\section{Introduction}

\smallskip
\subsection{The problem}
We consider the focusing quartic Klein-Gordon (NLKG) equation
\begin{equation}
\label{nlkg}
\partial_t^2 u - \partial_x^2 u + u = u^4,
\end{equation}
where $(t, x) \in \ensuremath{\mathbb{R}} \times \ensuremath{\mathbb{R}}$ are time and space variables, and $u(t,x) \in \mathbb{R}$.
Our results apply to more general power nonlinearities as well but we restrict
our discussion and proof to the case $p=4$ for concreteness; see Remark \ref{remp}.
The Hamiltonian associated with \eqref{nlkg} is
\begin{equation}\label{nlkgHam}
\calH(u,u_t) = \frac{1}{2} \int_\mathbb{R} \left[ (\partial_t u)^2 + (\partial_x u)^2 + u^2 \right]dx
  - \frac{1}{5}\int_\mathbb{R} u^5 dx,
\end{equation}
and for initial data in the energy space, $(u(0),u_t(0)) \in H^1(\mathbb{R})\times L^2(\mathbb{R})$,
the equation is locally well-posed, and globally well-posed for small solutions.
{Global existence for small energy initial data follows from standard energy estimates, see for example \cite{Caz85}. It is also known that finite time blow-up occurs for solutions with negative energy initial data \cite{Levine} and for large positive initial data \cite{Wang, YX18}, which follows from a convexity argument.}

\eqref{nlkg} admits an explicit time-independent static soliton solution
\begin{equation}
\label{soliton}
Q(x) = (\alpha+1)^{\frac{1}{2 \alpha}} \ensuremath{\text{sech}}^{1/\alpha} (\alpha x), \quad
  \alpha := \frac{3}{2}.
\end{equation}
In this paper we study the dynamics of certain classes
of global-in-time solutions ($t\geq 0$) in the vicinity of $Q$
and, in particular, their behavior on the whole real line and their decay towards the soliton.
Letting $u = Q + v$ denote the perturbed soliton for a real-valued and sufficiently small $v$,
we see that
\begin{equation}
\label{nlkg-lin}
v_{tt} + \l v = N(v),
\end{equation}
where
\begin{equation}
\label{estimates-lin}
\l := -\partial_x^2 + 1 - 4Q^3,
\qquad
N(v) :=  6Q^2 v^2 + 4Q v^3 + v^4.
\end{equation}
The spectral properties of the linearized operator $\l$ were described in details in \cite{CGNT07},
also in the case of general powers $u^p$ in \eqref{nlkg}.
In particular, it is known that $\l$ has continuous spectrum $[1, \infty)$ and exactly two isolated eigenvalues:

\setlength{\leftmargini}{1.5em}
\begin{itemize}
\item[-] The first eigenvalue is $\lambda_0 = - \alpha (\alpha+2) = -\Omega^2$
($\Omega>0$) with corresponding even normalized eigenfunction
\begin{equation}
\label{even-efunction}
\rho(x) := c_0 (\ensuremath{\text{sech}}(\alpha x))^{\frac{\alpha+1}{\alpha}}, \qquad \l \rho = -\Omega^2 \rho,
\end{equation}
where the positive $c_0$ is chosen to satisfy $\|\rho\|_{L^2} = 1$;

\item[-] The other eigenvalue is $\lambda_1 = 0$ with corresponding odd eigenfunction $Q'(x)$;
in particular, $\l$ has no gap eigenvalues (internal mode) in $(0,1)$.

\item[-] Moreover, the operator $\l$ has no resonance at the edge of the continuous spectrum,
i.e., there does not exist a bounded solution $\varphi$ to $\l \varphi = 0$.
Equivalently, the potential $V=  -4Q^3$ is `generic'.
\end{itemize}

To avoid the translation mode and having to modulate the soliton,
we restrict our analysis to even perturbations, and decompose $v$ as
\begin{equation}
\label{v-decomp}
v(t, x) = a(t) \rho(x) + \chi(t, x), \quad \lr{\chi,\rho}_{L_x^2} = 0,\quad \chi = P_c v,
\end{equation}
where $P_c:L_x^2 \to L_x^2$ is the projection onto the continuous spectrum of $\l$.

\smallskip
\subsection{Stable manifold
and some literature
Before stating our main result we review the definition of the
stable manifold around the soliton $Q$ for even solutions of \eqref{nlkg},
following Kowalczyk, Martel and Mu\~noz in \cite{KMM21} (see also Bates-Jones \cite{BJ89}),
as well as the main result in \cite{KMM21} concerning\footnote{The results in \cite{KMM21}
cover the more general case of nonlinearities $|u|^{p-1}u$ with $p>3$.} \eqref{nlkg}.
Rewrite the linear part of the equation \eqref{nlkg-lin} as the system
\begin{equation}
\label{linearized-system-v}
\left\{
\begin{array}{l}
\partial_t v_1 = v_2,
\\
\partial_t v_2 = -\mathcal{L}v_1,
\end{array}
\right.
\end{equation}
where $\vec v := (v_1, v_2) = (v, v_t)$ and the operator $\mathcal{L}$ is defined in \eqref{estimates-lin}.
The exponentially unstable and stable solutions of \eqref{linearized-system-v}
are $\vec v_+ (t,x) = e^{\Omega t} Y_+(x)$ and $\vec v_- (t,x) = e^{-\Omega t} Y_-(x)$, respectively, where
\begin{equation}\label{Ypm}
Y_+ = \begin{pmatrix} \rho,  \Omega \rho \end{pmatrix}
  \quad \text{and} \quad Y_- = \begin{pmatrix} \rho, -\Omega \rho \end{pmatrix}.
\end{equation}
We let
\begin{equation}\label{Zpm}
Z_+ = \begin{pmatrix} \rho,  \Omega^{-1} \rho \end{pmatrix}

\end{equation}
so that
\lr{Y_-, Z_+} = 0$.

For every $\delta_0>0$, define
\begin{equation}
\label{A-0}
\mathcal{A}(\delta_0) := \left\{  \bm{\gamma} \in H^1_x(\mathbb{R}) \times L^2_x (\mathbb{R}) : \quad \bm{\gamma} ~ \textrm{is even},~ \|\bm{\gamma}\|_{H^1_x \times L^2_x} < \delta_0 ~ \text{and}~ \lr{\bm{\gamma}, Z_+} = 0 \right\}.
\end{equation}

Then, the following stability result was obtained in \cite{BJ89,KMM21}.

\begin{theorem}[\cite{BJ89,KMM21}]
\label{thmSM}
There exist $C, \delta_0>0$ and a Lipschitz function $h: \mathcal{A}(\delta_0) \to \mathbb{R}$
with $h(0)=0$ and $|h(\bm{\gamma})|\leq C\n{\bm{\gamma}}^{3/2}_{H_x^1 \times L^2_x}$
such that, denoting
\begin{equation}
\label{center-manifold}
\mathcal{M}(\delta_0) := \{(Q, 0) + \bm{\gamma} + h(\bm{\gamma}) Y_+: ~ \bm{\gamma} \in \mathcal{A}(\delta_0)\}
\end{equation}
the following holds:

\begin{itemize}
\item[1.] If $(u_0, u_1) \in \mathcal{M}(\delta_0)$, then the solution of \eqref{nlkg}
with  initial data $(u_0, u_1)$ is global and satisfies
\begin{equation}
\label{global-bound-KMM}
\big\|(u(t)-Q, \partial_t u(t))\big\|_{H_x^1 \times L^2_x}
  \leq C \big\|(u_0-Q, u_1)\big\|_{H_x^1 \times L^2_x} \quad \text{for all}~ t\geq 0.
\end{equation}

\smallskip
\item[2.] If a global even solution $(u, \partial_t u)(t)$ of \eqref{nlkg} satisfies
\begin{equation*}
\big\|(u(t)-Q, \partial_t u(t))\big\|_{H_x^1 \times L^2_x} < \frac{1}{2}\delta_0 \quad \text{for all}~ t\geq 0,
\end{equation*}
then $(u, \partial_t u)(t) \in \mathcal{M}(\delta_0)$ for all $t\geq 0$.
\end{itemize}
\end{theorem}

The above classical theorem provides a stable manifold $\mathcal{M}=\mathcal{M}(\delta_0)$ such that data
originating in $\mathcal{M}$ is global,
and stays globally-in-time close to the soliton in the energy norm.
Moreover, any solution that is globally sufficiently close to $Q$, has to lie in $\mathcal{M}$.
Note that, in view of the decomposition $u = Q+v = Q + a(t)\rho + \chi$, see \eqref{v-decomp},
if one chooses sufficiently small data $a(0), \dot{a}(0)$
and $(\chi(0,x), \chi_t(0,x)) \in H^1_x\times L^2_x$ small enough,
so that $(u(0),u_t(0)) \in \mathcal{M}(\delta_0)$,
then the global bound \eqref{global-bound-KMM} implies that, for every $t\geq 0$,
we have $\n{v(t)}_{H_x^1 \times L_x^2} \lesssim \delta_0$ and
\begin{equation}
\label{global-bound-KMM-a-chi}
|a(t)| + |\dot a(t)| + \normm{\chi(t)}{x} + \normm{\partial_t \chi(t)}{x} \lesssim \delta_0.
\end{equation}

A natural question to ask is whether the above global solutions
are also attracted to the soliton, besides remaining close to the it for all times,
and what is their asymptotic behavior as $t \rightarrow \infty$.

For nonlinear equations with large enough power nonlinearities these types
of asymptotic stability result have been achieved by several authors.
{The literature is ample so we will only mention a few recent results,
and refer the readers to the cited works for more discussions and references.
For the Klein-Gordon equation with power nonlinearity $u^p$ with $p>5$
asymptotic stability (on the stable manifold) was proven by Krieger-Nakanishi-Schlag \cite{KNS12}
for data in the energy space.
See also \cite{NSbook,NS3dnr} and reference therein, for a similar problem
and more general results in higher dimensions.
See also the review
by Kowalczyk-Martel-Mu\~noz \cite{KMMreview}.
For the nonlinear Schr\"{o}dinger (NLS) equation
Krieger-Schlag \cite{KriSchNLS} treated the case of a $|u|^{2\sigma}u$ nonlinearity with $\sigma>2$ in $1$d
and Cuccagna-Pelinovsky \cite{CP14} the cubic case.
See also the seminal work of Schlag \cite{Sch09} for the cubic NLS in $3$d,
the recent works and survey by Cuccagna-Maeda \cite{CM21B,CM21}
and Maeda-Yamazaki \cite{MaedaYamazaki} and references therein.
}

For low power nonlinearities instead (typically the energy sub-critical ones)
one cannot expect similar asymptotic stability result
in the energy space $H^1_x(\mathbb{R}) \times L^2_x(\mathbb{R})$, essentially because Strichartz estimates
are not very effective in controlling the nonlinearity.
There are then two natural different and complementary ways to look at this question:
(a) studying the asymptotics for energy solutions locally in the space variable $x$,
or
(b) studying asymptotics on the whole real line in a smaller space, e.g.,
a weighted Sobolev space.
The first approach is usually based on delicate virial type estimates, and has been
very successfully employed in several problems;
see for example the already mentioned \cite{KMM21}, the recent work of Li-L\"uhrmann \cite{LiLu22} on the soliton of quadratic KG, Cuccagna-Maeda-Scrobogna \cite{CMS22}
on cubic KG, and the works on kink solutions for relativistic scalar fields \cite{KowMarMun,KowMarMunVDB,CM22}.
The second approach has instead been pursued mostly in the context of
kinks (non-localized solitons) which naturally lead to nonlinear PDEs with potentials and
low power non-localized nonlinearities;
we refer the reader to Delort-Masmoudi \cite{DMKink}, Germain and the second author \cite{GP20} and Zhang
\cite{KGVSim},
L\"uhrmann-Schlag \cite{LSch}, Lindblad-L\"uhrmann-Soffer \cite{LLS20},
and Lindblad-L\"uhrmann-Schlag-Soffer \cite{LLSS}.

\smallskip
Concerning the problem we are interested in,
\cite{KMM21} showed the following asymptotic stability result

\begin{theorem}[{\cite[Theorem 1]{KMM21}}]\label{thmKMM}
Consider the Klein-Gordon equation
\begin{align}\label{nlkgalpha}
\partial_t^2 u - \partial_x^2 u + u = |u|^{2\alpha}u, \qquad \alpha > 1,
\end{align}
and denote its static soliton by
\begin{equation}\label{solitonalpha}
Q(x) := (\alpha+1)^{\frac{1}{2 \alpha}} \ensuremath{\text{sech}}^{1/\alpha} (\alpha x).
\end{equation}
Then, there exists a constant $\delta >0$ such that if a global even solution $(u,u_t)$ of \eqref{nlkgalpha}
satisfies
\begin{align}\label{KMMas}
\big\|(u(t)-Q, \partial_t u(t))\big\|_{H_x^1(\mathbb{R}) \times L^2_x(\mathbb{R})} < \delta \quad \text{for all}~ t\geq 0,
\end{align}
then, for any bounded interval $I$,
\begin{align}\label{KMMconc}
\lim_{t\rightarrow \infty} \big\|(u(t)-Q, \partial_t u(t))\big\|_{H_x^1(I) \times L^2_x(I)} = 0.
\end{align}
Moreover,
\begin{align}\label{KMMconcint}
\int_0^\infty \big\|(u(t)-Q, \partial_t u(t))\big\|_{H_x^1(I) \times L^2_x(I)}^2 \, dt < \infty.
\end{align}
\end{theorem}

Note that a global solution satisfying \eqref{KMMas} for $\delta$ small enough
is on the stable manifold and therefore stays close to the soliton in view of Theorem \ref{thmSM}.
Theorem \ref{thmKMM} then gives that these solutions actually converge as $t \rightarrow \infty$ to
$Q$ locally-in-space.
Our main theorem will give a complementary asymptotic stability result;
we will assume more localization on the (well-prepared)
initial data and prove convergence to $Q$ on the whole real-line
with optimal time-decay rates, and local-decay estimates as well.
In particular, we will show that, see \eqref{pointwise-decay-main-theorem},
\begin{align}\label{concint01}
& \int_0^\infty \big\|
  u(t)-Q
  \big\|_{L^\infty_x(\mathbb{R})}^{2+} \, dt < \infty,
\\
\label{concint02}
& \int_0^\infty \big\|\langle x \rangle^{-2} (
  u(t)-Q
  )
  \big\|_{L^\infty_x(\mathbb{R})}^{1+} \, dt < \infty.
\end{align}

\iffalse
UNUSED STUFF--------------------------------------------
We introduce the decomposition of $a(t)$ via the functions $a_\pm(t)$ as
$$
a_\pm (t) = \frac{1}{2} \left( a(t) \pm \frac{1}{\Omega} \dot a(t) \right), \quad
a(t) = a_+(t) + a_-(t),
$$
and applying the stability condition (\ref{stability-cond}), we obtain the integral equations
\begin{equation}
\label{a-pm-integral-eqns}
\left\{ \begin{array}{l}
a_+(t) = -\frac{1}{2\Omega} \int_t^\infty
e^{-\Omega(s-t)} F(v)(s) ds, \\
a_-(t) = e^{-\Omega t} a_-(0) + \frac{1}{2\Omega} \int_0^t
e^{-\Omega(t-s)} F(v)(s) ds
\end{array} \right.
\end{equation}

\begin{lemma}
For every $t > 0$ and $s\geq 0$,
\begin{eqnarray}
\nonumber
|a(t)| & \lesssim & |a_-(0)| +
\|a\|_{L_t^\infty (0, \infty)}^2 +
\int_0^\infty e^{-\Omega |t-s|}
\|
\chi(s, \cdot)\|_{L_x^\infty}^2\\
\nonumber
& \phantom{t} & +
\|a\|_{L_t^\infty (0, \infty)}^4 +
\int_0^\infty e^{-\Omega |t-s|} \| \chi(s, \cdot)\|_{L_x^\infty}^4 ds.
\end{eqnarray}
\end{lemma}

\begin{proof}
Since
$$
|F(v)| =
|\ip{6Q^2 v^2 + 4Q v^3 + v^4, \rho}_{L^2_x}|
\lesssim \ip{Q^2 v^2, \rho}_{L^2_x} +
\ip{v^4, \rho}_{L^2_x},
$$
we need to obtain the estimates for the upper bound terms, which are the following
\begin{equation*}
\left\{ \begin{array}{l}
\ip{Q^2 v^2, \rho}_{L^2_x} \lesssim
a^2(t) + \ip{Q^2 \rho, \chi^2(t)}_{L^2_x}, \\
\ip{v^4, \rho}_{L^2_x} \lesssim
a^4(t) + \ip{\rho, \chi^4(t)}_{L^2_x},
\end{array} \right.
\end{equation*}
and so
\begin{equation}
\label{F-upper-bound}
|F(v)(t)| \lesssim a^2(t) +
\ip{Q^2 \rho, \chi^2(t)}_{L^2_x} + h.o.t.
\end{equation}
Substituting this estimate into the integral equations (\ref{a-pm-integral-eqns}),
\begin{equation}
\nonumber
\left\{ \begin{array}{l}
|a_+(t)| \lesssim \int_t^\infty
e^{-\Omega(s-t)} |F(v)(s)| ds, \\
|a_-(t)| = |a_-(0)| + \int_0^t
e^{-\Omega(t-s)} |F(v)(s)| ds,
\end{array} \right.
\end{equation}
which can be combined into
\begin{equation}
\label{a-pm-estimates}
|a_\pm(t)| \lesssim |a_-(0)| +
\int_0^\infty e^{-\Omega |t-s|} |F(a, \chi)(s)| ds.
\end{equation}
Now, the assertion of the lemma follows from the estimates (\ref{F-upper-bound}) applied to (\ref{a-pm-estimates}).
\end{proof}
------------------------------------------------------------
\fi

\smallskip
\subsection{Main result}
Recall that we write $u(t,x) = Q(x) + v(t,x) = Q(x) + a(t)\rho(x) + \chi(t,x)$,
where $\rho$ is given in \eqref{even-efunction},
$Y_\pm := (\rho, \pm \Omega \rho)$ and $\mathcal{M}$ is defined in \eqref{center-manifold} with \eqref{Zpm}.
We will consider well-prepared initial data in a subset of $\mathcal{M}$ as we now describe.
Let $\bm{\gamma}(x) = (\gamma_1(x), \gamma_2(x))$
be given in the form (see \eqref{A-0})
\begin{equation*}
\bm{\gamma}(x) = b Y_-(x) + \bm{\zeta}(x), \quad \bm{\zeta} = (\zeta_1, \zeta_2),
\end{equation*}
where $b$ is a scalar constant, and $\bm{\zeta}$ is even in $x$, with $\lr{\rho, \zeta_j} = 0$ for $j=1,2$.
In other words, $\bm{\gamma}$ has components
\begin{equation}
\label{gamma-12-form}
\begin{aligned}
\gamma_1(x) = b \rho(x) + \zeta_1(x), \quad \gamma_2(x) = -\Omega b \rho(x) + \zeta_2(x).
\end{aligned}
\end{equation}
We define (see \eqref{estimates-lin})
\begin{align}\label{defH}
H := P_c (-\partial_x^2 - 4Q^3), \quad H+1 = P_c \l,
\end{align}
where $P_c$ denotes the projection onto the continuous spectrum,
and require that
\begin{equation}
\label{initial-data-gamma}
|b| + \n{(\sqrt{H+1}~ \zeta_1, \zeta_2)}_{H_x^2} + \n{\lr{x}(\sqrt{H+1}~ \zeta_1, \zeta_2)}_{L_x^2} \leq c\epsilon_0
\end{equation}
for sufficiently small $\epsilon_0>0$, with $c$ a small absolute constant ($c=1/10$, say).
Then, $\bm{\gamma}$ belongs to a strict subset of $\mathcal{A}(\epsilon_0)$.
We then define the following subset of $\mathcal{M}(\epsilon_0)$:
\begin{equation}
\label{center-manifold-subset}
\mathcal{M}_0(\epsilon_0) := \{(Q,0) + \bm{\gamma} + h(\bm{\gamma}) Y_+: \quad
  \bm{\gamma}~ \text{is even and satisfies \eqref{gamma-12-form}--\eqref{initial-data-gamma}} \}
  \subseteq \mathcal{M}(\epsilon_0),
\end{equation}
where $h$ is the Lipschitz function from Theorem \ref{thmSM}.
We are now ready to state our main result:

\begin{theorem}\label{main-theorem
There exists sufficiently small $\epsilon_0>0$ such that, for every $(u_0, u_1) \in \mathcal{M}_0(\epsilon_0)$,
the equation \eqref{nlkg} with initial data $(u, \partial_t u)(0) = (u_0, u_1)$
admits a unique global solution of the form
\begin{equation}
\label{main-theorem-solution}
u(t,x) = Q(x) + a(t) \rho(x) + \chi (t,x), \quad \lr{\rho, \chi}_{L_x^2} = 0,
\end{equation}
that satisfies, for all $t\geq 0$, \eqref{global-bound-KMM} and the
following global-in-time decay bounds:

\begin{align}
\label{pointwise-decay-main-theorem}
|a(t)| \lesssim \epsilon_0 \lr{t}^{-2}, \qquad \n{\chi}_{L_x^\infty} \lesssim \epsilon_0 \lr{t}^{-1/2},
\qquad
\n{ \lr{x}^{-2} \chi}_{L_x^\infty} \lesssim \epsilon_0 \lr{t}^{-1}.
\end{align}
Moreover, $\chi$ behaves like a free solution of the Klein-Gordon equation as $t \rightarrow \infty$.

\end{theorem}

\medskip
Here are a few remarks about our result:

\begin{remark}[About the initial data]\label{remdata2
The choice of the initial data can also be interpreted as follows.
We can express $(u_0, u_1)$ as
\begin{align*}
& u_0(x) = Q(x) + a_0 \rho(x) + \chi_0(x),
\\
& u_1(x) = a_1 \rho(x) + \chi_1(x),
\end{align*}
with $\lr{\chi_j,\rho}_{L^2_x}=0$, $j=1,2$,
$(a_0,a_1) = (a(0),\dot{a}(0))$, $(\chi_0,\chi_1) = (\chi(0),\partial_t\chi(0))$.
Then,
\begin{equation*}
(u_0, u_1) = (Q, 0) + \frac{1}{2} \Big( a_0 - \frac{1}{\Omega} a_1 \Big) Y_-
  + \frac{1}{2} \Big( a_0 + \frac{1}{\Omega} a_1 \Big) Y_+
  + (\chi_0, \chi_1),
\end{equation*}
which, compared with \eqref{gamma-12-form} would give
$b = (1/2)(a_0 - \Omega^{-1} a_1)$, $(\zeta_1,\zeta_2) = (\chi_0,\chi_1)$.
Then $(u_0, u_1) \in \mathcal{M}_0(\epsilon_0)$
if we impose the stability condition
\begin{equation*}
\frac{1}{2} \Big( a_0 + \frac{1}{\Omega} a_1 \Big)
= h\Big( \frac{1}{2} \Big( a_0 - \frac{1}{\Omega} a_1 \Big) Y_- + (\chi_0, \chi_1) \Big),
\end{equation*}
that $\chi_0,\chi_1$ are even, and
\begin{equation}
\label{initial-data-chi}
|a_0| + |a_1| + \n{(\sqrt{H+1}~ \chi_0, \chi_1)}_{H_x^2} + \n{\lr{x}(\sqrt{H+1}~ \chi_0, \chi_1)}_{L_x^2}
  \leq c \epsilon_0.
\end{equation}
for some absolute $c>0$ small enough.
\end{remark}

\begin{remark}[Linearized operator]\label{remlinop}
The potential appearing in linearized operator \eqref{estimates-lin} is
$V = -4Q^3 = -10 \cosh^{-2}(3x/2)$.
The corresponding Schr\"odinger operator is, up to a conjugation, given by
$H = -\partial_x^2 + c \cosh^{-2}(x)$ with $c= -40/9$.
In particular, $V$ is not in the class of P\"oschel-Teller potentials
$V_\ell(x) = - \ell(\ell+1)\cosh^{-2}(x)$, $\ell=0,1,2,\dots$
that can be conjugated to the flat $V=0$ operator;
see the discussion in \cite{LSch} and references therein.
\end{remark}

\begin{remark}[More general powers]\label{remp}
Consider the pure power Klein-Gordon equation
\begin{align}\label{nlkgp}
\partial_t^2 u - \partial_x^2 u + u = |u|^{2\alpha} u,
\end{align}
and its soliton
\begin{equation}\label{solitonp}
Q_\alpha(x) = (\alpha+1)^{\frac{1}{2 \alpha}} \ensuremath{\text{sech}}^{1/\alpha} (\alpha x),
\end{equation}
Recall that Theorem \ref{thmKMM} applies to all powers $\alpha > 1$.
In the borderline cubic case, $\alpha=1$, the linearized operator
has an even resonance. Then, the results of \cite{KMM21}
do not apply, and no asymptotic stability results are available to date.

In the quadratic case, $\alpha = 1/2$, the linearized operator has an internal mode (a gap eigenvalue
in $(0,1)$).
This is a more difficult case since stability does not hold at the linearized level
because of the presence of periodic-in-time localized bound states.
Nevertheless, a stability result like the one in Theorem \ref{thmKMM} was recently proven
by Li and L\"uhrmann \cite{LiLu22}.
See this latter, \cite{KowMarMun}, and the work of L\'eger and the second author \cite{LP22}
for more references about internal mode\ dynamics.

Our main result in Theorem \ref{main-theorem}, which corresponds to the case $2\alpha=3$,
can be easily extended to any $\alpha$ such that $2\alpha$ is an integer larger $3$.
Non-integer powers with $\alpha \geq 3/2$ can also be treated with minor modifications.
The treatment of powers $\alpha \in (1, 3/2)$ may instead require additional arguments
since our approach is not well suited for (relatively low) fractional powers.
However, it is natural to conjecture that the same result would hold,
consistently with the predictions of \cite{BiChSz11}; see
also the next remark for more about this.
\end{remark}

\begin{remark}[Pointwise and integrated decay on the real line
Our main Theorem \ref{main-theorem} complements known results in two ways.
First, it complements \cite{KMM21} by showing that asymptotic stability holds on the whole real line.
Second, it complements results for larger powers, such as \cite{KNS12},
by obtaining (also for lower power nonlinearities)
exact rates of convergence to the soliton for a large class of solutions trapped by it.
As already mentioned, we need to require stronger assumptions on the data, the most relevant of which
is the smallness of the weighted norm in \eqref{initial-data-chi}.

Also note that the first and second estimate in \eqref{pointwise-decay-main-theorem}
immediately imply the integrated decay \eqref{concint01}, while the first and third
give \eqref{concint02}.\footnote{Similar estimates with $\partial_t u$ replacing $Q-u$ can also be obtained,
but one would need to change slightly the assumptions on the data in terms of derivatives;
for simplicity, we do not pursue this here.}
\end{remark}

\begin{remark}[About local decay and rates of convergence
Algebraic rates of decay towards the soliton
on the stable manifold had been predicted numerically already in \cite{BiChSz11}.
In particular, Fig. 2 in \cite{BiChSz11} depicts the expected decay (or growth, for larger data)
in the quartic case \eqref{nlkg}.
Note that the data used for these numerical results is a Gaussian.

The predictions of \cite{BiChSz11} actually give a local-in-space rate of decay of $t^{-3/2}$,
as opposed to the rate of $t^{-1}$ which we obtain from \eqref{pointwise-decay-main-theorem}.
The discrepancy here is just due to the fact that we only require smallness of $xu(0) \in L^2$.
For such data, even at a the level of the linear operator $e^{it\sqrt{H+1}}$,
one cannot obtain more than $t^{-1}$ local decay.
Assuming the stronger localization $xu(0) \in L^1$, the linear decay rate is indeed $t^{-3/2}$,
and one can expect to obtain this same rate of convergence to $Q$
also for our nonlinear solutions.
\end{remark}

\smallskip
\subsection{Strategy of the proof
We begin by following the standard strategy
of substituting the decomposition \eqref{v-decomp} into \eqref{nlkg-lin}-\eqref{estimates-lin},
and separating the discrete and continuous modes as follows:
\begin{equation}
\label{system-eq}
\left\{ \begin{array}{l}
a_{tt} - \Omega^2 a = \lr{N(v), \rho}_{L_x^2} := F[v],
\\
\chi_{tt} + \l \chi = P_c N(v).
\end{array} \right.
\end{equation}
Following the Duhamel's principle, \eqref{system-eq} is equivalent to the system of integral equations
\begin{equation}
\label{integral-eq-0}
\left\{ \begin{array}{l}
a(t) = \cosh(\Omega t) a(0) + \dfrac{1}{\Omega} \sinh(\Omega t) \dot a (0)
+ \dfrac{1}{\Omega} \displaystyle{\int_0^t} \sinh(\Omega(t-s)) F[v](s) ds,
\\
\chi(t) = \cos (\sqrt{H+1} t) \chi(0) + \dfrac{\sin(\sqrt{H+1} \, t)}{\sqrt{H+1}} \dot \chi (0) +
  \displaystyle{\int_0^t} \dfrac{\sin(\sqrt{H+1} (t-s))}{\sqrt{H+1}} P_c N (v(s)) ds.
\end{array} \right.
\end{equation}
Imposing the \textit{stability condition}
\begin{equation}
\label{stability-cond}
\dot a (0) = -\Omega a (0) - \int_0^\infty e^{-\Omega s} F[v](s) ds,
\end{equation}
which ensures that the contribution of the exponentially
increasing term $e^{\Omega t}$ of $a(t)$ vanishes as $t \to \infty$,
\eqref{integral-eq-0} reduces to
\begin{equation}
\label{integral-eq}
\left\{ \begin{array}{l}
a(t) = e^{-\Omega t} \left[ a (0) + \dfrac{1}{2\Omega} \displaystyle{\int_0^\infty} e^{-\Omega s} F[v](s) ds \right] -
  \dfrac{1}{2\Omega} \displaystyle{\int_0^\infty} e^{-\Omega |t-s|} F[v](s) ds,
\\
\chi(t) = \cos (\sqrt{H+1} t) \chi(0) + \dfrac{\sin(\sqrt{H+1} \, t)}{\sqrt{H+1}} \dot \chi (0) +
  \displaystyle{\int_0^t} \dfrac{\sin(\sqrt{H+1} (t-s))}{\sqrt{H+1}} P_c N(v(s)) ds.
\end{array}
\right.
\end{equation}
Note that \eqref{stability-cond} holds for solutions on the stable manifold,
in view of \eqref{global-bound-KMM}.

