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\AtBeginDocument{{\noindent\small
\sc{Ilirias Journal of Mathematics}\newline
ISSN: 2334-6574, URL: www.ilirias.com/ijm\newline
Volume 6 Issue y(2017), Pages 1-x.}

\thanks{\copyright 2017 Ilirias Research Institute, Prishtin\"e, Kosov\"e.}
\vspace{9mm}}

\begin{document}
\title[ Well-posedness for perturbations of the KdV]
{Well-posedness for some perturbations of the KdV equation with
low regularity data}

\author[X. Carvajal, M. Panthee]
{Xavier Carvajal, Mahendra Panthee}  % in alphabetical order

\address{Xavier Carvajal \newline
 Instituto de Matem\'atica - UFRJ
 Av. Hor\'acio Macedo, Centro de Tecnologia
 Cidade Universit\'aria, Ilha do Fund\~ao,
 Caixa Postal 68530
 21941-972 Rio de Janeiro,  RJ,  Brasil}
\email{carvajal@im.ufrj.br}

\address{Mahendra Panthee \newline
Centro de An\'alise Matem\'atica, Geometria e Sistemas
 Din\^amicos, Departamento de Matem\'atica, Instituto Superior T\'ecnico,
1049-001 Lisboa, Portugal.\newline
Department of Mathematics,
Central Department of Mathematics, Tribhuvan University, Kirtipur,
Kathmandu, Nepal}
\email{mpanthee@math.ist.utl.pt}




\thanks{Submitted August 1, 2007. Published January 2, 2008.}

\thanks{X. C. was  supported by the grant ABC.}


\subjclass[2000]{35A07, 35Q53}
\keywords{Bourgain spaces; KdV equation; local smoothing effect}

\begin{abstract}
 We  study  some well-posedness issues of  the initial value problem
 associated with the equation
 \[
 u_t+u_{xxx}+\eta Lu+uu_x=0, \quad x \in \mathbb{R}, \; t\geq 0,
 \]
 where $\eta>0$, $\widehat{Lu}(\xi)=-\Phi(\xi)\hat{u}(\xi)$ and
 $\Phi \in \mathbb{R}$ is  bounded above.
 Using the theory developed by Bourgain and Kenig, Ponce and Vega,
 we prove that the initial value problem is locally well-posed for
 given data in Sobolev spaces $H^s(\mathbb{R})$ with regularity below $L^2$.
 Examples of this model are the Ostrovsky-Stepanyams-Tsimring equation
 for $\Phi(\xi)=|\xi|-|\xi|^3$, the derivative
 Korteweg-de Vries-Kuramoto-Sivashinsky equation for
 $\Phi(\xi)=\xi^2-\xi^4$, and  the Korteweg-de Vries-Burguers equation
 for $\Phi(\xi)=-\xi^2$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem*{remark}{Remark}





\section{Introduction}

In this paper we consider the initial value problem (IVP)
\begin{equation}\label{eq:hs}
 \begin{gathered}
  u_t+u_{xxx}+\eta Lu+uu_x=0, \quad x \in \mathbb{R}, \;t\geq 0,\\
     u(x,0)=u_0(x),
 \end{gathered}
\end{equation}
where $\eta>0$ is a constant and the linear operator $L$ is defined
via the Fourier transform by $\widehat{Lu}(\xi)=-\Phi(\xi)\hat{u}(\xi)$.
The Fourier symbol
\begin{align}\label{phi}
\Phi(\xi)=\sum_{j=0}^{n}\sum_{i=0}^{2m}c_{i,j}\xi^i |\xi|^j,
\quad c_{i,j} \in \mathbb{R},\; c_{2m,n}=-1.
\end{align}
is a real valued function which is bounded above; i.e., there is a constant
$C$ such that $\Phi(\xi) < C$. Without loss of generality, we  suppose
that $\Phi(\xi) < 1$. For this, let us perform  the following scale change
 $$
v(x,t) = \frac1{\lambda^2} u\big(\frac x{\lambda} , \frac t{\lambda^3}\big).
$$
Then $v$ satisfies the equation
\begin{equation}\label{scale.1}
\lambda^3 v_t + \lambda^3 v_{xxx} + \eta T v + \lambda^3 vv_x = 0,
 \end{equation}
where
 $$
\widehat{Tv}(\xi) = \Phi(\lambda \xi) \hat{v}(\xi).
$$
If we take $\lambda^3=C$, where $C$ is as earlier, then the
Fourier symbol of the new operator $T$ in \eqref{scale.1} is
bounded above by $1$. Finally, inverting the scale change, we
obtain well-posedness result for the original IVP (\ref{eq:hs})
from that of \eqref{scale.1}. So, throughout this work we consider
the IVP (\ref{eq:hs}) with $\Phi(\xi)$ in (\ref{phi}) satisfying
$\Phi(\xi) < 1$.

Our interest here is to obtain well-posedness results for
\eqref{eq:hs} with given data $u_0$ in the Sobolev spaces
$H^s(\mathbb{R})$ with regularity below $L^2$. The $L^2$-based Sobolev
space $H^s(\mathbb{R})$ is defined by
$$
H^s(\mathbb{R}) := \{f\in \mathcal{S}'(\mathbb{R}) : \|f\|_{H^s} < \infty\},
$$
where
$$
\|f\|_{H^s}^2 = \int_{\mathbb{R}} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi,
$$
and $\hat f(\xi)$ is the usual Fourier transform given by
$$
\hat f(\xi) \equiv \mathcal{F}(f)(\xi)
= \frac 1{\sqrt{2\pi}}\int_{\mathbb{R}}e^ {-ix\xi} f(x)\, dx.
$$
However, from here onwards, we will neglect the factor $2\pi$
in the definition of the Fourier transform because it does not alter
our analysis.

Also, we consider the homogeneous Sobolev space $\dot H^s(\mathbb{R})$
defined via the norm
$$
\|f\|_{\dot H^s}^2 = \int_{\mathbb{R}} |\xi|^{2s}|\hat f(\xi)|^2d\xi.$$

Before stating the main results of this work, we give some examples that
belong to the class considered in (\ref{eq:hs}).

The first example of this type is the generalized
 Ostrovsky-Stepanyams-Tsimring (OST) equation.
\begin{equation}\label{eqhs}
   \begin{gathered}
     u_t+u_{xxx}-\eta(\mathcal{H}u_x+\mathcal{H}u_{xxx})+u^ku_x=0,
 \quad x \in \mathbb{R}, \; t\geq  0, \; k\in \mathbb{Z}^+,\\
     u(x,0)=u_0(x),
   \end{gathered}
\end{equation}
where $\mathcal{H}$ denotes the Hilbert transform:
\begin{align*}
\mathcal{H}g(x)=\text{ P.V.}
\frac{1}{\pi}\int\frac{g(x-\xi)}{\xi}d\xi,
\end{align*}
$u=u(x,t)$ is a real-valued function and $\eta>0$ is a constant.

Equation (\ref{eqhs}) with $k=1$ was derived by Ostrovsky et al.
in \cite{O:O} to describe the radiational  instability of long
waves in a stratified shear flow.  Recently,  Carvajal and Scialom
in \cite{Cv-Sc} considered the IVP (\ref{eqhs}) and proved the
local well-posedness results for given data  in $H^s$, $s \geq 0$ when
$k=1,2,3$. They also obtained the global well-posedness result for
data  in $L^2$ when $k=1$. The earlier well-posedness results for (\ref{eqhs})
with $k=1$  can be found in \cite{pa:ba1}, where for given data in $H^s(\mathbb{R})$,
local result when $s>1/2$ and global result when $s\geq 1$ have been obtained.

Another model that fits in the class (\ref{eq:hs}) is the derivative
Korteweg-de Vries-Kuramoto Sivashinsky equation
\begin{equation}\label{1eqhs}
   \begin{gathered}
     u_t+u_{xxx}+\eta(u_{xx}+u_{xxxx})+uu_x=0, \quad x \in \mathbb{R}, \;
t\geq 0,\\
     u(x,0)=u_0(x),
   \end{gathered}
\end{equation}
where $u=u(x,t)$ is a real-valued function and $\eta>0$ is a constant.

