\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 142, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2010/142\hfil Well-posedness and ill-posedness]
{Remark on well-posedness and ill-posedness  for the KdV
equation}

\author[Takamori Kato\hfil EJDE-2010/142\hfilneg]
{Takamori Kato}

\address{Takamori Kato \newline
 Graduate School of Mathematics, Nagoya University,
Chikusa-ku, Nagoya, 464-8602, Japan}
\email{d08003r@math.nagoya-u.ac.jp}

\thanks{Submitted August 19, 2010. Published October 8, 2010.}
\subjclass[2000]{35Q55}
\keywords{KdV equation; well-posedness; ill-posedness; Cauchy problem;
\hfill\break\indent  Fourier restriction norm; low regularity}

\begin{abstract}
 We consider the Cauchy problem for the KdV equation with low
 regularity initial data given in the space $H^{s,a}(\mathbb{R})$,
 which is defined by the norm
 \[
 \| \varphi \|_{H^{s,a}}=\| \langle \xi \rangle^{s-a}
 |\xi|^a \widehat{\varphi} \|_{L_{\xi}^2}.
 \]
 We obtain the local well-posedness in $H^{s,a}$ with
 $s \geq \max\{-3/4,-a-3/2\} $,  $-3/2< a \leq 0$ and
 $(s,a) \neq (-3/4,-3/4)$.
 The proof is based on Kishimoto's work \cite{Ki09} which proved
 the sharp well-posedness in the Sobolev space $H^{-3/4}(\mathbb{R})$.
 Moreover we prove ill-posedness when
 $s< \max\{-3/4,-a-3/2\}$, $a\leq -3/2$ or $a >0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

We consider the Cauchy problem of the Korteweg-de Vries equation
as follows;
\begin{equation} \label{KdV}
\begin{gathered}
 \partial_t u + \partial_{x}^3 u- 3 \partial_x (u)^2 =0,
\quad  (t,x) \in [0,T] \times \mathbb{R}, \\
 u(0, x)=u_0(x), \quad   x \in \mathbb{R}.
\end{gathered}
\end{equation}
Here the given data $u_0$ and an unknown function $u$ are
real-valued. We consider \eqref{KdV} with
 initial data given in the space $H^{s,a}(\mathbb{R})$, which is defined by
the norm
\begin{align*}
\| \varphi \|_{H^{s,a}}:=\| \langle \xi \rangle^{s-a} |\xi|^a
\widehat{\varphi} (\xi)  \|_{L_{\xi}^2},
\end{align*}
where $\langle \xi \rangle:= (1+ | \xi|^2)^{1/2}$ and
$\widehat{u}$ is the Fourier transform of $u$.
The KdV equation was originally derived by Korteweg and de Vries \cite{KV}
as a model for the propagation of shallow water waves along a canal.
This equation is completely integrable in the sense that there are
Lax formulations, which have an infinite number of conservation laws
as follows;
\[
\int u^2 dx,\quad \int (\partial_x u)^2+2 u^3 dx,\quad
\int (\partial_x^2 u)^2+ 5 \partial_x(\partial_x u)^2 +\frac{5}{2} u^4 dx, \quad
\text{etc.}
\]
Our main aim is to prove the local well-posedness (LWP for short)
for \eqref{KdV} with low regularity initial data given
in $H^{s,a}(\mathbb{R})$. The main tool is the Fourier restriction
norm method introduced by Bourgain \cite{Bo}.

We recall some known results of LWP for \eqref{KdV} with initial
data given in the Sobolev space $H^s(\mathbb{R})$. The viscosity
method was applied to establish LWP for \eqref{KdV} with $s>3/2$,
(see \cite{BS75}). Kenig, Ponce and Vega \cite{KPV93} proved LWP
for $s>3/4$ by the iterative approach exploiting the local
smoothing effect for the Airy operator $e^{-t \partial_x^3}$. Bourgain
\cite{Bo} established the Fourier restriction norm method and
showed LWP for $s \geq 0$ by this method, which was improved to
$s>-3/4$ by Kenig, Ponce and Vega \cite{KPV96}. In \cite{KPV01},
they also proved that the data-to-solution map fails to be
uniformly continuous as a map from $H^s$ to $C([0,T];H^s)$ for $s
<-3/4$, (see also \cite{CCT}). Kishimoto \cite{Ki09} showed LWP
and the global well-posedness for \eqref{KdV} at the critical
regularity $s=-3/4$, (see also \cite{Gu}). In \cite{Tz}, Tzvetkov
proved the flow map $\dot{H}^s \ni u_0 \mapsto u(t) \in \dot{H}^s$
cannot be $C^2$ for $s<-3/4$.

Under the following assumptions we obtain the following
well-posedness result which is generalization of \cite{Ki09}.
\begin{equation}
\label{co_op} s \geq \max \bigl\{ -\frac{3}{4}, -a-\frac{3}{2}\bigr\},\quad
 -\frac{3}{2}< a \leq 0, \quad
(s,a) \neq (-\frac{3}{4},-\frac{3}{4})\,.
\end{equation}

\begin{theorem} \label{thm_well}
 Let $s,a$ satisfy \eqref{co_op}. Then
\eqref{KdV} is locally well-posed in $H^{s,a}$.
\end{theorem}

We put $s_a=-a-3/2$ and $B_r(\mathcal{X}):= \{ u \in \mathcal{X}; \| u \|_{\mathcal{X}} \leq r   \}$ for a Banach space $\mathcal{X}$.
We obtain ill-posedness for \eqref{KdV} in the following sense when
$s< \max\{-3/4,-a-3/2 \}$, $a \leq -3/2$ or $a >0$.

\begin{theorem} \label{thm_ill}
\begin{itemize}
\item[(i)] Let $r>1$ and $-3/2< a <-3/4$. Then,
from Proposition~\ref{prop_well} below, there exist $T>0$ and the flow map
for \eqref{KdV} $B_r(H^{s_a,a}) \ni u_0 \mapsto u(t) \in H^{s_a,a} $ for any $t \in (0,T]$.
The flow map is discontinuous on $B_r(H^{s_a,a})$ (with $H^{s,a}$ topology) to $H^{s_a,a}$ (with $H^{s,a}$ topology) for any $s<s_a$.

\item[(ii)] Let $s < s_a$, $a \leq -3/2$ or $0< a$. Then there is
no $T>0$ such that the flow map for \eqref{KdV}, $u_0 \mapsto u(t) $,
is $C^2$ as a map from $B_r(H^{s,a})$ to $H^{s,a}$ for $t \in (0,T]$.

\item[(iii)]  Let $s <-3/4$ and $a \in \mathbb{R}$. Then there is
no $T>0$ such that the flow map for \eqref{KdV}, $u_0 \mapsto u(t)$,
is $C^3$ as a map from $B_r(H^{s,a})$ to $H^{s,a}$ for any
$t \in (0,T]$.
\end{itemize}
\end{theorem}

We consider \eqref{KdV} with initial data given in the homogeneous
Sobolev space $\dot{H}^s(\mathbb{R})$.
Noting $\dot{H}^{s}(\mathbb{R})=H^{s,a}(\mathbb{R})$ if $s=a$,
we immediately obtain the following results.

\begin{corollary} \label{dot_well}
Let $-3/4 < s \leq 0$. Then \eqref{KdV} is well-posed in $\dot{H}^{s}$.
\end{corollary}

\begin{corollary} \label{dot_ill}
\begin{itemize}
\item[(i)] Let $r>1$, $s_s-s-3/2$ and $-3/2<s <-3/4$.
Then, from Theorem~\ref{dot_well}, there exists $T>0$ and the flow map for \eqref{KdV} $B_r(H^{s_s,s})
\ni u_0 \mapsto u(t ) \in H^{s_s,s}$ for any $t \in (0,T]$. The flow map
is discontinuous on $B_r(H^{s_s,s})$ (with $\dot{H}^{s}$ topology) to
$H^{s_s,s}$ (with $\dot{H}^{s}$ topology).

\item[(ii)] Let $s>0$ or $s \leq -3/2$. Then there is no $T>0$
such that the flow map for \eqref{KdV},
$u_0 \mapsto u(t)$, is $C^2$ as a map from
$B_r(\dot{H}^s)$ to $\dot{H}^s$ for $t \in (0, T]$.
\end{itemize}
\end{corollary}


\noindent\textbf{Remark.}
We do not know whether LWP for \eqref{KdV} holds or not in
$H^{-3/4,-3/4}$.
In the present paper, we only prove LWP when
$s \geq \max\{-3/4, -a-3/2\}$, $-3/2<a<0$ and
$(s,a) \neq (-3/4,-3/4)$ because
the case $a=0$ is proved in \cite{Ki09}.

The main idea is how to define the function space to construct
the solution of \eqref{KdV}. The bilinear estimates of the
 nonlinear term $\partial_x(u)^2$ play an important role to prove
Theorem~\ref{thm_well}. Here the Bourgain space $\hat{X}^{s,a,b}$
is defined by
\[
\hat{X}^{s,a,b}:=\{ f \in \mathcal{Z}'(\mathbb{R}^2);
\| f \|_{\hat{X}^{s,a,b}}:=\| \langle \xi \rangle^{s-a}
|\xi|^a \langle \tau-\xi^3 \rangle^b f
\|_{L_{\tau,\xi}^2} < \infty \}.
\]
Here $\mathcal{Z}'(\mathbb{R}^n)$ denotes the dual space of
\begin{align*}
\mathcal{Z}(\mathbb{R}^n) :=\{ f \in \mathcal{S}(\mathbb{R}^n);
D^{\alpha} \mathcal{F}
 f(0)=0 \text{ for every multi-index } \alpha \}.
\end{align*}
For details on $\mathcal{Z}(\mathbb{R}^n)$, see e.g.
\cite[pp. 237]{Tr}.