To rewrite the evolution equation for $\chi(t,x)$ in \eqref{system-eq}
in a convenient way for our analysis, we introduce the profile function $g(t,x)$ as
\begin{equation}
\label{g-profile}
g(t, x) := e^{it \sqrt{H+1}}  ( \partial_t - i \sqrt{H+1} ) \chi(t, x).
\end{equation}
Then the real-valued function $\chi$ is given in terms of $g$ and $\bar g$ by
\begin{equation}
\label{chi-in-terms-of-g}
\chi(t, x) = \frac{1}{2i \sqrt{H+1}} (e^{it \sqrt{H+1}} \bar g(t,x) -  e^{-it \sqrt{H+1}}  g(t, x)).
\end{equation}
We then want to look at Duhamel's formula in terms of $g$ on the distorted Fourier side.
We refer the reader to Section \ref{secdFT} for the definition of the distorted Fourier
Transform (dFT) and its properties.
Applying the distorted Fourier transform to \eqref{g-profile}
gives $\widetilde g(t, k) = e^{it \lr{k}} ( \partial_t - i \lr{k} ) \widetilde \chi(t,k)$,
and the second equation in \eqref{integral-eq} becomes
\begin{equation}
\label{wg-evolutionintro}
\partial_t \widetilde g (t, k) = e^{it\lr{k}} \widetilde{\mathcal{F}} P_c N(Q, v),
  \qquad N(Q, v) := P_c(6Q^2 v^2 + 4 Q v^3 + v^4).
\end{equation}
Integrating \eqref{wg-evolutionintro} over time gives
\begin{equation}
\label{wg-duhamelintro}
\widetilde g (t,k) = \widetilde g (0,k) + \int_0^t e^{is\lr{k}} \widetilde{\mathcal{F}} P_c N (s, k) ds.
\end{equation}
Equation \eqref{wg-duhamelintro} is the starting point for our analysis of the
projection on the continuous spectrum of the solution.

To obtain the desired nonlinear estimates that will imply our main result,
we introduce the following norm
\begin{equation}
\label{main-normintro}
\|v\|_{X} := \|\lr{t}^2 a(t)\|_{L^\infty_t (0, \infty)} +
\|\ensuremath{\partial}_k \widetilde g \|_{L^\infty_t ((0, \infty): L_k^2)} + \|\lr{k}^2 \widetilde g \|_{L^\infty_t ((0, \infty): L_k^2)},
\end{equation}
The first norm above is used to control the evolution of the solution in the direction of the
even eigenfunction, while the other two control the evolution of the continuous spectral component.
The proof of our result will follow from a priori estimates for this norm:

\begin{proposition}\label{propmain}
For $\n{v}_X$ sufficiently small we have
\begin{equation*}
\| v \|_X \lesssim  |a(0)| + \normm{\partial_k \widetilde g (0)}{k} + \normm{\lr{k}^2 \widetilde g (0)}{k} + \|v\|_X^2.
\end{equation*}
\end{proposition}

Proposition \ref{propmain} follows from the three Propositions \ref{a(t)-norm-estimates},
\ref{proposition-partial-k-L2-norm} and \ref{proposition-k2-wg-norm-bound}.
With Proposition \ref{propmain} we can prove the main theorem.

\iffalse
\begin{proposition}
\label{a(t)-norm-estimates}
Given $\n{v}_X \ll 1$, for every $t \geq 0$ we have
\begin{equation}
\label{a-bound}
\lr{t}^2 |a(t)| \lesssim |a(0)| + \|v\|_X^2.
\end{equation}
\end{proposition}

\begin{proposition}
\label{proposition-partial-k-L2-norm}
Given $\n{v}_X \ll 1$, for every $t \geq 0$ we have \begin{equation}
\label{prtl-k-wg-norm-bound}
\normm{\partial_k \widetilde g (t)}{k} \lesssim \normm{\partial_k \widetilde g (0)}{k} + \|v\|_X^2.
\end{equation}
\end{proposition}

\begin{proposition}
\label{proposition-k2-wg-norm-bound}
For every $t \geq 0$ we have
\begin{equation}
\label{k2-wg-norm-bound}
\normm{\lr{k}^2 \widetilde g (t)}{k} \lesssim \normm{\lr{k}^2 \widetilde g (0)}{k} + \|v\|_X^2.
\end{equation}
\end{proposition}
\fi

\smallskip
\subsection{Proof of Theorem \ref{main-theorem}}
First, given initial data $(u_0,u_1) \in \mathcal{M}_0(\epsilon_0)$
with \eqref{gamma-12-form}-\eqref{initial-data-gamma},
Theorem \ref{thmSM} gives a global solution trapped by the soliton for all $t\geq 0$
provided $\epsilon_0$ is chosen small enough depending on $\delta_0$.

The smallness condition
\eqref{initial-data-chi} on the initial data
with (recall \eqref{g-profile})
$(-\sqrt{H+1}\chi_0,\chi_1) = (\Im(g(0)), \Re(g(0)))$
and the estimates
\eqref{derivative-in-fourier-bound} and \eqref{k-s-h-tilde-bound}, imply that
\begin{align}
|a(0)| + \normm{\partial_k \widetilde g (0)}{k} + \normm{\lr{k}^2 \widetilde g (0)}{k} \leq C \epsilon_0
\end{align}
for some absolute constant $C>0$.
Then, Proposition \ref{propmain} gives, for a possibly different $C>0$,
$\| v \|_X \leq C\epsilon_0 + C \|v\|_X^2$.

Therefore, we see that
the map defined by
the first equation on the right-hand side of \eqref{integral-eq}
and by the right-hand side of \eqref{wg-duhamelintro}
takes a ball of radius $2C\epsilon_0$, in $X$, which we denote by $B_{2C \epsilon_0}^X$,
into itself for some absolute $C>0$.
Moreover, repeating the same arguments one can verify
that the map is a contraction,
hence, there is a unique fixed point
in $B_{2C \epsilon_0}^X$.

Finally, the bound $\| v \|_X \lesssim \epsilon_0$
implies directly the first bound in \eqref{pointwise-decay-main-theorem}.
The other two estimates in \eqref{pointwise-decay-main-theorem}
are obtained as a direct consequence
of the pointwise and local decay estimates in Lemmas \ref{lemma-dispersive-estimates}
and \ref{local-l-infty-decay-lemma}. $\hfill \Box$

\iffalse
\comment{
Given the definition \eqref{g-profile}, the integral equations in \eqref{integral-eq} are equivalent to
\begin{equation}
\label{integral-eq-proof}
\left\{ \begin{array}{l}
a(t) = e^{-\Omega t} \left[ a (0) +
\frac{1}{2\Omega} \int_0^\infty e^{-\Omega s} F[v](s) ds \right] -
\frac{1}{2\Omega} \int_0^\infty e^{-\Omega |t-s|} F[v](s) ds, \\[3pt]
\widetilde g (t,k) = \widetilde g (0,k) + \int_0^t e^{is\lr{k}} \widetilde{\mathcal{F}} P_c N (s, k) ds.
\end{array} \right.
\end{equation}
Using \eqref{g-profile}, the initial condition \eqref{initial-data-main-theorem} implies that
\begin{equation}
\label{initial-data-X}
|a(0)|+ \normm{\partial_k \widetilde{g}(0)}{k} + \normm{\lr{k}^2 \widetilde{g}(0)}{k} \leq \epsilon_0.
\end{equation}
From \eqref{a-bound}, \eqref{prtl-k-wg-norm-bound} and \eqref{k2-wg-norm-bound},
we get an estimate on $\n{v}_X$ defined in \eqref{main-norm} as follows
\begin{equation}
\label{v-bootstrap}
\|v\|_X \lesssim |a(0)|+ \normm{\partial_k \widetilde{g}(0)}{k} + \normm{\lr{k}^2 \widetilde{g}(0)}{k} + \n{v}_X^2.
\end{equation}
We can see that, given \eqref{v-bootstrap}, the map defined by
the right hand side of \eqref{integral-eq-proof} takes a ball $B_{C_3 \epsilon_0} (Q,0; X)$
into itself for some uniform $C_3>0$. Moreover, repeating the same arguments we can verify
that the map is a contraction on $B_{C_3 \epsilon_0} (Q,0; X)$.
Hence, there is a unique fixed point of \eqref{integral-eq-proof} in $B_{C_3 \epsilon_0} (Q,0; X)$.
Now, the bounds in \eqref{pointwise-decay-main-theorem} follow from \eqref{a-bound}
and \eqref{dispersive-decay-for-chi} (with $N=1$).
}
\fi

\smallskip
\subsection{Notations}
\label{subsection-notations}
We let $\lr{x} := (1+x^2)^{\frac{1}{2}}$.
We write $a \lesssim b$ is there is a uniform constant $C$ such that $a \leq Cb$.
Indicator functions in variable $y$ will be denoted as $\mathbf{1}_I(y)$
meaning the function is equal to $1$ whenever $y\in I$ and vanishes otherwise.
To distinguish between the function $h$ and its complex conjugate, in a number of occasions,
we use $h^{(+)}(x) := h(x)$ to denote the function itself and $h^{(-)}(x) := \overline{h}(x)$
to denote the complex conjugate of $h$.

\iffalse
\subsubsection*{Littlewood-Paley decomposition}
Let $\varphi: \mathbb{R} \to [0,1]$ be a smooth even cutoff function supported in $(-2,2)$
and equal to $1$ on $[-1,1]$. For every integer $m$ we define
$\varphi_m(k) := \varphi(2^{-m}k) - \varphi(2^{-m+1}k)$, and the family
$(\varphi_m(k))_{m \in \mathbb{Z}}$ forms a partition of unity,
\begin{equation*}
\sum_{m\in \mathbb{Z}} \varphi_m(k) =1 \quad \text{for every  } k \neq 0.
\end{equation*}
We also define the following cutoff functions:  \cfp{May not be using this anymore}

\begin{itemize}
\item a generic smooth cutoff function supported at $|k| \approx 2^m$ denoted by $\varphi_{\sim m}$~;

\item for a given $m_0 \in \mathbb{Z}$, we set
\begin{equation}
\label{cutoff-around-root}
\varphi_m^{(m_0)}(k) = \left\{\begin{array}{l}
 \varphi_m (k) \quad \text{if  } m > m_0,\\
 \varphi_{\leq m_0} (k) \quad \text{if  } m = m_0,
\end{array} \right.
\end{equation}
where $\varphi_{\leq m_0} (k)$ is defined, for every $k$, as
\begin{equation}
\label{cut-off-m0}
\varphi_{\leq m_0}(k) =
1-\sum_{m > m_0}\varphi_m(k);
\end{equation}
\end{itemize}
By construction, each $\varphi_m$ is supported in the compact region around $2^m$, and $\varphi_{m_0}^{(m_0)}$ is supported in a compact region containing $k=0$.
\fi

\smallskip
\subsection{Acknowledgments}
A. K. was supported in part by NSERC grant RGPIN 2018-04536,
and thanks the Fields Institute and McMaster University for its support.

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

\medskip
\section{Linear Theory}\label{secdFT}

In this section, we consider
the second order differential operator $H := -\partial_x^2 + V$,
where $V(x)$ is a potential satisfying the following assumption

\begin{assumption}
\label{V-assumption}
The potential $V$ is a smooth function exponentially decaying at infinity,
\begin{equation}
\label{potential-decay}
\lim_{|x| \to \infty} e^{\alpha |x|} V(x) = 0 \quad \text{for some } \alpha>0,
\end{equation}
and $H = -\ensuremath{\partial}_{xx} + V$ has no point spectrum, and no zero energy resonance,
i.e. there is no bounded nontrivial solution in the kernel of $H$
or, equivalently, $V$ is `generic' in the sense of Definition \ref{defGen}.

\end{assumption}

In our application to the equation \eqref{nlkg-lin}, the role of the operator $H$
is played by $P_c \mathcal{L}$.
For an operator $H$ as above we present the basic facts about Jost functions,
reflection and transmission coefficients,
and the distorted Fourier transform.
This theory is based on classical results of Deift-Trubowitz \cite{DT79} and Faddeev \cite{F59,F64}.
See also the book of Yafaev \cite{Y10}.

\begin{remark}
As it was proven in \cite{CGNT07},
the potential $-4Q^3$ given in \eqref{estimates-lin} is generic.
\end{remark}

\subsection{Jost functions}
We define $f_+(x, k)$ and $f_-(x, k)$ as solutions to the differential equation
\begin{equation}
\label{ode-Jost}
H f_\pm (x, k) = k^2 f_\pm (x, k)
\end{equation}
which satisfy the asymptotic boundary conditions
\begin{equation}
\label{f-pm-asymptotics}
\lim_{x \to \infty} |e^{-ikx} f_+(x, k) - 1| = 0 \quad \text{and} \quad \lim_{x \to -\infty} |e^{ikx} f_-(x, k) - 1| = 0
\end{equation}
We further define $m_\pm (x, k) := e^{\mp ikx} f_\pm (x, k)$. For fixed $x$, the functions $m_\pm (x,k)$ are analytic in $\Im k >0$ and continuous in $\Im k \geq 0$. The following lemma provides with the primary estimates on  $m_\pm$, which will be used later.
\begin{lemma}
\label{bounds-m-pm}
For every $s = 0,1,2$ and every positive integer $N$, we have
\begin{equation}
\label{estimates-partial_k^s_m}
\begin{aligned}
&|\partial_k^s (m_\pm (x, k) - 1)| \lesssim \lr{x}^{-12} \lr{k}^{-1}, \quad \qquad   \pm x \geq -1, \\
&|\partial_k^s (m_\pm (x, k) - 1)| \lesssim  \lr{x}^{s+1} \lr{k}^{-1}, \quad \qquad \pm x \leq 1.
\end{aligned}
\end{equation}
Moreover, in the presence of the $x$ derivatives, we have
\begin{equation}
\label{estimates-partial_k_and_x_m}
\begin{aligned}
&|\partial_x^N \partial_k^s m_\pm (x, k)| \lesssim \lr{x}^{-12} \lr{k}^{-1+N}, \quad \qquad   \pm x \geq -1, \\
&|\partial_x^N \partial_k^s m_\pm (x, k)| \lesssim \lr{x}^{s+1} \lr{k}^{-1+N}, \qquad \qquad \pm x \leq 1.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
The estimates follow by iterating the Volterra integral equations for $m_\pm$ given by
\begin{equation*}
m_\pm (x, k) = 1 \pm \int_{x}^{\pm \infty} D_k(\pm (y-x)) V(y) m_\pm(y,k) dy, \quad \text{where  } D_k(y) = \frac{e^{2iky}-1}{2ik}
\end{equation*}
as in Deift-Trubowitz \cite{DT79} or Appendix A in \cite{GPR18} under the assumption \eqref{potential-decay} on $V$.
\end{proof}

\begin{remark}
Results similar to those in Lemma \ref{bounds-m-pm} can be derived for any $s \geq 0$, and in such case some of the bounds should be modified, see for example \cite{CP21, GP20}. However, we will be working only with up to second derivatives of $m_\pm$, and so are the choices of $s$.
\end{remark}

The Jost functions $f_\pm$ are related through so-called transmission and reflection coefficients, $T(k)$ and $R_\pm(k)$, respectively \cite{DT79}. That is,
\begin{equation}
\label{jost-functions-via-scat-par}
\begin{aligned}
T(k) f_+(x,k) & = R_-(k) f_-(x,k) + f_-(x, -k),
\\
T(k) f_-(x,k) & = R_+(k) f_+(x,k) + f_+(x, -k),
\end{aligned}
\end{equation}
and using the boundary conditions \eqref{f-pm-asymptotics}, we obtain
\begin{equation}
\begin{aligned}
T(k) f_+(x,k) & \sim R_-(k) e^{-ikx} + e^{ikx} \quad \text{as } x\to -\infty
\\
T(k) f_-(x,k) & \sim R_+(k) e^{ikx} + e^{-ikx} \quad \text{as } x\to \infty.
\end{aligned}
\end{equation}
The coefficients $T(k)$ and $R_\pm(k)$ can be written explicitly through
\begin{equation}
\label{T-R-coefficients-explicit}
\begin{aligned}
\frac{1}{T(k)} & = 1 - \frac{1}{2ik} \int V(x) m_\pm(x,k) dx,
\\
\frac{R_\pm(k)}{T(k)} & = \frac{1}{2ik} \int e^{\mp 2ikx} V(x) m_\mp(x,k) dx,
\end{aligned}
\end{equation}
and satisfy the following relations at every $k$:
\begin{equation*}
\begin{aligned}
& T(-k) = \overline{T(k)}, \quad R_\pm (-k) = \overline{R_\pm(k)},
\\
& |T(k)|^2 + |R_\pm(k)|^2 = 1, \quad T(k) \overline{R_-(k)} + \overline{T(k)} R_+(k) = 0.
\end{aligned}
\end{equation*}
One distinguishes
two possible asymptotics of $T(k)$ for $k \approx 0$,
depending on the value of the integral $\int_{-\infty}^\infty V(x) m_+(x,0) dx$,
which is zero whenever the potential $V$ is \textit{generic}.
Here are some equivalent definitions of generic potentials.

\begin{definition}\label{defGen}
The potential $V$ is generic if one of the following, equivalent, conditions hold:
\begin{itemize}
\item $\int_{-\infty}^\infty V(x) m_+(x,0) dx = 0$;
\item $T(0) = 0$ and $R_\pm (0) = -1$;
\item $H = -\partial_x^2 + V$ has no zero resonance.
\end{itemize}
\end{definition}

For generic potentials, $T(k)$ and $R_\pm (k)$ have the following asymptotics around $k \approx 0$.

\begin{lemma}
\label{lemma-T-R-properties}
For $V$ satisfying Assumption \ref{V-assumption},
the coefficients $T(k)$ and $R_\pm(k)$ have the following properties:
\begin{itemize}
\item as $k\to 0$,
\begin{equation}
\label{T-behaviour-around-zero}
\begin{aligned}
T(k) & = \alpha k + \calO (k^2) \quad \text{for some } \alpha \neq 0,
\\
1 + R_\pm(k) & = \alpha_\pm k + o(k).
\end{aligned}
\end{equation}
In particular, $T(0) = 0$ and $R_\pm (0) = -1$.
\item For every $k\in \mathbb{R}$ we have
\begin{equation}
\label{uniform-estimate-T-R}
\begin{aligned}
|T(k)| + |R(k)| &\lesssim 1,\\
|\partial_k T(k)| + |\partial_k R_\pm(k)|& \lesssim \lr{k}^{-1}.
\end{aligned}
\end{equation}
\item For every $k \in \mathbb{R}$ we have
\begin{equation}
\label{bound-on-T-over-k}
\left| \frac{T(k)}{k} \right| + \lr{k} \left| \partial_k \left(\frac{T(k)}{k}\right) \right| \lesssim \frac{1}{\lr{k}},
\end{equation}
and
\begin{equation}
\label{bound-on-R-over-k}
\left| \frac{1+R_\pm(k)}{k} \right| + \left| \partial_k \left(\frac{1+R_\pm(k)}{k}\right) \right| \lesssim \frac{1}{\lr{k}}.
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
The detailed proof of the estimates \eqref{T-behaviour-around-zero}--\eqref{uniform-estimate-T-R}
can be found in Deift-Trubowitz \cite{DT79}, so we only prove the
estimates \eqref{bound-on-T-over-k}--\eqref{bound-on-R-over-k}.
It is worth pointing out that Assumption \ref{V-assumption} on $V$
is stronger than needed for this lemma to be true, and it suffices to have $\lr{x}^2 V \in L_x^1$.

The asymptotic of $T(k)$ around $k \approx 0$ imply that there exists a positive constant $k^*$ such that
\begin{equation*}
\left| \frac{T(k)}{k} \right| \lesssim \frac{1}{\lr{k}} \quad \text{for every   } |k| \leq k^*.
\end{equation*}
On the other hand, for $|k|>k^*$, we use uniform boundedness of $|T(k)|$ to get
\begin{equation*}
\left| \frac{T(k)}{k} \right| \lesssim \frac{|T(k)|}{\lr{k}}
  \lesssim \frac{1}{\lr{k}}.
\end{equation*}
Combining these estimates together we obtain the first part of \eqref{bound-on-T-over-k}. To get the bound on the derivative part, we note that
\begin{equation*}
\ensuremath{\partial}_k \left( \frac{T(k)}{k} \right) = \left( \frac{T(k)}{k}\right)^2 \left(\frac{1}{2i} \int V(x) \partial_k m_\pm (x,k) dx - 1 \right),
\end{equation*}
which leads to a desired estimate in the view of \eqref{estimates-partial_k^s_m} and assumptions on $V$.
\end{proof}

\subsection{Pseudo-differential bounds}
We provide some $L^2 \to L^2$ bounds for pseudodifferential
operators whose symbols include $m_\pm(x,k)$ and their derivatives.
Under milder assumptions than ours, which in particular do not require any regularity for $V$,
these types of bounds are given in \cite{CP21}.

\begin{lemma}
\label{lemma-pseudo-bound-1}
For every function $a\in L_k^\infty$ we have
\begin{equation}
\label{pseudo-bound-1}
\normm{\int \mathbf{1}_{[-1,\infty)} e^{ikx} a(k) (m_+(x,k) - 1) h(x) dx}{k} \lesssim \normm{\lr{x}^{-11} h}{x},
\end{equation}
and
\begin{equation}
\label{pseudo-bound-2}
\normm{\int \mathbf{1}_{[-1,\infty)} e^{ikx} a(k) \partial_k m_+(x,k) h(x) dx}{k} \lesssim \normm{\lr{x}^{-10} h}{x}.
\end{equation}
Similar estimates hold for $\mathbf{1}_{(-\infty, 1]}(m_-(x,k) - 1)$
and $\mathbf{1}_{(-\infty, 1]}\partial_k m_-(x,k)$, respectively.
\end{lemma}

\begin{proof}
Using the estimates \eqref{estimates-partial_k^s_m},
we have that the left-hand side of \eqref{pseudo-bound-1} is bounded by
\begin{equation*}
\begin{aligned}
\normm{\int \lr{x}^{-12} \lr{k}^{-1} |h(x)| dx}{k}
  \lesssim \normm{\lr{x}^{-11} h}{x} \normm{\lr{k}^{-1}}{k} \lesssim \normm{\lr{x}^{-11} h}{x},
\end{aligned}
\end{equation*}
where we used the Cauchy-Schwartz inequality for $dx$ integral.
The estimate \eqref{pseudo-bound-2} follows in a similar fashion using \eqref{estimates-partial_k_and_x_m}.
\end{proof}

\subsection{Distorted Fourier transform}
In this subsection, we present the generalization of the standard Fourier transform
which we apply later to the perturbed Schr\"{o}dinger operator $H = -\partial_x^2 +V$.
Throughout the section and further in the paper,
we use the notation
\begin{align*}
\widetilde D := \sqrt{-\partial_x^2+V}, \qquad D := \sqrt{-\partial_x^2} = |\partial_x|.
\end{align*}
The standard Fourier transform on the line is defined for every $\phi \in L^2$ as
$$
\widehat{\mathcal{F}} \phi(k) = \widehat{\phi}(k) := \frac{1}{\sqrt{2\pi}} \int e^{-ikx} \phi(k)\, dx,
$$
with
$$
\widehat{\mathcal{F}}^{-1} \phi(x) := \frac{1}{\sqrt{2\pi}} \int e^{ikx} \phi(k)\, dx.
$$
When dealing with the operator $H$, the function $e^{-ikx}$
inside the standard Fourier transform can be replaced by a generalized
eigenfunction $\psi(x, k)$ given in terms of the
Jost functions $f_\pm$ from \eqref{ode-Jost} in the following way:
\begin{equation}
\label{wave-function}
\psi(x,k) := \frac{1}{\sqrt{2\pi}} \left\{\begin{array}{l}
\begin{aligned}
&T(k) f_+(x,k) \quad & \text{for } k \geq 0, \\
&T(-k) f_-(x, -k) \quad & \text{for } k < 0
\end{aligned}
\end{array} \right.
\end{equation}
and we now introduce the \textit{distorted} Fourier transform for $h \in \mathcal{S}$ by
\begin{equation}
\label{distorted-fourier}
\widetilde{\mathcal{F}} h(k) = \widetilde{h}(k) := \int \overline{\psi(x,k)} h(x) \, dx.
\end{equation}

\begin{lemma}
\label{lemma-distorted-fourier}
In above settings, $\widetilde{\mathcal{F}}: L^2_x \to L^2_k$ is an isometry,
\begin{equation}
\label{isometry}
\|\widetilde{\mathcal{F}} h \|_{L^2_k} = \|P_c h \|_{L^2_x},\quad \text{for all  } h \in L^2.
\end{equation}
The inverse of the distorted Fourier transform is given by
$$
\widetilde{\mathcal{F}}^{-1} g (x) = \int \psi(x,k) g (k) dk.
$$
Setting $\widetilde D := \sqrt{-\ensuremath{\partial}_{xx} +V}$, we have
$$
m(\widetilde D)P_c = \widetilde{\mathcal{F}}^{-1} m(k) \widetilde{\mathcal{F}}.
$$
Moreover, the distorted Fourier transform satisfies the following properties:
\begin{itemize}
\item $\widetilde{\mathcal{F}} \widetilde{\mathcal{F}}^{-1} = \text{\rm Id}_{L_k^2}$ and $\widetilde{\mathcal{F}}^{-1} \widetilde{\mathcal{F}} = P_c$;
\item if $h \in L^1_x$, then $\widetilde{h}$ is a continuous, bounded function;
\item since $V$ is generic, $\widetilde{h}(0) = 0$;
\item we have
\begin{equation}
\label{derivative-in-fourier-bound}
\|\ensuremath{\partial}_k \widetilde{h}\|_{L^2_k} \lesssim \|\lr{x} h\|_{L^2_x},
\end{equation}
and, for $s \geq 0$
\begin{equation}
\label{k-s-h-tilde-bound}
{\big\| \lr{k}^s \widetilde{h} \big\|}_{L^2_k} = \n{P_c h}_{H_x^s}
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
The proofs can be found in Agmon \cite{A75}, Dunford-Schwartz \cite{DS88}, Germain-Pusateri-Rousset \cite{GPR18} and Yafaev \cite{Y10}.
\end{proof}

\subsection{Decomposition of the generalized eigenfunctions}
By the definition of the generalized eigenfunction we have
\begin{equation}
\label{gen-eigenfunction-1}
\begin{aligned}
&\text{for  } k\geq 0, \qquad \sqrt{2\pi} \psi(x, k) = T(k) f_+(x, k) = T(k) m_+(x, k) e^{ikx} \\[2pt]
&\text{for  } k<0, \qquad \sqrt{2\pi} \psi(x, k) = T(-k) f_-(x, -k) = T(-k) m_-(x, -k) e^{ikx}
\end{aligned}
\end{equation}
We introduce smooth versions of the indicator functions, $\sigma_+$ and $\sigma_-$,
\begin{equation}
\label{sigma-functions}
\sigma_+(x) = \int_{-\infty}^x \varphi(y) dy, \quad \text{   and   } \quad \sigma_+(x) + \sigma_-(x) = 1,
\end{equation}
where $\varphi$ is a smooth, nonnegative function, which is equal to $1$ in a neighborhood of $0$, vanishes outside of $[-2,2]$ and $\int \varphi ~dx = 1$. By construction, $\sigma_+(x)$ vanishes for large negative values of $x$ while $\sigma_-(x)$ vanishes for large positive values of $x$. These functions allow to decompose expressions in \eqref{gen-eigenfunction-1} as
\begin{equation*}
\begin{aligned}
&\text{for  } k\geq0, \quad \sqrt{2\pi} \psi(x, k) = \sigma_+(x) T(k) f_+(x, k) + \sigma_-(x) \left[ f_-(x, -k) +R_-(k) f_-(x,k) \right], \\[2pt]
&\text{for  } k<0, \quad \sqrt{2\pi} \psi(x, k) = \sigma_-(x) T(-k) f_-(x, -k) + \sigma_+(x) \left[ f_+(x, k) +R_+(-k) f_+(x,-k) \right],
\end{aligned}
\end{equation*}
where we used the relations \eqref{jost-functions-via-scat-par}.
We further rewrite the generalized eigenfunction above using $m_\pm (x,k) = f_\pm (x,k) e^{\mp ikx}$,
\begin{equation}
\label{gen-eigenfunction-2}
\begin{aligned}
&\text{for  } k\geq0, \qquad
\begin{aligned}
\sqrt{2\pi} \psi(x, k) = &~ \sigma_+(x) T(k) m_+(x, k) e^{ikx} \\
& + \sigma_-(x) \left[ m_-(x, -k) e^{ikx} + R_-(k) m_-(x,k) e^{-ikx} \right]
\end{aligned}\\
&\text{for  } k<0, \qquad
\begin{aligned}
\sqrt{2\pi} \psi(x, k) =&~ \sigma_-(x) T(-k) m_-(x, -k) e^{ikx}\\
& + \sigma_+(x) \left[ m_+(x, k)e^{ikx} +R_+(-k) m_+(x,-k) e^{-ikx} \right],
\end{aligned}
\end{aligned}
\end{equation}

\begin{remark}
Based on the boundedness of $\sigma_{\pm} m_{\pm}$ in \eqref{estimates-partial_k^s_m}, $T(k)$ and $R_\pm(k)$ in \eqref{uniform-estimate-T-R},
\begin{equation}
\label{psi-bound}
|\psi(x, k)| \lesssim 1.
\end{equation}
\end{remark}

