text:
\section{Introduction}
We consider the following Hartree type Dirac equation:
\begin{eqnarray}\label{maineq}
\left\{\begin{array}{l}
(-i\partial_t + \al\cdot D + m\beta) \psi = (V* \left< \psi,\beta\psi\right>)\beta\psi \;\; \mathrm{in}\;\; \mathbb{R}^{1+2},\\
\psi(0) = \psi_0 \in H^s(\rt),
\end{array} \right.
\end{eqnarray}
where $\psi:\mathbb{R}^{1+2} \to \ct$ is the spinor field represented by a column vector $\left(
                                                                                \begin{array}{c}
                                                                                \psi_1 \\
                                                                                \psi_2 \\
                                                                                \end{array}
                                                                                \right)$, $D = -i\nabla$, and $\al = (\al^1, \al^2), \beta$ are the Dirac matrices defined by
\begin{align*}
 \alpha^1 = \left(\begin{array}{ll} 0 & \;\;1 \\ 1 & \;\;0 \end{array} \right) \quad \alpha^2 = \left(\begin{array}{ll} 0 & -i \\ i & \;\;0 \end{array} \right),\quad \beta = \left(\begin{array}{ll} 1 & \;\,\,\,0 \\ 0 & -1 \end{array} \right).
\end{align*}
The constant $m \ge 0$ is a physical mass parameter and the symbol $*$ denotes convolution in $\rt$. The potential $V$ is of Coulomb type and defined by $c|x|^{-\gamma}\;(0 < \gamma < 2, c \in \mathbb R \setminus \{0\})$. Throughout the paper $\left< , \right>$ denotes the inner product on $\mathbb C^2$ and $\left< \psi, \phi\right>_{L_x^2} := \int_{\mathbb R^2} \left< \psi, \phi\right>\,dx$. When $\gamma = 1$, \eqref{maineq} can be regarded as a simplified model of Chern-Simons-Dirac system in Coulomb gauge \cite{boucanma}.

It is well-known that every smooth solution to \eqref{maineq} satisfies the mass conservation: $\|\psi(t)\|_{L_x^2} = \|\psi_0\|_{L_x^2}$ for any $t$ within the existence time interval. For instance see \cite{chagla}. In the massless case ($m=0$) the equation \eqref{maineq} has the scaling invariance structure in $\dot{H}^{\frac{\gam-1}2}$. That is, the $\dot{H}^{\frac{\gam-1}2}$-scaled function $\psi_\lambda$ for $\lam > 0$ defined by $\psi_\lam(t,x) = \lam^\frac{3-\gam}{2} \psi (\lam t,\lam x)$ is also the solution to the equation \eqref{maineq}. By this reason, the \eqref{maineq} is said to be mass-critical (supercritical, subcritical) when $\gamma = (>, <)\;1$, respectively.

In this paper we use the representation of solution based on the massive Klein-Gordon equation. Thus we only consider the massive case ($m > 0$) and for simplicity we assume that $m = 1$.

Let us define the energy projection operators $\Pi_\pm(D)$ by
$$
\Pi_{\pm}(D) := \frac12 \left( I \pm \frac1{\brad}[\alpha \cdot D + \beta] \right),
$$
where $\left< D\right> = (1 - \Delta)^\frac12$.
Then we get
\begin{align}
\al\cdot D + \beta = \brad \left(\Pi_{+}(D) - \Pi_{-}(D)\right),
\end{align}
and
\begin{align}\label{proj-commu}
\Pi_{\pm}(D)\Pi_{\pm}(D) = \Pi_{\pm}(D), \;\; \Pi_{\pm}(D)\Pi_{\mp}(D) = 0.
\end{align}
We denote $\Pi_\pm(D)\psi $ by $\psi_\pm$. Then the equation \eqref{maineq} becomes the following system of semi-relativistic Hartree equations:
\begin{align}\label{maineq-decom}
 (-i\partial_t \pm \brad)\psi_\pm = \Pi_{\pm}(D)[(V* \left<\psi,\beta\psi\right>)\beta\psi]
\end{align}
with initial data $\psi_{\pm}(0,\cdot) = \psi_{0, \pm} := \Pi_{\pm}(D)\psi_0$. The free solutions of \eqref{maineq-decom} are $e^{\mp it\brad}\psi_{0, \pm}$, respectively, where
$$
e^{\mp it\langle D\rangle}f(x) = \mathcal F^{-1} e^{\mp it \langle \xi \rangle}\mathcal F f = \frac1{(2\pi)^2}\int_{\mathbb R^2} e^{i(x\cdot \xi \mp t\langle \xi \rangle)}\widehat f(\xi)\,d\xi.
$$
Here $\langle \xi \rangle := (1 + |\xi|^2)^\frac12$ and $\mathcal F, \mathcal F^{-1}$ are Fourier transform, its inverse, respectively.
Then by Duhamel's principle the Cauchy problem \eqref{maineq-decom} is equivalent to solving the integral equations:
\begin{align}\label{inteq0}
\psi_\pm(t) = e^{\mp it\langle D\rangle}\psi_{0, \pm} + i\int_0^t e^{\mp i(t-t')\langle D\rangle}\Pi_{\pm}(D)[(V*\langle \psi, \beta \psi\rangle) \beta \psi](t')\,dt'.
\end{align}
We say that the solution $\psi$ scatters forward (or backward) in a Hilbert space $\mathcal H$ if there exist $\psi^\ell$,  linear solutions to $(-i\partial_t + \alpha \cdot D + \beta)\psi = 0$, such that
\begin{align}\label{sense1}
\|\psi(t) - \psi^\ell(t)\|_{\mathcal H} \to 0\;\;\mbox{as}\;\;t \to +\infty\; (-\infty,\;\;\mbox{respectively}).
\end{align}
In most cases we may take $\mathcal H = H^s$.
Equivalently, $\psi$ is said to scatter forward ( or backward) in $\mathcal H$ if there exist $\psi_{\pm }^\ell := e^{\mp it\brad}\varphi_{\pm } \;(\varphi_{\pm} \in \mathcal H)$ such that
\begin{align}\label{sense2}
\|\psi_\pm(t) - \psi_{\pm}^\ell(t)\|_{\mathcal H} \to 0\;\;\mbox{as}\;\;t \to \pm \infty.
\end{align}

We are concerned with the low-regularity global well-posedness (GWP) and scattering with small initial data for \eqref{maineq} and we equivalently consider  \eqref{inteq0}.
For these questions the Dirac equation with Yukawa potential was studied in \cite{yang, tes2d, tes3d} and \cite{geosha}. Yang \cite{yang} and Tesfahun \cite{tes3d} showed independently scattering results on $H^s(\mathbb R^3)$ for $s>0$ with mass $m \ge 0$.  Tesfahun \cite{tes2d} established scattering in $H^s(\mathbb R^2)$ for $s > 0$ for the massive case. One can find results about power type nonlinearity in \cite{behe, behe2d}. As related topics we refer the readers to \cite{mats1, mats2}, \cite{huhoh, boucanma, holee}, and \cite{choz, coss,  hete, pusa} for the massless Dirac equation with nonlinearity $(|x|^{-\gamma}*|\psi|^{p-1})\psi$, Chern-Simons-Dirac equation in Coulomb gauge, and massive semi-relativistic equation with Coulomb potential, respectively.

Until now not much has been known about the low-regularity issue and the  corresponding scattering theory of Dirac equation \eqref{maineq} with Coulomb type potential. In this paper we suggest a 2d result on the mass-supercritical case as follows.

\begin{thm}\label{mainthm}
Let $1 < \gamma < 2$ and $s > \gam-1$. Then there exists $\ve > 0$ such that for all $\psi_0 \in H^s(\rt)$ satisfying
$$
\|\psi_0\|_{H^s} < \ve,
$$
there exists a unique global solution $\psi \in C(\mathbb{R}; H^s(\rt))$ to \eqref{maineq}. In particular, the global solution $\psi$ scatters both forward and backward in $H^s$.
\end{thm}

For the proof of Theorem \ref{mainthm} we apply the frequency localization to potential $V$ which is two-fold. One is to handle the high-frequency part of $V$ by the null-form and modulation estimates of \cite{tes2d}. The other is to deal with the low-frequency part of $V$ via localized bilinear estimates arising from null-structure described in \cite{behe2}. Since the low-frequency part of Yukawa type potential is bounded, we do not need to handle it. However, the low-frequency part of Coulomb type potential contains significant singularity which is the main obstacle for GWP. To control it we establish the 2d localized bilinear estimates, which are adapted from 3d versions in \cite{heya, yang}. For details see Section \ref{low-esti} below. Our argument can also be applied to the 3d problem in the same way. We leave it to the readers.

In view of the scaling structure there seems to be still a room for GWP and scattering when $\frac{\gamma-1}2 \le s \le \gamma-1$. This issue would be worth pursuing as a future work.

\begin{rem}\label{assumption}
The potential $V$ can be generalized as follows. $\widehat V \in C^\infty(\mathbb R^2 \setminus \{0\})$  and there exist $\gamma_0, \gamma$ such that $1 < \gamma_0 \le 2$, $1\le \gamma < 2$, and
\newcommand{\pgam}{{\gam_0})}
\begin{align*}
\left||\xi|^{k} \partial^k\widehat{V}(\xi) \right| \les   |\xi|^{-(2-\pgam)}\chi_{\{|\xi| \le 1\}} + |\xi|^{-(2-\gam)}\chi_{\{|\xi| > 1 \}}\;(0 \le k \le 4).
\end{align*}
The prototype of $V$ with the property as above is $\mathcal F^{-1}(m_0^2+|\xi|^2)^{-\frac{\gam}2}$ $(m_0 \ge 0, 1 < \gam \le 2)$, which includes the Coulomb type potential $c|x|^{-\gam} \,(m_0 = 0, 1 < \gam < 2)$ and Yukawa type $\mathcal F^{-1}(m_0^2+|\xi|^2)\;(m_0 > 0)$. $V$ does not include two critical potentials: mass-critical case $c|x|^{-1}\;(m_0 = 0, \theta = 1)$, 2d Coulomb potential $-\frac1{2\pi}\ln |x| = \mathcal F^{-1}(|\xi|^{-2})\;(m_0 = 0)$. However, it allows a slight modification of low frequency part of the critical potentials $(\mathcal F^{-1}(|\xi|^{-(1-\varepsilon)}(1+|\xi|)^{-\theta}), \;0 < \varepsilon \ll 1, \theta = \varepsilon$ or $ 1+\varepsilon)$. For this type potential we can get exactly the same conclusion as Theorem \ref{mainthm}. At this point, the low-frequency index $\gamma_0$ is not involved in the regularity because the summation of low-frequency part is carried out on $|\xi| \le 1$ and results in a constant (see Section \ref{lowfrequency} below).
\end{rem}

The mass-supercritical potential can be referred to as a short-range one in the context of scattering theory. In the mass-subcritical (long-range) regime it is usually expected that a number of nontrivial smooth solutions do not behave like a linear solution in the sense of \eqref{sense1} or \eqref{sense2} as time goes on. This turns out to be true for our problem with $0 < \gamma \le 1$. To be more precise, let
$$
\mathcal V(\psi, \phi)(t) = \int  \Big (|\cdot|^{-\gamma}*\left<\psi, \phi\right>\Big)(t, x)  \left<\psi,  \phi\right> (t, x) \;dx.
$$
Then we have the following.

\begin{thm}\label{non-scat}
Let $0 < \gamma < 1$. Suppose that $\psi$ is a smooth global solution to \eqref{maineq} which scatters forward in $L_x^2$ to a smooth linear solution $\psii$.
Suppose that for some $0 < \theta < 1$ and $t_* > 0$ the linear solution $\psii$ satisfies that
	\begin{align}\label{assu-scat}
	 |\mathcal V(\psii, \beta\psii)(t)| \ge \theta \mathcal V(\psii, \psii)(t)
	\end{align}
for any $t > t_*$. Then $\psi, \psii =0$  in $L_x^2$.
Furthermore, if $\gamma = 1$ and $\psi$ is a smooth global solution to \eqref{maineq} which scatters forward in $H^{s, \delta}$ for some $0 < s, \delta \ll 1$ to a smooth linear solution $\psii$ satisfying \eqref{assu-scat}, then $\psi, \psii =0$  in $L_x^2$.
\end{thm}
\noindent Here $H^{s, \delta}$ denotes the space $\{f \in L_x^2 : \|f\|_{H^{s, \delta}} := \|f\|_{H^s} + \|(1+|x|)^\delta f\|_{L_x^2}  < \infty \}$. As for the backward scattering, exactly the same statement holds with the only change $t < t_* < 0$. If $\psi$ scatters in $L_x^2$ or $H^{s, \delta}$ to $\psii$ satisfying \eqref{assu-scat}, then by Lemmas \ref{infty-esti} and \ref{infty-weight} below one can find $0 < \theta' < 1$ and $t_{**} > 0$ such that
\begin{align}\label{trans-psi}
\mathcal V(\psi, \beta \psi )(t) \ge \theta' \mathcal V(\psi, \psi)(t)
\end{align}
for any $t > t_{**}$.

A sufficient condition for \eqref{assu-scat} is that there exists $0 < \theta'' < 1$ such that
$$
\theta''|{\psii}_{, 1}(t, x)| \ge |{\psii}_{, 2}(t, x)|\;\;{\rm or}\;\; \theta''|{\psii}_{, 2}(t, x)| \ge |{\psii}_{, 1}(t, x)|
$$
for any $(t, x) \in \mathbb R^{1 + 2}$ with $t > t_*$, where $\psii = \left(
                                                                        \begin{array}{c}
                                                                          {\psii}_{, 1} \\
                                                                          {\psii}_{, 2} \\
                                                                        \end{array}
                                                                      \right)
$.

In Section \ref{sec-nonsca}, we prove Theorem \ref{non-scat}. Here we explain how the condition \eqref{assu-scat} and $H^{s, \delta}$ assumption come into play. In the proof it is essential to handle the lower bound of functional $H(t)  = {\rm sgn}(c) {\rm Im} \left< \psi(t), \psii(t) \right>_{L_x^2} $ by $\|\psii(0)\|_{L_x^2}$. We will show
$$\left|\frac{d}{dt}H(t)\right| \ge |c| |\mathcal V(\psii, \beta\psii)(t)| + o(t^{-\gamma}).$$ $H^{s, \delta}$ assumption is needed for the remainder term $o(t^{-\gamma})$ when $\gamma = 1$. Due to the matrix $\beta$ the value of $\mathcal V(\psii, \beta\psii)$ may vanish. To avoid this we assume \eqref{assu-scat} and hence obtain
$|\frac{d}{dt}H(t)| \gtrsim t^{-\gamma}\|\psii(0)\|_{L_x^2}^2 + o(t^{-\gamma})$ for sufficiently large $t$.
Therefore, if $0 < \gamma \le 1$ and $\|\psii(0)\|_{L_x^2} > 0$, then the lower bound eventually will lead us to contradiction to the uniform boundedness of $H(t)$.

In the papers \cite{chagla, ozya}, a class of initial data was considered for the nonlinear spinor interaction $\left< \psi, \beta\psi\right>$ to be destructive and hence for the nonlinear term to vanish. This argument can be applied to the equation \eqref{maineq}. In this case the solution to \eqref{maineq} is just linear and is referred to {\it scatter trivially}. As an example we can take the initial data $\psi_0$ satisfying that $A\psi_0 = \overline{\psi_0}$, where $A := \left(\begin{array}{ll} 0 & a_1 \\ a_2 & 0 \end{array} \right)$ for any $a_1,a_2 \in \mathbb{C}$ with $|a_1| = |a_2|=1$. This is the reason why some condition such as (1.8) (consequently \eqref{trans-psi}) is necessary for the nonexsitence of scattering.

In view of Theorem \ref{non-scat} the mass-critical index $\gamma = 1$ is actually scattering-critical. Hence as in many scattering-critical problems we expect a modified scattering theory.

\section{Notation and function space}

\subsection{Notations}\label{nota}
Throughout the paper we use the following notations.

