)\|v\|_{L^{2}}^{2j+2},$

for $j=2,3,\dots$. Note that these constants $C_{0}(j)$, $C_{1}(j)$, and
$C_{2}(j)$ are independent of the cutoff size $N\in\mathbb{N}\cup\\{\infty\\}$
and $t\in\mathbb{R}$.

Define the modified energy $\mathcal{E}_{N,t}(v)$ by

$\displaystyle\mathcal{E}_{N,t}(v):=\frac{1}{2}\|v(t)\|_{H^{s}}^{2}-\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0,N}(v)(t).$
(3.8)

Then, the following energy estimate holds:

$\displaystyle\bigg{|}\frac{d}{dt}\mathcal{E}_{N,t}(v)\bigg{|}\leq
C_{s}(\|v\|_{L^{2}})$ (3.9)

for any solution $v\in C(\mathbb{R};H^{s}(\mathbb{T}))$ to (3.6), uniformly in
$N\in\mathbb{N}\cup\\{\infty\\}$ and $t\in\mathbb{R}$.

In the remaining part of this section, we continue with the proof of Theorem
1.2, assuming Proposition 3.4. We present the proof of Proposition 3.4 in
Section 4. See Lemmas 4.10 and 4.11. In the following, we simply denote
$\mathcal{E}_{\infty,t}$ by $\mathcal{E}_{t}$ and drop the subscript
$N=\infty$ from the multilinear forms, when $N=\infty$. For example, we write
$\mathcal{N}^{(j)}_{0}$ for $\mathcal{N}^{(j)}_{0,\infty}$.

###  3.4. Weighted Gaussian measures

Let $s>\frac{1}{2}$. As in [32], we would like to define the weighted Gaussian
measures associated with the modified energies $\mathcal{E}_{N,t}(v)$ and
$\mathcal{E}_{t}(v)$:111111Noting that we have $\mathbf{P}_{\leq N}v=v$ for
all solutions to (3.6), we have $\mathcal{E}_{N,t}(\mathbf{P}_{\leq
N}v)=\mathcal{E}_{N,t}(v)$. In the following, we explicitly insert
$\mathbf{P}_{\leq N}$ for clarity. A similar comment applies to
$\mathcal{N}^{(j)}_{0,N}$.

$\displaystyle d\rho_{s,N,r,t}$
$\displaystyle=\text{``}Z_{s,N,r}^{-1}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{-\mathcal{E}_{N,t}(\mathbf{P}_{\leq N}v)}dv\text{''}$
$\displaystyle=Z_{s,N,r}^{-1}F_{N,r,t}d\mu_{s}$ (3.10)

and

$\displaystyle d\rho_{s,r,t}$
$\displaystyle=\text{``}Z_{s,r}^{-1}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{-\mathcal{E}_{t}(v)}dv\text{''}$
$\displaystyle=Z_{s,r}^{-1}F_{r,t}d\mu_{s},$ (3.11)

where $\mathcal{E}_{N,t}$ is the modified energy defined in (3.8) and
$F_{N,r,t}$ and $F_{r,t}$ are given by

$\displaystyle F_{N,r,t}(v)$ $\displaystyle:=\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}\exp\bigg{(}\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0,N}(\mathbf{P}_{\leq
N}v)(t)\bigg{)},$ (3.12) $\displaystyle F_{r,t}(v)$
$\displaystyle=F_{r,\infty,t}(v):=\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}\exp\bigg{(}\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0}(v)(t)\bigg{)}.$

It follows from Proposition 3.4 that

$\displaystyle\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}\exp\bigg{(}\sum_{j=2}^{\infty}\big{|}\mathcal{N}^{(j)}_{0,N}(\mathbf{P}_{\leq
N}v)\big{|}\bigg{)}$
$\displaystyle\leq\exp\bigg{(}\sum_{j=2}^{\infty}C_{0}(j)r^{2j}\bigg{)}\leq
C(s,r),$

uniformly in $N\in\mathbb{N}\cup\\{\infty\\}$ and $t\in\mathbb{R}$. Hence, we
have

$\displaystyle Z_{s,N,r}=\int_{H^{s-\frac{1}{2}-}}F_{N,r,t}(v)d\mu_{s}\leq
C(s,r),$

uniformly in $N\in\mathbb{N}\cup\\{\infty\\}$ and $t\in\mathbb{R}$. See also
Remark 3.6 below. This shows that $\rho_{s,N,r,t}$ and $\rho_{s,r,t}$ in
(3.10) and (3.11) are well defined probability measures on
$H^{s-\frac{1}{2}-\varepsilon}(\mathbb{T})$, $\varepsilon>0$. Moreover, the
following lemma immediately follows from the computation above as in [32,
Proposition 6.2 and Corollary 6.3]. See Subsection 4.7.

######  Lemma 3.5.

Let $s>\frac{1}{2}$ and $r>0$.

(i) Given any finite $p\geq 1$, $F_{N,r,t}(v)$ converges to $F_{r,t}(v)$ in
$L^{p}(\mu_{s})$, uniformly in $t\in\mathbb{R}$, as $N\to\infty$.

(ii) For any $\gamma>0$, there exists $N_{0}\in\mathbb{N}$ such that

$|\rho_{s,N,r,t}(A)-\rho_{s,r,t}(A)|<\gamma$

for any $N\geq N_{0}$ and any measurable set $A\subset L^{2}(\mathbb{T})$,
uniformly in $t\in\mathbb{R}$.

######  Remark 3.6.

The normalizing constants $Z_{s,N,r}$ and $Z_{s,r}$ a priori depend on
$t\in\mathbb{R}$. It is, however, easy to see that they are indeed independent
of $t\in\mathbb{R}$ by (i) noticing that the correction terms
$\big{\\{}\mathcal{N}^{(j)}_{0,N}\big{\\}}_{j=2}^{\infty}$ defined in
Proposition 3.4 is in fact autonomous in terms of $\widetilde{u}(t)=S(t)v(t)$
and (ii) the invariance of $\mu_{s}$ under $S(t)$ (Lemma 3.2). See also Remark
4.9 below. The same comment applies to the normalizing constant
$\widehat{Z}_{s,N,r}$ defined in (3.13).

###  3.5. A change-of-variable formula

Next, we go over an important global aspect of the proof of Theorem 1.2. Given
$N\in\mathbb{N}$, let $dL_{N}=\prod_{|n|\leq N}d\widehat{u}_{n}$ denote the
Lebesgue measure on $E_{N}\cong\mathbb{C}^{2N+1}$. Then, from (3.10) and
(3.12) with (2.1), we have

$\displaystyle d\rho_{s,N,r,t}$
$\displaystyle=Z_{s,N,r}^{-1}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0,N}(\mathbf{P}_{\leq
N}v)}d\mu_{s}$
$\displaystyle=\widehat{Z}_{s,N,r}^{-1}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{-\mathcal{E}_{N,t}(\mathbf{P}_{\leq N}v)}dL_{N}\otimes
d\mu_{s,N}^{\perp},$

where $\widehat{Z}_{s,N,r}$ is a normalizing constant defined by

$\displaystyle\widehat{Z}_{s,N,r}=\int_{L^{2}}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{-\mathcal{E}_{N,t}(\mathbf{P}_{\leq N}v)}dL_{N}\otimes
d\mu_{s,N}^{\perp}.$ (3.13)

Then, proceeding as in [32] and exploiting invariance of $L_{N}$ under the map
$\widetilde{\Psi}_{N}(t,\tau)$ for (3.6), we have the following change-of-
variable formula.

######  Lemma 3.7.

Let $s>\frac{1}{2}$, $N\in\mathbb{N}$, and $r>0$. Then, we have

$\displaystyle\rho_{s,N,r,t}(\Psi_{N}(t,\tau)(A))$
$\displaystyle=Z_{s,N,r}^{-1}\int_{\Psi_{N}(t,\tau)(A)}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0,N}(\mathbf{P}_{\leq
N}v)}d\mu_{s}(v)$
$\displaystyle=\widehat{Z}_{s,N,r}^{-1}\int_{A}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{-\mathcal{E}_{N,t}(\mathbf{P}_{\leq
N}\Psi_{N}(t,\tau)(v))}dL_{N}\otimes d\mu_{s,N}^{\perp}$

for any $t,\tau\in\mathbb{R}$ and any measurable set $A\subset
L^{2}(\mathbb{T})$. Here, $\Psi_{N}(t,\tau)$ is the solution map to (3.5)
defined in (3.7).

###  3.6. On the measure evolution property and the proof of Theorem 1.2

In this subsection, we use the energy estimate (Proposition 3.4) and the
change-of-variable formula (Lemma 3.7) to establish a growth estimate on the
truncated weighted Gaussian measure $\rho_{s,N,r,t}$ under
$\Psi_{N}(t)=\Psi_{N}(t,0)$ for (3.5). Thanks to the improved energy estimate,
the following estimates are simpler than those presented in [32].

######  Lemma 3.8.

Let $\frac{1}{2}<s<1$. Then, given $r>0$, there exists $C=C(r)>0$ such that

$\displaystyle\frac{d}{dt}\rho_{s,N,r,t}(\Psi_{N}(t)(A))\leq
C\rho_{s,N,r,t}(\Psi_{N}(t)(A))$ (3.14)

for any $N\in\mathbb{N}$, any $t\in\mathbb{R}$, and any measurable set
$A\subset L^{2}(\mathbb{T})$. As a consequence, we have the following
estimate; given $t\in\mathbb{R}$ and $r>0$, there exists $C=C(t,r)>0$ such
that

$\displaystyle\rho_{s,N,r,t}(\Psi_{N}(t)(A))\leq C\rho_{s,N,r,t}(A)$ (3.15)

for any $N\in\mathbb{N}$ and any measurable set $A\subset L^{2}(\mathbb{T})$.

###### Proof.

As in [44, 45, 43, 32], the main idea of the proof of Lemma 3.8 is to reduce
the analysis to that at $t=0$ in the spirit of the classical Liouville theorem
on Hamiltonian dynamics. Let $t_{0}\in\mathbb{R}$. By the definition of
$\Psi(t,\tau)$, Lemma 3.7, and Proposition 3.4, we have

$\displaystyle\frac{d}{dt}\rho_{s,N,r,t}$
$\displaystyle(\Psi_{N}(t)(A))\bigg{|}_{t=t_{0}}$
$\displaystyle=\frac{d}{dt}\rho_{s,N,r,t_{0}+t}\big{(}\Psi_{N}(t_{0}+t,t_{0})(\Psi_{N}(t_{0})(A))\big{)}\bigg{|}_{t=0}$
$\displaystyle=\widehat{Z}_{s,N,r}^{-1}\frac{d}{dt}\int_{\Psi_{N}(t_{0})(A)}\mathbf{1}_{\\{\|v\|_{L^{2}}\leq
r\\}}e^{-\mathcal{E}_{N,t_{0}+t}(\mathbf{P}_{\leq
N}\Psi_{N}(t_{0}+t,t_{0})(v))}dL_{N}\otimes d\mu_{s,N}^{\perp}\bigg{|}_{t=0}$
$\displaystyle=-\int_{\Psi_{N}(t_{0})(A)}\frac{d}{dt}\mathcal{E}_{N,t_{0}+t}\big{(}\mathbf{P}_{\leq
N}\Psi_{N}(t_{0}+t,t_{0})(v)\big{)}\bigg{|}_{t=0}d\rho_{s,N,r,t_{0}}$
$\displaystyle\leq C_{r}\rho_{s,N,r,t_{0}}(\Psi_{N}(t_{0})(A)).$

This proves (3.14). The second estimate (3.15) follows from a direct
integration of (3.14). ∎

As in [32], we can upgrade Lemma 3.8 to the untruncated measure
$\rho_{s,r,t}$.

######  Lemma 3.9.

Let $\frac{1}{2}<s<1$. Then, given $t\in\mathbb{R}$ and $r>0$, there exists
$C=C(t,r)>0$ such that

$\displaystyle\rho_{s,r,t}(\Psi(t)(A))\leq C\rho_{s,r,t}(A)$

for any measurable set $A\subset L^{2}(\mathbb{T})$.

This lemma follows from the approximation properties of $\Psi_{N}(t)$ to
$\Psi(t)$ (Lemma 3.3) and $\rho_{s,N,r,t}$ to $\rho_{s,r,t}$ (Lemma 3.5 (ii))
along with some limiting argument. See [32, Lemma 6.10] for the details of the
proof.

Once we have Lemma 3.9, the proof of Theorem 1.2 follows just as in [32]. We
present its proof for the convenience of readers. Recall that in view of (3.4)
and Lemmas 3.2, that it suffices to prove that $\mu_{s}$ is quasi-invariant
under $\Psi(t)$, i.e. under the dynamics of (3.2).

Fix $t\in\mathbb{R}$. Let $A\subset L^{2}(\mathbb{T})$ be a measurable set
such that $\mu_{s}(A)=0$. Then, for any $r>0$, we have

$\mu_{s,r}(A)=0.$

By the mutual absolute continuity of $\mu_{s,r}$ and $\rho_{s,r,t}$, we obtain

$\rho_{s,r,t}(A)=0$

for any $r>0$. Then, by Lemma 3.9, we have

$\rho_{s,r,t}(\Psi(t)(A))=0.$

By invoking the mutual absolute continuity of $\mu_{s,r}$ and $\rho_{s,r,t}$
once again, we have

$\mu_{s,r}(\Psi(t)(A))=0.$

Then, the dominated convergence theorem yields

$\mu_{s}\big{(}\Psi(t)(A)\big{)}=\lim_{r\to\infty}\mu_{s,r}\big{(}\Psi(t)(A)\big{)}=0.$

This proves Theorem 1.2, assuming Proposition 3.4. In the next section, we
implement an infinite iteration of normal form reductions and prove the
improved energy estimate (Proposition 3.4).