We consider the bilinear estimate in the Bourgain space
$\hat{X}^{s,a,b}$ as follows
\begin{equation} \label{BE-3}
 \| \xi f * g \|_{\hat{X}^{s,a,b-1}}
\leq C \| f \|_{\hat{X}^{s,a,b}} \| g \|_{\hat{X}^{s,a,b}}.
\end{equation}
However, \eqref{BE-3} fails to hold for any $b \in \mathbb{R}$ when
\begin{gather} \label{co_cr1}
 s=-\frac{3}{4},\quad -\frac{3}{4} < a \leq 0, \\
 \label{co_cr2}
s=-\frac{3}{4}+\varepsilon_1,\quad
a=-\frac{3}{4} , \quad \text{or}\quad
 s=-a-\frac{3}{2},\quad  -\frac{16}{15} <a< -\frac{3}{4},
\end{gather}
where $\varepsilon_1$ is a sufficiently small number such that
$0< \varepsilon_1 \leq s+3/4$.
Therefore, the standard argument by using the Fourier restriction
norm method does not work for \eqref{co_cr1}--\eqref{co_cr2}.
To overcome this difficulty, we modify the Bourgain space to
establish bilinear estimates for \eqref{co_cr1}--\eqref{co_cr2}.
An idea of a modification of
the Bourgain space is used by Bejenaru-Tao \cite{BT} to prove LWP
at the critical regularity $s=-1$ for the quadratic Schr\"{o}dinger
equation with nonlinear term
$u^2$. We consider counterexamples of \eqref{BE-3} to find a suitable
function space in the case \eqref{co_cr1}.
Noting Example \ref{exa3} in the appendix, we make a modification to the Besov
type space as follows:
\begin{align*}
\| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
:= \bigl\|  \bigl\{ \| \langle \xi \rangle^s  \langle \tau-\xi^3 \rangle^{1/2}
f \|_{L_{\tau,\xi}^2(A_j \cap B_k)} \bigr\}_{j \geq 0, k \geq 0}
 \bigr\|_{l_j^2 l_k^1},
\end{align*}
where $A_j$, $B_k$ are two dyadic decompositions defined by
\begin{gather*}
A_j:=\{ (\tau,\xi) \in \mathbb{R}^2; 2^{j} \leq \langle \xi \rangle
< 2^{j+1} \}, \\
B_k:=\{ (\tau,\xi) \in \mathbb{R}^2; 2^{k} \leq \langle \tau-\xi^3 \rangle
< 2^{k+1} \},
\end{gather*}
for $j,k \in \mathbb{N} \cup \{0 \}$.  For a normed space
$\mathcal{X}$ and a set $\Omega \subset \mathbb{R}^n$,
$\| \cdot  \|_{\mathcal{X}(\Omega)}$ is defined by
$\| f \|_{\mathcal{X}(\Omega)} := \|\chi_{\Omega} f \|_{\mathcal{X}}$
where $\chi_{\Omega}$ is the characteristic function of $\Omega$.

 From Examples \ref{exa1} and \ref{exa2} in the appendix,
we have to take $b=a/3+1/2$ on the domain
\begin{align*}
D_0:=\bigl\{ (\tau,\xi) \in \mathbb{R}^2 ~; ~
|\xi| \leq 1 \text{ and } |\tau| \sim |\xi|^{-3} \bigr\}
\end{align*}
to obtain \eqref{BE-3} for \eqref{co_cr1}. Therefore,
we make a modification on the Bourgain norm in the
low frequency part $\{ |\xi| \leq 1  \}$ as follows:
\[
 \| f \|_{\hat{X}_L^{a}}:=
\begin{cases}
\| f \|_{\hat{X}_L^{a,a/3+1/2}(A_0)} & \text{ for }
-3/4 <a< 0 , \\
\| f \|_{\hat{X}_L^{-3/4,1/4+\varepsilon_1/2} (A_0)} & \text{ for } a=-3/4, \\
 \| f \|_{\hat{X}_L^{a,1/4+\varepsilon_2/2 }(A_0)} &  \text{ for }
-3/2< a< -3/4.
\end{cases}
\]
where $\varepsilon_2$ is a sufficiently small number
 satisfying $0< \varepsilon_2 \leq -(a+3/4)$ and $\hat{X}_L^{a,b} $
is equipped with the norm
\[
 \| f \|_{\hat{X}_L^{a,b}}:=\| |\xi|^a \langle \tau-\xi^3 \rangle^b
f  \|_{L_{\tau,\xi}^2(A_0)}.
\]
Following the above argument, we define
the function space
\[
\hat{Z}^{s,a}:=\bigl\{f \in \mathcal{Z}'(\mathbb{R}^2);
\|f\|_{\hat{Z}^{s,a}} :=\| p_h f \|_{\hat{X}_{(2,1)}^{s,1/2}}+
\| p_l f \|_{\hat{X}_L^{a}} < \infty \bigr\},
\]
where $p_h$, $p_l$ are the projection operators such that
$(p_h f) (\xi):=f(\xi)|_{|\xi| \geq 1}$ and
$(p_l f) (\xi) :=f(\xi)|_{|\xi| \leq 1}$. Using the function space above,
we obtain the following estimates which are the main estimates
in this article.

\begin{proposition} \label{BE-ES-1}
Let $s,a$ satisfy \eqref{co_op}. Then
\begin{gather}
\label{BE-1}
 \| \langle \tau-\xi^3 \rangle^{-1} ~ \xi~f*g
\|_{\hat{Z}^{s,a}}
\leq C \| f \|_{\hat{Z}^{s,a}} \| g \|_{\hat{Z}^{s,a}}, \\
\label{BE-2}
 \| \langle \xi \rangle^{s-a} |\xi|^{a+1} 
 \langle \tau-\xi^3 \rangle^{-1} f*g  \|_{L_{\xi}^2 L_{\tau}^1}
\leq C \| f \|_{\hat{Z}^{s,a}} \| g \|_{\hat{Z}^{s,a}}.
\end{gather}
\end{proposition}


We will use $A \lesssim B$ to denote $A \leq CB$ for some positive
constant $C$ and write $A \sim B$ to mean $A \lesssim B$ and
$B \lesssim A$.
The rest of this paper is organized as follows.
In Section 2, we give some preliminary lemmas.
In Section 3, we prove the bilinear estimates. In Section 4,
We give the proofs of Theorem~\ref{thm_well} and~\ref{thm_ill}.

\section{Preliminaries}

In this section, we prepare some lemmas to show the main theorems
and the bilinear estimates. When we use the variables
$(\tau,\xi)$, $(\tau_1,\xi_1)$ and $(\tau_2,\xi_2)$, we always
assume the relation
\[
(\tau,\xi)=(\tau_1,\xi_1)+(\tau_2,\xi_2).
\]
We state the smoothing estimates for the KdV equation.

\begin{lemma} \label{lem_dy_1}
Suppose that $f$ and $g$ are supported on a single $A_j$ for $j \geq 0$.
 If
\begin{align*}
K:=\inf \{ |\xi_1-\xi_2|;\exists \tau_1,\tau_2 \text{ s.t. }
(\tau_1,\xi_1) \in \operatorname{supp}f,~(\tau_2,\xi_2)
\in \operatorname{supp}g  \}>0,
\end{align*}
then we have
\begin{equation} \label{es_dy_1-2}
\| |\xi|^{1/2}~f*g \|_{L_{\tau,\xi}^{2}} \lesssim K^{-1/2}
\| f \|_{\hat{X}_{(2,1)}^{0,1/2}} \| g \|_{\hat{X}_{(2,1)}^{0,1/2}}.
\end{equation}
\end{lemma}

\begin{lemma} \label{lem_dy_2}
Assume that $f$ is supported on $A_{j}$ and $g $ is an arbitrary
test function for $j \geq 0$.
If a non-empty set $\Omega \subset \mathbb{R}^2$ satisfies
\begin{align*}
K:=\inf \{|\xi+\xi_1|;\exists \tau,\tau_1 \text{ s.t. } (\tau,\xi)
\in \Omega,~(\tau_1,\xi_1) \in \operatorname{supp}f  \}>0,
\end{align*}
then
\begin{equation} \label{es_dy_2-2}
\| f* g \|_{L_{\xi, \tau}^2(\Omega \cap B_k)}
\lesssim  2^{k/2}~K^{-1/2}~
 \| f \|_{\hat{X}_{(2,1)}^{0,1/2} } \| |\xi|^{-1/2} g  \|_{L_{\tau,\xi}^2}.
\end{equation}
\end{lemma}

For the proof of these lemmas, refer the reader to
 \cite[Lemmas 3.2 and 3.3]{Ki09}.
Here we put $U(t):=\exp (-t \partial_x^3)$ and a smooth cut-off
function $\varphi (t)$ satisfying
$\varphi (t)= 1$  for $|t|<1$ and  $\varphi (t)=0$ for $|t|>2 $.
For a Banach space $\mathcal{X}$,
$\| \cdot \|_{\mathcal{X}}$ denotes
$\| u \|_{\mathcal{X}}=\| \widehat{u} \|_{\hat{\mathcal{X}}} $.
We mention the linear estimates below.