\noindent $(1)$ $\|\cdot\|$ denotes $\|\cdot\|_{L_{t,x}^2}$.\\

\noindent$(2)$ (Mixed-normed spaces) 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 $\|u\|_{L_I^qX} := \|\|u(t)\|_X\|_{L_I^q} < \infty$. Especially, we denote  $L_I^qL_x^r = L_t^q(I; L_x^r(\rt))$, $L_{I, x}^q = L_I^qL_x^q$, $L_t^qL_x^r = L_{\mathbb R}^qL_x^r$.\\

\noindent$(3)$ (Littlewood-Paley operators) Let $\rho$ be a Littlewood-Paley function such that $\rho \in C^\infty_0(B(0, 2))$ with $\rho(\xi) = 1$  for $|\xi|\le 1$ and define $\rho_{\lam}(\xi):= \rho\left(\frac {\xi}{\lam}\right) - \rho\left(\frac{2\xi}{\lam}\right)$ for $\lam \in 2^\mathbb{Z}$. Then we define the frequency projection $P_\lam$ by $\mathcal{F}(P_\lam f)(\xi) = \rho_{\lam}(\xi)\widehat{f}(\xi)$, and also $P_{\le \mu} := I - \sum_{\lam > \mu}P_{\lam}$. In addition $P_{\mu \le \cdot \le \nu} := \sum_{\mu \le \lambda \le  \nu}P_\lam$ and $P_{\sim \nu} := \sum_{\lambda \sim \nu}P_{\lambda}$. For $ \lam \in 2^\mathbb{Z}$ we denote $\widetilde{\rho_\lam} = \rho_{\frac{\lam}{2}} + \rho_\lam + \rho_{2\lam}$. In particular, $\widetilde{P_\lam}P_\lam = P_\lam\widetilde{P_\lam} = P_\lam$ where $\widetilde{P_\lam} = \mathcal{F}^{-1}\widetilde{\rho_\lam}\mathcal{F}$. Next we define a Fourier localization operators $P_\lam^1$ as follow:
$$
P_\lam^1 f =  \left\{ \begin{array}{ll} 0  &\;\;\;\mbox{if}\;\; \lam < 1 ,\\ P_{\le 1}f &\;\;\;\mbox{if}\;\; \lam = 1 ,\\ P_\lam f & \;\;\;\mbox{if}\;\; \lam > 1.   \end{array} \right .
$$
Especially, we denote $P_\lam^1 f $ by $f_{\lam}$ for any measurable function $f$.\\

\noindent$(4)$ As usual different positive constants depending only on $\gamma_0, \gamma$ are denoted by the same letter $C$, if not specified. $A \lesssim B$ and $A \gtrsim B$ mean that $A \le CB$ and
$A \ge C^{-1}B$, respectively for some $C>0$. $A \sim B$ means that $A \lesssim B$ and $A \gtrsim B$.

\subsection{$U^p - V^p$ and adapted function space}
We explain concisely $U^p-V^p$ spaces. For more details we refer the readers to \cite{ haheko,haheko2, kota, kotavi}.
Let $1 \le p < \infty$ and $\mathcal{I}$ be the collection of finite partitions $\{ t_0 , \cdots, t_N\}$ satisfying $-\infty < t_0 < \cdots < t_N \le \infty$. If $t_N =\infty$, by convention, $u(t_N) := 0$ for any $u : \mathbb R \to L_x^2(\mathbb R^2)$.
Let us define a $U^p$-atom by a step function $a : \mathbb{R} \to L_x^2$ of the form
$$
a(t) = \sum_{k=1}^N \chi_{[t_{k-1},t_k)}\phi_k(t) \;\;\mbox{with}\;\; \sum_{k=1}^N \|\phi_k\|_{L_x^2}^p=1.
$$
Then the $U^p$ space is defined by
$$
U^p = \left\{ u = \sum_{j=1}^\infty \lam_j a_j : \mbox{$a_j$ are $U^p$-atoms and $\{\lam_j\} \in \ell^1$ },  \|u\|_{U^p}< \infty \right\},
$$
where the $U^p$-norm is defined by
$$
\|u\|_{U^p}:= \inf_{\mbox{representation of $u$}} \;\;\sum_{j=1}^\infty|\lam_j|.
$$
We next define $V^p$ as the space of all right-continuous functions $v : \mathbb{R} \to L_x^2$ satisfying that $\underset{t \to -\infty}{\lim}v(t) =0$ and the norm
$$
\|v\|_{V^p} := \sup_{\{t_k\} \in \mathcal{I}} \left( \sum_{k=1}^N \|v(t_k)  - v(t_{k-1})\|_{L_x^2}^p \right)^{\frac1p}
$$
is finite.

We introduce some properties of $U^p$ and $V^p$.
\begin{lem}[\cite{haheko}]\label{embedd} Let $1 \le p < q <\infty$. Then the following holds.
	\item[(i)] $U^p$ and $V^p$ are Banach spaces.
	\item[(ii)] The embeddings $U^p \hookrightarrow V^p \hookrightarrow U^q \hookrightarrow L^{\infty}(\mathbb{R};L_x^2) $ are continuous.
\end{lem}
These spaces have the useful duality property.
\begin{lem}[Corollary of \cite{kotavi}]\label{duality} Let $u \in U^p$ be absolutely continuous with $1 < p < \infty$. Then
	$$
	\|u\|_{U^p} = \sup \left\{ \left| \int \left<u', v \right>_{L_x^2} dt \right| : v \in C_0^{\infty}, \;\; \|v\|_{V^{p'}}=1 \right\}.
	$$
\end{lem}

Now let us define the adapted function spaces $U_{\pm}^p,\; V_{\pm}^p$ as follows:
$$
\|u\|_{U_{\pm}^p} : =\|\epm u\|_{U^p} \;\;\mbox{and}\;\;  \|u\|_{V_{\pm}^p} : =\|\epm u\|_{V^p}.
$$

\begin{lem}[Logarithm interpolation, Lemma 4.12 of \cite{kotavi}]\label{log} Let $p > 2$ and $v \in V_{\pm}^2$. Then there exist a constant $\kappa(p) > 0$ and functions $u \in U_\pm^2$ and $w \in U_{\pm}^p$ such that
	$$
	v = u +w
	$$
	and for all $M\ge1$
	$$
	\frac{\kappa}{M} \|u\|_{U_\pm^2} + e^M\|w\|_{U_{\pm}^p} \les \|v\|_{V_{\pm}^2}.
	$$
\end{lem}

\begin{prop}[Transfer principle, Proposition 2.19 of \cite{haheko}]\label{transfer} Let
	$$
	T : L_x^2 \times L_x^2 \times \cdots \times L_x^2 \to L_{loc}^1
	$$
	be a multilinear operator. If
	$$
	\left\|T\left( e^{\pm_1 it\brad}f_1 , e^{\pm_2 it\brad}f_2 , \cdots ,e^{\pm_k it\brad}f_k \right)\right\|_{L_t^qL_x^r} \les \prod_{j=1}^{k}\|f_j\|_{L_x^2}
	$$
	for some $1 \le q,r \le \infty$, then we have
	$$
	\|T(u_1, u_2,\cdots, u_k)\|_{L_t^q L_x^r} \les \prod_{j=1}^k \|u_j\|_{U_{\pm_j}^q},
	$$
	where $u_j$ are arbitrary functions in $U_{\pm_j}^q$.
\end{prop}

\newcommand{\etahat}{\widehat{\eta}}
\newcommand{\sighat}{\widehat{\sigma}}
\newcommand{\brae}{\left<\eta \right>}
\newcommand{\bras}{\left<\sigma \right>}

\section{Null-form and bilinear estimates}

 The null structure of \eqref{maineq} was revealed concretely in \cite{tes2d} and can be described as follows:
\begin{align*}
\left<  \Pi_{+}(D) \psi_1 , \beta \Pi_{\pm}(D)\psi_2 \right>
= \sum_{j=1}^3 Q_j^\pm(\psi_1,\psi_2)    + \sum_{j=1}^4 B_j^\pm (\psi_1, \psi_2),
\end{align*}
where the null forms $Q_j^\pm$ and bilinear forms $B_j^\pm$ are given by
\begin{align*}
 Q_j^\pm(\psi_1,\psi_2)&=  \mathcal{F}^{-1} \iint_{\eta-\sigma = \xi} \left< q_j^\pm (\eta,\sigma)   \widehat{\psi_1}(\eta) ,  \beta \widehat{\psi_2}(\sigma)  \right> \,d\eta d\sigma    ,\\
 B_j^\pm (\psi_1, \psi_2)&= \mathcal{F}^{-1} \iint_{\eta-\sigma = \xi} \left<  b_j^\pm (\eta,\sigma)  \widehat{\psi_1}(\eta) ,  \beta \widehat{\psi_2}(\sigma)  \right> \,d\eta d\sigma
\end{align*}
 for $j= 1,2,3$, and
$$
 B_4^\pm (\psi_1, \psi_2)= \pm  \iint_{\eta-\sigma = \xi}  \left<\eta\right>^{-1} \left<  \widehat{\psi_1^+}(\eta) ,  \widehat{\psi_2^\pm}(\sigma) \right>\,d\eta d\sigma,
$$
where
\begin{align*}
q_1^{\pm}(\eta, \sigma) &:=  (|\etahat||\sighat| \pm \etahat\cdot \sighat)I,   \qquad\qquad   b_1^{\pm}(\eta, \sigma) :=  (1- |\etahat||\sighat|)I,\\
q_2^{\pm}(\eta, \sigma) &:=  -(\etahat|\sighat| \pm \sighat|\etahat|)\cdot\al, \qquad\quad   b_2^{\pm}(\eta, \sigma) :=  - \frac{[\eta(\bras - |\sigma|) \pm \sigma(\brae - |\eta|)]\cdot\al}{\brae\bras},\\
q_3^{\pm}(\eta, \sigma) &:=  \pm(\etahat_1\sighat_2 - \etahat_2 \sighat_1)\al^1\al^2,   \;\;\;\;\;\;   b_3^{\pm}(\eta, \sigma) :=  \pm \frac{[(\eta-\sigma)\cdot\al]\beta - (\brae \pm \bras)\beta + I}{\brae \bras}.\\
\end{align*}
Here we used the normalization $\widehat{\xi} = \frac{\xi}{\left<\xi \right>}, \widehat{\xi}_j = \frac{\xi_j}{\left<\xi \right>}\; (j =  1, 2 )$.

\medskip

We now provide localized null-form estimates and bilinear estimates.

\subsection{High-frequency estimates}

\begin{lem}[Null-form estimates, Corollary 1 of \cite{tes2d}]\label{null-esti}
Let $\mu,\lam_1,\lam_2 \in 2^{\mathbb{Z}}$ satisfy $\mu, \lam_1, \lam_2 \ge 1$ and $\lammin$ be minimum value of $\lam_1$ and $\lam_2$.  Then the following holds:
\item[(i)]
\begin{align*}
\left\|P_\mu Q_j^+ (\psi_{\lam_1} , \phi_{\lam_2})\right\| \les \left\{\begin{array}{ll}
\lammin^{\frac12} \|\psi_{\lam_1}\|_{U_+^2}\|\phi_{\lam_2}\|_{U_+^2} & \mbox{for}\;\; \lam_1 \nsim \lam_2, \\ \mu\lam_{1}^{-\frac12} \|\psi_{\lam_1}\|_{U_+^2}\|\phi_{\lam_2}\|_{U_+^2} & \mbox{for}\;\; \lam_1 \sim \lam_2
\end{array}\right.
\end{align*}
for any $\psi_{\lam_1}, \phi_{\lam_2} \in U_+^2$.
\item[(ii)]  \begin{align*}
\left\|P_\mu Q_j^- (\psi_{\lam_1} , \phi_{\lam_2})\right\| \les \left\{\begin{array}{ll}
\lammin^{\frac12} \|\psi_{\lam_1}\|_{U_+^2}\|\phi_{\lam_2}\|_{U_-^2} & \mbox{for}\;\; \lam_1 \nsim \lam_2, \\ \mu^{\frac12} \|\psi_{\lam_1}\|_{U_+^2}\|\phi_{\lam_2}\|_{U_-^2} & \mbox{for}\;\; \lam_1 \sim \lam_2
\end{array}\right.
\end{align*}
for any $\psi_{\lam_1} \in U_+^2, \phi_{\lam_2} \in U_-^2$.
\end{lem}

\begin{lem}[Lemma 10 of \cite{tes2d}]\label{bilinear} Let $\psi_{\lam_1},\phi_{\lam_2} \in V_{\pm}^2$ and  $Q,B$ be any one of $Q_j^\pm$'s and $B_j^\pm$'s, respectively. Then we have
\begin{align*}
\|P_\mu Q(\psi_{\lam_1} , \phi_{\lam_2})\| &\les \lam_1^{1-\frac{2}{r}} \lam_2^{\frac2r} \|\psi_{\lam_1}\|_{U_\pm^{\frac{2r}{r-2}}} \|\phi_{\lam_1}\|_{U_\pm^{r}} \quad\mbox{for all}\;\; 2<r<\infty, \\
\|P_\mu B(\psi_{\lam_1} , \phi_{\lam_2})\| &\les \frac{\lam_1^{1-s} \lam_2^{s}}{\lammin} \|\psi_{\lam_1}\|_{V_\pm^2} \|\phi_{\lam_1}\|_{V_\pm^2} \qquad\mbox{for all}\;\; 0<s<1.
\end{align*}
In particular
\begin{align}\label{endpoint}
\|P_\mu B(\psi_{\lam_1} , \phi_{\lam_2})\| &\les \frac{\lam_1^{1-s} \lam_2^{s}}{\lammin} \|\psi_{\lam_1}\|_{U_\pm^2} \|\phi_{\lam_1}\|_{U_\pm^2}  \qquad\mbox{for} \;s=0,1.
\end{align}
\end{lem}

\subsection{Low-frequency estimates}\label{low-esti}
We introduce bilinear estimates for free waves, which are crucial for the low-frequency estimates of potential.  In \cite{yang, heya} one can find a 3d versions Corollary \ref{bilinear2} below. We adopt a similar argument to extend the 3d estimates to the 2d ones.