##  4\. Proof of Proposition 3.4: Normal form reductions

In this section, we present the proof of Proposition 3.4 by implementing an
infinite iteration scheme of normal form reductions. This procedure allows us
to construct an infinite sequences of correction terms and thus build the
desired modified energies $\mathcal{E}_{N,t}(v)$ and $\mathcal{E}_{t}(v)$ in
(3.8).

Our main goal is to obtain an effective estimate on the growth of the
$H^{s}$-norm of a solution $v$ to the truncated equation (3.6), independent of
$N\in\mathbb{N}$. For simplicity of presentation, however, we work on the
equation (3.2) without the frequency cutoff $\mathbf{1}_{|n|\leq N}$ in the
following. We point out that the same normal form reductions and estimates
hold for the truncated equation (3.6), uniformly in $N\in\mathbb{N}$, with
straightforward modifications: (i) set $\widehat{v}_{n}=0$ for all $|n|>N$ and
(ii) the multilinear forms for (3.6) are obtained by inserting the frequency
cutoff $\mathbf{1}_{|n|\leq N}$ in appropriate places.121212Using the bi-trees
introduced in Subsection 4.2 below, it follows from (3.6) that we simply need
to insert the frequency cutoff $\mathbf{1}_{|n^{(j)}|\leq N}$ on the parental
frequency $n^{(j)}$ assigned to each non-terminal node $a\in\mathcal{T}^{0}$.
In the following, we introduce multilinear forms such as
$\mathcal{N}_{0}^{(j)}$, $\mathcal{N}_{1}^{(j)}$, and $\mathcal{R}^{(j)}$ for
the untruncated equation (3.2). With a small modification, these multilinear
forms give rise to $\mathcal{N}_{0,N}^{(j)}$, $\mathcal{N}_{1,N}^{(j)}$, and
$\mathcal{R}_{N}^{(j)}$, $N\in\mathbb{N}$, for the truncated equation (3.6),
appearing in Proposition 3.4. See Subsection 4.7.

###  4.1. First few steps of normal form reductions

In the following, we describe the first few steps of normal form reductions.
We keep the following discussion only at a formal level since its purpose is
to show the complexity of the problem and the necessity of effective book-
keeping notations that we introduce in the next subsection. We will present
the full procedure in Subsections 4.4 and 4.5.

Let $v\in C(\mathbb{R};H^{\infty}(\mathbb{T}))$ be a global solution to
(3.2).131313While we work with (3.2) without a frequency cutoff in the
following, it follows from the uniform boundedness of the frequency truncation
operator $\mathbf{P}_{\leq N}$ that our argument and estimates also hold for
(3.6), uniformly in $N\in\mathbb{N}$. Noting that any solution to (3.6) (for
some $N\in\mathbb{N}$) is smooth, the following computation can be easily
justified for solutions to (3.6).  With $\phi(\bar{n})$ and $\Gamma(n)$ as in
(1.9) and (3.3), we have141414Recall our convention of using $v_{n}$ to denote
the Fourier coefficient $\widehat{v}_{n}$.

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}$
$\displaystyle=-\operatorname*{Re}i\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\langle
n\rangle^{2s}e^{-i\phi(\bar{n})t}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}}$
$\displaystyle=:\mathcal{N}^{(1)}(v)(t).$ (4.1)

In view of Lemma 3.1 with (3.3), we have $|\phi(\bar{n})|\geq 1$ in the
summation above. Then, by performing a normal form reduction, namely,
differentiating by parts as in (1.11), we obtain

$\displaystyle\mathcal{N}^{(1)}(v)(t)$
$\displaystyle=\operatorname*{Re}\frac{d}{dt}\bigg{[}\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}}\bigg{]}$
$\displaystyle\hphantom{X}-\operatorname*{Re}\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}\partial_{t}\big{(}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}}\big{)}$
$\displaystyle=\operatorname*{Re}\frac{d}{dt}\bigg{[}\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}}\bigg{]}$
$\displaystyle\hphantom{X}-\operatorname*{Re}\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}\Big{\\{}\mathcal{R}(v)_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}}$
$\displaystyle\hphantom{XXXXX}+v_{n_{1}}\overline{\mathcal{R}(v)_{n_{2}}}v_{n_{3}}\overline{v_{n}}+v_{n_{1}}\overline{v_{n_{2}}}\mathcal{R}(v)_{n_{3}}\overline{v_{n}}+v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{\mathcal{R}(v)_{n}}\Big{\\}}$
$\displaystyle\hphantom{X}-\operatorname*{Re}\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\frac{e^{-i\phi(\bar{n})t}}{\phi(\bar{n})}\langle
n\rangle^{2s}\Big{\\{}\mathcal{N}(v)_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}}$
$\displaystyle\hphantom{XXXXX}+v_{n_{1}}\overline{\mathcal{N}(v)_{n_{2}}}v_{n_{3}}\overline{v_{n}}+v_{n_{1}}\overline{v_{n_{2}}}\mathcal{N}(v)_{n_{3}}\overline{v_{n}}+v_{n_{1}}\overline{v_{n_{2}}}v_{n_{3}}\overline{\mathcal{N}(v)_{n}}\Big{\\}}$
$\displaystyle=:\partial_{t}\mathcal{N}_{0}^{(2)}(v)(t)+\mathcal{R}^{(2)}(v)(t)+\mathcal{N}^{(2)}(v)(t).$
(4.2)

In the second equality, we applied the product rule and used the equation
(3.2) to replace $\partial_{t}v_{n_{j}}$ (and $\partial_{t}v_{n}$,
respectively) by the resonant part $\mathcal{R}(v)_{n_{j}}$ (and
$\mathcal{R}(v)_{n}$, respectively) and the non-resonant part
$\mathcal{N}(v)_{n_{j}}$ (and $\mathcal{N}(v)_{n}$, respectively).

As we see below, we can estimate the boundary term $\mathcal{N}_{0}^{(2)}$ and
the contribution $\mathcal{R}^{(2)}$ from the resonant part in a
straightforward manner.

######  Lemma 4.1.

Let $\mathcal{N}^{(2)}_{0}$ and $\mathcal{R}^{(2)}$ be as in (4.2). Then, we
have

$\displaystyle|\mathcal{N}^{(2)}_{0}(v)|$
$\displaystyle\lesssim\|v\|_{L^{2}}^{4},$
$\displaystyle|\mathcal{R}^{(2)}(v)|$
$\displaystyle\lesssim\|v\|_{L^{2}}^{6}.$

See Lemma 4.10 (with $J=1$) for the proof.

It remains to treat the last term $\mathcal{N}^{(2)}$ in (4.2). For an
expository purpose, we only consider the first term among the four terms in
$\mathcal{N}^{(2)}$ in the following. A full consideration is given in
Subsection 4.4 once we introduce proper notations in the next subsection. With
(3.2), we have

$\displaystyle\mathcal{N}^{(2)}(v)(t)$
$\displaystyle=\operatorname*{Re}i\sum_{n\in\mathbb{Z}}\sum_{\Gamma(n)}\sum_{\begin{subarray}{c}n_{1}=m_{1}-m_{2}+m_{3}\\\
n_{1}\neq
m_{1},m_{3}\end{subarray}}\frac{e^{-i(\phi_{1}+\phi_{2})t}}{\phi_{1}}\langle
n\rangle^{2s}(v_{m_{1}}\overline{v_{m_{2}}}v_{m_{3}})\overline{v_{n_{2}}}v_{n_{3}}\overline{v_{n}},$
(4.3)

where $\phi_{1}=\phi(\bar{n})$ and
$\phi_{2}=\phi(m_{1},m_{2},m_{3},n_{1})=m_{1}^{4}-m_{2}^{4}+m_{3}^{4}-n_{1}^{4}$
denote the phase functions from the first and second “generations”.151515In
the next subsection, we make a precise definition of what we mean by
“generation”. It turns out that we can not establish a direct 6-linear
estimate on $\mathcal{N}^{(2)}$ in (4.3).

We divide the frequency space in (4.3) into

$C_{1}=\big{\\{}|\phi_{1}+\phi_{2}|\leq 6^{3}\big{\\}}$ (4.4)

and its complement $C_{1}^{c}$.161616Clearly, the number $6^{3}$ in (4.4) does
not make any difference at this point. However, we insert it to match with
(4.11). See also (4.7). We then write $\mathcal{N}^{(2)}$ as

$\mathcal{N}^{(2)}=\mathcal{N}^{(2)}_{1}+\mathcal{N}_{2}^{(2)},$ (4.5)

where $\mathcal{N}^{(2)}_{1}$ is the restriction of $\mathcal{N}^{(2)}$ onto
$C_{1}$ and $\mathcal{N}_{2}^{(2)}:=\mathcal{N}^{(2)}-\mathcal{N}^{(2)}_{1}$.
On the one hand, we can estimate the contribution $\mathcal{N}^{(2)}_{1}$ from
$C_{1}$ in an effective manner (Lemma 4.11 with $J=1$) thanks to the frequency
restriction on $C_{1}$. On the other hand, $\mathcal{N}_{2}^{(2)}$ can not be
handled as it is and thus we apply the second step of normal form reductions
to $\mathcal{N}^{(2)}_{2}$. After differentiation by parts with (3.2) as in
(4.2), we arrive at

$\displaystyle\mathcal{N}^{(2)}_{2}(v)$
$\displaystyle=\partial_{t}\mathcal{N}^{(3)}_{0}(v)+\mathcal{R}^{(3)}(v)+\mathcal{N}^{(3)}(v),$

where $\mathcal{N}^{(3)}_{0}$ is a 6-linear form and $\mathcal{R}^{(3)}$ and
$\mathcal{N}^{(3)}$ are 8-linear forms, corresponding to the contributions
from the resonant part $\mathcal{R}(v)$ and the non-resonant part
$\mathcal{N}(v)$ upon the substitution of (3.2). See (4.19) below for the
precise computation.

As in the previous step, we can estimate $\mathcal{N}^{(3)}_{0}$ and
$\mathcal{R}^{(3)}$ in a straightforward manner (Lemma 4.10 with $J=2$). On
the other hand, we can not estimate $\mathcal{N}^{(3)}$ as it is and hence we
need to split it as

$\mathcal{N}^{(3)}=\mathcal{N}^{(3)}_{1}+\mathcal{N}^{(3)}_{2},$ (4.6)

where $\mathcal{N}^{(3)}_{1}$ is the restriction of $\mathcal{N}^{(3)}$ onto

$C_{2}=\big{\\{}|\phi_{1}+\phi_{2}+\phi_{3}|\leq 8^{3}\big{\\}}$ (4.7)

and $\mathcal{N}_{2}^{(3)}:=\mathcal{N}^{(3)}-\mathcal{N}^{(3)}_{1}$. Here,
$\phi_{j}$, $j=1,2,3$, denotes the phase function from the $j$th “generation”.
As we see below, $\mathcal{N}^{(3)}_{1}$ satisfies a good 8-linear estimate
(Lemma 4.11 with $J=2$) thanks to the frequency restriction on $C_{2}$. On the
other hand, $\mathcal{N}_{2}^{(3)}$ can not be handled as it is and thus we
apply the third step of normal form reductions to $\mathcal{N}^{(3)}_{2}$. In
this way, we iterate normal form reductions in an indefinite manner.

As we iteratively apply normal form reductions, the degrees of the multilinear
terms increase linearly. After $J$ steps, we obtain the multilinear terms
$\mathcal{N}_{0}^{(J+1)}$ of degree $2J+2$ and $\mathcal{R}^{(J+1)}$ and
$\mathcal{N}^{(J+1)}$ of degree $2J+4$. See (4.4). As in the first and second
steps described above, we also divide $\mathcal{N}^{(J+1)}$ into “good” and
“bad” parts and apply another normal form reduction to the bad part of degree
$2J+4$, where time differentiation can fall on any of the $2J+4$ factors. An
easy computation shows that the number of terms grows factorially (see (4.9))
and hence we need to introduce an effective way to handle this combinatorial
complexity. In the next subsection, we introduce indexing notation by bi-
trees, which allows us to denote a factorially growing number of multilinear
terms in a concise manner.

###  4.2. Notations: index by ordered bi-trees

In [18], the first author with Guo and Kwon implemented an infinite iteration
of normal form reductions to study the cubic NLS on $\mathbb{T}$, where
differentiation by parts was applied to the evolution equation satisfied by
the interaction representation. In [18], (ternary) trees and ordered trees
played an important role for indexing various multilinear terms and
frequencies arising in the general steps of the Poincaré-Dulac normal form
reductions.

Our main goal here is to implement an infinite iteration scheme of normal form
reduction applied to the $H^{s}$-energy functional171717More precisely, to the
evolution equation satisfied by the $H^{s}$-energy functional.
$\|v(t)\|_{H^{s}}^{2}$, as we saw above. In particular, we need tree-like
structures that grow in two directions. For this purpose, we introduce the
notion of bi-trees and ordered bi-trees in the following. Once we replace
trees and ordered trees by bi-trees and ordered bi-trees, other related
notions can be defined in a similar manner as in [18] with certain differences
to be noted.