In the view of \eqref{estimates-partial_k^s_m}, we even further decompose $\psi$ into a singular and regular parts as follows
\begin{equation}
\label{generalized-eigenf-decomp}
\sqrt{2\pi} \psi(x,k) = \psi_S(x, k) + \psi_R(x,k).
\end{equation}
The singular part $\psi_S$ is given by
\begin{equation}
\label{psi-singular-part-decomposition}
\psi_S(x,k) := \sigma_+(x) \psi_{S, +}(x,k) + \sigma_-(x) \psi_{S, -}(x,k),
\end{equation}
where, setting the indicator functions $\mathbf{1}_+ := \mathbf{1}_{[0, \infty)}$ and $\mathbf{1}_- := \mathbf{1}_{(-\infty, 0)}$,
\begin{equation}
\label{singular-part-pm}
\left\{ \begin{array}{l}
\psi_{S,+}(x,k) := \mathbf{1}_+(k) T(k) e^{ikx} + \mathbf{1}_-(k) \left[ e^{ikx} + R_+(-k) e^{-ikx} \right],\\[4pt]
\psi_{S,-}(x,k) := \mathbf{1}_+(k) \left[ e^{ikx} + R_-(k) e^{-ikx} \right] + \mathbf{1}_-(k) T(-k) e^{ikx}.
\end{array} \right.
\end{equation}
The rest is a regular part $\psi_R(x,k)$ given by
\begin{equation}
\label{regular-part}
\psi_R(x,k) := \left\{ \begin{array}{l}
\begin{aligned}
& \sigma_+(x) T(k) (m_+(x, k)-1) e^{ikx}\\[-2pt]
&+ \sigma_-(x) [ (m_-(x, -k)-1) e^{ikx} + R_-(k) (m_-(x,k)-1) e^{-ikx} ], \qquad k\geq0
\end{aligned}\\
\phantom{t}\\
\begin{aligned}
& \sigma_-(x) T(-k) (m_-(x, -k)-1) e^{ikx} \\[-2pt]
&+ \sigma_+(x) [ (m_+(x, k)-1) e^{ikx} +R_+(-k) (m_+(x,-k)-1) e^{-ikx} ], \qquad k<0.
\end{aligned}
\end{array}
\right.
\end{equation}
Note that the regular part is localized around $x=0$ due to the estimates \eqref{estimates-partial_k^s_m}.
On the other hand, the singular part $\psi_S$
is a combination of oscillatory terms given by exponential functions with bounded coefficients,
see \eqref{singular-part-pm}.
From \eqref{singular-part-pm}, we can further rewrite the elements of the singular part as
\begin{equation}
\label{singular-part-pm-2}
\left\{ \begin{array}{l}
\psi_{S,+}(x,k) := a_{+,+}(k) e^{ikx} + a_{+,-}(k) e^{-ikx},
\\[4pt]
\psi_{S,-}(x,k) := a_{-,+}(k) e^{ikx} + a_{-,-}(k) e^{-ikx},
\end{array} \right.
\end{equation}
where
\begin{equation}
\label{a-functions}
\begin{aligned}
& a_{+,+}(k) = \mathbf{1}_+(k) T(k) + \mathbf{1}_-(k), \qquad a_{+,-}(k) = \mathbf{1}_-(k) R_+(-k),
\\
& a_{-,+}(k) = \mathbf{1}_+(k) + \mathbf{1}_-(k) T(-k), \qquad a_{-,-}(k) = \mathbf{1}_+(k) R_-(k).
\end{aligned}
\end{equation}
As it would be useful later, we also point out that, given $h \in P_c L^2_x$,
the decomposition \eqref{generalized-eigenf-decomp} allows to rewrite
\begin{equation}
\label{h-sing-reg-decomp}
h = h_S + h_R, \quad \text{where  } h_* := \frac{1}{\sqrt{2\pi}} \int \psi_*(x, k) \widetilde{h}(k) dk \quad \text{  for  } * \in \{S, R\}.
\end{equation}

\begin{remark}
We point out that the outcomes of Lemma \ref{lemma-pseudo-bound-1}
also apply to $\psi_R$. In other words, based on the construction of $\psi_R$ in \eqref{regular-part}, we can verify that
\begin{equation}
\label{pseudo-psi-regular-bound}
\normm{\int \overline{\psi_R(x,k)} h(x) dx}{k} \lesssim \normm{\lr{x}^{-11} h}{x},
\end{equation}
and
\begin{equation}
\label{pseudo-dk-psi-regular-bound}
\normm{\int \partial_k \overline{\psi_R(x,k)} h(x) dx}{k} \lesssim \normm{\lr{x}^{-10} h}{x}.
\end{equation}
Moreover, the function $\psi_R$ decays as
\begin{equation}
\label{psi-regular-decay}
|\psi_R(x,k)| \lesssim \lr{x}^{-12} \lr{k}^{-1}.
\end{equation}
\end{remark}

\iffalse
\subsubsection{Some estimates for low frequencies}
Here, we consider the function
\begin{equation}
\label{mu-1-expression}
\mu_{\alpha, \beta, \gamma}(k,l,m)
:= \int V(x) \frac{\overline{\psi(x,k)}}{k^\alpha}  \frac{\psi(x,l)}{l^\beta} \frac{\psi(x,m)}{m^\gamma} dx
\end{equation}
with the potential $V(x)$ satisfying Assumption \ref{V-assumption}.
\fp{We will see later in Section \ref{section-quadratic},
that the functions of type $\mu_{\alpha, \beta, \gamma}$ will appear (integrated over $k, l$ and $m$).
We now show that
$\mu_{\alpha, \beta, \gamma}$ is uniformly bounded.
}

\begin{lemma}
\label{lemma-mu-1}
For every $\nu, \theta, \alpha, \beta, \gamma \in \{0, 1\}$, we have a uniform bound
\begin{equation}
\label{mu-1-bounds}
|\partial_l^\nu \partial_m^\theta \mu_{\alpha, \beta, \gamma}(k,l,m)| \lesssim \frac{1}{\lr{k}^\alpha \lr{l}^{\beta} \lr{m}^{\gamma}}
\end{equation}
for every $k,l,m \in \mathbb{R}$.
\end{lemma}

\begin{proof}
The proof of \eqref{mu-1-bounds} is based on the following claim: for every $\nu, \beta \in \{0,1\}$
\begin{equation}
\label{mu1-bound-claim} \left|\partial_\xi^{\nu} \left( \frac{\psi(x, \xi)}{\xi^\beta} \right)\right| \lesssim \frac{\lr{x}^{\nu+1}}{\lr{\xi}^{\beta}} \quad \text{whenever} \quad \xi \neq 0.
\end{equation}
Then, using \eqref{psi-bound}, Assumption \ref{V-assumption} and assuming \eqref{mu1-bound-claim}, we have
\begin{equation*}
|\partial_l^\nu \partial_m^\theta \mu_{\alpha, \beta, \gamma}(k,l,m)|
\lesssim \int V(x) \frac{\lr{x}^{\nu+ \theta+2}}{\lr{k}^\alpha \lr{l}^{\beta} \lr{m}^{\gamma}} dx
\lesssim \frac{1}{\lr{k}^\alpha \lr{l}^{\beta} \lr{m}^{\gamma}},
\end{equation*}
which implies \eqref{mu-1-bounds}. Therefore, it suffices to prove \eqref{mu1-bound-claim}
uniformly in $x$ and $\xi \neq 0$.

From \eqref{gen-eigenfunction-1}, we can write
$$
\sqrt{2 \pi} \frac{\psi(x, \xi)}{\xi} = \mathbf{1}_+(\xi) \frac{T(\xi)}{\xi} m_+(x,\xi) e^{ix\xi}
+  \mathbf{1}_-(\xi) \frac{T(-\xi)}{\xi} m_-(x,-\xi) e^{ix\xi}
$$
as a piecewise function defined differently on intervals $(-\infty, 0)$ and $(0, \infty)$.
For both intervals, the claim \eqref{mu1-bound-claim} is verified using \eqref{estimates-partial_k^s_m},
\eqref{uniform-estimate-T-R} and \eqref{bound-on-T-over-k}.
\end{proof}
\fi

\subsection{Linear estimates and local decay
In this section we provide linear and local decay estimates for the term
\begin{equation}
\label{g-profile-linear-term}
\lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c f,
\end{equation}
where $\lr{\widetilde D} = \sqrt{H+1} = \widetilde{\mathcal{F}}^{-1} \lr{k} \widetilde{\mathcal{F}}$.
These estimates will be used in later sections to bound
the term $\chi(t,x)$ in \eqref{system-eq}.

\begin{lemma}
\label{lemma-dispersive-estimates}
Under Assumption \ref{V-assumption} on $V$, for every $t\geq 0$ and every $N=0,1,2$ we have
\begin{equation}
\label{dispersive-estimates}
\left\|
\partial_x^N \lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c f \right\|_{L_x^\infty} \lesssim \lr{t}^{-\frac{1}{2}+\frac{N}{5}} \left( \|\partial_k \widetilde f\|_{L^2_k} + \|\lr{k}^2 \widetilde f\|_{L^2_k} \right)
\end{equation}
\end{lemma}

The proof of Lemma \ref{lemma-dispersive-estimates}
is standard and follows from similar, in fact easier, arguments to those in \cite[Proposition 3.10]{GP20}
since here we are not dealing with any bad frequency.
In particular, using
also the bound $\norm{\lr{k} h}{k}{\infty} \lesssim \norm{\partial_k h}{k}{2} + \norm{\lr{k}^2 h}{k}{2}$,
one can deduce from \cite[Proposition 3.10]{GP20} that
\begin{equation}
\label{dispersive-estimates0}
\left\|
\lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c f \right\|_{L_x^\infty}
  \lesssim \lr{t}^{-\frac{1}{2}} \left( \|\partial_k \widetilde f\|_{L^2_k} + \|\lr{k}^2 \widetilde f\|_{L^2_k} \right).
\end{equation}
The estimate \eqref{dispersive-estimates} with $N=1,2$ can be proven with the same argument,
or also using Gagliardo-Nirenberg-Sobolev interpolation and
\eqref{k-s-h-tilde-bound}.

\iffalse
\begin{lemma}
\label{lemma-stationary-phase}
Given $X \in \ensuremath{\mathbb{R}}$, $x \in \ensuremath{\mathbb{R}}$ and $t \geq 0$. Consider the integral
\begin{equation}
\label{integral-stat}
I_{\iota, \nu}(t, X, x) := \int_{\ensuremath{\mathbb{R}}_\iota} e^{it (\nu \lr{k} - k X)} a(x, k) h(k) dk, \qquad \iota, \nu \in \{+, -\},
\end{equation}
and let the following assumption hold
\begin{equation}
\label{a-assumption}
\sup_{x\in \ensuremath{\mathbb{R}}, k \in \ensuremath{\mathbb{R}}_\iota}  (|a(x, k)| + \lr{k} |\partial_k a(x, k)|) \lesssim \lr{k}^{-1+N}.
\end{equation}
Then we have the linear estimate
\begin{equation}
\label{linear-estimate-1}
\begin{aligned}
|I_{\iota, \nu}(t, X, x)| \lesssim &~ \lr{t}^{-\frac{1}{2}+\frac{N}{5}} \left( \|\lr{k} h\|_{L^\infty_k} + \|\partial_k h\|_{L^2_k} + \|\lr{k}^2 h\|_{L^2_k} \right).
\end{aligned}
\end{equation}
\end{lemma}

\begin{lemma}
\label{lemma-sobolev}
For a Schwartz function $h$, we have
\begin{equation}
\label{sobolev}
\norm{\lr{k} h}{k}{\infty} \lesssim \norm{\partial_k h}{k}{2} + \norm{\lr{k}^2 h}{k}{2}.
\end{equation}
\end{lemma}

The above lemmas are not new, and similar versions of it appeared previously in the literature.
In particular, Lemma \ref{lemma-stationary-phase} is a stationary phase lemma based on Proposition 3.10 in \cite{GP20},
where only the case with $N=0$ was considered and the effect of "bad frequencies" was taken in the account.
In our case, we avoid dealing with such "bad frequencies" \textbf{because what??}
\fi
Note that in \cite{GP20} the decay rates (for $N=0$) associated to the $L^2$ norms
on the right-hand side of \eqref{dispersive-estimates} are faster,
and needed to compensate the small growth of these norms
over the time.
In our case, we do not have such a growth, so \eqref{dispersive-estimates}
suffices for our purposes.

\iffalse
First, we show how to use the lemmas above to prove Lemma \ref{lemma-dispersive-estimates}.

\begin{proof}[Proof of Lemma \ref{lemma-dispersive-estimates}]
From the definition of the distorted Fourier transform, we have
\cak{correct this!!!!}
\cfp{look at this}
\begin{equation*}
\begin{aligned}
\partial_x^N \lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c f & = \int_{\mathbb{R}} \partial_x^N \psi(x,k) \lr{k}^{-1} e^{\pm it\lr{k}} \widetilde{f}(k) dk \\
& = \int_{\mathbb{R}_+} \partial_x^N \psi(x,k) \lr{k}^{-1} e^{\pm it\lr{k}} \widetilde{f}(k) dk + \int_{\mathbb{R}_-} \partial_x^N \psi(x,k) \lr{k}^{-1} e^{\pm it\lr{k}} \widetilde{f}(k) dk\\
& =: I_+ + I_-.
\end{aligned}
\end{equation*}
Both terms $I_\pm$ are treated similarly, so we only consider $I_+$. Using \eqref{gen-eigenfunction-2} and setting $X := -x/t$, we rewrite
\begin{equation*}
\begin{aligned}
\sqrt{2 \pi} I_+  = &~ \int_{\mathbb{R}_+} e^{it( \pm \lr{k} - kX)} \lr{k}^{-1} T(k) \partial_X^N [ \sigma_+(x) m_+(x,k) ] \widetilde{f}(k) dk\\[2pt]
& + \int_{\mathbb{R}_+} e^{it( \pm \lr{k} - kX)} \lr{k}^{-1} \partial_X^N [ \sigma_-(x) m_-(x,-k)]  \widetilde{f}(k) dk\\[2pt]
& + \int_{\mathbb{R}_+} e^{it( \pm \lr{k} + kX)} \lr{k}^{-1} R_-(k) \partial_X^N [ \sigma_-(x) m_-(x,k) ] \widetilde{f}(k) dk.
\end{aligned}
\end{equation*}
Each of the above integrals is of form \eqref{integral-stat}, and the estimate \eqref{dispersive-estimates} is obtained by using Lemma \ref{lemma-stationary-phase} and Lemma \ref{lemma-sobolev}.
\end{proof}

Below we provide with the proof of Lemma \ref{lemma-stationary-phase} and Lemma \ref{lemma-sobolev}.

\begin{proof}[Proof of Lemma \ref{lemma-dispersive-estimates}]
The proof follows the steps provided in \cite{GP20}.
We consider only the case when $\iota = \nu = +$ and $X \geq 0$, with all the other cases being treated similarly. We split the analysis into two cases, when $t \lesssim 1$ and $t \gtrsim 1$.
When $t\lesssim 1$, one can directly obtain
$$
|I_{++}| \lesssim \|\lr{k}^{-1+N} h\|_{L^2_k} \lesssim \lr{t}^{-1/2} \|\lr{k}^2 h\|_{L^2_k}.
$$
For the case $t \gtrsim 1$, we split the region of integration dyadically, using the Littlewood-Paley decomposition, as follows:
$$
I_{++} = \sum_{\substack{m \in \ensuremath{\mathbb{Z}}
}} I_m, \quad \text{where  }
I_m(t, X, x) := \int_{\mathbb{R}_+} e^{it (\lr{k} - k X)} a(x, k) h(k) \varphi_m(k) dk.
$$
Note that
\begin{equation*}
|I_m (t, X, x)| \lesssim \int_{\mathbb{R}_+} \lr{k}^{-1+N} |h(k)| \varphi_m(k) dk \lesssim \min \left( 2^m \norm{\lr{k} h}{k}{\infty}, 2^{m\left( \frac{2N-5}{2} \right)} \norm{\lr{k}^2 h}{k}{2} \right).
\end{equation*}
Then, the total contribution by frequencies satisfying $2^m \lesssim t^{-1/2}$ (low frequencies) and $2^m \gtrsim t^{1/5}$ (high frequencies) are bounded as in \eqref{dispersive-estimates}. Hence, from now on we may assume
\begin{equation}
\label{dyadic-interval}
t^{-1/2} \lesssim 2^m \lesssim t^{1/5}.
\end{equation}
We introduce a new function to represent the phase in $I_m$,
\begin{equation}
\label{new-function}
\begin{aligned}
& \phi(k) := \lr{k} - kX, \\
& \phi'(k) = \frac{k}{\lr{k}} - X, \quad \phi''(k) = \frac{1}{\lr{k}^3}.
\end{aligned}
\end{equation}
Note that $\phi'$ has no roots in case $X \geq 1$, while for $X \in [0,1)$ it has a unique, non-degenerate root at some positive $k_0$. Direct computations help to find $k_0 = X/ \sqrt{1-X^2}$. Further we assume that the stationary point $k_0$ exists, otherwise analysis becomes even easier.

For the integral $I_m$, the variable $k$ in the support of $\varphi_m$ satisfies $k \approx 2^m$ and $\lr{k} \approx 2^{m^+}$, where $m^+ := {\rm max}(m, 0)$. Let $n\in \mathbb{Z}_+$ be such an integer that $t \in [2^{n-1}, 2^{n}]$ and let $q_0 \in \mathbb{Z}$ be the smallest integer that satisfies $2^{q_0} \geq 2^{(3/2)m^+} 2^{-n/2}$.
Note that $2^{q_0} \approx 2^{(3/2)m^+} t^{-1/2} \approx \lr{k}^{3/2} t^{-1/2}$.

We may assume that the stationary point $k_0$ belongs to the support of $\varphi_m$, and so $k_0 \approx 2^m$, otherwise the proof is easier. For a fixed $k_0$, there are only finitely many (up to three) indices $m$ for which $k_0 \in {\rm supp}(\varphi_m)$. We can call such set of indices to be $M$, and $|M|\leq 3$.
Under these assumptions, we further decompose the integral $I_m$ for $m \in M$ as follows
\begin{equation*}
I_m = \sum_{q \in [q_0, \infty) \cap \ensuremath{\mathbb{Z}}} I_{m, q}, \quad \text{where  }
I_{m, q}(t, X, x) :=  \int_{\mathbb{R}_+} e^{it \phi(k)} a(x, k) h(k) \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk.
\end{equation*}

The contribution of $I_{m, q_0}$ to the sum over $m\in M$ is immediate.
For $q > q_0$ and $k$ in the support of $\varphi_m$, we have the relation
\begin{equation}
\label{tangent-identity}
|\phi'_X(k)| \approx |k-k_0|  \lr{k}^{-3}.
\end{equation}
Noting that $e^{it \phi} = \frac{\partial_k e^{i t\phi}}{it \phi'}$, we apply the integration by parts to $I_{m, q}$ and obtain
$$
\begin{aligned}
I_{m, q} = & ~  \frac{1}{it} \int_{\ensuremath{\mathbb{R}}^+} e^{it \phi(k)} \frac{\phi''}{(\phi')^2} a(x, k) h(k) \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk\\[3pt]
& - \frac{1}{it} \int_{\ensuremath{\mathbb{R}}^+} e^{it \phi(k)} \frac{1}{\phi'} \partial_k a(x, k) h(k) \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk\\[3pt]
& - \frac{1}{it} \int_{\ensuremath{\mathbb{R}}^+} e^{it \phi(k)} \frac{1}{\phi'} a(x, k) \partial_k h(k) \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk\\[3pt]
& - \frac{1}{it} \int_{\ensuremath{\mathbb{R}}^+} e^{it \phi(k)} \frac{1}{\phi'} a(x, k) h(k) \partial_k [\varphi_m(k) \varphi_q^{(q_0)}(k-k_0)] dk\\[3pt]
=: &~ \frac{1}{it} \left( J^{(1)}_{m, q} + J^{(2)}_{m, q} + J^{(3)}_{m, q} + J^{(4)}_{m, q} \right).
\end{aligned}
$$
To successfully bound the integrals above, we need the following estimate
\begin{equation}
\label{linear-estimate-lemma-change-of-variables-integral}
\begin{aligned}
\sum_{q>q_0, m\in M} \left( \int_{\mathbb{R}^+} \frac{\lr{k}^{3/2}}{|k - k_0|^2} \varphi_{q}^{(q_0)} ( k-k_0 ) \varphi_m(k) dk \right)^s & \approx
\sum_{p\geq -n/2} \left( \int_{\mathbb{R}^+} \frac{1}{y^2} \varphi_{\sim p} (y) dy \right)^s \approx \lr{t}^{s/2},
\end{aligned}
\end{equation}
where we applied a change of variables $y = (k - k_0)2^{-(3/2)m^+}$ and change of the index of summation $q \mapsto p + (3/2)m^+$.
Then, using (\ref{linear-estimate-lemma-change-of-variables-integral}), the total contribution of the term $J^{(1)}_{m, q}$ is
$$
\begin{aligned}
\sum_{q>q_0, m\in M} |J^{(1)}_{m, q}| & \lesssim \sum_{q>q_0, m\in M} \int_{\ensuremath{\mathbb{R}}^+}  \frac{\lr{k}^{2+N}}{|k-k_0|^2} |h(k)| \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk\\[3pt]
& \lesssim \|\lr{k} h\|_{L^\infty_k} \sum_{q>q_0, m\in M} \int_{\ensuremath{\mathbb{R}}^+}  \frac{t^{N/5}\lr{k}^{3/2}}{|k-k_0|^2} \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk \lesssim \lr{t}^{\frac{1}{2}+\frac{N}{5}} \|\lr{k} h\|_{L_k^\infty},
\end{aligned}
$$
where we also used that $\lr{k} \lesssim t^{1/5}$.
The total contribution of the term $J^{(2)}_{m, q}$ is found using the Cauchy-Schwartz inequality as
$$
\begin{aligned}
\sum_{q>q_0, m\in M} |J^{(2)}_{m, q}| & \lesssim \sum_{q>q_0, m\in M} \int_{\ensuremath{\mathbb{R}}^+}  \frac{\lr{k}^{1+N}}{|k-k_0|} |h(k)| \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk\\[3pt]
& \lesssim \lr{t}^{\frac{1}{4}+\frac{N}{5}} \|\lr{k}^2 h\|_{L_k^2}.
\end{aligned}
$$
The total contribution of the term $J^{(3)}_{m, q}$ is also through the Cauchy-Schwartz inequality,
$$
\begin{aligned}
\sum_{q>q_0, m} |J^{(3)}_{m, q}| & \lesssim \sum_{q>q_0, m \in M} \int_{\ensuremath{\mathbb{R}}^+}  \frac{\lr{k}^{2+N}}{|k-k_0|} |\partial_k h| \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk\\[3pt]
& \lesssim t^{\frac{N}{5}}\| \partial_k h\|_{L^2_k} \sum_{q>q_0, m\in M}
\left( \int_{\ensuremath{\mathbb{R}}^+} \frac{t^{\frac{1}{2}} \lr{k}^{3/2}}{|k-k_0|^2} \varphi_m(k) \varphi_q^{(q_0)}(k-k_0) dk \right)^{1/2}  \lesssim \lr{t}^{\frac{1}{2}+\frac{N}{5}} \| \partial_k h\|_{L^2_k}.
\end{aligned}
$$
The same approach is used to compute the contribution of the term $J^{(4)}_{m, q}$, as its integrand is supported according to $\partial_k [\varphi_m(k) \varphi_q^{(q_0)}(k-k_0)] = \varphi_{\sim m}(k) \varphi_{\sim q}(k-k_0)$, and so
$$
\begin{aligned}
\sum_{q>q_0, m\in M} |J^{(4)}_{m, q}| & \lesssim \lr{t}^{\frac{1}{2}+\frac{N}{5}} \|\lr{k}^2 h\|_{L^2_k}.
\end{aligned}
$$
The upper bounds of the terms above, after multiplication by $t^{-1}$, are bounded by the right hand side of (\ref{linear-estimate-1}), which concludes the proof.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lemma-sobolev}]

The proof simply relies on the fact that
\begin{equation}
|h(k)|^2 = 2 \Re \int_{-\infty}^k h(\xi) \partial_\xi h(\xi) d\xi = - 2 \Re \int_k^{\infty} h(\xi) \partial_\xi h(\xi) d\xi.
\end{equation}
Depending on sign of $k$, we use one or another formula, and the estimates are very similar in both cases. In what follows, we only consider the case $k>0$. Then, we have
\begin{equation*}
\begin{aligned}
\lr{k}^2 |h(k)|^2 & \lesssim \lr{k}^2 \int_k^\infty |h(\xi)| |\partial_\xi h(\xi)| d\xi \lesssim  \int_k^\infty \lr{\xi}^2 |h(\xi)| |\partial_\xi h(\xi)| d\xi  \lesssim \norm{\partial_\xi h}{\xi}{2}\norm{\lr{\xi}^2 h}{\xi}{2},
\end{aligned}
\end{equation*}
which implies \eqref{sobolev}.
\end{proof}

\fi

\begin{remark}
Lemma \ref{lemma-dispersive-estimates} can be repeated for $V=0$.
In such case, $\widetilde D$ is replaced by $D$ and the distorted Fourier
transform becomes the standard Fourier transform. The corresponding
estimate is
\begin{equation}
\label{dispersive-estimates-standard-fourier}
\left\|
\lr{D}^{-1} e^{\pm it \lr{D}} f \right\|_{L_x^\infty} \lesssim \lr{t}^{-1/2} \left( \|\partial_k \widehat f\|_{L^2_k} + \|\lr{k}^2 \widehat f\|_{L^2_k} \right).
\end{equation}
\end{remark}

\smallskip
Next, we state the local
decay estimates which will be used in later sections.
We start with the estimates for the standard Fourier transform under additional assumption at $k=0$.

\begin{lemma}
Suppose $\hat h \in H_k^1$ and $\hat h (0) = 0$, then for every $t \geq 0$ we have
\begin{equation}
\label{local-l-infty-decay-standard-Fourier}
\left\| \lr{x}^{-1} \lr{D}^{-1} e^{\pm it \lr{D}} h \right\|_{L^\infty_x}
  \lesssim \lr{t}^{-3/4} \| \widehat{h} \|_{H^1_k}.
\end{equation}
\end{lemma}

The proof of \eqref{local-l-infty-decay-standard-Fourier} is standard,
and can be found in \cite[Lemma 3.6]{CP21} in the completely analogous case of the Schr\"odinger equation.

