\documentclass[reqno]{amsart}
\usepackage{amssymb}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 71, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/71\hfil Power series solution]
{Power series solution for the modified KdV equation}

\author[T. Nguyen\hfil EJDE-2008/71\hfilneg]
{Tu Nguyen}

\address{Tu Nguyen \newline
Department of Mathematics, University of Chicago,
5734 S. University Ave., Chicago, IL 60637, USA}
\email{tu@math.uchicago.edu}

\thanks{Submitted April 10, 2008. Published May 13, 2008.}
\subjclass[2000]{35Q53}
\keywords{Local well-posedness; KdV equation}

\begin{abstract}
 We use the method developed by Christ \cite{MR2333210} to prove
 local well-posedness  of a modified Korteweg de Vries equation
 in $\mathcal{F}L^{s,p}$ spaces.
\end{abstract}

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

\section{Introduction}

The modified Korteweg de Vries (mKdV) equation on a torus $\mathbb{T}$
has the form
 \begin{equation}
 \begin{gathered}
\partial_{t}u+\partial_{x}^{3}u+u^{2}\partial_{x}u=0 \\
u(\cdot,0)=u_{0}
\end{gathered}\label{eq:mKdV}
\end{equation}
where $(x,t)\in\mathbb{T}\times\mathbb{R}$, $u$ is a real-valued function.
If $u$ is a smooth solution of \eqref{eq:mKdV}, then
$\| u(\cdot,t)\|_{L^{2}(\mathbb{T})}=\| u_{0}\|_{L^{2}(\mathbb{T})}$
for all $t$; therefore,
$\widetilde{u}(x,t)=u(x+\frac{1}{2\pi}\| u_{0}\|_{L^{2}(\mathbb{T})}^{2}t,t)$
is a solution of
\begin{equation}
\begin{gathered}
\partial_{t}u+\partial_{x}^{3}u+\Big(u^{2}-\frac{1}{2\pi}
\int_{\mathbb{T}}u^{2}(x,t)dx\Big)\partial_{x}u=0\\
u(\cdot,0)=u_{0}
\end{gathered}\label{eq:modified mKdV}
\end{equation}
Thus, \eqref{eq:modified mKdV} and \eqref{eq:mKdV} are essentially
equivalent. Using Fourier restriction norm method,
Bourgain \cite{MR1215780}
proved that \eqref{eq:modified mKdV} is locally well-posed for initial data $u_{0}\in H^{s}(\mathbb{T})$ when  $s\geq1/2$, and the solution map is uniformly continuous.
In \cite{MR1466164}, he also showed that the solution
map is not $C^{3}$ in $H^{s}(\mathbb{T})$ when $s<1/2$. Takaoka and Tsutsumi \cite{MR2097834} proved
local-wellposedness of \eqref{eq:modified mKdV} when $1/2>s>3/8$, and they showed that solution map is not uniformly continuous for this range of $s$. For
\eqref{eq:mKdV}, Kappeler and Topalov \cite{MR2131061} used inverse
scattering method to show wellposedness when $s\geq0$ and Christ,
Colliander and Tao \cite{MR2018661} showed that uniformly continuous
dependence on the initial data does not hold when $s<1/2$. Thus,
there is a gap between known local well-posedness results and the
space $H^{-1/2}(\mathbb{T})$ suggested by the standard scaling argument.

Recently, Gr\"unrock and Vega \cite{MR0003} showed local well-posedness
of the mKdV equation on $\mathbb{R}$ with initial data in
\[
\widehat{H_{s}^{r}}(\mathbb{R}):=\{f\in\mathcal{D}'(\mathbb{R})
:\| f\|_{\widehat{H_{s}^{r}}}:=\| \langle \cdot\rangle ^{s}
\hat{f}(\cdot)\|_{L^{r'}}<\infty\},
\]
when $2\geq r>1$ and $s\geq\frac{1}{2}-\frac{1}{2r}$.
(for $r>\frac{4}{3}$, this was obtained by Gr\"unrock \cite{MR2096258}).
This is an extension of the result of Kenig, Ponce and Vega \cite{MR1211741}
that local-wellposedness holds in $H^{s}(\mathbb{R})$ when $s\geq1/4$.
Furthermore, as $\widehat{H_{s}^{r}}$
scales like $H^{\sigma}$ with $\sigma=s+\frac{1}{2}-\frac{1}{r}$,
this result covers spaces that have scaling exponent $-\frac{1}{2}+$.

There is also a related recent work of Gr\"unrock and Herr \cite{MR0002}
on the derivative nonlinear Schr\"odinger equation on $\mathbb{T}$.
Both \cite{MR0003} and \cite{MR0002} used a version of Bourgain's
method.

