# Scattering for 2d semi-relativistic Hartree equations with short range
potential

Changhun Yang  Department of Mathematics, Chungbuk National University,
Chungdae-ro1, Seowon-gu, Cheongju-si, Chungcheongbuk-do, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

We study the long time behavior of small solutions to the semi-relativistic
Hartree equations in two dimension. The nonlinear term is convolved with the
singular potential $|x|^{-\gamma}$ for $1<\gamma<2$, which is referred to as
short-range interaction potential in the sense of scattering phenomenon. We
establish the scattering results for small solutions in a weighted space, in
other words, we prove that the nonlinear solutions exist globally and behave
asymptotically like a linear solution whenever the initial data is
sufficiently small. To achieve this, we should obtain time decay estimates for
the nonlinear term which is integrable. A key observation is that the loss in
time in the course of weighted energy estimates can be recovered by the space
resonance method with the help of null structure.

##  1\. Introduction

We consider the semi-relativistic Hartree equations

(1.1) $\displaystyle\left\\{\begin{aligned}
-i\partial_{t}u+\sqrt{m^{2}-\Delta}u&={\lambda}\left(|x|^{-\gamma}*|u|^{2}\right)u&&\mathrm{in}\;\;\mathbf{\mathbb{R}}\times\mathbb{R}^{d},\\\
u(0,\cdot)&=u_{0}&&\mathrm{in}\;\;\mathbb{R}^{d},\end{aligned}\right.$

where the unknown $u:\mathbb{R}^{1+d}\to\mathbb{C}$, some fixed constant
${\lambda}\in\mathbf{\mathbb{R}}$, and $0<\gamma<d$. The nonlocal differential
operator $\sqrt{m^{2}-\Delta}$ is defined as a Fourier multiplier operator
associated to the symbol $\sqrt{m^{2}+|\xi|^{2}}$ and $\ast$ denotes the
convolution in $\mathbb{R}^{d}$. $m>0$ represents the mass which will be
normalized to $1$ by scaling throughout the paper. For three dimensional case
$d=3$, (1.1) with Coulomb potential ($\gamma=1$) is often referred to as Boson
star equation which describes the dynamics and collapse of relativistic Boson
stars. It was rigorously derived as the mean-field limit of large systems of
bosons. See [17, 7, 15] and references therein. (1.1) with generalized
potential in all dimension has been also widely studied by many researchers in
the mathematical viewpoint. The singular potential $|x|^{-\gamma}$ is called
the long range (or, short range) interaction when $0<\gamma\leq 1$ (or,
$1<\gamma<d$, respectively). The mass and energy of solution to (1.1) are
defined by

(1.2) $\displaystyle M(u)(t)$
$\displaystyle=\|u(t)\|_{L^{2}(\mathbf{\mathbb{R}}^{d})},$ $\displaystyle
E(u)(t)$
$\displaystyle=\frac{1}{2}\int_{\mathbf{\mathbb{R}}^{d}}\overline{u}\sqrt{m^{2}-\Delta}udx+\frac{\lambda}{4}\int_{\mathbf{\mathbb{R}}^{d}}\left(|x|^{-\gamma}\ast|u|^{2}\right)|u|^{2}dx,$

respectively and conserved as time evolves. From the conservations laws, one
sees that $H^{\frac{1}{2}}$ is the energy space. Furthermore, in massless case
$m=0$, we have the scaling symmetry. If $u$ is solution to (1.1), $u_{\alpha}$
defined by $u_{\alpha}(t,x)=\alpha^{\frac{1-\gamma+d}{2}}u(\alpha t,\alpha x)$
is also solution, so the scaling invariant space is
$H^{\frac{\gamma-1}{2}}(\mathbb{R}^{d})$. Thus (1.1) is often called
$H^{\frac{\gamma-1}{2}}(\mathbb{R}^{d})$-critical.

There have been numerous results on global solutions to semi-relativistic
Hartree equations (1.1). Lenzmann [14] proved that the Boson star equation
($\gamma=1$ and $d=3$) was globally well-posed in the energy space
$H^{\frac{1}{2}}(\mathbf{\mathbb{R}}^{3})$. Cho and Ozawa [3] extended this
result to (1.1) under the suitable assumption on $\gamma$ and $d$.
Furthermore, they verified the asymptotic behavior of solutions when $d\geq 3$
and $2<\gamma<d$ by showing the small data scattering in
$H^{s}(\mathbf{\mathbb{R}}^{d})$ for $s>\frac{\gamma}{2}-\frac{d-2}{2d}$. By a
scattering in $H^{s}(\mathbf{\mathbb{R}}^{d})$, we mean that solutions to
nonlinear PDEs converge to solutions to the linear equation in
$H^{s}(\mathbf{\mathbb{R}}^{d})$ as time goes to infinity. On the other hand,
they computed the time decay of $L^{2}$ norm of the nonlinearity, computed on
a solution of the linear equation, as follows:

(1.3)
$\displaystyle\left\|\Big{(}|x|^{-\gamma}*\big{|}e^{-it\sqrt{m^{2}-\Delta}}u_{0}\big{|}^{2}\Big{)}e^{-it\sqrt{m^{2}-\Delta}}u_{0}\right\|_{L^{2}(\mathbf{\mathbb{R}}^{d})}\sim|t|^{-\gamma}\quad\text{
for }|t|\gg 1.$

The computation is based on the well-known dispersive property of the linear
solutions (see [16, Lemma 3])

$\displaystyle\|e^{-it\sqrt{m^{2}-\Delta}}u_{0}\|_{L^{\infty}(\mathbf{\mathbb{R}}^{d})}\lesssim(1+|t|)^{-\frac{d}{2}}\quad\text{for}\quad
u_{0}\in C_{0}^{\infty}(\mathbf{\mathbb{R}}^{d}).$

Based on (1.3), they further proved the nonexistence of scattering state when
the long range interaction $(0<\gamma\leq 1\text{ for }d\geq 3\text{ and
}0<\gamma<1$ for  $d=2)$ was concerned where the time decay of nonlinear term
fails to be integrable. In contrast, the global solutions to (1.1) with the
short range interaction $(1<\gamma<d)$ would be expected to scatter since the
nonlinear term decays sufficiently fast to be integrable. For $d=3$, Pusateri
[18] confirmed this anticipation for sufficiently small initial data in a
weighted spaces by closing the gap corresponding to $1<\gamma\leq 2$ by
proving the scattering result of (1.1). Inspired by the work [18], we focus on
the two dimensional case with full short range interaction, $1<\gamma<2$.

Besides, there have been also a lot of results on local well-posedness of
(1.1) given rough initial data and global well-posedness under the various
assumptions. We refer to [4, 5, 6, 11, 13] and references therein for
interested readers.

######  Remark 1.1.

Concerning (1.1) with $\gamma=1$ which corresponds to the scattering-critical
equation where the decay of nonlinear term barely fails to be integrable, the
modified scattering results have been established by Pusateri [18] for three
dimensional case $d=3$ and by Kwon, Lee, and the author of the paper [12] for
two dimensional case $d=2$.

######  Remark 1.2.

Let us introduce the fractional Schrödinger Hartree equation

(1.4) $\displaystyle
i\partial_{t}u-(-\Delta)^{\alpha}u={\lambda}\left(|x|^{-\gamma}*|u|^{2}\right)u,\quad\mathrm{in}\;\;\mathbf{\mathbb{R}}\times\mathbb{R}^{d},$

where $u:\mathbf{\mathbb{R}}^{1+d}\rightarrow\mathbf{\mathbb{C}}$,
$0<\alpha<1$, $\alpha\neq 1$, $0<\gamma<d$ and
$\lambda\in\mathbf{\mathbb{R}}\setminus\\{0\\}$. The fractional Laplacian
$(-\Delta)^{\alpha}$ is the nonlocal differential operator associated to the
symbol $|\xi|^{\alpha}$. Cho [1] established the small data scattering result
for (1.4) with the short range interaction in a weighted space when $d\geq 3$.
We expect that the argument in this paper can be applied to (1.4) with two
dimension.

We now state our main theorem for the two dimensional semi-relativistic
Hartree equations (1.1) with the short range interaction.

######  Theorem 1.3.

Let $s\geq 30$ and $1<\gamma<2$. There exists $\epsilon_{0}>0$ satisfying the
following:

If the initial data $u_{0}$ satisfies

$\|u_{0}\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}u_{0}\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\leq\varepsilon,$

for some $0<\epsilon\leq\epsilon_{0}$, there exists a unique global solutions
to (1.1) such that

$\sup_{t\in\mathbf{\mathbb{R}}}\left(\|u(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}e^{it\langle
D\rangle}u(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\right)\lesssim\varepsilon,$

and

(1.5)
$\displaystyle\|u(t)\|_{W^{7,\infty}(\mathbf{\mathbb{R}}^{2})}\lesssim(1+|t|)^{-1}\varepsilon.$

Furthermore, there exist $\delta>0$ and asymptotic state $v_{+}$ with

$\displaystyle\|v_{+}\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}v_{+}\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim\varepsilon,$

satisfying

$\displaystyle\Big{\|}e^{it\sqrt{m^{2}-\Delta}}u(t,x)-v_{+}\Big{\|}_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\Big{\|}\langle
x\rangle^{2}\Big{(}e^{it\sqrt{m^{2}-\Delta}}u(t,x)-v_{+}\Big{)}\Big{\|}_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim(1+t)^{-\delta}\varepsilon.$

######  Remark 1.4.

The required regularity $s$ depends on $\gamma$ but we are not interested in
the dependence as well as their optimality in this paper.