######  Definition 4.2.

Given a partially ordered set $\mathcal{T}$ with partial order $\leq$, we say
that $b\in\mathcal{T}$ with $b\leq a$ and $b\neq a$ is a child of
$a\in\mathcal{T}$, if $b\leq c\leq a$ implies either $c=a$ or $c=b$. If the
latter condition holds, we also say that $a$ is the parent of $b$.

As in [18], our trees in this paper refer to a particular subclass of usual
trees with the following properties.

######  Definition 4.3.

(i) A tree $\mathcal{T}$ is a finite partially ordered set satisfying the
following properties:

  1. (a)

Let $a_{1},a_{2},a_{3},a_{4}\in\mathcal{T}$. If $a_{4}\leq a_{2}\leq a_{1}$
and $a_{4}\leq a_{3}\leq a_{1}$, then we have $a_{2}\leq a_{3}$ or $a_{3}\leq
a_{2}$,

  2. (b)

A node $a\in\mathcal{T}$ is called terminal, if it has no child. A non-
terminal node $a\in\mathcal{T}$ is a node with exactly three ordered181818For
example, we simply label the three children as $a_{1},a_{2}$, and $a_{3}$ by
moving from left to right in the planar graphical representation of the tree
$\mathcal{T}$. As we see below, we assign the Fourier coefficients of the
interaction representation $v$ at $a_{1}$ and $a_{3}$, while we assign the
complex conjugate of the Fourier coefficients of $v$ at the second child
$a_{2}$. children denoted by $a_{1},a_{2}$, and $a_{3}$,

  3. (c)

There exists a maximal element $r\in\mathcal{T}$ (called the root node) such
that $a\leq r$ for all $a\in\mathcal{T}$,

  4. (d)

$\mathcal{T}$ consists of the disjoint union of $\mathcal{T}^{0}$ and
$\mathcal{T}^{\infty}$, where $\mathcal{T}^{0}$ and $\mathcal{T}^{\infty}$
denote the collections of non-terminal nodes and terminal nodes, respectively.

(ii) A bi-tree $\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2}$ is a disjoint
union of two trees $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, where the root
nodes $r_{j}$ of $\mathcal{T}_{j}$, $j=1,2$, are joined by an edge. A bi-tree
$\mathcal{T}$ consists of the disjoint union of $\mathcal{T}^{0}$ and
$\mathcal{T}^{\infty}$, where $\mathcal{T}^{0}$ and $\mathcal{T}^{\infty}$
denote the collections of non-terminal nodes and terminal nodes, respectively.
By convention, we assume that the root node $r_{1}$ of the tree
$\mathcal{T}_{1}$ is non-terminal, while the root node $r_{2}$ of the tree
$\mathcal{T}_{2}$ may be terminal.

(iii) Given a bi-tree $\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2}$, we
define a projection $\Pi_{j}$, $j=1,2$, onto a tree by setting

$\displaystyle\Pi_{j}(\mathcal{T})=\mathcal{T}_{j}.$ (4.8)

In Figure 1, $\Pi_{1}(\mathcal{T})$ corresponds to the tree on the left under
the root node $r_{1}$, while $\Pi_{2}(\mathcal{T})$ corresponds to the tree on
the right under the root node $r_{2}$.

Note that the number $|\mathcal{T}|$ of nodes in a bi-tree $\mathcal{T}$ is
$3j+2$ for some $j\in\mathbb{N}$, where $|\mathcal{T}^{0}|=j$ and
$|\mathcal{T}^{\infty}|=2j+2$. Let us denote the collection of trees in the
$j$th generation (namely, with $j$ parental nodes) by $BT(j)$, i.e.

$BT(j):=\\{\mathcal{T}:\mathcal{T}\text{ is a bi-tree with
}|\mathcal{T}|=3j+2\\}.$

$r_{1}$$r_{2}$$\underline{j=1}$ $r_{1}$$r_{2}$$\underline{j=2}$
$r_{1}$$r_{2}$$\underline{j=3}$
Figure 1. Examples of bi-trees of the $j$th generation, $j=1,2,3$.

Next, we introduce the notion of ordered bi-trees, for which we keep track of
how a bi-tree “grew” into a given shape.

######  Definition 4.4.

(i) We say that a sequence $\\{\mathcal{T}_{j}\\}_{j=1}^{J}$ is a chronicle of
$J$ generations, if

  1. (a)

$\mathcal{T}_{j}\in BT(j)$ for each $j=1,\dots,J$,

  2. (b)

$\mathcal{T}_{j+1}$ is obtained by changing one of the terminal nodes in
$\mathcal{T}_{j}$ into a non-terminal node (with three children),
$j=1,\dots,J-1$.

Given a chronicle $\\{\mathcal{T}_{j}\\}_{j=1}^{J}$ of $J$ generations, we
refer to $\mathcal{T}_{J}$ as an ordered bi-tree of the $J$th generation. We
denote the collection of the ordered trees of the $J$th generation by
$\mathfrak{BT}(J)$. Note that the cardinality of $\mathfrak{BT}(J)$ is given
by $|\mathfrak{BT}(1)|=1$ and

$|\mathfrak{BT}(J)|=4\cdot 6\cdot 8\cdot\cdots\cdot 2J=2^{J-1}\cdot
J!=:c_{J},\quad J\geq 2.$ (4.9)

(ii) Given an ordered bi-tree $\mathcal{T}_{J}\in\mathfrak{BT}(J)$ as above,
we define projections $\pi_{j}$, $j=1,\dots,J-1$, onto the previous
generations by setting

$\displaystyle\pi_{j}(\mathcal{T}_{J})=\mathcal{T}_{j}\in\mathfrak{BT}(j).$

We stress that the notion of ordered bi-trees comes with associated
chronicles. For example, given two ordered bi-trees $\mathcal{T}_{J}$ and
$\widetilde{\mathcal{T}}_{J}$ of the $J$th generation, it may happen that
$\mathcal{T}_{J}=\widetilde{\mathcal{T}}_{J}$ as bi-trees (namely as planar
graphs) according to Definition 4.3, while
$\mathcal{T}_{J}\neq\widetilde{\mathcal{T}}_{J}$ as ordered bi-trees according
to Definition 4.4. In the following, when we refer to an ordered bi-tree
$\mathcal{T}_{J}$ of the $J$th generation, it is understood that there is an
underlying chronicle $\\{\mathcal{T}_{j}\\}_{j=1}^{J}$.

Given a bi-tree $\mathcal{T}$, we associate each terminal node
$a\in\mathcal{T}^{\infty}$ with the Fourier coefficient (or its complex
conjugate) of the interaction representation $v$ and sum over all possible
frequency assignments. In order to do this, we introduce index functions,
assigning integers to all the nodes in $\mathcal{T}$ in a consistent manner.

######  Definition 4.5.

(i) Given a bi-tree $\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2}$, we
define an index function ${\bf n}:\mathcal{T}\to\mathbb{Z}$ such that

  * (a)

$n_{r_{1}}=n_{r_{2}}$, where $r_{j}$ is the root node of the tree
$\mathcal{T}_{j}$, $j=1,2$,

  * (b)

$n_{a}=n_{a_{1}}-n_{a_{2}}+n_{a_{3}}$ for $a\in\mathcal{T}^{0}$, where
$a_{1},a_{2}$, and $a_{3}$ denote the children of $a$,

  * (c)

$\\{n_{a},n_{a_{2}}\\}\cap\\{n_{a_{1}},n_{a_{3}}\\}=\emptyset$ for
$a\in\mathcal{T}^{0}$,

where we identified ${\bf n}:\mathcal{T}\to\mathbb{Z}$ with
$\\{n_{a}\\}_{a\in\mathcal{T}}\in\mathbb{Z}^{\mathcal{T}}$. We use
$\mathfrak{N}(\mathcal{T})\subset\mathbb{Z}^{\mathcal{T}}$ to denote the
collection of such index functions ${\bf n}$ on $\mathcal{T}$.

Given $N\in\mathbb{N}$, we define a subcollection
$\mathfrak{N}_{N}(\mathcal{T})\subset\mathfrak{N}(\mathcal{T})$ by imposing
$|n_{a}|\leq N$ for any $a\in\mathcal{T}$. We also define
$\mathfrak{N}_{N}^{0}(\mathcal{T})\subset\mathfrak{N}(\mathcal{T})$ by
imposing $|n_{a}|\leq N$ for any non-terminal nodes $a\in\mathcal{T}^{0}$.

(ii) Given a tree $\mathcal{T}$, we also define an index function ${\bf
n}:\mathcal{T}\to\mathbb{Z}$ by omitting the condition (a) and denote by
$\mathfrak{N}(\mathcal{T})\subset\mathbb{Z}^{\mathcal{T}}$ the collection of
index functions ${\bf n}$ on $\mathcal{T}$.

######  Remark 4.6.

(i) In view of the consistency condition (a), we can refer to
$n_{r_{1}}=n_{r_{2}}$ as the frequency at the root node without ambiguity. We
shall simply denote it by $n_{r}$ in the following.

(ii) Just like index functions for (ordered) trees considered in [18], an
index function ${\bf n}=\\{n_{a}\\}_{a\in\mathcal{T}}$ for a bi-tree
$\mathcal{T}$ is completely determined once we specify the values
$n_{a}\in\mathbb{Z}$ for all the terminal nodes $a\in\mathcal{T}^{\infty}$. An
index function ${\bf n}$ for a bi-tree
$\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2}$ is basically a pair $({\bf
n}_{1},{\bf n}_{2})$ of index functions ${\bf n}_{j}$ for the trees
$\mathcal{T}_{j}$, $j=1,2$, (omitting the non-resonance condition in [18,
Definition 3.5 (iii)]), satisfying the consistency condition (a):
$n_{r_{1}}=n_{r_{2}}$.

(iii) Given a bi-tree $\mathcal{T}\in\mathfrak{BT}(J)$ and $n\in\mathbb{Z}$,
consider the summation of all possible frequency assignments $\\{{\bf
n}\in\mathfrak{N}(\mathcal{T}):n_{r}=n\\}$. While
$|\mathcal{T}^{\infty}|=2J+2$, there are $2J$ free variables in this
summation. Namely, the condition $n_{r}=n$ reduces two summation variables. It
is easy to see this by separately considering the cases
$\Pi_{2}(\mathcal{T})=\\{r_{2}\\}$ and $\Pi_{2}(\mathcal{T})\neq\\{r_{2}\\}$.

Given an ordered bi-tree $\mathcal{T}_{J}$ of the $J$th generation with a
chronicle $\\{\mathcal{T}_{j}\\}_{j=1}^{J}$ and associated index functions
${\bf n}\in\mathfrak{N}(\mathcal{T}_{J})$, we would like to keep track of the
“generations” of frequencies. In the following, we use superscripts to denote
such generations of frequencies.

Fix ${\bf n}\in\mathfrak{N}(\mathcal{T}_{J})$. Consider $\mathcal{T}_{1}$ of
the first generation. Its nodes consist of the two root nodes $r_{1}$,
$r_{2}$, and the children $r_{11},r_{12},$ and $r_{13}$ of the first root node
$r_{1}$. See Figure 1. We define the first generation of frequencies by

$\big{(}n^{(1)},n^{(1)}_{1},n^{(1)}_{2},n^{(1)}_{3}\big{)}:=(n_{r_{1}},n_{r_{11}},n_{r_{12}},n_{r_{13}}).$

From Definition 4.5, we have

$n^{(1)}=n_{r_{2}},\quad n^{(1)}=n^{(1)}_{1}-n^{(1)}_{2}+n^{(1)}_{3},\quad
n^{(1)}_{2}\neq n^{(1)}_{1},n^{(1)}_{3}.$

Next, we construct an ordered bi-tree $\mathcal{T}_{2}$ of the second
generation from $\mathcal{T}_{1}$ by changing one of its terminal nodes
$a\in\mathcal{T}^{\infty}_{1}=\\{r_{2},r_{11},r_{12},r_{13}\\}$ into a non-
terminal node. Then, we define the second generation of frequencies by setting

$\big{(}n^{(2)},n^{(2)}_{1},n^{(2)}_{2},n^{(2)}_{3}\big{)}:=(n_{a},n_{a_{1}},n_{a_{2}},n_{a_{3}}).$

Note that we have $n^{(2)}=n^{(1)}$ or $n_{k}^{(1)}$ for some
$k\in\\{1,2,3\\}$,

$n^{(2)}=n^{(2)}_{1}-n^{(2)}_{2}+n^{(2)}_{3},\quad n^{(2)}_{2}\neq
n^{(2)}_{1},n^{(2)}_{3},$

where the last identities follow from Definition 4.5. This extension of
$\mathcal{T}_{1}\in\mathfrak{BT}(1)$ to $\mathcal{T}_{2}\in\mathfrak{BT}(2)$
corresponds to introducing a new set of frequencies after the first
differentiation by parts, where the time derivative may fall on any of $v_{n}$
and $v_{n_{j}}$, $j=1,2,3$.191919The complex conjugate signs on $v_{n}$ and
$v_{n_{j}}$ do not play any significant role. Hereafter, we drop the complex
conjugate sign, when it does not play any important role.