This equation arises as a model for long waves in a viscous fluid flowing
down an inclined plane and also describes drift waves in a plasma
(see \cite{CKTR, TK}).
The equation (\ref{1eqhs}) is a particular case of Benney-Lin
equation \cite{B,TK}; i.e.,
\begin{equation}\label{2eqhs}
   \begin{gathered}
    u_t+u_{xxx}+\eta(u_{xx}+u_{xxxx})+\beta u_{xxxxx}+uu_x=0, \quad
x \in \mathbb{R},\; t\geq 0,    \\
  u(x,0)=u_0(x),
 \end{gathered}
\end{equation}
when $\beta=0$.

The IVP associated to (\ref{1eqhs}) was studied by Biagioni,
Bona, Iorio and Scialom in \cite{BBIS}.
They also determined the
limiting behavior of solutions as the dissipation tends to zero. Biagioni and
Linares proved global well-posedness for the IVP
(\ref{2eqhs}) for initial data in
$L^2$ in \cite{BL}.

Another example is the Korteweg-de Vries-Burgers equation
\begin{align}\label{M-R}
   \begin{gathered}
    u_t+u_{xxx}-\eta u_{xx}+uu_x=0, \quad x \in \mathbb{R}, \;
t\geq 0,\;\eta>0,\\
  u(x,0)=u_0(x),
 \end{gathered}
\end{align}

Recently, Molinet and Ribaud considered the IVP (\ref{M-R}) in \cite{M-R1}
and proved that it  is locally well-posed for given data in $H^s$, $s>-1$.
The equation (\ref{M-R}) is also
known as the parabolic regularization of the KdV equation with
$\eta>0$. Some years ago, when the interest was to obtain local
results for given data in larger Sobolev spaces, this regularization was
used to obtain  well-posedness results for $\eta >0$ and then pass the
limit $\eta \downarrow 0$. However, this limit is a delicate matter.

Now, we state the main results of this work. The first result deals with the
local well-posedness for given data in the Sobolev spaces of negative index.

\begin{theorem}\label{teorp}
The IVP (\ref{eq:hs}) with $\eta>0$ and $\Phi(\xi)$ given by
(\ref{phi})  is locally well-posed for any data $u_0 \in
H^s(\mathbb{R})$, $s>-3/4$.
\end{theorem}

To prove this theorem we follow  the theory developed by Bourgain
\cite{bou:bou} and Kenig, Ponce and Vega \cite{kpv2:kpv2}. The
main ingredients in the proof  are estimates
in the integral equation associated to an extended IVP that is defined
for all $t\in \mathbb{R}$ (see IVP (\ref{eq:hs2}) below).
The proof we presented here does not use the Bourgain type space
associated to the linear part of the IVP \eqref{eq:hs}; instead it
uses the usual Bourgain space associated to  the KdV equation.
To carry out this scheme, the Proposition \ref{prop3} plays a fundamental
role which permits us to use  a bilinear estimate for $\partial_x(u^2)$
(see \cite{kpv2:kpv2}), that is a central part of our arguments.

The result of the Theorem \ref{teorp} improves the known local
well-posedness results for the IVP (\ref{eqhs}) and (\ref{1eqhs})
described above. Note that, the value $s>-3/4$, in the case  of
the Korteweg-de Vries (KdV) equation, is sharp in the sense that
for $s < -3/4$, the IVP associated to the KdV equation is
ill-posed. We should mention that, the lack of conserved
quantities in the spaces with regularity below $L^2$, prevents us
to get global solution using the usual technique.

The second result is concerned with the particular case of the IVP
(\ref{eq:hs}) for given data in the homogeneous Sobolev space when
the Fourier symbol is of the form $\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$,
$k\in \mathbb{Z}^+$.

\begin{theorem}\label{teorp1}
The IVP (\ref{eq:hs}) with $\eta>0$ and $\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$,
$k\in \mathbb{Z}^+$, is locally well-posed
for any data $u_0 \in \dot{H}^s(\mathbb{R})$, $s>-1/2$.
\end{theorem}

Although this theorem does not improve the result obtained in
Theorem \ref{teorp}, it is interesting on its own   because
the proof we present here uses different tools, that are simpler
than the ones used in the proof of Theorem \ref{teorp}. The main
ingredients in the proof are the refined local smoothing effect (see
(\ref{x32}) in Corollary \ref{corx2} below), and a Strichartz type
estimate (see Proposition \ref{propx0} below). Using these
estimates we are able to apply fixed point argument to obtain a
local well-posedness result in the homogeneous Sobolev spaces of
negative order without the use of Bourgain type spaces.


Now we introduce  function spaces that will  be used for proving
Theorem \ref{teorp}.  We consider the following IVP associated to
the Linear KdV equation
\begin{equation}\label{eq:hs0}
   \begin{gathered}
     u_t+u_{xxx}=0, \quad x, \; t\in \mathbb{R},\\
     u(0)=u_0.
   \end{gathered}
\end{equation}
The solution to  (\ref{eq:hs0}) is given by $u(x,t)=U(t)u_0(x)$,
where the unitary group $U(t)$ is defined as
\begin{align}\label{gU}
\widehat{U(t)u_0}(\xi)=e^{it\xi^3}\widehat{u_0}(\xi).
\end{align}
For $s,b\in \mathbb{R}$, we define the space $X_{s,b}$ as  the
completion of the Schwartz space $S(\mathbb{R}^2)$ with respect to
the norm
\begin{equation}\label{xsb-norm}
 \begin{split}
  \|u\|_{X_{s,b}} \equiv  \|U(-t)u\|_{H_{s,b}}
&:= \|\langle \tau \rangle^{b}
  \langle \xi \rangle^{s}  \widehat{U(-t)u}(\xi,\tau) \|_{L_{\tau}^2L_{\xi}^2} \\
&=\|\langle \tau-\xi^3 \rangle^{b}
  \langle \xi \rangle^{s}  \widehat u(\xi,\tau) \|_{L_{\tau}^2L_{\xi}^2 {,}}
 \end{split}
\end{equation}
where $\widehat u(\xi,\tau)$
is the Fourier transform of $u$ in both space and time variables.
The space $X_{s,b}$ is the usual
Bourgain space for the KdV equation (see \cite{bou:bou}).

 Note that (\ref{eq:hs}) is defined only for $t \ge 0$. To use
Bourgain's type space, we should be able to write the IVP
(\ref{eq:hs}) for all $t \in \mathbb{R}$. For this, we define
\begin{equation}\label{eta}
\eta (t)\equiv \eta \mathop{\rm sgn}(t)= \begin{cases}
 \eta & \text{if } t \geq 0 ,\\
 -\eta & \text{if } t <0
\end{cases}
\end{equation}
and write (\ref{eq:hs}) in the form
\begin{equation}\label{eq:hs2}
   \begin{gathered}
     u_t+u_{xxx}+\eta(t)Lu+uu_x=0, \quad x, \; t \in \mathbb{R} ,\\
     u(0)=u_0.
   \end{gathered}
\end{equation}
Now we consider the IVP associated to the linear part of
(\ref{eq:hs2}),
\begin{equation}\label{eq:hs1}
   \begin{gathered}
     u_t+u_{xxx}+\eta(t)Lu=0, \quad x, \, t\in \mathbb{R},\\
     u(0)=u_0.
   \end{gathered}
\end{equation}
The solution to (\ref{eq:hs1}) is given by $u(x,t)=V(t)u_0(x)$
where the  semigroup $V(t)$ is defined as
\begin{equation}\label{gV}
\widehat{V(t)u_0}(\xi)=e^{it\xi^3+\eta
|t|\Phi(\xi)}\widehat{u_0}(\xi).
\end{equation}

Observe that, defining  $\widetilde{U}(t)$ by
$\widehat{\widetilde{U}(t)u_0}(\xi)=e^{\eta |t|\Phi(\xi)}\widehat{u_0}(\xi)$,
the semigroup $V(t)$ can be written as
$V(t)=U(t)\widetilde{U}(t)$ where $U(t)$ is the unitary group
associated to the KdV equation (see (\ref{gU})).