\begin{proposition} \label{prop_linear1}
Let $s,a \in \mathbb{R}$ and $u(t)=\varphi (t) U(t) u_0$.
Then the following estimate holds.
\[
\| u\|_{Z^{s, a}} +\| u \|_{L_t^{\infty}( \mathbb{R};H_{x}^{s,a})}
\lesssim \| u_0 \|_{H^{s,a}}.
\]
\end{proposition}

\begin{proposition}\label{prop_linear2}
Let $s,a \in \mathbb{R}$ and
\[
u(t)=\varphi (t) \int_{0}^{t} U(t-s) F(s) ds .
\]
Then
\[
\| u \|_{Z^{s, a}}+ \| u \|_{L_t^{\infty} (\mathbb{R}; H_x^{s,a})}
\lesssim \|\mathcal{F}_{\tau,\xi}^{-1} \langle \tau-\xi^3 \rangle^{-1}
\widehat{F}  \|_{Z^{s,a}}+
\| \langle \xi \rangle^{s-a} ~|\xi|^{a}~  \langle \tau-\xi^3 \rangle^{-1}
\widehat{F}  \|_{L_{\xi}^2 L_{\tau}^1}.
\]
\end{proposition}

The proofs of these two propositions are given
in \cite{GTV}.

\section{Proof of the bilinear estimates}

In this section, we give the proof of the bilinear estimates
\eqref{BE-1} and \eqref{BE-2}. We use the following notation
for simplicity,
\begin{align*}
 A_{<j_1}:=\cup_{j <j_1} A_{j}, \quad
B_{[k_1,k_2)}:=\cup_{k_1 \leq k <k_2} B_{k},
\quad  \text{etc.}
\end{align*}
We now prove the key bilinear estimates.

\begin{proposition} \label{prop_BE-2}
Let $s,a$ satisfy \eqref{co_op}. Suppose that $f$ and $g$ are
restricted on $A_{j_1}$ and $A_{j_2}$ for
$j_1, j_2 \in \mathbb{N} \cup \{0\}$. For $j \geq 0$, we obtain
\begin{gather} \label{BE-X}
 \| \langle \tau-\xi^3 \rangle^{-1}\xi f*g \|_{\hat{Z}^{s,a}(A_j)}
 \lesssim C(j,j_1,j_2) \| f \|_{\hat{Z}^{s,a}} \| g \|_{\hat{Z}^{s,a}},
 \\
\label{BE-Y}
\bigl\|~\langle \xi \rangle^{s-a}~
|\xi|^{a+1}~ \langle \tau-\xi^3 \rangle^{-1} f*g
\bigr\|_{L_{\xi}^2 L_{\tau}^1 (A_j)}
\lesssim C(j,j_1,j_2) \| f \|_{\hat{Z}^{s,a}} \| g\|_{\hat{Z}^{s,a}}
\end{gather}
in the following five cases.
\begin{itemize}
\item[(i)] At least two of $j,j_1,j_2$ are less than $20$ and 
$C(j,j_1,j_2) \sim 1$.

\item[(ii)] $j_1,j_2 \geq 20$, $|j_1-j_2| \leq 1$, $0<j <j_1-10$ and 
$C(j,j_1,j_2) \sim 2^{-\delta j}$ for some $\delta>0$.

\item[(iii)] $j,j_2 \geq 20$, $|j-j_1| \leq 10$, $0< j_2 < j+11$ and 
$C(j,j_1,j_2) \sim 2^{-\delta j_2}+2^{-\delta (j-j_2) }$ for some $\delta>0$.

\item[(iv)] $j_1,j_2 \geq 20$, $j=0$ and $C(j,j_1,j_2) \sim 1$.

\item[(v)] $j,j_1 \geq 20$, $j_2=0$ and $C(j,j_1,j_2) \sim 1$.
\end{itemize}
\end{proposition}

We remark that the cases (iii), (v) are also true with $j_1$
and $j_2$ exchanged because of symmetry.
Using this proposition and
$ \| f \|_{\hat{Z}^{s,a}}^2 \sim
\sum_{j} \|f \|_{\hat{Z}^{s,a}(A_j)}^2$,
we obtain \eqref{BE-1} and \eqref{BE-2} in the same manner
as the proof inc \cite[Theorem 2.2]{KiS}.

\begin{proof}
We only prove \eqref{BE-X}--\eqref{BE-Y} in the case
$s \geq \max\{-3/4,-a-3/2  \}$, $-3/2<a<0$ and
$(s,a) \neq (-3/4,-3/4)$, because the case $a=0$ is shown
in \cite{Ki09}. In the same manner as
\cite[Proposition 3.4 (ii) and (iii)]{Ki09},
we obtain the desired estimates in
the cases (ii) and (iii). Therefore we omit the proof of these cases.

Here we put $2^{k_{\rm max}}=\max\{ 2^{k},~2^{k_1},~2^{k_2}  \}$.
Then we have $2^{k_{\rm max}} \gtrsim |\xi \xi_1 (\xi-\xi_1)|$.
 From the definition, we easily obtain
\begin{equation} \label{imb}
\hat{X}^{s,a,1/2+\varepsilon} \hookrightarrow \hat{Z}^{s,a}
\hookrightarrow \hat{X}^{s,a,1/4},
\end{equation}
where $\varepsilon>0$ is a sufficiently small number.
First, we prove \eqref{BE-X}.


(I) Estimate for (i). In this case, we can assume $j,j_1,j_2 \leq 30$.
 From \eqref{imb}, the left hand side of \eqref{BE-X} is bounded
by $C\| |\xi|^{a+1}~\langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} f*g
  \|_{L_{\xi, \tau}^2 }$. We use the H\"{o}lder inequality and the
Young inequality to obtain
\begin{align*}
 \| |\xi|^{a+1} ~\langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} f*g  
 \|_{L_{\xi,\tau}^2} 
& \lesssim
\| f*g \|_{L_{\xi}^{\infty} L_{\tau}^4 } \\
& \lesssim \| f \|_{L_{\xi}^2 L_{\tau}^{8/5}}\| g \|_{L_{\xi}^2 L_{\tau}^{8/5}}
\lesssim \| f \|_{\hat{X}^{s,a,1/4}} \| g \|_{\hat{X}^{s,a,1/4}}.
\end{align*}

(II) Estimate for (iv). We prove
\begin{equation} \label{es_hhl}
\| \langle \tau-\xi^3 \rangle^{-1}~\xi~f*g \|_{\hat{X}_L^a (A_0)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{equation}

  (IIa) We consider the estimate \eqref{es_hhl} in the case
$|\xi| \leq 2^{-2j_1}$. In this case, the left hand side of
\eqref{es_hhl}  is bounded by $ C \| |\xi|^{a+1} \langle \tau
\rangle^{-1/2+\varepsilon} f*g  \|_{L_{\tau,\xi}^2}$ from \eqref{imb}.
We use H\"{o}lder's inequality and Young's inequality to have
\begin{align*}
\| |\xi|^{a+1} \langle \tau \rangle^{-1/2+\varepsilon}
f*g  \|_{L_{\tau,\xi}^2}
& \lesssim 2^{-2sj_1} \| |\xi|^{a+1} \|_{L_{\xi}^2 (|\xi| \leq 2^{-2j_1})} \| 
(\langle \xi \rangle^s f) * (\langle \xi \rangle^s g)
\|_{L_{\xi}^{\infty} L_{\tau}^4} \\
& \lesssim 2^{-2(s+a+3/2) j_1}
\| \langle \xi \rangle^s  f \|_{L_{\tau,\xi}^2}
\| \langle \xi \rangle^s g \|_{L_{\xi}^2 L_{\tau}^{4/3}} \\
& \lesssim 2^{-2(s+a+3/2)j_1} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}
We prove only the case $2^{-2 j_1} \leq |\xi|  \leq 1$ below.

(IIb) In the case $2^{k_{\rm max}}=2^{k_2}$, we have
$2^{k_2} \gtrsim |\xi| 2^{j_1}$. Since $|\xi|^{a+1} \leq |\xi|^{-s-1/2}$ and 
$2^{-k_2/2} \lesssim 2^{-k/4} (|\xi| 2^{j_1})^{-1/4}$,
we use \eqref{es_dy_2-2} with $K_2 \sim 2^{j_1}$ to have
\begin{align*}
(\text{L.H.S.}) & \lesssim 2^{-2sj_1} \sum_{k \geq 0} 2^{-k/2}
 \| |\xi|^{a+1}~
(\langle \xi \rangle^s f) * (\langle \xi  \rangle^s g) \|_{L_{\tau,\xi}^2 (B_k)}\\
& \lesssim 2^{j_1}  \sum_{k \geq 0} 2^{-k/2} \| \langle |\xi| 2^{2j_1}
\rangle^{-s-1/2}(\langle \xi \rangle^s f) *(\langle \xi \rangle^s g)
\|_{L_{\tau,\xi}^2 (B_k)} \\
& \lesssim  2^{j_1} \sum_{k \geq 0} 2^{-3k/4} \| \langle |\xi| 2^{2j_1} \rangle^{-(s+3/4)}
(\langle \xi \rangle^s f)
*(\langle \xi \rangle^s \langle \tau-\xi^3 \rangle^{1/2} g) \|_{L_{\tau,\xi}^2 (B_k)} \\
& \lesssim \sum_{k \geq 0} 2^{-k/4} \|f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}

In the same manner as above, we obtain the desired estimate
in the case $2^{k_{\rm max}}=2^{k_1} $.