In general, we construct an ordered bi-tree $\mathcal{T}_{j}$ of the $j$th
generation from $\mathcal{T}_{j-1}$ by changing one of its terminal nodes
$a\in\mathcal{T}^{\infty}_{j-1}$ into a non-terminal node. Then, we define the
$j$th generation of frequencies by

$\big{(}n^{(j)},n^{(j)}_{1},n^{(j)}_{2},n^{(j)}_{3}\big{)}:=(n_{a},n_{a_{1}},n_{a_{2}},n_{a_{3}}).$

As before, it follows from Definition 4.5 that

$n^{(j)}=n^{(j)}_{1}-n^{(j)}_{2}+n^{(j)}_{3},\quad n^{(j)}_{2}\neq
n^{(j)}_{1},n^{(j)}_{3}.$

Given an ordered bi-tree $\mathcal{T}$, we denote by
$B_{j}=B_{j}(\mathcal{T})$ the set of all possible frequencies in the $j$th
generation.

We denote by $\phi_{j}$ the phase function for the frequencies introduced at
the $j$th generation:

$\displaystyle\phi_{j}$
$\displaystyle=\phi_{j}\big{(}n^{(j)},n^{(j)}_{1},n^{(j)}_{2},n^{(j)}_{3}\big{)}:=\big{(}n_{1}^{(j)}\big{)}^{4}-\big{(}n_{2}^{(j)}\big{)}^{4}+\big{(}n_{3}^{(j)}\big{)}^{4}-\big{(}n^{(j)}\big{)}^{4}.$

Note that we have $|\phi_{1}|\geq 1$ in view of Definition 4.5 and Lemma 3.1.
We also denote by $\mu_{j}$ the phase function corresponding to the usual
cubic NLS (at the $j$th generation):

$\displaystyle\mu_{j}$
$\displaystyle=\mu_{j}\big{(}n^{(j)},n^{(j)}_{1},n^{(j)}_{2},n^{(j)}_{3}\big{)}:=\big{(}n_{1}^{(j)}\big{)}^{2}-\big{(}n_{2}^{(j)}\big{)}^{2}+\big{(}n_{3}^{(j)}\big{)}^{2}-\big{(}n^{(j)}\big{)}^{2}$
$\displaystyle=-2\big{(}n^{(j)}-n_{1}^{(j)}\big{)}\big{(}n^{(j)}-n_{3}^{(j)}\big{)}.$

Then, by Lemma 3.1, we have

$\displaystyle|\phi_{j}|\sim(n^{(j)}_{\text{max}})^{2}\cdot|\big{(}n^{(j)}-n_{1}^{(j)}\big{)}\big{(}n^{(j)}-n_{3}^{(j)}\big{)}|\sim(n^{(j)}_{\text{max}})^{2}\cdot|\mu_{j}|,$
(4.10)

where
$n^{(j)}_{\text{max}}:=\max\big{(}|n^{(j)}|,|n_{1}^{(j)}|,|n_{2}^{(j)}|,|n_{3}^{(j)}|\big{)}$.

Given an ordered bi-tree $\mathcal{T}\in\mathfrak{BT}(J)$ for some
$J\in\mathbb{N}$, define $C_{j}\subset\mathfrak{N}(\mathcal{T})$ by

$C_{j}=\big{\\{}|\widetilde{\phi}_{j+1}|\leq(2j+4)^{3}\big{\\}},$ (4.11)

where $\widetilde{\phi}_{j}$ is defined by

$\displaystyle\widetilde{\phi}_{j}=\sum_{k=1}^{j}\phi_{k}.$ (4.12)

In Subsection 4.4, we perform normal form reductions in an iterative manner.
At each step, we divide multilinear forms into “nearly resonant” part
(corresponding to the frequencies belonging to $C_{j}$) and highly non-
resonant part (corresponding to the frequencies belonging to $C_{j}^{c}$) and
apply a normal form reduction only to the highly non-resonant part.

###  4.3. Arithmetic lemma

As we see in the next subsection, normal form reductions generate multilinear
forms of higher and higher degrees, where we need to sum over all possible
ordered bi-trees in $\mathfrak{BT}(J)$. The main issue is then to control the
rapidly growing cardinality $c_{J}=|\mathfrak{BT}(J)|$ defined in (4.9). On
the one hand, we utilize the divisor counting estimate (see (4.15) below) as
in [18]. On the other hand, we split the argument into two parts. The
following lemma shows the heart of the matter in the multilinear estimates
presented in the next subsection. This allows us to show that there is a
sufficiently fast decay at each step of normal form reductions.

######  Lemma 4.7.

Let $s<1$ and $J\in\mathbb{N}$. Then, the following estimates hold:

(i)
$\displaystyle\qquad\sup_{\mathcal{T}_{J}\in\mathfrak{BT}(J)}\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})\\\
{n}_{r}=n\end{subarray}}\mathbf{1}_{\bigcap_{j=1}^{J-1}C_{j}^{c}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\prod_{j=2}^{J}\frac{1}{|\widetilde{\phi}_{j}|^{2}}\lesssim\frac{1}{\prod_{j=2}^{J}(2j+2)^{3-}},$
(4.13) (ii)
$\displaystyle\qquad\sup_{\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)}\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})\\\
{n}_{r}=n\end{subarray}}\mathbf{1}_{(\bigcap_{j=1}^{J-1}C_{j}^{c})\cap
C_{J}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\prod_{j=2}^{J}\frac{1}{|\widetilde{\phi}_{j}|^{2}}\lesssim\frac{J^{3+}}{\prod_{j=2}^{J}(2j+2)^{3-}}.$
(4.14)

Before proceeding further, let us recall the following arithmetic fact [20].
Given $n\in\mathbb{N}$, the number $d(n)$ of the divisors of $n$ satisfies

$\displaystyle d(n)\leq C_{\delta}n^{\delta}$ (4.15)

for any $\delta>0$. This divisor counting estimate will be used iteratively in
the following proof.

###### Proof of Lemma 4.7.

(i) We first consider the case $J=1$. In this case, from Lemma 3.1 with $s\leq
1$, we have

$\displaystyle\text{LHS of
}\eqref{sum1}\leq\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}n_{1},n_{3}\in\mathbb{Z}\\\
n_{1},n_{3}\neq n\\\ |\phi_{1}|\geq 1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\lesssim\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}n_{1},n_{3}\in\mathbb{Z}\\\
n_{1},n_{3}\neq n\end{subarray}}\frac{1}{|(n-n_{1})(n-n_{3})|^{2}}\lesssim 1.$
(4.16)

Next, we consider the case $J\geq 2$. Fix
$\mathcal{T}_{J}\in\mathfrak{BT}(J)$. For simplicity of notations, we drop the
supremum over $\mathcal{T}_{J}\in\mathfrak{BT}(J)$ in the following with the
understanding that the implicit constants are independent of
$\mathcal{T}_{J}\in\mathfrak{BT}(J)$. A similar comment applies to the proof
of the estimate (4.14) presented in (ii) below.

The main idea is to apply the divisor counting argument in an iterative
manner. It follows from the divisor counting estimate (4.15) with the
factorization of $\phi_{j}$ (Lemma 3.1) that for fixed $n^{(j)}$ and
$\phi_{j}$, there are at most $O(|\phi_{j}|^{0+})$ many choices for
$n^{(j)}_{1}$, $n^{(j)}_{2}$, and $n^{(j)}_{3}$ on $B_{j}$. Also, note that
$\phi_{j}$ is determined by $\widetilde{\phi}_{1},\dots,\widetilde{\phi}_{j}$
and

$|\phi_{j}|\leq\max(|\widetilde{\phi}_{j-1}|,|\widetilde{\phi}_{j}|).$ (4.17)

since $\phi_{j}=\widetilde{\phi}_{j}-\widetilde{\phi}_{j-1}$. In the
following, we apply the divisor counting argument to sum over the frequencies
in $B_{J}$, $B_{J-1}$, …, $B_{2}$. From Definition 4.4 (ii) and (4.11), we
have

$\displaystyle\text{LHS of }\eqref{sum1}$
$\displaystyle=\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\pi_{1}(\mathcal{T}_{J}))\\\ {n}_{r}=n\\\ |\phi_{1}|\geq
1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\sum_{\begin{subarray}{c}\psi_{2}\in\mathbb{Z}\\\
|\psi_{2}|>6^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{2}\\\
\widetilde{\phi}_{2}=\psi_{2}\end{subarray}}\frac{1}{|\psi_{2}|^{2}}\,\cdots\sum_{\begin{subarray}{c}\psi_{J}\in\mathbb{Z}\\\
|\psi_{J}|>(2J+2)^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{J}\\\
\widetilde{\phi}_{J}=\psi_{J}\end{subarray}}\frac{1}{|\psi_{J}|^{2}}$ By
applying the divisor counting argument in $B_{J}$ with (4.17), we have
$\displaystyle\lesssim\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\pi_{1}(\mathcal{T}_{J}))\\\ {n}_{r}=n\\\ |\phi_{1}|\geq
1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\sum_{\begin{subarray}{c}\psi_{2}\in\mathbb{Z}\\\
|\psi_{2}|>6^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{2}\\\
\widetilde{\phi}_{2}=\psi_{2}\end{subarray}}\frac{1}{|\psi_{2}|^{2}}\,\cdots$
$\displaystyle\hphantom{XX}\sum_{\begin{subarray}{c}\psi_{J-1}\in\mathbb{Z}\\\
|\psi_{J-1}|>(2J)^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{J-1}\\\
\widetilde{\phi}_{J-1}=\psi_{J-1}\end{subarray}}\frac{1}{|\psi_{J-1}|^{2}}\sum_{\begin{subarray}{c}\psi_{J}\in\mathbb{Z}\\\
|\psi_{J}|>(2J+2)^{3}\end{subarray}}\frac{1}{|\psi_{J}|^{2}}|\psi_{J-1}|^{0+}|\psi_{J}|^{0+}$
By iteratively applying the divisor counting argument in $B_{J-1}$, …,
$B_{2}$, we have
$\displaystyle\lesssim\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\pi_{1}(\mathcal{T}_{J}))\\\ {n}_{r}=n\\\ |\phi_{1}|\geq
1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2-}}\sum_{\begin{subarray}{c}\psi_{2},\dots,\psi_{J}\in\mathbb{Z}\\\
|\psi_{j}|>(2j+2)^{3}\\\
j=2,\dots,J\end{subarray}}\prod_{j=2}^{J}\frac{1}{|\psi_{j}|^{2-}}$
$\displaystyle\lesssim\frac{1}{\prod_{j=2}^{J}(2j+2)^{3-}},$

where the last inequality follows from (4.16).

(ii) Fix $\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)$. We proceed with the divisor
counting argument as in (i). From (4.11), we have
$|\phi_{J+1}|\lesssim|\widetilde{\phi}_{J}|+J^{3}$ on $C_{J}$ and thus for
fixed $n^{(J+1)}$ and $\phi_{J+1}$, there are at most
$O(J^{0+}|\widetilde{\phi}_{J}|^{0+})$ many choices for $n^{(J+1)}_{1}$,
$n^{(J+1)}_{2}$, and $n^{(J+1)}_{3}$ on $B_{J+1}$. Also, on $C_{J}$, there are
at most $O\big{(}J^{3}\big{)}$ many choices for $\widetilde{\phi}_{J+1}$.
Hence, for fixed $\widetilde{\phi}_{J}$, there are also at most
$O\big{(}J^{3}\big{)}$ many choices for
$\phi_{J+1}=\widetilde{\phi}_{J+1}-\widetilde{\phi}_{J}$ on $C_{J}$. Then, the
contribution to (4.14) in this case is estimate by

$\displaystyle\text{LHS of }\eqref{sum2}$
$\displaystyle=\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\pi_{1}(\mathcal{T}_{J+1}))\\\ {n}_{r}=n\\\ |\phi_{1}|\geq
1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\sum_{\begin{subarray}{c}\psi_{2}\in\mathbb{Z}\\\
|\psi_{2}|>6^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{2}\\\
\widetilde{\phi}_{2}=\psi_{2}\end{subarray}}\frac{1}{|\psi_{2}|^{2}}\,\cdots$
$\displaystyle\hphantom{XX}\sum_{\begin{subarray}{c}\psi_{J}\in\mathbb{Z}\\\
|\psi_{J}|>(2J+2)^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{J}\\\
\widetilde{\phi}_{J}=\psi_{J}\end{subarray}}\frac{1}{|\psi_{J}|^{2}}\sum_{\begin{subarray}{c}B_{J+1}\\\
|\phi_{J+1}+\psi_{J}|\leq(2J+4)^{3}\end{subarray}}1$ By applying the divisor
counting argument in $B_{J+1}$, we have $\displaystyle\lesssim
J^{3+}\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\pi_{1}(\mathcal{T}_{J+1}))\\\ {n}_{r}=n\\\ |\phi_{1}|\geq
1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\sum_{\begin{subarray}{c}\psi_{2}\in\mathbb{Z}\\\
|\psi_{2}|>6^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{2}\\\
\widetilde{\phi}_{2}=\psi_{2}\end{subarray}}\frac{1}{|\psi_{2}|^{2}}\,\cdots$
$\displaystyle\hphantom{XX}\sum_{\begin{subarray}{c}\psi_{J}\in\mathbb{Z}\\\
|\psi_{J}|>(2J+2)^{3}\end{subarray}}\sum_{\begin{subarray}{c}B_{J}\\\
\widetilde{\phi}_{J}=\psi_{J}\end{subarray}}\frac{1}{|\psi_{J}|^{2}}|\psi_{J}|^{0+}$
By iteratively applying the divisor counting argument in $B_{J}$, …, $B_{2}$
and then applying (4.16), we have $\displaystyle\lesssim
J^{3+}\sup_{n\in\mathbb{Z}}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\pi_{1}(\mathcal{T}_{J+1}))\\\ {n}_{r}=n\\\ |\phi_{1}|\geq
1\end{subarray}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2-}}\sum_{\begin{subarray}{c}\psi_{J}\in\mathbb{Z}\\\
|\psi_{J}|>(2j+2)^{3}\\\
j=2,\dots,J\end{subarray}}\prod_{j=2}^{J}\frac{1}{|\psi_{j}|^{2-}}$
$\displaystyle\lesssim\frac{J^{3+}}{\prod_{j=2}^{J}(2j+2)^{3-}}.$

This proves (4.14). ∎

######  Remark 4.8.

In [18], the authors applied the divisor counting argument even to the
frequencies of the first generation. On the other hand, we did not apply the
divisor counting argument to the frequencies of the first generation in the
proof of Lemma 4.7 above. Instead, we simply used (4.16) to control the first
generation. By using only the factor
$\mu_{1}=-2(n^{(1)}-n_{1}^{(1)})(n^{(1)}-n_{3}^{(1)})$ (and not the entire
$\phi_{1}$) for the summation, (4.16) allows us to exhibit the required
smoothing in Proposition 3.4.