\iffalse
\begin{proof}
Using the definition of the standard Fourier transform,
\begin{equation*}
\lr{D}^{-1} e^{\pm it \lr{D}} h(x) = \frac{1}{\sqrt{2\pi}} \int \lr{k}^{-1} e^{\pm it \lr{k}} \hat{h} (k) e^{ikx} ~dk.
\end{equation*}
The estimates for $t\leq 1$ are straightforward, so we may assume $t>1$. Then, we split the region of integration into two parts: $|k|\leq t^{-1/2}$ and $|k|> t^{-1/2}$ - and consider each of them separately. In other words, we set
\begin{equation*}
\begin{aligned}
\lr{D}^{-1} e^{\pm it \lr{D}} h(x) & = \frac{1}{\sqrt{2\pi}} \int_{|k|\leq t^{-1/2}} \lr{k}^{-1} e^{\pm it \lr{k}} \hat{h} (k) e^{ikx} ~dk\\[3pt]
& \quad + \frac{1}{\sqrt{2\pi}} \int_{|k|> t^{-1/2}} \lr{k}^{-1} e^{\pm it \lr{k}} \hat{h} (k) e^{ikx} ~dk\\[3pt]
& =: I(x) + II(x).
\end{aligned}
\end{equation*}
Using the fundamental theorem of Calculus, we have
\begin{equation}
\label{ftc-outcome}
|\hat h(k)| = \left| \int_0^k \partial_{k'} \hat h(k')~ dk \right| \lesssim |k|^{1/2} \normm{\partial_k \hat h}{k},
\end{equation}
where we used that $\hat h(0) = 0$.
Then, we can bound $I$ as
\begin{equation*}
|I(x)| \lesssim \int_{|k|\leq t^{-1/2}} |k|^{1/2} \normm{\partial_k \hat h}{k} dk \lesssim \lr{t}^{-3/4} \| \widehat{h} \|_{H^1_k}.
\end{equation*}
For the term $II$ we only consider the case $k>t^{-1/2}$, as the case $k<-t^{-1/2}$ is treated similarly. Then, using the relation $\lr{k}^{-1} e^{it\lr{k}} = (itk)^{-1} \partial_k e^{it\lr{k}}$, we integrate by parts to obtain
\begin{equation*}
\begin{aligned}
\sqrt{2\pi} II(x) & = \frac{1}{it}  \int_{t^{-1/2}}^\infty \partial_k e^{it\lr{k}} \left(\frac{1}{k} e^{ikx} \hat{h} (k) \right) ~dk\\[3pt]
& = -\frac{1}{it^{1/2}} e^{it\lr{t^{-1/2}}} e^{ixt^{-1/2}} \hat{h}(t^{-1/2}) + \frac{1}{it}  \int_{t^{-1/2}}^\infty e^{it\lr{k}} \frac{1}{k^2} e^{ikx} \hat{h} (k)  ~dk\\[3pt]
& \quad - \frac{1}{it}  \int_{t^{-1/2}}^\infty e^{it\lr{k}} \frac{1}{k} \partial_k ( e^{ikx} \hat{h} (k))  ~dk\\[3pt]
& =: II_1(x) + II_2(x) + II_3(x).
\end{aligned}
\end{equation*}
From \eqref{ftc-outcome}, we directly get the bound $|II_1(x)| \lesssim \lr{t}^{-3/4} \| \widehat{h} \|_{H^1_k}$. For the term $II_2$ we also  use  \eqref{ftc-outcome},
\begin{equation*}
|II_2(x)| \lesssim \frac{1}{t} \int_{t^{-1/2}}^\infty k^{-3/2} \| \widehat{h} \|_{H^1_k} dk \lesssim \lr{t}^{-3/4} \| \widehat{h} \|_{H^1_k}.
\end{equation*}
We point out the the term $II_3$ contains a factor $x$ after $\partial_k$ hits $e^{ikx}$, so to overcome the growth in $x$, we multiply $II_3$ by $\lr{x}^{-1}$. Then, with the help of Cauchy-Schwartz inequality, we have
\begin{equation*}
|\lr{x}^{-1} II_3(x)| \lesssim \frac{1}{t} \| \widehat{h} \|_{H^1_k}  \left( \int_{t^{-1/2}}^\infty \frac{1}{k^2} dk\right)^{1/2} \lesssim \lr{t}^{-3/4} \| \widehat{h} \|_{H^1_k}.
\end{equation*}
Combining the above estimates, we derive \eqref{local-l-infty-decay-standard-Fourier}.
\end{proof}
\fi

Estimates for the distorted Fourier transform have better decays rates under genericity assumption on $V$.

\begin{lemma}
\label{local-l-infty-decay-lemma}
Suppose $V$ satisfies Assumption \ref{V-assumption}. Then, for every $t \geq 0$, we have
\begin{equation}
\label{local-l-infty-decay}
\left\| \lr{x}^{-2} \lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c h \right\|_{L^\infty_x}
  \lesssim \lr{t}^{-1} \| \widetilde{h} \|_{H^1_k}.
\end{equation}
\end{lemma}

The proof of this result is also standard and can be found in \cite{KGVSim}.
An analogous estimate for the Schr\"odinger flow is given in \cite[Lemma 3.6]{CP21};
see also the references in \cite{KGVSim,CP21} for other related and more general estimates.

\iffalse
\begin{proof}
From the definition of the distorted Fourier transform,
\begin{equation}
\label{local-l-infty-decay-rel1}
\lr{\widetilde D}^{-1} e^{it \lr{\widetilde D}} P_c h(x) = \int_\ensuremath{\mathbb{R}} \lr{k}^{-1} \psi(x, k) e^{it \lr{k}} \widetilde h(k) dk.
\end{equation}

For $0\leq t \leq 1$, we have
\begin{equation*}
\begin{aligned}
\left| \lr{x}^{-2} \lr{\widetilde D}^{-1} e^{it \lr{\widetilde D}} P_c h(x) \right| & \lesssim  \int_\ensuremath{\mathbb{R}} \lr{k}^{-1} |\lr{x}^{-2} \psi(x, k)| |\widetilde h(k)| dk \lesssim \|\widetilde h\|_{L_k^2},
\end{aligned}
\end{equation*}
where we used that $|\lr{x}^{-1} \psi(x, k)| \lesssim 1$ uniformly for all $x$ and $k$. This estimate is consistent with a desired bound in \eqref{local-l-infty-decay}.
It remains to consider the case $t \geq 1$. Recall that $\psi(x,k)$ in the right hand side integral in \eqref{local-l-infty-decay-rel1} is a piecewise function defined in \eqref{wave-function}. We consider only $\ensuremath{\mathbb{R}}^+$ region of integration, and the estimates for $\ensuremath{\mathbb{R}}^-$ region are similar. Hence, from \eqref{gen-eigenfunction-1} we have
\begin{equation}
\label{local-decay-proof-term-I}
\sqrt{2\pi}
\int_{\ensuremath{\mathbb{R}}^+} \lr{k}^{-1} \psi(x, k) e^{it \lr{k}} \widetilde h(k)~ dk = \int_{\mathbb{R}^+} \lr{k}^{-1} T(k) m_+(x, k) e^{ixk} e^{it\lr{k}} \widetilde h(k)~dk :=I
\end{equation}
This term is treated
using the integration by parts via $e^{it \lr{k}} = (itk)^{-1} \lr{k} \partial_k e^{it \lr{k}}$, and recalling $\widetilde{h}(0)=0$
\begin{equation*}
\begin{aligned}
it I
& = -
\int_{\mathbb{R}^+} \partial_k \left(k^{-1} T(k) \widetilde{h}(k)\right) e^{ixk} m_+(x, k) e^{it\lr{k}}~dk\\[2pt]
& \quad -i x
\int_{\mathbb{R}^+} k^{-1} T(k) e^{ixk} m_+(x, k) \widetilde h(k) e^{it\lr{k}}~dk\\[2pt]
& \quad -
\int_{\mathbb{R}^+} k^{-1} T(k) e^{ixk} \partial_k m_+(x, k) \tilde h(k) e^{it\lr{k}}~dk\\[2pt]
& = : A_1 + A_2 + A_3.
\end{aligned}
\end{equation*}
Note that the presence of $T(k)$ allows to compensate the factor $k^{-1}$. Indeed, for the first term we have
\begin{equation*}
\lr{x}^{-1}|A_1| \lesssim \int_{\mathbb{R}_+} \left| \partial_k \left(k^{-1} T(k) \widetilde{h}(k)\right) \right| dk \lesssim \n{\widetilde h}_{H_k^1},
\end{equation*}
where we used $\lr{x}^{-1} |m_+(x,k)| \lesssim 1$ from \eqref{estimates-partial_k^s_m}, estimates in \eqref{bound-on-T-over-k} and Cauchy -Schwartz inequality. In similar fashion, we bound two other terms, and obtain
\begin{equation*}
\lr{x}^{-2} \left( |A_2| + |A_3| \right) \lesssim \n{\widetilde h}_{H_k^1}.
\end{equation*}

\iffalse
This term is treated exactly as in Chen-Pusateri, and more details will be added, and we have
\begin{equation}
\label{I-plus-term-local-estimates}
\lr{x}^{-1} |I_+| \lesssim t^{-1} \n{\widetilde h}_{H^1_k}.
\end{equation}

For the term $I_-$, we apply similar estimates with slight modifications to avoid the vanishing denominator at $k=0$.
We use smooth cutoff functions to consider $|k|\gtrsim 1$ and $|k|\lesssim 1$ regions separately, and we write
\begin{equation}
\begin{aligned}
I_- = &~ \sigma_-(x) \int_{\mathbb{R}^+} \frac{1}{\lr{k}} \varphi_{\gtrsim 1}(k) (m_-(x, -k) e^{ixk} + R_-(k) m_-(x, k) e^{-ikx}) e^{it\lr{k}} \widetilde h(k)~dk\\[3pt]
& + \sigma_-(x) \int_{\mathbb{R}^+} \frac{1}{\lr{k}} \varphi_{\lesssim 1}(k) (R_-(k) +1) m_-(x, k) e^{-ikx} e^{it\lr{k}} \widetilde h(k)~dk\\[3pt]
&+ \sigma_-(x) \int_{\mathbb{R}^+} \frac{1}{\lr{k}} \varphi_{\lesssim 1}(k) (m_-(x, -k) - m_-(x, k)) e^{ikx} e^{it\lr{k}} \widetilde h(k)~dk\\[3pt]
&+ \sigma_-(x) \int_{\mathbb{R}^+} \frac{1}{\lr{k}} \varphi_{\lesssim 1}(k) (e^{ixk} - e^{-ikx}) m_-(x, k) e^{it\lr{k}} \widetilde h(k)~dk.
\end{aligned}
\end{equation}
Using $\lr{k}^{-1} e^{it\lr{k}} = (itk)^{-1} \partial_k e^{it\lr{k}}$ we proceed with similar estimates as in Chen-Pusateri, and obtain
$$
\lr{x}^{-2} |I_-| \lesssim t^{-1} \|\widetilde h\|_{H^1_k}.
$$

\fi
\end{proof}

\fi

We can extend the result of the above lemma to cases when we deal with the singular
and regular parts of the function $\lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c h$
according to the decomposition in \eqref{h-sing-reg-decomp}.
First, we have the following analogue of Lemma \ref{local-l-infty-decay-lemma}:

\begin{lemma}
\label{lemma-local-l-infty-decay-singular}
Suppose $V$ satisfies Assumption \ref{V-assumption}. Then, for every $t \geq 0$, we have
\begin{equation}
\label{local-l-infty-decay-singular}
\left\| \lr{x}^{-2} \left( \lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c h \right)_S \right\|_{L^\infty_x}
  \lesssim \lr{t}^{-1} \| \widetilde{h} \|_{H^1_k},
\end{equation}
\end{lemma}

\iffalse
\begin{proof}
As previously, the proof in case $0\leq t \leq 1$ is straightforwards, so we can assume that $t >1$.
From the definition in \eqref{h-sing-reg-decomp}, we have
\begin{equation*}
\sqrt{2\pi} \left( \lr{\widetilde D}^{-1} e^{it \lr{\widetilde D}} P_c h\right) _S (x)
  = \int_\ensuremath{\mathbb{R}} \lr{k}^{-1} \psi_S(x, k) e^{it \lr{k}} \widetilde h(k) dk.
\end{equation*}
Recall from \eqref{psi-singular-part-decomposition} that $\psi_S(x,k)$
is a piecewise continuous function, which is defined differently for $k<0$
and $k \geq 0$. Hence, we only focus on the integral corresponding to $k\geq 0$, as the analysis for $k<0$ part is identical.
Then, we have
\begin{equation*}
\begin{aligned}
&\int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} \psi_S(x, k) e^{it \lr{k}} \widetilde h(k) dk = I + II,\\[3pt]
& I := \sigma_+(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} T(k) e^{ikx} e^{it\lr{k}} \widetilde{h} (k)dk,\\[3pt]
& II := \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} (e^{ikx} + R_-(k) e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk.
\end{aligned}
\end{equation*}
The term $I$ is treated exactly as \eqref{local-decay-proof-term-I}. In fact, it is even easier since we do not have to deal with $m_+$ function, and the computations lead to $\lr{x}^{-1}|I| \lesssim \n{\widetilde h}_{H_k^1}$. The term $II$, we rewrite as
\begin{equation*}
\begin{aligned}
II &= \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} (e^{ikx} - e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk\\[3pt]
& \quad + \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} (1+R_-(k)) e^{-ikx} e^{it\lr{k}} \widetilde{h} (k)dk\\[3pt]
& =: II_1 + II_2.
\end{aligned}
\end{equation*}
Note that $1+R_-(k)$ behaves very similarly to $T(k)$ as it can be seen from \eqref{T-behaviour-around-zero} and \eqref{bound-on-R-over-k}. Hence, we treat $II_2$ exactly as $I$ (or \eqref{local-decay-proof-term-I}), and obtain $\lr{x}^{-1}|II_2| \lesssim \n{\widetilde h}_{H_k^1}$.

For $II_1$, we split the region of integration using the smooth cut-off functions to distinguish $k \lesssim 1$ and $k \gtrsim 1$. Recall the function $\varphi_{\leq 1}(k)$ from \eqref{cut-off-m0} and let $\varphi_{> 1}(k) = 1 - \varphi_{\leq 1}(k)$. Then,
\begin{equation*}
\begin{aligned}
II_1 &= \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \varphi_{\leq 1}(k) \lr{k}^{-1} (e^{ikx} - e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk\\[3pt]
& \quad + \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \varphi_{> 1}(k) \lr{k}^{-1} (e^{ikx} - e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk\\[3pt]
& =: II_1^{(1)} + II_1^{(2)}.
\end{aligned}
\end{equation*}
We integrate $II_1^{(1)}$ by parts, using $\lr{k}^{-1}e^{it\lr{k}} = (itk)^{-1} \partial_k e^{it\lr{k}}$ and keeping in mind that $\widetilde{h}(0)=0$. Then, $II_1^{(1)}$ satisfies
\begin{equation*}
\begin{aligned}
|II_1^{(1)}| \lesssim \frac{1}{t} \int_{\ensuremath{\mathbb{R}}_+} \left| \partial_k(\varphi_{\leq 1}(k) k^{-1} (e^{ikx} - e^{-ikx}) \widetilde{h}(k)) \right|dk.
\end{aligned}
\end{equation*}
Note that in $II_1^{(1)}$ we assume $k \lesssim 1$, for which the following estimate holds:
\begin{equation}
\label{exp-bound-1}
\lr{x}^{-1} \left|\frac{e^{ikx} - e^{-ikx}}{k}\right| + \lr{x}^{-2} \left|\partial_k \left( \frac{e^{ikx} - e^{-ikx}}{k}\right)\right| \lesssim 1.
\end{equation}
Then, using \eqref{exp-bound-1} and the Cauchy-Schwartz inequality, we obtain $\lr{x}^{-2} |II_1^{(1)}| \lesssim \lr{t}^{-1} \n{\widetilde{h}}_{H_k^1}$. For $II_1^{(2)}$, the computations are quite similar. We have $k\gtrsim 1$, where
\begin{equation}
\label{exp-bound-2}
\left|\frac{e^{ikx} - e^{-ikx}}{k}\right| + \lr{x}^{-1} \left|\partial_k \left( \frac{e^{ikx} - e^{-ikx}}{k}\right)\right| \lesssim \frac{1}{k}
\end{equation}
holds. Then, integrating by parts in $k$, using \eqref{exp-bound-2} and Cauchy-Schwartz inequality, we get $\lr{x}^{-1} |II_1^{(2)}| \lesssim \lr{t}^{-1} \n{\widetilde{h}}_{H_k^1}$. Putting the above estimates together, we derive \eqref{local-l-infty-decay-singular}.
\end{proof}
\fi

For the regular part, the function $\psi_R$ is already decaying according
to \eqref{psi-regular-decay}, which allows to get estimates with
growing weight.

\begin{lemma}
Suppose $V$ satisfies Assumption \ref{V-assumption}. Then, for every $t \geq 0$, we have
\begin{equation}
\label{local-l-infty-decay-regular}
\left\| \lr{x}^{10} \left( \lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c h \right)_R \right\|_{L^\infty_x}
  \lesssim \lr{t}^{-1} \| \widetilde{h} \|_{H^1_k}.
\end{equation}
\end{lemma}

\iffalse
\begin{proof}
The proof of \eqref{local-l-infty-decay-regular} follows similarly to the proof of
\eqref{local-l-infty-decay-singular}, and one needs to apply certain modifications
due to the presence of $m_\pm$ functions.

From the definition in \eqref{h-sing-reg-decomp}, we have
\begin{equation*}
\sqrt{2\pi} \left( \lr{\widetilde D}^{-1} e^{it \lr{\widetilde D}} P_c h\right) _S (x) = \int_\ensuremath{\mathbb{R}} \lr{k}^{-1} \psi_R(x, k) e^{it \lr{k}} \widetilde h(k) dk.
\end{equation*}
Then focusing on the case $t>1$ and $k\geq 0$, from \eqref{regular-part} we have
\begin{equation*}
\begin{aligned}
&\int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} \psi_R(x, k) e^{it \lr{k}} \widetilde h(k) dk = I + II + III + IV + V,
\end{aligned}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
& I := \sigma_+(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} T(k) (m_+(x,k)-1) e^{ikx} e^{it\lr{k}} \widetilde{h} (k)dk,\\
& II := \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} \varphi_{>1}(k) ((m_-(x, -k)-1) e^{ikx} + R_-(k) (m_-(x,k)-1) e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk,\\
& III := \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} \varphi_{\leq 1}(k) (1 + R_-(k)) (m_-(x,k)-1) e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk,\\
& IV := \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} \varphi_{\leq 1}(k) \big(m_-(x, -k)- m_-(x,k)\big) e^{ikx} e^{it\lr{k}} \widetilde{h} (k)dk,\\
& V := \sigma_-(x) \int_{\ensuremath{\mathbb{R}}_+} \lr{k}^{-1} \varphi_{\leq 1}(k) \big((m_-(x, k)-1\big) (e^{ikx} - e^{-ikx}) e^{it\lr{k}} \widetilde{h} (k)dk.
\end{aligned}
\end{equation*}
Estimates for the terms $I, II, III$ and $V$ follow the same spirit of the proof of Lemma \ref{lemma-local-l-infty-decay-singular}, taking into account \eqref{estimates-partial_k^s_m}. For $IV$ we use
\begin{equation*}
m_-(x, -k)- m_-(x,k) = - k \int_{-1}^1 \partial_2 m_-(x,zk) ~dz,
\end{equation*}
where $\partial_2 m_-$ denotes the partial derivative with respect to the second variable. Then, estimates follow by integrating $IV$ by parts in $k$ and using \eqref{estimates-partial_k^s_m}.
\end{proof}
\fi

\iffalse
\subsubsection{Local $L^2 \to L^2$ bound}

\begin{lemma}
\label{lemma-local-l2-l2}
Suppose $V$ satisfies Assumption \ref{V-assumption}. Then, for every $t\geq 0$ we have
\begin{equation}
\label{local-l2-l2}
\left\| \lr{x}^{-3} \partial_x \lr{\widetilde D}^{-1} e^{\pm it \lr{\widetilde D}} P_c h \right\|_{L^2_x} \lesssim \lr{t}^{-1} \left( \|\partial_k \widetilde{h} \|_{L^2_k} + \normm{\lr{k} \widetilde{h}}{k} \right).
\end{equation}
\end{lemma}

\begin{proof}
As previously, the proof in case $0\leq t\leq 1$ is straightforward, so we assume $t>1$.
From the definition of the distorted Fourier transform, we have
\begin{equation*}
\partial_x \lr{\widetilde D}^{-1} e^{it \lr{\widetilde D}} P_c h(x) = \int_\mathbb{R} ( \partial_x \psi(x,k) ) \lr{k}^{-1} e^{it \lr{k}} \widetilde{h}(k) dk
\end{equation*}
We consider only the contribution of $k\geq 0$ interval, as $k<0$ is treated similarly.
Then, $\psi(x,k) = e^{ikx}  m_+(x,k)T(k)$, and we write
\begin{equation*}
\begin{aligned}
\int_{\mathbb{R}_+} ( \partial_x \psi(x,k) ) \lr{k}^{-1} e^{it \lr{k}} \widetilde{h}(k) dk &= \int_{\mathbb{R}_+} e^{ikx}  (\partial_x m_+(x,k) ) T(k) \lr{k}^{-1} e^{it \lr{k}} \widetilde{h}(k) dk\\
& \quad + i \int_{\mathbb{R}_+} k e^{ikx} m_+(x,k) T(k) \lr{k}^{-1} e^{it \lr{k}} \widetilde{h}(k) dk\\[3pt]
& =: \mathcal{I}_1(t,x) + \mathcal{I}_2(t,x).
\end{aligned}
\end{equation*}
We consider each of the above integral separately. Since $\widetilde{h}(0) = 0$, we use the relation $\lr{k}^{-1} e^{it\lr{k}} = (itk)^{-1} \partial_k e^{it\lr{k}}$ to integrate $\mathcal{I}_1$ by parts,
\begin{equation}
\label{I-1-in-local-l2-l2}
\begin{aligned}
\mathcal{I}_1(t,x) &= - \frac{1}{it}\int_{\mathbb{R}_+} e^{it \lr{k}} \partial_k \left( \frac{T(k)}{k} \widetilde{h}(k) \right) e^{ikx}  (\partial_x m_+(x,k) ) dk\\
& \quad - \frac{x}{t} \int_{\mathbb{R}_+} e^{it \lr{k}} \frac{T(k)}{k} \widetilde{h}(k) e^{ikx}  (\partial_x m_+(x,k) ) dk\\
& \quad - \frac{1}{it} \int_{\mathbb{R}_+} e^{it \lr{k}} \frac{T(k)}{k} \widetilde{h}(k) e^{ikx}  (\partial_k \partial_{x} m_+(x,k) ) dk\\[3pt]
& =: \mathcal{I}_{11}(t,x) + \mathcal{I}_{12}(t,x) + \mathcal{I}_{13}(t,x).
\end{aligned}
\end{equation}
Applying Minkowski inequality to each of the integrals, and using  \eqref{estimates-partial_k_and_x_m} and \eqref{bound-on-T-over-k}, we get $\normm{\lr{x}^{-3} \mathcal{I}_{1j}(t)}{x}\lesssim \lr{t}^{-1}  \| \widetilde{h} \|_{H^1_k}$ for every $j=1,2,3$. Hence, we obtain the desired bound for $\normm{\lr{x}^{-3} \mathcal{I}_{1}(t)}{x}$.
We illustrate these estimates for the term $\mathcal{I}_{13}$ in \eqref{I-1-in-local-l2-l2}, and the other terms are treated similarly. From Minkowski inequality, we have
\begin{equation*}
\begin{aligned}
\normm{\lr{x}^{-3} \mathcal{I}_{12}}{x} \lesssim \frac{1}{t} \int_{\mathbb{R}_+} \left| \frac{T(k)}{k} \widetilde{h}(k) \right| \normm{\lr{x}^{-3} \partial_k \partial_x m_+}{x} dk.
\end{aligned}
\end{equation*}
From \eqref{estimates-partial_k_and_x_m}, $\normm{\lr{x}^{-3} \partial_k \partial_x m_+}{x} \lesssim 1$, and so \begin{equation*}
\begin{aligned}
\normm{\lr{x}^{-3} \mathcal{I}_{12}}{x} \lesssim \frac{1}{t} \int_{\mathbb{R}_+} \left| \frac{T(k)}{k} \widetilde{h}(k) \right|  dk \lesssim \lr{t}^{-1} \| \widetilde{h} \|_{L^2_k},
\end{aligned}
\end{equation*}
where we used \eqref{bound-on-T-over-k} and Cauchy-Schwartz inequality.

For the term $\mathcal{I}_2$, we also integrate it by part, and get
\begin{equation}
\label{I-2-in-local-l2-l2}
\begin{aligned}
\mathcal{I}_2(t,x) &= - \frac{1}{t}\int_{\mathbb{R}_+} e^{it \lr{k}} \partial_k \left( {T(k)} \widetilde{h}(k) \right) e^{ikx}  m_+(x,k) dk\\
& \quad - \frac{ix}{t} \int_{\mathbb{R}_+} e^{it \lr{k}} T(k) \widetilde{h}(k) e^{ikx}  m_+(x,k) dk\\
& \quad - \frac{1}{t} \int_{\mathbb{R}_+} e^{it \lr{k}} T(k) \widetilde{h}(k) e^{ikx}  (\partial_k m_+(x,k) ) dk\\[3pt]
& =: \mathcal{I}_{21}(t,x) + \mathcal{I}_{22}(t,x) + \mathcal{I}_{23}(t,x).
\end{aligned}
\end{equation}
The last two terms can be treated similarly to terms in \eqref{I-1-in-local-l2-l2}, and we obtain for every $j=2,3$ that $\normm{\lr{x}^{-3} \mathcal{I}_{2j}(t)}{x} \lesssim \lr{t}^{-1} \normm{\lr{k}\widetilde{h}}{k}$. The first term $\mathcal{I}_{21}$ splits into two, the contributions from $(\partial_k T(k) ) \widetilde{h}$ and $ T(k) \partial_k \widetilde{h}$. The contribution from $(\partial_k T(k) ) \widetilde{h}$ is easy to estimate using Minkowski inequality and \eqref{uniform-estimate-T-R}. For the remaining term, we have
\begin{equation*}
\begin{aligned}
\frac{1}{t}\int_{\mathbb{R}_+} e^{it \lr{k}} {T(k)} (\partial_k \widetilde{h}(k) ) e^{ikx}  m_+(x,k) dk &= \frac{1}{t}\int_{\mathbb{R}_+} e^{it \lr{k}} {T(k)} (\partial_k \widetilde{h}(k) ) e^{ikx}  (m_+(x,k)-1) dk\\
& \quad + \frac{1}{t}\int_{\mathbb{R}_+} e^{it \lr{k}} {T(k)} (\partial_k \widetilde{h}(k) ) e^{ikx} dk\\
& =: J_1(t,x) + J_2(t,x).
\end{aligned}
\end{equation*}
The term $J_1$ is again treated by Minkowski inequality in a view of $|m_+(x,k)-1|\lesssim \lr{x}\lr{k}^{-1}$ from \eqref{estimates-partial_k_and_x_m}.
For the term $J_2$, we use the isometry of the standard Fourier transform, and get
\begin{equation*}
\begin{aligned}
\normm{J_2(t)}{x} \lesssim \frac{1}{t} \normm{\mathbf{1}_+(k) T (\partial_k \widetilde{h})}{k} \lesssim \lr{t}^{-1} \normm{\partial_k \widetilde{h}}{k}.
\end{aligned}
\end{equation*}
This completes the proof.
\end{proof}
\fi

\medskip
\section{Set-up and bootstrap
In this section we set-up
the upcoming analysis of the evolution equations
for $a(t)$ and $\chi(t,x)$ in \eqref{system-eq}, and state our main bootstrap propositions
that will imply Proposition \ref{propmain}.

We let $H:= -\ensuremath{\partial}_{xx} + V(x)$ with $V(x) :=-4Q^3$,
where $Q(x)$ is the soliton \eqref{soliton}.
Then the operator \eqref{estimates-lin} is $\calL = H+1$.
Recall the definition of
$g(t, x) := e^{it \sqrt{H+1}}  ( \partial_t - i \sqrt{H+1} ) \chi(t, x)$
in \eqref{g-profile}, so that
$\chi$ is given as in \eqref{chi-in-terms-of-g}
Recall also the notation for our main norm  \eqref{main-normintro},
\begin{equation}
\label{main-norm}
\|v\|_{X} := \|\lr{t}^2 a(t)\|_{L^\infty_t (0, \infty)} +
\|\ensuremath{\partial}_k \widetilde g \|_{L^\infty_t ((0, \infty): L_k^2)} + \|\lr{k}^2 \widetilde g \|_{L^\infty_t ((0, \infty): L_k^2)}.
\end{equation}

\smallskip
\begin{remark}
\label{remark-chi-estimates}
From Lemma \ref{lemma-dispersive-estimates} and Lemma \ref{local-l-infty-decay-lemma}
the following estimates for $\chi$ follow
\begin{equation}
\label{dispersive-decay-for-chi}
\|\partial_x^N \chi(t)\|_{L_x^\infty} \lesssim \lr{t}^{-\frac{1}{2}+\frac{N}{5}}
  \left( \|\partial_k \widetilde g\|_{L^2_k} + \|\lr{k}^2 \widetilde g\|_{L^2_k} \right)
  \lesssim \lr{t}^{-\frac{1}{2}+\frac{N}{5}} {\| v \|}_X, \qquad N=0,1,2,
\end{equation}
and
\begin{equation}
\label{local-decay-for-chi}
\left\| \lr{x}^{-2} \chi(t) \right\|_{L^\infty_x} \lesssim \lr{t}^{-1} \| \widetilde g \|_{H^1_k} \lesssim \lr{t}^{-1} {\| v \|}_X.
\end{equation}
\end{remark}

Applying the distorted Fourier transform to the relation \eqref{g-profile},
we obtain
$\widetilde g(t, k) = e^{it \lr{k}} ( \partial_t - i \lr{k} ) \widetilde \chi(t, k)$
and, after substitution into the second equation in \eqref{system-eq},
\begin{equation}\label{wg-evolution}
\partial_t \widetilde g (t, k) = e^{it\lr{k}} \widetilde{\mathcal{F}} P_c N(Q, v),
\end{equation}
where $N_c(Q, v) = P_c(6Q^2 v^2 + 4 Q v^3 + v^4)$ and we recall that the term $v$
is decomposed as $v(t,x) = a(t) \rho(x) + \chi (t, x)$.
Integrating the equation \eqref{wg-evolution} over time give
\begin{equation}
\label{wg-duhamel}
\widetilde g (t,k) = \widetilde g (0,k) + \int_0^t e^{is\lr{k}} \widetilde{\mathcal{F}} P_c N (s, k) ds,
\end{equation}
and applying the partial derivative with respect to $k$, we obtain
\begin{equation}
\label{prtl-wg-duhamel}
\ensuremath{\partial}_k \widetilde g (t,k) = \ensuremath{\partial}_k \widetilde g (0,k) + \int_0^t \ensuremath{\partial}_k \left[ e^{is\lr{k}} \widetilde{\mathcal{F}} P_c N (s, k) \right] ds.
\end{equation}

In the course of our estimates we will need to deal with $\|\partial_t \widetilde g (t)\|_{L_k^2}$,
which can be bounded using the following
\begin{lemma}
\label{lemma-prtl-t-wg}
Given $\n{v}_X \ll 1$, we have
\begin{equation}
\label{prtl-t-wg-estimates}
\|\ensuremath{\partial}_t \widetilde g\|_{L_k^2} \lesssim \lr{t}^{-3/2} \|v\|_X^2
\end{equation}
\end{lemma}

\begin{proof}
From the evolution equation \eqref{wg-evolution}, since $\widetilde{\mathcal{F}}$ is an $L_x^2 \to L_k^2$ isometry, we have
\begin{equation*}
\|\ensuremath{\partial}_t \widetilde g\|_{L_k^2} \lesssim \|Q^2v^2\|_{L_x^2} + \|Q v^3\|_{L_x^2} + \|v^4\|_{L_x^2}.
\end{equation*}
We substitute the decomposition $v(t, x) = a(t) \rho(x)+ \chi(t,x)$,
and bound the first two terms using the local decay \eqref{local-decay-for-chi} as
\begin{equation*}
\|Q^2v^2\|_{L_x^2} \lesssim |a(t)|^2 + \|\lr{x}^{-2} \chi\|^2_{L_x^\infty} \lesssim \lr{t}^{-2} \|v\|_X^2,
\end{equation*}
with a similar estimate for $\|Qv^3\|_{L_x^2}$.
For the remaining term we have
\begin{equation*}
\|v^4\|_{L_x^2} \lesssim |a(t)|^4 + \| \chi\|^4_{L_x^2} \lesssim \lr{t}^{-8} \|v\|_X^4 + \| \chi\|_{L_x^2} \| \chi\|^3_{L_x^\infty}.
\end{equation*}
The term $\| \chi\|_{L_x^\infty}$ is bounded by \eqref{dispersive-decay-for-chi},
while $\|\chi\|_{L_x^2} \lesssim \|\widetilde g \|_{L_k^2} \leq \|v\|_X$, using \eqref{isometry}.
Then, we have $\|v^4\|_{L_x^2} \lesssim \lr{t}^{-3/2} \|v\|^4_X,$
and combining all the estimates we obtain the desired bound \eqref{prtl-t-wg-estimates}.
\end{proof}

In the following subsections, we give estimates on the three norms in \eqref{main-norm}.