(IIc) We consider the estimate \eqref{es_hhl} in the case
$2^{k_{\rm max}}=2^{k}$. If $2^{k_{\rm max}} \gg |\xi| 2^{2j_1}$,
then we have $2^{k_{\rm max}} \sim 2^{k_1}$ or
$2^{k_{\rm max}} \sim 2^{k_2}$. Thus we only consider the
case $2^{k_{\rm max}} \sim |\xi| 2^{2 j_1}$.

(IIc-1) In the case $-3/4<a<0$, we prove
\begin{align} \label{es_hhl-1}
\| |\xi|^{a+1}~ \langle \tau-\xi^3 \rangle^{a/3-1/2} f*g
\|_{L_{\tau,\xi}^2(A_0)} \lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} 
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align}

(i) We consider \eqref{es_hhl-1} when $f*g$ is supported on the domain
\begin{align*}
D_1:=\bigl\{ (\tau,\xi) \in \mathbb{R}^2; |\tau | \geq |\xi|^{-3}
\text{ and } |\xi| \leq 1 \bigr\}.
\end{align*}
In this case, $2^{-j_1/2} \lesssim |\xi| \leq 1$ and
$2^{3j_1/2} \lesssim |\tau| \lesssim 2^{4j_1}$.  From $|\xi| \sim 2^{k-2j_1}$, 
we use \eqref{es_dy_1-2} with $K \sim 2^{j_1}$ to obtain
\begin{align*}
\text{(L.H.S.)} \lesssim & \sum_{k \geq 3j_1/2+ O(1)} 2^{(a/3-1/2) k}
\| |\xi|^{a+1} f*g  \|_{L_{\tau,\xi}^2 (B_k)}  \\
\lesssim & 2^{(-2s-2a-1)j_1} \sum_{k \geq 3j_1/2+O(1)} 2^{4ak /3}
\| |\xi|^{1/2}~(\langle \xi \rangle^s f)* (\langle \xi \rangle^s g) \|_{L_{\tau,\xi}^2} \\
\lesssim & 2^{-2(s+3/4) j_1} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}

(ii) We consider \eqref{es_hhl-1} when $f*g$ is restricted to
the domain
\begin{align*}
D_2:= \bigl\{ (\tau,\xi) \in \mathbb{R}^2; |\tau| \leq |\xi|^{-3} \text{ and }|\xi | 
\leq 1 \bigr\}.
\end{align*}
In the present case, we have
$2^{-2j_1} \leq |\xi| \lesssim 2^{-j_1/2}$ and
$ 1 \lesssim |\tau| \lesssim 2^{3j_1/2}$.
We use the H\"{o}lder inequality and the Young inequality to have
\begin{align*}
&\text{(L.H.S.)} \\
&\lesssim  \sum_{k \leq 3j_1/2+ O(1)} 2^{(a/3-1/2) k}
\| |\xi|^{a+1} f*g  \|_{L_{\tau,\xi}^2 (B_k)} \\
&\lesssim  2^{-2sj_1} \sum_{k \leq 3j_1/2+O(1)} 2^{(a /3-1/2)k}
\| |\xi|^{a+1} \|_{L_{\xi}^2(|\xi| \sim 2^{k-2j_1})}
\| (\langle \xi \rangle^s f)* (\langle \xi \rangle^s g) \|_{L_{\xi}^{\infty}
L_{\tau}^2} \\
&\lesssim  2^{(-2s-2a-3) j_1} \sum_{k \leq 3j_1/2+O(1)} 2^{4(a+3/4)/3}
\| \langle \xi \rangle^s f \|_{L_{\xi}^2 L_{\tau}^1}
\| \langle \xi \rangle^s g \|_{L_{\xi}^2 L_{\tau}^2} \\
&\lesssim  2^{-2(s+3/4)j_1} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}

(IIc-2) In the case $a=-3/4$, we prove
\begin{align} \label{es_hhl-2}
\| |\xi|^{1/4} \langle \tau \rangle^{-3/4+ \varepsilon_1/2}
f*g \|_{L_{\tau,\xi}^2 (A_0)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align}
 From $|\tau| \sim |\xi| 2^{2 j_1} \geq 1$,
we use H\"{o}lder's inequality and Young's inequality to obtain
\begin{align*}
\text{(L.H.S.)} \lesssim & 2^{(-2s-3/2+\varepsilon_1) j_1}
\| |\xi|^{-1/2+\varepsilon_1/2} \|_{L_{\xi}^2 ( |\xi| \leq 1)}
\| (\langle \xi \rangle^s f)* (\langle \xi \rangle^s g) \|_{L_{\xi}^{\infty}
L_{\tau}^{2} } \\
\lesssim & 2^{(-2s-3/2+\varepsilon_1) j_1}
\| \langle \xi \rangle^s f \|_{L_{\xi}^2 L_{\tau}^1}
\| \langle \xi \rangle^s g  \|_{L_{\tau,\xi}^2} \\
\lesssim & 2^{(-2s-3/2+\varepsilon_1) j_1} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}
Since $-2s-3/2+\varepsilon_1 \leq -\varepsilon_1$ in present case,
we have \eqref{es_hhl-2}.

(IIc-3) In the case $-3/2<a<-3/4$, we estimate
\begin{equation} \label{es_hhl-3}
\| |\xi|^{a+1}~ \langle \tau \rangle^{-3/4+ \varepsilon_2/2}
f*g \|_{L_{\tau,\xi}^2 (A_0)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{equation}
 From the assumption, $ -s-3/4 +\varepsilon_2 \leq 0  $ and
$|\xi|^{a+1} \leq |\xi|^{-s-1/2}$. Now we use the H\"{o}lder
inequality and the Young inequality to have
\begin{align*}
\text{(L.H.S.)} &\lesssim
 2^{-2sj_1} \| |\xi|^{-s-1/2} \langle \tau \rangle^{-3/4+\varepsilon_2/2} 
 (\langle \xi \rangle^s f)*
(\langle \xi \rangle^s g) \|_{L_{\tau,\xi}^2} \\
&\lesssim 2^{- \varepsilon_2 j_1 }
\| |\xi|^{-1/2-\varepsilon_2/2} \|_{L_{\xi}^2(|\xi|
\geq 2^{-2 j_1})} \|(\langle \xi \rangle^s f)*
(\langle \xi \rangle^s g) \|_{L_{\xi}^{\infty} L_{\tau}^2} \\
&\lesssim  \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}



(III) Estimate for (v). We prove
\begin{equation} \label{es_hlh}
2^{j} \sum_{k \geq 0} 2^{-k/2}~
\| (\langle \xi \rangle^s f )*g \|_{L_{\tau,\xi}^2(B_k)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a}(A_0)}.
\end{equation}
In the case $|\xi_2| \leq 2^{-2j} $, we use H\"{o}lder's inequality and
Young's inequality to have
\begin{align*}
\text{(L.H.S.)} \lesssim & 2^j
\| (\langle \xi \rangle^s f)* g \|_{L_{\tau,\xi}^2 }
\lesssim  2^{j} \| \langle \xi \rangle^s f  \|_{L_{\xi}^2 L_{\tau}^1}
\| g \|_{L_{\xi}^1 L_{\tau}^2 (|\xi| \leq 2^{-2j})} \\
\lesssim & 2^{j} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\||\xi|^{-a} \|_{L_{\xi}^2(|\xi| \leq 2^{-2j}) }
\| |\xi|^a g \|_{L_{\tau,\xi}^2}\\
\lesssim & 2^{2aj} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,0}}.
\end{align*}
Therefore we only consider the case $2^{-2j} \leq |\xi_2| \leq 1$.

(IIIa) We consider the estimate \eqref{es_hlh} in the case
$2^{k_{\rm max}}=2^{k}$.  From $2^{k} \geq 2^{k_2}$, we
use \eqref{es_dy_1-2} with $K \sim 2^{j}$ to obtain
\begin{align*}
\text{(L.H.S.)} \lesssim & 2^{j} \| \langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} 
(\langle \xi \rangle^s f) *g  \|_{L_{\tau,\xi}^2 } \\
\lesssim & \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| \langle \tau \rangle^{-1/2+\varepsilon}
 g\|_{\hat{X}^{0,1/2,1}}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,1/4}}.
\end{align*}

(IIIb) We consider the estimate \eqref{es_hlh} in the case
$2^{k_{\rm max}}=2^{k_1}$. The left hand side of \eqref{es_hlh} is bounded 
by $ C 2^{j} \| \langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} 
(\langle \xi \rangle^s f) *g   \|_{L_{\tau,\xi}^2}$.  From
$|\xi_2| \lesssim 2^{k-2j_1}$, we use the H\"{o}lder inequality
and the Young inequality to have
\begin{align*}
 2^{j}  \| \langle \tau-\xi^3  \rangle^{-1/2+\varepsilon}
(\langle \xi \rangle^s f)* g \|_{L_{\tau,\xi}^2}
&\lesssim 2^{j} \| (\langle \xi \rangle^s f)* g   \|_{L_{\xi}^2 L_{\tau}^6} \\
&\lesssim 2^{j } \| \langle \xi \rangle^s f \|_{L_{\tau,\xi}^2}
\| g \|_{L_{\xi}^1 L_{\tau}^{3/2} (|\xi| \lesssim 2^{k_1-2j})}\\
&\lesssim 2^{j} \sum_{k_1}
\| \langle \xi \rangle^s f \|_{L_{\tau,\xi}^2(B_{k_1})}
2^{k_1/2-j} \| g \|_{L_{\xi}^2 L_{\tau}^{3/2}},
\end{align*}
which is bounded by the right hand side of \eqref{es_hlh}.