###  4.4. Normal form reductions

With the notations introduced in Subsection 4.2, let us revisit the discussion
in Subsection 4.1 and then discuss the general $J$th step. We first implement
a formal infinite iteration scheme of normal form reductions without
justifying switching of limits and summations. We justify formal computations
at the end of this subsection. Let $v\in C(\mathbb{R};H^{\infty}(\mathbb{T}))$
be a global solution to (3.2). Using the notations introduced in Subsection
4.2, we write (4.1) as

$\displaystyle\frac{d}{dt}\bigg{(}\frac{1}{2}\|v(t)\|_{H^{s}}^{2}\bigg{)}$
$\displaystyle=-\operatorname*{Re}i\sum_{\mathcal{T}_{1}\in\mathfrak{BT}(1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{1})}\langle
n_{r}\rangle^{2s}e^{-i\phi_{1}t}\prod_{a\in\mathcal{T}^{\infty}_{1}}v_{n_{a}}=:\mathcal{N}^{(1)}(v)(t).$

By performing a normal form reduction, we then obtain

$\displaystyle\mathcal{N}^{(1)}(v)(t)$
$\displaystyle=\operatorname*{Re}\partial_{t}\bigg{[}\sum_{\mathcal{T}_{1}\in\mathfrak{BT}(1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{1})}\frac{\langle
n_{r}\rangle^{2s}e^{-i\phi_{1}t}}{\phi_{1}}\prod_{a\in\mathcal{T}_{1}^{\infty}}v_{n_{a}}\bigg{]}$
$\displaystyle\hphantom{X}-\operatorname*{Re}\sum_{\mathcal{T}_{1}\in\mathfrak{BT}(1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{1})}\frac{\langle
n_{r}\rangle^{2s}e^{-i\phi_{1}t}}{\phi_{1}}\partial_{t}\bigg{(}\prod_{a\in\mathcal{T}^{\infty}_{1}}v_{n_{a}}\bigg{)}$
$\displaystyle=\operatorname*{Re}\partial_{t}\bigg{[}\sum_{\mathcal{T}_{1}\in\mathfrak{BT}(1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{1})}\frac{\langle
n_{r}\rangle^{2s}e^{-i\phi_{1}t}}{\phi_{1}}\prod_{a\in\mathcal{T}^{\infty}_{1}}v_{n_{a}}\bigg{]}$
$\displaystyle\hphantom{X}-\operatorname*{Re}\sum_{\mathcal{T}_{1}\in\mathfrak{BT}(1)}\sum_{b\in\mathcal{T}_{1}^{\infty}}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{1})}\frac{\langle
n_{r}\rangle^{2s}e^{-i\phi_{1}t}}{\phi_{1}}\mathcal{R}(v)_{n_{b}}\prod_{a\in\mathcal{T}^{\infty}_{1}\setminus\\{b\\}}v_{n_{a}}$
$\displaystyle\hphantom{X}+\operatorname*{Re}i\sum_{\mathcal{T}_{2}\in\mathfrak{BT}(2)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{2})}\frac{\langle
n_{r}\rangle^{2s}e^{-i(\phi_{1}+\phi_{2})t}}{\phi_{1}}\prod_{a\in\mathcal{T}^{\infty}_{2}}v_{n_{a}}$
$\displaystyle=:\partial_{t}\mathcal{N}_{0}^{(2)}(v)(t)+\mathcal{R}^{(2)}(v)(t)+\mathcal{N}^{(2)}(v)(t).$
(4.18)

Compare (4.18) with (4.2). In the second equality, we applied the product rule
and used the equation (3.2) to replace $\partial_{t}v_{n_{b}}$ by the resonant
part $\mathcal{R}(v)_{n_{b}}$ and the non-resonant part
$\mathcal{N}(v)_{n_{b}}$. In substituting the non-resonant part
$\mathcal{N}(v)_{n_{b}}$, we turned the terminal node
$b\in\mathcal{T}^{\infty}_{1}$ into a non-terminal node with three children
$b_{1},b_{2},$ and $b_{3}$, which corresponds to extending the tree
$\mathcal{T}_{1}\in\mathfrak{BT}(1)$ (and ${\bf
n}\in\mathfrak{N}(\mathcal{T}_{1})$) to $\mathcal{T}_{2}\in\mathfrak{BT}(2)$
(and to ${\bf n}\in\mathfrak{N}(\mathcal{T}_{2})$, respectively).

######  Remark 4.9.

(i) Strictly speaking, the phase factor appearing in $\mathcal{N}^{(2)}(v)$
may be $\phi_{1}-\phi_{2}$ when the time derivative falls on the terms with
the complex conjugate. In the following, however, we simply write it as
$\phi_{1}+\phi_{2}$ since it does not make any difference in our analysis.
Also, we often replace $\pm 1$ and $\pm i$ by $1$ for simplicity when they do
not play an important role. Lastly, for notational simplicity, we drop the
real part symbol on multilinear forms with the understanding that all the
multilinear forms appear with the real part symbol.

(ii) Due to the presence of $e^{-i\phi_{1}t}$ in their definitions, the
multilinear forms such as $\mathcal{N}_{0}^{(2)}(v)$ are non-autonomous in
$t$. Therefore, strictly speaking, they should be denoted as
$\mathcal{N}_{0}^{(2)}(t)(v(t))$. In the following, however, we establish
nonlinear estimates on these multilinear forms, uniformly in $t\in\mathbb{R}$,
by simply using $|e^{-i\phi_{1}t}|=1$. Hence, we simply suppress such
$t$-dependence when there is no confusion. The same comment applies to other
multilinear forms. Note that this convention was already used in Proposition
3.4.

It is worthwhile to note that the multilinear forms introduced in this section
are non-autonomous when they are expressed in terms of the interaction
representation $v$, solving (3.2). When they are expressed in terms of the
original solution $u$ to (1.1) (or $\widetilde{u}$ to (3.1)), however, it is
easy to see that these multilinear terms are indeed autonomous.

Thanks to Lemma 4.7, the terms $\mathcal{N}_{0}^{(2)}$ and $\mathcal{R}^{(2)}$
can be estimated in a straightforward manner; see Lemma 4.10 below. We split
$\mathcal{N}^{(2)}$ as in (4.5). As mentioned above, the good part
$\mathcal{N}^{(2)}_{1}$ is handled in an effective manner (Lemma 4.11) thanks
to the frequency restriction on $C_{1}$. We then apply the second step of
normal form reductions to $\mathcal{N}^{(2)}_{2}$ and obtain

$\displaystyle\mathcal{N}^{(2)}_{2}(v)$
$\displaystyle=\partial_{t}\bigg{[}\sum_{\mathcal{T}_{2}\in\mathfrak{BT}(2)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{2})}\mathbf{1}_{C_{1}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i(\phi_{1}+\phi_{2})t}}{\phi_{1}(\phi_{1}+\phi_{2})}\prod_{a\in\mathcal{T}^{\infty}_{2}}v_{n_{a}}\bigg{]}$
$\displaystyle\hphantom{X}-\sum_{\mathcal{T}_{2}\in\mathfrak{BT}(2)}\sum_{b\in\mathcal{T}^{\infty}_{2}}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{2})}\mathbf{1}_{C_{1}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i(\phi_{1}+\phi_{2})t}}{\phi_{1}(\phi_{1}+\phi_{2})}\,\mathcal{R}(v)_{n_{b}}\prod_{a\in\mathcal{T}^{\infty}_{2}\setminus\\{b\\}}v_{n_{a}}$
$\displaystyle\hphantom{X}-\sum_{\mathcal{T}_{3}\in\mathfrak{BT}(3)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{3})}\mathbf{1}_{C_{1}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i(\phi_{1}+\phi_{2}+\phi_{3})t}}{\phi_{1}(\phi_{1}+\phi_{2})}\,\prod_{a\in\mathcal{T}^{\infty}_{3}}v_{n_{a}}$
$\displaystyle=:\partial_{t}\mathcal{N}^{(3)}_{0}(v)+\mathcal{R}^{(3)}(v)+\mathcal{N}^{(3)}(v).$
(4.19)

As in the previous step, we can estimate $\mathcal{N}^{(3)}_{0}$ and
$\mathcal{R}^{(3)}$ in a straightforward manner (Lemma 4.10), while we split
$\mathcal{N}^{(3)}$ into the good part $\mathcal{N}^{(3)}_{1}$ and the bad
part $\mathcal{N}^{(3)}_{2}$ as in (4.6), where $\mathcal{N}^{(3)}_{1}$ is the
restriction of $\mathcal{N}^{(3)}$ onto $C_{2}$ defined in (4.11). We then
apply the third step of normal form reductions to the bad part
$\mathcal{N}^{(3)}_{2}$. In this way, we iterate normal form reductions in an
indefinite manner.

After the $J$th step, we have

$\displaystyle\mathcal{N}^{(J)}_{2}(v)$
$\displaystyle=\partial_{t}\bigg{[}\sum_{\mathcal{T}_{J}\in\mathfrak{BT}(J)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})}\mathbf{1}_{\bigcap_{j=1}^{J-1}C_{j}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i\widetilde{\phi}_{J}t}}{\prod_{j=1}^{J}\widetilde{\phi}_{j}}\,\prod_{a\in\mathcal{T}^{\infty}_{J}}v_{n_{a}}\bigg{]}$
$\displaystyle\hphantom{X}-\sum_{\mathcal{T}_{J}\in\mathfrak{BT}(J)}\sum_{b\in\mathcal{T}^{\infty}_{J}}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})}\mathbf{1}_{\bigcap_{j=1}^{J-1}C_{j}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i\widetilde{\phi}_{J}t}}{\prod_{j=1}^{J}\widetilde{\phi}_{j}}\,\mathcal{R}(v)_{n_{b}}\prod_{a\in\mathcal{T}^{\infty}_{J}\setminus\\{b\\}}v_{n_{a}}$
$\displaystyle\hphantom{X}-\sum_{\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})}\mathbf{1}_{\bigcap_{j=1}^{J-1}C_{j}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i\widetilde{\phi}_{J+1}t}}{\prod_{j=1}^{J}\widetilde{\phi}_{j}}\,\prod_{a\in\mathcal{T}^{\infty}_{J+1}}v_{n_{a}}$
$\displaystyle=:\partial_{t}\mathcal{N}^{(J+1)}_{0}(v)+\mathcal{R}^{(J+1)}(v)+\mathcal{N}^{(J+1)}(v),$
(4.20)

where $\widetilde{\phi}_{J}$ is as in (4.12). In the following, we first
estimate $\mathcal{N}^{(J+1)}_{0}$ and $\mathcal{R}^{(J+1)}$ by applying
Cauchy-Schwarz inequality and then applying the divisor counting argument
(Lemma 4.7).

######  Lemma 4.10.

Let $\mathcal{N}^{(J+1)}_{0}$ and $\mathcal{R}^{(J+1)}$ be as in (4.4). Then,
for any $s<1$, we have

$\displaystyle|\mathcal{N}^{(J+1)}_{0}(v)|\lesssim\frac{c_{J}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\|v\|_{L^{2}}^{2J+2},$
(4.21) $\displaystyle|\mathcal{R}^{(J+1)}(v)|\lesssim\frac{(2J+2)\cdot
c_{J}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\|v\|_{L^{2}}^{2J+4}.$ (4.22)

###### Proof.

We split the proof into the following two cases:

$\text{(i)
}\Pi_{2}(\mathcal{T}_{J})=\\{r_{2}\\}\qquad\text{and}\qquad\text{(ii)
}\Pi_{2}(\mathcal{T}_{J})\neq\\{r_{2}\\},$

where $\Pi_{2}$ denotes the projection defined in (4.8).