\smallskip
\subsection{Estimates for the first norm in \eqref{main-norm}}

\begin{proposition}
\label{a(t)-norm-estimates}
Given $\n{v}_X \ll 1$, for every $t \geq 0$ we have
\begin{equation}
\label{a-bound}
\lr{t}^2 |a(t)| \lesssim |a(0)| + \|v\|_X^2.
\end{equation}
\end{proposition}

\begin{proof}
Using the integral equation for $a(t)$ in \eqref{integral-eq}, with $F[v]$ as in \eqref{system-eq}, we have
\begin{equation*}
\begin{aligned}
\lr{t}^2 a(t) &= \lr{t}^2 e^{-\Omega t} \left[ a (0) + \frac{1}{2\Omega} \int_0^\infty e^{-\Omega s} F[v](s) ds \right]\\[3pt]
& \quad -
\frac{\lr{t}^2 e^{-\Omega t}}{2\Omega} \int_0^t e^{\Omega s} F[v](s) ds -
\frac{\lr{t}^2}{2\Omega} \int_t^\infty e^{-\Omega (s-t)} F[v](s) ds\\[3pt]
& := \mathcal{I}_1 (t) + \mathcal{I}_2(t) + \mathcal{I}_3(t).
\end{aligned}
\end{equation*}
To bound $F[v](s)$
we use the definition
in \eqref{estimates-lin} and the decomposition of $v$ in \eqref{v-decomp
to get
\begin{equation}
\label{F-v-bound}
|F[v](s)| \lesssim |\lr{v^2(s), \rho}|+ |\lr{v^3(s), \rho}| + |\lr{v^4(s), \rho}| \lesssim \norm{\lr{x}^{-2}v(s)}{x}{\infty}^2 \lesssim \lr{s}^{-2} \n{v}_X^2,
\end{equation}
where for the last inequality we used \eqref{local-decay-for-chi}.
Then, using \eqref{F-v-bound}, direct computations show that
\begin{equation*}
|\mathcal{I}_1(t)|\lesssim |a(0)| +\n{v}_X^2, \quad \text{and} \quad |\mathcal{I}_2(t)|, |\mathcal{I}_3(t)|\lesssim \n{v}_X^2.
\end{equation*}
Combining these estimates, we obtain the desired result.
\end{proof}

\smallskip
\subsection{Estimates for the second norm in \eqref{main-norm}}
Here, we give the $L_k^2$ bound for $\partial_k \widetilde g$.

\begin{proposition}
\label{proposition-partial-k-L2-norm}
Given $\n{v}_X \ll 1$, for every $t \geq 0$ we have \begin{equation}
\label{prtl-k-wg-norm-bound}
\normm{\partial_k \widetilde g (t)}{k} \lesssim \normm{\partial_k \widetilde g (0)}{k} + \|v\|_X^2.
\end{equation}
\end{proposition}

The proof is based on the estimates for the terms in the Duhamel expression
for $\partial_k \widetilde g$ in \eqref{prtl-wg-duhamel}. Recall that, according to the decomposition $v(s,x) = a(s) \rho(x) + \chi(s,x)$, we have
\begin{equation}
\label{FN_c-decomposition}
\widetilde{\mathcal{F}} N_c (s, k) = \widetilde{\mathcal{F}} P_c \left(6Q^2 (a(s)\rho + \chi(s))^2 +4 Q (a(s)\rho + \chi(s))^3 + (a(s)\rho + \chi(s))^4 \right).
\end{equation}
Then the $L_k^2$ bound for $\partial_k \widetilde g$ is given by
\begin{equation}
\label{partial-k-wg-intermediate-bound}
\normm{\partial_k \widetilde g(t)}{k} \lesssim \normm{\partial_k \widetilde g(0)}{k} + \normm{A_1(t)}{k} + \normm{A_2(t)}{k} + \normm{A_3(t)}{k},
\end{equation}
where
\begin{equation}
\label{A-j-functions}
\begin{aligned}
& A_1(t,k) := \int_0^t \partial_k \left[e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q^2 (a(s)\rho + \chi(s))^2(k) \right] ds,\\
& A_2(t,k) := \int_0^t \partial_k \left[e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q (a(s)\rho + \chi(s))^3(k) \right] ds,\\
& A_3(t,k) := \int_0^t \partial_k \left[e^{is\lr{k}} \widetilde{\mathcal{F}} P_c \left(a(s)\rho + \chi(s)\right)^4(k) \right] ds.
\end{aligned}
\end{equation}
To prove Proposition \ref{proposition-partial-k-L2-norm}, it suffices to show that $\normm{A_j(t)}{k} \lesssim \n{v}_X$ for every $j=1,2,3$. We will verify these estimates in the following lemmas as well as in Section \ref{section-quadratic} and Section \ref{section-quartic}.

\begin{lemma}
\label{lemma-A-1-L2-norm-bound}
Given $\n{v}_X\ll 1$, for every $t\geq 0$ we have
\begin{equation}
\label{A-1-L2-norm-bound}
\normm{A_1(t)}{k} \lesssim \normm{\int_0^t
s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q^2 \chi^2(s) ds}{k} + \n{v}_X^2.
\end{equation}
\end{lemma}

\begin{proof}
Distributing $\partial_k$
and using Minkowski inequality, we obtain
\begin{equation*}
\begin{aligned}
\normm{A_1(t)}{k} \lesssim&~ \int_0^t \normm{ \partial_k \widetilde{\mathcal{F}} P_c Q^2 (a(s)\rho + \chi(s))^2 }{k}ds\\
& + \normm{\int_0^t s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q^2 (a(s)\rho + \chi(s))^2 ds}{k}\\
& =: A_{11} + A_{12}.
\end{aligned}
\end{equation*}
The first term $A_{11}$ is easily bounded using \eqref{derivative-in-fourier-bound},
followed by
the local decay estimates \eqref{local-decay-for-chi}:
\begin{equation*}
\begin{aligned}
A_{11} &\lesssim \int_0^t \normm{\lr{x} Q^2 (a(s) \rho + \chi(s))^2}{x}ds
\\
& \lesssim \int_0^t a^2(s) ds + \int_0^t \norm{\lr{x}^{-2} \chi(s)}{x}{\infty}^2 ds \lesssim \n{v}_X^2.
\end{aligned}
\end{equation*}
For the second term $A_{12}$ we bound it similarly, term by term, as follows
\begin{equation*}
\begin{aligned}
A_{12} & \lesssim \int_0^t s\; a^2(s) \normm{\widetilde{\mathcal{F}} P_c Q^2 \rho^2}{k} ds + \int_0^t s\; a(s) \normm{\widetilde{\mathcal{F}} P_c Q^2 \rho \chi(s)}{k} ds\\
& \quad + \normm{\int_0^t
s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q^2 \chi^2(s)  ds}{k}\\
& := B_1 + B_2 + B_3.
\end{aligned}
\end{equation*}
Using \eqref{isometry} and \eqref{local-decay-for-chi} we obtain
\begin{equation*}
\begin{aligned}
& B_1 \lesssim \n{v}_X^2 \int_0^t \frac{s}{\lr{s}^4} ds \lesssim \n{v}_X^2,
\\
& B_2 \lesssim \int_0^t s\; a(s) \normm{\lr{x}^{-2} \chi(s)}{x} ds \lesssim \n{v}_X^2 \int_0^t \frac{s}{\lr{s}^3} ds \lesssim \n{v}_X^2.
\end{aligned}
\end{equation*}
This completes the proof. \end{proof}

\begin{remark}
Note that in the proof of Lemma \ref{lemma-A-1-L2-norm-bound}, the term $B_3$ can not be bounded simply
 using Minkowski's inequality, as the corresponding integral in $ds$ would diverge when $t \to \infty$.
 Indeed, we would have
\begin{equation}
\label{divergent-B3}
B_3 \lesssim \int_0^t s \normm{\lr{x}^{-2} \chi(s)}{k}^2 ds \lesssim \n{v}_X^2 \int_0^t \frac{s}{\lr{s}^2} ds \approx \n{v}_X^2 \log t.
\end{equation}
To get a $O(\n{v}_X^2)$ bound for the term $\normm{A_1(t)}{k}$,
we will need some more refined analysis, which will be provided in Section \ref{section-quadratic}.
\end{remark}

\begin{lemma}
\label{lemma-A-2-L2-norm-bound}
Given $\n{v}_X\ll 1$, for every $t\geq 0$ we have
\begin{equation}\label{A-2-L2-norm-bound}
\normm{A_2(t)}{k} \lesssim  \n{v}_X^3.
\end{equation}
\end{lemma}

\begin{proof}
The proof is similar to the proof of Lemma \ref{lemma-A-1-L2-norm-bound}.
However, unlike in \eqref{divergent-B3}, each term, which appears after applying $\partial_k$ inside the $A_2(t)$ integral
in \eqref{A-j-functions} and expanding the cubic term $(a(s)\rho + \chi(s))^3$, is directly bounded by $\n{v}_X^3$.
\end{proof}

\begin{lemma}
\label{lemma-A-3-L2-norm-bound}
Given $\n{v}_X\ll 1$, for every $t\geq 0$ we have
\begin{equation}
\label{A-3-L2-norm-bound}
\normm{A_3(t)}{k} \lesssim \normm{\int_0^t \partial_k \left[ e^{is\lr{k}} \widetilde{\mathcal{F}} P_c \chi^4(s) \right] ds}{k} + \n{v}_X^4.
\end{equation}
\end{lemma}

\begin{proof}
The proof is similar to the proof of previous lemmas, using the fact that $\rho(x)$ is an exponentially decaying function
and \eqref{local-decay-for-chi}.
\end{proof}

\begin{remark}
Since there are no (exponentially) decaying function in front of $\chi^4$
we cannot just rely on the local decay estimates \eqref{local-decay-for-chi}
to bound the quartic terms on the left-hand side of \eqref{A-3-L2-norm-bound}.
We will bound these terms in Section \ref{section-quartic} ,
using some multilinear analysis in the spirit of \cite{CP21,GP20}.
\end{remark}

The proof of Proposition \ref{proposition-partial-k-L2-norm} is now
straightforward assuming that both the quadratic term in \eqref{A-1-L2-norm-bound} and the
quartic term in \eqref{A-3-L2-norm-bound} are bounded by $\n{v}_X^2$.
These two bounds will be proven, respectively, in the upcoming Sections \ref{section-quadratic} and \ref{section-quartic}.

\begin{proof}[Proof of Proposition \ref{proposition-partial-k-L2-norm}]
Under the above assumption, Lemmas \ref{lemma-A-1-L2-norm-bound}--\ref{lemma-A-3-L2-norm-bound}
imply
\begin{equation*}
\normm{A_1(t)}{k} + \normm{A_2(t)}{k} + \normm{A_3(t)}{k} \lesssim \n{v}_X^2.
\end{equation*}
This bound applied to \eqref{partial-k-wg-intermediate-bound} completes the proof.
\end{proof}

\smallskip
\subsection{Estimates for the third norm in \eqref{main-norm}}
Here we give the $L_k^2$ bound for $\normm{\lr{k}^2 \widetilde g}{k}$.

\begin{proposition}
\label{proposition-k2-wg-norm-bound}
For every $t \geq 0$ we have
\begin{equation}
\label{k2-wg-norm-bound}
\normm{\lr{k}^2 \widetilde g (t)}{k} \lesssim \normm{\lr{k}^2 \widetilde g (0)}{k} + \|v\|_X^2.
\end{equation}
\end{proposition}

To prove the above statement, we use the Duhamel's expression \eqref{wg-duhamel}
apply Minkowski, and recall
\eqref{estimates-lin},
to get
\begin{equation}
\label{k2-wg-bound-decomposition}
\begin{aligned}
\normm{\lr{k}^2 \widetilde g (t)}{k} & \lesssim \normm{\lr{k}^2 \widetilde g (0)}{k} + \int_0^t \normm{ \lr{k}^2 \widetilde{\mathcal{F}} P_c Q^2 v^2(s)}{k} ds+ \int_0^t \normm{ \lr{k}^2 \widetilde{\mathcal{F}} P_c Q v^3(s)}{k} ds\\
& \quad + \int_0^t \normm{ \lr{k}^2 \widetilde{\mathcal{F}} P_c v^4(s)}{k} ds\\
& := \normm{\lr{k}^2 \widetilde g (0)}{k} + \mathcal{I}_1(t) + \mathcal{I}_2(t) + \mathcal{I}_3(t).
\end{aligned}
\end{equation}
Therefore, it suffices to show the following:

\begin{lemma}
\label{lemma-k2-wg-I-j-bound}
For every $t \geq 0$ we have
\begin{equation}
\label{k2-wg-I-j-bound}
\mathcal{I}_j (t) \lesssim \n{v}_X^2 \quad \text{for every  } j=1,2,3.
\end{equation}
\end{lemma}

\begin{proof}
Starting with the quadratic terms, we recall the definition of the norm \eqref{main-norm},
and estimate using \eqref{k-s-h-tilde-bound}
followed by the pointwise and local decay estimates \eqref{dispersive-decay-for-chi} and \eqref{local-decay-for-chi}:
\begin{align*}
\normm{ \lr{k}^2 \widetilde{\mathcal{F}} P_c Q^2 v^2(s)}{k} & \lesssim {\| Q^2 v^2(s) \|}_{H_x^2}
  \lesssim {\| Q v(s) \|}_{H_x^2} {\| Q v(s) \|}_{L_x^\infty}
\\
& \lesssim \sup_{j=0,1,2} {\| \langle x\rangle^{-5} \partial_x^j v(s) \|}_{L_x^2} {\| Q v(s) \|}_{L_x^\infty}
  \lesssim  \langle s \rangle^{-1/10}\n{v}_X \langle s \rangle^{-1} \n{v}_X,
\end{align*}
which proves the desired bound for $j=1$.
\iffalse
\fi
The estimates for the cubic term $\mathcal{I}_2$ is simpler so we can skip it.
For $\mathcal{I}_3$
and use the decay \eqref{dispersive-decay-for-chi} to see that
\begin{align*}
\normm{ \lr{k}^2 \widetilde{\mathcal{F}} P_c v^4(s)}{k} \lesssim {\big\| v^4(s) \big\|}_{H_2^x}
  \lesssim {\big\| v(s) \big\|}_{H_x^2} {\big\| v(s) \big\|}_{L^\infty_x}^3
  \lesssim \langle s \rangle^{-3/2} \n{v}_X^4.
\end{align*}
\end{proof}

\medskip
\section{Quadratic term analysis}
\label{section-quadratic}
In this section we prove the following bound for the quadratic
terms on the right-hand side of \eqref{A-1-L2-norm-bound} that were not treated in the previous section.

\begin{lemma}
\label{lemma-quadratic-term-bound}
Given $\n{v}_X \ll 1$, for every $t \geq 0$ we have
\begin{equation*}
\normm{\int_0^t s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q^2 \chi^2(s) ds}{k} \lesssim \n{v}_X^2.
\end{equation*}
\end{lemma}

We split the integral into two integrals over separate regions $0 \leq s\leq 1$ and $s >1$, and obtain
\begin{equation}\label{quadC12}
\begin{aligned}
\normm{\int_0^t s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} P_c Q^2 \chi^2(s) ds}{k}
& \lesssim \left\| \int_0^1
s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} Q^2 \chi^2(s)  ds \right\|_{L^2_k}
\\
& \quad + \left\| \int_1^t   s \frac{k}{\lr{k}} e^{is\lr{k}} \widetilde{\mathcal{F}} Q^2 \chi^2(s)  ds \right\|_{L^2_k}
  =: C_1 + C_2.
\end{aligned}
\end{equation}
Of course, if $t \leq 1$ then the term $C_2$ is not present and the term $C_1$
corresponds to the integration over the region $0 < s < t$
and can be treated in the same way below.
Using the local decay \eqref{local-decay-for-chi} we can bound
\begin{equation}
\label{A-3-bound}
C_1 \lesssim \int_0^1 s  \|\widetilde{\mathcal{F}} Q^2 \chi^2(s)\|_{L_k^2} ds
  \lesssim \int_0^1 s \|\lr{x}^{-2} \chi(s) \|^2_{L_x^\infty} \lesssim \|v\|_X^2,
\end{equation}

To get a bound for the term $C_2$ we use the following smoothing-type estimate:

\iffalse
we consider the multilinear integrals,
which appear after the use of the distorted Fourier transform formula together
with \eqref{chi-in-terms-of-g}. Then, the integral corresponding to the term $C_2$ becomes
\begin{equation}
\label{quadratic-term-integral}
\begin{aligned}
\int_1^t s & \frac{k}{\lr{k}} e^{is\lr{k}} \big(\widetilde{\mathcal{F}} Q^2 \chi^2 \big) (k) ds
\\
&= -\frac{1}{4} \sum_{\iota_1, \iota_2 \in \{+, -\}} \iota_1 \iota_2
  \int_1^t s \int \frac{e^{is \Phi_{\iota_1 \iota_2}(k,l,m)}}{\lr{l}\lr{m}}
  \widetilde g^{(\iota_1)}(l) \widetilde g^{(\iota_2)}(m) \mu_{\iota_1 \iota_2} (k,l,m) dl dm ds,
\end{aligned}
\end{equation}
\cfp{Only need the first expression if the new proof works}
where
\begin{equation}
\label{Phi-mu-functions-quadratic}
\begin{aligned}
& \Phi_{\iota_1 \iota_2}(k,l,m) = \lr{k} - \iota_1 \lr{l} - \iota_2 \lr{m} \\
& \mu_{\iota_1 \iota_2} (k,l,m) = \int \frac{k}{\lr{k}} Q^2(x)  \overline{\psi(x, k)} \psi^{(\iota_1)} (x,l) \psi^{(\iota_2)} (x,m) dx
\end{aligned}
\end{equation}

The expression on the right-hand side of \eqref{quadratic-term-integral} is the sum of four terms,
each corresponding to a certain sign pair $(\iota_1, \iota_2)$.
Each of these four terms can be treated similarly,
so we only concentrate on the term with $\iota_1 = \iota_2 = +$.
Moreover, for simplicity, we drop the $++$ subscripts from $\Phi$ and $\mu$
functions in \eqref{Phi-mu-functions-quadratic}.
We also drop the term $\frac{k}{\lr{k}}$ from the $\mu$ integral in \eqref{Phi-mu-functions-quadratic}
as it is uniformly bounded by $1$ and does not affect the analysis.
In other words, we set
\begin{equation*}
\begin{aligned}
& \Phi(k,l,m) := \lr{k} - \lr{l} - \lr{m},
\\
& \mu (k,l,m) := \int Q^2(x) \overline{\psi(x, k)} \psi (x,l) \psi (x,m) dx,
\end{aligned}
\end{equation*}
and deal with the integral
\begin{equation}
\label{D-integral}
D(t,k) := \int_1^t s \int \frac{e^{is \Phi(k,l,m)}}{\lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \mu (k,l,m) dl dm ds,
\end{equation}
and, to prove Lemma \eqref{lemma-quadratic-term-bound},
it suffices to show that $\|D\|_{L_k^2} \lesssim \n{v}_X^2$.
\fi

\def\lr{x}{\lr{x}}

\begin{lemma}\label{lemsmoothing}
Given a function $\mathcal{Q}: \mathbb{R}_x\times \mathbb{R}_k \mapsto \mathbb{C}$ such that, for some $\beta \in \mathbb{R}$,
\begin{equation*
\sup_{x\in\mathbb{R},\,k\in\mathbb{R}} \big| \lr{x}^{\beta} \mathcal{Q}(x,k) \big| < \infty,
\end{equation*}
and $\phi: \mathbb{R} \mapsto \mathbb{C}$ such that $|\phi(k)|\lesssim |k|/\lr{k}$,
then, for all $t\geq0$,
\begin{equation}\label{smoothinghom}
{\Big\| \lr{x}^{\beta} \int_\mathbb{R} \mathcal{Q}(x,k)
  \phi(k) e^{i\lr{k}s} h(k)\,dk \Big\|}_{L_x^\infty L_s^{2}([0,t])} \lesssim {\| h \|}_{L^2}.
\end{equation}
Under the same assumptions, the following inhomogeneous estimate holds:
\begin{align}\label{smoothinginhom}
{\Big\| \int_0^t \left[ \int_\mathbb{R} e^{-i \lr{k}s} \overline{\phi}(k)
  \overline{\mathcal{Q}}(x,k)F(s,x)\,dx \right] \,ds \Big\|}_{L_k^2}
\lesssim & {\Big\|\lr{x} ^{-\beta}F \Big\|}_{L_x^1 L_s^2([0,\infty])}.
\end{align}
\end{lemma}

\smallskip
We now complete the proof of Lemma \ref{lemma-quadratic-term-bound} using Lemma \ref{lemsmoothing} above.

\begin{proof}[Proof of Lemma \ref{lemma-quadratic-term-bound}]
Applying \eqref{smoothinginhom}
to the term $C_2$ in \eqref{quadC12}, that is,
letting $\beta=0$, $\mathcal{Q}(y,x) = \psi(x,k)$, $\phi(k) = k/\lr{k}$,
and $F(s,x) = s \,Q^2(x) \chi^2(s,x)$, we get
\begin{align}\label{Destimate}
\begin{split}
C_2
  & = {\Big\| \int_0^t \left[ \int_\mathbb{R} e^{i \lr{k} s} \frac{k}{\lr{k}}
  \overline{\psi(x,k)} F(s,x) \,dx \right] \,ds \Big\|}_{L_{k}^{2}}
  \\
  & \lesssim {\| s \, Q^2(\cdot) \chi^2(s,\cdot) \|}_{L_x^1(\mathbb{R}) L_s^2([0,\infty])}
  \\
  & \lesssim {\| s \, Q(\cdot) \chi^2(s,\cdot) \|}_{L_s^2([0,\infty]) L^\infty_x(\mathbb{R})}
  \\
  & \lesssim {\| v \|}_X^2 {\| s \lr{s}^{-2} \|}_{L_s^2([0,\infty])} \lesssim {\| v \|}_X^2,
\end{split}
\end{align}
having used the local decay estimate \eqref{local-decay-for-chi}.
\end{proof}

\smallskip
Observe that Lemma \ref{lemsmoothing} does not use the genericity of the potential.
The proof is similar to the one for the analogous
case of the Schr\"odinger flow $e^{itH}$ given in \cite[Lemma 3.5]{CP22}.
We provide the details here for completeness.

\begin{proof}[Proof of Lemma \ref{lemsmoothing}]
Without loss of generality we may assume $\beta=0$
and restrict the integral in \eqref{smoothinghom} to $k\in[0,\infty)$.
Making a change of variable $\lr{k} = \lambda$,
so that $k\lr{k}^{-1} dk = d\lambda$, $dk = \frac{\lambda}{\sqrt{\lambda^2-1}} d\lambda$ one has
\begin{align*}
I(s,x) & := \int_0^\infty \mathcal{Q}(x,k)\phi(k)e^{i \lr{k} s}h(k)\,dk
  \\
  & = \int_1^{\infty}\mathcal{Q}(x,\sqrt{\lambda^2-1})\phi(\sqrt{\lambda^2-1})
  e^{is\lambda} h(\sqrt{\lambda^2-1}) \frac{\lambda}{\sqrt{\lambda^2-1}}\,d\lambda.
\end{align*}

Taking the $L^2_s$ norm and applying Plancherel's theorem we obtain
\begin{align*}
{\| I(\cdot,x) \|}_{L^2_s([0,t])}^2
  & \lesssim \left\| \int_\mathbb{R} \mathcal{Q}(x,\sqrt{\lambda^2-1}) \, \mathbf{1}_{[1,\infty)}(\lambda)
  \phi(\sqrt{\lambda^2-1})e^{is\lambda} h(\sqrt{\lambda^2-1}) \frac{\lambda}{\sqrt{\lambda^2-1}}
   \,d\lambda \right\|_{L^2_s(\mathbb{R})}^2
\\
& \lesssim \int_1^{\infty} \left|  \mathcal{Q}(x,\sqrt{\lambda^2-1})
  \frac{\lambda \, \phi(\sqrt{\lambda^2-1})}{\sqrt{\lambda^2-1}}
  h(\sqrt{\lambda^2-1}) \right|^{2} \,d\lambda
\\
  & \lesssim \int_1^{\infty} \big| h(\sqrt{\lambda^2-1}) \big|^2 \,d\lambda
  \lesssim \int_0^{\infty} | h (k) |^{2} \frac{k}{\lr{k}} \,dk
  \lesssim {\| h \|}_{L^2}^2 .
\end{align*}

To obtain the inhomogeneous estimate \eqref{smoothinginhom}
we test the expression on the left-hand side against an arbitrary function $h \in L^2$:
\begin{align*}
& \left \langle  h,\int_0^t \left[ \int_\mathbb{R} e^{-i \lr{k}s} \overline{\phi}(k)
\overline{\mathcal{Q}}(y,k)F(s,y)\,dy\right] ds \right \rangle_{L^2_k}
\\
& = \int_0^t \int_\mathbb{R} \Big( \int_\mathbb{R} e^{i \lr{k}s}\phi(k)\mathcal{Q}(y,k) h(k)\,dk \Big)  \overline{F}(s,y)\,dyds
\end{align*}
Applying H\"older's inequality followed by the homogeneous estimate \eqref{smoothinghom}:
\begin{align*}
& \left| \int_0^t \int_\mathbb{R} \Big( \int e^{i\lr{k}s}\phi(k)\mathcal{Q}(y,k) h(k)\,dk \Big) \overline{F}(s,y) \,dyds\right|
\\
& \lesssim {\Big\|
  \int\mathcal{Q}(x,k)\phi(k)e^{i\lr{k} s}h(k)\,dk \Big\|}_{L_x^\infty L_s^2([0,t])}
  \big\| F \big\|_{L_x^1 L_s^2([0,t])}
\\
& \lesssim {\| h \|}_{L^2} {\big\| F \big\|}_{L_x^1 L_s^2([0,t])}
\end{align*}
which implies \eqref{smoothinginhom} by duality.
\end{proof}

\iffalse
The main difficulty here is to get an uniformly bounded integral for every $t \geq 0$,
unlike in \eqref{divergent-B3}, where the bound in $t$ was logarithmic.
Therefore, instead of direct application of Minkowski inequality as
in \eqref{divergent-B3}, we split the regions of integration
with respect to the size of the phase $\Phi$, and estimate each integral separately.
This way, modulo the presence of other terms, the integral would
be bounded by something like $\int_1^t s^{-1-\alpha} ds \lesssim 1$ with sufficiently small positive $\alpha>0$.