We first make a decomposition on the unit circle as in \cite{ste,yang}. For a fixed collection $ \Omega_\ka :=\{ \xi_{\ka}^\nu\}_\nu$ of unit vectors satisfying the conditions:
\begin{enumerate}
	\item[$(1)$] $|\xi_{\ka}^\nu - \xi_{\ka}^{\nu'}| \ge \ka^{-1}$ if $\nu \neq \nu'$,
	\item[$(2)$] For any unit vector $\xi$, there exists a $\xi_{\ka}^\nu$ such that $|\xi - \xi_{\ka}^\nu| < \ka$,
\end{enumerate}
we set $k_\ka^\nu := \sigma_\ka^\nu \left( \sum_{\ka} \sigma_{\ka}^\nu\right)^{-1}$ and $K_\ka^\nu := \mathcal{F}^{-1} k_{\ka}^\nu \mathcal{F}$ and $\sigma_\ka^\nu = \rho\left( \ka (\frac{\xi}{|\xi|} - \xi_{\ka}^\nu)\right)$, where $\rho$ is the cut-off function as stated in notation (3).  Then we get the following lemma.

\begin{lem}[Lemma 3.3 of \cite{behe2}] For any $\xi_{\ka}^{\nu},\xi_{\ka}^{\nu'} \in \Omega_\ka$ with $|\pm_1\xi_{\ka}^{\nu} - \pm_2 \xi_{\ka}^{\nu'}|\le \ka^{-1}$, we obtain
	$$
	\left| \left< \Pi_{\pm_1}(\lam_1 \xi_{\ka}^{\nu})v , \beta \Pi_{\pm_2}(\lam_2 \xi_{\ka}^{\nu'})w \right> \right| \les \ka^{-1} |v||w|
	$$
	for all $v,w \in \mathbb{C}^2$.
\end{lem}

Next we localize the Klein-Gordon Strichartz estimates. For this purpose let us define a frequency localization operator $\Gam$. Let $\gam \in C_0^\infty(-\frac23,\frac23)$ with $\gam(s) = 1$ if $|s| \le \frac13$ and $B_{\lam'} = \lam'\mathbb{Z}^2$ for $\lam' \in 2^{\mathbb{Z}}$. We define a cut-off function $\gam_{\lam',n}$ such as $\gam_{\lam',n}(\xi) = \gam \left(\frac{\xi_1 - n_1}{\lam'}\right) \gam \left(\frac{\xi_2 - n_2}{\lam'}\right)$, where $\xi = (\xi_1, \xi_2)$ and $n = (n_1,n_2)$ for $\xi \in \rt$ and $n \in B_{\lam'}$ and we prescribe the frequency localization operator $\Gam_{\lam',n} = \mathcal{F}^{-1}\gam_{\lam',n}\mathcal{F}$. Then we get the following.

\begin{lem} Let  $(q,r)$ satisfy that $\frac1q + \frac1r = \frac12$. Then
	\begin{align}\label{kg-stri2}
	\|\Gam_{\lam',n}\epm f_\lam\|_{L_t^q L_x^r} \les (\lam')^{\frac2q}\|f_\lam\|_{L_x^2}
	\end{align}
	for all $\lam,\lam' \in 2^\mathbb{Z}(\lam' \le \lam)$.
\end{lem}
This lemma follows immediately from the standard Strichartz estimate:
\begin{lem}[Klein-Gordon Strichartz estimates \cite{chozxi}]\label{KG-stri} Let $(q,r)$ satisfy that $\frac1q + \frac1r = \frac12$. Then
\begin{align}\label{KG-stri}
\|\epm f_\lam\|_{L_t^q L_x^r} \les \lam^{\frac2q}\|f\|_{L_x^2}
\end{align}
for all $\lam \in 2^\mathbb{Z}$.
\end{lem}
Now we extend the localized Klein-Gordon Strichartz estimates globally.
\begin{lem}
	Let $(q,r)$ satisfy that $\frac1q + \frac1r = \frac12$ and $2<q< \infty$. Then
	\begin{align}\label{localizing-stri}
	\left( \sum_{\nu\in \Omega_\ka}\sum_{n \in B_{\lam'}} \|K_l^\nu \Gam_{\lam',n}  \psi_\lam\|_{L_t^qL_x^r}^2 \right)^{\frac12} \les ( \lam')^\frac2q \|\psi_\lam\|_{V_{\pm}^2}.
	\end{align}
\end{lem}
\begin{proof}
By Proposition \ref{transfer} and \eqref{kg-stri2}, we get
\begin{align*}
\|\gam_{\lam',n}\epm \psi_\lam\|_{L_t^q L_x^r} \les ( \lam')^\frac2q \|\psi_\lam\|_{U_{\pm}^q} \les ( \lam')^\frac2q \|\psi_\lam\|_{V_{\pm}^2}.
\end{align*}
From the definition of operators $P_\lam^1, \,\Gam_{\lam',n} $ we see that
\begin{align*}
\| K_{\ka}^\nu \Gam_{\lam',n} \psi_{\lam}  \|_{L_t^qL_x^r} \les  ( \lam')^\frac2q \|K_{\ka}^\nu \Gam_{\lam',n} \psi_{\lam} \|_{V_{\pm}^2}.
\end{align*}
Then, by the orthogonality with respect to $n,\nu$, we obtain
\begin{align*}
\left( \sum_{\nu\in \Omega_\ka}\sum_{n \in B_{\lam'}} \|K_l^\nu \Gam_{\lam',n}  \psi_\lam\|_{L_t^qL_x^r}^2 \right)^{\frac12} &\les (\lam')^{\frac2q} \left( \sum_{\nu\in \Omega_\ka}\sum_{n \in B_{\lam'}} \|K_l^\nu \Gam_{\lam',n}  \psi_\lam\|_{V_{\pm}^2}^2 \right)^{\frac12}\\
&\les (\lam')^{\frac2q} \|\psi_\lam\|_{V_{\pm}^2}.
\end{align*}
\end{proof}

\begin{prop}\label{non-resonance}
Let  $\mu,\lam_1,\lam_2 \in 2^{\mathbb{Z}}$ satisfy $\mu \ll \lam_1 \sim \lam_2$ and $\lam_1,\lam_2 \ge 1$. Suppose $\pm_1 = \pm_2$.  Assume that $\psi_{j,\lam_j} = \Pi_{\pm_j}(D)\psi_{j,\lam_j} \in V_{\pm_j}^2$ for $j = 1,2$. Then
\begin{align*}
\left\|P_\mu \left< \psi_{1,\lam_1}, \beta\psi_{2,\lam_2} \right> \right\| \les \mu\|\psi_{1,\lam_1}\|_{V_{\pm_1}^2} \|\psi_{2,\lam_2}\|_{V_{\pm_2}^2}.
\end{align*}
\end{prop}

\begin{proof}
We decompose $P_\mu \left< \psi_{1,\lam_1}, \beta\psi_{2,\lam_2} \right>$ with respect to $n,n' \in B_{\mu}$, $\nu,\nu' \in \Omega_\ka$ with $\ka \sim \mu^{-1}\lam_1$. Only those $n' \in B_{\mu}$  with $|n-n'|\le \mu$ are chosen for each $n \in B_\mu$ and we carry out the summation under a support condition $\left|\pm_1 \xi_\ka^\nu - \pm_2 \xi_\ka^{\nu'} \right| = |\xi_1 - \xi_2| \sim \ka^{-1}$. Then we estimate
\begin{align*}
\|P_\mu \left< \psi_{1,\lam_1}, \beta\psi_{2,\lam_2} \right> \| &\les  \sum_{\xi_\ka^\nu , \xi_\ka^{\nu'} \in \Omega_\ka} \sum_{n,n' \in B_\mu} \left\|P_\mu \left< K_{\ka}^\nu \Gam_{\mu,n}\psi_{1,\lam_1}, \beta K_{\ka}^{\nu'} \Gam_{\mu,n} \psi_{2,\lam_2} \right> \right\|\\
&\les  \sum_{\xi_\ka^\nu , \xi_\ka^{\nu'} \in \Omega_\ka} \sum_{n,n' \in B_\mu} \ka^{-1} \left\|K_{\ka}^\nu \Gam_{\mu,n}\psi_{1,\lam_1}\right\|_{L_{t,x}^4}  \left\| K_{\ka}^{\nu'} \Gam_{\mu,n} \psi_{2,\lam_2} \right\|_{L_{t,x}^4}.
\end{align*}
Therefore, using Cauchy-Schwarz inequality with $n,n'$ and $\nu, \nu'$ and \eqref{localizing-stri}, we obtain
$$
\|P_\mu \left< \psi_{1,\lam_1}, \beta\psi_{2,\lam_2} \right> \| \les \mu (\lam_1)^{-1} \mu^{\frac12}\mu^{\frac12}\|\psi_{1,\lam_1}\|_{V_{\pm_1}^2}\|\psi_{2,\lam_2}\|_{V_{\pm_2}^2} \les \mu \|\psi_{1,\lam_1}\|_{V_{\pm_1}^2}\|\psi_{2,\lam_2}\|_{V_{\pm_2}^2}.
$$
\end{proof}

\begin{lem}\label{l2-lem} Let  $\mu,\lam_1,\lam_2 \in 2^{\mathbb{Z}}$ satisfy $\mu \ll \lam_1 \sim \lam_2$ and $\lam_1,\lam_2 \ge 1$.  Assume that $f_{\lam_1},g_{\lam_2} \in L_x^2$. Then
   \begin{align}\label{l2-esti}
	\left\|P_\mu \left( e^{\pm_1it\brad}f_{\lam_1}\cdot e^{\pm_2it\brad}g_{\lam_2} \right) \right\|_{L_x^2} \les \mu^{\frac12}\|f_{\lam_1}\|_{L_x^2} \|g_{\lam_2}\|_{L_x^2}
   \end{align}
 with $\pm_1 = \pm_2$.
\end{lem}
\begin{proof}
We assume that $\pm_1 = \pm_2 = +$. Then
	\begin{align*}
	\left\|P_{\mu} \left( e^{it\brad} f_{\lam_1}\cdot e^{it\brad} g_{\lam_2} \right) \right\| \les \sum_{\begin{subarray} ||n_i| \sim \lam_i \\ |n_1 + n_2|  \sim \mu\end{subarray}} \|I_{n_1,n_2}\|,
	\end{align*}
	where
	$$
	I_{n_1,n_2}(t,x) = \left( \Gam_{\mu,n_1}e^{it\brad} f_{\lam_1}\cdot \Gam_{\mu,n_2}e^{it\brad} g_{\lam_2} \right).
	$$
	Then $I_{n_1,n_2}$ is written as
	\begin{align*}
	I_{n_1,n_2}(t,x) = I_{n_1,n_2}^1 (t,x) +  I_{n_1,n_2}^2 (t,x),
	\end{align*}
	where
	\begin{align*}
	I_{n_1,n_2}^i = \int_{A_i} \int e^{i(t,x)\cdot(\left< \eta \right>+\left< \xi \right>,\, \xi+\eta)}  \left(\rho_{\lam_1}(\xi) \gam_{\mu,n_1}(\xi)\widehat{f}(\xi)  \cdot \rho_{\lam_2}(\eta) \gam_{\mu,n_2}(\eta)\widehat{g}(\xi)  \right) d\eta d\xi \;\;\;\; (i=1,2).
 	\end{align*}
 	Here we denote $A_i$ by
 	$$
 	A_i \subset \{  \xi = (\xi_1 , \xi_2)\in \mathbb{R}^2 : |\xi| \sim |\xi_i|  \}\;\mbox{for}\;i=1,2 \;\;\;\mbox{and}\;\;\; A_1\cup A_2 = \mathbb{R}^2.
 	$$
 	We make the change of variables $(\xi_i, \eta) \rightarrow (\left< \eta \right>+\left< \xi \right>,\xi+\eta)= \zeta = (\zeta_1,\zeta_2,\zeta_3)$ with
 	\begin{align}\label{yacobi}
 	d\xi_i d\eta = \left|\frac{\partial(\zeta_1,\zeta_2,\zeta_3)}{\partial(\eta_1,\eta_2,\xi_i)} \right|^{-1}d\zeta.
 	\end{align}
 	Since $|n_1 + n_2 | \sim \mu$, $\left|\frac{\partial(\zeta_1,\zeta_2,\zeta_3)}{\partial(\eta_1,\eta_2,\xi_i)} \right| \sim 1$. Now we fix $i=1$. The other cases can be estimated similarly. We may estimate by Minkowski's inequality
 	\begin{align*}
 	\|I_{n_1,n_2}^1\| \les \int_{|\xi_2| \les \mu} \left\| \iint e^{i(t,x)\cdot(\left< \eta \right>+\left< \xi \right>,\xi+\eta)}  \left(  \rho_{\lam_1}(\xi) \gam_{\mu,n_1}(\xi)\widehat{f}(\xi)  \cdot \rho_{\lam_2}(\eta) \gam_{\mu,n_2}(\eta)\widehat{g}(\xi)  \right)   d\eta d\xi_1\right\| d\xi_2.
 	\end{align*}
 	Using Plancherel's theorem with respect to  $(t,x)$, the change of variables $\zeta \to (\xi_1,\eta)$, and \eqref{yacobi}, we get
 	$$
 	\|I_{n_1,n_2}^1\| \les \int_{|\xi_2| \les \mu} \left\|  \left( \rho_{\lam_1}(\xi) \gam_{\mu,n_1}(\xi)\widehat{f}(\xi)  \cdot \rho_{\lam_2}(\eta) \gam_{\mu,n_2}(\eta)\widehat{g}(\xi)  \right)  \right\|_{L_{\eta,\xi_1}^2} d\xi_2.
 	$$
 	And also we have
 	\begin{align*}
 	\|I_{n_1,n_2}^2\| \les \mu^{\frac12} \left\|\left( \rho_{\lam_1}(\xi) \gam_{\mu,n_1}(\xi)\widehat{f}(\xi)  \cdot \rho_{\lam_2}(\eta) \gam_{\mu,n_2}(\eta)\widehat{g}(\xi)  \right)  \right \|_{L_{\eta,\xi}^2} \les \mu^{\frac12} \|\Gam_{\mu,n_1} f_{\lam_1}\|_{L_x^2} \|\Gam_{\mu,n_2} g_{\lam_2}\|_{L_x^2}.
 	\end{align*}
 	These estimates yield that
 	$$
 	\left\|P_{\mu} \left( e^{it\brad} f_{\lam_1}\cdot e^{it\brad} g_{\lam_2} \right) \right\| \les \sum_{\begin{subarray} ||n_i| \sim \lam_i \\ |n_1 + n_2|  \sim \mu\end{subarray}}  \mu^{\frac12} \|\Gam_{\mu,n_1} f_{\lam_1}\|_{L_x^2} \|\Gam_{\mu,n_2} g_{\lam_2}\|_{L_x^2} \les \mu^{\frac12} \|f_{\lam_1}\|_{L_x^2} \|g_{\lam_2}\|_{L_x^2}.
 	$$
 	This completes the proof of Lemma \ref{l2-lem}.
 \end{proof}

\begin{prop}\label{resonance} Let  $\mu,\lam_1,\lam_2 \in 2^{\mathbb{Z}}$ satisfy $\mu \ll \lam_1 \sim \lam_2$ and $\lam_1,\lam_2 \ge 1$. Suppose $\pm_1 \neq \pm_2$.  Assume that $\psi_{j,\lam_j} = \Pi_{\pm_j}(D)\psi_{j,\lam_j} \in V_{\pm_j}^2$ for $j = 1,2$. Then
	\begin{align}\label{eq-bilinear2}
	\left\|P_\mu \left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right> \right\| \les \mu^{\frac12}\|\psi_{1,\lam_1}\|_{V_{\pm_1}^2} \|\psi_{2,\lam_2}\|_{V_{\pm_2}^2}.
	\end{align}
\end{prop}