In Theorem 1.3, we prove the global existence of solutions to (1.1) when the
initial data is sufficiently small in a weighted space. Also, we show that the
small nonlinear solutions decays as time evolves with the optimal decay rate
in the sense that they decay with the same rate as the linear solution.
Furthermore, the solution scatters, i.e., behaves like a linear solution in
the weighted spaces as time goes to infinity with the convergence rate
$t^{-\delta}$.

Our proof is based on the standard bootstrap argument in a weighted Sobolev
spaces. We adapt the weighted space so that we obtain and utilize the optimal
time decay of solutions from the well-known dispersive estimates (see (2.4)).
Then, we perform the weighted energy estimates to prove the bounds of the
energy norms under the a priori assumption. The strategy is based on the
method of space-time resonance which was developed by Germain-Masmoudi-Shata
[8, 9, 10].

Let us explain in detail the ideas of proof in the weighted energy estimates.
We introduce the interaction representation of solutions $u(t)$ so as to track
the scattering states

$\displaystyle f(t,x):=e^{it\sqrt{m^{2}-\Delta}}u(t,x).$

Then we can express $f$ via Duhamel’s representation

(1.6) $\displaystyle\begin{aligned}
\widehat{f}(t,\xi)&=\widehat{u_{0}}(\xi)+i{\lambda}\mathcal{I}(t,\xi),\\\
\mathcal{I}(t,\xi)&=\int_{0}^{t}\int_{\mathbb{R}^{2}}e^{it^{\prime}\phi(\xi,\eta)}|\eta|^{-2+\gamma}\widehat{f}(t^{\prime},\xi-\eta)\widehat{|u|^{2}}(t^{\prime},\eta)d\eta
dt^{\prime}\end{aligned}$

with the resonance function

$\displaystyle\phi(\xi,\eta)=\langle\xi\rangle-\langle\xi-\eta\rangle.$

In the course of weighted energy estimates, we have to bound $x^{2}f$ in
$H^{10}(\mathbf{\mathbb{R}}^{2})$ which is converted to
$\nabla^{2}\widehat{f}$ in the fourier space. The most delicate term in
$\nabla^{2}\widehat{f}$ occurs when two derivatives both fall on
$e^{it^{\prime}\phi(\xi,\eta)}$ to give us

(1.7)
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{2}}|t^{\prime}|^{2}\big{(}\nabla_{\xi}\phi(\xi,\eta)\big{)}^{2}e^{is\phi(\xi,\eta)}|\eta|^{-2+\gamma}\widehat{f}(t^{\prime},\xi-\eta)\widehat{|u|^{2}}(t^{\prime},\eta)d\eta
dt^{\prime},$

where we have to compensate the loss in time, $|t^{\prime}|^{2}$. Here, we
encounter the main difficulty from two dimensional nature, i.e., the weaker
time decay $|t^{\prime}|^{-1}$ of solutions (1.5) compared to three or higher
dimensional problem because if we just take $L^{2}$-norm of cubic nonlinearity
in (1.7), we only manage to obtain time decay $|t^{\prime}|^{-2}$ at best
which is not sufficient to not only recover the time loss but also for the
nonlinear term to be integrable in time. A key observation to overcome this is
to exploit the structure in the quadratic term $|u|^{2}$, in other words, to
exploit the space resonance. Writing it it in terms of $f$ as

(1.8)
$\displaystyle\widehat{|u|^{2}}(t^{\prime},\eta)=\frac{1}{(2\pi)^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{-it^{\prime}\phi(\sigma,\eta)}\widehat{f}(t^{\prime},\sigma-\eta)\overline{\widehat{f}(t^{\prime},\sigma})d\sigma,$

we find that whenever we preform an integration by parts by using the
following relation

$\displaystyle
e^{it^{\prime}\phi}=-\frac{i}{t^{\prime}}\,\frac{\nabla_{\sigma}\phi\cdot\nabla_{\sigma}e^{it^{\prime}\phi}}{|\nabla_{\sigma}\phi|^{2}},$

we can obtain a time decay $|t^{\prime}|^{-1}$. At the same time, however, it
yields a singularity in $\eta$ variable because

$\displaystyle\nabla_{\sigma}\phi(\sigma,\eta)|_{\eta=0}=\nabla_{\sigma}\Big{(}\langle\sigma\rangle-\langle\sigma-\eta\rangle\Big{)}\Big{|}_{\eta=0}=0.$

Nevertheless, in order to recover the time loss, $|t^{\prime}|^{2}$, in (1.7),
we indeed integrate by parts (1.8) twice, which will be possible because we a
priori assume $x^{2}f\in H^{10}(\mathbf{\mathbb{R}}^{2})$.

The resulting multiplier which is singular near the origin, or more
specifically comparable to $|\eta|^{-2}$, can be bounded with the help of null
structure in (1.7). Indeed, as already observed above, we have

$\displaystyle\nabla_{\xi}\phi(\xi,\eta)|_{\eta=0}=0,$

so we can think that the following multiplier behaves like the constant

$|\nabla_{\xi}\phi(\xi,\eta)|^{2}|\nabla_{\sigma}\phi(\sigma,\eta)|^{-2}\approx
1,$

as far as estimates are concerned111See (3.9) for precise inequality.. Thus,
we can finally estimate (1.7) similarly to (1.6) under the assumption
$x^{2}f\in H^{10}(\mathbf{\mathbb{R}}^{2})$. We remark that the regularity of
weighted function is required when we apply the multiplier inequality in this
procedure.

### Organization

In Section 2, we provide the proof of main theorem relying on the standard
bootstrapping argument. The rest sections are devoted to proving the weighted
energy estimates which played a key role in Section 2. In section 3, we
discuss some auxiliary estimates and establish the time decay estimates for
the frequently localized quadratic term by exploiting the space resonance. In
section 4, we finally prove the weighted energy estimates with the help of
null structures.

### Notations

$\bullet$ (Fourier transform)
$\mathcal{F}g(\xi)=\widehat{g}(\xi):=\int_{\mathbf{\mathbb{R}}^{2}}e^{-ix\cdot\xi}g(x)dx$
and
$g(x)=\frac{1}{(2\pi)^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{ix\cdot\xi}\widehat{g}(\xi)d\xi$.

$\bullet$ (Japaneses bracket) $\langle x\rangle:=(1+|x|^{2})^{\frac{1}{2}}$
for $x\in\mathbb{R}^{2}$.

$\bullet$ (Mixed-normed space) For a Banach space $X$ and an interval $I$,
$u\in L_{I}^{q}X$ iff $u(t)\in X$ for a.e. $t\in I$ and
$\|u\|_{L_{I}^{q}X}:=\|\|u(t)\|_{X}\|_{L_{I}^{q}}<\infty$. Especially, we
denote $L_{I}^{q}L_{x}^{r}=L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{2}))$,
$L_{I,x}^{q}=L_{I}^{q}L_{x}^{q}$,
$L_{t}^{q}L_{x}^{r}=L_{\mathbb{R}}^{q}L_{x}^{r}$.

$\bullet$ As usual different positive constants are denoted by the same letter
$C$, if not specified. $A\lesssim B$ and $A\gtrsim B$ means that $A\leq CB$
and $A\geq C^{-1}B$, respectively for some $C>0$. $A\sim B$ means that
$A\lesssim B$ and $A\gtrsim B$.

$\bullet$ (Fourier multiplier) $D=-i\nabla$. For
$m:\mathbf{\mathbb{R}}^{2}\rightarrow\mathbf{\mathbb{R}}$,
$m(D)f:=\mathcal{F}^{-1}\big{(}m(\xi)\widehat{f}(\xi)\big{)}$.

$\bullet$ (Littlewood-Paley operator) Let $\chi$ be a bump function such that
$\chi\in C_{0}^{\infty}(B(0,2))$ with $\chi(\xi)=1$ for $|\xi|\leq 1$ and
define

$\chi_{L}(\xi):=\chi\big{(}\tfrac{\xi}{L}\big{)}-\chi\big{(}\tfrac{2\xi}{L}\big{)}\text{
for }L\in 2^{\mathbb{Z}}.$

We define the projection operator $P_{L}$ for $L\in 2^{\mathbf{\mathbb{Z}}}$
via $\mathcal{F}(P_{L}f)(\xi)=\chi_{L}(\xi)\widehat{f}(\xi)$. Then, we
immediately have $f=\sum_{L\in 2^{\mathbf{\mathbb{Z}}}}P_{L}f$.

We also introduce an another partition of unity $\rho_{N}$ for $N\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}$ such that

$\displaystyle\rho_{1}=\chi\text{ and }\rho_{N}=\chi_{N}\text{ for }N\geq 2.$

We define the projection operator $S_{N}$ for $N\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}$ by
$\mathcal{F}(S_{N}f)(\xi)=\rho_{N}(\xi)\widehat{f}(\xi)$. So,
$S_{1}=\sum_{L\leq 1}P_{L}$ and $S_{N}=P_{N}$ for $N\geq 2$. By orthogonality,
we have

$\displaystyle\|f\|_{H^{s}}^{2}\approx\sum_{L\in
2^{\mathbf{\mathbb{Z}}}}\langle
L\rangle^{2s}\|P_{L}f\|_{L^{2}}^{2}\approx\sum_{N\in
2^{\mathbf{\mathbb{N}}}}N^{2s}\|S_{N}f\|_{L^{2}}^{2}.$

$\bullet$ (Commutator) The commutator of $A$ and $B$, denoted by $[A,B]$, is
defined as $[A,B]=AB-BA$.