In other words, given positive $c, \delta \ll 1$, we define the following regions for every $s \in (1, t)$ inside the $ds$ integral in \eqref{D-integral}
\begin{itemize}
\item \textbf{Region 1}: $|\Phi(k,l,m)| \lesssim s^{-\delta}$ and
$|l|, |m| \lesssim s^{-c \delta}$;
\item \textbf{Region 2}: $|\Phi(k,l,m)| \lesssim s^{-\delta}$ and $ |l|, |m| \gtrsim s^{-c \delta}$;
\item \textbf{Region 3}: $|\Phi(k,l,m)| \gtrsim s^{-\delta}$.
\end{itemize}
To incorporate this bounds into the integrals, we recall the smooth even cutoff function $\varphi$ supported in $[-2,2]$ from Notation subsection \ref{subsection-notations}. We also use throughout this section the function
$$
\phi(x):= 1-\varphi(x) \quad \text{for every  } x\in \mathbb{R}.
$$
Then, for example, the integral for Region 1 will contain the terms $\varphi(s^\delta \Phi(k,l,m))$, $\varphi(s^{c\delta} l)$ and $\varphi(s^{c \delta} m)$.
In what follows, we split the integral $D$ as follows
\begin{equation}
\label{D-decomposition}
\begin{aligned}
D & =  \int_1^t s \int \frac{e^{is \Phi(k,l,m)}}{\lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \varphi(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) \mu (k,l,m) dl dm ds\\
&\quad + \int_1^t s \int \frac{e^{is \Phi(k,l,m)}}{\lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \varphi(s^\delta \Phi(k,l,m)) \phi(s^{c \delta} l) \phi(s^{c \delta} m) \mu (k,l,m) dl dm ds\\
& \quad + \int_1^t s \int \frac{e^{is \Phi(k,l,m)}}{\lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \phi(s^\delta \Phi(k,l,m)) \mu (k,l,m) dl dm ds\\[3pt]
& =: D_1(t,k) + D_2(t,k) + D_3(t,k),
\end{aligned}
\end{equation}
where each $D_j$ corresponds to the integral defined in Region $j$. Below, we show that each of these integrals satisfies $\normm{D_j(t)}{k} \lesssim \n{v}_X^2$ for every $t \geq 0$, which would complete the proof of Lemma \ref{lemma-quadratic-term-bound}.

\subsection{Estimates in Region 1}
Here we estimate the $L_k^2$ norm of the term $D_1$.

\begin{lemma}
\label{lemma-D-1-bound}
For every $t\geq 0$, we have \begin{equation*}
\normm{D_1(t)}{k} \lesssim \n{v}_X^2.
\end{equation*}
\end{lemma}

Below we give a proof of this estimate. The main idea is to come up with a sufficiently negative power of $s$ in the integral $D_1$ so that the integral over $ds$ will be bounded for every positive $t$. First step is to use the relation
$\frac{e^{is\Phi}}{\lr{l}\lr{m}} = -\frac{1}{s^2 l m} \partial_l \partial_m e^{is \Phi}$ to integrate $D_1$ by part in $l$ and $m$. Then, we obtain
\begin{equation*}
\begin{aligned}
D_1(t,k) & = - \int_1^t s^{-1} \int e^{is\Phi} \varphi(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) \partial_l \partial_m \left[ \widetilde g(l) \widetilde g(m) \mu_1(k,l,m) \right] dldm ds\\
& \quad - \int_1^t s^{-1} \int e^{is\Phi} \frac{\widetilde g(l) \widetilde g(m) \mu(k,l,m)}{lm}
\partial_l \partial_m \left[ \varphi(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) \right] dldm ds\\
& =: D_{11}(t,k) + D_{12}(t,k),
\end{aligned}
\end{equation*}
where $\mu_1(k,l,m) :=  (lm)^{-1} \mu(k,l,m)$. The term $(lm)^{-1}$ is treated differently in the integrals $D_{11}$ and $D_{12}$. In $D_{11}$, it is absorbed into $\mu_1(k,l,m)$, as we already have estimates for such term in Lemma \ref{lemma-mu-1}. On the other hand, we keep $(lm)^{-1}$ in $D_{12}$ as it is, because it later (partially) cancels by contributions from the term $\partial_l \partial_m(\varphi \dots)$, which would allow to perform the integration by parts again.

\subsubsection{Estimates for the term $D_{11}$}

The term $\partial_l \partial_m \left[ \widetilde g(l)\widetilde g(m)\mu_1 (k,l,m) \right]$ inside the integral, after applying the derivatives, becomes a sum of several terms. All these terms can be treated similarly, so we only show how to treat one term, for example, $\widetilde g(l) [\partial_m \widetilde g(m)] \partial_l \mu_1(k,l,m)$. The corresponding integral is
\begin{equation*}
D_{11}^{(1)}(t,k) = - \int_1^t s^{-1} \int e^{is\Phi} \varphi(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) \widetilde g(l) [\partial_m \widetilde g(m)] \partial_l \mu_1(k,l,m) dl dm ds,
\end{equation*}
and we can bound it as
\begin{equation*}
\|D_{11}^{(1)}\|_{L_k^2} \lesssim \int_1^t s^{-1} \int \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) |\widetilde g(l)| |\partial_m \widetilde g(m)| \left[ \| \varphi(s^\delta \Phi(k,l,m)) \partial_l \mu_1(k,l,m) \|_{L_k^2} \right] dl dm ds.
\end{equation*}
Note that for fixed $l$ and $m$, the function $F(k) := \varphi(s^\delta \Phi(k,l,m))$ is supported in the finite region of length $\lesssim s^{-\delta/4}$. In what follows, since $|\partial_l \mu_1(k,l,m)| \lesssim 1$ according to Lemma \ref{lemma-mu-1}, we have
\begin{equation*}
\| \varphi(s^\delta \Phi(k,l,m)) \partial_l \mu_1(k,l,m) \|_{L_k^2} \lesssim s^{-\delta/8}.
\end{equation*}
Then, the $L_k^2$ norm of $D_{11}^{(1)}$ is bounded by
\begin{equation}
\begin{aligned}
\|D_{11}^{(1)}\|_{L_k^2} & \lesssim \int_1^t s^{-1-\delta/8} \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) |\widetilde g(l)| |\partial_m \widetilde g(m)| dl dm ds\\
& \lesssim \int_1^t s^{-1-\delta/8 - c\delta} \normm{\widetilde g}{l} \normm{\partial_m \widetilde g}{m} dl dm ds \lesssim \n{v}_X^2,
\end{aligned}
\end{equation}
where we used the Cauchy-Schwartz inequality for $dl$ and $dm$ integrals.
As such estimates can be repeated for every term of $D_{11}$, we derive
\begin{equation}
\label{D-11-bound}
\normm{D_{11}(t)}{k} \lesssim \n{v}_X^2.
\end{equation}

\subsubsection{Estimates for the term $D_{12}$}
First, we expand
\begin{equation}
\label{expansion-varphi-product}
\begin{aligned}
\partial_l \partial_m \left[ \varphi(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) \right] & = s^{2c\delta} \varphi(s^\delta \Phi(k,l,m)) \varphi'(s^{c \delta} l) \varphi'(s^{c \delta} m)\\
& \quad - s^{\delta+ c\delta} \frac{l}{\lr{l}} \varphi'(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi'(s^{c \delta} m)\\
& \quad - s^{\delta+ c\delta} \frac{m}{\lr{m}} \varphi'(s^\delta \Phi(k,l,m)) \varphi'(s^{c \delta} l) \varphi(s^{c \delta} m)\\
& \quad + s^{2\delta} \frac{l m}{\lr{l}\lr{m}} \varphi''(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m).
\end{aligned}
\end{equation}
We point out that the functions $\varphi'(y)$ and $\varphi''(y)$ are both supported inside the regions $|y| \leq 2$, which would imply the integration over small regions whenever dealing with the rhs terms in \eqref{expansion-varphi-product}. To see this, we first substitute \eqref{expansion-varphi-product} into the expression for $D_{12}$ to get
\begin{equation*}
\begin{aligned}
D_{12}(t,k) & = - \int_1^t s^{-1+2c\delta} \int e^{is\Phi} \widetilde g(l) \widetilde g(m) \mu_1(k,l,m)
\varphi(s^\delta \Phi(k,l,m)) \varphi'(s^{c \delta} l) \varphi'(s^{c \delta} m) dldm ds\\
& \quad + \int_1^t s^{-1 + \delta + c\delta} \int e^{is\Phi} \frac{\widetilde g(l) \widetilde g(m) \mu(k,l,m)}{m\lr{l}}
\varphi'(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi'(s^{c \delta} m) dldm ds\\
& \quad + \int_1^t s^{-1 + \delta + c\delta} \int e^{is\Phi} \frac{\widetilde g(l) \widetilde g(m) \mu(k,l,m)}{l\lr{m}}
\varphi'(s^\delta \Phi(k,l,m)) \varphi'(s^{c \delta} l) \varphi(s^{c \delta} m) dldm ds\\
& \quad - \int_1^t s^{-1 + 2\delta} \int e^{is\Phi} \frac{\widetilde g(l) \widetilde g(m) \mu(k,l,m)}{\lr{l}\lr{m}}
\varphi''(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \varphi(s^{c \delta} m) dldm ds\\[2pt]
& =: D_{12}^{(1)}(t,k) + D_{12}^{(2)}(t,k) + D_{12}^{(3)}(t,k) + D_{12}^{(4)}(t,k).
\end{aligned}
\end{equation*}
The estimates for the first term $D_{12}^{(1)}$ are very similar to the estimates for $D_{11}^{(1)}$, and we end up with $\normm{D_{12}^{(1)}(t)}{k} \lesssim \n{v}_X^2$. For the rest of the term, such estimates are not enough as the integral over $ds$ does not have a good finite bound. To overcome such issue, we integrate by parts one more time to get sufficiently large negative power in $s$. For each of the last three terms the procedure is analogous, so we show how to perform it only for the second term $D_{12}^{(2)}$.

Using that $\frac{1}{\lr{l}} e^{is\Phi} = -\frac{1}{isl} \partial_l e^{is\Phi}$, we integrate $D_{12}^{(2)}$ by parts in $l$ to obtain
\begin{equation*}
\begin{aligned}
D_{12}^{(2)}(t,k) = \frac{1}{i} \int_1^t &s^{-2+\delta+c\delta}\\
&\times \int e^{is\Phi} \widetilde g(m) \varphi'(s^{c \delta} m) \partial_l \left[ \widetilde g(l) \mu_1(k,l,m) \varphi'(s^\delta \Phi(k,l,m)) \varphi(s^{c \delta} l) \right]  dldm ds,
\end{aligned}
\end{equation*}
where $(lm)^{-1}$ is absorbed into $\mu_1(k,l,m)$. Now $s$ has sufficiently large negative power, and we can repeat the steps applied to the integral $D_{11}$ above. We leave all these estimates to the reader, as they are straightforward and lead to the bound $\normm{D_{12}^{(2)}(t)}{k} \lesssim \n{v}_X^2$. The terms $D_{12}^{(3)}$ and $D_{12}^{(4)}$ are treated similarly. Combining these estimates, we have
\begin{equation}
\label{D-12-bound}
\normm{D_{12}(t)}{k} \lesssim \n{v}_X^2.
\end{equation}

\subsubsection{Proof of Lemma \ref{lemma-D-1-bound}}
The proof follows from the bounds \eqref{D-11-bound} and \eqref{D-12-bound}.

\subsection{Estimates in Region 2}
Here we estimate the $L_k^2$ norm of $D_2$ defined in \eqref{D-decomposition}.
\begin{lemma}
\label{lemma-D-2-bound}
For every $t\geq 0$, we have \begin{equation*}
\normm{D_2(t)}{k} \lesssim \n{v}_X^2.
\end{equation*}
\end{lemma}

\begin{proof}
The analysis is very similar to the analysis in Region 1, with slight modifications due to the fact that both $l$ and $m$ are now bounded away from zero. First, we integrate $D_2$ by parts in $l$ and $m$ to get
\begin{equation*}
\begin{aligned}
D_2(t,k) & = - \int_1^t s^{-1} \int e^{is\Phi} \varphi(s^\delta \Phi(k,l,m)) \phi(s^{c \delta} l) \phi(s^{c \delta} m) \partial_l \partial_m \left[ \widetilde g(l) \widetilde g(m) \frac{\mu(k,l,m)}{lm} \right] dldm ds\\
& \quad - \int_1^t s^{-1} \int e^{is\Phi} \frac{\widetilde g(l) \widetilde g(m) \mu(k,l,m)}{lm}
\partial_l \partial_m \left[ \varphi(s^\delta \Phi(k,l,m)) \phi(s^{c \delta} l) \phi(s^{c \delta} m) \right] dldm ds\\
& =: D_{21}(t,k) + D_{22}(t,k),
\end{aligned}
\end{equation*}
Unlike in the estimates for the Region 1, we do not absorb $(lm)^{-1}$ into $\mu_1$. Instead, we use it to obtain a satisfactory bound for the integrals over $l$ and $m$ after use of the Cauchy-Schwartz inequality. To show the details, let us only deal with $(lm)^{-1} \widetilde g(l) [\partial_m \widetilde g(m)] [\partial_l \mu]$ term of $\partial_l \partial_m [\dots]$ in $D_{21}(t,k)$, as the rest of the terms can be treated similarly. In other words, we consider the term
\begin{equation*}
\begin{aligned}
D_{21}^{(1)}(t,k) & = - \int_1^t s^{-1} \int e^{is\Phi} \varphi(s^\delta \Phi(k,l,m)) \phi(s^{c \delta} l) \phi(s^{c \delta} m) \widetilde g(l) [\partial_m \widetilde g(m)] \frac{\partial_l \mu(k,l,m)}{lm} dl dm ds.
\end{aligned}
\end{equation*}
Similarly, as for the term $D_{11}^{(1)}$ above, since $|\partial_l \mu (k,l,m)|\lesssim 1$ according to \eqref{mu-1-bounds}
\begin{equation*}
\begin{aligned}
\|D_{21}^{(1)}\|_{L_k^2} & \lesssim \int_1^t s^{-1-\delta/8} \phi(s^{c \delta} l) \phi(s^{c \delta} m) \frac{|\widetilde g(l)| |\partial_m \widetilde g(m)|}{|l||m|} dl dm ds\\
& \lesssim \int_1^t s^{-1-\delta/8 + c \delta} \normm{\widetilde g}{l} \normm{\partial_m \widetilde g}{m} dl dm ds \lesssim \n{v}_X^2,
\end{aligned}
\end{equation*}
and therefore,
\begin{equation}
\label{D-21-bound}
\normm{D_{21}(t)}{k} \lesssim \n{v}_X^2.
\end{equation}
The term $D_{22}(t,k)$ can be estimated using combination of steps used to treat $D_{12}(t,k)$ and $D_{21}(t,k)$ above, and leads to the same bound $\n{v}_X^2$.
\end{proof}

\subsection{Estimates in Region 3}
It remains to show the following bound for $L_k^2$ norm of $D_3$ from \eqref{D-decomposition}.
\begin{lemma}
\label{lemma-D-3-bound}
For every $t\geq 0$, we have \begin{equation*}
\normm{D_3(t)}{k} \lesssim \n{v}_X^2.
\end{equation*}
\end{lemma}
Using that $e^{is \Phi} = -\frac{i \partial_s e^{is \Phi}}{\Phi}$, we integrate $D_3(t,k)$ by part in $s$ and obtain
\begin{equation}
\label{D3-decomposition}
\begin{aligned}
D_3(t,k) & = -i \int s \frac{e^{is \Phi(k,l,m)}}{\Phi \lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \phi(s^\delta \Phi(k,l,m)) \mu (k,l,m) dl dm \Big|_{s=1}^{s=t}\\
& \quad + i \int_1^t \int \frac{e^{is \Phi(k,l,m)}}{\Phi \lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \phi(s^\delta \Phi(k,l,m)) \mu (k,l,m) dl dm ds\\
& \quad + i \int_1^t s \int \frac{e^{is \Phi(k,l,m)}}{\Phi \lr{l}\lr{m}} \partial_s [\widetilde g(l) \widetilde g(m)] \phi(s^\delta \Phi(k,l,m)) \mu (k,l,m) dl dm ds\\
& \quad + i \delta \int_1^t s^{\delta} \int \frac{e^{is \Phi(k,l,m)}}{\lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \phi'(s^\delta \Phi(k,l,m)) \mu (k,l,m) dl dm ds\\[2pt]
& =: D_{31}(t,k)+ D_{32}(t,k)+ D_{33}(t,k) + D_{34}(t,k).
\end{aligned}
\end{equation}
We show that $L_k^2$ norm of each integral above is bounded by $\n{v}_X^2$. The main idea is to apply the integration by parts in $l$ and $m$ to ensure that the integrals over $s$ which would appear later in computations are uniformly bounded for every $t\geq 1$. However, there is no integration over $s$ in the term $D_{31}$, and it suffices to show that it is uniformly bounded in $s$. For the term $D_{33}$, we get better decay in $s$ inside the integral due to the presence of $\partial_s \widetilde g$, which satisfy estimates given by \eqref{prtl-t-wg-estimates}.

\subsubsection{Estimates for the term $D_{31}$}
Abusing notations, we set
\begin{equation*}
D_{31}(s,k) := \int s \frac{e^{is \Phi(k,l,m)}}{\Phi \lr{l}\lr{m}} \widetilde g(l) \widetilde g(m) \phi(s^\delta \Phi(k,l,m)) \mu (k,l,m) dl dm,
\end{equation*}
and show that the desired bound holds for every $s \geq 1$, which would be sufficient in this case. Using $\frac{e^{is\Phi}}{\lr{l}\lr{m}} = - \frac{1}{s^2 lm} \partial_l \partial_m e^{is\Phi}$, we integrate $D_{31}$ by part twice, and have
\begin{equation}
\label{new-D31-term}
\begin{aligned}
D_{31}(s,k) = \frac{1}{s} \int e^{is \Phi} \partial_l \partial_m \left[ \widetilde g(l) \widetilde g(m) \mu_{0,1,1} (k,l,m) \frac{\phi(s^\delta \Phi)}{\Phi} \right] dl dm
\end{aligned}
\end{equation}
where $\mu_{0,1,1}$ is defined through \eqref{mu-1-expression}. Expanding $\partial_l \partial_m [\dots]$ inside the integral above, we get several terms, which are all treated similarly. In what follows, we only show how to get estimates for the term obtained when $\partial_l$ hits $\mu_{0,1,1}$ and $\partial_m$ hits $\phi(s^\delta \Phi)$, namely
\begin{equation*}
\begin{aligned}
D_{31}^{(1)}(s,k) = - s^{-1+\delta} \int \frac{e^{is \Phi}}{\lr{m}}  \widetilde g(l) \widetilde g(m) \partial_l \mu_{0,1,0} \frac{\phi'(s^\delta \Phi)}{\Phi} dl dm,
\end{aligned}
\end{equation*}
where we used that $m\mu_{0,1,1} = \mu_{0,1,0}$. Applying the Minkowski inequality to get the bound for $L_k^2$ norm of $D_{31}^{(1)}$, we have
\begin{equation*}
\begin{aligned}
\normm{D_{31}^{(1)}(s)}{k} \lesssim s^{-1+\delta} \int \frac{1}{\lr{m}}  |\widetilde g(l)|| \widetilde g(m)| \normm{\partial_l \mu_{0,1,0} \frac{\phi'(s^\delta \Phi)}{\Phi}}{k} dl dm.
\end{aligned}
\end{equation*}
The $L_k^2$ norm inside the above integral is bounded as follows:
\begin{lemma}
We have
\begin{equation}
\label{bound-for-factor-in-D3}
\normm{\partial_l \mu_{0,1,0} \frac{\phi'(s^\delta \Phi)}{\Phi}}{k} \lesssim \frac{s^{\delta/2}}{\lr{l}}.
\end{equation}
\end{lemma}
\begin{proof}
We apply the substitution $y=\Phi$ inside the integral. Then, $dk = \frac{\lr{k}}{k} dy$ and
\begin{equation*}
\normm{\partial_l \mu_{0,1,0} \frac{\phi'(s^\delta \Phi)}{\Phi}}{k}^2
  \lesssim \int_{s^{-\delta}}^\infty \frac{\lr{k}}{y^2 |k|} |\partial_l \mu_{0,1,0}|^2 dk
  \lesssim \frac{1}{\lr{l}^2} \int_{s^{-\delta}}^\infty \frac{1}{y^2} dk \lesssim \frac{s^{\delta}}{\lr{l}^2},
\end{equation*}
where we used that $k^{-1}\mu_{0,1,0} = \mu_{1,1,0}$, and both distribution functions
are bounded according to \eqref{mu-1-bounds}.
\end{proof}
Using the estimate \eqref{bound-for-factor-in-D3}, the $L_k^2$ norm of $D_{31}^{(1)}$ is bounded by
\begin{equation*}
\begin{aligned}
\normm{D_{31}^{(1)}(s)}{k} \lesssim s^{-1+3\delta/2} \int \frac{1}{\lr{l}\lr{m}}  |\widetilde g(l)|| \widetilde g(m)|
  dl dm \lesssim \normm{\widetilde g}{l} \normm{\widetilde g}{m} \lesssim \n{v}_X^2
\end{aligned}
\end{equation*}
for every $s \geq 1$. The rest of the terms in \eqref{new-D31-term} are treated in a similar fashion,
and have the same bound. In what follows, the $L_k^2$ norm of $D_{31}(t,k)$ is bounded by $\n{v}_X^2$ as desired.

\subsubsection{Estimates for the terms $D_{32}, D_{33}$ and $D_{34}$}
Here, we explain how the remaining terms in \eqref{D3-decomposition} can be treated.
However, we do not get into full details of the computations, as computations are very
similar to what have been done before.

For the term $D_{32}$, we again apply integration by parts twice as before, and each
term will be bounded by something like $\n{v}_X^2 \int_1^t s^{-2+\epsilon} ds \lesssim \n{v}_X^2$
with sufficiently small positive $\epsilon$.

The term $D_{33}$ contains $\partial_s \widetilde g$, whose $L^2$ norm decays as in \eqref{prtl-t-wg-estimates}.
In that follows, it suffices to integrate by part only once as follows:
if $\partial_s$ hits $\widetilde g(l)$ we integrate by parts in $m$ and vice versa.
Then, each term in $D_{33}$ will be bounded by something like
$\n{v}_X^3 \int_1^t s^{-3/2+\epsilon} ds \lesssim \n{v}_X^2$ with sufficiently small positive $\epsilon$.

The term $D_{34}$ does not have $\Phi$ in the denominator, and $\phi'(s^\delta \Phi)$
means that the integration goes over the regions with $\Phi \approx s^{-\delta}$. In what follows,
integrating by parts in $l$ and $m$, we obtain that every term in $D_{34}$ will be bounded by something like $\n{v}_X^2 \int_1^t s^{-2+\epsilon} ds \lesssim \n{v}_X^2$ with sufficiently small positive $\epsilon$.

\subsubsection{Proof of Lemma \ref{lemma-D-3-bound}}
The proof follows from the estimates above, which show that every term in
\eqref{D3-decomposition} is bounded by $\n{v}_X^2$.

\fi

\medskip
\section{Quartic term analysis}
\label{section-quartic}
In this section we provide a proof of the following lemma:

\begin{lemma}
\label{lemma-quartic-term-bound}
Given $\n{v}_X \ll 1$, for every $t \geq 0$ we have
\begin{equation}
\label{quartic-term-bound}
\left\| \int_0^t \partial_k \left( e^{is\lr{k}} \widetilde{\mathcal{F}}
  \chi^4(s) \right) ds \right\|_{L_k^2} \lesssim \n{v}_X^4.
\end{equation}
\end{lemma}

This lemma gives a bound for the quartic term in \eqref{A-3-L2-norm-bound}
and, together with Lemma \ref{lemma-A-1-L2-norm-bound} and \ref{lemma-quadratic-term-bound},
and Lemma \ref{lemma-A-3-L2-norm-bound},
will thus complete the proof of Proposition \ref{proposition-partial-k-L2-norm}.
This is turn give the main theorem.

\medskip
The term inside the integral in \eqref{quartic-term-bound} can be written as
\begin{equation}
\label{quartic-integral}
\begin{aligned}
& 16 e^{is\lr{k}} \widetilde{\mathcal{F}}
  \chi^4(s,k) \\
&=
\sum_{\iota_1, \iota_2, \iota_3, \iota_4 \in \{+, -\}} \iota_1 \iota_2 \iota_3 \iota_4 \int \frac{e^{is \Phi_{\iota_1 \iota_2 \iota_3 \iota_4}
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g^{(\iota_1)}(l) \widetilde g^{(\iota_2)}(m) \widetilde g^{(\iota_3)}(n) \widetilde g^{(\iota_4)}(p)
\mu_{\iota_1 \iota_2 \iota_3 \iota_4}
dlmnp
\end{aligned}
\end{equation}
where we used the short-hand notation $dlmnp := dldmdndp$,
the definition of the distorted Fourier transform
and the relation \eqref{chi-in-terms-of-g}, with the functions
$\Phi_{\iota_1 \iota_2 \iota_3 \iota_4}$ and $\mu_{\iota_1 \iota_2 \iota_3 \iota_4}$,
depending on the variables $k,l,m,n,p$, defined as follows
\begin{equation}
\label{Phi-quartic}
\Phi_{\iota_1 \iota_2 \iota_3 \iota_4} (k, l, m, n, p) = \lr{k} - \iota_1 \lr{l} - \iota_2 \lr{m} - \iota_3 \lr{n} - \iota_4 \lr{p},
\end{equation}
and
\begin{equation}
\label{spectral-distribution}
\mu_{\iota_1 \iota_2 \iota_3 \iota_4} (k, l, m, n, p) =
\int \overline{\psi(x, k)} \psi^{(\iota_1)}(x, l) \psi^{(\iota_2)}(x, m)
  \psi^{(\iota_3)}(x, n) \psi^{(\iota_4)}(x, p) dx.
\end{equation}
We call $\mu_{\iota_1 \iota_2 \iota_3 \iota_4} (k, l, m, n, p)$ as \textit{the nonlinear spectral distribution}.
We further decompose this spectral distribution into singular
and regular parts according to \eqref{generalized-eigenf-decomp},
and consider each part separately.

\smallskip
\subsection{Decomposition of the nonlinear spectral distribution}
Using the decomposition \eqref{generalized-eigenf-decomp},
we rewrite the nonlinear spectral distribution (\ref{spectral-distribution}) as
$$
(2 \pi)^{5/2} \mu_{\iota_1 \iota_2 \iota_3 \iota_4} (k, l, m, n, p) = \mu_{\iota_1 \iota_2 \iota_3 \iota_4}^S(k, l, m, n, p)
  + \mu_{\iota_1 \iota_2 \iota_3 \iota_4}^R(k, l, m, n, p),
$$
where $\mu^S$ and $\mu^R$ denote the `Singular' and `Regular' parts, respectively.
Below, we give the exact construction of each part.
The singular part is the sum of integrals of functions which oscillate
at either $-\infty$ or $+ \infty$. The regular part consists instead of integrals
localized or compactly supported functions.

Recalling \eqref{psi-singular-part-decomposition},
we define the singular part by
\begin{equation}
\label{spectral-singular-decomp}
\begin{aligned}
& \mu_{\iota_1 \iota_2 \iota_3 \iota_4}^S(k, l, m, n, p)
  = \mu_{\iota_1 \iota_2 \iota_3 \iota_4}^{+}(k, l, m, n, p)
  + \mu_{\iota_1 \iota_2 \iota_3 \iota_4}^{-}(k, l, m, n, p),
  \\[2pt]
& \mu_{\iota_1 \iota_2 \iota_3 \iota_4}^{*}(k, l, m, n, p) =
  \int \nu_*(x) \overline{\psi_{S,*}(x,k)} \psi^{(\iota_1)}_{S,*}(x,l)
  \psi^{(\iota_2)}_{S,*}(x,m) \psi^{(\iota_3)}_{S,*}(x,n) \psi^{(\iota_4)}_{S,*}(x,p) dx,
\end{aligned}
\end{equation}
where $\nu_*(x) = \sigma_*^5(x)$ for $* \in \{+, -\}$.

For the regular part, we define the set
\begin{equation*}
Z_R := \{(A_1, A_2, A_3, A_4, A_5): \exists j = 1,2,3,4,5 \text{ s.t. } A_j = R\} \subseteq \{S, R\}^5,
\end{equation*}
and write $\mu^R$ as the sum of two parts as follows:
\begin{equation*}
\mu^R_{\iota_1 \iota_2 \iota_3 \iota_4} (k,l,m,n,p)
  = \mu^{R,1}_{\iota_1 \iota_2 \iota_3 \iota_4} (k,l,m,n,p)
  + \mu^{R,2}_{\iota_1 \iota_2 \iota_3 \iota_4} (k,l,m,n,p)
\end{equation*}
where
\begin{equation}
\label{mu-r-1}
\mu^{R,1}_{\iota_1 \iota_2 \iota_3 \iota_4} = \sum_{(A,B,C,D,E) \in Z_R}
\int \overline{\psi_{A}(x,k)} \psi^{(\iota_1)}_{B}(x,l)
\psi^{(\iota_2)}_{C}(x,m) \psi^{(\iota_3)}_{D}(x,n)
\psi^{(\iota_4)}_{E}(x,p)dx,
\end{equation}
and
\begin{equation}
\label{mu-r-2}
\mu^{R,2}_{\iota_1 \iota_2 \iota_3 \iota_4} = \int \overline{\psi_{S}(x,k)} \psi^{(\iota_1)}_{S}(x,l)
\psi^{(\iota_2)}_{S}(x,m) \psi^{(\iota_3)}_{S}(x,n)
\psi^{(\iota_4)}_{S}(x,p)dx - \mu^S_{\iota_1 \iota_2 \iota_3 \iota_4}.
\end{equation}
Notice that $\mu^{R,1}$ is the sum of integrals of functions localized in $x$ around $0$,
and $\mu^{R,2}$ consists of integrals of functions compactly supported around $0$
due to the presence of product of the form $\sigma_+(x) \sigma_-(x)$.