\begin{proof}
	Since $\epm \psi = \overline{\emp \overline{\psi}}$, $\|\psi\|_{V_{\pm}^2} = \|\overline{\psi}\|_{V_{\mp}^2}$. Hence it suffices to prove that
	\begin{align}\label{realproduct}
	\left\|P_\mu \left( \psi_{1,\lam_1} \cdot \psi_{2,\lam_2} \right) \right\| \les \mu^{\frac12}\|\psi_{1,\lam_1}\|_{V_{\pm_1}^2} \|\psi_{2,\lam_2}\|_{V_{\pm_2}^2}.
	\end{align}
	for $\pm_1 = \pm_2 =+$.	We first show that  for any $\mu \les \lam_1$
	\begin{align}\label{claim-bilinear}
	\|P_{\le\mu} \left( P_{\sim \lam_1}\psi \cdot P_{\sim \lam_1}\psi \right)\| \les \mu^{\frac12} \|\psi\|_{V_{+}^2}^2
	\end{align}
	for real-valued function $\psi \in V_+^2$. To this end let $\omega_\mu = \left[\mathcal{F}^{-1}\left(\rho\left(\frac{\cdot}{2\mu}\right)\right)\right]^2$. Then
	$$
	\|P_{\le\mu}\left( P_{\sim \lam_1}\psi \cdot P_{\sim \lam_1}\psi\right)\| \les \|\omega_\mu*\left( P_{\sim \lam_1}\psi \cdot P_{\sim \lam_1}\psi\right)\| .
	$$
	Hence it suffices to show that
	\begin{align*}
	\|\omega_\mu * \left( P_{\sim \lam_1}\psi \cdot P_{\sim \lam_1}\psi\right)\| \les \mu^{\frac12}\|\psi\|_{V_{+}^2}^2.
	\end{align*}
	 We define
	$$
	n(f) := \| \omega_\mu* \left( P_{\sim \lam_1}f \cdot P_{\sim \lam_1}f\right) \|_{L_x^2}^2.
	$$
	Then we get $n(f+g) \le n(f) + n(g)$. Using \eqref{l2-esti} we see that
	\begin{align}\label{n-linear}
	\|n(e^{-it\left<D\right>}f)\|_{L_t^4} \les \left\|\omega_\mu* \left( e^{-it\left<D\right>}f \cdot e^{-it\left<D\right>}f \right) \right\|_{L_{t,x}^2}^{\frac12} \les \mu^{\frac14} \|f\|_{L_x^2}
	\end{align}
	for all $f\in L_x^2$. Let $\psi \in U_{+}^4$ be with atomic decomposition
	$$
	e^{it\brad}\psi = \sum c_ja_j,\;\; \sum|c_j| \le 2 \|\psi\|_{U_{+}^4},
	$$
	where $a_j$ are $U^4$-atoms. Then we get
	\begin{align}
	\|n(\psi)\|_{L_t^4} \le \sum_{j} |c_j|\|n(e^{-it\brad}a_j)\|_{L_t^4} \les \mu^{\frac14}\|\psi\|_{U_{+}^4},
	\end{align}
	provided that for any $U^4$-atom $a$ the estimate
	$$
	\|n(e^{-it\brad}a)\|_{L_t^4} \les \mu^{\frac14}
	$$
	holds true. Indeed, let $a(t) = \sum_{k}\chi_{I_k}(t) \phi_k $, for some time partition $I_k \subset \mathbb{R}$ and $\phi_k \in L_x^2$ satisfying $\sum_{k}\|\phi_k\|^4\le 1$. Then by \eqref{n-linear} we obtain
	\begin{align*}
	\|n(e^{-it\brad} a)\|_{L_t^4} &\le \left\|\sum_{k}\chi_{I_k}(t) n(e^{-it\left<D\right>}\phi_k)\right\|_{L_t^4} \le \left(\sum_{k}\left\| n(e^{-it\left<D\right>}\phi_k)\right\|_{L_t^4}^4\right)^{\frac14}\\
	&\les \mu^{\frac14} \left(\sum_{\mu}\left\| \phi_\mu\right\|_{L_x^2}^4\right)^{\frac14} \les \mu^{\frac14}
	\end{align*}
	and hence
	$$
	\|\omega_\mu * \left( P_{\sim \lam_1}\psi \cdot P_{\sim \lam_1}\psi \right)\| = \|n(\psi(t))\|_{L_t^4}^2 \les \mu^{\frac12}\|\psi\|_{U_{+}^4}^2 \les \mu^{\frac12}\|\psi\|_{V_{+}^2}^2.
	$$
	This leads us to \eqref{claim-bilinear}.

	We now consider  \eqref{realproduct} for $\psi_{j,\lam_j} \in V_{+}^2$ satisfying $\|\psi_{j,\lam_j}\|_{V_{+}^2}=1$ for $j=1,2$. Let us define the functions $\psi_p, \, \psi_m$ by $\psi_{p} := \psi_{1,\lam_1} + \psi_{2,\lam_2}$ $\psi_{m} := \psi_{1,\lam_1} - \psi_{2,\lam_2}$. In addition we readily get $P_{\sim \lam_1} \psi_{p} =  \psi_{p},\; P_{\sim \lam_1} \psi_{m} =  \psi_{m}$ and $ \|\psi_{p}\|_{V_{+}^2} \les 1, \; \|\psi_{m}\|_{V_{+}^2}\les 1$. For $c\in \mathbb{C}^2$ let ${\rm Re} \,c$ and ${\rm Im} \,c$ denote $\left( \begin{aligned}[l]  {\rm Re} \,c_1 \\ {\rm Re}\, c_2 \end{aligned} \right)$ and $\left( \begin{aligned}[l]  {\rm Im} \,c_1 \\ {\rm Im}\, c_2 \end{aligned} \right)$, respectively. Then we see that
	\begin{align*}
	{\rm Re}\,(\left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right>) &= \frac14 \Big(\left< {\rm Re}\, \psi_p , {\rm Re}\,\psi_p \right> - \left<{\rm Re}\, \psi_m , {\rm Re}\, \psi_m \right> + \left< {\rm Im}\, \psi_p , {\rm Im}\, \psi_p \right> - \left< {\rm Im} \,\psi_m , {\rm Im} \,\psi_m \right>\Big),\\
	{\rm Im}\, (\left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right>) &= { \rm Re}\,(-i\left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right>)
	\end{align*}
	which implies
	\begin{align}\label{appen1}
	\|P_{\le\mu} \left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right>\| \les \|P_{\le\mu} {\rm Re}\left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right>\| + \|P_{\le\mu} {\rm Re} \left< -i\psi_{1,\lam_1}, \psi_{2,\lam_2} \right>\|.
	\end{align}
	Since $P_{\sim \lam_1} ({\rm Re}\,\psi_p )=  \psi_p $ and $P_{\sim \lam_1} ({\rm Im}\,\psi_\pm) = P_{\sim \lam_1}\psi_\pm $, we obtain
	\begin{align*}
	\|P_{\le\mu} {\rm Re} \,\left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right>\| &\les \|P_{\le\mu}\left< {\rm Re} \,\psi_p, {\rm Re} \psi_p \right>\| + \|P_{\le\mu}\left< {\rm Re}\, \psi_m, {\rm Re}\, \psi_m \right>\|\\
	&\qquad\qquad + \|P_{\le\mu}\left< {\rm Im}\,\psi_p, {\rm Im}\psi_p \right>\| + \|P_{\le\mu}\left< {\rm Im}\,\psi_m, {\rm Im}\,\psi_m \right>\|\\
	&\les \mu^{\frac12} \left( \| {\rm Re}\, \psi_p\|_{V_{\pm}^2}^2  + \|{\rm Re}\,  \psi_m\|_{V_{\pm}^2}^2 + \|{\rm Im}\,\psi_p\|_{V_{\pm}^2}^2 + \|{\rm Im}\,\psi_m\|_{V_{\pm}^2}^2    \right)\\
	&\les \mu^{\frac12} \left( \|\psi_p\|_{V_{\pm}^2}^2  + \|\psi_m\|_{V_{\pm}^2}^2 \right) \les \mu^{\frac12}.
	\end{align*}
	where in second inequality we used the estimate \eqref{claim-bilinear}. The second term of right-hand side of \eqref{appen1} is obtained similarly as above.
		Therefore we get
	$$
		\left\|P_\mu \left< \psi_{1,\lam_1}, \psi_{2,\lam_2} \right> \right\| \les \mu^{\frac12}\|\psi_{1,\lam_1}\|_{V_{+}^2} \|\psi_{2,\lam_2}\|_{V_{+}^2}
	$$
	for any $\psi_{j,\lam_j}\in V_+^2$. This completes the proof of \eqref{eq-bilinear2}.
\end{proof}

By Propositions \ref{non-resonance} and \ref{resonance} we reach the following corollary.
\begin{cor}\label{bilinear2} Let  $\mu,\lam_1,\lam_2 \in 2^{\mathbb{Z}}$ satisfy $\mu \ll \lam_1 \sim \lam_2$, $0<\mu \le 1$, and $\lam_1,\lam_2 \ge 1$.  Assume that $\psi_{j,\lam_j} = \Pi_{\pm_j}(D)\psi_{j,\lam_j} \in V_{\pm_j}^2$ for $j = 1,2$. Then
	\begin{align}
	\left\|P_\mu \left< \psi_{1,\lam_1}, \beta\psi_{2,\lam_2} \right> \right\| \les \mu^{\frac12}\|\psi_{1,\lam_1}\|_{V_{\pm_1}^2} \|\psi_{2,\lam_2}\|_{V_{\pm_2}^2}.
	\end{align}
\end{cor}

\section{Proof of Theorem \ref{mainthm}}

\newcommand{\x}{X_{\pm}^s}

In this section we prove Theorem \ref{mainthm} by contraction argument. At first we denote time-restricted space $\left\{ \psi = \chi_{\left[0, \infty\right)} \phi : \phi \in U_{\pm}^2\right\}$ by $U_{\pm}^2(\left[0, \infty\right))$. And let us define Banach spaces $\x, X^s$ as
$$
\x := \left\{ \psi \in C(\mathbb{R} ; L_x^2)\cap U_{\pm}^2(\left[0, \infty\right)) : \|\psi\|_{\x} := \left( \sum_{\lam\ge1} \lam^{2s}\|P_\lam^1 \psi\|_{U_{\pm}^2}^2\right)^{\frac12} < \infty \right\},
$$
 \begin{align*}
X^s := \left\{ \psi : \psi_\pm : =\Pi_{\pm}(D)\psi \in \x  \;\;\mbox{and}\;\;  \|\psi\|_{X^s}:= \|\psi_+\|_{X_+^s}  +  \|\psi_-\|_{X_-^s} < \infty   \right\}.
\end{align*}
Let us consider a complete metric space $(X^s(\delta), d)$ and a map $\mathcal{H}$ on $X^s(\de)$ given by
 \begin{align*}
 X^s(\de) = \left\{ \psi \in X^s : \|\psi\|_{X^s} \le \de  \right\}  \;\;\mbox{with metric}\;\;  d(\psi, \phi) := \|\psi - \phi \|_{X^s},
 \end{align*}
\begin{align}\label{mapping}
\mathcal{H}(\psi) &=  \sum_{\pm_0 \in \{ \pm\}}e^{\mp_0 it\brad}\Pi_{\pm_0}(D)\psi_0 + i \sum_{\pm_j \in \{\pm\}} \mathcal{N}_{\pm_0}(\psi_{\pm_1},\psi_{\pm_2},\psi_{\pm_3})(t),
\end{align}
where
\begin{align*}
\mathcal{N}_{\pm_0}(\psi_{\pm_1},\psi_{\pm_2},\psi_{\pm_3})(t) = \int_0^t e^{\mp_0 i(t-t')\brad}\Pi_{\pm_0}(D)[(V*\left< \psi_{\pm_1}, \beta\psi_{\pm_2}\right>)\beta\psi_{\pm_3}] d\,t'.
\end{align*}
Then we will show that $\mathcal{H}$ is a contraction mapping on $X^s(\de)$.
Indeed, the linear part of \eqref{mapping} can be handled as follows:
\begin{align}\label{linear-est}
\left\|e^{\mp_0 it\brad} \Pi_{\pm_0}(D) \psi_0 \right\|_{X_{\pm_0}^s}^2 = \sum_{\lam \ge 1} \lambda^{2s}\left\|\chi_{[0,\infty)} P_\lam^1 \Pi_{\pm_0}(D) \psi_0 \right\|_{U_{\pm_0}^2}^2 \le C\|\psi_0\|_{H^s}^2.
\end{align}
For the nonlinear parts $\mathcal{N}_{\pm_0}(\psi_{\pm_1},\psi_{\pm_2},\psi_{\pm_3})$ we will show the following trilinear estimates.
\begin{prop}\label{nonlinear}
	Let $\psi_{j,\pm_j} \in X_{\pm_j}^s$ for $s > \gamma-1$. Then we have
	\begin{align*}
	\|\mathcal{N}_{\pm_0}(\psi_{1,\pm_1},\psi_{2,\pm_2},\psi_{3,\pm_3})\|_{X_{\pm_0}^s} \les \prod_{j=1}^3\|\psi_{j,\pm_j}\|_{X_{\pm_j}^s}.
	\end{align*}
\end{prop}

Once Proposition \ref{nonlinear} has been proved, this trilinear estimate together with linear estimate \eqref{linear-est} yields
\begin{align*}
\|\mathcal{H}(\psi)\|_{X^s} = \sum_{\pm_0 \in \{\pm\}}\|\Pi_{\pm_0}\mathcal H(\psi)\|_{X_{\pm_0}^s} \le C(\|\psi_0\|_{H^s} + \|\psi\|_{X^s}^3).
\end{align*}
If $\de$ is small enough so that $C\delta^3 \le \frac\delta8$ and $\psi_0$ satisfies $C\|\psi_0\|_{H^s} \le \frac\delta2$,
$\mathcal H$ is a self-mapping on $X^s(\delta)$. In particular, we get
\begin{align*}
\|\mathcal{H}(\psi) - \mathcal{H}(\phi)\|_{X^s} &\le C\left(\|\psi\|_{X^s}+ \|\phi\|_{X^s}\right)^2 \|\psi- \phi\|_{X^s} \le 4C\de^2\|\psi- \phi\|_{X^s} \le \frac12\|\psi- \phi\|_{X^s}.
\end{align*}
Hence $\mathcal H: X^s(\de) \to X^s(\de)$ is a contraction mapping for sufficiently small $\de$ and then we get a unique solution $\psi_p \in L^\infty([0, \infty); H^s)$ to \eqref{maineq}. The time continuity and continuous dependency on data follow readily from the Duhamel's formula and Proposition \ref{nonlinear}. By the time symmetry of \eqref{maineq} we also obtain a unique solution $\psi_n \in C((-\infty, 0], H^s)$ with the continuous dependency on data. Defining $\psi = \psi_p + \psi_n$, we get the global well-posedness of \eqref{maineq}.