$\bullet$ For $\alpha=(\alpha_{1},\alpha_{2})$, where
$\alpha_{i}\in\mathbf{\mathbb{N}}\cup\\{0\\}$ for $i=1,2$, we define
$\partial_{x}^{\,\alpha}=\partial_{x_{1}}^{\,\alpha_{1}}\partial_{x_{2}}^{\,\alpha_{2}}$
and $|\alpha|=\alpha_{1}+\alpha_{2}$.

$\bullet$ For the distinction between a vector and a scalar, we use the bold
letter for a vector-valued function and the normal letter for a scalar-valued
function.

$\bullet$ Let
$\textbf{v}=(v_{1},v_{2}),\textbf{w}=(w_{1},w_{2})\in\mathbf{\mathbb{C}}^{2}$.
Then $\textbf{v}\otimes\textbf{w}$ denotes the usual tensor product such that
$(\textbf{v}\otimes\textbf{w})_{ij}=v_{i}w_{j}$ for $i,j=1,2$. We also denote
a tensor product of $\textbf{v}\in\mathbf{\mathbb{C}}^{n}$ and
$\textbf{w}\in\mathbf{\mathbb{C}}^{m}$ by a matrix
$\textbf{v}\otimes\textbf{w}=(v_{i}w_{j})_{\begin{subarray}{c}i=1,\cdots,n\\\
i=1,\cdots,m\end{subarray}}$. For simplicity, we use the simplified notation

$\mathbf{v}^{k}=\overbrace{\mathbf{v}\otimes\cdots\otimes\mathbf{v}}^{k\;\text{times}},\qquad\nabla^{k}=\overbrace{\nabla\otimes\cdots\otimes\nabla}^{k\;\text{times}}.$

The product of $\mathbf{v}$ and $f\in\mathbf{\mathbb{C}}$ is given by
$\mathbf{v}f=\mathbf{v}\otimes f$.

### Acknowledgement

The author was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (No. 2021R1C1C1005700).

##  2\. Proof of Theorem 1.3

Throughout the paper, we fix $m=1$ by scaling. In this section, we prove the
Theorem 1.3. Let $u(t)$ be a local solution on a time interval $[0,T]$, i.e.
$u\in C([0,T]:H^{s}(\mathbf{\mathbb{R}}^{2}))$, which can be constructed by
the standard fixed point argument.222For example, see [3, Proposition 2.1].
For given a solution to (1.1), let us define

$\displaystyle f(t,x):=e^{it\sqrt{1-\Delta}}u(t,x).$

Using the Duhamel’s formula, we write (1.1) as

(2.1) $\displaystyle
f(t)=u_{0}+i\lambda\int_{0}^{t}e^{it^{\prime}\sqrt{1-\Delta}}N_{\gamma}(u,u,u)(t^{\prime})dt^{\prime},$

where

$\displaystyle
N_{\gamma}(u,v,w):=\big{(}|x|^{-\gamma}\ast(u\overline{v})\big{)}w,\qquad
1<\gamma<2.$

We solve the integral equation (2.1) in the space defined via the following
norm

$\displaystyle\sup_{t\in\mathbf{\mathbb{R}}}\left(\|u(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}f(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\right).$

Given a local solution $u$ on a time interval $[0,T]$, we assume that the
following norm is a priori small

(2.2)
$\displaystyle\|u\|_{X_{T}}:=\sup_{t\in[0,T]}\left(\|u(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}f(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\right)\leq\varepsilon_{1},$

for some $\varepsilon_{1}>0$. In order to prove the existence of global
solution, by the standard bootstrapping argument, we suffice to show that
under the a priori assumptions, one has

(2.3) $\displaystyle\|u\|_{X_{T}}\leq\varepsilon_{0}+C\varepsilon_{1}^{3},$

where $\varepsilon_{0}$ is the size of the initial data

$\|u_{0}\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}u_{0}\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\leq\varepsilon_{0}.$

Let us recall from [16, Lemma 3] the following frequency localized dispersive
estimates

(2.4) $\displaystyle\|e^{it\langle
D\rangle}S_{N}\phi\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\langle
t\rangle^{-1}N^{2}\|\phi\|_{L^{1}(\mathbf{\mathbb{R}}^{2})}\quad\text{ for
}N\geq 1.$

Then, one verifies that under the a priori assumption, our solution $u$
satisfies

(2.5)
$\displaystyle\|u(t)\|_{W^{7,\infty}(\mathbf{\mathbb{R}}^{2})}\lesssim\langle
t\rangle^{-1}\varepsilon_{1}.$

Indeed, using the dispersive estimates (2.4), we estimate

$\displaystyle\|\langle D\rangle^{7}$ $\displaystyle
u(t)\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}\leq\sum_{N\geq 1}\langle
N\rangle^{7}\|e^{it\langle
D\rangle}S_{N}f(t)\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\langle t\rangle^{-1}\sum_{N\geq 1}\langle
N\rangle^{9}\|\langle
x\rangle^{2}S_{N}f(t)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\langle
t\rangle^{-1}\big{(}\|xf(t)\|_{H^{9}(\mathbf{\mathbb{R}}^{2})}+\|\langle
x\rangle^{2}f(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\big{)},$

where we bounded the weighted norm as follows

$\displaystyle\|\langle x\rangle^{2}S_{N}f\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\leq\|[\langle
x\rangle^{2},S_{N}]f\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}+\|S_{N}\langle
x\rangle^{2}f\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$ $\displaystyle\lesssim
N^{-1}\|S_{\sim N}xf\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}+\|S_{N}\langle
x\rangle^{2}f\|_{L^{2}(\mathbf{\mathbb{R}}^{2})},$

where $S_{\sim N}$ denotes $\displaystyle\sum_{\\{M\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}\,:\,\frac{N}{4}\leq M\leq 4N\\}}S_{M}$.
Then,

(2.6) $\displaystyle\begin{aligned}
\|xf(t)\|_{H^{9}(\mathbf{\mathbb{R}}^{2})}&\approx\sum_{|\alpha|\leq
9}\|\partial^{\alpha}(xf)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\\\
&\lesssim\sum_{|\alpha|\leq
9}\sum_{\beta+\beta^{\prime}=\alpha}\|\partial^{\beta}(x\langle
x\rangle^{-2})\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}\big{\|}\partial^{\beta^{\prime}}\big{(}\langle
x\rangle^{2}f\big{)}\big{\|}_{L^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\|\langle
x\rangle^{2}f\|_{H^{9}(\mathbf{\mathbb{R}}^{2})}.\end{aligned}$

Using the time decay of solutions under the a priori assumption, we compute
the time decay of Hartree nonlinear terms.

######  Proposition 2.1 (Weighted Energy Estimates).

Assume that $u\in C([0,T]:H^{s}(\mathbf{\mathbb{R}}^{2}))$ satisfies the a
priori assumptions (2.2). Then, we have

(2.7)
$\displaystyle\left\|N_{\gamma}(u,u,u)(t)\right\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim(1+|t|)^{-\gamma}\varepsilon_{1}^{3},$ (2.8)
$\displaystyle\left\|x^{2}e^{it\langle
D\rangle}N_{\gamma}(u,u,u)(t)\right\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim(1+|t|)^{-1-\frac{\gamma-1}{3}}\varepsilon_{1}^{3}.$

The poof of Proposition 2.1 will be provided in Section 4. One sees that the
time decay of nonlinear term is integrable, which immediately implies that

$\displaystyle\sup_{t\in[0,T]}\left\|\int_{0}^{t}e^{it^{\prime}\langle
D\rangle}N_{\gamma}(u,u,u)(t^{\prime})dt^{\prime}\right\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\epsilon_{1}^{3},$
$\displaystyle\sup_{t\in[0,T]}\left\|\langle
x\rangle^{2}\int_{0}^{t}e^{it^{\prime}\langle
D\rangle}N_{\gamma}(u,u,u)(t^{\prime})dt^{\prime}\right\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\epsilon_{1}^{3},$

which completes the proof of (2.3). Furthermore, once we set a linear profile
$v_{+}$ as

$\displaystyle
v_{+}\,:=\lim_{t\to\infty}f(t)=u_{0}+i\lambda\lim_{t\to\infty}\int_{0}^{t}e^{it^{\prime}\langle
D\rangle}N_{\gamma}(u,u,u)(t^{\prime})dt^{\prime},$

we have from (2.7)

$\displaystyle\left\|e^{it\langle
D\rangle}u(t,x)-v_{+}\right\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}\lesssim\left\|\int_{t}^{\infty}e^{it^{\prime}\langle
D\rangle}N_{\gamma}(u,u,u)(t^{\prime})dt^{\prime}\right\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}\lesssim(1+t)^{-\gamma+1}\varepsilon_{1}^{3},$

and similarly, from (2.8)

$\displaystyle\left\|\langle x\rangle^{2}\left(e^{it\langle
D\rangle}u(t,x)-v_{+}\right)\right\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim(1+t)^{-\frac{\gamma-1}{3}}\varepsilon_{1}^{3}.$

##  3\. Time decay estimates

In this section, we establish the time decay estimates for the nonlinear terms
and frequency localized solutions under the a priori assumption.