We decompose accordingly \eqref{quartic-integral} and write
\begin{equation*}
16 e^{is\lr{k}} \widetilde{\mathcal{F}} \chi^4(s,k)
  = \mathcal{N}_+ (s,k) + \mathcal{N}_{-}(s,k) + \mathcal{N}_{R,1}(s,k) + \mathcal{N}_{R,2}(s,k),
\end{equation*}
where
\begin{equation}
\label{general-N-term}
\begin{aligned}
&(2 \pi)^{5/2}\mathcal{N}_*(s,k)
\\
&= \sum_{\iota_1, \iota_2, \iota_3, \iota_4 \in \{+, -\}} \iota_1 \iota_2 \iota_3 \iota_4 \int \frac{e^{is \Phi_{\iota_1 \iota_2 \iota_3 \iota_4}
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g^{(\iota_1)}(l) \widetilde g^{(\iota_2)}(m) \widetilde g^{(\iota_3)}(n) \widetilde g^{(\iota_4)}(p)
\mu_{\iota_1 \iota_2 \iota_3 \iota_4}^*
dlmnp.
\end{aligned}
\end{equation}
Hence, to prove Lemma \ref{lemma-quartic-term-bound} it suffices to show that bound
in \eqref{quartic-term-bound} holds for every term in the right-hand side of the following inequality:
\begin{equation}
\label{integral-decomposition-in-N}
\begin{aligned}
\left\| \int_0^t \partial_k \left( e^{is\lr{k}} \widetilde{\mathcal{F}} \chi^4(s) \right) ds \right\|_{L_k^2}
\lesssim &~ \normm{\int_0^t \partial_k \mathcal{N}_+(s,k) ds}{k}
  + \normm{\int_0^t \partial_k \mathcal{N}_-(s,k) ds}{k}
  \\[3pt]
& + \normm{\int_0^t \partial_k \mathcal{N}_{R,1}(s,k) ds}{k}
+ \normm{\int_0^t \partial_k \mathcal{N}_{R,2}(s,k) ds}{k}.
\end{aligned}
\end{equation}
In Subsection \ref{subsection-singular-N}, we prove the bound for the first
two terms in \eqref{integral-decomposition-in-N} corresponding to the singular part.
In Subsection \ref{subsection-regular-N}, we prove the bound for the rest of
the terms corresponding to the regular part.

\smallskip
\subsection{Estimates for the singular part}
\label{subsection-singular-N}
Our goal here is to show that for every $t \geq 0$, \begin{equation}
\label{quartic-singular-part-goal}
\normm{\int_0^t \partial_k \mathcal{N}_\pm (s,k) ds}{k} \lesssim \|v\|_X^4.
\end{equation}
We will provide the proof for $\mathcal{N}_+$ only, as $\mathcal{N}_-$ case can be treated similarly.
We first expand $\psi_{S, +}$ functions in \eqref{spectral-singular-decomp} as the linear combination in \eqref{singular-part-pm-2}, and rewrite $\mu^{S, +}$ as follows
\begin{equation}
\label{mu-S-plus-1}
\begin{aligned}
\mu_{\iota_1 \iota_2 \iota_3 \iota_4}^{S, +}(k, l, m, n, p)= \sqrt{2\pi}
\sum_{\epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3,\epsilon_4 \in \{+,-\}}& \overline{a_{+, \epsilon_0}(k)} ~ a_{+, \epsilon_1}^{(\iota_1)}(l) ~ a_{+, \epsilon_2}^{(\iota_2)}(m) ~ a_{+, \epsilon_3}^{(\iota_3)}(n) ~ a_{+, \epsilon_4}^{(\iota_4)}(p) \\
&\times \widehat{\nu_+} (\epsilon_0 k - \iota_1 \epsilon_1 l - \iota_2 \epsilon_2 m - \iota_3 \epsilon_3 n - \iota_4 \epsilon_4 p),
\end{aligned}
\end{equation}
where $\widehat{\nu_+}$ represents the (standard) Fourier transform of $\nu_+$.
Then, substituting the expression \eqref{mu-S-plus-1} into \eqref{general-N-term},
and introducing the multilinear form
{\small \begin{equation}
\label{multilinear-form}
\begin{aligned}
\mathcal{T}_{\b, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}^{\iota_1, \iota_2, \iota_3, \iota_4} (f_1, f_2, f_3, f_4) (k) := ( \iota_1 \iota_2 \iota_3 \iota_4) &\int  \frac{e^{is \Phi_{\iota_1 \iota_2 \iota_3 \iota_4}
}}{\lr{l} \lr{m} \lr{n} \lr{p}} f_1^{(\iota_1)} (l) f_2^{(\iota_2)} (m) f_3^{(\iota_3)} (n) f_4^{(\iota_4)} (p)
\\[3pt]
&\times \b (\epsilon_0 k - \iota_1 \epsilon_1 l - \iota_2 \epsilon_2 m - \iota_3 \epsilon_3 n - \iota_4 \epsilon_4 p) dlmnp
\end{aligned}
\end{equation}}
with $\iota_j, \epsilon_j \in \{+,-\}$ and a distribution $\b$,
the formula for $\mathcal{N}_+$ is equivalent to
\begin{equation}
\label{N-plus-in-multilinear}
\begin{aligned}
4 \pi^2 \mathcal{N}_+(s,k) =  \sum_{\substack{\iota_1, \iota_2, \iota_3, \iota_4 \in \{+, -\} \\
\epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3,\epsilon_4 \in \{+,-\}}}  \overline{a_{+, \epsilon_0}(k)}
~
\mathcal{T}_{\widehat{\nu_+}, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}^{\iota_1, \iota_2, \iota_3, \iota_4} (a_{+, \epsilon_1} \widetilde g, a_{+, \epsilon_2} \widetilde g, a_{+, \epsilon_3} \widetilde g, a_{+, \epsilon_4} \widetilde g)(k)
\end{aligned}
\end{equation}
To carry on the analysis of $\mathcal{N}_+$
we first analyze the properties of the multilinear form $\mathcal{T}$ in \eqref{multilinear-form}.
For simplicity, we consider the particular form given by
\begin{equation}
\label{particular-multilinear-form}
\mathcal{T}_{\b, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}(f_1, f_2, f_3, f_4)(k)
  := \mathcal{T}_{\b, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}^{+,+,+,+} (f_1, f_2. f_3, f_4) (k)
\end{equation}
as for the remaining cases, when at least one of
\iota_1, \iota_2, \iota_3, \iota_4$ is `$-$',
the analysis is the same.
We will also drop the dependence on $\epsilon_0$ when this is not relevant
(for example in Lemmas \ref{lemma-der-multilinear} and \ref{lemma-multilinear-in-physical})
with the understanding that we will be considering $\epsilon_0=+$.

The following algebraic lemma establishes an alternative representation
of $\partial_k \mathcal{T}$, which will be used later to derive the desired estimate \eqref{quartic-singular-part-goal}.

\begin{lemma}
\label{lemma-der-multilinear}
For $f_j \in \mathcal{S}$,
we have
\begin{equation}
\label{multilinear-form-derivative}
\begin{aligned}
\lr{k} \partial_k \mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)(k) =& ~i s \mathcal{T}_{q\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)(k)\\[1.5pt]
& + \epsilon_1 \mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (\lr{\cdot} \partial_l f_1, f_2, f_3, f_4)(k)\\[1.5pt]
& + \epsilon_2 \mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, \lr{\cdot} \partial_m f_2, f_3, f_4)(k)\\[1.5pt]
& + \epsilon_3 \mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, \lr{\cdot} \partial_n  f_3, f_4)(k)\\[1.5pt]
& + \epsilon_4 \mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, \lr{\cdot} \partial_p f_4)(k)\\[2pt]
& + \int \frac{\Phi~ e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} f_1(l) f_2(m) f_3(n) f_4(p) \partial_q b(q) dlmnp,
\end{aligned}
\end{equation}
where $\Phi := \Phi_{+,+,+,+}$, see \eqref{Phi-quartic},
and $q = k-\epsilon_1 l-\epsilon_2 m-\epsilon_3 n - \epsilon_4 p$.
\end{lemma}

\begin{proof}
Throughout the proof we drop the $\epsilon_1, \epsilon_2,\epsilon_3,\epsilon_4$ indices for simplicity.
Then, given $\epsilon_0 = \iota_1= \iota_2=\iota_3=\iota_4=+$
in \eqref{multilinear-form} from the choice of $\mathcal{T}_\b$,
we differentiate \eqref{multilinear-form} with respect to $k$ and obtain
\begin{equation}
\label{partial-k-T-1}
\lr{k} \partial_k \mathcal{T}_{\b} (f_1, f_2, f_3, f_4)(k) = is k \mathcal{T}_{\b} (f_1, f_2, f_3, f_4)(k) + \lr{k} \mathcal{T}_{\partial_q \b} (f_1, f_2, f_3, f_4)(k).
\end{equation}
Then, the expression \eqref{partial-k-T-1} can be further rewritten as
\begin{equation}
\label{partial-k-T-2}
\begin{aligned}
\lr{k} \partial_k \mathcal{T}_{\b} (f_1, f_2, f_3, f_4)(k)
= & is \mathcal{T}_{q \b} (f_1, f_2, f_3, f_4)(k) + \lr{k} \mathcal{T}_{\partial_q \b} (f_1, f_2, f_3, f_4)(k)
\\
& + \epsilon_1 is \mathcal{T}_{\b} (\cdot f_1, f_2, f_3, f_4)(k)
+ \epsilon_2is \mathcal{T}_{\b} (f_1, \cdot f_2, f_3, f_4)(k)\\
& + \epsilon_3is \mathcal{T}_{\b} (f_1, f_2, \cdot f_3, f_4)(k) + \epsilon_4 is\mathcal{T}_{\b} (f_1, f_2, f_3, \cdot f_4)(k),
\end{aligned}
\end{equation}
where $\cdot f$ denotes the multiplication of $f$ by its variable, $(\cdot f)(x) = x f(x)$.
Using $is l \lr{l}^{-1} e^{is \Phi} = - \partial_l (e^{is \Phi})$
and $q = k-\epsilon_1 l-\epsilon_2 m-\epsilon_3 n - \epsilon_4 p$,
we integrate the third term in \eqref{partial-k-T-2} by parts in $l$ to obtain
\begin{equation}
\label{T-b-l-1}
\begin{aligned}
is\mathcal{T}_\b (\cdot f_1, f_2, f_3,f_4)(k) &
= \int  \frac{is l \, e^{is \Phi}}{\lr{l} \lr{m} \lr{n} \lr{p}}
f_1 (l) f_2 (m) f_3 (n) f_4 (p) \b (q) dlmnp
\\
&= \int  \frac{ e^{is \Phi}}{\lr{m} \lr{n} \lr{p}} [\partial_l f_1 (l)] f_2 (m) f_3 (n) f_4 (p)\b ( q) dlmnp\\
& \quad + \int  \frac{ e^{is \Phi}}{\lr{m} \lr{n} \lr{p}}  f_1 (l) f_2 (m) f_3 (n) f_4 (p)\partial_l \b ( q) dlmnp\\[3pt]
& = \mathcal{T}_\b (\lr{\cdot} \partial_l f_1, f_2, f_3,f_4)(k) -\epsilon_1 \mathcal{T}_{\partial_q \b} (\lr{\cdot} f_1, f_2, f_3,f_4)(k),
\end{aligned}
\end{equation}
where we used the relation $\partial_l \b (q) = -\epsilon_1 \partial_q \b (q)$.
The last three terms in \eqref{partial-k-T-2} are manipulated similarly.
Then, we substitute to the last four terms in \eqref{partial-k-T-2}
their alternative expressions as in \eqref{T-b-l-1},
and obtain \eqref{multilinear-form-derivative}.
This way the terms corresponding to $\mathcal{T}_{\partial_q \b}$
are combined into the last line in \eqref{multilinear-form-derivative}.
\end{proof}

The following lemma will be used to transfer estimates
for the multilinear forms into a physical space.

\begin{lemma}
\label{lemma-multilinear-in-physical}
Given $f_j \in \mathcal{S}$, we have
\begin{equation}
\label{multi-linear-form-inverse-fourier}
\widehat{\mathcal{F}}^{-1} \left[ e^{-is\lr{k}}  \mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}
  (f_1, f_2, f_3, f_4)(k)\right] (x) = \widehat{\mathcal{F}}^{-1}[\b](x) \prod_{j =1,2,3,4} u_j(s, \epsilon_j x),
\end{equation}
where $u_j(s,x) = \lr{D}^{-1}e^{-is\lr{D}} \reallywidecheck{f_j}(x)$ with
$\lr{D} = \sqrt{-\partial_x^2+1}$ and $\reallywidecheck{f}$ the inverse Fourier transform.
\end{lemma}

The proof relies on explicit computations, and is left to the reader.

A direct application of Lemma \ref{lemma-multilinear-in-physical}
leads to
\begin{equation}
\label{L-2-multilinear-singular-part-bound-1}
\normm{\mathcal{T}_{\b, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)}{k}
  \lesssim \Big\| \widehat{\mathcal{F}}^{-1}[\b] \prod_{j =1,2,3,4} u_j(s, \epsilon_j \cdot) \Big\|_{L^2_x}
\end{equation}

\begin{remark}
Throughout the computations for the singular part,
the distribution $\b$ is replaced by $\widehat{\nu_+}$,
which is given by the formula
\begin{equation}
\label{nu-plus-fourier}
\widehat{\nu_+}(k) = \sqrt{\frac{\pi}{2}}\delta_0 + \frac{\widehat{\zeta}(k)}{ik}+\widehat{\overline{\omega}} (k),
\end{equation}
where $\zeta$ is an even $C_c^\infty$ function with integral $1$
and $\overline{\omega}$ denotes a (generic) $C_c^\infty$ function.
See for example \cite{GPR18}
\end{remark}

From the formula \eqref{N-plus-in-multilinear} we see that $\mathcal{N}_+$
is a sum of several terms.
The analysis for all these terms is similar and, therefore, in what follows
we consider only the terms with fixed $\iota_1,\iota_2,\iota_3,\iota_4=+$
and drop these from our notation.
Then, to prove \eqref{quartic-singular-part-goal}, it suffices to show that,
for every $\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4\in\{+,-\}$ we have
\begin{equation}
\label{quartic-singular-part-goal-2}
\Big\| \int_0^t \partial_k
  \sum_{\epsilon_0 \in \{+,-\}} \overline{a_{+, \epsilon_0}}(k) \,
  \mathcal{T}_{\widehat{\nu_+}, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)(k) \,
  ds \Big\|_{L^2_k} \lesssim \|v\|_X^4,
\end{equation}
where we set
\begin{equation*}
f_j(s,\cdot) = a_{+, \epsilon_j}(\cdot) \widetilde g (\cdot) \quad \text{for  } j=1,2,3,4.
\end{equation*}

Notice that the coefficients $a_{+, \epsilon_0}(k)$ have jump discontinuities
at $k=0$, see \eqref{a-functions},
which may, in principle, result in a singular $\delta$-type contribution when differentiated.
However, using the formulas \eqref{a-functions} and the vanishing at $k=0$
of $T(k)$ and $R_\pm(k)+1$, see \eqref{T-behaviour-around-zero}, one can see that this is not the case
for the summation in \eqref{quartic-singular-part-goal-2}.
More precisely, we first write
\begin{align*}
& \sum_{\epsilon_0 \in \{+,-\}} \overline{a_{+, \epsilon_0}}(k) \,
  \mathcal{T}_{\widehat{\nu_+}, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)
\\
& = \big[ \overline{a_{+,+}}(k) + \overline{a_{+,-}}(k) \big] \,
  \mathcal{T}_{\widehat{\nu_+}, +, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)
\\
& + \overline{a_{+,-}}(k) \,
  \Big[ \mathcal{T}_{\widehat{\nu_+}, -, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)
  - \mathcal{T}_{\widehat{\nu_+}, +, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4) \Big],
\end{align*}
and observe that the difference in the last line vanishes at $k=0$,
see  \eqref{particular-multilinear-form} and \eqref{multilinear-form}.
Then, we use that, for $\iota_0 \in \{+,-\}$,
$\partial_k (a_{\iota_0, +}(k) + a_{\iota_0,-}(k)) = \mathbf{1}_{\iota_0}(k) \partial_k T(\iota_0k)
  + \mathbf{1}_{-\iota_0}(k) \partial_k R_{\iota_0}(-\iota_0k)$.
Therefore, using also \eqref{uniform-estimate-T-R} and Minkowski's inequality we can bound
\begin{align*}
\Big\| \int_0^t \partial_k \sum_{\epsilon_0 \in \{+,-\}} \overline{a_{+, \epsilon_0}}
  ~ \mathcal{T}_{\widehat{\nu_+}, \epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4) ds \Big\|_{L^2_k}
  \lesssim \int_0^t \normm{\mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)}{k}ds
  \\
  + \Big\|  \int_0^t
  ~ \partial_k\mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4) ds \Big\|_{L^2_k}
:= E_1(t) + E_2(t).
\end{align*}
Below we give estimates for $E_1(t)$ and $E_2(t)$,
and show that each of them is bounded by $\n{v}_X^4$,
which would imply the bound \eqref{quartic-singular-part-goal}.

\subsubsection{Estimates for $E_1$}
We show that $E_1(t) \lesssim \n{v}_X^4$ for every $t \geq 0$.
Using \eqref{L-2-multilinear-singular-part-bound-1},
\begin{align}\label{E1}
\begin{split}
E_1(t) & \lesssim \int_0^t \normm{\nu_+ u_1(s,\epsilon_1 \cdot) u_2(s,\epsilon_2 \cdot) u_3(s,\epsilon_3 \cdot)
  u_4(s,\epsilon_4 \cdot)}{x} ds
  \\
& \lesssim \int_0^t \normm{u_1(s, \cdot)}{x} \|u_2(s, \cdot)\|_{L_x^\infty} \|u_3(s, \cdot)\|_{L_x^\infty}
  \|u_4(s, \cdot)\|_{L_x^\infty} ds,
\end{split}
\end{align}
where $u_j(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} (a_{+, \epsilon_j} \widetilde g)$.
The
formulas \eqref{a-functions} and the bounds in \eqref{uniform-estimate-T-R}
imply that for every $s \geq 0$ we have
\begin{equation}
\label{u-j-norm-for-E1}
\normm{u_j(s)}{x}
  \lesssim \normm{\widetilde g(s)}{k} \lesssim \n{v}_X.
\end{equation}
{Using \eqref{dispersive-estimates-standard-fourier} we can bound
\begin{align}\label{u-j-infty-norm}
\begin{split}
\norm{u_j(s)}{x}{\infty}
  \lesssim \lr{s}^{-1/2} \left( \big\| \partial_k \big( a_{+, \epsilon_j} \widetilde g(s) \big) \big\|_{L^2_k}
    + \|\lr{k}^2
    \widetilde g(s) \|_{L^2_k} \right);

\end{split}
\end{align}
using that $\widetilde g(s,0)=0$, we see that no singularity arises when differentiating $a_{+,\epsilon_j}$
and, therefore,
\begin{align}\label{dkawtg}
\big\| \partial_k \big( a_{+, \epsilon_j} \widetilde g(s) \big) \big\|_{L^2_k}
  \lesssim \big\| \partial_k \widetilde g(s) \big\|_{L^2_k};
\end{align}
then, from the definition \eqref{main-norm} and \eqref{u-j-infty-norm}-\eqref{dkawtg} we have
\begin{equation}
\label{u-j-infty-norm-for-E1}
\norm{u_j(s)}{x}{\infty} \lesssim \lr{s}^{-1/2} \n{v}_X.
\end{equation}
Using \eqref{u-j-norm-for-E1} and \eqref{u-j-infty-norm-for-E1} in \eqref{E1} we obtain}
\begin{equation*}
E_1(t) \lesssim \n{v}_X^4 \int_0^t \lr{s}^{-3/2} ds\lesssim \n{v}_X^4.
\end{equation*}

\subsubsection{Estimates for $E_2$}
We want to show that
for every $t \geq 0$,
\begin{equation}
\label{E2-bound-goal}
\normm{\int_0^t
~ \partial_k\mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}
  (f_1, f_2, f_3, f_4) ds}{k} \lesssim \n{v}_X^4.
\end{equation}
From Lemma \ref{lemma-der-multilinear}, the left-hand side of the above expression can be written as
\begin{equation}
\label{E2-decomposition}
\begin{aligned}
\Big\| \int_0^t
~ \partial_k & \mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4}  (f_1, f_2, f_3, f_4) ds \Big\|_{L_k^2}\\
& = \normm{\lr{k}^{-1} \int_0^t s
~ \mathcal{T}_{q\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4) ds}{k}\\
&\quad  + \normm{\lr{k}^{-1} \int_0^t \mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (\lr{\cdot}\partial_l f_1, f_2, f_3, f_4) ds}{k}\\
&\quad + \normm{\lr{k}^{-1} \int_0^t  \mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, \lr{\cdot} \partial_m f_2, f_3, f_4) ds}{k}\\
& \quad + \normm{\lr{k}^{-1} \int_0^t  \mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, \lr{\cdot} \partial_n f_3, f_4) ds}{k}\\
& \quad + \normm{\lr{k}^{-1} \int_0^t  \mathcal{T}_{\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, \lr{\cdot} \partial_p f_4) ds}{k}\\
& \quad + \normm{\lr{k}^{-1}\int_0^t \int \frac{\Phi~ e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} f_1(l) f_2(m) f_3(n) f_4(p) \partial_q \widehat{\nu_+} (q) dlmnpds}{k},\\[3pt]
& =: E_2^{(1)}(t) + E_2^{(2)}(t) + E_2^{(3)}(t) + E_2^{(4)}(t) + E_2^{(5)}(t) + E_2^{(6)}(t),
\end{aligned}
\end{equation}
where we recall that $q = k -\epsilon_1 l - \epsilon_2 m - \epsilon_3 n - \epsilon_4 p$.
Therefore, to prove \eqref{E2-bound-goal} it suffices to show that every $E_2^{(j)}$
in \eqref{E2-decomposition} is bounded by $\n{v}_X^4$.
This will be established in the lemmas below.

\begin{lemma}
\label{lemma-E2-1-bound}
For every $t \geq 0$, we have
\begin{equation}
\label{E2-1-bound}
E_2^{(1)}(t) \lesssim \n{v}_X^4.
\end{equation}
\end{lemma}

\begin{proof}
First, according to the Fourier representation of $\widehat{\nu_+}$ in \eqref{nu-plus-fourier}, we have
\begin{equation*}
q \widehat{\nu_+} = -i \widehat{\zeta}(q) + q \widehat{\overline{\omega}}(q) = \widehat{\Psi}(q)
\end{equation*}
with $\Psi$ being a Schwartz function.
In the estimates below we extract several $\lr{x}^{-2}$
factors from $\Psi$, and use them to apply the local $L_x^\infty$ decay estimates
\eqref{local-l-infty-decay-standard-Fourier}.
Then, using Minkowski inequality and \eqref{L-2-multilinear-singular-part-bound-1}, we have
\begin{equation*} \begin{aligned}
E_2^{(1)} & \lesssim \int_0^t s
\normm{\mathcal{T}_{q\widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} (f_1, f_2, f_3, f_4)}{k} ds\\
& \lesssim \int_0^t s \normm{\Psi u_1(s,\epsilon_1 \cdot) u_2(s,\epsilon_2 \cdot) u_3(s,\epsilon_3 \cdot) u_4(s,\epsilon_4 \cdot)}{x} ds\\
& \lesssim \int_0^t s
\|\lr{x}^{-2}u_1(s)\|_{L_x^\infty}
\|\lr{x}^{-2}u_2(s)\|_{L_x^\infty} \|\lr{x}^{-2}u_3(s)\|_{L_x^\infty} \|\lr{x}^{-2}u_4(s)\|_{L_x^\infty} ds,
\end{aligned}
\end{equation*}
where $u_j(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} (a_{+, \epsilon_j} \widetilde g)$
for every $j=1,2,3,4$.
Applying the local decay estimates \eqref{local-l-infty-decay-standard-Fourier},
using \eqref{a-functions} and the bounds in \eqref{uniform-estimate-T-R}
together with $\widetilde{g}(0)=0$,
we get
\begin{equation*}
E_2^{(1)} \lesssim \n{v}_X^4 \int_0^t s \lr{s}^{-3} ds \lesssim  \n{v}_X^4.
\end{equation*}

\end{proof}

\begin{lemma}
\label{lemma-E2-j-bound}
For every $t \geq 0$ and $j=2,3,4,5$, we have
\begin{equation}
\label{E2-j-bound}
E_2^{(j)}(t) \lesssim \n{v}_X^4.
\end{equation}
\end{lemma}

\begin{proof}
The proofs for $j=2,3,4,5$ are identical so we only prove \eqref{E2-j-bound} for $j=2$.
Using \eqref{L-2-multilinear-singular-part-bound-1}
we can estimate as follows:
\begin{equation*}
\begin{aligned}
E_2^{(2)}(t) \lesssim \int_0^t \normm{u_1(s, \cdot)}{x} \|u_2(s, \cdot)\|_{L_x^\infty}
  \|u_3(s, \cdot)\|_{L_x^\infty} \|u_4(s, \cdot)\|_{L_x^\infty} ds,
\end{aligned}
\end{equation*}
where
\begin{equation*}
\left\{
\begin{array}{l}
u_1(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} (\lr{\cdot} \partial_l [a_{+, \epsilon_1} \widetilde g]),
\\
u_r(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} (a_{+, \epsilon_r} \widetilde g) \quad \text{ for  } r=2,3,4.
\end{array}
\right.
\end{equation*}
For $u_1$ we have
$\normm{u_1(s)}{x} \lesssim \n{\widetilde g}_{H_l^1} \lesssim \n{v}_X$,
while the $L_x^\infty$ norms of $u_r$ for $r=2,3,4$
are bounded similarly to \eqref{u-j-infty-norm-for-E1}.
Combining these estimates, we get the desired result
\begin{equation*}
E_2^{(2)} \lesssim \n{v}_X^4 \int_0^t \lr{s}^{-3/2} ds \lesssim \n{v}_X^4.
\end{equation*}
\end{proof}

The next lemma will complete the proof of \eqref{E2-bound-goal}.

\begin{lemma}
\label{lemma-E2-6-bound}
For every $t \geq 0$, we have
\begin{equation}
\label{E2-6-bound}
E_2^{(6)}(t) \lesssim \n{v}_X^4.
\end{equation}
\end{lemma}

\begin{proof}
We first consider the integral
\begin{equation*}
\mathcal{I} := \int_0^t \int \frac{\Phi~ e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} f_1(l) f_2(m) f_3(n) f_4(p) \partial_q \widehat{\nu_+} (q) dlmnp~ds,
\end{equation*}
where $q = k -\epsilon_1 l - \epsilon_2 m -\epsilon_3 n -\epsilon_4 p$.
Using that $\Phi e^{is\Phi} = -i \partial_s e^{is\Phi}$, we integrate $\mathcal{I}$ by parts in $s$ to get
\begin{equation}
\label{I-in-E2-6-decomposition}
\begin{aligned}
\mathcal{I} & = - i \int \frac{ e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} f_1(l) f_2(m) f_3(n) f_4(p) \partial_q \widehat{\nu_+} (q) dlmnp \Bigg|_{s=0}^{s=t}\\
& \quad + i \int_0^t \int \frac{e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} \partial_s [f_1(l) f_2(m) f_3(n) f_4(p)] \partial_q \widehat{\nu_+} (q) dlmnp ~ds\\[5pt]
&=: -i \mathcal{I}_1(t,k) + i \mathcal{I}_2(t,k).
\end{aligned}
\end{equation}
Both integrals contain the term $\partial_q \widehat{\nu_+} (q)$, which we treat using the relations
\begin{equation}
\label{prtl-nu-plus}
\partial_q \widehat{\nu_+} (q) = - \epsilon_1 \partial_l \widehat{\nu_+} (q)
  = - \epsilon_2 \partial_m \widehat{\nu_+} (q) = - \epsilon_3 \partial_n \widehat{\nu_+} (q)
  = - \epsilon_4 \partial_p \widehat{\nu_+} (q)
\end{equation}
to integrate by parts and deal with $\widehat{\nu_+} (q)$ only.

We look at
$\mathcal{I}_1$ first.
From \eqref{prtl-nu-plus}, we integrate by parts in $l$, and obtain
\begin{equation}
\label{I1-in-E2-6-decomposition}
\begin{aligned}
\mathcal{I}_1 & =\epsilon_1 \int \frac{ e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} \partial_l f_1(l) f_2(m) f_3(n) f_4(p) \widehat{\nu_+} (q) dlmnp \Bigg|_{s=0}^{s=t}\\
& \quad - \epsilon_1 \int \frac{l  e^{is\Phi}}{\lr{l}^3\lr{m}\lr{n} \lr{p}} f_1(l) f_2(m) f_3(n) f_4(p) \widehat{\nu_+} (q) dlmnp \Bigg|_{s=0}^{s=t}\\
&\quad - i \epsilon_1  \int \frac{s l  e^{is\Phi} }{\lr{l}^2\lr{m}\lr{n} \lr{p}} f_1(l) f_2(m) f_3(n) f_4(p) \widehat{\nu_+} (q) dlmnp \Bigg|_{s=0}^{s=t}\\[3pt]
&=: \mathcal{I}_1^{(1)}(t,k) + \mathcal{I}_1^{(2)}(t,k) +\mathcal{I}_1^{(3)}(t,k).
\end{aligned}
\end{equation}
The integrals above are similar in structure, and, in fact, can be treated in the same way.
We only show how to deal with $\mathcal{I}_1^{(3)}$ as it contain the potentially
problematic factor $s$.
Also, we drop the endpoint values of $s$ and, abusing notations, only work with
\begin{equation*}
\mathcal{I}_1^{(3)}(s,k) := - i \epsilon_1  \int \frac{s l  e^{is\Phi} }{\lr{l}^2\lr{m}\lr{n} \lr{p}}
  f_1(l) f_2(m) f_3(n) f_4(p) \widehat{\nu_+} (q) dlmnp.
\end{equation*}
From \eqref{particular-multilinear-form}, we have
\begin{equation*}
\mathcal{I}_1^{(3)}(s,k) = -i \epsilon_1 s \mathcal{T}_{ \widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} \left(\frac{\cdot}{\lr{\cdot}} f_1, f_2, f_3, f_4 \right).
\end{equation*}
Then, using \eqref{L-2-multilinear-singular-part-bound-1},
\begin{equation*}
\begin{aligned}
\normm{\mathcal{I}_1^{(3)}(s)}{k} & \lesssim s \normm{\nu_+ u_1(s, \epsilon_1\cdot) u_2(s, \epsilon_2 \cdot) u_3(s, \epsilon_3 \cdot) u_4(s, \epsilon_4 \cdot)}{x}\\
& \lesssim s \normm{u_1(s, \cdot)}{x} \|u_2(s, \cdot)\|_{L_x^\infty} \|u_3(s, \cdot)\|_{L_x^\infty} \|u_4(s, \cdot)\|_{L_x^\infty}
\end{aligned}
\end{equation*}
where
\begin{equation*}
\left\{
\begin{array}{l}
u_1(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} \left(\frac{\cdot}{\lr{\cdot}} a_{+, \epsilon_1} \widetilde g\right),\\
u_r(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} (a_{+, \epsilon_r} \widetilde g) \quad \text{ for  } r=2,3,4.
\end{array}
\right.
\end{equation*}
Then,
the dispersive decay estimates \eqref{dispersive-estimates-standard-fourier} imply
\begin{equation*}
\normm{\mathcal{I}_1^{(3)}(s)}{k} \lesssim s \lr{s}^{-3/2} \n{v}_X^4 \lesssim \n{v}_X^4.