This paper is organized as follows: In Section 2, we prove Theorem
\ref{teorp}. In Section 3, we present a  refined local smoothing
effect when $\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$, $k \in \mathbb{Z}^+$,
in (\ref{phi}). In Section 4, we to obtain some Strichartz type
estimates. In Section 5, we prove Theorem \ref{teorp1}.


\section{Local Well-posedness in $H^s$ for $s>-3/4$}

This section is devoted to supply the proof of the  Theorem
\ref{teorp}. We start by proving some preliminary results.


\subsection{Preliminary estimates}

\begin{proposition}\label{prop4}
Let $s>-3/4$. There exist $b' \in (-\frac{1}{2},0)$ and
$\epsilon_s >0$ such that for any $b \in(\frac{1}{2}, b'+1]$ with
$1-b+b'\le \epsilon_s$, and $u, v \in X_{s,b}$
\begin{equation*}
 \|(uv)_x\|_{ X_{s,b'}} \leq c \|u\|_{
 X_{s,b}} \,\|v\|_{ X_{s,b}}.
\end{equation*}
\end{proposition}

The proof of the above proposition can be found in
\cite{kpv2:kpv2}.

 We consider a cut-off function $\psi  \in C^{\infty}(\mathbb{R})$,
such that $0\leq \psi(t) \leq 1$,
\begin{equation}\label{psi}
 \psi(t)= \begin{cases}
 1 & \text{if \,$ |t|  \leq 1$},\\
 0 & \text{if \,$ |t| \ge 2$}.
\end{cases}
\end{equation}
Let us define $\psi_{T}(t)\equiv\psi(\frac{t}{T})$ and
$\tilde{\psi}_{T}(t)= \mathop{sgn}(t)\psi_{T}(t)$.

The following Proposition  plays a central role in  the proof of
our first main result,  Theorem \ref{teorp}. This Proposition allows
us to work in the usual  $X_{s,b}$ space associated to the KdV
equation instead of the Bourgain space associated to the IVP
(\ref{eq:hs2}).

\begin{proposition}\label{prop3}
Let $-1/2<b' \le 0$, $T \in [0,1]$. Then we have
\begin{equation}\label{eq1}
  \|\psi(t)V(t)u_{0}\|_{ X_{s,b}} \le  c
  \|u_{0}\|_s.
\end{equation}
If $1/2 < b \le b'/3+2/3$, $s\in \mathbb{R}$ then
\begin{equation}\label{eq2}
  \| \psi_{T}(t)\int_{0}^{t}V(t-t')(uu_x)(t')dt'\|_{ X_{s,b}}
  \leq c  T^{1+b'/2-3b/2}
  \|uu_x\|_{ X_{s,b'}},
\end{equation}
where $c$ is a constant.
\end{proposition}

Before proving this proposition, we record the following results.

\begin{lemma}\label{lpsi}
Let $0<T\leq 1$, $1/2<b<1$ and $a<1$. Then we have
\begin{gather}\label{ineq1}
\|\psi_{T}(t)\|_{H_t^{b}} \le  c (T^{1/2}+T^{1/2-b}), \\
\label{ineq2} \|\psi_{T}(t)\,e^{a|t|}\|_{H_t^{b}} \le
 T^{1/2} \langle \frac{1}{T^b}\rangle. \\
\label{ineq3}
|\mathcal{F}(|t|\,\psi_{T}(t)\,e^{a|t|})(\tau)| \le
\frac{c T^{2}}{1+(\tau^2+a^2)T^2}, \\
\label{ineq4}
|\mathcal{F}(|t|\,\tilde{\psi}_{T}(t)\,e^{a|t|})(\tau)| \le
\frac{c T^{2}}{1+(\tau^2+a^2)T^2},
\end{gather}
where $c$ is a constant independent of $T$ and $a$.
\end{lemma}

\begin{proof}
Using the definition of the space $H^b$, we have
\begin{align*} \|\psi_{T}(t)\|_{H_t^{b}} \le c
 \|\psi_{ T}\|_{L^2}+c \|D_t^{b}\psi_{T}\|_{L^2}=
cT^{1/2}\|\psi\|_{L^2}+cT^{1/2-b}\|D_t^{b}\psi\|_{L^2},
\end{align*}
where we used the fact that $\widehat{h(t/T)}(\tau)= T\hat{h} (T\, \tau)$.
To prove \eqref{ineq2}, we exploit the compact support of $\psi_T$ to get
\[
\|\psi_{T}(t)\,e^{a|t|}\|_{H_t^{b}}
\le c T^{1/2}\|h\|_{L^{2}}+c\|D_t^b
(\psi_{T}(t)\,e^{a|t|})\|_{L^{2}}
\le c  (T^{1/2} +T^{1/2-b}\|D_t^b h\|_{L^{2}}),
\]
where $h(t)=\psi(t)e^{aT|t|}$. Integrating by parts twice we obtain
\[
|\hat{h}(\tau)| \le \frac{c}{\langle \tau
\rangle^2},
\]
where, in the case when $a<0$, we have used the fact that
$|x|e^{x}\leq e, \;\forall\,x\leq 1$, in particular for $ x=aT$.
This proves inequality (\ref{ineq2}).

 To prove  inequality (\ref{ineq3}) we have
\begin{equation}\label{eq.8}
\mathcal{F}(|t|\,\psi_{T}(t)\,e^{a|t|})(\tau)= T^2\hat{p}(T\tau)
\end{equation}
where $p(t)=|t|\psi(t)e^{aT|t|}$.
Integrating by parts, we obtain
\[
|\hat{p}(\tau)| \le \frac{c}{|\tau-iaT|^{k}}, \quad k=0,1,2.
\]
Therefore,
\begin{equation}\label{eq.9}
|\hat{p}(\tau)| \le \frac{c}{1+ \tau^2+a^2 T^2}.
\end{equation}
Combining \eqref{eq.8} and \eqref{eq.9} yields the desired
inequality (\ref{ineq3}). The proof of (\ref{ineq4}) is similar.
\end{proof}

\begin{remark} \rm It's not possible to obtain similar inequalities
 as (\ref{ineq1}) and (\ref{ineq2}) for $\tilde{\psi}_T(t)$
because of the discontinuity.
\end{remark}

In the following estimates, without loss of generality, we suppose that $\eta=1$.

\begin{lemma}\label{prop2}
Let $-1/2<b' \leq 0$, $1/2< b \leq b'/3+2/3$, $T \in (0,1]$,
$a<1$. Then
\begin{align}
  \| \psi_{T}(t)\int_{0}^{t}e^{|t-t'|a}f(t')dt'\|_{H_t^{b}}
  \leq & \, c \, T^{1+b'/2-3b/2}
  \|f\|_{H^{b'}}, \label{eq0}
\end{align}
where $c$ is a constant independent of $a$, $f$ and $T$.
\end{lemma}

\begin{proof}
It is sufficient to prove  Lemma \ref{prop2} when $|a|\le 1$. In fact, let us
suppose that Lemma \ref{prop2} has been established in the case
$|a|\le 1$. Then when $a<-1$, we use the change of variable $t'a\equiv t'$,
to obtain
\begin{align}\label{IJ}
\psi_{T}(t)I_a(t):=\psi_{T}(t)\int_0^t e^{|t-t'|a}f(t')dt'=
\frac{1}{a}\psi_{aT}(at)\int_0^{at}
e^{|at-t'|}f_a(t')dt'=\frac{1}{a}J(at).
\end{align}
where $f_a(t')=f(t'/a)$ and $J(t)=\psi_{aT}(t)\int_0^{t}
e^{|t-t'|}f_a(t')dt'$.
Note that for $a<-1$,
\begin{align}\label{IJ1}
\|J(t)\|_{H^b}\le c  |aT|^{1+b'/2-3b/2}\|f_a\|_{H^{b'}}\le c
|a|^{3/2-b'/2-3b/2} T^{1+b'/2-3b/2}\|f\|_{H^{b'}}.
\end{align}
Since $b'+b>0$ and $|a|>1$, from (\ref{IJ}) and (\ref{IJ1}) we
obtain
\begin{align*}
\|\psi_{T}(t)I_a(t)\|_{H^b}
&= \frac{1}{|a|}\|J(at)\|_{H^b} \\
&\le c \frac{\langle a \rangle^b}{|a|^{3/2}}\|J(t)\|_{H^b}\\
&\le \frac{c}{|a|^{(b'+b)/2}}T^{1+b'/2-3b/2}\|f\|_{H^{b'}}.
\end{align*}
Hence, we arrived at (\ref{eq0}) in this case too.