Now we move onto the scattering property of \eqref{maineq-decom}. Since the backward scattering can be treated similarly to the forward one, we omit its proof. For $\lambda \ge 1$ let us define
$$
\vp_{\lam,\pm}:=  P_\lambda^1 \epm  \mathcal{N}_{\pm}(\psi),
$$
where $\mathcal{N}_{\pm}(\psi) = \sum_{\pm_j \in \{\pm\}} \mathcal{N}_{\pm}(\psi_{\pm_1},\psi_{\pm_2},\psi_{\pm_3})(t)$. Then Lemma \ref{embedd} shows that
$$
\vp_{\lam,\pm} \in V_{\pm}^2
$$
for all $\lam\ge1$. Since $\sum_{\lam \ge1} \lam^{2s} \|\vp_{\lam,\pm}\|_{V_\pm^2} \les 1$, we have
$$
\phi_\pm :=\psi_{0,\pm} + \lim_{t \to \infty} \sum_{\lam\ge1}\vp_{\lam,\pm} \in H^s
$$
and
$$
\|\psi_\pm(t) - \emp \phi_\pm\|_{H^s}  \xrightarrow{t \to \infty} 0.
$$
This completes the proof of scattering of \eqref{maineq-decom}.
\subsection {Proof of Proposition \ref{nonlinear}}
By Lemma \ref{duality} we obtain
\begin{align*}
& \left\|P_{\lam_4}^1 \int_0^t e^{\mp i(t-t')\brad}\Pi_{\pm}(D)[(V*\left< \psi_{1,\pm_1}, \beta\psi_{2,\pm_2}\right>)\beta\psi_{3,\pm_3}] dt' \right\|_{U_{\pm\brad}^2}\\
& \qquad\qquad =\left\|P_{\lam_4}^1 \int_0^t e^{\pm it'\brad}\Pi_{\pm}(D)[(V*\left< \psi_{1,\pm_1}, \beta\psi_{2,\pm_2}\right>)\beta\psi_{3,\pm_3}] dt' \right\|_{U^2}\\
& \qquad\qquad = \sup_{\begin{subarray}{ll} \|\phi\|_{V^2}=1 \\ \;\;\phi \in C_0^\infty  \end{subarray}}\left| \iint (V*\left< \psi_{1,\pm_1}, \beta\psi_{2,\pm_2}\right>)\left<\beta\psi_{3,\pm_3}, \Pi_{\pm}(D)P_{\lam_4}^1 e^{\mp it\brad}\phi\right> dt dx\right|\\
& \qquad\qquad = \sup_{\|\psi_4\|_{V_{\pm\brad}^2}=1}\left| \iint (V*\left< \psi_{1,\pm_1}, \beta\psi_{2,\pm_2}\right>)\left<\beta\psi_{3,\pm_3}, \Pi_{\pm}(D)P_{\lam_4}^1 \psi_4\right> dt dx\right|.
\end{align*}
This induces that
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1,\pm_1},\psi_{2,\pm_2},\psi_{3,\pm_3})\|_{X_\pm^s}^2 \les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sup_{\|\psi_4\|_{V_{\pm\brad}^2}=1}\left| \sum_{\lam_i \ge 1}\iint (V*\left< \psi_{1,\pm_1}, \beta\psi_{2,\pm_2}\right>)\left<\beta\psi_{3,\pm_3},  \psi_{4,\pm_4}\right> dt dx\right| \right)^2.
\end{align*}
The following lemma to be proved in the next section plays a crucial role in the proof of Proposition \ref{nonlinear}.
\begin{lem}\label{non-esti} Let $\de > \gam -1$. Then we have for all $\psi_{j,\pm_j}\in U_{\pm_j}^2,\; \psi_{4,\pm_4} \in V_{\pm_4}^2$
	\begin{align}\label{non-integral}
	\left| \iint (V*\left< \psi_{1,\pm_1}, \beta\psi_{2,\pm_2}\right>)\left<\beta\psi_{3,\pm_3},  \psi_{4,\pm_4}\right> dt dx\right| \les (\lammed)^\de \prod_{j=1}^{3}\|\psi_{j,\pm_j}\|_{U_{\pm_j}^2}\|\psi_{4,\pm_4}\|_{V_{\pm_4}^2},
	\end{align}
	where $\lammed$ is median of $\lam_1,\lam_2,\lam_3$.
\end{lem}
For the sake of simplicity we denote $\psi_{\pm_j}$ by $\psi_j$. Then Lemma \ref{non-esti} implies that for $\gam -1 <\de < s$
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1},\psi_{2},\psi_{3})\|_{X_\pm^s}^2 \les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sum_{\lam_i \ge 1} (\lammed)^{\de} \prod_{j=1}^{3}\|\psi_j\|_{U_{\pm_j}^2}\right)^2.
\end{align*}
We divide the cases into (i) $\lam_3 \sim \lam_4$, (ii) $\lam_3 \ll \lam_4$, and (iii) $\lam_3 \gg \lam_4$.
\paragraph{(i) The case: $\lam_3 \sim \lam_4$} By Cauchy-Schwarz we obtain
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1},\psi_{2},\psi_{3})\|_{X_\pm^s}^2 &\les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sum_{\lam_i \ge 1} (\lammed)^{\de} \prod_{j=1}^{3}\|\psi_j\|_{U_{\pm_j}^2}\right)^2\\
                                                    &\les \sum_{ \lam_4 \ge 1} \lam_4^{2s}  \left(\sum_{\lam_3 \sim \lam_4} \lam_1^{\de-s}\lam_2^{\de-s} \lam_1^s\|\psi_1\|_{U_{\pm_1}^2}\lam_2^s\|\psi_2\|_{U_{\pm_2}^2}\|\psi_3\|_{U_{\pm_3}^2} \right)^2\\
                                                    &\les \|\psi_1\|_{X_{\pm_1}^s}^2\|\psi_2\|_{X_{\pm_2}^s}^2\sum_{ \lam_4 \ge 1} \left(\sum_{\lam_3 \sim \lam_4} \lam_4^{s}\|\psi_3\|_{U_{\pm_3}^2}\right)^2\\
                                                    &\les \prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_j}^s}^2.
\end{align*}

\paragraph{(ii) The case: $\lam_3 \ll \lam_4$} We further divide the case into $\lam_1 \sim \lam_2$ and $\lam_1 \nsim \lam_2$.
If $\lam_1 \sim \lam_2$, by Cauchy-Schwarz inequality and Young's convolution inequality  we get
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1},\psi_{2},\psi_{3})\|_{X_\pm^s}^2 &\les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sum_{\lam_i \ge 1} (\lammed)^{\de} \prod_{j=1}^{3}\|\psi_j\|_{U_{\pm_j}^2}\right)^2\\
&\les \sum_{ \lam_4 \ge 1} \lam_4^{2s}  \left(\sum_{\lam_3 \ll \lam_4} \lam_1^{\de-s}\lam_3^{-s} \lam_1^s\|\psi_1\|_{U_{\pm_1}^2}\|\psi_2\|_{U_{\pm_2}^2}  \lam_3^s\|\psi_3\|_{U_{\pm_3}^2} \right)^2\\
&\les \|\psi_1\|_{X_{\pm_1}^s}^2\|\psi_3\|_{X_{\pm_3}^s}^2\sum_{ \lam_4 \ge 1} \left(\sum_{\lam_4 \ll \lam_2} \left(\frac{\lam_4}{\lam_2}\right)^{s} \lam_2^s\|\psi_2\|_{U_{\pm_2}^2}\right)^2\\
&\les \prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_j}^s}^2.
\end{align*}
Next if $\lam_1 \ll \lam_2$, we have
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1},\psi_{2},\psi_{3})\|_{X_\pm^s}^2 &\les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sum_{\lam_i \ge 1} (\lammed)^{\de} \prod_{j=1}^{3}\|\psi_j\|_{U_{\pm_j}^2}\right)^2\\
&\les \sum_{ \lam_4 \ge 1} \lam_4^{2s}  \left(\sum_{\lam_3 \ll \lam_4} \lam_1^{\de-s}\lam_3^{\de-s} \lam_1^s\|\psi_1\|_{U_{\pm_1}^2}\|\psi_2\|_{U_{\pm_2}^2}  \lam_3^s\|\psi_3\|_{U_{\pm_3}^2} \right)^2\\
&\les \|\psi_1\|_{X_{\pm_1}^s}^2\|\psi_3\|_{X_{\pm_3}^s}^2\sum_{ \lam_4 \ge 1} \left(\sum_{\lam_4 \sim \lam_2}  \lam_2^s\|\psi_2\|_{U_{\pm_2}^2}\right)^2\\
&\les \prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_j}^s}^2.
\end{align*}
In the case $\lam_1 \gg \lam_2$, a similar estimate to the above can be obtained.

\smallskip

\paragraph{(iii) The case: $\lam_3 \gg \lam_4$} We also divide this case into $\lam_1 \sim \lam_2$ and $\lam_1 \nsim \lam_2$.
If $\lam_1 \sim \lam_2$, by Cauchy-Schwarz inequality and Young's convolution inequality we get
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1},\psi_{2},\psi_{3})\|_{X_\pm^s}^2 &\les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sum_{\lam_i \ge 1} (\lammed)^{\de} \prod_{j=1}^{3}\|\psi_j\|_{U_{\pm_j}^2}\right)^2\\
&\les \sum_{ \lam_4 \ge 1} \lam_4^{2s}  \left(\sum_{\lam_3 \gg \lam_4} \lam_1^{\de-s}\lam_2^{\de-s} \lam_1^s\|\psi_1\|_{U_{\pm_1}^2}\lam_2^s\|\psi_2\|_{U_{\pm_2}^2}\|\psi_3\|_{U_{\pm_3}^2} \right)^2\\
&\les \|\psi_1\|_{X_{\pm_1}^s}^2\|\psi_2\|_{X_{\pm_2}^s}^2\sum_{ \lam_4 \ge 1} \left(\sum_{\lam_3 \gg \lam_4} \left( \frac{\lam_4}{\lam_3}\right)^s \lam_3^{s}\|\psi_3\|_{U_{\pm_3}^2}\right)^2\\
&\les \prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_j}^s}^2.
\end{align*}
Next if $\lam_1 \ll \lam_2$, we have
\begin{align*}
\|\mathcal{N}_{\pm}(\psi_{1},\psi_{2},\psi_{3})\|_{X_\pm^s}^2 &\les \sum_{ \lam_4 \ge 1} \lam_4^{2s} \left( \sum_{\lam_i \ge 1} (\lammed)^{\de} \prod_{j=1}^{3}\|\psi_j\|_{U_{\pm_j}^2}\right)^2\\
&\les \sum_{ \lam_4 \ge 1} \lam_4^{2s}  \left(\sum_{\lam_3 \gg \lam_4} \lam_1^{\de-s}\lam_2^{\de-s} \lam_1^s\|\psi_1\|_{U_{\pm_1}^2}\lam_2^s\|\psi_2\|_{U_{\pm_2}^2}  \|\psi_3\|_{U_{\pm_3}^2} \right)^2\\
&\les \|\psi_1\|_{X_{\pm_1}^s}^2\|\psi_3\|_{X_{\pm_3}^s}^2\sum_{ \lam_4 \ge 1} \left(\sum_{\lam_3 \gg \lam_4}  \left( \frac{\lam_4}{\lam_3}\right)^s \lam_3^s\|\psi_3\|_{U_{\pm_3}^2}\right)^2\\
&\les \prod_{j=1}^{3}\|\psi_{j}\|_{X_{\pm_j}^s}^2.
\end{align*}
The case $\lam_1 \gg \lam_2$ can be treated similarly as above. Therefore we get the desired results.

\section{Proof of Lemma \ref{non-esti}}
\newcommand{\pmu}{P_{\mu}}
\newcommand{\utwo}{U_{\pm}^2}
\newcommand{\utwoj}{U_{\pm_j}^2}
\newcommand{\vtwo}{V_{\pm}^2}
\newcommand{\vtwoj}{V_{\pm_j}^2}
\newcommand{\ufour}{U_{\pm}^4}
\newcommand{\ufourj}{U_{\pm_j}^4}
In this section we prove the Lemma \ref{non-esti}. We decompose the left-hand side of \eqref{non-integral} into low-high frequency parts as follows:
\begin{align*}
J({\lam_1,\lam_2,\lam_3,\lam_4}) &:= \left| \iint V*\left< \psi_1, \beta\psi_2\right>\left<\beta\psi_3, \psi_4\right>  dxdt  \right|\\
                                 &= \sum_{\mu >0}  \left| \iint P_\mu \left( V*\left< \psi_1, \beta\psi_2\right>\right)\cdot  \left( \widetilde{P_\mu} \left<\beta\psi_3, \psi_4\right>  \right) dxdt  \right|\\
                                 &= \sum_{0<\mu \le 1}\left| \iint P_\mu \left( V*\left< \psi_1, \beta\psi_2\right>\right)\cdot  \left( \widetilde{P_\mu} \left<\beta\psi_3, \psi_4\right>  \right) dxdt  \right| \\
                                 & \qquad  +  \sum_{\mu > 1}\left| \iint P_\mu \left( V*\left< \psi_1, \beta\psi_2\right>\right)\cdot  \left( \widetilde{P_\mu} \left<\beta\psi_3, \psi_4\right>  \right) dxdt  \right|\\
                                 &=: J^l + J^h.
\end{align*}

This section is composed of three subsections. In the first one we deal with the low-frequency parts by using fully the estimates of Section \ref{low-esti}. In the next one we reveal the modulation lemma useful to treat high-frequency parts. And the detailed proof of high-frequency parts is provided in the third subsection. From now on we may assume that $\lam_1 \le \lam_2$, $\lam_3 \le \lam_4$, and $\pm_1 = \pm_3 = +$ thanks to the symmetry.

\subsection{Low-frequency parts}\label{lowfrequency}
Since $0 < \mu \le 1$ and $\lam_j\ge1(j=1,\cdots,4)$, the support condition yields that $\lam_1 \sim \lam_2$ and $\lam_3 \sim \lam_4$. To prove Lemma \ref{non-esti} for $J^l$ we use the Corollary \ref{bilinear2}. In fact, we get
\begin{align*}
J^l &= \sum_{0<\mu \le 1}\left| \iint P_\mu \left( V*\left< \psi_1, \beta\psi_2\right>\right)\cdot  \left( \widetilde{P_\mu} \left<\beta\psi_3, \psi_4\right>  \right) dxdt  \right| \\
&\les  \sum_{0<\mu \le 1} \left\| P_\mu \left( V*\left< \psi_1, \beta\psi_2\right>\right) \right\| \left\| \widetilde{P_\mu} \left<\beta\psi_3, \psi_4\right>  \right\|\\
&\les \sum_{0<\mu \le 1} \mu^{-1+ \gam} \prod_{j=1}^3\|\psi_j\|_{V_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}\\
&\les \prod_{j=1}^3\|\psi_j\|_{V_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
\end{align*}

\subsection{Modulation estimates}
Let us define the modulation projections by
$$
\Lambda_{\lam}^\pm u = \mathcal{F}_{t,x}^{-1} \left( \beta_\lam(|\tau \pm \left< \xi \right> |) \mathcal{F}_{t,x} u \right).
$$
and
$$
\Lambda_{\ge \lam}^\pm u = \sum_{\sigma \ge \lam}\Lambda_{\sigma}^\pm u, \qquad  \Lambda_{<\lam}^\pm u = 1- \Lambda_{\ge\lam}^\pm u.
$$

\begin{lem}[Corollary 2.18 of \cite{haheko}]\label{modul} Let $\lam \in 2^{\mathbb{Z}}$. Then
$$
\| \Lambda_{\ge \lam}^\pm u \| \les \lam^{-\frac12}\|u\|_{\vtwo}.
$$
\end{lem}

\begin{lem}
Let $\pm_4=-$. Suppose $\lam_j \in 2^{\mathbb{Z}}$ satisfy $\lam_1 \le \lam_2 \ll \lam_3 \sim \lam_4$ and $|\xi| \ge 1$, where $\xi$ is the frequency of $V$. Then
\begin{align}
\left| \iint V* B(\psi_1,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| \les \lam_2^{\de}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2},  \label{j3-esti}\\
\left| \iint V* Q(\psi_1,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| \les \lam_2^{\de}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}, \label{j4-esti}
\end{align}
for any $\de >0$.
\end{lem}
\begin{proof}
To prove  \eqref{j3-esti} and \eqref{j4-esti} we decompose $\psi_j$ into  low and high modulation parts
$$
\psi_j = \Lambda_{<\frac{\lam_3}8}^{\pm_j} \psi_j + \Lambda_{\ge\frac{\lam_3}8}^{\pm_j} \psi_j =: \psi_j^l + \psi_j^h.
$$
 Since $\tau_1 - \tau_2 + \tau_3 -\tau_4 =0$ and $\xi_1 - \xi_2 + \xi_3 -\xi_4 =0$, using $\pm_1 = \pm_3 = +,\pm_4 =-$, we have
$$
\left| \iint V* Y(\psi_1^l,\psi_2^l)\cdot Q(\psi_3^l,\psi_4^l)  dxdt  \right| = 0,
$$
where $Y$ is the one of $B$ or $Q$. Therefore we do not consider the case $\psi_j = \psi_j^l$ for all $j = 1,\cdots,4$.

We first prove the \eqref{j3-esti}. By Bernstein's and H\"older's inequalities and Lemma \ref{null-esti}, we have
\begin{align*}
\left| \iint V* B(\psi_1^h,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| &\les \sum_{\mu \ge 1}\left\|\pmu \left(V*B(\psi_1^h,\psi_2)\right) \right\|_{L_t^2 L_x^{\frac1{1-\ve}}}  \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|_{L_t^2L_x^{\frac1\ve}}\\
&\les  \sum_{\mu \ge 1} \mu ^{-1 +\gam -2\ve}\|\psi_1^h\|_{L_{t,x}^2}   \|\psi_2\|_{L_t^\infty L_x^\frac2{1-2\ve}}  \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|_{L_{t,x}^2}\\
&\les  \sum_{\mu \ge 1}  \mu^{-\frac12+\gam -\de-2\ve} \lam_2^{\de +2\ve} \lam_3^{-\frac12} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}\\
&\les  \lam_2^{\de} \lam_3^{-\frac{\de+1 -\gam}2}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}
\end{align*}
for some $0< \ve \ll1$. By \eqref{endpoint}, we also obtain
\begin{align*}
\left| \iint V* B(\psi_1^h,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| &\les \sum_{\mu \ge 1}\left\|\pmu \left(V*B(\psi_1^h,\psi_2)\right) \right\|  \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|\\
&\les \sum_{\mu \ge 1}  \mu^{-2+\gam}   \lam_3  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}\\
&\les  \lam_3  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}.
\end{align*}
Applying Lemma \ref{log} to the above two estimates, we get
$$
\left| \iint V* B(\psi_1^h,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| \les  \lam_2^{\de}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
$$
If $\psi_2 = \psi_2^h$, we may obtain the desired result in a similar manner.