In this paper, we  apply the new method of solution developed by Christ
\cite{MR2333210} to investigate the local well-posedness of
\eqref{eq:modified mKdV}
with initial data in
\[
\mathcal{F}L^{s,p}(\mathbb{T})
:=\{f\in\mathcal{D}'(\mathbb{T}):\| f\|_{\mathcal{F}L^{s,p}}
:=\| \langle \cdot\rangle ^{s}\hat{f}(\cdot)\|_{l^{p}}<\infty\}.
\]
Let $B(0,R)$ be the ball of radius $R$ centered at $0$ in
$\mathcal{F}L^{s,p}(\mathbb{T})$.
Our main result is the following.

\begin{theorem} \label{thm1.1}
Suppose $s\geq1/2$, $1\leq p < \infty$ and $p'(s+1/4)>1$. Let
$W$ be the solution map for smooth initial data of
\eqref{eq:modified mKdV}.
Then for any $R>0$ there is $T>0$ such that the solution map $W$
extends to a uniformly continuous map from $B(0,R)$ to
$C([0,T],\mathcal{F}L^{s,p}(\mathbb{T}))$.
\end{theorem}

We note that the $\mathcal{F}L^{s,p}(\mathbb{T})$ spaces that are
covered by Theorem 1.1 have scaling index $\frac{1}{4}+$. The restriction
$s\geq1/2$ is due to the presence of the derivative in the nonlinear
term, and is only used to bound the operator $S_{2}$ in section 3.
The same restriction on $s$ is also required in the work on the derivative
nonlinear Schr\"odinger equation on $\mathbb{T}$ by Gr\"unrock
and Herr \cite{MR0002}. We believe that the range of $p$ in
Theorem 1.1 is not sharp.

Concerning \eqref{eq:mKdV}, we have the following result.

\begin{corollary} \label{coro1.2}
Suppose $s\geq1/2$, $1\leq p < \infty$ and $p'(s+1/4)>1$. Let
$\widetilde{W}$ be the solution map for smooth initial data of \eqref{eq:mKdV}.
Then for any $R>0$ there is $T>0$ such that for any $c>0$, the
solution map $\widetilde{W}$ extends to a uniformly continuous map
from $B(0,R)\cap\left\{ \varphi:\| \varphi\|_{L^{2}}=c\right\} \subset\mathcal{F}L^{s,p}(\mathbb{T})$
to $C([0,T],\mathcal{F}L^{s,p}(\mathbb{T}))$.
\end{corollary}

As in \cite{MR2333210}, the solution map $W$ obtained in
Theorem 1.1 gives a weak solution of \eqref{eq:modified mKdV}
in the following sense. Let $T_{N}$ be defined by
$T_{N}u=(\chi_{[-N,N]}\widehat{u})^{\vee}$.
Let $\mathcal{N}u:=\left(u^{2}-\frac{1}{2\pi}\int_{\mathbb{T}}u^{2}(x,t)dx\right)\partial_{x}u$
be the limit in $C([0,T],\mathcal{D}'(\mathbb{T}))$ of $\mathcal{N}(T_{N}u)$
as $N\rightarrow\infty$, provided it exists.

\begin{proposition} \label{prop1.3}
Let $s$ and $p$ be as in Theorem 1.1. Let $\varphi\in\mathcal{F}L^{s,p}$
and $u:=W\varphi\in C([0,T],\mathcal{F}L^{s,p})$. Then $\mathcal{N}u$
exists and $u$ satisfies \eqref{eq:modified mKdV} in the sense of
distribution in $(0,T)\times\mathbb{T}$.
\end{proposition}

To prove these results, we formally expand the solution map into
a sum of multilinear operators. These multilinear operators are described
in the section 2. Then we will show that if $u(\cdot,0)\in\mathcal{F}L^{s,p}$
then the sum of these operators converges in $\mathcal{F}L^{s,p}$
for small time $t$, when $s$ and $p$ satisfy the conditions of
Theorem 1.1. Furthermore, this gives a weak solution of
\eqref{eq:modified mKdV},
justifying our formal derivation.

\section{Multilinear operators}

We rewrite \eqref{eq:modified mKdV} as a system of ordinary differential
equations of the spatial Fourier series of $u$
(see \cite[formula (1.9)]{MR2097834}, and \cite[Lemma 8.16]{MR1215780}).
\begin{equation}  \label{eq:first}
\begin{aligned}
&\frac{d\hat{u}(n,t)}{dt}-in^{3}\hat{u}(n,t)\\
& = -i\sum_{n_{1}+n_{2}+n_{3}=n}\hat{u}(n_{1},t)\hat{u}(n_{2},t)n_{3}
\hat{u}(n_{3},t)
 +i\sum_{n_{1}}\hat{u}(n_{1},t)\hat{u}(-n_{1},t)n\hat{u}(n,t)
\\
 & =  \frac{-in}{3}\sum_{n_{1}+n_{2}+n_{3}=n}^{*}\hat{u}(n_{1},t)
  \hat{u}(n_{2},t)\hat{u}(n_{3},t)
 +in\hat{u}(n,t)\hat{u}(-n,t)\hat{u}(n,t),
\end{aligned}
\end{equation}
where the star means the sum is taken over the triples satisfying
$n_{j}\ne n$, $j=1,2,3$. We note that these are precisely the triples with
$\sigma(n_{1},n_{2},n_{3})\ne0$.

Let $a(n,t)=e^{-in^{3}t}\hat{u}(n,t)$, then $a_{n}(t)$ satisfy
\begin{align*}
\frac{da(n,t)}{dt}
&=-\frac{in}{3}\sum_{n_{1}+n_{2}+n_{3}=n}^{*}e^{i\sigma(n_{1},
 n_{2},n_{3})t}a(n_{1},t)a(n_{2},t)a(n_{3},t)\\
&\quad +ina(n,t)a(-n,t)a(n,t),
\end{align*}
where
\[
\sigma(n_{1},n_{2},n_{3})=n_{1}^{3}+n_{2}^{3}+n_{3}^{3}-
(n_{1}+n_{2}+n_{3})^{3}=-3(n_{1}+n_{2})(n_{2}+n_{3})(n_{3}+n_{1}).
\]
Or, in integral form,
\begin{equation} \label{eq:a integral sum}
\begin{aligned}
a(n,t)
& =  a(n,0)-\frac{in}{3}\int_{0}^{t}\sum_{n_{1}+n_{2}+n_{3}=n}^{*}
 e^{i\sigma(n_{1},n_{2},n_{3})s}a(n_{1},s)a(n_{2},s)a(n_{3},s)ds\\
&\quad   +in\int_{0}^{t} |a(n,s) |^{2}a(n,s)ds.
\end{aligned}
\end{equation}