###  3.1. Preliminaries

We begin with estimating quadratic terms convolved with the short range
potential.

######  Lemma 3.1.

Let $1<\gamma<2$. For any $\mathbb{C}$-valued functions $u\in
L^{2}(\mathbf{\mathbb{R}}^{2})\cap L^{\infty}(\mathbf{\mathbb{R}}^{2})$, we
get

(3.1)
$\displaystyle\left\||x|^{-\gamma}*|u|^{2}\right\|_{L^{\infty}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\|u\|_{L^{2}}^{2-\gamma}\|u\|_{L^{\infty}}^{\gamma}.$

###### Proof.

The proof is quite standard, but we provide it for completeness. We estimate

$\displaystyle\||x|^{-\gamma}\ast|u|^{2}\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle=\sup\int_{\mathbf{\mathbb{R}}^{2}}|y|^{-\gamma}|u|^{2}(x-y)dy$
$\displaystyle\leq\sup\int_{|y|<R}|y|^{-\gamma}|u|^{2}(x-y)dy+\sup\int_{|y|\geq
R}|y|^{-\gamma}|u|^{2}(x-y)dy$
$\displaystyle\leq\|u\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}^{2}R^{2-\gamma}+R^{-\gamma}\|u\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}^{2}.$

Then taking
$R=\|u\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}^{-1}\|u\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
gives (3.1). ∎

The next lemma is useful when we handle the Fourier multiplier operator.

######  Lemma 3.2 (Coifman-Meyer operator estimates).

Assume that a multiplier $\textbf{m}(\xi,\eta)$ satisfies that

(3.2) $\displaystyle
C(\textbf{m}):=\left\|\iint_{\mathbb{R}^{2}\times\mathbf{\mathbb{R}}^{2}}\textbf{m}(\xi,\eta)e^{ix\cdot\xi}e^{iy\cdot\eta}\,d\eta
d\xi\right\|_{L_{x,y}^{1}(\mathbb{R}^{2}\times\mathbf{\mathbb{R}}^{2})}<\infty.$

Then, for $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$,

(3.3)
$\displaystyle\left\|\int_{\mathbb{R}^{2}}\textbf{m}(\xi,\eta)\widehat{u}(\xi\pm\eta)\widehat{v}(\eta)\,d\eta\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim
C(\textbf{m})\|u\|_{L^{p}(\mathbf{\mathbb{R}}^{2})}\|v\|_{L^{q}(\mathbf{\mathbb{R}}^{2})}.$

###### Proof.

See, for example, [18, Lemma B.1]. ∎

###  3.2. Time decay estimates

We find time decay of the frequency localized quadratic terms under the a
priori assumption.

######  Lemma 3.3.

Let $u$ satisfy the a priori assumption (2.2). For a dyadic number $N\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}$, we have

(3.4)
$\displaystyle\left\|S_{N}\big{(}|u(t)|^{2}\big{)}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\min\big{(}N^{-7}\langle
t\rangle^{-2},N^{2-s}\big{)}\varepsilon_{1}^{2},$

###### Proof.

We write

$\displaystyle S_{N}\big{(}|u(t)|^{2}\big{)}(x)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{ix\cdot\xi}\rho_{N}(\xi)\langle\xi\rangle^{-7}\mathcal{F}\big{\\{}\langle
D\rangle^{7}(|u(t)|^{2})\big{\\}}(\xi)d\xi$
$\displaystyle=\frac{1}{(2\pi)^{2}}\mathcal{F}^{-1}\big{(}\rho_{N}(\xi)\langle\xi\rangle^{-7}\big{)}\ast\langle
D\rangle^{7}(|u(t)|^{2})(x).$

By Young’s inequality, we have

$\displaystyle\left\|S_{N}\big{(}|u(t)|^{2}\big{)}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\|\mathcal{F}^{-1}(\rho_{N}(\xi)\langle\xi\rangle^{-7})\|_{L^{1}(\mathbf{\mathbb{R}}^{2})}\|\langle
D\rangle^{7}(|u(t)|^{2})\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim
N^{-7}\|u(t)\|_{W^{7,\infty}(\mathbf{\mathbb{R}}^{2})}^{2},$

which completes the proof of the first bound in (3.4) by (2.5).

We consider the second bound. Using the Sobolev embedding, we immediately
obtain that

$\displaystyle\left\|S_{N}\big{(}|u(t)|^{2}\big{)}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim
N\left\|S_{N}\big{(}|u(t)|^{2}\big{)}\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}.$

The Plancherel’s identity yields that

$\displaystyle\|S_{N}\big{(}|u(t)|^{2}\big{)}\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}=\frac{1}{2\pi}\|\rho_{N}\mathcal{F}\big{(}|u(t)|^{2}\big{)}\|_{L^{2}(\mathbf{\mathbb{R}}^{2})},$

where

$\displaystyle\rho_{N}(\eta)\mathcal{F}\big{(}|u(t)|^{2}\big{)}(\eta)$
$\displaystyle=\int_{\mathbf{\mathbb{R}}^{2}}\rho_{N}(\eta)\widehat{u}(\sigma)\overline{\widehat{u}(\sigma+\eta)}d\sigma$
$\displaystyle=\int_{\mathbf{\mathbb{R}}^{2}}\frac{\rho_{N}(\eta)}{\langle\sigma\rangle^{s}\langle\sigma+\eta\rangle^{s}}\widehat{\langle
D\rangle^{s}u}(\sigma)\overline{\widehat{\langle
D\rangle^{s}u}(\sigma+\eta)}d\sigma,$

which implies that

(3.5)
$\displaystyle\|S_{N}|u(t)|^{2}\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim
N\langle N\rangle^{-s}\|u\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}^{2}.$

∎

In view of (2.5) and (3.4) in the previous lemma, it seems that time decay
$\langle t\rangle^{-2}$ for $L^{\infty}(\mathbf{\mathbb{R}}^{2})$ norm of
quadratic term is optimal. However, by exploiting structure from bilinear
interaction, we can obtain an extra time decay under the a priori assumption
at the cost of a derivative for the low frequency part. In other words, we
employ the space resonance method.

######  Lemma 3.4.

Let $u$ satisfy the a priori assumption (2.2). For a dyadic number $L\in
2^{\mathbf{\mathbb{Z}}}$, we have

(3.6)
$\displaystyle\left\|P_{L}\big{(}|u(t)|^{2}\big{)}\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\min\big{(}L^{-2}{\langle t\rangle}^{-3},L\langle
L\rangle^{-s}\big{)}\varepsilon_{1}^{2}.$

######  Remark 3.5.

By the Sobolev embedding, we immediately have that

(3.7)
$\displaystyle\left\|P_{L}\big{(}|u(t)|^{2}\big{)}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\min\big{(}L^{-1}{\langle t\rangle}^{-3},L^{2}\langle
L\rangle^{-s}\big{)}\varepsilon_{1}^{2}.$

###### Proof.

The second bound in (3.6) can be proved similarly to the proof of (3.5).

We consider the first bound in (3.6). We write

$\displaystyle\mathcal{F}\big{(}|u(t)|^{2}\big{)}(\eta)=\frac{1}{(2\pi)^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it({\langle\sigma+\eta\rangle}-{\langle\sigma\rangle})}\widehat{f}(t,\sigma)\overline{\widehat{f}(t,\eta+\sigma})d\sigma,$

where $f(t,x)=e^{it\sqrt{1-\Delta}}u(t,x)$. Using the relation

(3.8) $\displaystyle
e^{it({\langle\sigma+\eta\rangle}-{\langle\sigma\rangle})}=-\frac{i}{t}\,\frac{\nabla_{\sigma}({\langle\sigma+\eta\rangle}-{\langle\sigma\rangle})\cdot\nabla_{\sigma}e^{it({\langle\sigma+\eta\rangle}-{\langle\sigma\rangle})}}{|{\langle\sigma+\eta\rangle}-{\langle\sigma\rangle}|^{2}},$

we perform an integration by parts twice to write

$\displaystyle\mathcal{F}\big{(}|u(t)|^{2}\big{)}(\eta)=J_{1}+J_{2}+J_{3}+J_{4},$

where

$\displaystyle J_{1}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}\nabla_{\sigma}\cdot\big{(}\mathbf{m}(\eta,\sigma)\,\nabla_{\sigma}\cdot\mathbf{m}(\eta,\sigma)\big{)}\widehat{u}(t,\sigma)\overline{\widehat{u}(t,\eta+\sigma)}d\sigma,$
$\displaystyle J_{2}(t,\eta)$
$\displaystyle=\frac{2}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}\nabla_{\sigma}\cdot\mathbf{m}(\eta,\sigma)\,\mathbf{m}(\eta,\sigma)\cdot\nabla_{\sigma}\Big{(}\widehat{f}(t,\sigma)\overline{\widehat{f}(t,\eta+\sigma})\Big{)}d\sigma,$
$\displaystyle J_{3}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}\nabla_{\sigma}\otimes\mathbf{m}(\eta,\sigma)\cdot\mathbf{m}(\eta,\sigma)\otimes\nabla_{\sigma}\Big{(}\widehat{f}(t,\sigma)\overline{\widehat{f}(t,\eta+\sigma})\Big{)}d\sigma,$
$\displaystyle J_{4}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\cdot\nabla_{\sigma}\otimes\nabla_{\sigma}\Big{(}\widehat{f}(t,\sigma)\overline{\widehat{f}(t,\eta+\sigma})\Big{)}d\sigma,$

where

$\displaystyle\mathbf{m}(\eta,\sigma):=\frac{\nabla_{\sigma}(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}{|\nabla_{\sigma}(\langle\sigma+\eta\rangle-\langle\sigma\rangle)|^{2}}.$