 From here onwards, we consider $|a|\leq 1$.
Now let $b>1/2$, then we have
 \begin{align*}
I_a(t)&:=  \int_0^t e^{|t-t'|a}f(t')dt'
=\int_0^t e^{(t-t')\mathop{sgn}(t)a}f(t')dt'\\
&=  e^{a|t|}\int_0^te^{-\mathop{sgn}(t)at'}
\int_{\mathbb{R}}e^{it'\tau}\hat f(\tau)d\tau dt'\\
&=  e^{a|t|}\int_{\mathbb{R}}\hat f(\tau)\int_0^te^{(i\tau-\mathop{sgn}(t)a)t'}dt'd\tau \\
&=  e^{a|t|}\int_{\mathbb{R}}\hat f(\tau)\frac{e^{(i\tau-\mathop{sgn}(t)a)t} -1}{i\tau-\mathop{sgn}(t)a}d\tau \\
&= \int_{\mathbb{R}}\hat f(\tau)\frac{e^{i\tau t}
-e^{a|t|}}{i\tau-\mathop{sgn}(t)a}d\tau.
\end{align*}
We have
$$
\frac1{\mathop{sgn}(t)a-i\tau}=\mathop{sgn}(t)\frac{a}{a^2+{\tau}^2}
+i\frac{\tau}{a^2+{\tau}^2}.
$$
If we define
$$
p_a(t) = \frac{a}{a^2+t^2}, \quad  q_a(t) = \frac{t}{a^2+t^2}
$$
and replace $\tau$ by $t'$ we obtain
\begin{equation}\label{I1.1}
\begin{split}
 I_a(t)& =  \mathop{sgn}(t)\int_{\mathbb{R}}p_a(t')
 \big[e^{a|t|}-e^{it't}\big]\widehat{f}(t')dt'
+ic\int_{\mathbb{R}}q_a(t')\big[e^{a|t|}-e^{it't}\big]\widehat{f}(t')dt'\\
&:=I_{a,1}(t)+I_{a,2}(t).
\end{split}
\end{equation}

\subsection*{Estimate for $I_{a,1}$}
 We consider two cases.

\noindent {\bf Case 1: $|t'|>1/T$.} Let $\widehat{f}(t')\equiv
\widehat{f}(t') \chi_{\{|t'|>1/T\}}$. From the definition of
$I_{a,1}$ we have
\begin{equation}\label{H0}
\psi_{T}(t)I_{a,1}(t)
= a\mathop{\rm sgn}(t)\psi_T(t)
\int_{\mathbb{R}}\frac{\widehat{f}(t')}{a^2+ {t'}^2}\big[e^{a|t|}
-e^{itt'}\big]dt'
=  ah\Big(\frac{t}{T}\Big),
\end{equation}
where $h(t)=\mathop{sgn}(t)\psi(t)
\int_{\mathbb{R}}\{\widehat{f}(t')/(a^2+
{t'}^2)\}\big[e^{aT|t|}-e^{iTtt'}\big]dt'$.
We have
\begin{align}\label{H}
\widehat{h(t)}(\tau)= \int_{\mathbb{R}}\frac{\widehat{f}(t')}{a^2+
{t'}^2}K(a,T,\tau,t')dt',
\end{align}
where
\begin{align*}
K(a,T,\tau,t')=\int_{\mathbb{R}}\mathop{sgn}(t)\psi(t)\big[e^{aT|t|}
-e^{iTtt'}\big]e^{-it\tau}dt.
\end{align*}
Integrating by parts,
\begin{align*}
|K(a,T,\tau,t')|\le c \frac{\langle t'\rangle}{\langle \tau
\rangle}, \quad \text{and} \quad |K(a,T,\tau,t')|\le c
\frac{\langle t'\rangle}{\langle \tau \rangle^2}+ c\frac{\langle
t'\rangle^2}{\langle \tau \rangle^2}\le c \frac{\langle
t'\rangle^2}{\langle \tau \rangle^2}.
\end{align*}
Hence
\begin{align*}
|K(a,T,\tau,t')|\le c \frac{\langle t'\rangle^{2b}}{\langle \tau
\rangle^{2b}}.
\end{align*}
Therefore, from (\ref{H}) we obtain
\begin{align*}
|\widehat{h(t)}|(\tau) \le \frac{c}{\langle \tau
\rangle^{2b}} \int_{|t'|>1/T}\frac{|\widehat{f}(t')|}{a^2+
{t'}^2}\langle t'\rangle^{2b}dt'\le c \frac{T^{3/2+b'-2b}}{\langle
\tau \rangle^{2b}}\|f\|_{H^{b'}.}
\end{align*}
Now, using (\ref{H0}) we have
\begin{align*}
\|\psi_{T}(t)I_a(t)\|_{H^b}=
|a|\,\|h\Big(\frac{t}{T}\Big)\|_{H^b}\le
|a|T^{1/2-b}\|h(t)\|_{H^b} \le
cT^{3/2+b'-2b}T^{1/2-b}\|f\|_{H^{b'}.}
\end{align*}
Hence
\begin{align*}
\|\psi_{T}(t)I_a(t)\|_{H^b}\le cT^{2+b'-3b}\|f\|_{H^{b'}}\le
cT^{1+b'/2-3b/2}\|f\|_{H^{b'}.}
\end{align*}

\noindent {\bf Case 2: $|t'|\leq 1/T$. }
Let $\widehat{f}(t')\equiv
\widehat{f}(t') \chi_{\{|t'| \leq 1/T\}}$ and as earlier
$\tilde{\psi}_{T}(t)=\mathop{sgn}(t)\psi_{T}(t)$. We have
\begin{align*}
\mathcal{F}(\psi_{T}(t)
I_{a,1}(t))(\tau)&= \int_{\mathbb{R}}e^{-it\tau}\tilde{\psi}_{T}(t)\int_{\mathbb{R}}p_a(t')\big[e^{a|t|}-e^{it't}\big]\hat f(t')dt' dt\\
&= \int_{\mathbb{R}}p_a(t')\hat f(t')\int_{\mathbb{R}}\tilde{\psi}_{T}(t)e^{-it\tau}\big[e^{a|t|}-e^{it't}\big]dtdt'\\
&= \int_{\mathbb{R}}p_a(t')\widehat{f}(t')\{\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau)-\mathcal{F}\big(\tilde{\psi}_{T}(t)\big)(\tau-t')\}dt'\\
&= \int_{\mathbb{R}}p_a(t')\widehat{f}(t')\{\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau)-\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau-t')\}dt'\\
&\quad
+ \int_{\mathbb{R}}p_a(t')\widehat{f}(t')\{\mathcal{F}\big(\tilde{\psi}_{T}(t)\,e^{a|t|}\big)(\tau-t')-\mathcal{F}\big(\tilde{\psi}_{T}(t)\big)(\tau-t')\}dt'\\
&:=I_{a,11}(\tau)+I_{a,12}(\tau).
\end{align*}
Since $|p_a(t')|\le 1/ |t'|$, we can estimate the
term $I_{a,11}(\tau)$ as in \cite{gtv:gtv}. Therefore we will
estimate only the term $I_{a,12}(\tau)$.

Let us define $h(t',\tau):=
\mathcal{F}\big(\tilde{\psi}_{T}(t)[e^{a|t|}-1]\big)(\tau-t')$,
then we have
\begin{align}\label{ht}
h(t',\tau)=\int_0^a \mathcal{F}\big(|t|\,\tilde{\psi}_{
T}(t)\,e^{s|t|}\big)(\tau-t')ds.
\end{align}
 From (\ref{ineq4}) we have that
\begin{align}\label{eq3}
|\mathcal{F}\big(|t|\,\tilde{\psi}_{T}(t)\,e^{s|t|}\big)(\tau-t')| \le
\frac{c T^{2}}{(1+(|\tau-t'|+|s|)T)^{2}},
\end{align}
where $c$ is independent of $s$, $\tau$, $t'$ and $T$.