If, $a$ is sufficiently nice, say $a\in C([0,T],l^{1})$ (which is
the case if $u\in C([0,T],H^{s}(\mathbb{T}))$ for large $s$) then
we can exchange the order of the integration and summation to obtain
\begin{equation} \label{eq:a sum integral}
\begin{aligned}
a(n,t) & =  a(n,0)-\frac{in}{3}\sum_{n_{1}+n_{2}
+n_{3}=n}^{*}\int_{0}^{t}e^{i\sigma(n_{1},n_{2},n_{3})s}a(n_{1},s)
a(n_{2},s)a(n_{3},s)ds\\
&\quad +in\int_{0}^{t}|a(n,s)|^{2}a(n,s)ds.
 \end{aligned}
\end{equation}
Replacing the $a(n_{j},s)$ in the right hand side by their equations
obtained using \eqref{eq:a sum integral}, we get
\begin{equation} \label{eq:a sum integral first step}
\begin{aligned}
a(n,t)
& =  a(n,0)-\frac{in}{3}\sum_{n_{1}+n_{2}+n_{3}=n}^{*}
 a(n_{1},0)a(n_{2},0)a(n_{3},0)\int_{0}^{t}e^{i\sigma(n_{1},n_{2},n_{3})s}ds\\
& \quad +in|a(n,0)|^{2}a(n,0)\int_{0}^{t}ds
 +\textrm{ additional terms }\\
& =  a(n,0)-\frac{n}{3}\sum_{n_{1}+n_{2}+n_{3}=n}^{*}
 \frac{a(n_{1},0)a(n_{2},0)a(n_{3},0)}{\sigma(n_{1},n_{2},n_{3})}
 (e^{i\sigma(n_{1},n_{2},n_{3})t}-1)\\
& \quad +int|a(n,0)|^{2}a(n,0)+\textrm{ additional terms }
\end{aligned}
\end{equation}
The additional terms are those which depend not only on $a(\cdot,0)$.
An example of the additional terms is
\begin{align*}
&-\frac{nn_{3}}{9}\!\sum_{n_{1}+n_{2}+n_{3}=n}^{*}a(n_{1},0)a(n_{2},0)
\!\sum_{m_{1}+m_{2}+m_{3}=n_{3}}^{*}\int_{0}^{t}
e^{i\sigma(n_{1},n_{2},n_{3})s}\int_{0}^{s}
e^{i\sigma(m_{1},m_{2},m_{3})s'}\\
&\times a(m_{1},s')a(m_{2},s')a(m_{3},s')ds'ds
\end{align*}
Then we can again use \eqref{eq:a sum integral}
for each appearance of $a(m,\cdot)$ in the additional terms, and
obtain more and more complicated additional terms. We refer to section 2 of \cite{MR2333210} for more detailed description
of these additional terms. Continuing this process indefinitely,
we get a formal expansion of $a(n,t)$ as a sum of multilinear operators
of $a(\cdot,0)$.

We will now describe these operators and then show that their sum
converges. Again, we refer to section 3 of \cite{MR2333210} for more
detailed explanations. Each of our multilinear operators will be associated
to a tree, which has the property that each of its node has either
zero or three children. We will only consider trees with this property.
If a node $v$ of $T$ has three children, they will be denoted by
$v_{1},v_{2},v_{3}$. We denote by $T^{0}$ the set of non-terminal
nodes of $T$, and $T^{\infty}$ the set of terminal nodes of $T$.
Clearly, if $|T|=3k+1$ then $|T^{0}|=k$ and
$|T^{\infty}|=2k+1$.

\begin{definition} \label{def2.1} \rm
Let $T$ be a tree. Then $\mathcal{J}(T)$ is the set of
$j\in\mathbb{Z}^{T}$ such that if $v\in T^{0}$ then
\[
j_{v}=j_{v_{1}}+j_{v_{2}}+j_{v_{3}},
\]
and either $j_{v_{i}}\ne j_{v}$ for all $i$, or
$j_{v_{1}}=-j_{v_{2}}=j_{v_{3}}=j_{v}$.
We will denote by $v(T)$ be the root of $T$ and $j(T)=j(v(T))$.
For $j\in\mathcal{J}(T)$ and $v\in T^{0}$,
\[
\sigma(j,v):=\sigma(j(v_{1}),j(v_{2}),j(v_{3})).
\]
Also define
\[
\mathcal{R}(T,t)=\{s\in\mathbb{R}_{+}^{T^{0}}:\text{ if }v<w
\text{ then }0\leq s_{v}\leq s_{w}\leq t\}.
\]
\end{definition}

Using the above definitions, we can rewrite
(\ref{eq:a sum integral first step}) as
\begin{align*}
a(n,t) & =  a(n,0)+\sum_{|T|=4}\omega_{T}\sum_{j\in\mathcal{J}(T),j(T)=n}
 na(j(v(T)_{1}),0) a(j(v(T)_{2}),0)\\
&\quad \times a(j(v(T)_{3}),0)\int_{\mathcal{R}(T,t)}c(j,v(T),s)ds
  +\text{additional terms},
\end{align*}
here $c(j,v,s)=e^{i\sigma(j,v)s}$, and $\omega_{T}$ is a constant
with $|\omega_{T}|\leq1$.