Since

(3.9)
$\displaystyle\frac{|\eta|}{\max(\langle\eta+\sigma\rangle,\langle\sigma\rangle)\min(\langle\eta+\sigma\rangle,\langle\sigma\rangle)^{2}}\lesssim\left|\frac{\eta+\sigma}{{\langle\eta+\sigma\rangle}}-\frac{\sigma}{{\langle\sigma\rangle}}\right|\lesssim\frac{|\eta|}{\max(\langle\eta+\sigma\rangle,\langle\sigma\rangle)},$

one can easily verify that for multi index $\alpha$,

(3.10)
$\displaystyle\left|\partial_{\sigma}^{\alpha}\mathbf{m}(\eta,\sigma)\right|\lesssim|\eta|^{-1}\max(\langle\eta+\sigma\rangle,\langle\sigma\rangle)\min(\langle\eta+\sigma\rangle,\langle\sigma\rangle)^{2+|\alpha|}.$

First, we consider $J_{1}$. By dyadic decomposition, we write $J_{1}$ as

$\displaystyle\chi_{L}(\eta)J_{1}(t,\eta)$
$\displaystyle=\sum_{N_{1},N_{2}\geq 1}J_{1}^{(L,N_{1},N_{2})}(t,\eta),$
$\displaystyle J_{1}^{(L,N_{1},N_{2})}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}m_{1}^{(L,N_{1},N_{2})}(\eta,\sigma)\widehat{S_{N_{1}}u}(t,\sigma)\overline{\widehat{S_{N_{2}}u}(t,\eta+\sigma)}d\sigma$
$\displaystyle m_{1}^{(L,N_{1},N_{2})}(\eta,\sigma)$
$\displaystyle=\nabla_{\sigma}\cdot\big{(}\mathbf{m}(\eta,\sigma)\,\nabla_{\sigma}\cdot\mathbf{m}(\eta,\sigma)\big{)}\chi_{L}(\eta)\rho_{N_{1}}(\eta+\sigma)\rho_{N_{2}}(\sigma),$

where, by the support condition, the summation actually runs over

$\displaystyle\mathcal{A}_{L}=\\{(N_{1},N_{2})\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}\;:\;N_{1}\sim N_{2}\gtrsim L,\text{ or
}\min(N_{1},N_{2})\ll\max(N_{1},N_{2})\sim L\\}.$

Indeed, the integral $J_{1}^{(L,N_{1},N_{2})}$ is vanishing for
$(N_{1},N_{2})\notin\mathcal{A}_{L}$.

Using the operator inequality (3.3) with
$C\big{(}m_{1}^{(L,N_{1},N_{2})}\big{)}$ satisfying

$\displaystyle
C\big{(}m_{1}^{(L,N_{1},N_{2})}\big{)}:=\left\|\iint_{\mathbb{R}^{2}\times\mathbf{\mathbb{R}}^{2}}m_{1}^{(L,N_{1},N_{2})}(\eta,\sigma)e^{ix\cdot\eta}e^{iy\cdot\sigma}\,d\eta
d\sigma\right\|_{L_{x,y}^{1}(\mathbb{R}^{2}\times\mathbf{\mathbb{R}}^{2})}<\infty,$

we have

$\displaystyle\left\|J_{1}^{(L,N_{1},N_{2})}(t,\eta)\right\|_{L_{\eta}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim|t|^{-2}C\big{(}m_{1}^{(L,N_{1},N_{2})}\big{)}\left\|S_{N_{1}}{u}(t)\,\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\left\|S_{N_{2}}|u(t)|^{2}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}.$

Using (3.9) and (3.10), a direct computation yields that333 For the detailed
computation, we refer to [2, Section 5.2]

$\displaystyle\big{|}m_{1}^{(L,N_{1},N_{2})}(\eta,\sigma)\big{|}\lesssim
L^{-2}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{6}$

and

(3.11) $\displaystyle C\big{(}{m_{1}^{(L,N_{1},N_{2})}}\big{)}\lesssim
L^{-2}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{14},$

thus, by Lemma 3.3, we have under the a priori assumption

$\displaystyle\left\|\chi_{L}(\eta)J_{1}(t,\eta)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\sum_{N_{1},N_{2}\geq
1}\left\|J_{1}^{(L,N_{1},N_{2})}(t)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\;\lesssim|t|^{-2}L^{-2}\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{14}\|S_{N_{1}}u(t)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\|S_{N_{2}}u(t)\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\;\lesssim|t|^{-2}L^{-2}\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{14}N_{1}^{-s}N_{2}^{-7}\langle
t\rangle^{-1}\varepsilon_{1}^{2}$ $\displaystyle\;\lesssim\langle
t\rangle^{-3}L^{-2}\varepsilon_{1}^{2}.$

Next, we consider $J_{2}$. By dyadic decomposition as in the previous case, we
write $J_{2}$ as

$\displaystyle\rho_{L}(\eta)J_{2}(t,\eta)$
$\displaystyle=\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}J_{2}^{(L,N_{1},N_{2})}(t,\eta),$
$\displaystyle J_{2}^{(L,N_{1},N_{2})}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}\mathbf{m}_{2}^{(L,N_{1},N_{2})}(\eta,\sigma)\cdot
e^{-it\langle\sigma\rangle}\widehat{S_{N_{1}}xf}(t,\sigma)\overline{\widehat{S_{N_{2}}u}(t,\eta+\sigma)}d\sigma$
$\displaystyle\mathbf{m}_{2}^{(L,N_{1},N_{2})}(\eta,\sigma)$
$\displaystyle=\nabla_{\sigma}\cdot\mathbf{m}(\eta,\sigma)\,\mathbf{m}(\eta,\sigma)\rho_{L}(\eta)\rho_{N_{1}}(\eta+\sigma)\rho_{N_{2}}(\sigma).$

A direct computation with (3.9) yields that

(3.12) $\displaystyle C\big{(}\mathbf{m}_{2}^{(L,N_{1},N_{2})}\big{)}\lesssim
L^{-2}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{13}.$

Applying the operator inequality (3.3) with (3.12), we obtain

$\displaystyle\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\left\|J_{2}^{(L,N_{1},N_{2})}(t)\right\|_{L^{2}}$
$\displaystyle\;\lesssim|t|^{-2}L^{-2}\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{13}\|S_{N_{1}}xf(t)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\|S_{N_{2}}u(t)\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\;\lesssim|t|^{-3}L^{-2}\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{13}N_{1}^{-10}N_{2}^{-7}\varepsilon_{1}^{2}$
$\displaystyle\;\lesssim|t|^{-3}L^{-2}\varepsilon_{1}^{2},$

where we used that

(3.13)
$\displaystyle\|xf(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim\epsilon_{1},$

which holds under the a priori assumption, as explained in (2.6).

$J_{3}$ can be estimated similarly as $J_{2}$. We omit the proof.

Lastly, we consider $J_{4}$. We write

$\displaystyle J_{4}=J_{4,1}+J_{4,2}$

where

$\displaystyle J_{4,1}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it\langle\sigma+\eta\rangle}\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\cdot\Big{(}\widehat{u}(t,\sigma)\overline{\widehat{x^{2}f}(t,\eta+\sigma})\Big{)}d\sigma$
$\displaystyle\qquad+\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{-it\langle\sigma\rangle}\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\cdot\Big{(}\widehat{x^{2}f}(t,\sigma)\overline{\widehat{u}(t,\eta+\sigma})\Big{)}d\sigma,$
$\displaystyle J_{4,2}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\cdot\Big{(}\widehat{xf}(t,\sigma)\otimes\overline{\widehat{xf}(t,\eta+\sigma})\Big{)}d\sigma.$

For $J_{4,1}$, we only consider the first term. The other term can be dealt
with analogously. As before, we perform dyadic decomposition

$\displaystyle\chi_{L}(\eta)J_{4,1}(t,\eta)$
$\displaystyle=\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}J_{4,1}^{(L,N_{1},N_{2})}(t,\eta),$
$\displaystyle J_{4,1}^{(L,N_{1},N_{2})}(t,\eta)$
$\displaystyle=\frac{1}{t^{2}}\int_{\mathbf{\mathbb{R}}^{2}}\mathbf{m}_{4,1}^{(L,N_{1},N_{2})}(\eta,\sigma)\cdot
e^{-it\langle\sigma\rangle}\widehat{S_{N_{1}}x^{2}f}(t,\sigma)\overline{\widehat{S_{N_{2}}u}(t,\eta+\sigma)}d\sigma,$
$\displaystyle\mathbf{m}_{4,1}^{(L,N_{1},N_{2})}(\eta,\sigma)$
$\displaystyle=\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\chi_{L}(\eta)\rho_{N_{1}}(\eta+\sigma)\rho_{N_{2}}(\sigma).$