 Observe that $0\le s\le a$ if $a\geq 0$  and $a \le s \le 0$ if $a \le
0$. Thus we obtain
\begin{align*}
|h(t',\tau)|\leq & c T^{2}\int_0^{|a|}\frac{1}{(1+(|\tau-t'|+|s|)T)^{2}} ds\\
&=c T^{2}\frac{|a|}{(1+|\tau-t'|T)(1+|\tau-t'|T+ |a|T)}.
\end{align*}
 As
$|Tt'|\le 1$, we have
\begin{align*}
\frac{1}{1+|\tau-t'|T}\le \frac{2}{1+|\tau|T}.
\end{align*}
Hence
\begin{align*}
|h(t',\tau)|\leq  c T^{2}\frac{|a|}{(1+|\tau|T)^2}.
\end{align*}
Using the H\"older's inequality we obtain
\begin{equation}\label{I2}
\begin{split}
|I_{a,12}(\tau)| &=|\int_{\mathbb{R}}p_a(t')\hat
f(t')h(t',\tau)dt'|\\
&\le \|f\|_{H^{b'}}\,\Big(\int_{|t'|
\le 1/T}\frac{\langle
t' \rangle^{-2b'}|h(t',\tau)|^2}{(a^2+{t'}^2)^2}dt'\Big)^{1/2}\\
& \le \frac{c T^{2}}{(1+|\tau|T)^2}\, \|f\|_{H^{b'}}\Big(\int_{|t'| \le
1/T}(1+|t'|^{-2b'})dt'\Big)^{1/2}\\
& \le \frac{c T^{3/2+b'}}{(1+|\tau|T)^2}\, \|f\|_{H^{b'}.}
\end{split}
\end{equation}
Finally, we arrive at
\begin{align*}
\Big(\int_{\mathbb{R}}(1+|\tau|)^{2b}|I_{a,12}(\tau)|^2 \,d\tau \Big)^{1/2}
&\leq c  T^{3/2+b'}\,\|f\|_{H^{b'}}\Big(\int_{\mathbb{R}}
 \frac{1+|\tau|^{2b}}{(1+|\tau|T)^4}d\tau\Big)^{1/2} \\
&\leq  c T^{3/2+b'}\,\|f\|_{H^{b'}}\Big(\frac{1}{T^{1/2}}
 +\frac{1}{T^{b+1/2}}\Big) \\
&\leq  c T^{1-b+b'}\,\|f\|_{H^{b'}.}
\end{align*}
Therefore, in this case we have
\[
\|\psi_T I_{a,1}\|_{H^b}\le c T^{1-b+b'}\,\|f\|_{H^{b'}}.
\]

\subsection*{Estimate for $I_{a,2}$}
 The estimate for $I_{a,2}$ is
similar to  that of $I_{a,1}$, exchanging $p_a$ by $q_a$ and
$\tilde{\psi}_T(t)$ by $\psi_T(t)$. So, we omit its calculation.
\end{proof}

In the following remark we present improvement of the estimate obtained
in the Lemma \ref{prop2} in some particular cases. Although, this
improvement does not help to improve our main result, it will be of
interest on its own.

\begin{remark} \rm (1) The proof in the case $|t'|\le 1/T$ is valid for all
$a <1$.

\noindent (2) We know that
\[
\widehat{\mathcal{H}g}(\eta)=-i \mathop{\rm
sgn}(\eta)\,\widehat{g}(\eta)\quad \text{and}\quad
\widehat{q_a}(t)=-i \mathop{\rm sgn}(t)\,e^{a|t|}.
\]
Thus
\begin{align*}
I_{a,2}(t)
&=-\tilde{\psi_T}(t)\widehat{q_a}(t)\int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt'
+ \sqrt{2\pi}\tilde{\psi}_T(t)\mathcal{F}^{-1}\mathcal{H}(q_a\widehat{f}\,)(t)\\
&= \tilde{\psi}_T(t)\mathcal{F}^{-1}(q_a)(t)\int_{\mathbb{R}}q_a(t')\widehat{f}
(t')dt'+ \sqrt{2\pi}\tilde{\psi}_T(t)\mathcal{F}^{-1}\mathcal{H}(q_a\widehat{f}
 \,)(t),
\end{align*}
where $\tilde{\psi_T}(t)=\mathop{sgn}(t)\psi_T(t)$. Consequently,
\begin{align*}
\widehat{I_{a,2}}(\tau)&=\widehat{\tilde{\psi}_T}\star q_a(\tau)
\int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt'+\sqrt{2\pi}
\widehat{\tilde{\psi}_T}\star \mathcal{H}(q_a\widehat{f}\,)(\tau)\\
&= \widehat{\tilde{\psi}_T}\star q_a(\tau)\int_{\mathbb{R}}q_a(t')
\widehat{f}(t')dt'+\sqrt{2\pi}\mathcal{H}(\widehat{\tilde{\psi}_T})
\star (q_a\widehat{f}\,)(\tau).
\end{align*}
Similarly,
\begin{align*}
\widehat{I_{a,1}}(\tau)=\frac{1}{i} \widehat{\psi_T}\star q_a(\tau)
\int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt'
+\frac{\sqrt{2\pi}}{i}\mathcal{H}(\widehat{\psi_T})
\star (q_a\widehat{f}\,)(\tau).
\end{align*}

\noindent (3) If $1/2<b < b'/3+2/3$, then
$1-b+b'> 3/4+b'/2-b> 1+b'/2-3b/2>0$.

\noindent (4) If $|a|>1$, then $|q_a(t')| \le  c/\langle t'\rangle$, hence
\begin{align*}
\int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt' \le c \int_{\mathbb{R}}
\frac{|\widehat{f}(t')|\, \langle t'\rangle^{1-b}}{\langle
t'\rangle\, \langle t'\rangle^{1-b}}dt' \le c \|f\|_{H^{b-1}} \,
\Big\| \frac{1}{\langle t'\rangle^{b}}\Big\|_{ L^2},
\end{align*}
and therefore we obtain a more refined estimate than (\ref{ga}).