If $\psi_3 = \psi_3^h$, then we get
\begin{align*}
\left| \iint V* B(\psi_1,\psi_2)\cdot Q(\psi_3^h,\psi_4)  dxdt  \right| &\les \sum_{\mu \ge 1}\left\|\pmu \left(V*B(\psi_1,\psi_2)\right) \right\|_{L_t^2 L_x^\infty}  \left\| \widetilde{\pmu}Q(\psi_3^h,\psi_4)\right\|_{L_t^2L_x^1}\\
&\les  \sum_{\mu \ge 1} \mu^{-1+\gam}\left\| \pmu B(\psi_1,\psi_2)\right\|_{L_{t,x}^2}  \|\psi_3^h\|_{L_{t,x}^2}   \|\psi_4\|_{L_t^\infty L_x^2} \\
&\les  \sum_{\mu \ge 1}  \mu^{-1+\gam}  \lam_3^{-\frac12} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}\\
&\les  \lam_2^{\de} \sum_{\mu \ge 1} \mu^{-\frac32+\gam -\de}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
\end{align*}
The case $\psi_4 = \psi_4^h$ can also be treated similarly.

We next consider the \eqref{j4-esti}. The proof of \eqref{j4-esti} is very similar to the one of \eqref{j3-esti} except changing $B(\psi_1, \psi_2)$ with $Q(\psi_1,\psi_2)$. For the convenience of readers we brief on the details. By the consecutive use of Bernstein's inequality, Lemma \ref{null-esti}, and Lemma \ref{modul} we have
\begin{align}
\begin{aligned}\label{qq-u2}
\left| \iint V* Q(\psi_1^h,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| &\les \sum_{\mu \ge 1}\left\|\pmu \left(V*Q(\psi_1^h,\psi_2)\right) \right\|_{L_t^2 L_x^{\frac1{1-\ve}}}  \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|_{L_t^2L_x^{\frac1\ve}}\\
&\les  \sum_{\mu \ge 1} \mu^{-1+\gam -2\ve}  \|\psi_1^h\|_{L_{t,x}^2}   \|\psi_2\|_{L_t^\infty L_x^{\frac2{1-2\ve}}}  \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|_{L_{t,x}^2}\\
&\les  \sum_{\mu \ge 1}  \mu^{-\frac12+\gam-\de-2\ve} \lam_2^{\de + 2\ve} \lam_3^{-\frac12} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}\\
&\les  \lam_2^{\de} \lam_3^{-\frac{\de+1 -\gam}2}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}
\end{aligned}
\end{align}
for some $0 <\ve \ll 1$. In a similar way we get
\begin{align}
\begin{aligned}\label{qq-u4}
\left| \iint V* Q(\psi_1^h,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right|  &\les  \sum_{\mu \ge 1}  \mu^{-1+\gam -2\ve} \lam_2^{2\ve}\|\psi_1^h\|_{L_{t,x}^2}   \|\psi_2\|_{L_t^\infty L_x^2}  \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|_{L_{t,x}^2}\\
&\les \sum_{\mu \ge 1} \mu^{-1+\gam - 2\ve}    \lam_3^{\frac12 + 2\ve}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}\\
&\les   \lam_3^{\frac12 + 2\ve}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}.
\end{aligned}
\end{align}
The logarithm interpolation(Lemma \ref{log}) between \eqref{qq-u2} and \eqref{qq-u4} gives us that
$$
\left| \iint V* Q(\psi_1^h,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt  \right| \les  \lam_2^{\de}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
$$
The case $\psi_2 = \psi_2^h$ can be estimated similarly.

We now treat the case $\psi_3 = \psi_3^h$ as follows.
\begin{align*}
\left| \iint V* Q(\psi_1,\psi_2)\cdot Q(\psi_3^h,\psi_4)  dxdt  \right| &\les \sum_{\mu \ge 1}\left\|\pmu \left(V*Q(\psi_1,\psi_2)\right) \right\|_{L_t^2 L_x^\infty}  \left\| \widetilde{\pmu}Q(\psi_3^h,\psi_4)\right\|_{L_t^2L_x^1}\\
&\les  \sum_{\mu \ge 1} \mu^{-1+\gam} \left\| \pmu Q(\psi_1,\psi_2)\right\|_{L_{t,x}^2}  \|\psi_3^h\|_{L_{t,x}^2}   \|\psi_4\|_{L_t^\infty L_x^2} \\
&\les  \sum_{\mu \ge 1}  \mu^{-\frac12+\gam}    \lam_3^{-\frac12} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}\\
&\les  \lam_2^{\de} \sum_{\mu \ge 1} \mu^{-1 +\gam -\de}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
\end{align*}
This completes the proof of \eqref{j4-esti}.
\end{proof}

\subsection{High-frequency parts}
For the high-frequency parts we utilize the null-form and bilinear estimates. To this end we further decompose the high-frequency parts of nonlinear term into bilinear terms as follows:

\begin{align*}
J^h &= \sum_{\mu\ge1}\left| P_\mu \left(\iint V*\left< \psi_1, \beta\psi_2\right>\right)\cdot  \left( \widetilde{P_\mu} \left<\beta\psi_3, \psi_4\right>  \right) dxdt  \right|\\
&\les J_1^h + J_2^h + J_3^h + J_4^h,
\end{align*}
where
\begin{align*}
J_1^h = \sum_{\mu\ge1} \left| \iint V* B(\psi_1,\psi_2)\cdot B(\psi_3,\psi_4)  dxdt\right|,\\
J_2^h = \sum_{\mu\ge1} \left| \iint V* Q(\psi_1,\psi_2)\cdot B(\psi_3,\psi_4)  dxdt\right|,\\
J_3^h = \sum_{\mu\ge1} \left| \iint V* B(\psi_1,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt\right|,\\
J_4^h = \sum_{\mu\ge1} \left| \iint V* Q(\psi_1,\psi_2)\cdot Q(\psi_3,\psi_4)  dxdt\right|.
\end{align*}

\subsubsection{ Estimates for $J_1^h$} By Lemma \ref{bilinear} and H\"older's inequality we get
\begin{align*}
J_1^h  &\les \sum_{\mu \ge 1}\mu^{-2+\gam}\left\|\pmu B(\psi_1,\psi_2)\right\| \left\| \widetilde{\pmu}B(\psi_3,\psi_4)\right\|   \les \sum_{\mu \ge 1}\mu^{-2+\gam} \left( \frac{\lam_2}{\lam_1}\right)^{\frac{2-\gam}4} \left( \frac{\lam_4}{\lam_3}\right)^{\frac{2-\gam}4} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}\\
      &\les  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
\end{align*}

\subsubsection{Estimates for $J_2^h$} It follows from Lemma \ref{null-esti} that
\begin{align*}
J_2^h &\les \sum_{\mu \ge 1}\mu^{-2+\gam}\left\|\pmu Q(\psi_1,\psi_2)\right\| \left\| \widetilde{\pmu}B(\psi_3,\psi_4)\right\|  \les \sum_{ \mu \ge 1} \mu^{-\frac32+\gam}  \left( \frac{\lam_4}{\lam_3}\right)^{\frac14} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}\\
      &\les (\lammed)^{\de} \sum_{ \mu \ge 1} \mu^{-\frac32+\gam -\de}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2} \les   (\lammed)^{\de} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
\end{align*}
\subsubsection{Estimates for $J_3^h$} Let us consider the subcases: (i) $\lam_3 \ll \lam_4$, (ii) $\lam_3 \sim \lam_4 \les \lam_2$, and (iii) $\lam_2 \ll \lam_3 \sim \lam_4$.
\paragraph{(i) The case: $\lam_3 \ll \lam_4$}

By Lemma \ref{bilinear} we get
\begin{align*}
J_3^h &\les \sum_{\mu \ge 1}\mu^{-2+\gam}\left\|\pmu B(\psi_1,\psi_2)\right\| \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\| \les \sum_{\mu \ge 1}\mu^{-2+\gam} \left( \frac{\lam_2}{\lam_1}\right)^{\frac{2-\gam}4} \lam_3^{\frac{\de}2} \lam_4^{\frac{2-\gam}4}\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}\\
      &\les (\lam_3)^{\de}\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}    \les (\lammed)^{\de}\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
\end{align*}

\paragraph{(ii) The case: $\lam_3 \sim \lam_4 \les \lam_2$}
By Lemmas \ref{null-esti} and \ref{bilinear} we have
\begin{align}
\begin{aligned}\label{j3-u2}
J_3^h  &\les \sum_{\mu \ge 1}\mu^{-2+\gam}\left\|\pmu B(\psi_1,\psi_2)\right\| \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\|  \les  \sum_{\mu \ge 1}\mu^{-\frac32+\gam} \left( \frac{\lam_2}{\lam_1}\right)^{\frac{1}4} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}\\
&\les (\lammed)^{\de} \sum_{\mu \ge 1}\mu^{-\frac54+\gam - \de}  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}  \les   (\lammed)^{\de}\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}
\end{aligned}
\end{align}
and
\begin{align}
\begin{aligned}\label{j3-u4}
J_3^h &\les \sum_{\mu \ge1}\mu^{-1} \left( \frac{\lam_2}{\lam_1}\right)^{\de} \lam_3^\frac12 \lam_4^\frac12\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4} \les \lam_3\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}.
\end{aligned}
\end{align}
Using logarithm interpolation(Lemma \ref{log}) between \eqref{j3-u2} and \eqref{j3-u4}, we get
$$
J_3^h \les (\lam_3)^{\de}\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
$$
\paragraph{(iii) The case: $\lam_2 \ll \lam_3 \sim \lam_4$} Since we have \eqref{j3-esti} with $\pm_4 =-$,  we  deal only with the case $\pm_4 = + $. By Lemmas \ref{null-esti} and \ref{bilinear} we get
\begin{align*}
J_3^h &\les \sum_{\mu \ge 1} \mu^{-2+\gam}  \left( \frac{\lam_2}{\lam_1}\right)^{\varepsilon} \mu \lam_3^{-\frac12}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2} \les  \lam_3^{-\frac14} \sum_{\mu \ge 1} \mu^{-1+\gam}\lam_2^{-\de +\varepsilon} \lam_2^{\de } \mu^{-\frac14}     \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2} \\
&\les \lam_2^{\de} \lam_3 ^{-\frac14}    \sum_{\mu \ge 1} \mu^{-\frac54+\gam - \de + \varepsilon}    \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2} \les  \lam_2^{\de} \lam_3 ^{-\frac14} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}
\end{align*}
for $0< \varepsilon < \min\{\de , \frac14\}$. In the same way as in \eqref{j3-u4} we also have
\begin{align*}
J_3^h &\les \lam_3  \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}.
\end{align*}
The logarithm interpolation(Lemma \ref{log}) yields that
$$
J_3^h \les (\lammed)^{\de}\prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{V_{\pm_4}^2}.
$$

\subsubsection{Estimates for $J_4^h$} We consider the subcases: (i) $ \lam_4 \les \lam_2$ and (ii) $\lam_2 \ll \lam_4$.
\medskip
\paragraph{(i) The case: $\lam_4 \les \lam_2$} By H\"older's inequality and Lemma \ref{null-esti} we obtain
\begin{align*}
J_4^h &\les \sum_{\mu \ge 1}\mu^{-2 + \gam}\left\|\pmu Q(\psi_1,\psi_2)\right\| \left\| \widetilde{\pmu}Q(\psi_3,\psi_4)\right\| \les \sum_{\mu \ge 1} \mu^{-1 + \gam} \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}\\
&\les   \sum_{\mu \ge 1} \mu^{-1 + \gam - \frac{\de + \gam-1}2} \lammed^{\frac{\de + \gam-1}2}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2} \les  (\lammed)^{\frac{\de + \gam-1}2}     \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}
\end{align*}
and by Lemma \ref{bilinear}
\begin{align*}
J_4^h &\les \sum_{\mu \ge 1} \mu^{-2 + \gam} \mu^{\frac12}\lam_3^{1-\de}\lam_4^\de \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}  \les (\lammed)^{\frac32}     \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}.
\end{align*}
By logarithm interpolation(Lemma \ref{log}) we get the desired estimates.
\medskip

\paragraph{(ii) The case: $\lam_2 \ll \lam_4$} We may consider subcases: (a) $\lam_3 \ll \lam_4 \sim \mu$ and (b) $\mu \ll \lam_3 \sim \lam_4$. But since $\lam_2 \ll \lam_4$, the case (a) does not occur. Hence we have only to consider the case (b). Due to \eqref{j4-esti} it remains to prove the estimates for the case $\pm_4 =+$. By Lemma \ref{null-esti} we get
\begin{align*}
J_4^h &\les \sum_{\mu \ge 1}   \mu^{-2+\gam} \mu^{\frac32} \lam_3^{-\frac12}      \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2} \les  (\lammed)^{\de-\varepsilon} \lam_3^{-\varepsilon}  \sum_{\mu \ge 1}   \mu^{-1+\gam -\de +2\varepsilon}       \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2} \\
      &\les (\lammed)^{\de-\varepsilon}    \lam_3^{-\varepsilon}   \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^2}
\end{align*}
for $0< \varepsilon <\frac{\de -\gam +1}{2}$. By Lemma \ref{bilinear} we also get
\begin{align*}
J_4^h &\les \sum_{\mu \ge 1}   \mu^{-2+\gam}  (\lam_1 \lam_2 \lam_3 \lam_4)^{\frac12}     \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4} \les \lammed \lam_3    \prod_{j=1}^3\|\psi_j\|_{U_{\pm_j}^2}\|\psi_4\|_{U_{\pm_4}^4}.
\end{align*}
Then Lemma \ref{log} gives us the desired estimates.

\section{Nonexistence of scattering}\label{sec-nonsca}

In this section we prove Theorem \ref{non-scat}. Since the proof for backward scattering is similar to the forward scattering, we only consider the forward scattering. We proceed by contradiction, assuming that $\|\psii(0)\|_{L_x^2} > 0$.

Let us define functionals $H(t), H_\varepsilon (t) (\varepsilon > 0)$ by
	\begin{align*}
	H(t) = {\rm sgn}(c) {\rm Im} \left< \psi, \psii \right>_{L_x^2},\;\;H_\varepsilon(t) = {\rm sgn}(c) {\rm Im} \left< (1-\varepsilon\Delta)^{-1}\psi,  (1-\varepsilon\Delta)^{-1}\psii \right>_{L_x^2}.
	\end{align*}
It is clear that $H(t)$ is uniformly bounded from the mass conservation and that for each $t > 0$, $\lim_{\varepsilon \to 0}H_\varepsilon(t) = H(t)$.
Now by taking the time derivative and using the self-adjointness of Dirac operator we have
	\begin{align*}
&	\frac{d}{dt}H_\varepsilon(t)\\
 &= {\rm sgn}(c) {\rm Im} \left( \left< \partial_t(1-\varepsilon\Delta)^{-1}\psi, (1-\varepsilon\Delta)^{-1}\psii \right>_{L_x^2} +  \left<(1-\varepsilon\Delta)^{-1} \psi, \partial_t(1-\varepsilon\Delta)^{-1}\psii \right>_{L_x^2} \right)\\
	&= {\rm sgn} (c) {\rm Im} \left(  \left< -i(\al\cdot D + \beta)\psi + i V*\left<\psi, \beta \psi\right> \beta \psi, (1-\varepsilon\Delta)^{-2}\psii \right>_{L_x^2}   + \left< \psi, -i(\al\cdot D + \beta)(1-\varepsilon\Delta)^{-2}\psii \right>_{L_x^2}    \right)\\
	&= {\rm sgn}(c) {\rm Im} \left(  i \left<  V * \left<\psi, \beta \psi\right> \beta \psi, (1-\varepsilon\Delta)^{-2} \psii \right>_{L_x^2}       \right)\\
	&= |c| {\rm Re} \left<  |\cdot |^{-\gamma}*\left<\psi, \beta \psi\right> \beta \psi, (1-\varepsilon\Delta)^{-2} \psii \right>_{L_x^2}.
		\end{align*}
Integrating over $[t_*, t^*]$, we get
$$
H_\varepsilon(t^*) - H_\varepsilon(t_*) = |c| {\rm Re} \int_{t_*}^{t^*} \left<  |\cdot |^{-\gamma}*\left<\psi, \beta \psi\right> \beta \psi, (1-\varepsilon\Delta)^{-2} \psii \right>_{L_x^2}\,dt,
$$
If $\psi \in C(\mathbb R; H^\frac\gamma2)$, then Hardy-Sobolev inequality yields $|\cdot|^{-\gamma}*\left<\psi, \beta \psi\right> \beta \psi \in C([t_*, t^*]; L_x^2)$ and thus by taking the limit as $\varepsilon \to 0$ we finally get
$$
H(t^*) - H(t_*) =  \int_{t_*}^{t^*}Y(t)\,dt,
$$
where $Y(t) = |c| {\rm Re} \left<  |\cdot |^{-\gamma}*\left<\psi, \beta \psi\right> \beta \psi,  \psii \right>_{L_x^2}$.