A direct computation with (3.9) yields that

(3.14) $\displaystyle
C\big{(}\mathbf{m}_{4,1}^{(L,N_{1},N_{2})}\big{)}\lesssim
L^{-2}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{12},$

so, applying the operator inequality (3.3) with (3.14), we obtain

$\displaystyle\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\left\|J_{4,1}^{(L,N_{1},N_{2})}(t)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\;\lesssim|t|^{-2}L^{-2}\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{12}\|S_{N_{1}}x^{2}f(t)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\|S_{N_{2}}u(t)\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\;\lesssim|t|^{-2}L^{-2}\sum_{(N_{1},N_{2})\in\mathcal{A}_{L}}\max(N_{1},N_{2})^{2}\min(N_{1},N_{2})^{12}N_{1}^{-10}N_{2}^{-7}\langle
t\rangle^{-1}\varepsilon_{1}^{2}$ $\displaystyle\;\lesssim\langle
t\rangle^{-3}L^{-2}.$

For $J_{4,2}$, we integrate by parts once again using the relation (3.8) to
obtain

$\displaystyle J_{4,2}(t,\eta)$
$\displaystyle=J_{4,2,1}(t,\eta)+J_{4,2,2}(t,\eta),$ $\displaystyle
J_{4,2,1}(t,\eta)$
$\displaystyle=\frac{i}{t^{3}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}\mathbf{m}\otimes\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\cdot\nabla\otimes\Big{(}\widehat{xf}(t,\sigma)\otimes\overline{\widehat{xf}(t,\eta+\sigma})\Big{)}d\sigma,$
$\displaystyle J_{4,2,2}(t,\eta)$
$\displaystyle=\frac{i}{t^{3}}\int_{\mathbf{\mathbb{R}}^{2}}e^{it(\langle\sigma+\eta\rangle-\langle\sigma\rangle)}\nabla_{\sigma}\cdot\Big{(}\mathbf{m}(\eta,\sigma)\,\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\Big{)}\cdot\Big{(}\widehat{xf}(t,\sigma)\otimes\overline{\widehat{xf}(t,\eta+\sigma})\Big{)}d\sigma.$

Using the following pointwise bounds of the multiplier

$\displaystyle\left|\mathbf{m}\otimes\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\right|\lesssim|\eta|^{-3}\min(\langle\sigma\rangle,\langle\sigma+\eta\rangle)^{3}\max(\langle\sigma\rangle,\langle\sigma+\eta\rangle)^{6},$

and

$\displaystyle\left|\nabla_{\sigma}\cdot\big{(}\mathbf{m}(\eta,\sigma)\,\mathbf{m}\otimes\mathbf{m}(\eta,\sigma)\big{)}\right|\lesssim|\eta|^{-3}\min(\langle\sigma\rangle,\langle\sigma+\eta\rangle)^{3}\max(\langle\sigma\rangle,\langle\sigma+\eta\rangle)^{7},$

we obtain by the Hölder inequality

$\displaystyle\left\|J_{4,2,1}(t)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim|t|^{-3}\left\||\eta|^{-3}\chi_{L}(\eta)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\|x^{2}f(t)\|_{H^{10(\mathbf{\mathbb{R}}^{2})}}\|xf(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim|t|^{-3}L^{-2}\varepsilon_{1}^{2},$

and

$\displaystyle\left\|J_{4,2,2}(t)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim|t|^{-3}\left\||\eta|^{-3}\chi_{L}(\eta)\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\|xf(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\|xf(t)\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim|t|^{-3}L^{-2}\varepsilon_{1}^{2},$

respectively. ∎

##  4\. Weighted Energy estimate : Proof of Proposition 2.1

In this section, we prove Proposition 2.1 and complete the proof of our main
theorem.

###### Proof of (2.7).

Recall that under the a priori assumption (2.2), one has

$\displaystyle\|u(t)\|_{W^{7,\infty}(\mathbf{\mathbb{R}}^{2})}\lesssim\langle
t\rangle^{-1}\varepsilon_{1}.$

Interpolating this with the conservation law (1.2)

$\displaystyle\|u(t)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}=\|u_{0}\|_{L^{2}(\mathbf{\mathbb{R}}^{2})},$

we obtain for $2\leq p\leq\infty$

(4.1) $\displaystyle\|u(t)\|_{L^{p}(\mathbf{\mathbb{R}}^{2})}\lesssim\langle
t\rangle^{-2(\frac{1}{2}-\frac{1}{p})}\varepsilon_{1}.$

Then, by using (3.1), Leibnitz rule, and Hardy-Littlewood-Sobolev inequality,
we estimate

$\displaystyle\|N_{\gamma}(u,u,u)(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\||x|^{-\gamma}\ast|u(t)|^{2}\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}\|u(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\||x|^{-\gamma}\ast|u(t)|^{2}\|_{H_{\frac{4}{\gamma}}^{s}(\mathbf{\mathbb{R}}^{2})}\|u(t)\|_{L^{\frac{4}{2-\gamma}}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\|u(t)\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}^{\gamma}\|u_{0}\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}^{2-\gamma}\|u(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}+\|u(t)\|_{H^{s}(\mathbf{\mathbb{R}}^{2})}\|u(t)\|_{L^{\frac{4}{2-\gamma}}(\mathbf{\mathbb{R}}^{2})}^{2}$
$\displaystyle\lesssim\langle t\rangle^{-\gamma}\epsilon_{1}^{3}.$

∎

###### Proof of (2.8).

By the Plancherel’s theorem, we have

$\|x^{2}e^{it\langle
D\rangle}u\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\sim\|\langle\xi\rangle^{10}\mathcal{F}(x^{2}e^{it\langle
D\rangle}u)\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\sim\left\|\langle\xi\rangle^{10}\nabla_{\xi}^{\,2}\widehat{f}\,\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}.$

The Duhamel’s formula (1.6) implies that $\nabla^{2}\widehat{f}$ can be
represented by

$\displaystyle\langle\xi\rangle^{10}\nabla^{2}\widehat{f}(t,\xi)$
$\displaystyle=\langle\xi\rangle^{10}\nabla^{2}\widehat{u_{0}}(\xi)+i{\lambda}\sum_{j=1}^{4}\int_{0}^{t}\mathcal{I}_{j}(t^{\prime},\xi)dt^{\prime},$

where, by abusing the notation,

$\displaystyle\mathcal{I}_{1}(t^{\prime},\xi)$
$\displaystyle=\langle\xi\rangle^{10}\int_{\mathbb{R}^{2}}e^{it^{\prime}\phi(\xi,\eta)}|\eta|^{-2+\gamma}\widehat{x^{2}f}(t^{\prime},\xi-\eta)\mathcal{F}(|u(t^{\prime})|^{2})(\eta)d\eta,$
$\displaystyle\mathcal{I}_{2}(t^{\prime},\xi)$
$\displaystyle=2it^{\prime}\langle\xi\rangle^{10}\int_{\mathbb{R}^{2}}e^{it^{\prime}\phi(\xi,\eta)}|\eta|^{-2+\gamma}\nabla_{\xi}\phi(\xi,\eta)\cdot\widehat{xf}(t^{\prime},\xi-\eta)\mathcal{F}(|u(t^{\prime})|^{2})(\eta)d\eta,$
$\displaystyle\mathcal{I}_{3}(t^{\prime},\xi)$
$\displaystyle=it^{\prime}\langle\xi\rangle^{10}\int_{\mathbb{R}^{2}}\nabla_{\xi}\otimes\nabla_{\xi}\phi(\xi,\eta)(\xi,\eta)e^{it^{\prime}\phi(\xi,\eta)}|\eta|^{-2+\gamma}\widehat{f}(t^{\prime},\xi-\eta)\mathcal{F}(|u(t^{\prime})|^{2})(\eta)d\eta,$
$\displaystyle\mathcal{I}_{4}(t^{\prime},\xi)$
$\displaystyle=-(t^{\prime})^{2}\langle\xi\rangle^{10}\int_{\mathbb{R}^{2}}\big{(}\nabla_{\xi}\phi\otimes\nabla_{\xi}\phi\big{)}(\xi,\eta)e^{it^{\prime}\phi(\xi,\eta)}|\eta|^{-2+\gamma}\widehat{f}(t^{\prime},\xi-\eta)\mathcal{F}(|u(t^{\prime})|^{2})(\eta)d\eta,$

where

(4.2) $\displaystyle\phi(\xi,\eta)$
$\displaystyle=\langle\xi\rangle-\langle\xi-\eta\rangle,\text{ and
}\nabla_{\xi}\phi(\xi,\eta)=\frac{\xi}{\langle\xi\rangle}-\frac{\xi-\eta}{\langle\xi-\eta\rangle}.$

We prove that

$\sum_{j=1}^{4}\left\|\mathcal{I}_{j}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\langle
t^{\prime}\rangle^{-1-\frac{\gamma-1}{3}}\varepsilon_{1}^{3}.$

Estimates for $\mathcal{I}_{1}$. We write

$\displaystyle\left\|\mathcal{I}_{1}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\leq\left\|\mathcal{I}_{1,1}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}+\left\|\mathcal{I}_{1,2}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})},$

where

$\displaystyle\mathcal{I}_{1,1}(t^{\prime},\xi)$
$\displaystyle=\int_{\mathbb{R}^{2}}e^{it^{\prime}{\phi}(\xi,\eta)}|\eta|^{-2+\gamma}\langle\xi-\eta\rangle^{10}\widehat{x^{2}f}(t^{\prime},\xi-\eta)\mathcal{F}(|u(t^{\prime})|^{2})(\eta)\,d\eta,$
$\displaystyle\mathcal{I}_{1,2}(t^{\prime},\xi)$
$\displaystyle=\int_{\mathbb{R}^{2}}e^{it^{\prime}{\phi}(\xi,\eta)}\big{(}\langle\xi\rangle^{10}-\langle\xi-\eta\rangle^{10}\big{)}|\eta|^{-2+\gamma}\widehat{x^{2}f}(t^{\prime},\xi-\eta)\mathcal{F}(|u(t^{\prime})|^{2})(\eta)\,d\eta.$