Continuing this replacement process, it leads to
\begin{align*}
a(n,t)
& =  a(n,0)+\sum_{|T|<3k+1}\omega_{T}\sum_{j\in\mathcal{J}(T),j(T)=n}
\prod_{u\in T^{0}}j_{u}\prod_{v\in T^{\infty}}a(j_{v},0)
\int_{\mathcal{R}(T,t)}c(j,s)ds\\
 &\quad +\text{additional terms}
\end{align*}
where
\[
c(j,s)=\prod_{v\in T^{0}}c(j,v,s_{v})
\]
We will show that the series
\[
a(n,0)+\sum_{T}\omega_{T}\sum_{j\in\mathcal{J}(T),j(T)=n}
\prod_{u\in T^{0}}j_{u}\prod_{v\in T^{\infty}}a(j_{v},0)
\int_{\mathcal{R}(T,t)}c(j,s)ds
\]
converges in $C([0,T],l^{p})$ when $a(\cdot,0)\in l^{p}$.


\section{$l^{p}$ convergence}

%\begin{definition}
Let $T$ be a tree and $j\in\mathcal{J}(T)$. We define
\[
I_{T}(t,j)=\int_{\mathcal{R}(T,t)}c(j,s)ds,
\]
 and \[
S_{T}(t)(a_{v})_{v\in T^{\infty}}(n)=\omega_{T}
\sum_{j\in\mathcal{J}(T),j(T)=n}\prod_{u\in T^{0}}j_{u}
\prod_{v\in T^{\infty}}a_{v}(j_{v})I_{T}(t,j).
\]
We first give an estimate for $I_{T}(t,j)$ which allows us to bound
$S_{T}$.

\begin{lemma} \label{lem3.2}
For $0\leq t\leq1$,
\[
|I_{T}(j,t)|\leq(Ct)^{|T^{0}|/2}\prod_{v\in T^{0}}\langle \sigma(j,v)
\rangle ^{-1/2}.
\]
\end{lemma}

\begin{proof}
For $v\in T^{0}$, define the level
of $v$, denoted $l(v)$, to be the length of the unique path connecting
$v(T)$ and $v$. Let $O$ be the set of $v\in T^{0}$ for which
$l(v)$ is odd, and $E$ those $v$ for which $l(v)$ is even.

First we fix the variables $s_{v}$ with $v\in E$, and take the integration
in the variables $s_{v}$ with $v\in O$. For each $v\in O$, the
result of the integration is
\[
\frac{1}{\sigma(j,v)}\left(e^{i\sigma(j,v)s_{\tilde{v}}}
-e^{i\sigma(j,v)\max\{s_{v(1)},s_{v(2)},s_{v(3)}\}}\right)
\]
if $\sigma(j,v)\ne0$, and
\[
s_{\tilde{v}}-\max\{s_{v(1)},s_{v(2)},s_{v(3)}\}.
\]
if $\sigma(j,v)=0$. Here $\widetilde{v}$ is the parent of $v$.
Thus, we obtain the factor
\[
\prod_{v\in O}\langle \sigma(j,v\rangle ^{-1}
\]
and an integral in $s_{v}$, $v\in E$ where the integrand is bounded
by $2^{|O|}$. As the domain of integration in $s_{v}$
with $v\in E$ has measure less than $t^{|E|}$, we see
that
 \[
|I_{T}(j,t)|\leq2^{|T^{0}|}t^{|E|}\prod_{v\in O}\langle \sigma(j,v)
\rangle ^{-1}.
\]
By switching the role of $O$ and $E$, we get
\[
|I_{T}(j,t)|\leq2^{|T^{0}|}t^{|O|}\prod_{v\in E}\langle
 \sigma(j,v)\rangle ^{-1}.
\]
Combining these two estimates, we obtain the lemma.
\end{proof}

By Lemma \ref{lem3.2},
\[
|S_{T}(t)(a_{v})_{v\in T^{\infty}}(n)|\leq(Ct)^{|T^{0}|/2}
\sum_{j\in\mathcal{J}(T):j(T)=n}\prod_{u\in T^{0}}\langle
\sigma(j,u)\rangle ^{-1/2}|j_{u}|\prod_{v\in T^{\infty}}|a_{v}(j_{v})|.
\]
Let
\[
\widetilde{S}_{T}(a_{v})_{v\in T^{\infty}}(n)
=\sum_{j\in\mathcal{J}(T):j(T)=n}\prod_{u\in T^{0}}\langle
\sigma(j,u)\rangle ^{-1/2}|j_{u}|\prod_{v\in T^{\infty}}|a_{v}(j_{v})|,
\]
and
\[
\widetilde{S}(a_{1},a_{2},a_{3})(n)
=\sum_{n_{1}+n_{2}+n_{3}=n}^{*}|n|\langle \sigma(n_{1},n_{2},n_{3})
\rangle ^{-1/2}\prod_{i=1}^{3}|a_{i}(n_{i})|+|n|\prod_{i=1}^{3} |a_{i}(n)|.
\]
It is clear that
\[
\widetilde{S}_{T}(a_{v})_{v\in T^{\infty}}
=\widetilde{S}(\widetilde{S}_{T_{1}}(a_{v})_{v\in T_{1}^{\infty}},
\widetilde{S}_{T_{2}}(a_{v})_{v\in T_{2}^{\infty}},
\widetilde{S}_{T_{3}}(a_{v})_{v\in T_{3}^{\infty}}).
\]
where $T_{i}$ is the subtree of $T$ that contains all nodes $u$
such that $u\leq v(T)_{i}$ (recall that $v(T)$ is the root of $T$).
Hence, to bound $S_{T}$, it suffices to bound $\widetilde{S}$. For
this purpose, we will use the following simple lemma.