(IIIc) We consider the estimate \eqref{es_hlh} in the case
$2^{k_{\rm max}} =2^{k_2}$. If $2^{k_{\rm max}}=2^{k_2} \gg |\xi_2| 2^{2j}$,
 then we have $2^{k_{\rm max}} \sim
2^{k}$ or $2^{k_{\rm max}} \sim 2^{k_1}$. Therefore we only consider the case
$2^{k_2} \sim |\xi_2| 2^{2j}$.


(IIIc-1) In the case $a=-3/4$, we prove
\begin{equation} \label{es_hlh-1}
2^{j} \sum_{k \geq 0} 2^{-k/2} \| (\langle \xi \rangle^s f)*g
\|_{L_{\tau,\xi}^2(B_k)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_L^{-3/4,1/4+\varepsilon_1/2}}.
\end{equation}
 From $2^{(-1/4- \varepsilon_1/2) k_2} \lesssim (|\xi_2| 2^{2j})^{-1/4}
2^{-\varepsilon_1 k/2}$, we use \eqref{es_dy_2-2} with
$K_2 \sim 2^{j}$ to have
\begin{align*}
\text{(L.H.S.)} \lesssim &
 2^{j/2} \sum_{k \geq 0} 2^{(-1/2-\varepsilon_1/2) k }
\|(\langle \xi \rangle^s f)* (|\xi|^{-1/4}
\langle \tau \rangle^{1/4+\varepsilon_1/2} g) \|_{L_{\tau,\xi}^2 (B_k)} \\
\lesssim & \sum_{k \geq 0} 2^{-\varepsilon_1 k /2} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_L^{-3/4,1/4+\varepsilon_1/2}}.
\end{align*}

(IIIc-2) In the case $-3/2< a< -3/4$, we prove
\begin{equation} \label{es_hlh-2}
2^{j} \sum_{k \geq 0} 2^{-k/2} \| (\langle \xi \rangle^s f)*
g\|_{L_{\tau,\xi}^2(B_k)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_L^{a,1/4+\varepsilon_2/2}}.
\end{equation}
 From $|\xi_2|^{-(a+1/2)} \langle \tau_2 \rangle^{-1/4-\varepsilon_2/2}
\lesssim 2^{-j/2} 2^{-\varepsilon_2 k /2}$, we use \eqref{es_dy_2-2} with 
$K_2 \sim 2^{j}$ to obtain
\begin{align*}
\text{(L.H.S.)} &\lesssim  2^{j/2}
 \sum_{k \geq 0} 2^{(-1/2-\varepsilon_2/2)k}
 \| (\langle \xi \rangle^s f)* (|\xi|^{a+1/2}
 \langle \tau \rangle^{1/4+\varepsilon_2/2} g )
 \|_{L_{\tau,\xi}^2(B_k)} \\
& \lesssim  \sum_{k \geq 0} 2^{-\varepsilon_2 k /2} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\|  g \|_{\hat{X}_L^{a,1/4+\varepsilon_2/2}}.
\end{align*}

(IIIc-3) In the case $-3/4<a<0$, we prove
\begin{equation} \label{es_hlh-3}
2^{j} \sum_{k \geq 0} 2^{-k/2} \| (\langle \xi \rangle^s f)*g
 \|_{L_{\tau,\xi}^2(B_k)}
 \lesssim  \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{equation}


(i) We consider \eqref{es_hlh-3} when $g$ is restricted to $ D_2$.
In the present case, $2^{-2j} \leq |\xi_2| \lesssim 2^{-j/2}$ and
$ 1 \lesssim |\tau_2| \lesssim 2^{3j/2}$.
 From $|\xi|^{-a} \langle \tau \rangle^{-a/3-1/2} \sim
|\xi|^{-4a/3-1/2} 2^{-2aj/3-j} $, we use the H\"{o}lder inequality and
the Young inequality to obtain
\begin{align*}
\text{(L.H.S.)} \lesssim & 2^{j}~
\| (\langle \xi \rangle^s f)* g   \|_{L_{\xi}^2 L_{\tau}^4}
\lesssim 2^j
\| \langle \xi \rangle^s f \|_{L_{\xi}^2 L_{\tau}^{4/3}}
\| g \|_{L_{\xi}^1 L_{\tau}^2 (|\xi| \lesssim 2^{-j/2} )} \\
\lesssim & 2^{-2aj/3} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| |\xi_2|^{-4a/3-1/2} \|_{L_{\xi_2}^2 (|\xi_2| \lesssim 2^{-j/2} ) }
\| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{align*}
Since $\| |\xi|^{-4a/3-1/2} \|_{L_{\xi_2}(|\xi_2| \lesssim 2^{-j/2})}
 \lesssim 2^{2aj/3}$, we have \eqref{es_hlh-3}.

(ii) We consider \eqref{es_hlh-3} when $g$ is supported on $ D_1$.
In this case, we have $2^{-j/2} \lesssim |\xi_2| \leq 1$ and
$2^{3j/2} \lesssim |\tau| \lesssim 2^{2j}$.

(iia) Firstly, $g$ is restricted to $B_{[3j/2,3j/2+\alpha]}$ with
$0 \leq \alpha \leq j/2$.  From $2^{-j/2} \lesssim |\xi_2| \lesssim
 2^{-j/2+\alpha}$ and $|\xi_2|^{-a} \langle \tau_2 \rangle^{-a/3-1/2}
\sim |\xi_2|^{-4a/3-1/2} 2^{-2aj/3 -j}$, we use H\"{o}lder's
inequality and Young's inequality to obtain
\begin{align*}
\| \xi~f*g \|_{\hat{X}_{(2,1)}^{s,-1/2}(B_{\geq 2\alpha} )} \sim &
2^{j} \sum_{k \geq 2\alpha} 2^{-k/2} \| (\langle \xi \rangle^s f)* g
 \|_{L_{\tau,\xi}^2 (B_k)} \\
\lesssim & 2^{j} \sum_{k \geq 2\alpha} 2^{-k/2} \| \langle \xi \rangle^s f 
\|_{L_{\xi}^{2} L_{\tau}^1}
\| g \|_{L_{\xi}^{1} L_{\tau}^2 (|\xi| \lesssim 2^{-j/2+\alpha})} \\
\lesssim & 2^{j}
2^{-\alpha} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| |\xi|^{-4a/3-1/2} 
\|_{L_{\xi}^2(  |\xi| \lesssim 2^{-j/2+\alpha}  )} 
\| g  \|_{\hat{X}_L^{a,a/3+1/2}} \\
\lesssim & 2^{-\frac{4}{3}(a+\frac{3}{4}) \alpha} 
\| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{align*}
We put a sufficiently small number $\varepsilon_3$ satisfying
$0<\varepsilon_3 \leq 4(a+3/4)/3 $.  From the above estimate, we have
\begin{equation} \label{D_1-l}
\| \xi f*g \|_{\hat{X}_{(2,1)}^{s,-1/2}(B_{\geq 2\alpha})}
\lesssim 2^{- \varepsilon_3 \alpha} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{equation}

(iib) Secondly, $g$ is restricted to $B_{[3j/2+\gamma, 2j]}$ with
$0 \leq \gamma \leq 2^{j/2}$.  From
$2^{-j/2+\gamma} \lesssim |\xi| \leq 1$, we use \eqref{es_dy_2-2}
with $K_2 \sim 2^{j}$ to obtain
\begin{align*}
&\| \xi f*g  \|_{\hat{X}_{(2,1)}^{s,-1/2}(B_{ \leq 2\alpha }) } \sim
2^{j } \sum_{k \leq 2\alpha} 2^{-k/2}
\| (\langle \xi \rangle^s f)* g \|_{L_{\tau,\xi}^2 (B_k)} \\
\lesssim & 2^{j/2} \sum_{k \leq 2 \alpha}~1~ \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| |\xi|^{-1/2} g \|_{L_{\tau,\xi}^2 ( 2^{-j/2+\gamma} \lesssim |\xi|) } \\
\lesssim & \alpha 2^{-2aj/3-j/2} \| |\xi|^{-\frac{4}{3}(a+\frac{3}{4})} \|_{L_{\xi}^2 
( 2^{-j/2+\gamma} \lesssim |\xi| )} \| f \|_{\hat{X}_{(2,1)}^{s,1/2}}
\| g \|_{\hat{X}_L^{a,a/3+1/2}} \\
\lesssim & \alpha 2^{-\frac{4}{3}(a+\frac{3}{4}) \gamma }
\| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{align*}
 From the definition of $\varepsilon_3$, we have
\begin{align} \label{D_1-h}
\| \xi f*g \|_{\hat{X}_{(2,1)}^{s,-1/2}(B_{\leq 2\alpha} )}
\lesssim  \alpha 2^{- \gamma \varepsilon_3 }
\| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{align}
If $g$ is restricted to $B_{[3j/2+\gamma, 3j/2+\alpha]}$ with
 $\gamma < \alpha$, from \eqref{D_1-l} and \eqref{D_1-h}, we have
\begin{align} \label{es_hlh_A_0}
\| \xi f*g  \|_{\hat{X}_{(2,1)}^{s,-1/2}} \lesssim
\bigl( 2^{-\varepsilon_3 \alpha} +\alpha 2^{-\varepsilon_3 \gamma}  \bigr)
\| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,a/3+1/2}}.
\end{align}
Let be the decreasing sequence $\{ a_n \}_{n=0}^{N}$ defined by
\[
a_0=\frac{j}{2}, \quad  a_{n+1}=  \frac{1}{2} a_{n}, \quad
0 < a_N \leq \frac{1}{2},
\]
where $N$ is a minimum integer such that $N \geq \log_2 j$.
We first apply with $\alpha=a_0$ and $\gamma=a_1$, next apply with
$\alpha=a_1$ and $\gamma=a_2$. Repeating this procedure at the end
we apply with $\alpha=a_N$ and $\gamma=0$.
  From \eqref{es_hlh_A_0}, we obtain
\[
\| \xi f*g  \|_{\hat{X}_{(2,1)}^{s,-1/2}}
\lesssim \bigl( 1+ \sum_{n=0}^{N} \frac{1}{a_n} \bigr)
\| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_L^{a,a/3+1/2}(D_1)},
\]
which shows the claim since $ \sum_{n=0}^N \frac{1}{a_n}$
is bounded uniformly in $j$.