$\bullet$ Case (i): We first consider the case
$\Pi_{2}(\mathcal{T}_{J})=\\{r_{2}\\}$. Recall that for general
$J\in\mathbb{N}$, we need to control the rapidly growing cardinality
$c_{J}=|\mathfrak{BT}(J)|$ defined in (4.9). By Cauchy-Schwarz inequality and
Lemma 4.7, we have

$\displaystyle|\mathcal{N}^{(J+1)}_{0}(v)|$
$\displaystyle\lesssim\|v\|_{L^{2}}\sum_{\begin{subarray}{c}\mathcal{T}_{J}\in\mathfrak{BT}(J)\\\
\Pi_{2}(\mathcal{T}_{J})=\\{r_{2}\\}\end{subarray}}\Bigg{\\{}\sum_{n\in\mathbb{Z}}\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})\\\
{n}_{r}=n\end{subarray}}\mathbf{1}_{\bigcap_{j=1}^{J-1}C_{j}^{c}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\prod_{j=2}^{J}\frac{1}{|\widetilde{\phi}_{j}|^{2}}\bigg{)}$
$\displaystyle\hphantom{XXXXXX}\times\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})\\\
{n}_{r}=n\end{subarray}}\prod_{a\in\mathcal{T}^{\infty}_{J}\setminus\\{r_{2}\\}}|v_{n_{a}}|^{2}\bigg{)}\Bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\lesssim\frac{c_{J}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\|v\|_{L^{2}}^{2J+2}.$

$\bullet$ Case (ii): Next, we consider the case
$\Pi_{2}(\mathcal{T}_{J})\neq\\{r_{2}\\}$. In this case, we need to modify the
argument above since the frequency $n_{r}=n$ does not correspond to a terminal
node. Note that
$\mathcal{T}_{J}^{\infty}=\Pi_{1}(\mathcal{T}_{J})^{\infty}\cup\Pi_{2}(\mathcal{T}_{J})^{\infty}$
and

$\displaystyle\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})\\\
{n}_{r}=n\end{subarray}}\prod_{a\in\mathcal{T}^{\infty}_{J}}|v_{n_{a}}|^{2}=\prod_{j=1}^{2}\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\Pi_{j}(\mathcal{T}_{J}))\\\
{n}_{r_{j}}=n\end{subarray}}\prod_{a_{j}\in\Pi_{j}(\mathcal{T}_{J})^{\infty}}|v_{n_{a_{j}}}|^{2}\bigg{)}.$
(4.23)

Then, by Cauchy-Schwarz inequality and Lemma 4.7 with (4.23), we have

$\displaystyle|\mathcal{N}^{(J+1)}_{0}(v)|$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}\mathcal{T}_{J}\in\mathfrak{BT}(J)\\\
\Pi_{2}(\mathcal{T}_{J})=\\{r_{2}\\}\end{subarray}}\sum_{n\in\mathbb{Z}}\Bigg{\\{}\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})\\\
{n}_{r}=n\end{subarray}}\mathbf{1}_{\bigcap_{j=1}^{J-1}C_{j}^{c}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\prod_{j=2}^{J}\frac{1}{|\widetilde{\phi}_{j}|^{2}}\bigg{)}$
$\displaystyle\hphantom{XXXXXX}\times\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J})\\\
{n}_{r}=n\end{subarray}}\prod_{a\in\mathcal{T}^{\infty}_{J}}|v_{n_{a}}|^{2}\bigg{)}\Bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\lesssim\frac{c_{J}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\sup_{\begin{subarray}{c}\mathcal{T}_{J}\in\mathfrak{BT}(2)\\\
\Pi_{2}(\mathcal{T}_{J})\neq\\{r_{2}\\}\end{subarray}}\sum_{n\in\mathbb{Z}}\Bigg{\\{}\prod_{j=1}^{2}\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\Pi_{j}(\mathcal{T}_{J}))\\\
{n}_{r_{j}}=n\end{subarray}}\prod_{a_{j}\in\Pi_{j}(\mathcal{T}_{J})^{\infty}}|v_{n_{a_{j}}}|^{2}\bigg{)}\Bigg{\\}}^{\frac{1}{2}}$
By Cauchy-Schwarz inequality in $n$,
$\displaystyle\lesssim\frac{c_{J}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\sup_{\begin{subarray}{c}\mathcal{T}_{J}\in\mathfrak{BT}(2)\\\
\Pi_{2}(\mathcal{T}_{J})\neq\\{r_{2}\\}\end{subarray}}\prod_{j=1}^{2}\Bigg{\\{}\bigg{(}\sum_{n}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\Pi_{j}(\mathcal{T}_{J}))\\\
{n}_{r_{j}}=n\end{subarray}}\prod_{a_{j}\in\Pi_{j}(\mathcal{T}_{J})^{\infty}}|v_{n_{a_{j}}}|^{2}\bigg{)}\Bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\lesssim\frac{c_{J}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\|v\|_{L^{2}}^{2J+2}.$

This proves the first estimate (4.21).

The second estimate (4.22) follows from (4.21) and
$\ell^{2}_{n}\subset\ell^{6}_{n}$, noting that, given
$\mathcal{T}_{J}\in\mathfrak{BT}(J)$, we have
$\\#\\{b:b\in\mathcal{T}^{\infty}_{J}\\}=2J+2$. ∎

Next, we treat $\mathcal{N}^{(J+1)}$ in (4.4). As before, we write

$\mathcal{N}^{(J+1)}=\mathcal{N}^{(J+1)}_{1}+\mathcal{N}^{(J+1)}_{2},$ (4.24)

where $\mathcal{N}^{(J+1)}_{1}$ is the restriction of $\mathcal{N}^{(J+1)}$
onto $C_{J}$ defined in (4.11) and
$\mathcal{N}^{(J+1)}_{2}:=\mathcal{N}^{(J+1)}-\mathcal{N}^{(J+1)}_{1}$. In the
following lemma, we estimate the first term in (4.26):

$\displaystyle\mathcal{N}^{(J+1)}_{1}(v)=-\sum_{\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})}\mathbf{1}_{(\bigcap_{j=1}^{J-1}C_{j}^{c})\cap
C_{J}}\frac{\langle
n_{r}\rangle^{2s}e^{-i\widetilde{\phi}_{J+1}t}}{\prod_{j=1}^{J}\widetilde{\phi}_{j}}\,\prod_{a\in\mathcal{T}^{\infty}_{J+1}}v_{n_{a}}.$
(4.25)

Then, we apply a normal form reduction once again to the second term
$\mathcal{N}^{(J+1)}_{2}$ as in (4.4). In Subsection 4.5, we show that the
error term $\mathcal{N}^{(J+1)}_{2}$ tends to $0$ as $J\to\infty$.

######  Lemma 4.11.

Let $\mathcal{N}^{(J+1)}_{1}$ be as in (4.25). Then, for any $s<1$, we have

$\displaystyle|\mathcal{N}^{(J+1)}_{1}(v)|\lesssim\frac{J^{\frac{3}{2}+}\cdot
c_{J+1}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\|v\|_{L^{2}}^{2J+4}.$ (4.26)

###### Proof.

We only discuss the case $\Pi_{2}(\mathcal{T}_{J+1})=\\{r_{2}\\}$ since the
modification for the case $\Pi_{2}(\mathcal{T}_{J+1})\neq\\{r_{2}\\}$ is
straightforward as in the proof of Lemma 4.10. By Cauchy-Schwarz inequality
and Lemma 4.13, we have

$\displaystyle|\mathcal{N}^{(J+1)}_{1}(v)|$
$\displaystyle\lesssim\|v\|_{L^{2}}\sum_{\begin{subarray}{c}\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)\\\
\Pi_{2}(\mathcal{T}_{J+1})=\\{r_{2}\\}\end{subarray}}\Bigg{\\{}\sum_{n\in\mathbb{Z}}\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})\\\
{n}_{r}=n\end{subarray}}\mathbf{1}_{(\bigcap_{j=1}^{J-1}C_{j}^{c})\cap
C_{J}}\frac{\langle
n\rangle^{4s}}{|\phi_{1}|^{2}}\prod_{j=2}^{J}\frac{1}{|\widetilde{\phi}_{j}|^{2}}\bigg{)}$
$\displaystyle\hphantom{XXXXXX}\times\bigg{(}\sum_{\begin{subarray}{c}{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})\\\
{n}_{r}=n\end{subarray}}\prod_{a\in\mathcal{T}^{\infty}_{J+1}\setminus\\{r_{2}\\}}|v_{n_{a}}|^{2}\bigg{)}\Bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\lesssim\frac{J^{\frac{3}{2}+}\cdot
c_{J+1}}{\prod_{j=2}^{J}(2j+2)^{\frac{3}{2}-}}\|v\|_{L^{2}}^{2J+4}.$

This proves (4.26). ∎

######  Remark 4.12.

A notable difference from [18] appears in our definition of $C_{j}$ in (4.11);
on the one hand, the cutoff size on $\widetilde{\phi}_{j+1}$ in [18] depended
on $\widetilde{\phi}_{j}$ and $\phi_{1}$. On the other hand, our choice of the
cutoff size on $\widetilde{\phi}_{j+1}$ in (4.11) is independent of
$\widetilde{\phi}_{j}$ or $\phi_{1}$, thus providing simplification of the
argument.

Another difference appears in the first step of the normal form reductions. On
the one hand, we simply applied the first normal form reduction in (4.18)
without introducing a cutoff on the phase function $\phi_{1}$. On the other
hand, in [18], a cutoff of the form202020In [18], this parameter was denoted
by $N$. Here, we use $K$ to avoid the confusion with the frequency truncation
parameter $N\in\mathbb{N}$. $|\phi_{1}|>K$ was introduced to separate the
first multilinear term into the nearly resonant and non-resonant parts. The
use of this extra parameter $K=K(\|u_{0}\|_{L^{2}})$ allowed the authors to
show that the local existence time can be given by
$T\sim\|u_{0}\|_{L^{2}}^{-\alpha}$ for some $\alpha>0$. See [18] for details.
Since our argument only requires the summability (in $J$) of the multilinear
forms, we do not need to introduce this extra parameter.

We conclude this subsection by briefly discussing how to justify all the
formal steps performed in the normal form reductions. In particular, we need
to justify

  * (i)

the application of the product rule and

  * (ii)

switching time derivatives and summations

Suppose that a solution $v$ to (3.2) lies in
$C(\mathbb{R};H^{\frac{1}{6}}(\mathbb{T}))$. Then, from (3.2), we have

$\|\partial_{t}v_{n}\|_{C_{T}\ell^{\infty}_{n}}\leq\big{\|}\mathcal{F}^{-1}(|\widehat{v}_{n}|)\big{\|}_{C_{T}L^{3}_{x}}^{3}\lesssim\big{\|}\mathcal{F}^{-1}(|\widehat{v}_{n}|)\big{\|}_{C_{T}H^{\frac{1}{6}}_{x}}^{3}=\|v\|_{C_{T}H^{\frac{1}{6}}_{x}}^{3}$

for each $T>0$, where $C_{T}B_{x}=C([-T,T];B_{x})$. Hence,
$\partial_{t}v_{n}\in C([-T,T];\ell^{\infty}_{n})$, justifying (i) the
application of the product rule. Note that given $N\in\mathbb{N}$, any
solution $v$ to (3.6) belongs to $C(\mathbb{R};H^{\infty}(\mathbb{T}))$ and
hence (i) is justified. Moreover, the summations over spatial frequencies in
the normal form reductions applied to solutions to (3.6) are all finite and
therefore, (ii) the switching time derivatives and summations over spatial
frequencies trivially hold true for (3.6). In general, the proof of Lemma 4.10
shows that the summation defining $\mathcal{N}_{0}^{(j)}$ converges
(absolutely and uniformly in time). Then, Lemma 5.1 in [18] allows us to
switch the time derivative with the summations as temporal distributions, thus
justifying differentiation by parts.

###  4.5. On the error term $\mathcal{N}_{2}^{(J+1)}$

In this subsection, we prove that $\mathcal{N}_{2}^{(J+1)}$ in (4.24) tends to
0 as $J\to\infty$ under some regularity assumption on $v$. From (4.4), we have

$\displaystyle\mathcal{N}_{2}^{(J+1)}(v)$
$\displaystyle=-\sum_{\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})}\mathbf{1}_{\bigcap_{j=1}^{J}C_{j}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i\widetilde{\phi}_{J+1}t}}{\prod_{j=1}^{J}\widetilde{\phi}_{j}}\,\prod_{a\in\mathcal{T}^{\infty}_{J+1}}v_{n_{a}}.$
(4.27)

######  Lemma 4.13.

Let $\mathcal{N}^{(J+1)}_{2}$ be as in (4.27). Then, given $v\in
H^{s}(\mathbb{T})$, $s>\frac{1}{2}$, we have

$\displaystyle|\mathcal{N}^{(J+1)}_{2}(v)|\longrightarrow 0,$ (4.28)

as $J\to\infty$.

We point out that one can actually prove Lemma 4.13 under a weaker regularity
assumption $s\geq\frac{1}{6}$. See [37]. For our purpose, however, we only
need to prove the vanishing of the error term $\mathcal{N}^{(J+1)}_{2}$ for
sufficiently regular functions; our main objective is to obtain an energy
estimate (on the modified energy $\mathcal{E}_{N,t}$ defined in (3.8)) for
solutions to the truncated equation (3.6). Given $N\in\mathbb{N}$, any
solution $v$ to (3.6) belongs to $C(\mathbb{R};H^{\infty}(\mathbb{T}))$.
Therefore, while the convergence speed in (4.28) depends on $N\in\mathbb{N}$,
the final energy estimate (3.9) holds with an implicit constant independent of
$N\in\mathbb{N}$.