\begin{lemma} \label{lem3.3}
Let $S$ be the multilinear operator defined by
\[
S(a_{1},a_{2},a_{3})(n)=\sum_{n_{1}+n_{2}+n_{3}=n}
m(n_{1},n_{2},n_{3})\prod_{j=1}^{3}a_{j}(n_{j}),
\]
Let $1\leq p\leq\infty$. Then for any pair of indices
$i\ne j\in\{1,2,3\}$,
\[
\| S(a_{1},a_{2},a_{3})\|_{l^{p}}\leq
\sup_{n}\| m(n_{1},n_{2},n_{3})\|_{l_{i,j}^{p'}}
\prod_{k=1}^{3}\| a_{k}\|_{l^{p}}.
\]
\end{lemma}

\begin{proof}
By H\"older's inequality, for any $n$,
\begin{align*}
|S(a_{1},a_{2},a_{3})(n)|
&\leq\| m(n_{1},n_{2},n_{3})\|_{l_{i,j}^{p'}}\|
\prod_{k=1}^{3}a_{k}\|_{l_{i,j}^{p}} \\
&\leq\sup_{n}\| m(n_{1},n_{2},n_{3})\|_{l_{i,j}^{p'}}\|
\prod_{k=1}^{3}a_{k}\|_{l_{i,j}^{p}}
\end{align*}
 Taking $l^{p}$-norm in $n$ we obtain the lemma.
\end{proof}

Showing that $\widetilde{S}$ is a bounded multilinear map on $l^{s,p}:=\{a:\left\langle \cdot\right\rangle ^{s}a\in l^{p}\}$ is equivalent to showing that $S$ is bounded on $l^{p}$ where $S$ is the operator with kernel
\[
m(n_{1},n_{2},n_{3})
=\frac{\langle n\rangle ^{s}|n|}{\langle
\sigma(n_{1},n_{2},n_{3})\rangle ^{1/2}\prod_{k=1}^{3}\langle n_{k}
\rangle ^{s}}
\]
where $n_{1}+n_{2}+n_{3}=n$.
We split $S$ into sum of two operators $S_{1}$ and $S_{2}$ where
$S_{1}$ has kernel
 \[
m_{1}(n_{1},n_{2},n_{3})
=\frac{\langle n\rangle ^{s}|n|}{\prod_{k=1}^{3}\langle n_{k}\rangle ^{s}
\langle n-n_{k}\rangle ^{1/2}}\quad
\text{if }n=n_{1}+n_{2}+n_{3},\;  n_{i}\ne n
\]
and $S_{2}$ has kernel
\[
m_{2}(n_{1},n_{2},n_{3})=n/\langle n\rangle ^{2s}\quad
\text{if }n_{1}=-n_{2}=n_{3}=n.
\]
Clearly, $S_{2}$ is bounded on $l^{p}$ if and only if $s\geq1/2$.

It remains to bound $S_{1}$, for which we have the following result.

\begin{proposition} \label{prop3.4}
$S_{1}$ is bounded in $l^{p}\times l^{p}\times l^{p}$ to $l^{p}$
when $s\geq1/4$ and $p'(s+\frac{1}{4})>1$.
\end{proposition}

\begin{proof}
In the proof, all the sums are taken over the triples $(n_{1},n_{2},n_{3})$
that satisfy the additional property that $n_{i}\ne n$, for all
$1\leq i\leq3$.
Clearly, we can assume $n>0$. Note that if say $|n_{1}|\geq5n$
then as $|n_{2}+n_{3}|=|n-n_{1}|\geq4n$, at
least one of $n_{2}$ and $n_{3}$ has absolute value bigger than
$2n$. Also, we cannot have $|n_{i}|\leq n/4$ for all
$i$. Thus, up to permutation, there are four cases.
\begin{enumerate}
\item $|n_{1}|,|n_{2}|,|n_{3}|\in[n/4,5n]$
\item $|n_{1}|,|n_{2}|\in[n/4,5n]$, $|n_{3}|\leq n/4$
\item $|n_{1}|\in[n/4,5n]$, $|n_{2}|,|n_{3}|\leq n/4$
\item $|n_{1}|,|n_{2}|\geq2n$
\end{enumerate}
By Lemma \ref{lem3.3}, it suffices to show that in each of these four
regions, for some $i\ne j$ the $l_{i,j}^{p'}$-norm of $m$ is bounded.