Next, we prove \eqref{BE-Y}.  From the triangle inequality and
the Schwarz inequality, we have
\begin{equation} \label{es-L-1}
\| f \|_{L_{\tau}^1} \lesssim \sum_{k \geq 0} \| f \|_{L_{\tau}^1 (B_k)}
\lesssim \sum_{k \geq 0} 2^{k/2}~\| f \|_{L_{\tau}^2 (B_k)}.
\end{equation}
 From \eqref{es-L-1}, we obtain
\[
\| \langle \xi \rangle^s \langle \tau-\xi^3 \rangle^{-1} \xi f*g 
\|_{L_{\xi}^2 L_{\tau}^1(A_j)} \lesssim
\| \xi~ f* g \|_{\hat{X}_{(2,1)}^{s,-1/2} (A_j)},
\]
for any $j>0$. Thus we only consider the case (i) and (iv).

(IV) Estimate of (i).
In this case, the left hand side of \eqref{BE-Y} is bounded by
$C \| |\xi|^{a+1} \langle \tau-\xi^3 \rangle^{-1/2+\varepsilon} f*g 
 \|_{L_{\tau,\xi}^2}$. In the same manner as (I), we have the desired estimate.

(V) Estimate of (vi).
We prove
\begin{align} \label{Y-vi}
 \| |\xi|^{a+1} \langle \tau \rangle^{-1} f*g  \|_{L_{\xi}^2 L_{\tau}^1 (A_0)}
\lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align}
We easily obtain \eqref{Y-vi} in the case $|\xi| \leq 2^{-2 j_1}$.
Therefore we only consider the case $2^{-2 j_1} \leq |\xi| \leq 1$
below.

(Va) We consider the estimate \eqref{Y-vi} in the case $2^{k_{\rm max}}=2^{k_1}$
or $2^{k_2}$. Note that the left hand side of \eqref{Y-vi} is bounded by
$ C  \sum_{k \geq 0} 2^{-k/2}
\||\xi|^{a+1} f*g  \|_{L_{\tau,\xi}^2(B_k)}$ from \eqref{es-L-1}.
In the same manner as (IIb), we obtain \eqref{Y-vi} in the case
$2^{k_{\rm max}}=2^{k_1}$ or $2^{k_2}$.

(Vb) We consider the estimate \eqref{Y-vi} in the case
$2^{k_{\rm max}}=2^{k}$.
 From $|\xi|^{a+1} \leq | \xi |^{-s-1/2} $, we have
$|\xi|^{a+1} \langle \tau \rangle^{-1} \lesssim |\xi|^{-3/4} 2^{2sj_1-j_1/2}$.
We use the H\"{o}lder inequality and the Young inequality to have
\begin{align*}
\text{(L.H.S.)}
& \lesssim 2^{ - j_1/2}
\| |\xi|^{-3/4} (\langle \xi \rangle^s f)* (\langle \xi \rangle^s g)
\|_{L_{\xi}^{2} L_{\tau}^1} \\
& \lesssim 2^{-j_1/2}
\| |\xi|^{-3/4} \|_{L_{\xi}^2 (|\xi| \geq 2^{-2j_1})}
\| \langle \xi \rangle^s f \|_{L_{\xi}^2 L_{\tau}^1}
\| \langle \xi \rangle^s g \|_{L_{\xi}^2 L_{\tau}^1} \\
& \lesssim \| f \|_{\hat{X}_{(2,1)}^{s,1/2}} \| g \|_{\hat{X}_{(2,1)}^{s,1/2}}.
\end{align*}
\end{proof}

\section{Proof of the main results}

In this section, we give the proofs of Theorem~\ref{thm_well}
and \ref{thm_ill}. Here
$Z_{T}^{s,a}$ is defined by the norm
\begin{align*}
\|u \|_{Z_T^{s,a}}:=\inf \bigl\{ \| v \|_{Z^{s,a}};u(t)=v(t)\text{ on }
t \in [0,T] \bigr\}.
\end{align*}
We obtain the following main result.

\begin{proposition} \label{prop_well}
Let $s,a$ satisfy \eqref{co_op} and $r>1$.

(Existence) For any $u_0 \in B_r(H^{s,a})$, there exist
$T \sim r^{-6/(3+2 \min\{s,a\})}$ and $u \in C([0,T];H^{s,a})
\cap Z^{s,a}_T$ satisfying the following integral form for \eqref{KdV};
\begin{align} \label{integral-1}
u(t)=  U(t) u_0 +3  \int_{0}^t U(t-s) \partial_x (u(s))^2 ds.
\end{align}
Moreover the data-to-solution map
$B_r(H^{s,a}) \ni u_0 \mapsto u \in C([0,T]; H^{s,a}) \cap Z^{s,a}_{T}$
 is Lipschitz continuous.

(Uniqueness) Assume that $u,v \in C([0,T];H^{s,a}) \cap Z_{T}^{s,a} $
satisfy \eqref{integral-1}. Then, $u=v$ on $t \in [0,T]$.
\end{proposition}

\begin{proof}
We first prove the existence of the solution to \eqref{integral-1}.
The KdV equation is scale invariant with respect to the transform
\[
u(t,x) \mapsto u_{\lambda}(t,x) :=\lambda^{-2} u(\lambda^{-3}t, \lambda^{-1}x),
\quad  \lambda  \geq 1.
\]
A simple calculation shows
\begin{align*}
\|u_{\lambda}(0,\cdot) \|_{H^{s,a}} \leq \lambda^{-3/2-\min \{s,a \}}
\| u_0 \|_{H^{s,a}}.
\end{align*}
Therefore, we can assume that initial data is small enough.
 From this, we use Propositions~\ref{BE-ES-1},~\ref{prop_linear1}
 and~\ref{prop_linear2} to prove the existence of the solution by
Banach's fixed point argument. For the details, see the proof in
\cite[Proposition 4.1]{KT}.

We next prove the uniqueness of solutions by the argument in \cite{MT}.
We define the space ${W}^{s,a}$ by the norm
\[
\|u \|_{W^{s,a}}:= \| u \|_{Z^{s, a}}
+ \|u \|_{L^{\infty}(\mathbb{R};H^{s,a})}.
\]
In the same manner as the proof in \cite[Theorem 2.5]{MT},
we obtain, for $1/2 \leq b <1$,
\begin{equation} \label{uni-lem}
w \in X_{(1,1),T_{\lambda}}^{s,a,b} ,\quad  w(0,x)=0  \quad
\Rightarrow \quad
\lim_{\delta \to +0}
\| w|_{[0,\delta]} \|_{X_{(1,1),\delta}^{s,a,b} }=0,
\end{equation}
where $T_{\lambda}:= \lambda^{3} T$, $ \lambda \geq 1$ and the
space $X_{(1,1)}^{s,a,b}$
defined by
\[
\| u \|_{X_{(1,1)}^{s,a,b}}
:=  \bigl\| \bigl\{ \| \langle \xi \rangle^{s-a}
|\xi|^a \langle \tau-\xi^3 \rangle^{b}
\widehat{u} \|_{L_{\tau,\xi}^2(A_j \cap B_k)} \bigr\}_{j ,k \geq 0}
\bigr\|_{l_{j,k}^{1}}.
\]
Let $u \in W^{s,a}$ satisfy $u(0,x)=0$ and $\varepsilon$ is an
arbitrary positive number.
Since $W^{s,a}$ contains $\mathcal{Z}$ densely,
we can choose $v \in \mathcal{Z}$ satisfying
$\| v-u  \|_{W^{s,a}} < \varepsilon$.  From the definition, we have
\begin{align*}
\|v(0) \|_{H^{s,a}} = \| v(0)-u(0) \|_{H^{s,a}}
\lesssim \| u-v \|_{W^{s,a}} < \varepsilon.
\end{align*}
Note that
\[
 \sup_{t \in \mathbb{R}} \| u \|_{H^{s,a}} \lesssim
\| u  \|_{W^{s,a}} \lesssim \| u \|_{X^{s,a,b}},
\]
for $1/2 < b <1$. By the above argument, we have
\begin{align*}
\| u \|_{W_{T}^{s,a}} \lesssim & \| u-v \|_{W_{T}^{s,a}}+
\| v-U(t)v(0)   \|_{W_{T}^{s,a}}+ \| U(t) v(0) \|_{X_T^{s,a,b}} \\
\lesssim & \varepsilon + \| v-U(t)v(0) \|_{W_{T}^{s,a}}+
\| v(0) \|_{H^{s,a}} \\
\lesssim & \varepsilon+\|v-U(t)v(0) \|_{W_{T}^{s,a}}.
\end{align*}
Since the second term tends to $0$ as $T \to 0$
 from \eqref{uni-lem}, we have
\begin{align} \label{uni_lem-2}
\lim_{T \to 0} \| u  \|_{W_T^{s,a}}=0.
\end{align}
By combining Propositions \ref{BE-ES-1}, \ref{prop_linear1},
\ref{prop_linear2} and \eqref{uni_lem-2}, we have uniqueness.
For the details, see \cite{Ki09}.
\end{proof}