\noindent(5) In the case
 $|t'|>1/T$ we can to obtain a better estimate for $I_{a,2}$ because $\psi_T$
is regular (using the inequalities (\ref{ineq1}) and (\ref{ineq2})).
In fact,
let $\widehat{f}(t')\equiv \widehat{f}(t')
\chi_{\{|t'|>1/T\}}$. We have that
\[
\| \psi_{T}(t)I_{a,2}(t)\|_{H_t^{b}}
\le \|\psi_{T}(t)e^{|t|a}\|_{H_t^{b}}
\Big| \int_{\mathbb{R}}q_a(t')\widehat{f}(t')dt'\Big|+
\|\psi_{T}(t)\|_{H_t^{b}}
\|\mathcal{F}^{-1}(q_a\widehat{f}\,)(t)\|_{H_t^{b}}.
\]
Since $|t'|>1/T$ implies  $|t'| \simeq \langle t'\rangle$, using the
Cauchy-Schwartz inequality we obtain
\begin{equation}\label{ga}
 \begin{split}
\int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt'
&\leq  \int_{|t'|>1/T}\frac{|\widehat{f}(t')|}{|t'|}dt'\\
\lesssim&\int_{|t'|>1/T} \frac{|\widehat{f}(t')|\, |\, t'|^{-b'}}{
|\,t'|\, |\,t'|^{-b'}}dt'  \\
&\leq  \|f\|_{H^{b'}} \, \Big( \int_{|t'|>1/T}
\frac{dt'}{|t'|^{2(1+b')}}\Big)^{1/2} \lesssim T^{1/2+b'}\,
\|f\|_{H^{b'}.}
\end{split}
\end{equation}
Similarly,
\begin{equation}
\begin{split}
\int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt'
&\leq  c \|f\|_{H^{b'}} \, \Big( \int_{|t'|>1/T}
\frac{dt'}{|t'|^{2b'}(a^2+{t'}^2)}\Big)^{1/2}\\
&\leq  c
\frac{1}{|a|^{1/2+b'}}\,\Big(\int_{\mathbb{R}}
\frac{dt'}{|t'|^{2b'}(1+{t'}^2)}\Big)^{1/2} \|f\|_{H^{b'}} \\
&\leq  c  \frac{1}{|a|^{1/2+b'}}\, \|f\|_{H^{b'}.}
\label{ga1}
\end{split}
\end{equation}
Hence from (\ref{ga}) and (\ref{ga1}) we obtain
\begin{align}\label{ga2}
\int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt'  \le
\,c \Big(\frac{T}{|a |}\Big)^{1/4+b'/2}  \,
\|f\|_{H^{b'}\text{,}}
\end{align}
and
\begin{align*}
\|\mathcal{F}^{-1}(q_a\widehat{f}\,)(t)\|_{ H_t^{b}}^2
\lesssim & \int_{|t'|>1/T}\frac{|\widehat{f}(t')|^2}{\langle
t'\rangle^{2(1-b)}}dt' \\
\lesssim & \int_{|t'|>1/T}\frac{|\widehat{f}(t')|^2}{\langle
t'\rangle^{-2b'}|t'|^{2(1-b+b')}}dt'\\
&\leq  T^{2(1-b+b')}\|f\|_{H^{b'}}^2.
\end{align*}
So from inequality (\ref{ineq1}) we have
\begin{align*}
\|\psi_{T}(t)\|_{H_t^{b}}\|\mathcal{F}^{-1}(q_a\widehat{f}\,)(t)\|_{
H_t^{b}} \le c T^{1/2-b}\,T^{(1-b+b')}\|f\|_{H^{b'}}\le
c T^{3/2-2b+b'}\|f\|_{H^{b'}.}
\end{align*}
On the other hand, if $a<-1$ from inequalities (\ref{ineq2}) and
(\ref{ga2}) we obtain
\begin{align*}
&\|\psi_{T}(t)\,e^{a|t|}\|_{H_t^{b}}\int_{\mathbb{R}}|q_a(t')
\widehat{f}(t')|dt' \\
&\leq  c  \Big(\frac{T}{|a|}\Big)^{1/2}\Big(1+
\frac{1}{T^{b}}\Big)(1+|a|^b)\Big(\frac{T}{|a|}\Big)^{1/4+b'/2}\|f\|_{H^{b'}}\\
&\leq  c  \Big(\frac{T}{|a|}\Big)^{3/4+b'/2}\Big(1+
\frac{1}{T^{b}}\Big)(1+|a|^b)\|f\|_{H^{b'}}\\
&\leq
c T^{3/4+b'/2-b}\Big(1+\frac{1}{|a|^{3/4+b'/2-b}}\Big)\|f\|_{H^{b'}}\\
&\leq  c T^{3/4+b'/2-b}\|f\|_{H^{b'}}.
\end{align*}
Now if $|a|<1$ using the inequality (\ref{ineq2}) we obtain:
\begin{align*}
\|\psi_{T}(t)\,e^{a|t|}\|_{H_t^{b}}\int_{\mathbb{R}}|q_a(t')\widehat{f}(t')|dt'
&\leq c  T^{1/2+b'}\|f\|_{H^{b'}}T^{1/2}\Big(1+
\frac{1}{|T|^b}\Big)\\
&\leq c
T^{1/2+b'}\,T^{1/2-b}\|f\|_{H^{b'}}\\
&\leq c  T^{1-b+b'}\|f\|_{H^{b'}}\\
&\leq  c T^{3/4+b'/2-b}\|f\|_{H^{b'}}.
\end{align*}
Therefore, in this case
\begin{align*}
 \|
 \psi_{T}(t)I_a(t)\|_{H_t^{b}}\le
 c  T^{3/4+b'/2-b}\, \|f\|_{H^{b'}},
\end{align*}
where $c$ is a constant independent of $a$, $f$ and $T$.
\end{remark}

Now we prove Proposition \ref{prop3} which plays a crucial role in the proof of the first main result of this work.

\begin{proof}[Proof of Proposition \ref{prop3}]
To prove (\ref{eq1}), we use the estimate (\ref{ineq2}) with $T=1$ and
$a= \Phi(\xi)<1$, to obtain
\begin{align*}
\|\psi(t)\,e^{\eta \Phi(\xi)|t|}\|_{H_t^{b}}^2 \le c(\eta),
\end{align*}
where $c(\eta)$ is a constant. Therefore,
\begin{align*}
&\|\psi_{T}(t)V(t)u_{0}\|_{ X_{s,b}} \le c(\eta)\Big(
\int_{\mathbb{R}}\langle \xi\rangle^{2s}|\widehat{u_0}(\xi)|^2
d\xi \Big)^{1/2} \le c(\eta)\|u_0\|_{H^s.}
\end{align*}
Now, we move to prove  (\ref{eq2}). From definition \eqref{xsb-norm} of the $X_{s, b}$ norm, we have
\begin{equation}\label{x2.2}
\begin{split}
 \|\psi_T(t)\int_0^tV(t&-t') (uu_x)(t') dt'\|_{X_{s, b}} \\
&= \|U(-t)\psi_T(t)\int_0^tV(t-t') (uu_x)(t') dt'\|_{H_{s, b}}\\
&= \|\langle \tau\rangle^b\langle\xi\rangle^s
 \mathcal{F}_{\xi\tau}{\big[\psi_T(t)\int_0^tU(-t')\tilde U(t-t')
 (uu_x)(t') dt'\big]}\|_{L_{\tau}^2L_{\xi}^2}\\
&= \|\langle\xi\rangle^s\|\psi_T(t)\int_0^t e^{-it'\xi^3} e^{|t-t'|
 \Phi(\xi)}\widehat{uu_x}(t', \xi) dt'\|_{H_t^b}\|_{L_{\xi}^2}.
\end{split}
\end{equation}

If we fix  the variable $\xi$ and suppose
$f_\xi(t')=e^{-it' \xi^3} \widehat{uu_x}(t', \xi)$ the
estimate \eqref{eq2} follows from \eqref{x2.2} using  (\ref{eq0}).
\end{proof}


\subsection{Proof of the Theorem \ref{teorp}}

\begin{proof}
As discussed in the introduction, we will use Bourgain's space associated
to the KdV group to prove well-posedness of (\ref{eq:hs});
therefore we need to consider that (\ref{eq:hs2})  is defined for all $t$.
Now consider  (\ref{eq:hs2}) in its equivalent integral form
\begin{equation}\label{int1}
u(t)=V(t)u_{0}- \int_{0}^{t}V(t-t')(uu_x)(t')dt',
\end{equation}
where
$V(t)$ is the semigroup associated with the linear part given by (\ref{gV}).
Note that, if for all $t\in \mathbb{R}$, $u(t)$ satisfies
\[
u(t)=\psi(t)V(t)u_{0}- \psi_{T}(t)
\int_{0}^{t}V(t-t')(uu_x)(t')dt',
\]
then $u(t)$ satisfies (\ref{int1}) in $[-T,T]$.
We define an application
\[
  \Psi(u)(t)= \psi(t)\,V(t)u_0-\psi_{T}(t)
  \int_0^t V(t-t')(uu_x)(t')dt'.
\]
Let $s>-3/4$, and $u_0\in H^s$.  Let $b$ and $b'$  be two numbers given
by Proposition \ref{prop4}, such that
 $\theta\equiv \min\{1+b'/2-3b/2,\, 3/4+s/3-b\}>0$.  For $M>0$, let us
define a ball
\begin{align*}
  X_{s,b}^M= \{u\in X_{s,b} :   \|u\|_{X_{s,b}}\leq M \}.
\end{align*}
We will prove that there exists $M$ such that the application $\Psi$ maps
$X_{s,b}^M$ into $X_{s,b}^M$ and is a contraction.
Let $u\in X_{s,b}^M$. Then using Propositions \ref{prop4}, \ref{prop3} and the
definition of $X_{s,b}^M$ we obtain
\[
  \|\Psi(u)\|_{X_{s,b}}\leq  c\|u_0\|_s+c T^\theta
  \|(uu_x)\|_{X_{s,b'}}
  \leq  \frac{M}{4}+ cT^\theta M^2\leq  \frac{M}{2},
\]
where we have chosen  $M=4c\|u_0\|_{H^s}$ and $0<T<1$ such that
$cT^\theta M=1/4$.
Therefore, $\Psi$ maps $X_{s,b}^M$ into itself. With a
similar argument we can prove that $\Psi$ is a contraction.
Hence $\Psi$ has a unique fixed point $u$ which is a  solution to
(\ref{eq:hs}) such that $u \in C([-T,T], H^{s})$.