\noindent\textbf{Case 1.}
As $3n=\sum(n-n_{i})$ for some index $i$, say $i=3$,
we must have $|n-n_{3}|\sim n$. Since we also have
$|n_{1}|,|n_{2}|\gtrsim n$,
\[
|m(n_{1},n_{2},n_{3})|\lesssim\frac{\langle n\rangle ^{1/2-s}}{\langle n_{3}
\rangle ^{s}|(n-n_{1})(n-n_{2})|^{1/2}}.
\]
We will use the  inequality
\[
|\frac{1}{n_{3}(n-n_{2})}|=|\frac{1}{n_{1}}
\Big(\frac{1}{n_{3}}-\frac{1}{n-n_{2}}\Big)|\leq\frac{1}{|n_{1}|}
\Big(\frac{1}{|n_{3}|}+\frac{1}{|n-n_{2}|}\Big).
\]

\noindent (1)
 If $1/4\leq s\leq1/2$: then $\langle n_{3}\rangle ^{p'(1/2-s)}
\lesssim\langle n\rangle ^{p'(1/2-s)}$,
so \begin{align*}
\| m\|_{l_{1,2}^{p'}}^{p'}
& \lesssim  \sum_{|n_{1}|\leq5n}\frac{\langle n\rangle ^{p'(1/2-s)}}
 {|n-n_{1}|^{p'/2}}\sum_{|n_{2}|\leq5n}
 \frac{\langle n_{3}\rangle ^{p'(1/2-s)}}
 {\left(\langle n_{3}\rangle |n-n_{2}|\right)^{p'/2}}\\
& \lesssim  \sum_{|n_{1}|\leq5n}\frac{\langle n\rangle ^{p'(1/2-s)}}
 {|n-n_{1}|^{p'/2}}\sum_{|n_{2}|\leq 5n}\!
 \frac{\langle n\rangle ^{p'(1/2-s)}}{|n_{1}|^{p'/2}}
 \Big(\frac{1}{|n-n_{2}|^{p'/2}}+\frac{1}{|n-n_{1}-n_{2}|^{p'/2}}\Big)\\
& \lesssim  \sum_{|n_{1}|\leq5n}\frac{\langle n\rangle ^{p'(1-2s)}A_{n}}
 {|(n-n_{1})n_{1}|^{p'/2}}\\
& \lesssim  \langle n\rangle ^{p'(1-2s)}A_{n}\sum_{|n_{1}|\leq5n}
\Big(\frac{1}{n}(\frac{1}{|n-n_{1}|}+\frac{1}{|n_{1}|})\Big)^{p'/2}\\
& \lesssim  \langle n\rangle ^{p'(1/2-2s)}A_{n}^{2}.
\end{align*}
where $\sum_{0<j<5n}j^{-p'/2}=A_{n}$. As
\[
A_{n}\lesssim  \begin{cases}
n^{1-p'/2} & \text{if }p'<2\\
\log\langle n\rangle  & \text{if }p'=2\\
1 & \text{if }p'>2
\end{cases}
\]
we easily check that $\langle n\rangle ^{(1/2-2s)p'}A_{n}^{2}$
is bounded by a constant, under our hypothesis on $s$ and $p'$.

\noindent(2) If $s>1/2$: then
$\langle n-n_{2}\rangle ^{p'(s-1/2)}\lesssim\langle n\rangle ^{p'(s-1/2)}$,
so \begin{align*}
\| m\|_{l_{1,2}^{p'}}^{p'}
 & \lesssim  \sum_{|n_{1}|\leq5n}\frac{\langle n\rangle ^{p'(1/2-s)}}
  {|n-n_{1}|^{p'/2}}\sum_{|n_{2}|\leq5n}\frac{\langle n-n_{2}
  \rangle ^{p'(s-1/2)}}{\left(\langle n_{3}\rangle |n-n_{2}|\right)^{p's}}\\
 & \lesssim  \sum_{|n_{1}|\leq5n}\frac{\langle n\rangle ^{p'(1/2-s)}}
  {|n-n_{1}|^{p'/2}}\sum_{|n_{2}|\leq5n}\frac{\langle n
  \rangle ^{p'(s-1/2)}}{|n_{1}|^{p's}}\Big(\frac{1}{|n-n_{2}|^{p's}}
  +\frac{1}{|n-n_{1}-n_{2}|^{p's}}\Big)\\
 & \lesssim  \sum_{|n_{1}|\leq5n}\frac{B_{n}}{|n-n_{1}|^{p'/2}|n_{1}|^{p's}}\\
 & \lesssim  B_{n}\sum_{|n_{1}|\leq5n}|n-n_{1}|^{p'(s-1/2)}
 \Big(\frac{1}{n}(\frac{1}{|n-n_{1}|}+\frac{1}{|n_{1}|})\Big)^{p's}\\
 & \lesssim  \langle n\rangle ^{-p'/2}B_{n}^{2}.
\end{align*}
where $B_{n}=\sum_{0<j<5n}j^{-p's}$. As
\[
B_{n}\lesssim \begin{cases}
n^{1-p's} & \text{if }p's<1\\
\log\langle n\rangle  & \text{if }p's=1\\
1 & \text{if }p's>1
\end{cases}
\]
 we easily check that $\langle n\rangle ^{-p'/2}B_{n}^{2}$
is bounded by a constant, under our hypothesis on $s$ and $p'$.

\noindent\textbf{Case 2}
This case can be treated in exactly the same way as
the first case, except when $n_{3}=0$. In the region $n_{3}=0$,
\begin{align*}
\| m\|_{l_{1,3}^{p'}}^{p'}
& \lesssim  \sum_{n_{1}}\frac{\langle n
 \rangle ^{p'(1/2-s)}}{|n_{1}(n-n_{1})|^{p'/2}}
 \leq\sum_{n_{1}}\langle n\rangle ^{-p's}
\Big(\frac{1}{|n_{1}|^{p'/2}}+\frac{1}{|n-n_{1}|^{p'/2}}\Big)\\
& \lesssim  \langle n\rangle ^{-p's}A_{n}\lesssim1
\end{align*}


\noindent\textbf{Case 3}
As $|n_{1}|,|n-n_{2}|,|n-n_{3}|\sim n$,
\[
|m(n_{1},n_{2},n_{3})|\lesssim\frac{1}{\langle n_{2}\rangle ^{s}
\langle n_{3}\rangle ^{s}|n_{2}+n_{3}|^{1/2}}.
\]
Without loss of generality, we assume that $|n_{3}|\geq|n_{2}|$.