\end{equation*}
Similar steps repeated for $\mathcal{I}_1^{(1)}(t,k)$ and $\mathcal{I}_1^{(2)}(t,k)$
from \eqref{I1-in-E2-6-decomposition} give the same bound $\n{v}_X^4$ for the integrals.
Combining such estimates we have $\normm{\mathcal{I}_1(t)}{k} \lesssim \n{v}_X^4$ for every $t\geq 0$, as desired.

It remains to verify $\normm{\mathcal{I}_2(t)}{k} \lesssim \n{v}_X^4$ for $\mathcal{I}_2$
given in \eqref{I-in-E2-6-decomposition}. As in previous computations,
based on \eqref{prtl-nu-plus} we use $\partial_q \widehat{\nu_+}$
to integrate by parts in one of the integration variables $l, m, n$ or $p$.
The choice of the variable
depends on what function $\partial_s$ hits;
more precisely,
when integrating by parts, we want
avoid hitting the function of the type $\partial_s f_j$.
Applying $\partial_s$ to the product of functions in the $\mathcal{I}_2$
expression in \eqref{I-in-E2-6-decomposition}, we obtain four terms which can all be treated
in the same way, so we only look at one of them:
\begin{equation}
\label{I2-in-E2-6-decomposition}
\begin{aligned}
\mathcal{I}_2^{(1)}(t,k)
  & := \int_0^t \int \frac{e^{is\Phi}}{\lr{l}\lr{m}\lr{n} \lr{p}} [\partial_s f_1(l)]
  f_2(m) f_3(n) f_4(p) \partial_q \widehat{\nu_+} (q) dlmnp~ds
.
\end{aligned}
\end{equation}
We use \eqref{prtl-nu-plus}
to integrate by parts in $m$, and using \eqref{particular-multilinear-form} we write
\begin{equation*}
\begin{aligned}
\mathcal{I}_2^{(1)}(t,k) & = \epsilon_2 \int_0^t \mathcal{T}_{ \widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} \left(\partial_s f_1, \partial_m f_2, f_3, f_4 \right) ds\\
& \quad - \epsilon_2 \int_0^t \mathcal{T}_{ \widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} \left(\partial_s f_1, \frac{\cdot}{\lr{\cdot}^2} f_2, f_3, f_4 \right) ds\\
& \quad - i \epsilon_2 \int_0^t s \mathcal{T}_{ \widehat{\nu_+}, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4} \left(\partial_s f_1, \frac{\cdot}{\lr{\cdot}} f_2, f_3, f_4 \right) ds
=: J_1(t,k) + J_2(t,k) + J_3(t,k).
\end{aligned}
\end{equation*}
Note that all three integrals contain the term $\partial_s f_1$,
and we eventually use it to get $\lr{s}^{-3/2}$ decay via \eqref{prtl-t-wg-estimates}
which allows to overcome the factor $s$ in the integral $J_3$ without any difficulties.

For the first integral we use
\eqref{L-2-multilinear-singular-part-bound-1},
\begin{equation}
\label{J1-in-E2-6-bound-1}
\begin{aligned}
\normm{J_1(t)}{k} & \lesssim \int_0^t \normm{\nu_+ u_1(s, \epsilon_1\cdot) u_2(s, \epsilon_2 \cdot) u_3(s, \epsilon_3 \cdot) u_4(s, \epsilon_4 \cdot)}{x} ds\\
& \lesssim \int_0^t \|u_1(s, \cdot)\|_{L_x^\infty} \normm{u_2(s, \cdot)}{x}  \|u_3(s, \cdot)\|_{L_x^\infty} \|u_4(s, \cdot)\|_{L_x^\infty} ds
\end{aligned}
\end{equation}
where
\begin{equation*}
\left\{
\begin{array}{l}
u_1(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} (a_{+, \epsilon_1} \partial_s \widetilde g),\\
u_2(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} ( \partial_m [a_{+, \epsilon_2} \widetilde g]),\\
u_r(s,x) = \lr{D}^{-1} e^{-is \lr{D}} \widehat{\mathcal{F}}^{-1} ( a_{+, \epsilon_r} \widetilde g) \quad \text{ for  } r=3,4.
\end{array}
\right.
\end{equation*}
To bound the $L_x^\infty$ norm of $u_1$
we use the simple relation $\norm{u_1(s)}{x}{\infty} \lesssim \normm{\lr{l} \widehat{u}_1}{l}$,
which
implies
\begin{equation}
\label{J1-u-for-prtl_s-bound}
\norm{u_1(s)}{x}{\infty} \lesssim \normm{\partial_s \widetilde g}{l} \lesssim \lr{s}^{-3/2} \n{v}_X^2
\end{equation}
according to the bound in \eqref{prtl-t-wg-estimates}.
From the dispersive decay estimates for the case of the standard Fourier transform
\eqref{dispersive-estimates-standard-fourier},
\eqref{uniform-estimate-T-R}, and
\eqref{dkawtg},
we have
\begin{equation}
\label{J1-other-u-bound}
\normm{u_2(s)}{x} \lesssim \n{v}_X \quad \text{and} \quad \|u_r(s)\|_{L_x^\infty}\lesssim \lr{s}^{-1/2} \n{v}_X.
\end{equation}
Using \eqref{J1-u-for-prtl_s-bound} and \eqref{J1-other-u-bound} into \eqref{J1-in-E2-6-bound-1}, we obtain
\begin{equation}
\label{J1-in-E2-6-bound}
\normm{J_1(t)}{k} \lesssim \n{v}_X^5 \int_0^t \lr{s}^{-5/2} ds \lesssim \n{v}_X^4.
\end{equation}

The remaining integrals $J_2$ and $J_3$ can be treated in the same way with appropriate
modifications for $u_2(s,x)$.
In both cases, the $L_x^2$ norm of $u_2(s)$ is bounded by $\n{v}_X$,
and this lead to a $\n{v}_X^4$ bound for both $\normm{J_2(t)}{k}$ and $\normm{J_3(t)}{k}$, as desired.
Combining these estimates, we obtain $\|\mathcal{I}_2(t)\|_{L^2_k} \lesssim \n{v}_X^4$.
Together with the previously obtained result for $\mathcal{I}_1$,
we get the desired estimate \eqref{E2-6-bound}.
\end{proof}

\smallskip
\subsection{Estimates for the regular part}
\label{subsection-regular-N}
Our goal here is to show that for every $t \geq 0$, \begin{equation}
\label{quartic-regular-part-goal}
\normm{\int_0^t \partial_k \mathcal{N}_{R, j} (s,k) ds}{k} \lesssim \|v\|_X^4 \quad \text{for  } j = 1,2.
\end{equation}
We separate the analysis for $j=1$ and $j=2$. Recall the regular parts of the nonlinear spectral distribution given by \eqref{mu-r-1}--\eqref{mu-r-2} and the expression for $\mathcal{N}_{R,j}$ from \eqref{general-N-term}.

\subsubsection{Estimates for $\mathcal{N}_{R, 1}$}
Based on \eqref{general-N-term}, the expression for $\mathcal{N}_{R, 1}$ reads
\begin{equation}
\label{N-r-1-explicit}
\begin{aligned}
&(2 \pi)^{5/2}\mathcal{N}_{R,1}(s,k) \\
&= \sum_{\iota_1, \iota_2, \iota_3, \iota_4 \in \{+, -\}} \iota_1 \iota_2 \iota_3 \iota_4 \int \frac{e^{is \Phi_{\iota_1 \iota_2 \iota_3 \iota_4}
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g^{(\iota_1)}(l) \widetilde g^{(\iota_2)}(m) \widetilde g^{(\iota_3)}(n) \widetilde g^{(\iota_4)}(p)
\mu_{\iota_1 \iota_2 \iota_3 \iota_4}^{R,1}
dlmnp.
\end{aligned}
\end{equation}
The analysis for each term in the above sum is similar,
therefore we may only consider the term with $\iota_1 = \iota_2 = \iota_3 = \iota_4 = +$.
Next, we recall that $\mu_{++++}^{R,1}$ defined in \eqref{mu-r-1} is the sum of multiple integrals.
Every integral in the sum \eqref{mu-r-1} has at least one of the indices in $(A,B,C,D,E)$ equal to $R$;
they are similar in nature, and are treated essentially in the same way.
In particular, it is sufficient to consider only two terms:
\begin{itemize}
    \item the term with $A=R$;
    \item the term with $A=S, B=R$.
\end{itemize}
For these terms, the analysis is independent on the rest of the indices.
so we do not specify them explicitly as $R$ and $S$. In other words, we consider
\begin{equation}
\label{mu-r-1-1}
\mu^{R,1,1} := \int \overline{\psi_{R}(x,k)} \psi_{M_1}(x,l)
\psi_{M_2}(x,m) \psi_{M_3}(x,n)
\psi_{M_4}(x,p)dx, \quad M_j \in \{S, R\},
\end{equation}
and
\begin{equation}
\label{mu-r-1-2}
\mu^{R,1,2} := \int \overline{\psi_{S}(x,k)} \psi_{M_1}(x,l)
\psi_{M_2}(x,m) \psi_{M_3}(x,n)
\psi_{R}(x,p)dx, \quad M_j \in \{S, R\}.
\end{equation}
The corresponding integrals from \eqref{N-r-1-explicit} are
\begin{equation*}
\begin{aligned}
\mathcal{N}_{R,1,1}(s,k) = \int \frac{e^{is \Phi
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g(l) \widetilde g(m) \widetilde g(n) \widetilde g(p)
\mu^{R,1,1}
dlmnp,
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\mathcal{N}_{R,1,2}(s,k) = \int \frac{e^{is \Phi
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g(l) \widetilde g(m) \widetilde g(n) \widetilde g(p)
\mu^{R,1,2}
dlmnp,
\end{aligned}
\end{equation*}
where $\Phi = \Phi_{++++}$ given by \eqref{Phi-quartic}.
After applying the $\partial_k$ differentiation to the above integral, we get for $j=1,2$
\begin{equation}
\label{mathcal-A-B}
\begin{aligned}
\partial_k \mathcal{N}_{R,1,j}(s,k) & = is \frac{k}{\lr{k}} \int \frac{e^{is \Phi
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g(l) \widetilde g(m) \widetilde g(n) \widetilde g(p)
\mu^{R,1,j}
dlmnp\\
&\quad + \int \frac{e^{is \Phi
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g(l) \widetilde g(m) \widetilde g(n) \widetilde g(p) \partial_k
\mu^{R,1,j}
dlmnp\\
& =: \mathcal{A}_j(s,k) + \mathcal{B}_j(s,k).
\end{aligned}
\end{equation}
To show the bound in \eqref{quartic-regular-part-goal}
it suffices to verify the bound for the $L_k^2$ norms of $ds$ integral
associated with $\mathcal{A}_j$ and $\mathcal{B}_j$ from \eqref{mathcal-A-B}.
We show these estimates in the following lemmas.

\begin{lemma}
\label{lemma-A1-bound}
For every $t\geq 0$ we have
\begin{equation*}
\normm{\int_0^t \mathcal{A}_1(s, k) ds}{k} \lesssim \n{v}_X^4.
\end{equation*}
\end{lemma}

\begin{proof}
We substitute $\mu^{R,1,1}$ from \eqref{mu-r-1-1} into the expression for $\mathcal{A}_1$, and expand the function $\Phi=\Phi_{++++}$ according to its definition in \eqref{Phi-quartic}. As a result, using \eqref{h-sing-reg-decomp}, we can rewrite $\mathcal{A}_1$ as
\begin{equation*}
\mathcal{A}_1(s,k) = 4i s \pi^2 \frac{k}{\lr{k}}e^{is \lr{k}} \int \overline{\psi_R(x,k)} u_{M_1}(x) u_{M_2}(x) u_{M_3}(x) u_{M_4}(x) dx,
\end{equation*}
where $u(x)= \lr{\widetilde D}^{-1} e^{-is\lr{\widetilde D}} g(s, x)$.
Then, using the estimate \eqref{pseudo-psi-regular-bound}, we have
\begin{equation*}
\begin{aligned}
\normm{\int_0^t \mathcal{A}_1(s, k) ds}{k} & \lesssim \int_0^t \normm{ \mathcal{A}_1(s, k)}{k} ds \lesssim \int_0^t s \normm{\lr{x}^{-11} u_{M_1} u_{M_2} u_{M_3} u_{M_4}}{x} ds\\
& \lesssim \int_0^t s \norm{\lr{x}^{-2} u_{M_1}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_2}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_3}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_4}}{x}{\infty} ds.
\end{aligned}
\end{equation*}
From \eqref{local-l-infty-decay-singular} and \eqref{local-l-infty-decay-regular}, each of the above  $L_x^\infty$ norms is bounded by $\lr{s}^{-1} \n{\widetilde{g}}_{H_k^1}$, and this completes the proof.
\end{proof}

\begin{lemma}
\label{lemma-B1-bound}
For every $t\geq 0$ we have
\begin{equation*}
\normm{\int_0^t \mathcal{B}_1(s, k) ds}{k} \lesssim \n{v}_X^4.
\end{equation*}
\end{lemma}

\begin{proof}
The proof is similar to the proof of Lemma \ref{lemma-A1-bound}.
We rewrite $\mathcal{B}_1$ as
\begin{equation*}
\mathcal{B}_1(s,k) = 4 \pi^2 e^{is \lr{k}} \int \partial_k \overline{\psi_R(x,k)} u_{M_1}(x) u_{M_2}(x) u_{M_3}(x) u_{M_4}(x) dx,
\end{equation*}
where $u(x)= \lr{\widetilde D}^{-1} e^{-is\lr{\widetilde D}} g(s, x)$. Then the desired estimate follows from \eqref{pseudo-dk-psi-regular-bound}, \eqref{local-l-infty-decay-singular} and \eqref{local-l-infty-decay-regular}.
\end{proof}

\begin{lemma}
\label{lemma-A2-bound}
For every $t\geq 0$ we have
\begin{equation*}
\normm{\int_0^t \mathcal{A}_2(s, k) ds}{k} \lesssim \n{v}_X^4.
\end{equation*}
\end{lemma}

\begin{proof}
Using \eqref{h-sing-reg-decomp}, we rewrite $\mathcal{A}_2$ as
\begin{equation*}
\mathcal{A}_2(s,k) = 4i s \pi^2 \frac{k}{\lr{k}}e^{is \lr{k}} \int \overline{\psi_S(x,k)} u_{M_1}(x) u_{M_2}(x) u_{M_3}(x) u_{R}(x) dx,
\end{equation*}
where $u(x)= \lr{\widetilde D}^{-1} e^{-is\lr{\widetilde D}} g(s, x)$.
From the definition of $\psi_S$ in \eqref{psi-singular-part-decomposition}, one can verify that
\begin{equation}
\label{psi-sing-transform-plancherel}
\normm{\int \overline{\psi_S(x,k)} h(x)~dx}{k} \lesssim \normm{h}{x}.
\end{equation}
Then, applying \eqref{psi-sing-transform-plancherel} and \eqref{local-l-infty-decay-regular} we get
\begin{equation*}
\begin{aligned}
\normm{\int_0^t \mathcal{A}_2(s, k) ds}{k} & \lesssim \int_0^t s \normm{u_{M_1} u_{M_2} u_{M_3} u_{R}}{x} ds\\
& \lesssim \int_0^t s \norm{\lr{x}^{-2} u_{M_1}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_2}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_3}}{x}{\infty} \norm{\lr{x}^{10} u_{R}}{x}{\infty} ds \lesssim \n{v}_X^4
\end{aligned}
\end{equation*}
as desired.
\end{proof}

\begin{lemma}
\label{lemma-B2-bound}
For every $t\geq 0$ we have
\begin{equation*}
\normm{\int_0^t \mathcal{B}_2(s, k) ds}{k} \lesssim \n{v}_X^4.
\end{equation*}
\end{lemma}

\begin{proof}
The proof is similar to the proof of Lemma \ref{lemma-A2-bound}.
We rewrite $\mathcal{B}_2$ as
\begin{equation*}
\mathcal{B}_2(s,k) = 4 \pi^2 e^{is \lr{k}} \int \partial_k \overline{\psi_S(x,k)} u_{M_1}(x) u_{M_2}(x) u_{M_3}(x) u_{R}(x) dx,
\end{equation*}
where $u(x)= \lr{\widetilde D}^{-1} e^{-is\lr{\widetilde D}} g(s, x)$.
Similarly to \eqref{psi-sing-transform-plancherel}, one can verify
\begin{equation}
\label{dk-psi-sing-transform-plancherel}
\normm{\int \partial_k \overline{ \psi_S(x,k)} h(x)~dx}{k} \lesssim \normm{\lr{x}h}{x}.
\end{equation}
Then, we obtain
\begin{equation*}
\begin{aligned}
\normm{\int_0^t \mathcal{B}_2(s, k) ds}{k} & \lesssim \int_0^t s \normm{\lr{x} u_{M_1} u_{M_2} u_{M_3} u_{R}}{x} ds\\
& \lesssim \int_0^t s \norm{\lr{x}^{-2} u_{M_1}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_2}}{x}{\infty} \norm{\lr{x}^{-2} u_{M_3}}{x}{\infty} \norm{\lr{x}^{10} u_{R}}{x}{\infty} ds \lesssim \n{v}_X^4.
\end{aligned}
\end{equation*}
\end{proof}
This completes the proof of \eqref{quartic-regular-part-goal} for $j=1$.

\smallskip
\subsubsection{Estimates for $\mathcal{N}_{R, 2}$}
Based on \eqref{general-N-term}, the expression for $\mathcal{N}_{R, 2}$ reads
\begin{equation}
\label{N-r-2-explicit}
\begin{aligned}
&(2 \pi)^{5/2}\mathcal{N}_{R,2}(s,k) \\
&= \sum_{\iota_1, \iota_2, \iota_3, \iota_4 \in \{+, -\}} \iota_1 \iota_2 \iota_3 \iota_4 \int \frac{e^{is \Phi_{\iota_1 \iota_2 \iota_3 \iota_4}
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g^{(\iota_1)}(l) \widetilde g^{(\iota_2)}(m) \widetilde g^{(\iota_3)}(n) \widetilde g^{(\iota_4)}(p)
\mu_{\iota_1 \iota_2 \iota_3 \iota_4}^{R,2}
dlmnp.
\end{aligned}
\end{equation}
As previously, we only consider the contribution from $\mu_{++++}^{R,2}$.
Then, using \eqref{psi-singular-part-decomposition}, we can rewrite \eqref{mu-r-2} as
\begin{equation*}
\begin{aligned}
&\mu_{++++}^{R,2}(k,l,m,n,p)
\\
& = \sum \int \sigma_{\epsilon_0}(x) \overline{\psi_{S, \epsilon_0}}(x,k) \sigma_{\epsilon_1}(x)
  \psi_{S, \epsilon_1}(x,l) \sigma_{\epsilon_2}(x)\psi_{S, \epsilon_2}(x,m) \sigma_{\epsilon_3}(x)
  \psi_{S, \epsilon_3}(x,n) \sigma_{\epsilon_4}(x)\psi_{S, \epsilon_4}(x,p) dx,
\end{aligned}
\end{equation*}
where the sum is over
\begin{equation*}
\epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4\in \{+,-\} \quad \text{with   }  (\epsilon_0, \epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4) \neq (+,+,+,+,+), (-,-,-,-,-).
\end{equation*}
Note that the integrand in each integral is compactly supported due
to the presence of a $\sigma_+(x)\sigma_-(x)$ factor.
The structure of the integrals is similar, and, therefore,
it suffices to consider only the measure
\begin{equation}
\label{mu-r-2-1}
\mu^{R,2,1}(k,l,m,n,p) := \int \sigma_{+}(x) \overline{\psi_{S, +}(x,k)} \sigma_{-}(x)\psi_{S, -}(x,l) \psi_{S}(x,m) \psi_{S}(x,n) \psi_{S}(x,p) dx,
\end{equation}
and the corresponding integral
\begin{equation*}
\begin{aligned}
\mathcal{N}_{R,2,1}(s,k) = \int \frac{e^{is \Phi
}}{\lr{l} \lr{m} \lr{n} \lr{p}} \widetilde g(l) \widetilde g(m) \widetilde g(n) \widetilde g(p)
\mu^{R,2,1}(k,l,m,n,p)
dlmnp \qquad \text{with   } \Phi= \Phi_{++++}.
\end{aligned}
\end{equation*}
Using \eqref{h-sing-reg-decomp} and \eqref{mu-r-2-1}, we can rewrite
\begin{equation}
\label{N-r-2-1}
\mathcal{N}_{R,2,1}(s,k) = 4\pi^2 e^{is\lr{k}} \int \sigma_+(x) \sigma_-(x)
  \overline{\psi_{S,+}(x,k)} u_-(x) u_S^3(x) dx,
\end{equation}
where $u(x)= \lr{\widetilde D}^{-1} e^{-is\lr{\widetilde D}} g(s, x)$ and $u_-$ stands for
\begin{equation*}
u_-(x) := \frac{1}{\sqrt{2\pi}} \int_\mathbb{R} \psi_{S,-}(x,k) \widetilde{u}(k) dk.
\end{equation*}
We apply the $\partial_k$ differentiation to \eqref{N-r-2-1} and obtain
\begin{equation*}
\begin{aligned}
\partial_k \mathcal{N}_{R,2,1}(s,k) &= 4is \pi^2 \frac{k}{\lr{k}} e^{is\lr{k}}
  \int \sigma_+(x) \sigma_-(x) \, \overline{\psi_{S,+}(x,k)} u_-(x) u_S^3(x) dx
  \\
&\quad + 4\pi^2 e^{is\lr{k}} \int \sigma_+(x) \sigma_-(x) \partial_k
  \,\overline{\psi_{S,+}(x,k)} u_-(x) u_S^3(x) dx
\\
& = : \mathcal{C}_1 + \mathcal{C}_2.
\end{aligned}
\end{equation*}
Both terms $\mathcal{C}_1$ and $\mathcal{C}_2$ are treated using
the presence of the compactly supported function $\sigma_+(x) \sigma_-(x)$
and the local decay estimates .
We also observe that, similarly to \eqref{psi-sing-transform-plancherel}
and \eqref{dk-psi-sing-transform-plancherel}, we have
\begin{equation*}
\normm{\int \partial_k^r \overline{\psi_{S,+}(x,k)} h(x)~dx}{k} \lesssim \normm{\lr{x}^r h}{x}, \qquad r =0,1.
\end{equation*}
Then, using \eqref{local-l-infty-decay-singular} and
an analogous \eqref{local-l-infty-decay-singular}-type estimate
for $u_-$, we have
\begin{equation*}
\begin{aligned}
\normm{\int_0^t \mathcal{C}_1(s, k) ds}{k} & \lesssim \int_0^t s \normm{\sigma_- \sigma_+ u_{-} u_S^3}{x} ds
\\
& \lesssim \int_0^t s \norm{\lr{x}^{-2} u_-}{x}{\infty} \norm{\lr{x}^{-2} u_{S}}{x}{\infty}^3 ds \lesssim \n{v}_X^4.
\end{aligned}
\end{equation*}
Similar steps ensure that $\normm{\int_0^t \mathcal{C}_2(s, k) ds}{k} \lesssim \n{v}_X^4$.
Combining the above estimates, we get
\begin{equation*}
\normm{\int_0^t \partial_k \mathcal{N}_{R,2,1}(s, k) ds}{k} \lesssim \n{v}_X^4
\end{equation*}
which completes the proof \eqref{quartic-regular-part-goal} for $j=2$.

\subsubsection{Proof of Lemma \ref{lemma-quartic-term-bound}}
The proof follows from \eqref{integral-decomposition-in-N} and the estimates
\eqref{quartic-singular-part-goal} and \eqref{quartic-regular-part-goal}.

\iffalse
\subsection{Global existence}
This part follows from Theorem \eqref{thmSM} as long as we verify that every even
$\bm{\gamma}$ satisfying \eqref{gamma-12-form}--\eqref{initial-data-gamma} is in $\mathcal{A}(C\epsilon_0)$ with some uniform $\epsilon_0$-independent constant $C$. Indeed, \eqref{gamma-12-form} implies that $\lr{\bm{\gamma}, Z_+} = 0$, and to show $\n{\bm{\gamma}}_{H_x^1 \times L_x^2} \lesssim \epsilon_0$ we use the estimates \eqref{initial-data-gamma} as follows:
\begin{equation}
\begin{aligned}
&\n{\gamma_1}_{H_x^1} \lesssim |b| + \n{\zeta_1}_{H_x^1} \lesssim |b| + \n{\sqrt{H+1} \zeta_1}_{L_x^2} \lesssim \epsilon_0,\\[2pt]
&\n{\gamma_2}_{L_x^2} \lesssim |b| + \n{\zeta_2}_{L_x^2} \lesssim \epsilon_0.
\end{aligned}
\end{equation}
Hence, the global existence result is proven, and the form of the solution follows from \eqref{v-decomp}.

Note that if we express the initial conditions $(u_0,u_1) \in \mathcal{M}_0(\epsilon_0)$ as
\begin{equation*}
\begin{array}{l}
u_0(x) = Q(x) + a(0) \rho (x) + \chi(0, x), \\
u_1(x) = \partial_t a(0) \rho (x) + \partial_t \chi(0, x),
\end{array}
\end{equation*}
then, in terms of $b$ and $\bm{\zeta}$ used in \eqref{gamma-12-form}, we can write
\begin{equation*}
\begin{aligned}
a(0) = b + h(\bm{\gamma}), \quad \partial_t a(0) = -\Omega (b - h(\bm{\gamma})), \quad \chi(0) = \zeta_1 \quad \text{and}\quad  \partial_t \chi(0) = \zeta_2.
\end{aligned}
\end{equation*}
Given the Lipschitz property of $h$, the condition \eqref{initial-data-gamma} implies that
\begin{equation}
\label{initial-data-main-theorem}
|a(0)| + |\partial_t a(0)| + \n{(\sqrt{H+1}~ \chi(0), \partial_t \chi(0))}_{H_x^2} + \n{\lr{x}(\sqrt{H+1}~ \chi(0), \partial_t \chi(0))}_{L_x^2} \leq \epsilon_0.
\end{equation}

\subsection{Verification of the stability condition}
Here we show that, if the global solution satisfies \eqref{global-bound-KMM},
then the stability condition \eqref{stability-cond} holds.

First, note that the bound \eqref{global-bound-KMM} implies that, for every $t\geq 0$, we have
$$\n{v(t)}_{L_x^\infty} \lesssim \n{v(t)}_{H_x^1} \lesssim \delta_0.$$
This, in particular, means that the term $F[v](t)$ defined in \eqref{system-eq} satisfies $|F[v](t)|\lesssim \delta_0^2$ for every $t\geq 0$.

We rewrite the integral equation for $a(t)$ in \eqref{integral-eq-0} as
\begin{equation*}
\begin{aligned}
a(t) & = \frac{1}{2}e^{\Omega t} \left( a(0) + \frac{1}{\Omega} \dot a (0) +\frac{1}{\Omega} \int_0^t e^{-\Omega s}F[v](s) ds \right)\\
&\quad + \frac{1}{2}e^{-\Omega t} \left( a(0) - \frac{1}{\Omega} \dot a (0) -\frac{1}{\Omega} \int_0^t e^{\Omega s}F[v](s) ds \right)\\
& =: e^{\Omega t} C_1(t) + e^{-\Omega t} C_2(t).
\end{aligned}
\end{equation*}
From the above estimates on $F[v]$, it is clear that $e^{-\Omega t}C_2(t)$ is uniformly bounded at all positive times $t$. Moreover, $C_1(t)$ is also uniformly bounded for all $t\geq 0$, and as $t \to \infty$, $C_1(t)$ converges to some constant $C_1^*$. Therefore, due to the presence of exponentially increasing function $e^{\Omega t}$, the bound \eqref{global-bound-KMM-a-chi} is satisfied only if $C_1^* = 0$, which is equivalent to the stability condition \eqref{stability-cond}.
\fi

\bigskip