We will show that
\begin{align}\label{lowerbound}
|Y(t)| \gtrsim t^{-\gamma}
\end{align}
for sufficiently large $t$ when $\|\psii\|_{L_x^2} > 0$.
Once \eqref{lowerbound} has been shown, since the sign of $Y$ is fixed for large time and $0< \gamma \le 1$, \eqref{lowerbound} leads us to a contradiction to the fact that $H(t)$ is uniformly bounded on time and hence completes the proof of Theorem \ref{non-scat}.

From now on we focus on the proof of \eqref{lowerbound}. To do so we reconstitute $Y$ as follows:
\begin{align*}
Y&= Y_1 + Y_2 + Y_3,\\
Y_1 &= |c|  \mathcal V(\psii, \beta\psii),\\
	Y_2 &= |c|  {\rm Re} \left<   |\cdot|^{-\gamma}*\left(\left<\psi, \beta \psi\right> - \left<\psii, \beta \psii\right>\right) \beta \psii, \psii \right>_{L_x^2},\\
	Y_3 &= |c|  {\rm Re} \left<   |\cdot|^{-\gamma}*\left<\psi, \beta \psi\right> \beta (\psi -\psii), \psii \right>_{L_x^2}.
	\end{align*}

	We first deal with the $Y_1$.  By the assumption \eqref{assu-scat} we have that for any $t > t_*$
	\begin{align}
	\begin{aligned}\label{y1-esti}
	|Y_1| &= |c| |\mathcal V(\psii, \beta\psii)(t)| \ge \theta |c|\mathcal V(\psii, \psii)\\
	&= \theta|c| \int\Big( |\cdot|^{-\gamma}* |\psii|^2\Big)(t, x)|\psii(t, x)|^2dx\\
	&\ge \theta|c|  (4At)^{-\gam}   \left(\int   \rho\left(\frac{x}{At}\right) \left|\psii(t, x)\right|^2 dx\right)^2.
\end{aligned}
	\end{align}

	Here $\rho$ is the same cut-off function defined in the Section \ref{low-esti}. Let us set $\psii(0) = \vps$ and for the simplicity sake let us denote $\Pi_{\pm}(D)\vps$ by $\vp_\pm$. Then  $\psii = \epd \vp_+ - \emd \vp_-$ and we obtain
	\begin{align*}
\int \rho\left(\frac{x}{At}\right)   \left|\psii(x)\right|^2 dx&= \int \rho\left(\frac{x}{At}\right) |\epd \vp_+ - \emd \vp_- |^2dx\\
	&= \int   \rho\left(\frac{x}{At}\right)  \left(|\epd \vp_+|^2  +    |\emd \vp_- |^2  -2  {\rm Re} \left<\epd \vp_+, \emd \vp_- \right> \right) dx.
	\end{align*}
	We handle the last term in the integrand as follows:
	\begin{align*}
	 \int \rho\left(\frac{x}{At} \right) \left<\epd \vp_+, \emd \vp_- \right>  dx &=  \left< \rho\left(\frac{x}{At} \right)\epd \vp_+, \emd\Pi_{-}(D) \vps\right>_{L_x^2}\\
	&=  \left<\Pi_{-}(D) \left(\rho\left(\frac{x}{At} \right)  \epd \vp_+ \right), \emd \vps\right>_{L_x^2}\\
	&=  \left<\rho\left(\frac{x}{At} \right)\epd \Pi_{-}(D) \Pi_+(D)\vps,  \emd \vps\right>_{L_x^2}\\
	&\qquad\;\; + \left<  \left[\Pi_{-}(D), \rho\left(\frac{x}{At} \right)\right]  \epd \vp_+, \emd \vps\right>_{L_x^2}\\
	&= - \frac12  \left<\left[\frac{\al\cdot D + \beta}{\left< D\right>},\rho\left(\frac{x}{At} \right)\right]\epd \vp_+,   \emd \vps\right>_{L_x^2}.
	\end{align*}
	Here $\left[A ,B \right] := AB - BA$. We used \eqref{proj-commu} for the last integral. Then by Plancherel's theorem we see that
	\begin{align*}
	\left|2{\rm Re}\int \left<\epd \vp_+ , \emd\vp_-\right> dx\right| &= {\rm Re} \left|\int \left<\left[\frac{\al\cdot D + \beta}{\left< D\right>},\rho\left(\frac{x}{At} \right)\right] \vp_+, \emd \vps\right> dx\right|\\
	&\les (At)^2 \iint \frac{|\xi-\eta|}{\left<\xi\right>} |\widehat{\rho}\left(At(\xi-\eta)\right)|  |\Pi_+(\eta)\widehat{\vps}(\eta)|    |\widehat{\vps}(-\xi)| d\eta d\xi\\
	&\les (At)^{-1} \|\vps\|_{L_x^2}^2.
	\end{align*}
	From this we can choose sufficiently large $t$ so that
	\begin{align}\label{era}
	\left|2{\rm Re}\int \left<\epd \vp_+ , \emd\vp_-\right> dx\right| \le \frac1{10}\|\vps\|_{L_x^2}.
	\end{align}

	Let us set $\vpsl := P_{\ka^{-1}<\cdot\le \ka}\left(\rho\left(\frac{\cdot}{r}\right) \vps\right)$, $\phi(\xi) := x\cdot\xi + t\left<\xi\right>$, and $L(\xi) := \nabla_{\xi} \left(\frac{\nabla_{\xi}\phi}{|\nabla_{\xi}\phi|^2}\right)$. If $|x| \ge At$, by using integration by parts twice we have
	\begin{align}
	\begin{aligned}\label{decay-space}
	&\left|\epd \Pi_{+}(D)\vpsl(x) \right|\\
	&=  C \left|\int \nabla_{\xi}\phi e^{i\phi} \frac{\nabla_{\xi}\phi}{|\nabla_{\xi}\phi|^2} \widehat{\vp_{\ka,r}^+}(\xi) d \xi \right|\\
	&= C  \left| \int  e^{i\phi} L(\xi) \widehat{\vp_{\ka,r}^+}(\xi) d \xi  + \int  e^{i\phi} \frac{\nabla_{\xi}\phi}{|\nabla_{\xi}\phi|^2} \nabla_{\xi} \widehat{\vp_{\ka,r}^+}(\xi) d \xi\right|\\
	&= C \left| \int  e^{i\phi} L(\xi)^2 \widehat{\vp_{\ka,r}^+}(\xi) + e^{i\phi} \nabla_{\xi} L(\xi) \frac{\nabla_{\xi}\phi}{|\nabla_{\xi}\phi|^2}  \widehat{\vp_{\ka,r}^+}(\xi) + e^{i\phi} L(\xi)^2 \nabla_{\xi}\widehat{\vp_{\ka,r}^+}(\xi) d \xi \right| \\
	&\qquad +  C\left|\int  e^{i\phi} \nabla_{\xi} \left(\frac{1}{|\nabla_{\xi}\phi|^2}\right) \nabla_{\xi} \widehat{\vp_{\ka,r}^+}(\xi)  +   e^{i\phi}  \frac{1}{|\nabla_{\xi}\phi|^2} \nabla_{\xi}^2 \widehat{\vp_{\ka,r}^+}(\xi) d \xi\right|\\
	&\le C(A,\ka,r)|x|^{-2}\|\vp_+\|_{L_x^2}.
\end{aligned}
	\end{align}
 Since for large $\ka$ and $r$, $\|\vps - \vpsl\| \le \frac1{10}\|\vps\|$, by \eqref{era} and \eqref{decay-space}, we have
	\begin{align*}
	\left\|\rho\left(\frac{\cdot}{At}\right) \psii\right\|_{L_x^2}^2 &\ge \left\|\rho\left(\frac{\cdot}{At}\right)  \epd \Pi_{+}(D)\vps \right\|_{L_x^2}^2 - \frac{1}{10}\|\vps\|_{L_x^2}^2\\
	&\ge \left\|\rho\left(\frac{\cdot}{At}\right) \epd \Pi_{+}(D)\vpsl \right\|_{L_x^2}^2 - \frac{11}{50}\|\vps\|_{L_x^2}^2 - \frac{1}{10}\|\vps\|_{L_x^2}^2\\
	&\ge \|\epd \Pi_{+}(D) \vpsl\|_{L_x^2}^2 - \int \left( 1- \rho\left(\frac{\cdot}{At}\right)  \right) |\epd\Pi_{+}(D) \vpsl|^2 dx - \frac{8}{25}\|\vps\|_{L_x^2}^2\\
	&\ge \frac{17}{25}\|\vps\|_{L_x^2}^2 - \frac9{10}C(M,r)\|\vps\|_{L_x^2}^2\int_{|x|\ge At} |x|^{-4}dx\\
	&\ge \frac35\|\vps\|_{L_x^2}^2.
	\end{align*}
	This together with \eqref{y1-esti} leads us to
	\begin{align}\label{y1}
	|Y_1|  \gtrsim |t|^{-\gam} .
	\end{align}

	Now let us turn to $Y_2,\;Y_3$. For this purpose we introduce time decay estimate for the linear solutions and $L_x^\infty$ estimates for the potential term.

	\begin{lem}[see Lemma 4.2 of \cite{choz}]\label{time-decay}
		Let $f \in B_{1,1}^2$. Then
		\begin{align*}
		\|\epm f\|_{L_x^\infty} \les t^{-1} \|f \|_{B_{1,1}^2}.
		\end{align*}
	\end{lem}
Here $B_{1,1}^2$ is the inhomogeneous Besov space defined by $\left\{ f : \|f\|_{B_{1,1}^2}:= \sum_{\lam\ge 1}\lam^2 \|P_{\lam} f\|_{L_x^1}  < \infty\right\}$.

\begin{lem}\label{infty-esti}
	Let $0< \gam < 1$. Then for any $\mathbb C$-valued functions $u_1 \in L_x^2,\,u_2 \in L_x^2 \cap L_x^\infty$
	\begin{align}
	\||x|^{-\gam} * (u_1 \overline{u_2}) \|_{L_x^\infty} \les \|u_1\|_{L_x^2} \|u_2\|_{L_x^2}^{1-\gam} \|u_2\|_{L_x^\infty}^{\gam}.
	\end{align}
\end{lem}
\begin{proof}
	Using H\"older's inequality we get for any  $r > 2$
	\begin{align*}
	\||x|^{-\gamma} * (u_1 \overline{u_2})\|_{L_x^\infty} &=  \sup_{x\in \mathbb{R}^2}\left|\int |x-y|^{-\gam} |u_1(y) \overline{u_2}(y)|  dy\right| \\
	&\les \||x|^{-\gam}\|_{L_x^2(|x|\le R)}\|u_1\|_{L_x^2} \|u_2\|_{L_x^\infty} + \||x|^{-\gam}\|_{L_x^\frac{r}\gamma(|x|\ge R)}\|u_1\|_{L_x^2} \|u_2\|_{L_x^\frac{2r}{r-2\gamma}}\\
	&\les R^{1- \gam}\|u_1\|_{L_x^2} \|u_2\|_{L_x^\infty} + R^{\frac{\gamma(2 - r)}{r}}\|u_1\|_{L_x^2}\|u_2\|_{L_x^2}^{1-\frac{2\gamma}{r}} \|u_2\|_{L_x^\infty}^{\frac{2\gamma}{r}}.
	\end{align*}
	By taking $R = \|u_2\|_{L_x^2} \|u_2\|_{L_x^\infty}^{-1}$ we get the desired result.
\end{proof}

If $u_1 = u_2$, then  we can cover the case $\gamma = 1$ as follows.
\begin{lem}\label{infty-esti-1}
For any $\mathbb C$-valued functions $u \in L_x^2 \cap L_x^\infty$ we have
	\begin{align}
	\||x|^{-1} *|u|^2 \|_{L_x^\infty} \les \|u\|_{L_x^2} \|u\|_{L_x^\infty}.
	\end{align}
\end{lem}
\begin{proof}
	For any $1 < r_1 < 2$ and $r_2 > 2$ we have
	\begin{align*}
	\||x|^{-1} * |u|^2\|_{L_x^\infty} &=  \sup_{x\in \mathbb{R}^2}\left|\int |x-y|^{-\gam} |u(y)|^2  dy\right| \\
	&\les \||x|^{-1}\|_{L_x^{r_1}(|x|\le R)}\|u\|_{L_x^{r_1'}} \|u\|_{L_x^\infty} + \||x|^{-1}\|_{L_x^{r_2}(|x|\ge R)}\|u\|_{L_x^2} \|u\|_{L_x^\frac{2r_2}{r_2-2}}\\
	&\les R^{\frac2{r_1}- 1}\|u\|_{L_x^2}^{2- \frac2{r_1}}\|u\|_{L_x^\infty}^\frac2{r_2} + R^{\frac2{r_2} - 1}\|u\|_{L_x^2}^{2-\frac{2\gamma}{r_2}} \|u\|_{L_x^\infty}^{\frac{2}{r_2}}.
	\end{align*}
	By taking $R = \|u_2\|_{L_x^2} \|u_2\|_{L_x^\infty}^{-1}$ we finish the proof.
\end{proof}

If $u_1 \neq u_2$, then we need a regularity and a space decay assumption. Then we have
\begin{lem}\label{infty-weight}
	Let $0 < s, \delta  \ll 1$. Then for any $\mathbb C$-valued functions $u_1 \in H^{s, \delta}$ and $u_2 \in L_x^\infty$
	\begin{align}
	\||x|^{-1} * (u_1 \overline{u_2}) \|_{L_x^\infty} \les \|u_1\|_{H^{s, \delta}} \|u_2\|_{L_x^\infty}.
	\end{align}
\end{lem}
\begin{proof}
By H\"older's inequality and Sobolev embedding we see that
	\begin{align*}
	\||x|^{-1} * (u_1 \overline{u_2})\|_{L_x^\infty} &\les \||x|^{-1}\|_{L_x^\frac2{1+s}(|x|\le 1)}\|u_1\|_{L_x^\frac{2}{1-s}} \|u_2\|_{L_x^\infty} + \||x|^{-1}\|_{L_x^\frac{2}{2-\frac\delta2}(|x|\ge 1)}\|u_1\|_{L_x^\frac2{1+\frac\delta2}(|x|\ge 1)} \|u_2\|_{L_x^\infty}\\
	&\les \|u_1\|_{H^s} \|u_2\|_{L_x^\infty} + \|(1+|x|)^\delta u_1\|_{L_x^2} \|u_2\|_{L_x^\infty} \les \|u_1\|_{H^{s, \delta}}\|u_2\|_{L_x^\infty}.
	\end{align*}
	\end{proof}

 If $0 < \gamma < 1$, then from Lemmas \ref{infty-esti}, \ref{time-decay} and mass conservation it follows that
	\begin{align*}
	|Y_2| &\les   \left| \left<   V*\Big(\left<\psi, \beta \psi\right> - \left<\psii, \beta \psii\right>\Big) \beta \psii, \psii \right>_{L_x^2}\right|\\
	&\les  \left| \int  \Big(\left<\psi, \beta \psi\right> - \left<\psii, \beta \psii\right>\Big) V*\left<\beta \psii, \psii \right> dx\right|\\
	&\les  \|\psi - \psii\|_{L_x^2} \left( \|\psi\|_{L_x^2} + \|\psii\|_{L_x^2} \right)\|\psii\|_{L_x^2}\|\psii\|_{L_x^2}^{1-\gam}\|\psii\|_{L_x^\infty}^{\gam}\\
	&\les \|\psi - \psii\|_{L_x^2} \left( \|\psi_0\|_{L_x^2} + \|\vps\|_{L_x^2} \right)\|\vps\|_{L_x^2}^{2-\gam}t^{-\gam}\left(\|\vp_+\|_{B_{1,1}^2} + \|\vp_-\|_{B_{1,1}^2}\right)^{\gam}
	\end{align*}
	and
	\begin{align*}
	|Y_3| &\les \left| \left<   V*\left<\psi, \beta \psi\right> \beta (\psi -\psii), \psii \right>_{L_x^2} \right|\\
	&\les \left| \int   \left<\psi, \beta \psi\right> \left(V* \left<\beta (\psi -\psii),\psii \right>\right)  dx\right|\\
	&\les \|\psi\|_{L_x^2}^2 \|\psi-\psii\|_{L_x^2}\|\psii\|_{L_x^2}^{1-\gam}\|\psii\|_{L_x^\infty}^\gam\\
	&\les \|\psi_0\|_{L_x^2}^{2}\|\psi-\psii\|_{L_x^2}\|\vps\|_{L_x^2}^{1-\gam}t^{-\gam}\left(\|\vp_+\|_{B_{1,1}^2} + \|\vp_-\|_{B_{1,1}^2} \right)^{\gam}.
	\end{align*}
	Therefore we get
	\begin{align}\label{yj}
	|Y_j| = o(t^{-\gam})
	\end{align}
	for $j=1,2$. Then \eqref{y1} and \eqref{yj} conclude \eqref{lowerbound} for the case $0 < \gamma < 1$.