\noindent (1) If $|n_{2}|<|n_{3}|/2$:
\begin{align*}
\| m\|_{l_{2,3}^{p'}}^{p'}
& \lesssim  \sum_{0\leq|n_{2}|\leq n/4}\frac{1}{\langle n_{2}
  \rangle ^{p's}}\sum_{n/4\geq|n_{3}|>2n_{2}}
  \frac{1}{\langle n_{3}\rangle ^{p'(s+1/2)}}\\
& \lesssim  \sum_{0\leq|n_{2}|\leq n/4}\frac{1}{\langle n_{2}
  \rangle ^{p'(2s+1/2)-1}}
 \lesssim  1
\end{align*}
if $(s+1/4)p'>1$.

\noindent(2) If $|n_{2}|\geq|n_{3}|/2$:
\begin{align*}
\| m\|_{l_{2,3}^{p'}}^{p'}
& \lesssim  \sum_{|n_{3}|\leq n/4}\frac{1}{\langle n_{3}
  \rangle ^{2p's}}\sum_{|n_{3}|\geq n_{2}\geq|n_{3}|/2}
  \frac{1}{\langle n_{3}+n_{2}\rangle ^{p'/2}}\\
& \lesssim  \sum_{|n_{3}|\leq n/4}\frac{1}{\langle n_{3}
  \rangle ^{2p's}}\max\{\log\langle n_{3}\rangle ,\langle n_{3}
  \rangle ^{-p'/2+1}\}\\
& \lesssim  \sum_{|n_{3}|\leq n/4}\frac{\log\langle n_{3}
  \rangle }{\langle n_{3}\rangle ^{2p's}}+\sum_{|n_{3}|
  \leq n/4}\frac{1}{\langle n_{3}\rangle ^{p'(2s+1/2)-1}}\lesssim1
\end{align*}
as $2p's\geq p'(s+1/4)>1$.

\noindent\textbf{Case 4} $|n_{1}|,|n_{2}|>2n$: Note
that in this case, $|n_{1}|\sim|n-n_{1}|$ and
$|n_{2}|\sim|n-n_{3}|$.

\noindent(1) If $|n_{3}|,|n-n_{3}|\geq n/2:$ we have
\[
|m(n_{1},n_{2},n_{3})|\lesssim\frac{\langle n\rangle ^{1/2}}
{\langle n_{1}\rangle ^{s+1/2}\langle n_{2}\rangle ^{s+1/2}},
\]
hence
\begin{align*}
\| m\|_{l_{1,2}^{p'}}^{p'}
& \lesssim  \langle n\rangle ^{p'/2}\sum_{|n_{1}|,|n_{2}|>2n}
  \frac{1}{\langle n_{1}\rangle ^{p'(s+1/2)}\langle n_{2}
  \rangle ^{p'(s+1/2)}}\\
& \lesssim  \frac{\langle n\rangle ^{p'/2}}{\langle 2n
  \rangle ^{p'(2s+1)-2}}\lesssim1.
\end{align*}

\noindent (2) If $|n_{3}|<n/2$: then $|n_{1}|\sim|n_{2}|$
and $|n-n_{3}|\geq n/2$, so
\[
|m(n_{1},n_{2},n_{3})|\lesssim\frac{n^{s+1/2}}{\langle n_{1}
\rangle ^{2s+1}\langle n_{3}\rangle ^{s}},
\]
hence
\[
\| m\|_{l_{1,3}^{p'}}^{p'}\lesssim B_{n}\sum_{|n_{1}|>2n}
\frac{n^{p'(s+1/2)}}{\langle n_{1}\rangle ^{p'(2s+1)}}
\lesssim\frac{B_{n}}{n^{p'(s+1/2)-1}}\lesssim1
\]

\noindent(3) If $|n-n_{3}|<n/2$: then $|n_{1}|\sim|n_{2}|$
and $|n_{3}|\sim n$. Hence,
 \[
|m(n_{1},n_{2},n_{3})|\lesssim\frac{n}{\langle n_{1}\rangle ^{2s+1}
\langle n-n_{3}\rangle ^{1/2}}.
\]
Therefore,
\begin{align*}
\| m\|_{l_{1,3}^{p'}}^{p'}
& \lesssim  \sum_{|n_{1}|\geq2n}\sum_{n/2<n_{3}<3n/2}\frac{n^{p'}}
 {\langle n_{1}\rangle ^{p'(2s+1)}\langle n-n_{3}\rangle ^{p'/2}}\\
& \lesssim  \sum_{|n_{1}|\geq2n}\frac{A_{n}n^{p'}}
 {\langle n_{1}\rangle ^{p'(2s+1)}}\lesssim\frac{A_{n}}{n^{2p's-1}}
 \lesssim1
\end{align*}
This concludes the proof of the proposition.
\end{proof}