The rest of the proof follows in an analogous way to \cite{kpv2:kpv2},
so we omit the details.
\end{proof}

\section{A refined local smoothing effect}

In this section we prove the following local smoothing effect for
the semigroup $V_k(t)$ defined by (\ref{gV}) with
$\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$.
Similar results can also be obtained for more general $\Phi$ as
in (\ref{phi}). Our proof follows the ideas of
\cite{Cv-Sc}.

\begin{theorem}\label{teox2}
Let $T >0$, $u_0 \in L^q$, $0\le s < (k+3-p)/p+1/p_1$ and $p\ge 2$, $p_1 \ge 2$,
then
\begin{align}\label{x5}
\|D_x^s V_k(t)u_0\|_{L_T^p L_x^{p_1}}
\leq \frac{c(\eta)}{(k+3-p(s+1)+p/p_1)^{1/p}}\,(T^{1/p}e^{2\eta T}
+T^{\epsilon})\, \|u_0\|_{L^q},
\end{align}
where $\epsilon=\epsilon(p,k,s,p_1)=(k+3-p(s+1))/(k+2)+p/((k+1)p_1)$ and
$1/p+1/q=1$.
\end{theorem}

\begin{corollary}\label{teox20}
Let $u_0 \in L^q$, $T >0$, $2\le p< k+3$, and $0\le s < (k+3-p)/p$,
then
\begin{equation}\label{x50}
\|D_x^s V_k(t)u_0\|_{L_T^p L_x^\infty}
\leq \frac{c(\eta)}{(k+3-p(s+1))^{1/p}}\,(T^{1/p}e^{2\eta T}+T^{\epsilon_0})\,
 \|u_0\|_{L^q},
\end{equation}
where $\epsilon_0=\epsilon(p,k,s)=(k+3-p(s+1))/(k+2)$ and $1/p+1/q=1$.
\end{corollary}

In particular, the case when $p=2$ is interesting, which is stated as follows.

\begin{corollary}\label{corx2}
{\bf (1)} If $u_0 \in L^{2}$, $p_1 \ge 2$, $0\le s< 1+ (k-1)/2+1/p_1$,
$0< T < 1$ and $\gamma= \min\{1/2, \epsilon(2,k,s,p_1)\}$ then
\begin{equation}\label{x33}
\|D_x^s V_k(t)u_0\|_{L_T^2 L_x^{p_1}} \leq
\frac{c(\eta)\,T^\gamma}{(1+(k-1)/2+1/p_1-s)^{1/2}}
\|u_0\|_{L^2}.
\end{equation}

\noindent{\bf (2)}
If $u_0 \in \dot{H}^{s}$, $-k/2<s \le 0$, $0< T < 1$ and
$\gamma= \min\{1/2, \epsilon(2,k,1-s,p_1)\}$ then
\begin{equation}\label{x32}
\|D_x V_k(t)u_0\|_{L_T^2 L_x^{p_1}} \leq
\frac{c(\eta)\,T^\gamma}{((k-1)/2+1/p_1+s)^{1/2}}
\|D^{s}u_0\|_{L^2\text{,}}
\end{equation}
in the following cases:

(i) when $-(k-1)/2\le s \le 0$ and $2\le p_1$.

(ii) when $-k/2<s <-(k-1)/2$ and $2 \le p_1 \le (-s-(k-1)/2)^{-1}$.
\end{corollary}

In the proof of Theorem \ref{teox2} we will use the following result.

\begin{proposition}\label{Px6}
Let $p \geq 2$, and $1/p+1/q=1$, then
\begin{equation}\label{x6}
\|\hat{u}\|_{L^p} \leq c \|u\|_{L^q},
\end{equation}
\end{proposition}

The proof of the above  corollary can be found in \cite[Corollary 1.43]{JD}.


\begin{proof}[Proof of Theorem \ref{teox2}]
We can assume that $u_0 \in S(\mathbb{R})$. We consider a cut-off function
$\varphi \in C(\mathbb{R}\setminus\{0\})$, $0\le \varphi \le 1$ defined by
\begin{equation}\label{psi1}
 \varphi(t)= \begin{cases}
 1& \text{if } 0 \leq t  \leq 1,\\
 0& \text{if $t < 0$  or $t \geq 2$}.
\end{cases}
\end{equation}
Let us define $\varphi_{ T}(t)\equiv\varphi(\frac{t}{T})$, then
\[
\|D_{x}^s V_k(t)u_{0}(x) \|_{L_T^{p}L_x^{p_1}} \leq
\|\varphi_{T}(t)D_{x}^s V_k(t)u_{0}(x) \|_{L_{t}^{p}L_x^{p_1}}.
\]
Let $1/p_1+1/q_1=1$, using duality it is enough to prove for $u_0$ in $L^q$
and $g$ in $L_t^q L_x^{q_1}$
\[
J\equiv\Big|\int_{\mathbb{R}^2}\varphi_{T}(t)D_x^sV_k(t)u_{0}(x)g(x,t)dx\,dt
\Big| \leq c \|u_0\|_{L^q} \|g\|_{L_t^q L_x^{q_1}}.
\]
Using (\ref{gV}), we have
\begin{align*}
D_{x}^sV_k(t)u_{0}(x)=i\int_{\mathbb{R}}\, |\xi|^s
\,e^{it\xi^3+\eta t\Phi(\xi)+i x \xi} \hat{u}_0(\xi)d\xi.
\end{align*}
Therefore, by Fubini's theorem, Proposition \ref{Px6} and H\"older's
inequality we obtain
\begin{align*}
J \le c \|u_0\|_{L^q} \|Lg\|_{L^q},
\end{align*}
where $L g(\xi)$ is defined by
\[
Lg(\xi)\equiv  |\xi|^s \big|\int_{\mathbb{R}^2}\varphi_{T}(t)
g(x,t)e^{it\xi^3+\eta t\Phi(\xi)+i x \xi}
dx\, dt\big|
\]
and
\begin{equation}
|Lg(\xi)|
\le |\xi|^s \int_{\mathbb{R}}\varphi_{T}(t)
e^{\eta t\Phi(\xi)}|\mathcal{F}^{-1}(g(\cdot,t))(\xi)|dt. \label{tg1}
\end{equation}
We have
\begin{equation}\label{tg2}
\|Lg(\xi)\|_{L^q(\mathbb{R})}  \le \|Lg(\xi)\|_{L^q(|\xi|\le 2)} +
\|Lg(\xi)\|_{L^q(|\xi|>2)} \equiv J_1+J_2.
\end{equation}
In $J_1$ by (\ref{tg1}), Minkowski and H\"older's inequalities and
Proposition (\ref{Px6}) we obtain
\begin{equation}
\begin{aligned}
J_1 \leq & c \int_{\mathbb{R}}\varphi_{T}(t) \|e^{\eta t
\Phi(\xi)}\|_{L^{r_1}(|\xi|\le 2)}\|\mathcal{F}^{-1}(g(\cdot,t))(\xi)
\|_{L^{p_1}(|\xi|\le 2)}dt \\
\le& c e^{2\eta T}\|\varphi_{T}\|_{L^p} \,\|g\|_{L_t^q L_x^{q_1}}
\le c e^{2\eta T}T^{1/p} \,\|g\|_{L_t^q
L_x^{q_1}},
\end{aligned} \label{j1}
\end{equation}
where $c$ is a constant, $1/q=1/r_1+1/p_1$ and $1/p_1+1/q_1=1$.
Similarly for $J_2$ we have
\begin{align*}
J_2 \leq \int_{\mathbb{R}}\varphi_{T}(t) \|\,\xi^s\,
e^{\eta t \Phi(\xi)}\|_{L^{r_1}(|\xi|> 2)}\|g(\cdot,t)\|_{L_x^{q_1}}dt .
\end{align*}
For $t>0$, we have
\begin{align*}
\| \,|\xi|^se^{-\eta t|\xi|^{k+2}/2}\|_{L^{r_1}(|\xi|>2)}\, \le
\,\frac{c(\eta)}{t^{s/(k+2)+1/((k+2)r_1)}}.
\end{align*}
Therefore, for $0 \le s<(k+3-p)/p+1/p_1$ we obtain
\begin{align}\label{j2}
J_2 \le
c(\eta)\Big\|\frac{\varphi_{T}(t)}{t^{s/(k+2)+1/((k+2)r_1)}}\Big\|_{L^p}
\,\|g\|_{L_t^q L_x^{q_1}}
\le  c(\eta)\, T^{\epsilon}\,\|g\|_{L_t^q L_x^{q_1}},
\end{align}
where $\epsilon=(k+3-p(s+1))/(k+2)+p/((k+1)p_1)$.
 From (\ref{tg2}), (\ref{j1}) and (\ref{j2}) we obtain
\[
\|Lg\|_{L^q} \le \frac{c(\eta)}{(k+3-p(s+1)+p/p_1)^{1/p}}\,
(T^{1/p}e^{2\eta T}+T^{\epsilon})\,\|g\|_{L_t^q L_x^{q_1}}.
\]
\end{proof}