Next, we prove Theorem \ref{thm_ill} (i)--(iii).
We first consider Theorem \ref{thm_ill} (i).
In \cite{BT}, Bejenaru and Tao, for the quadratic
Schr\"{o}dinger equation with nonlinear term $u^2$, proved
the discontinuity of the data-to-solution map for any $s<-1$.
We essentially follow their argument to obtain the following
proposition.

\begin{proposition} \label{prop_ill}
Let $s <s_a$, $-3/2<a < -7/8$ and $0 <\delta \ll 1$. Then there
exist $T=T(\delta)>0$ and a sequence of initial data
$\{ \phi_{N,\delta}  \}_{N=1}^{\infty} \in H^{\infty}$ satisfying
the following three conditions for any $t \in (0,T]$,
\begin{itemize}
\item[(1)] $ \| \phi_{N, \delta} \|_{H^{s_a,a}} \sim \delta $,

\item[(2)] $\| \phi_{N, \delta} \|_{H^{s,a}} \to 0$
as $N \to \infty$,

\item[(3)] $\| u_{N,\delta} (t) \|_{H^{s,a}} \gtrsim \delta^2$, \\
where $u_{N, \delta}(t)$ is the solution to \eqref{KdV}
obtained in Proposition~\ref{prop_well} with the initial data
$\phi_{N,\delta}$.
\end{itemize}
\end{proposition}

\begin{proof}
Let $N \gg 1$. We put the initial data $ \phi_{N,\delta}$ as follows;
\[
\phi_{N, \delta}(x)= \delta N^{a+5/2} \cos(N x) \int_{-\gamma}^{\gamma}
e^{i \xi x} d\xi,
\]
where $\gamma:=N^{-2}$.
By a simple calculation, we have
\begin{equation} \label{initial-Fo}
\widehat{\phi}_{N, \delta}(\xi) \sim \delta N^{a+5/2} \chi_{B^{+}}(\xi)
+\delta N^{a+5/2} \chi_{B^{-}}(\xi),
\end{equation}
where
\[
B^{\pm}:=[\pm N-\gamma,~ \pm N+\gamma].
\]
Therefore,
\begin{equation} \label{initial-norm}
\| \phi_{N,\delta}  \|_{H^{s,a}} \sim \delta N^{s+a+3/2}, \quad
\| U(t) \phi_{N, \delta} \|_{H^{s,a}}
=\| \phi_{N,\delta} \|_{H^{s,a}} \sim \delta N^{s+a+3/2 } .
\end{equation}
Since $\| \phi_{N,\delta} \|_{H^{s_a}} \sim \delta$, we have
$T=T(\delta)>0$ and the solution $u_{N, \delta}$ to \eqref{KdV} with 
the initial data
$\phi_{N,\delta}$ by Proposition~\ref{prop_well}.
Let $t \in (0, T]$.
A quadratic term $A_2$ of the Taylor expansion is defined by
\[
A_2(u_0)(t) := 3 \int_{0}^{t} U(t-s) \partial_x(U(s) u_0)^2 ds.
\]
A simple calculation shows that
\begin{equation} \label{qua-Fo}
\widehat{A}_2(u_0) (t)= \exp (i \xi^3 t) \int
\frac{1-\exp(-iq(\xi,\xi_1) t)}{q(\xi,\xi_1)} \widehat{u}_0 (\xi_1) 
\widehat{u}_0(\xi-\xi_1) d\xi_1,
\end{equation}
where $q(\xi,\xi_1):=3\xi \xi_1 (\xi-\xi_1)$.
By similar argument to the proof in \cite[Theorem 1.2]{KT},
we obtain
\begin{equation} \label{qua-norm}
\| A_2(u_0) (t) \|_{H^{s,a}} \gtrsim \delta^2.
\end{equation}
Now we put $v_{N,\delta}(t):= u_{ N,\delta}(t) -U(t)\phi_{N,\delta}-
A_2(\phi_{N ,\delta})(t)$.
Since the data-to-solution map is Lipschitz continuous for $s=s_a$,
we obtain
\begin{equation} \label{er-norm}
\| v_{N, \delta} (t)  \|_{H^{s_a,a}} \lesssim \delta^3,
\end{equation}
by using Propositions \ref{BE-ES-1}, \ref{prop_linear1}
 and \ref{prop_linear2}.
 From \eqref{initial-norm}, \eqref{qua-norm} and \eqref{er-norm},
we obtain
\[
\| u_{ N,\delta}(t) \|_{H^{s,a}} \geq
\| A_2 (\phi_{N ,\delta})(t) \|_{H^{s,a}}-
\|v_{N,\delta}(t) \|_{H^{s,a} } -
\| U(t) \phi_{N, \delta} \|_{H^{s,a}} \gtrsim \delta^2,
\]
for all $N \gg 1$. Since $\| \phi_{N,\delta} \|_{H^{s,a}} \to 0$
as $N \to \infty$, this shows the discontinuity of the flow map.

We next prove Theorem~\ref{thm_ill} (ii).
We only prove that the following estimate fails.
\begin{equation} \label{qua_br}
\|A_2 (u_0) (t)  \|_{H^{s,a}} \lesssim \| u_0 \|_{H^{s,a}}^2,
\end{equation}
for $|t|$ bounded by the general argument. For details, see \cite{Ho}.

Let $N \gg 1$. We put a smooth initial data as follows;
\[
\phi_N(x):= N^{-s+1} \cos(Nx) \int_{-\gamma}^{\gamma} e^{i \xi x} d\xi
+N^{2a+1} \cos(N^{-2}x ) \int_{-\gamma/2}^{\gamma/2} e^{i \xi x} d\xi.
\]
A straightforward computation shows that
\begin{equation} \label{initial-3}
\widehat{\phi}_N (\xi) \sim N^{-s+1} (\chi_{B^{+}}(\xi) + \chi_{B^{-}}(\xi))
+N^{2a+1} \chi_{C} (\xi),
\end{equation}
where $C:= [\gamma/2, 3\gamma/2]$. Clearly,
 $\| \phi_N \|_{H^{s,a}} \sim 1$.
Substituting \eqref{initial-3} into
\eqref{qua-Fo}, we have
\begin{align*}
|\widehat{A}_2 (\phi_N)(t)|& \lesssim
N^{-2s} |\xi|~ \chi_{[-\gamma/2,\gamma/2]}(\xi)
+ N^{-s+a} |\xi|~\chi_{ [\pm N, \pm N+ \gamma] }(\xi) \\
&\quad +  \text{(remainder terms)}.
\end{align*}
Therefore,
\begin{equation} \label{qu_norm-3}
\| A_{2}(\phi_N) (t) \|_{H^{s,a}} \gtrsim
N^{-2s} \Bigl( \int_{-\gamma/2}^{\gamma/2} |\xi|^{2a+2} d\xi
\Bigr)^{1/2}
+N^{-s+2a} \Bigl(
\int_{N}^{N +\gamma} \langle \xi \rangle^{2s+2} \Bigr)^{1/2}.
\end{equation}
If $a \leq -3/2$, the first term of the right hand side
of \eqref{qu_norm-3} diverges. When we assume $a \geq -3/2$,
the right hand side of \eqref{qu_norm-3} is greater than
$C (N^{-2(s+a+3/2)}+N^{2a} )$. In the case $0< a$ or $s< -a-3/2$,
we have $\| A_2(\phi_{N}) (t) \|_{H^{s,a}} \to \infty$ as
$N \to \infty$, which shows the claim since
$\| \phi_N \|_{H^{s,a}} \sim 1$.