\begin{proof}[Proof of Theorem 1.1]
Let $u_{0}\in\mathcal{F}L^{s,p}$ and $a(n)=\widehat{u_{0}}(n)$.
By Proposition \ref{prop3.4},
\[
\| S_{T}((a_{v})_{v\in T^{\infty}})\|_{l^{s,p}}
 \leq C^{|T^{0}|}t^{|T^{0}|/2}\prod_{v\in T^{\infty}}\| a_{v}\|_{l^{s,p}}.
\]
Hence,
\begin{equation}
\begin{aligned}
& \| a(n)+\sum_{T} \omega_{T}\sum_{j\in\mathcal{J}(T),j(T)=n}
 \prod_{u\in T^{0}}j_{u}\prod_{v\in T^{\infty}}a(j_{v})
 \int_{\mathcal{R}(T,t)}c(j,s)ds\|_{l^{s,p}}\\
&\leq \| a\|_{l^{s,p}} + \sum_{T}\| S_{T}(a,\ldots,a)\|_{l^{s,p}}\\
&\leq\sum_{k=0}^{\infty}(Ct)^{k/2}
\| a\|_{l^{s,p}}^{2k+1}=\frac{\| u_{0}\|_{\mathcal{F}L^{s,p}}}{1-\sqrt{Ct}
\| u_{0}\|_{\mathcal{F}L^{s,p}}^{2}}
\end{aligned}\label{eq:converge}
\end{equation}
for all $t\lesssim\min\{1,\| u_{0}\|_{\mathcal{F}L^{s,p}}^{-4}\}$.

Let $T\sim \min\{1,\| u_{0}\|_{\mathcal{F}L^{s,p}}^{-4}\}$, then for $t\in [0,T]$  we can define 
\[ a(n,t)= a(n)+\sum_{T} \omega_{T}\sum_{j\in\mathcal{J}(T),j(T)=n}
 \prod_{u\in T^{0}}j_{u}\prod_{v\in T^{\infty}}a(j_{v})
 \int_{\mathcal{R}(T,t)}c(j,s)ds \]
and the solution map $u=Wu_{0}$ by 
\[\widehat{u}(n,t)=e^{-in^{3}t}a(n,t).\] 
It follows from (\ref{eq:converge}) that $W$ is uniformly continuous. The same argument as that of \cite{MR2333210} shows that $u$ is limit of classical solutions.
\end{proof}

The proof of Proposition 1.2 is basically the same as that of
\cite[Proposition 1.4]{MR2333210}, hence we omit it.

\subsection*{Acknowledgement}
I would like to thank my advisor Prof. Carlos Kenig for suggesting
this topic and for his helpful conversations.
I would also like to thank Profs. Axel Gr\"unrock and Sebastian Herr
for their valuable comments and suggestions.

\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\begin{thebibliography}{10}

\bibitem{MR1215780} J. Bourgain;
\emph{Fourier transform restriction phenomena for certain lattice
  subsets and applications to nonlinear evolution equations. {II}. {T}he
  {K}d{V}-equation}, Geom. Funct. Anal. \textbf{3} (1993), no.~3, 209--262.
  MR 1215780 (95d:35160b)

\bibitem{MR1466164} J. Bourgain;
\emph{Periodic {K}orteweg de {V}ries equation with measures as initial
  data}, Selecta Math. (N.S.) \textbf{3} (1997), no.~2, 115--159. MR1466164
  (2000i:35173)

\bibitem{MR2333210} M. Christ;
 \emph{Power series solution of a nonlinear {S}chr\"odinger
  equation}, Mathematical aspects of nonlinear dispersive equations, Ann. of
  Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007,
  pp.~131--155. MR 2333210

\bibitem{MR2018661} Michael Christ, James Colliander, and Terrence Tao;
 \emph{Asymptotics,
  frequency modulation, and low regularity ill-posedness for canonical
  defocusing equations}, Amer. J. Math. \textbf{125} (2003), no.~6, 1235--1293.
  MR 2018661 (2005d:35223)

\bibitem{MR2096258} Axel Gr{\"u}nrock;
 \emph{An improved local well-posedness result for the
  modified {K}d{V} equation}, Int. Math. Res. Not. (2004), no.~61, 3287--3308.
  MR 2096258 (2006f:35242)

\bibitem{MR0002} Axel Gr{\"u}nrock and Sebastian Herr;
 \emph{Low regularity local well-posedness
  of the {D}erivative {N}onlinear {S}chr{\"o}dinger {E}quation with periodic
  initial data}, SIAM J. Math. Anal., to appear.

\bibitem{MR0003} Axel Gr{\"u}nrock and Luis Vega;
 \emph{Local well-posedness for the modified
  {K}d{V} equation in almost critical {$ \widehat{H \sp r \sb s} $}-spaces},
  Trans. AMS., to appear.

\bibitem{MR2131061} T. Kappeler and P. Topalov;
 \emph{Global well-posedness of m{K}d{V} in {$L^2(\mathbb{T},\mathbb{R})$}},
Comm. Partial Differential Equations \textbf{30} (2005),
  no.~1-3, 435--449. MR 2131061 (2005m:35256)

\bibitem{MR1211741}
Carlos E. Kenig, Gustavo Ponce, and Luis Vega;
 \emph{Well-posedness and  scattering results for the generalized
Korteweg-de Vries equation via the  contraction principle},
 Comm. Pure Appl. Math. \textbf{46} (1993), no.~4,
  527--620. MR 1211741 (94h:35229)

\bibitem{MR2097834} Hideo Takaoka and Yoshio Tsutsumi;
 \emph{Well-posedness of the {C}auchy problem
  for the modified KdV equation with periodic boundary condition}, Int.
  Math. Res. Not. (2004), no.~56, 3009--3040. MR 2097834 (2006e:35295)

\end{thebibliography}

\end{document}