If $\gamma = 1$, then by Lemma \ref{infty-esti-1} we have
\begin{align*}
	|Y_2| &\les  \left| \int  \Big(\left<\psi, \beta \psi\right> - \left<\psii, \beta \psii\right>\Big) |\cdot|^{-1}*\left<\beta \psii, \psii \right> dx\right|\\
	&\les  \|\psi - \psii\|_{L_x^2} \left( \|\psi\|_{L_x^2} + \|\psii\|_{L_x^2} \right)\|\psii\|_{L_x^2}\|\psii\|_{L_x^\infty}\\
	&\les \|\psi - \psii\|_{L_x^2} \left( \|\psi_0\|_{L_x^2} + \|\vps\|_{L_x^2} \right)\|\vps\|_{L_x^2}t^{-1}\left(\|\vp_+\|_{B_{1,1}^2} + \|\vp_-\|_{B_{1,1}^2}\right) = o(t^{-1})
	\end{align*}
and by Lemma \ref{infty-weight} and mass conservation
\begin{align*}
	|Y_3| &\les \left| \left<   |\cdot|^{-1}*\left<\psi, \beta \psi\right> \beta (\psi -\psii), \psii \right>_{L_x^2} \right|\\
	&\les \left| \int   \left<\psi, \beta \psi\right> \left(V* \left<\beta (\psi -\psii),\psii \right>\right)  dx\right|\\
	&\les \|\psi\|_{L_x^2}^2 \|\psi-\psii\|_{H^{s,\delta}}\|\psii\|_{L_x^\infty}\\
	&\les \|\psi_0\|_{L_x^2}^{2}\|\psi-\psii\|_{H^{s, \delta}}t^{-1}\left(\|\vp_+\|_{B_{1,1}^2} + \|\vp_-\|_{B_{1,1}^2} \right) = o(t^{-1}).
	\end{align*}
This completes the proof of Theorem \ref{non-scat}.

\section*{Acknowledgements}
This work was supported in part by NRF-2018R1D1A3B07047782(Republic of Korea).

redpajama_set_name: RedPajamaArXiv


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Slovenia is the only country in the world that has a word love in its name.
It is third most forested country in Europe. Almost 60% of its territory is covered by forest.
Sport is an important part of life. The most popular sports are winter sports (ski jumping,skiing), hiking, team sports (basketball, football). Slovenian athletes are among the best in the world in terms of medals won in major championships per capita.
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Fruit seller on Lighthouse Beach in Kovalam is quiet for a change.
"Traveling is like toilet paper: The closer you get to the end the quicker it runs out." — Chrissy after her friend Vanina.
What was in a sense my busiest day so far is the day I have the least to write about and is also the day I took the most pictures. Fewer words, more images. Trivandrum and Kovalam by rickshaw, an Om tattoo on the inner right ankle, ritual prayer (puja) to the Divine Mother, and I wore local clothing — a nice change from my daily costume of white T-shirt.
Our rickshaw driver took us for some cheap, tasty, spicy breakfast on the way to town.
A family at the beach parking lot avoid the camera.
Corinne and Lily carefully choose our first non-ashram lunch.
A tourist takes Lonely Planet's advice to sunbathe conservatively.
Corinne's favorite young man strikes a pose.
A word of caution:Old Man Jim is a ponraizilg figure i think everyone can agree there hardly exists a person whose feelings towards the former president could be described as lukewarm. I am naturally suspicious of anyone who employs reasoning that consists merely of: 1) person X is bad, ergo 2) anything they do is bad. Ignoring the people who are firing rockets from Gaza into Israel has hardly produced positive results, and the Israelis cracking down more and more severely in an effort to limit Hamas' power has yielded results that can, at best, be described as having the opposite effect: case in point, Jimmy Carter is now going to sit down with Hamas. All of this leads me to conclude that nothing that has been done so far has worked. If you insist on playing to the more extreme elements of ANY group, they will continue to trend towards extremism. If you play to the more moderate elements, and give them some sense of legitimacy, they must (from that point on) act in a way that makes them seem worthy of that attention, or risk returning to the previous status quo. Also, by engaging moderate elements (not giving into demands, not trading with them merely LISTENING), you quite possibly drive a wedge between the moderate and extremist elements. If the moderates then police or limit the extremist elements of their group, that is one less job/angle that either the US or Israel has to worry about. I fail to understand the so-called "wisdom" of refusing to merely listen to people we do not agree with you learn more that way than only listening to people with whom you agree. And before anyone decides that somehow I am advocating giving support of any kind to groups such as Hamas (they have done nothing yet to deserve it), merely listening and meeting with people does not give them power OR legitimacy; they gain attention, certainly, but that added attention means added scrutiny as well, and with more people paying attention, any further negative action will produce considerably more public backlash. I would argue that our most skeptical state department representatives should have been visiting with these "leaders" a long time ago to hear in their words what their complaints/issues/requests are. Know your enemy: what better way to know and understand them than actually getting it from the proverbial horse's mouth?
Dead pent subject material , thanks for entropy.


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text:
'The SpongeBob Movie: Sponge On The Run' Trailer: Keanu Reeves Is A Sage In More Ways Than One
By eelyajekiM | @ | November 14th, 2019 at 12:37 pm
Paramount Pictures has released the first trailer for The SpongeBob Movie: Sponge On The Run. In it, we see the titular character go on his first ever CGI feature film adventure to find his beloved pet snail, Gary. And apparently Keanu Reeves is a sage who also happens to be a sage. Oh, the irony. Check out the trailer below.
Topics: Movies, News, Sequels, Trailers
Tags: Bill Fagerbakke, Carolyn Lawrence, Clancy Brown, Jill Talley, Lori Alan, Mary Jo Catlett, Mr Lawrence, Nickelodeon, Paramount Animation, Paramount Pictures, Rodger Bumpass, SpongeBob SquarePants, The SpongeBob Movie: Sponge On The Run, Tim Hill, Tom Kenny
'Little Evil': Trailer For 'The Omen' Inspired Netflix Horror Comedy Released
By The Movie God | @ | August 22nd, 2017 at 10:10 pm
Last year we learned that a horror comedy from Tucker & Dale vs. Evil director Eli Craig titled Little Evil was coming to Netflix. Now the streaming service has released an official trailer for the movie, which stars Adam Scott and Evangeline Lilly.
The movie follows a man (played by Scott) who just got married (to Lilly's character)…only to find out that his new stepson happens to be the Antichrist. It's basically The Omen, but with a healthy dose of comedy mixed in.
You can check out the trailer below.
Tags: Adam Scott, Brad Williams, Bridget Everett, Chris D'Elia, Clancy Brown, Donald Faison, Eli Craig, Evangeline Lilly, Little Evil, Marcus Terrell Smith, Netflix, Owen Atlas, Tyler Labine
Blu-ray Review: Highlander: 30th Anniversary Edition
By Empress Eve | @ | September 27th, 2016 at 10:00 am
30th Anniversary Edition – Director's Cut
Directed Russell Mulcahy
Starring Christopher Lambert, Roxanne Hart, Clancy Brown, Sean Connery
Lionsgate Home Video
Russell Mulcahy's 1986 feature Highlander centers on Connor MacLeod (Christopher Lambert), a man born in the Scottish Highlands in the 1500s who discovers he's immortal after he miraculously recovers from a deadly wound he sustained during his first battle against a rival clan. The film seamlessly goes back and forth between MacLeod's past in 16th century Scotland and his current life as antiques dealer Russell Nash in modern-day New York City, with a few other brief past-life flashbacks thrown in. Under the tutelage of Sean Connery's Ramírez, a fellow Immortal, MacLeod learns to swordfight and use his strength in preparation for the eventual Gathering, when all Immortals will be summoned together to fight for "The Prize."
In modern-day, MacLeod comes under suspicion by the police after a man is found decapitated by an extremely rare broadsword, which was recovered at the crime scene by police forensics pathologist Brenda Wyatt (Roxanne Hart), who seeks to know more about the weapon and where it came from. Meanwhile, the Kurgan (Clancy Brown), the fierce warrior who dealt Connor that initial deadly blow back in Scotland, has arrived in New York City to seek the Highlander out and finish what he started so long ago.
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Tags: Allan Cameron, Christopher Lambert, Clancy Brown, Director's Cut, Gerry Fisher, Gregory Widen, Highlander, Larry Ferguson, Peter Bellwood, Queen, Roxanne Hart, Russell Mulcahy, Sean Connery, William N. Panser
'Highlander' Getting A 30th Anniversary Edition Blu-ray Release
By Empress Eve | @ | July 18th, 2016 at 3:37 pm
The cult classic film Highlander, starring Christopher Lambert and Sean Connery, is getting a 30th Anniversary Edition release on DVD and Blu-ray this Fall!
Liongate announced today that the 30th Anniversary Edition will be released on September 27, 2016 and will contain new bonus features, such as never-before-seen interviews, deleted scenes, a new "making of" documentary, and audio commentary with director Russell Mulcahy.
Check out the DVD and Blu-ray box art here below, along with a full list of bonus features.
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New 'Warcraft' Behind The Scenes Videos
With its June 10th release date fast approaching, Legendary has been releasing plenty of peeks and previews for Warcraft in a home stretch attempt to get as many people as possible who aren't already sold on the movie excited to see it when it arrives.
Among the videos that have been released are some behind the scenes featurettes that chat with the cast and crew and look at things like training, visual effects—including taking CG character facial expressions and digital hair to another level—and re-creating The Lion's Pride inn from the game the movie is based on, World of Warcraft.
You can check out all of these videos plus a collection of videos looking at the various individual characters below.
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redpajama_set_name: RedPajamaCommonCrawl


Aerial photography jobs send me all over the metroplex. It's not uncommon for me to find myself flying right over the new Dallas Cowboy's Texas Stadium. I've had the pleasure of seeing it's construction from a unique perspective and I'm posting to share it with you guys… my annonymous readers who remain unknown due to no comments *cough*.
That's alright, I still love you all.


A result of an eight week study found that Facebook's fake account detection mechanisms can be easily overcome by 80 percent of the time with the special help of automated tools. The study conducted by researchers from the University of British Columbia (UBC).
To create a user account on an OSN [online social network] the bots must perform three tasks: providing an active email addresses, creating a user profile, and a little bit of solving a CAPTCHA. […] They are arguing whether an adversary can fully computerize the account creation process.
The trick that the researchers did is not new since its been used before by Koobface have long used automatically created accounts in spamming malicious links. This persuaded Facebook in developing specialized detection mechanisms over the years.
According to UBC researchers, such defenses are not that effective in fighting those malwares. The social bots the researchers let lose onto Facebook targeted 5,053 arbitrarily selected users to whom they sent friend requests.
The rate at which the embattled individuals accepted these requests were 20 percent on average. The researchers found out that by using female profiles, the bots have a higher success than those bots that have male profiles. However, the rate tripled when the bots started targeting friends of those who had accepted requests.
Following the time that the bots has become a friend with new users the automated programs scraped their profiles, news feeds and wall posts so that it could gather all the pertinent information. The collected data included things like gender, birth date, place of employment, names of attended schools, home city, current city, mail address, email address, phone number, IM account IDs and marital status.
The researcher got forced to stop the test after their servers can not handle the traffic that it is getting from the said test.
During the time of the trial, Facebook's real-time protection system only manages to block 20 of the 100 fake profiles. However, when they investigate it thoroughly, they discovered that the profiles got flagged as spam by other users.
Malware experts acknowledge Facebook's efforts in blocking automated account creation efforts on its social network. According to antivirus vendor BitDefender, the number of threats, using the same techniques has decreased considerably in the course of the past two years.
The researchers said that the best way to protect a person's account is by not accepting any friend invitation from unidentified users. Also, it would be better if users would not click on any links that that users sent on them, even if it comes from someone you know.


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Create a SVG image, e.g. in svg-edit. If you open the XML editor in svg-edit you can copy the SVG markup.
As you can see the color of the gradient is white, so opening this document in a browser window would just display a white page. The gradient is used in a rectangel that fills the entire SVG image. The SVG document itself has no size, so it will just fill it's container.
Go to The data: URI kitchen and generate a URI using the type image/svg+xml;charset=utf-8 checkmarking base64 and pasting in the SVG.


The Royal Australian Navy has been very busy over the past three years as a member of the 31-nation Combined Maritime Forces group patrolling the oceans west and north of Australia.
HMAS Arunta recently arrived home after a long time at sea during which time its crew discovered and seized 570 kilograms of heroin and 800 kilograms of hashish.
The street value of these discoveries is more than $200,000,000.
Three million hits of the drug Ice have been taken off our streets following a successful raid by Border Force officials.
They discovered 370 kilograms of ephedrine secreted in sea cargo and the 'find' has led to two arrests so far.
A former manager of premiership winning AFL players will go on trial after allegedly being caught with $1 million worth of Ice during a routine traffic stop.
Gold Coast police have discovered nearly $750,000 worth of drugs and a shotgun after raiding homes of ex-bikie gang members.The drugs included methamphetamines, MDMA, cannabis and prescription drugs.
Police have found $35,000 in cash and drugs in a four wheel drive vehicle after it left the Castlereagh Highway and flipped several times. The occupants escaped with minor injuries.
A fight between two friends over heroin has need with one of them fighting for his life in hospital with stab wounds to the abdomen.
A Crime And Corruption Commission audit of hospitals has found 66 per cent of thefts related to missing drugs.
The State Opposition in Victoria is calling Premier Andrews' Ice Action Plan a $184 million dud.
HMAS Arunta is returning to port after a nine month deployment in which time it discovered over 1.3 tonnes of illegal drugs hidden inside foreign vessels bound for Australia.
The crew seized 800kg of hashish in March, 250kg of heroin in May and another 260kg of heroin in June. HMAS Newcastle will take over from Arunta in the same region.