Using (3.1) and time decay estimates (4.1), the first term is bounded by

$\displaystyle\left\|\mathcal{I}_{1,1}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\Big{\|}\big{(}|x|^{-\gamma}\ast|u(t^{\prime})|^{2}\big{)}\langle
D\rangle^{10}(x^{2}f)\left(t^{\prime}\right)\Big{\|}_{L^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\left\||x|^{-\gamma}\ast|u(t^{\prime})|^{2}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}\|x^{2}f(t^{\prime})\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\|u(t^{\prime})\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}^{\gamma}\|u(t^{\prime})\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}^{2-\gamma}\|x^{2}f(t^{\prime})\|_{H^{10}(\mathbf{\mathbb{R}}^{2})}\lesssim\langle
t^{\prime}\rangle^{-\gamma}\varepsilon_{1}^{3}.$

Next, for $I_{1,2}$, we observe that the singularity near the origin,
$|\eta|^{-2+\gamma}$, is removed by the multiplier as follows:

(4.3)
$\displaystyle\big{(}\langle\xi\rangle^{10}-\langle\xi-\eta\rangle^{10}\big{)}|\eta|^{-2+\gamma}\lesssim|\eta|^{\gamma-1}\max(\langle\xi\rangle^{9},\langle\xi-\eta\rangle^{9}).$

We perform dyadic decomposition

$\displaystyle\mathcal{I}_{1,2}(t^{\prime},\xi)$
$\displaystyle=\sum_{N_{1},N_{2}\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}}\int_{\mathbb{R}^{2}}e^{it^{\prime}{\phi}(\xi,\eta)}\big{(}\langle\xi\rangle^{10}-\langle\xi-\eta\rangle^{10}\big{)}|\eta|^{-2+\gamma}\widehat{S_{N_{1}}x^{2}f}(t^{\prime},\xi-\eta)\widehat{S_{N_{2}}|u(t^{\prime})|^{2}}(\eta)\,d\eta,$

and obtain that

$\displaystyle\left\|\mathcal{I}_{1,2}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim$
$\displaystyle\qquad\sum_{N_{0},N_{1},N_{2}\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}}}\left\|\int_{\mathbb{R}^{2}}e^{it^{\prime}{\phi}(\xi,\eta)}m_{1}^{(N_{0},N_{1},N_{2})}(\xi,\eta)\widehat{S_{N_{1}}x^{2}f}(t^{\prime},\xi-\eta)\widehat{S_{N_{2}}|u(t^{\prime})|^{2}}(\eta)\,d\eta\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})},$

where

$\displaystyle
m_{1}^{(N_{0},N_{1},N_{2})}(\xi,\eta):=\big{(}\langle\xi\rangle^{10}-\langle\xi-\eta\rangle^{10}\big{)}|\eta|^{-2+\gamma}\rho_{N_{0}}(\xi)\rho_{N_{1}}(\xi-\eta)\rho_{N_{2}}(\eta).$

Here, we observe that the summation actually runs over those indices
$(N_{0},N_{1},N_{2})$ in the following set

$\displaystyle\mathcal{A}:=\left\\{(N_{0},N_{1},N_{2})\in(2^{\mathbf{\mathbb{N}}\cup\\{0\\}})^{3}\;|\;N_{0}\lesssim
N_{1}\sim N_{2}\text{ or }N_{0}\sim\max(N_{1},N_{2})\right\\}.$

A direct computation with (4.3) yields that

$\displaystyle
C\big{(}m_{1}^{(N_{0},N_{1},N_{2})}\big{)}\lesssim\max(N_{0},N_{1})^{9}N_{2}^{\,-1+\gamma},$

thus, by using the operator inequality (3.3) with the above bound, we have

$\displaystyle\left\|\mathcal{I}_{1,2}(t^{\prime},\xi)\right\|_{{}_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}}$
$\displaystyle\lesssim\sum_{(N_{0},N_{1},N_{2})\in\mathcal{A}}\max(N_{0},N_{1})^{9}N_{2}^{\,-1+\gamma}\|S_{N_{1}}x^{2}f(t^{\prime})\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\|S_{N_{2}}|u(t^{\prime})|^{2}\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}.$

Then, by applying (3.4) to the quadratic term, we bound the sum over
$N_{2}\leq\langle t^{\prime}\rangle^{\frac{2}{s-9}}$ by

$\displaystyle\langle
t^{\prime}\rangle^{-2}\sum_{\begin{subarray}{c}(N_{0},N_{1},N_{2})\in\mathcal{A}\\\
N_{2}\leq\langle
t^{\prime}\rangle^{\frac{2}{s-9}}\end{subarray}}\max(N_{0},N_{1})^{9}N_{2}^{-1+\gamma}N_{1}^{-10}N_{2}^{-7}\epsilon_{1}^{3}$
$\displaystyle\lesssim\langle
t^{\prime}\rangle^{-2+\frac{2}{s-9}(\gamma+1)}\varepsilon_{1}^{3},$

which is bounded by $\langle
t^{\prime}\rangle^{-\frac{\gamma-1}{3}}\varepsilon_{1}^{3}$, since $s\geq 30$.
On the other hand, we use (3.7) to bound the remaining contribution by

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},N_{2})\in\mathcal{A}\\\
N_{2}\geq\langle
t^{\prime}\rangle^{\frac{2}{s-9}}\end{subarray}}\max(N_{0},N_{1})^{9}N_{1}^{-10}N_{2}^{1+\gamma-s}\varepsilon_{1}^{3}\lesssim\langle
t^{\prime}\rangle^{-2+\frac{2}{s-9}(\gamma+1)}\varepsilon_{1}^{3}.$

Estimates for $\mathcal{I}_{2}$. We decompose the frequency variables
$\xi,\xi-\eta$ into the dyadic pieces $|\xi|\sim N_{0}$ and $|\xi-\eta|\sim
N_{1}$ for $N_{0},N_{1}\in 2^{\mathbf{\mathbb{N}}\cup\\{0\\}}$, respectively.
On the other hand, for $\eta$ variable associated to the potential, we
decompose not only high frequency but also the low frequency part near the
origin, i.e., $|\eta|\sim L$ for $L\in 2^{\mathbf{\mathbb{Z}}}$. Then, we
write

$\displaystyle\mathcal{I}_{2}(t^{\prime},\xi)=\sum_{N_{0},N_{1}\in
2^{\mathbf{\mathbb{N}}\cup\\{0\\}},\,L\in
2^{\mathbf{\mathbb{Z}}}}\mathcal{I}_{2}^{(N_{0},N_{1},L)}(t^{\prime},\xi),$

where

$\displaystyle\mathcal{I}_{2}^{(N_{0},N_{1},L)}(t^{\prime},\xi)=it^{\prime}\int_{\mathbb{R}^{2}}e^{it^{\prime}\phi(\xi,\eta)}\mathbf{m}_{2}^{(N_{0},N_{1},L)}(\xi,\eta)\cdot\widehat{S_{N_{1}}xf}(t^{\prime},\xi-\eta)\widehat{P_{L}(|u(t^{\prime})|^{2})}(\eta)\,d\eta,$

where

$\displaystyle\mathbf{m}_{2}^{(N_{0},N_{1},L)}(\xi,\eta)=\langle\xi\rangle^{10}|\eta|^{-2+\gamma}\,\nabla_{\xi}\phi(\xi,\eta)\rho_{N}(\xi)\rho_{N_{1}}(\xi-\eta)\chi_{L}(\eta).$

Then,

$\displaystyle\left\|\mathcal{I}_{2}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\sum_{(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}}\left\|\mathcal{I}^{(N_{0},N_{1},L)}_{2}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})},$

where

(4.4)
$\displaystyle\widetilde{\mathcal{A}}=\left\\{(N_{0},N_{1},L)\in(2^{\mathbf{\mathbb{N}}\cup\\{0\\}})^{2}\times
2^{\mathbf{\mathbb{Z}}}\;|\;L\lesssim N_{1}\sim N_{2}\text{ or
}L\sim\max(N_{0},N_{1})\right\\}.$

From (3.9), one can readily verify that

(4.5) $\displaystyle C\big{(}\mathbf{m}_{2}^{(N_{0},N_{1},L)}\big{)}\lesssim
N_{0}^{10}L^{-1+\gamma}\max(N_{0},N_{1})^{-1}.$

Then, by the operator inequality (3.3) with the above bound, we obtain

(4.6) $\displaystyle\begin{aligned}
&\left\|\mathcal{I}_{2}^{(N_{0},N_{1},L)}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\\\
&\qquad|t^{\prime}|N_{0}^{10}L^{-1+\gamma}\max(N_{0},N_{1})^{-1}\left\|S_{N_{1}}{xf}(t^{\prime})\,\right\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}\left\|P_{L}|u(t^{\prime})|^{2}\right\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}.\end{aligned}$

We observe from (3.13) that under the a priori assumption

$\left\|S_{N_{1}}{xf}(t^{\prime})\,\right\|_{L^{2}}\lesssim
N_{1}^{-10}\epsilon_{1}.$