Finally, we consider Theorem \ref{thm_ill} (iii). Similar to the
proof of Theorem~\ref{thm_ill} (ii), we only prove that the following
estimate fails for $|t|$ bounded.
\begin{equation} \label{cub_br}
\| A_3 (u_0) (t)  \|_{H^{s,a}} \lesssim \| u_0 \|_{H^{s,a}}^3,
\end{equation}
where $A_3$ is the cubic term of the Taylor expansion.
We put the sequence of initial data
$\{ \psi \}_{N=1}^{\infty} \in H^{\infty}$
as follows;
\begin{align*}
\psi_{N}(x)= N^{-s+1/4}~\cos(Nx) \int_{-N^{-1/2}}^{N^{1/2}}
e^{i \xi x} d\xi.
\end{align*}
Similar to this data is used in \cite{Bo97}. In the same manner
as the argument in \cite{Bo97}, we prove \eqref{cub_br} fails.
\end{proof}


\section{Appendix}

We mention the typical counterexamples of \eqref{BE-3} in the case
\eqref{co_cr1}.

\begin{example}[high-high-low interaction] \label{exa1} \rm
We define the rectangles $P_1, P_2$ as follows;
\begin{gather*}
P_1:=  \bigl\{ (\tau,\xi) \in \mathbb{R}^2~ ;~
|\xi-N | \leq  N^{-1/2}, ~~ |\tau-( 3N^2 \xi -2 N^3) |
\leq 1/2  \bigr\}, \\
P_2:=  \bigl\{ (\tau,\xi) \in \mathbb{R}^2;(-\tau,-\xi)
 \in A_1 \bigr\}.
\end{gather*}
Here we put
\begin{equation} \label{rec-1}
f(\tau,\xi):= \chi_{P_1}(\tau,\xi), \quad
g(\tau,\xi):=\chi_{P_2}(\tau,\xi).
\end{equation}
Then
\begin{equation} \label{int-1}
 f*g(\tau,\xi) \gtrsim  N^{-1/2}~\chi_{R_1} (\tau,\xi),
\end{equation}
where
\[
 R_1:=  \bigl\{ (\tau,\xi) \in \mathbb{R}^2;
\xi \in [ 1/2 N^{-1/2},3/4 N^{-1/2} ], ~~
 |\tau- 3N^2 \xi | \leq 1/2 \bigr\}.
\]
Inserting \eqref{rec-1} and \eqref{int-1} into \eqref{BE-3},
the necessary condition for \eqref{BE-3} is $b \leq 4s/3+a/3+3/2$.
If \eqref{BE-3} for $s=-3/4$,
$b \leq a/3+1/2$.
\end{example}

\begin{example}[high-low-high interaction] \label{exa2} \rm
We define the rectangle
\[
Q:=\bigl\{ (\tau,\xi) \in \mathbb{R}^2 ;|\xi-2N^{-1/2}| \leq N^{-1/2},~~
|\tau-3N^2 \xi | \leq 1/2 \bigr\}.
\]
Here we put
\begin{equation} \label{rec-2}
f(\tau,\xi) = \chi_{P_1}(\tau,\xi), \quad
g(\tau,\xi)=\chi_{Q}(\tau,\xi).
\end{equation}
Then
\begin{align} \label{int-2}
f*g(\tau,\xi) \gtrsim N^{-1/2}~\chi_{R_2}(\tau,\xi),
\end{align}
where
\[
R_2:=  \bigl\{ (\tau,\xi) \in \mathbb{R}^2;
|\xi-N| \leq N^{-1/2}/2 , ~
 |\tau- (3N^2 \xi- 2 N^3 ) | \leq 1/2 \bigr\}.
\]
Substituting \eqref{rec-2} and \eqref{int-2} into \eqref{BE-3},
the necessary condition for \eqref{BE-3} is $b \geq a/3+1/2$.
\end{example}

\begin{example}[high-high-high interaction] \label{exa3} \rm
We put
\begin{equation} \label{rec-3}
f(\tau,\xi)= \chi_{P_1}(\tau,\xi), \quad
g(\tau,\xi)= \chi_{P_1}(\tau,\xi).
\end{equation}
Then
\begin{align} \label{int-3}
 f*g(\tau,\xi) \gtrsim  N^{-1/2}~\chi_{R_3} (\tau,\xi),
\end{align}
where
\[
 R_3:=  \bigl\{ (\tau,\xi) \in \mathbb{R}^2;|
\xi -2N | \leq N^{-1/2}/2 ,
~ | \tau- (3N^2 \xi -4 N^3) | \leq 1/2 \bigr\}.
\]
Inserting \eqref{rec-3} and \eqref{int-3} into \eqref{BE-3},
the necessary condition for \eqref{BE-3} is
$b \leq 1/2$ for $s=-3/4$.

On the other hand, we put
\begin{equation} \label{rec-4}
f(\tau,\xi)=\chi_{R_3}(\tau,\xi), \quad
g(\tau,\xi)=\chi_{P_2}(\tau,\xi).
\end{equation}
Then
\begin{align} \label{int-4}
f*g(\tau,\xi) \gtrsim N^{-1/2}~\chi_{R_4}(\tau,\xi),
\end{align}
where
\[
R_4:=  \bigl\{ (\tau,\xi) \in \mathbb{R}^2;
|\xi-N | \leq  N^{-1/2}/4, ~
 |\tau- (3N^2 \xi-2N^3 ) | \leq 1/2 \bigr\}.
\]
Substituting \eqref{rec-4} and \eqref{int-4} into \eqref{BE-3},
the necessary condition for \eqref{BE-3} is $b \geq 1/2$ for $s=-3/4$.
\end{example}

\subsection*{Acknowledgements}
The author would like to thank his adviser Kotaro Tsugawa
for many helpful conversation and encouragement.


\begin{thebibliography}{00}
\bibitem{BT} J. Bejenaru and T. Tao;
\emph{Sharp well-posedness and ill-posedness results for a
quadratic nonlinear Schr\"{o}dinger equation},
J. Funct. Anal. \textbf{233} (2006), 228--259.

\bibitem{BS75} J. L. Bona and R. Smith;
\emph{The initial value problem for the Korteweg-de Vries equation},
Philos. Trans. Roy. Soc. London Ser A \textbf{278} (1975),
no. 1287, 555--601.

\bibitem{Bo} J. Bourgain;
\emph{Fourier restriction phenomena for certain lattice subset
applications to nonlinear evolution equation},
Geometric and Functional Anal.
\textbf{3} (1993), 107--156, 209--262.

\bibitem{Bo97} J. Bourgain;
\emph{Periodic Korteweg de Vries equation with measures as initial
data}, Selecta Math. (N.S.)
\textbf{3} (1997), no. 2, 115--159.

\bibitem{CCT} M. Christ, J. Colliander and T. Tao;
\emph{Asymptotics, frequency modulation, and low regularity
ill-posedness for canonical defocusing equations},
 Amer. J. Math. \textbf{125} (2003), 1235--1293.

\bibitem{GTV} J. Ginibre, Y. Tsutsumi and G. Velo;
\emph{On the Cauchy problem for the Zaharov system}, J. Func. Anal.
\textbf{151} (1997), no. 2, 384--436.

\bibitem{Ho} J. Holmer;
\emph{Local ill-posed of the 1D Zaharov system},
Electron J. Differential Equations \textbf{2007} No. 24, 22 pp.

\bibitem{Gu} Z. Guo;
\emph{Global well-posedness of Korteweg-de Vries equation in
$H^{-3/4}(\mathbb{R})$}, J. Math. Pures Appl. (9) \textbf{91} (2009),
no. 6, 583--597.

\bibitem{KPV93} C. E. Kenig, G. Ponce, and L. Vega;
 \emph{Well-posed and scattering results for the generalized
Korteweg-de Vries equation via the contraction principle},
Comm. Pure Appl. Math. \textbf{46} (1993), no. 1, 527--620.

\bibitem{KPV96} C. E. Kenig, G. Ponce, and L. Vega;
\emph{A bilinear estimate with applications to the KdV equation},
J. Amer. Math. Soc, \textbf{9} (1996), no. 2, 573--603.

\bibitem{KPV01} C. E. Kenig, G. Ponce and L. Vega;
\emph{On the ill-posedness of some canonical dispersive equations},
 Duke Math. J. \textbf{106} (2001), no. 3, 617--633.

\bibitem{Ki09} N. Kishimoto;
\emph{Well-posedness of the Cauchy problem for the Korteweg-de
Vries equation at the critical regularity}, Differential
Integral Equations \textbf{22} (2009), 447--464.

\bibitem{KiS} N. Kishimoto;
\emph{Low-regularity bilinear estimates for a quadratic nonlinear
Sch\"{o}dinger equation}, J. Differential Equations,
\textbf{247} (2009), no. 5, 1397--1439.

\bibitem{KT} N. Kishimoto and K. Tsugawa;
 \emph{Local well-posedness for quadratic Schr\"{o}dinger equations
and "good'' Boussinesq equation}, Differential Integral
Equations \textbf{23} (2010), no. 5--6, 463--493.

\bibitem{KV} D. J. Korteweg and G. de Vries;
\emph{On the change of form of long waves advancing in a rectangular
canal, and on a new type of long stationary waves},
Philos. Mag. 5 \textbf{39} (1895), 422--443.

\bibitem{MT} T. Muramatu and S. Taoka;
 \emph{The initial value problem for the 1-D semilinear
Schr\"{o}dinger equation in Besov space}, J. Math. Soc.
Japan \textbf{56} (2004), no. 3, 853--888.

\bibitem{Tr} H. Triebel; ``Theory of Function Spaces'',
 Monographs in Mathematics, 78.
Birkh\"{a}user Verlag, Basel, 1983.

\bibitem{Tz} N. Tzvetkov;
\emph{Remark on the local ill-posedness for KdV equation},
 C. R. Acad. Sci. Paris S\'{e}r. I Math. \text{329} (1999), 1043--1047.

\end{thebibliography}

\end{document}