###### Proof.

Given ${\bf n}\in\mathfrak{N}(\mathcal{T}_{J+1})$, it follows from Definition
4.5 and the triangle inequality that there exists $C_{0}>0$ such that

$\displaystyle|n_{r}|\leq C_{0}^{J}|n_{b_{k}}|$ (4.29)

for (at least) two terminal nodes $b_{1},b_{2}\in\mathcal{T}_{J+1}^{\infty}$
Then, by Young’s inequality (placing $v_{n_{b_{k}}}$ in $\ell^{2}_{n}$,
$k=1,2$, and the rest in $\ell^{1}_{n}$) with (4.9), (4.11), and (4.29), we
have

$\displaystyle|\mathcal{N}_{2}^{(J+1)}(v)|$
$\displaystyle\lesssim\frac{C_{0}^{2sJ}\cdot
c_{J}}{\prod_{j=2}^{J}(2j+2)^{3}}\sup_{\mathcal{T}_{J+1}\in\mathfrak{BT}(J+1)}\sum_{{\bf
n}\in\mathfrak{N}(\mathcal{T}_{J+1})}\bigg{(}\prod_{k=1}^{2}\langle
n_{b_{k}}\rangle^{s}|v_{n_{b_{k}}}|\bigg{)}\bigg{(}\prod_{a\in\mathcal{T}^{\infty}_{J+1}\setminus{\\{b_{1},b_{2}\\}}}|v_{n_{a}}|\bigg{)}$
$\displaystyle\lesssim\frac{C_{0}^{2sJ}\cdot
c_{J}}{\prod_{j=2}^{J}(2j+2)^{3}}\|v\|_{H^{s}}^{2J+4}\longrightarrow 0,$

as $J\to\infty$. ∎

###  4.6. Improved energy bound

We are now ready to establish the improved energy estimate (3.9). Let $v$ be a
smooth global solution212121In fact, it suffices to assume that $v\in
C(\mathbb{R};H^{\frac{1}{6}}(\mathbb{T}))$. See [18, 37]. to (3.2). Then, by
applying the normal form reduction $J$ times, we obtain222222Once again, we
are replacing $\pm 1$ and $\pm i$ by 1 for simplicity since they play no role
in our analysis.

$\displaystyle\partial_{t}\bigg{(}\frac{1}{2}\|v\|_{H^{s}}^{2}\bigg{)}=\partial_{t}\sum_{j=2}^{J+1}\mathcal{N}^{(j)}_{0}(v)+\sum_{j=2}^{J+1}\mathcal{N}^{(j)}_{1}(v)+\sum_{j=2}^{J+1}\mathcal{R}^{(j)}(v)+\mathcal{N}^{(J+1)}_{2}.$

Thanks to Lemma 4.13, by taking the limit as $J\to\infty$, we obtain

$\displaystyle\partial_{t}\bigg{(}\frac{1}{2}\|v\|_{H^{s}}^{2}\bigg{)}=\partial_{t}\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0}(v)+\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{1}(v)+\sum_{j=2}^{\infty}\mathcal{R}^{(j)}(v).$

In other words, defining the modified energy $\mathcal{E}_{t}(v)$ by

$\displaystyle\mathcal{E}_{t}(v):=\frac{1}{2}\|v(t)\|_{H^{s}}^{2}-\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0}(v)(t),$

we have

$\displaystyle\partial_{t}\mathcal{E}_{t}(v)=\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{1}(v)(t)+\sum_{j=2}^{\infty}\mathcal{R}^{(j)}(v)(t).$

Suppose that $\|v\|_{C(\mathbb{R};L^{2})}\leq r$. Then, applying Lemmas 4.10
and 4.11 with (4.9), we obtain

$\displaystyle|\partial_{t}\mathcal{E}_{t}(v)|$
$\displaystyle\lesssim\sum_{j=2}^{\infty}\frac{c_{j-1}}{\prod_{k=2}^{j-1}(2k+2)^{\frac{3}{2}-}}r^{2j}+\sum_{j=2}^{\infty}\frac{j^{\frac{3}{2}+}\cdot
c_{j}}{\prod_{k=2}^{j-1}(2k+2)^{\frac{3}{2}-}}r^{2j+2}$
$\displaystyle+\sum_{j=2}^{\infty}\frac{j\cdot
c_{j-1}}{\prod_{k=2}^{j-1}(2k+2)^{\frac{3}{2}-}}r^{2j+2}$ $\displaystyle\leq
C(r).$

In view of the boundedness of the frequency projections and noting that any
solution to (3.6) is in $H^{\infty}(\mathbb{T})$, the same energy estimate
holds for solutions to the truncated equation (3.6), uniformly in
$N\in\mathbb{N}$.

###  4.7. On the proof of Lemma 3.5

We conclude this section with a brief discussion on the proof of Lemma 3.5.
First, note that Lemma 3.5 (ii) is an immediate corollary of Lemma 3.5 (i).
Moreover, Lemma 3.5 (i) follows from Egoroff’s theorem once we prove that

$\mathfrak{S}_{N}(v)=\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0,N}(v)$

converges almost surely to

$\mathfrak{S}_{\infty}(v)=\sum_{j=2}^{\infty}\mathcal{N}^{(j)}_{0}(v).$

See [32, Proposition 6.2]. In fact, one can show that $\mathfrak{S}_{N}(v)$
converges to $\mathfrak{S}_{\infty}(v)$ for any $v\in L^{2}(\mathbb{T})$.

Recall from (4.4) that $\mathcal{N}^{(j)}_{0}(v)$ consists of a sum of the
multilinear forms associated with ordered bi-trees
$\mathcal{T}_{j-1}\in\mathfrak{BT}(j-1)$. Given $N\in\mathbb{N}$, the
multilinear form $\mathcal{N}^{(j)}_{0,N}(v)$ is obtained in a similar manner
with the following modifications:

  * (i)

We set $v_{n}=0$ for all $|n|>N$. This corresponds to setting $v_{n_{a}}=0$
for all $|n|>N$ and all terminal nodes $a\in\mathcal{T}_{j-1}^{\infty}$.

  * (ii)

In view of (3.6), we also set $v_{n_{a}}=0$ for all $|n|>N$ and all parental
nodes in $\mathcal{T}_{j-1}$. This amounts to setting $v_{n_{a}}=0$ for all
$|n|>N$ and all non-terminal nodes $a\in\mathcal{T}_{j-1}^{0}$.

In particular, we have

$\displaystyle\mathcal{N}^{(j)}_{0,N}(v)=\sum_{\mathcal{T}_{j-1}\in\mathfrak{BT}(j-1)}\sum_{{\bf
n}\in\mathfrak{N}_{N}(\mathcal{T}_{j-1})}\mathbf{1}_{\bigcap_{k=1}^{j-2}C_{k}^{c}}\frac{\langle
n_{r}\rangle^{2s}e^{-i\widetilde{\phi}_{j-1}t}}{\prod_{k=1}^{j-1}\widetilde{\phi}_{k}}\,\prod_{a\in\mathcal{T}^{\infty}_{j-1}}v_{n_{a}},$
(4.30)

where $\mathfrak{N}_{N}(\mathcal{T}_{j-1})$ is as in Definition 4.5. Namely,
$\mathcal{N}^{(j)}_{0,N}(v)$ is obtained from $\mathcal{N}^{(j)}_{0}(v)$ by
simply truncating all the frequencies (including the “hidden”232323Namely, the
parental frequencies at the non-terminal nodes do not appear explicitly in the
sum in (4.30) but they implicitly appear through the relation in Definition
4.5. parental frequencies) by $N\in\mathbb{N}$.

Write

$\displaystyle\mathcal{N}^{(j)}_{0}(v)-\mathcal{N}^{(j)}_{0,N}(v)$
$\displaystyle=\Big{\\{}\mathcal{N}^{(j)}_{0}(v)-\widetilde{\mathcal{N}}^{(j)}_{0,N}(v)\Big{\\}}+\Big{\\{}\widetilde{\mathcal{N}}^{(j)}_{0,N}(v)-\mathcal{N}^{(j)}_{0,N}(v)\Big{\\}}$
$\displaystyle=:\hskip 1.42262pt\text{I}\hskip 1.42262pt_{j}+\text{I I}_{j},$
(4.31)

where $\widetilde{\mathcal{N}}^{(j)}_{0,N}(v)$ is obtained from
$\mathcal{N}^{(j)}_{0,N}(v)$ by replacing ${\bf
n}\in\mathfrak{N}_{N}(\mathcal{T}_{j-1})$ with ${\bf
n}\in\mathfrak{N}_{N}^{0}(\mathcal{T}_{j-1})$. i.e. we are truncating only the
parental frequencies at non-terminal nodes $a\in\mathcal{T}^{0}_{j-1}$. Then,
by writing

$\text{I
I}_{j}=\widetilde{\mathcal{N}}^{(j)}_{0,N}(v)-\widetilde{\mathcal{N}}^{(j)}_{0,N}(\mathbf{P}_{\leq
N}v),$

it follows from the multilinearity and the boundedness in $L^{2}(\mathbb{T})$
(Lemma 4.10) that the second term $\text{I I}_{j}$ tends to 0 as $N\to\infty$,
by simply writing the difference in a telescoping sum. More precisely, we
write I I as a telescoping sum, replacing $2j$ factors of $v_{n_{a}}$,
$a\in\mathcal{T}^{\infty}_{j-1}$, into $2j$ factors of $(\mathbf{P}_{\leq
N}v)_{n_{a}}$. This introduces $2j$ differences, each containing exactly one
factor of $v-\mathbf{P}_{\leq N}v$ (tending to 0 as $N\to\infty$). We then
simply apply Lemma 4.10 on each difference.

Similarly, we can show that $\hskip 1.42262pt\text{I}\hskip 1.42262pt_{j}$ in
(4.31) tends to 0 as $N\to\infty$ by writing the difference in a telescoping
sum. Namely, noting only the difference between $\mathcal{N}^{(j)}_{0}(v)$ and
$\widetilde{\mathcal{N}}^{(j)}_{0,N}(v)$ is the frequency cutoffs
$\mathbf{1}_{|n_{a}|\leq N}$ at each non-terminal node
$a\in\mathcal{T}^{0}_{j-1}$, we introduce $j-1$ differences by adding the
frequency cutoff $\mathbf{1}_{|n_{a}|\leq N}$ at each non-terminal node in a
sequential manner. By construction, each of the $j-1$ differences has one non-
terminal node $a_{*}\in\mathcal{T}^{0}_{j-1}$ with the restriction
$|n_{a_{*}}|>N$. Then, from Definition 4.5, we see that there exists at least
one terminal node $b\in\mathcal{T}^{\infty}_{j-1}$ which is a descendants of
$a_{*}$ such that

$\displaystyle|n_{b}|\geq C_{0}^{-J}|n_{a_{*}}|>C_{0}^{-J}N$

for some $C_{0}>0$ (compare this with (4.29)). This forces each of the $j-1$
differences in the telescoping sum to tend to 0 as $N\to\infty$, and hence
$\hskip 1.42262pt\text{I}\hskip 1.42262pt_{j}$ in (4.31) tends to 0 as $N\to
0$.

Therefore, $\mathcal{N}^{(j)}_{0,N}(v)$ converges to
$\mathcal{N}^{(j)}_{0}(v)$ as $n\to\infty$ for any $v\in L^{2}(\mathbb{T})$.
Finally, in view of the fast decay in $j$ in Lemma 4.10, the convergence of
$\mathfrak{S}_{N}(v)$ to $\mathfrak{S}_{\infty}(v)$ follows from the dominated
convergence theorem.

##  5\. Proof of Theorem 1.6: Non quasi-invariance under the dispersionless
model

In this section, we present the proof of Theorem 1.6. The basic ingredients
are the Fourier series representation of the (fractional) Brownian loops, the
law of the iterated logarithm, and the solution formula (1.20) to the
dispersionless model (1.19). More precisely, we show that, while the Gaussian
random initial data distributed according to $\mu_{s}$ satisfies the law of
the iterated logarithm, the solution given by (1.20) does not satisfy the law
of the iterated logarithm for any non-zero time. We divide the argument into
three cases: (i) $s=1$ corresponding to the Brownian/Ornstein-Uhlenbeck loop,
(ii) $\frac{1}{2}<s<\frac{3}{2}$, corresponding to the fractional Brownian
loop (and $s>\frac{3}{2}$ with $s\notin\frac{1}{2}+\mathbb{N}$), and (iii)
$s\in\frac{1}{2}+\mathbb{N}$: the critical case. For simplicity, we set $t=1$
in the following. The proof for non-zero $t\neq 1$ follows in a similar
manner.

###  5.1. Brownian/Ornstein-Uhlenbeck loop

We first consider the $s=1$ case. Under the law of the random Fourier series

$u(x)=u(x;\omega)=\sum_{n\in\mathbb{Z}}\frac{g_{n}(\omega)}{\langle
n\rangle}e^{inx}$ (5.1)

corresponding to the Gaussian measure $\mu_{1}$, $\operatorname*{Re}u$ and
$\operatorname*{Im}u$ are independent stationary Ornstein-Uhlenbeck (OU)
processes (in $x$) on $[0,2\pi)$. Recall that the law of this process can be
written as

$u(x)\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\mathbf{P}_{\neq
0}w(x)+g_{0}=w(x)-\fint_{0}^{2\pi}w(y)dy+g_{0},$ (5.2)

where $w$ is a complex OU bridge with $w(0)=w(2\pi)=0$ and $g_{0}$ is a
standard complex-valued Gaussian random variable (independent from $w$).

We now recall the law of the iterated logarithm for the Brownian motion (see
[41, I.16.1]):

######  Proposition 5.1.

Let $B(t)$ be a standard Brownian motion on $\mathbb{R}_{+}$. Then, for each
$t\geq 0$,

$\limsup_{h\downarrow 0}\frac{B(t+h)-B(t)}{\sqrt{2h\log\log\frac{1}{h}}}=1,$
(5.3)

almost surely.