Using this and (3.7), we bound the summation over $L\leq\langle
t^{\prime}\rangle^{-1}$ by

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\,\lesssim\,\langle
t^{\prime}\rangle^{-1}\end{subarray}}\left\|\mathcal{I}_{2}^{(N_{0},N_{1},L)}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim|t^{\prime}|\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\,\lesssim\,\langle
t^{\prime}\rangle^{-1}\end{subarray}}N_{0}^{10}L^{1+\gamma}\max(N_{0},N_{1})^{-1}N_{1}^{-10}\varepsilon_{1}^{3}$
$\displaystyle\lesssim\langle t^{\prime}\rangle^{-\gamma}\varepsilon_{1}^{3},$

the summation over $\langle t^{\prime}\rangle^{-1}\leq L\leq\langle
t^{\prime}\rangle^{\frac{2}{s-3}}$ by

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
\langle t^{\prime}\rangle^{-1}\,\lesssim L\,\lesssim\,\langle
t^{\prime}\rangle^{\frac{3}{s-3}}\end{subarray}}\left\|\mathcal{I}_{2}^{(N_{0},N_{1},L)}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\quad\lesssim\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
\langle t^{\prime}\rangle^{-1}\,\lesssim L\,\lesssim\,\langle
t^{\prime}\rangle^{\frac{2}{s-3}}\end{subarray}}|t^{\prime}|^{-2}N_{0}^{10}L^{-2+\gamma}\max(N_{0},N_{1})^{-1}N_{1}^{-10}\varepsilon_{1}^{3}$
$\displaystyle\quad\lesssim\langle
t^{\prime}\rangle^{-\gamma}\varepsilon_{1}^{3}+\langle
t^{\prime}\rangle^{-2+\frac{2}{s-3}(7+\gamma)}\varepsilon_{1}^{3},$

and finally the summation over $L\geq\langle
t^{\prime}\rangle^{\frac{2}{s-3}}$ by

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\,\geq\,\langle
t^{\prime}\rangle^{\frac{2}{s-3}}\end{subarray}}\left\|\mathcal{I}_{2}^{(N_{0},N_{1},L)}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim|t^{\prime}|\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\,\geq\,\langle
t^{\prime}\rangle^{\frac{2}{s-3}}\end{subarray}}N_{0}^{10}L^{1+\gamma-s}\max(N_{0},N_{1})^{-1}N_{1}^{-10}$
$\displaystyle\lesssim\langle
t^{\prime}\rangle^{-2+\frac{2}{s-3}(7+\gamma)}\varepsilon_{1}^{3},$

which is bounded by $\langle
t^{\prime}\rangle^{-\frac{\gamma-1}{3}}\varepsilon_{1}^{3}$, since $s\geq 30$.

Estimates for $\mathcal{I}_{3}$. $\mathcal{I}_{3}$ can be handled in an
anolugous manner to $\mathcal{I}_{2}$. Indeed, one finds that the multiplier
$\nabla_{\xi}^{2}\phi(\xi,\eta)$ in $\mathcal{I}_{3}$ verifies the smaller
bound than the one given in (3.9) satisfied by the multiplier
$\nabla_{\xi}\phi(\xi,\eta)$ in $\mathcal{I}_{2}$. More precisely, the
following bound holds

$\displaystyle\left|\nabla_{\xi}^{2}\phi(\xi,\eta)\right|\lesssim\frac{|\eta|}{\max(\langle\xi\rangle,\langle\xi-\eta\rangle)^{2}}.$

Thus, if we let

$\displaystyle\mathbf{m}_{3}^{(N_{0},N_{1},L)}(\xi,\eta)=\langle\xi\rangle^{10}|\eta|^{-2+\gamma}\nabla_{\xi}^{2}\phi(\xi,\eta)\rho_{N_{0}}(\xi)\chi_{L}(\eta)\rho_{N_{1}}(\xi-\eta),$

one can verify that

$\displaystyle C(\mathbf{m}_{3}^{(N_{0},N_{1},L)})$ $\displaystyle\lesssim
N_{0}^{10}\max(N_{0},N_{1})^{-2}L^{-1+\gamma}.$

Applying these bounds into (LABEL:I2:CM) instead of (4.5), one can obtain the
desired bounds.

Estimates for $\mathcal{I}_{4}$. It remains to estimate the main case
$\mathcal{I}_{4}$. As before, we decompose

$\displaystyle\mathcal{I}_{4}(t^{\prime},\xi)=\sum_{N_{0},N_{1}\in(2^{\mathbf{\mathbb{N}}})^{2},\,L\in
2^{\mathbf{\mathbb{Z}}}}\mathcal{I}_{4}^{(N_{0},N_{1},L)}(t^{\prime},\xi),$

where

$\displaystyle\mathcal{I}_{4}^{(N_{0},N_{1},L)}(t^{\prime},\xi)=-(t^{\prime})^{2}\int_{\mathbb{R}^{2}}\mathbf{m}_{4}^{(N_{0},N_{1},L)}(\xi,\eta)e^{it^{\prime}\phi(\xi,\eta)}\widehat{S_{N_{1}}f}(t^{\prime},\xi-\eta)\widehat{P_{L}(|u(t^{\prime})|^{2})}(\eta)\,d\eta,$

where

$\displaystyle\mathbf{m}_{4}^{(N_{0},N_{1},L)}(\xi,\eta)=\langle\xi\rangle^{10}|\eta|^{-2+\gamma}\big{(}\nabla_{\xi}\phi(\xi,\eta)\big{)}^{2}\rho_{N_{0}}(\xi)\rho_{N_{1}}(\xi-\eta)\chi_{L}(\eta).$

Then,

$\displaystyle\left\|\mathcal{I}_{4}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\sum_{(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}}\left\|\mathcal{I}^{(N_{0},N_{1},L)}_{4}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})},$

where $\widetilde{\mathcal{A}}$ is given in (4.4). From (3.9), one can readily
verify that

$\displaystyle C\big{(}\mathbf{m}_{4}^{(N_{0},N_{1},L)}\big{)}\lesssim
L^{\gamma}N_{0}^{10}\max(N_{0},N_{1})^{-2},$

which gives that

$\displaystyle\left\|\mathcal{I}_{4}^{(N_{0},N_{1},L)}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim|t^{\prime}|^{2}L^{\gamma}N_{0}^{10}\max(N_{0},N_{1})^{-2}\|S_{N_{1}}u(t^{\prime})\|_{L^{\infty}(\mathbf{\mathbb{R}}^{2})}\|P_{L}(|u(t^{\prime})|^{2})\|_{L^{2}(\mathbf{\mathbb{R}}^{2})}.$

Using (3.6), we estimate the summation over $L\leq\langle
t^{\prime}\rangle^{-1}$

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\leq\langle
t^{\prime}\rangle^{-1}\end{subarray}}\left\|\mathcal{I}^{(N_{0},N_{1},L)}_{4}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}\lesssim\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\leq\langle
t^{\prime}\rangle^{-1}\end{subarray}}|t^{\prime}|L^{\gamma+1}N_{0}^{10}\max(N_{0},N_{1})^{-2}N_{1}^{-7},$

the summation over $\langle t^{\prime}\rangle^{-1}\leq L\leq\langle
t^{\prime}\rangle^{\frac{3}{s-3}}$

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
\langle t^{\prime}\rangle^{-1}\leq L\leq\langle
t^{\prime}\rangle^{\frac{3}{s-3}}\end{subarray}}\left\|\mathcal{I}^{(N_{0},N_{1},L)}_{4}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\quad\lesssim\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
\langle t^{\prime}\rangle^{-1}\leq L\leq\langle
t^{\prime}\rangle^{\frac{3}{s-3}}\end{subarray}}\langle
t^{\prime}\rangle^{-2}L^{\gamma-2}N_{0}^{10}\max(N_{0},N_{1})^{-2}N_{1}^{-7}\varepsilon_{1}^{3}$
$\displaystyle\quad\lesssim\langle
t^{\prime}\rangle^{-2}\varepsilon_{1}^{3}+\langle
t^{\prime}\rangle^{-2+\frac{3(6+\gamma)}{s-3}}\varepsilon_{1}^{3},$

and finally the summation over $L\geq\langle
t^{\prime}\rangle^{\frac{3}{s-3}}$

$\displaystyle\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\geq\langle
t^{\prime}\rangle^{\frac{3}{s-3}}\end{subarray}}\left\|\mathcal{I}^{(N_{0},N_{1},L)}_{4}(t^{\prime},\xi)\right\|_{L_{\xi}^{2}(\mathbf{\mathbb{R}}^{2})}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}(N_{0},N_{1},L)\in\widetilde{\mathcal{A}}\\\
L\geq\langle
t^{\prime}\rangle^{\frac{3}{s-3}}\end{subarray}}|t^{\prime}|L^{\gamma+1-s}\langle
N_{0}\rangle^{10}\max(N_{0},N_{1})^{-2}N_{1}^{-7}\varepsilon_{1}^{3}$
$\displaystyle\lesssim\langle
t^{\prime}\rangle^{-2+\frac{3}{s-3}(\gamma+6)}\varepsilon_{1}^{3},$

which is bounded by $\langle
t^{\prime}\rangle^{-\frac{\gamma-1}{3}}\varepsilon_{1}^{3}$, since $s\geq 30$.
∎

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