It follows from the representation (5.2), the absolute continuity242424The
absolute continuity property claimed here can be easily seen by the Fourier
series representations of the Brownian motion/bridge (with (5.1) and (5.2))
and Kakutani’s theorem (Lemma 5.3 below). For example, the Brownian motion
$B(t)$ on $[0,2\pi)$ has the following Fourier-Wiener series
$B(t)=g_{0}t+\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\frac{g_{n}}{n}e^{int}.$ of
the Brownian bridge with respect to the Brownian motion on any interval
$[0,\gamma)$, $\gamma<2\pi$, and the absolute continuity of the OU bridge with
respect to the Brownian bridge also on any interval $[0,\gamma)$, that the
limit (5.3) also holds for $\operatorname*{Re}u$ and $\operatorname*{Im}u$ on
$[0,2\pi)$.

Define $\psi$ by

$\psi(h)=\sqrt{2h\log\log\frac{1}{h}},\quad 0<h<1.$

Let $0\leq x<2\pi$. As a corollary to Proposition 5.1, we have

$\displaystyle\limsup_{h\downarrow
0}\frac{\operatorname*{Re}u(x+h)-\operatorname*{Re}u(x)}{\psi(h)}=1$ (5.4)

almost surely.

In the following, by a direct calculation, we show that
$\operatorname*{Re}[e^{-i|u|^{2}}u]$ does not satisfy (5.4) with a positive
probability. This will show that the pushforward measure
$\widetilde{\Phi}(t)_{*}\mu_{s}$ under the dynamics of the dispersionless
model (1.19) is not absolutely continuous with respect to the Gaussian measure
$\mu_{s}$.

On the one hand, we have

$\displaystyle\operatorname*{Re}[$ $\displaystyle
e^{-i|u(y)|^{2}}u(y)]-\operatorname*{Re}[e^{-i|u(x)|^{2}}u(x)]$
$\displaystyle=(\operatorname*{Re}u(y)-\operatorname*{Re}u(x))\cos|u(y)|^{2}+(\cos|u(y)|^{2}-\cos|u(x)|^{2})\operatorname*{Re}u(x)$
$\displaystyle\hphantom{X}+(\operatorname*{Im}u(y)-\operatorname*{Im}u(x))\sin|u(y)|^{2}+(\sin|u(y)|^{2}-\sin|u(x)|^{2})\operatorname*{Im}u(x).$

On the other hand, by the Taylor expansion with
$\eta(x,y)=|u(y)|^{2}-|u(x)|^{2}$, we have

$\displaystyle\cos|u(y)|^{2}$
$\displaystyle=\cos|u(x)|^{2}-\sin|u(x)|^{2}\cdot\eta(x,y)+O\big{(}\eta^{2}(x,y)\big{)},$
$\displaystyle\sin|u(y)|^{2}$
$\displaystyle=\sin|u(x)|^{2}+\cos|u(x)|^{2}\cdot\eta(x,y)+O\big{(}\eta^{2}(x,y)\big{)}.$

Putting together, we obtain

$\displaystyle\operatorname*{Re}[e^{-i|u(y)|^{2}}$ $\displaystyle
u(y)]-\operatorname*{Re}[e^{-i|u(x)|^{2}}u(x)]$
$\displaystyle=(\operatorname*{Re}u(y)-\operatorname*{Re}u(x))\cos|u(y)|^{2}-\sin|u(x)|^{2}\cdot\eta(x,y)\operatorname*{Re}u(x)$
$\displaystyle\hphantom{X}+(\operatorname*{Im}u(y)-\operatorname*{Im}u(x))\sin|u(y)|^{2}+\cos|u(x)|^{2}\cdot\eta(x,y)\operatorname*{Im}u(x)$
$\displaystyle\hphantom{X}+O\big{(}\eta^{2}(x,y)\big{)}\cdot(|\operatorname*{Re}u(x)|+|\operatorname*{Im}u(x)|)$
$\displaystyle=(\operatorname*{Re}u(y)-\operatorname*{Re}u(x))\cos|u(y)|^{2}$
$\displaystyle\hphantom{X}-\sin|u(x)|^{2}\cdot\big{\\{}(\operatorname*{Re}u(y)-\operatorname*{Re}u(x))(\operatorname*{Re}u(y)+\operatorname*{Re}u(x))\big{\\}}\operatorname*{Re}u(x)$
$\displaystyle\hphantom{X}-\sin|u(x)|^{2}\cdot\big{\\{}(\operatorname*{Im}u(y)-\operatorname*{Im}u(x))(\operatorname*{Im}u(y)+\operatorname*{Im}u(x))\big{\\}}\operatorname*{Re}u(x)$
$\displaystyle\hphantom{X}+(\operatorname*{Im}u(y)-\operatorname*{Im}u(x))\sin|u(y)|^{2}$
$\displaystyle\hphantom{X}+\cos|u(x)|^{2}\cdot\big{\\{}(\operatorname*{Re}u(y)-\operatorname*{Re}u(x))(\operatorname*{Re}u(y)+\operatorname*{Re}u(x))\big{\\}}\operatorname*{Im}u(x)$
$\displaystyle\hphantom{X}+\cos|u(x)|^{2}\cdot\big{\\{}(\operatorname*{Im}u(y)-\operatorname*{Im}u(x))(\operatorname*{Im}u(y)+\operatorname*{Im}u(x))\big{\\}}\operatorname*{Im}u(x)$
$\displaystyle\hphantom{X}+O\big{(}\eta^{2}(x,y)\big{)}\cdot(|\operatorname*{Re}u(x)|+|\operatorname*{Im}u(x)|).$
(5.5)

Fix $0\leq x<2\pi$. Let $\\{h_{n}=h_{n}(\omega)\\}_{n\in\mathbb{N}}$ be a
(random) sequence achieving the limit supremum in (5.4) almost surely. Then,
for this sequence $\\{h_{n}\\}_{n\in\mathbb{N}}$, we have

$\displaystyle\limsup_{n\to\infty}\frac{|\operatorname*{Im}u(x+h_{n})-\operatorname*{Im}u(x)|}{\psi(h_{n})}\leq
1$ (5.6)

almost surely. Divide the expression in (5.5) by $\psi(h_{n})$, after
replacing $y$ by $x+h_{n}$. Then, by taking the limit as $n\to\infty$ and
applying (5.4) and (5.6), we have

$\displaystyle\limsup_{n\rightarrow\infty}$
$\displaystyle\frac{\operatorname*{Re}[e^{-i|u(x+h_{n})|^{2}}u(x+h_{n})]-\operatorname*{Re}[e^{-i|u(x)|^{2}}u(x)]}{\psi(h_{n})}$
$\displaystyle\geq-2\sin|u(x)|^{2}\cdot(\operatorname*{Re}u(x))^{2}$
$\displaystyle\phantom{X}-\big{|}\cos|u(x)|^{2}\big{|}-\big{|}\sin|u(x)|^{2}\big{|}-2\cdot\big{|}\sin|u(x)|^{2}\cdot\operatorname*{Im}u(x)\operatorname*{Re}u(x)\big{|}$
$\displaystyle\phantom{X}-2\cdot\big{|}\cos|u(x)|^{2}\cdot\operatorname*{Re}u(x)\operatorname*{Im}u(x)\big{|}-2\cdot\big{|}\cos|u(x)|^{2}\big{|}(\operatorname*{Im}u(x))^{2},$
(5.7)

almost surely.

Fix $M\gg 1$ by

$\displaystyle M^{2}=-\frac{\pi}{2}+2k\pi,$ (5.8)

for some large $k\in\mathbb{N}$ (to be chosen later). Given $\varepsilon>0$,
define the set

$A=\big{\\{}\omega\in\Omega:|\operatorname*{Re}u(x;\omega)-M|\leq\varepsilon,\
|\operatorname*{Im}u(x;\omega)|\leq\varepsilon\big{\\}}.$

Noting that under the law of the OU loop, $\operatorname*{Re}u(x)$ and
$\operatorname*{Im}u(x)$ are independent Gaussian random variables, we have

$P(A)\geq\delta(M,\epsilon)>0$

for any $\varepsilon>0$. By choosing $\varepsilon>0$ sufficiently small such
that $\varepsilon M\ll 1$, we have

$\displaystyle\big{|}|u(x)|^{2}-M^{2}\big{|}\leq
2\varepsilon(M+\varepsilon)=o(1).$ (5.9)

Then, from (5.8) and (5.9), we have

$\displaystyle\text{RHS of }\eqref{LIL5}$ $\displaystyle\geq
2\big{|}\sin|u(x)|^{2}\big{|}\cdot
M(M-3\varepsilon)-2(1+\varepsilon(M+2\varepsilon))$ $\displaystyle\geq M^{2}$

on $A$. By choosing $M\gg 1$, we see that the set

$A_{1}=\left\\{\limsup_{h\downarrow
0}\frac{\operatorname*{Re}[e^{-i|u(x+h)|^{2}}u(x+h)]-\operatorname*{Re}[e^{-i|u(x)|^{2}]}u(x)]}{\psi(h)}=1\right\\}$

does not have probability 1 under the law of $u$. Therefore, $\mu_{1}$ is not
quasi-invariant under the flow of the dispersionless model (1.19).

###  5.2. Fractional Brownian motion

In this subsection, we extend the previous result to the distribution of the
random Fourier series

$\displaystyle u_{s}(x)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\frac{g_{n}}{\langle
n\rangle^{s}}e^{inx},$ (5.10)

corresponding to $\mu_{s}$. For $\frac{1}{2}<s<\frac{3}{2}$, the series (5.10)
is related to a fractional Brownian motion. Recall that a fractional Brownian
motion with Hurst parameter $H$, $0<H<1$, is the Gaussian process
$B^{H}(t),t\geq 0$ with stationary increments and covariance

$\mathbb{E}[B^{H}(t_{1})B^{H}(t_{2})]=\frac{\rho(H)}{2}(t_{1}^{2H}+t_{2}^{2H}-|t_{2}-t_{1}|^{2H}),$
(5.11)

where

$\rho(H)=\mathbb{E}\big{[}\big{(}B^{H}(1)\big{)}^{2}\big{]}=-2\frac{\cos(\pi
H)}{\pi}\Gamma(-2H)\ \text{ when
}H\neq\tfrac{1}{2}\qquad\text{and}\qquad\rho\big{(}\tfrac{1}{2}\big{)}=1.$

When $H=\frac{1}{2}$, a fractional Brownian motion becomes the standard
Brownian motion. In the following, we only consider the case
$H\neq\frac{1}{2}$.

It is known that there is a subtle issue on building a series representation
for a fractional Brownian motion $B^{H}$. Instead, we consider the following
series252525As mentioned in Section 1, we drop the factor of $2\pi$.

$\widehat{B}^{H}(t)=\widetilde{g}_{0}t+\sqrt{2}\sum_{n\geq
1}\bigg{(}\widetilde{g}_{n}\frac{\cos(nt)-1}{n^{H+\frac{1}{2}}}+\widetilde{g}_{n}^{\prime}\frac{\sin(nt)}{n^{H+\frac{1}{2}}}\bigg{)},$

for $t\in[0,2\pi]$, where $\widetilde{g}_{n}$ and $\widetilde{g}_{n}^{\prime}$
are now independent real-valued standard Gaussians. Then, Picard [38] showed
the following result on the relation between $B^{H}$ and $\widehat{B}^{H}$.

######  Lemma 5.2 (Theorems 24 and 27 in Section 6 of [38]).

The processes $B^{H}(t)$ and $\widehat{B}^{H}(t)$ can be coupled in such a way
that

$B^{H}(t)-\widehat{B}^{H}(t)$

is a $C^{\infty}$-function on $(0,2\pi]$. Moreover, if $H\neq\frac{1}{2}$,
then the laws of $B^{H}$ and $\widehat{B}^{H}$ are equivalent on $[0,T]$ for
$T<2\pi$ (and mutually singular if $T=2\pi$).

Recall Kakutani’s criterion [22] in the Gaussian case.

######  Lemma 5.3.

Let $\\{g_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\widetilde{g}_{n}\\}_{n\in\mathbb{N}}$ be two sequences of centered
Gaussian random variables with variances
$\mathbb{E}[g_{n}^{2}]=\sigma_{n}^{2}>0$ and
$\mathbb{E}[\widetilde{g}_{n}^{2}]=\widetilde{\sigma}_{n}^{2}>0$. Then, the
laws of the sequences $\\{g_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\tilde{g}_{n}\\}_{n\in\mathbb{N}}$ are equivalent if and only if

$\sum_{n\in\mathbb{N}}\bigg{(}\frac{\widetilde{\sigma}^{2}_{n}}{\sigma^{2}_{n}}-1\bigg{)}^{2}<\infty.$

From (5.10), we have

$\displaystyle\operatorname*{Re}u_{s}(x)$
$\displaystyle=\operatorname*{Re}g_{0}+\sum_{n\geq
1}\bigg{(}\frac{\operatorname*{Re}g_{n}+\operatorname*{Re}g_{-n}}{\langle
n\rangle^{s}}\cos(nx)+\frac{-\operatorname*{Im}g_{n}+\operatorname*{Im}g_{-n}}{\langle
n\rangle^{s}}\sin(nx)\bigg{)}.$ (5.12)

Then, applying Lemma 5.3 to the sequences

$\bigg{\\{}\widetilde{g}_{0}t,\
\sqrt{2}\frac{\widetilde{g}_{n}}{n^{H+\frac{1}{2}}},\