\section{Some Strichartz type estimates}

\begin{proposition}\label{propx0}
Let $2\le p$, $k \ge 1$,
\,$c_{p,k}=\frac{p-2}{2p(k+2)}$, $0<T<1$, $s\le 0$ and
\[
\frac{1}{r}+\frac{s}{(k+2)} - c_{p,k}>0\,.
\]
 Then
\begin{align*}
\|V_k(t)u_0\|_{L_T^r L_x^p}\le
c(\eta,sqr_0)T^{\{\frac{1}{r}+\frac{s}{(k+2)}-c_{p,k}\}}\|u_0\|_{{H}^{s}{,}}
\end{align*}
where $1/q+1/p=1$, $r_0=2/(2-q)$.
\end{proposition}

\begin{proof}
Let $\Phi(\xi)= |\xi|^k-|\xi|^{k+2}$. By (\ref{x6}) we have
\begin{align*}
\|V_k(t)u_0\|_{L_T^r L_x^p}
&\leq   c \,\|\widehat{V_k(t)u_0}\|_{L_T^r L_\xi^q}\\
&\leq   c \, \|e^{\eta t\Phi(\xi) }\widehat{u_0}\|_{L_T^r L_\xi^q(|\xi| \le 2)}
+c \|e^{\eta t \Phi(\xi)}\widehat{u_0}\|_{L_T^r L_\xi^q(|\xi| > 2)}\\
&\equiv J_1+J_2.
\end{align*}
In $J_1$, using H\"older's inequality we have
\begin{align*}
J_1 \le c3^{-s}e^{\eta T}T^{1/r}.
\end{align*}
To estimate $J_2$, by H\"older's inequality we obtain
\begin{align*}
&\int_{|\xi| > 2}e^{q\eta t\Phi(\xi) }|\widehat{u_0}(\xi)|^q d\xi\\
&\le \int_{|\xi| > 2}e^{-q\eta t|\xi|^{k+2}/2}(1+|\xi|)^{-sq}(1+|\xi|)^{sq}|
 \widehat{u_0}(\xi)|^q d\xi \\
&\le \|e^{-q\eta t|\xi|^{(k+2)}/2}(1+|\xi|)^{-sq}\|_{L^{r_{0}}(|\xi|>2)}
 \|u_0\|_{H^{s}}^{q} \\
&\le  c(\eta)\Big[\frac{1}{t^{1/((k+2)r_{0})}}+\frac{1}{t^{1/((k+2)r_{0})
-sq/(k+2)}}\Big(\int_{\mathbb{R}}e^{-|y|^{(k+2)}}|y|^{-sqr_0}dy\Big)^{1/r_0}
\Big]\|u_0\|_{H^{s}}^{q},
\end{align*}
where $r_0=2/(2-q)$.
Therefore
\begin{align*}
J_2 \le c(\eta,sqr_0)\,
\big\| \frac{1}{t^{1/((k+2)r_{0}q)-s/(k+2)}}\big\|_{L_T^r}\|u_0\|_{H^{s}}
\le c(\eta, sqr_0)\,T^{\frac{1}{r}+\frac{s}{(k+2)}-c_{p,k}}\|u_0\|_{H^{s}},
\end{align*}
where $c_{p,k}=\frac{p-2}{2p(k+2)}$.
\end{proof}

\begin{corollary}\label{propx3}
Let $0<T<1$, $s \leq 0$, $1<r <2(k+2)/(1-2s)$. Then
\begin{align*}
\|V_k(t)u_0\|_{L_T^r L_x^\infty}\le c(\eta, s)T^{1/r
-(1-2s)/(2(k+2))}\|u_0\|_{H^s}.
\end{align*}
\end{corollary}

\begin{proposition}\label{propx2}
Let $u_0 \in L^2$. Then
\[
\|V_k(t)u_0\|_{L_T^\infty L_x^2}\le e^{\eta T}\|u_0\|_{L^2}.
\]
\end{proposition}

\begin{proof}
Using Plancherel identity
\begin{align*}
\|V_k(t)u_0\|_{L_x^2}=
\|e^{\eta t(|\xi|^k-|\xi|^{k+2})}\widehat{u_0}(\xi)\|_{L_\xi^2} \le
e^{\eta T} \|u_0\|_{L^2},
\end{align*}
where we used the estimate $e^{\eta t(|\xi|^k-|\xi|^{k+2})}\le e^{\eta T}$.
\end{proof}

In the following section we give an application of the above
results.

\section{Proof of Theorem \ref{teorp1}}

This section is devoted to give proof of the local well-posedness result
for given data in homogeneous Sobolev space with regularity below $L^2$.
We consider the IVP
\begin{equation}\label{y1}
 \begin{gathered}
 \partial_{t}u+ \partial^{3}_{x}u+\eta L_k(u)+u\partial_{x}u  =  0,
 \quad x \in \mathbb{R}, \; t \geq 0,   \\
u(x,0)  = u_{0}(x),
\end{gathered}
\end{equation}
which is a special case of  (\ref{eq:hs}) with Fourier symbol
$\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$.

To prove Theorem \ref{teorp1}, we need the following proposition.

\begin{proposition} \label{prop5.1}
Let $0\le -s<1/2$. If $u \in L^{2/(1-2s)}$, then
\begin{equation}\label{hls}
\|D_x^s (u)\|_{L^2}\le c \|u\|_{L^{2/(1-2s)}}.
\end{equation}
If $u \in L^1 \cap L^2$, then
\begin{equation}\label{hls1}
\|D_x^s (u)\|_{L^2}\le c (\|u\|_{L^{1}}+\|u\|_{L^{2}}).
\end{equation}
\end{proposition}

\begin{proof} The inequality (\ref{hls}) follows
using the Hardy, Sobolev, Littlewood inequality and
$$
\widehat{\big(\frac{1}{|\xi|^{1+s}}\big)}(\eta)=\frac{1}{|\eta|^{-s}}.
$$
The inequality (\ref{hls1}) follows from
\[
\|D_x^s (u)\|_{L^2}^2
=\int_{|\eta|<1}\frac{|\widehat{u}(\eta)|^2}{|\eta|^{-2s}}d\eta +
\int_{|\eta|\geq 1}\frac{|\widehat{u}(\eta)|^2}{|\eta|^{-2s}}d\eta.
\]
\end{proof}

Now we are in position to supply proof of our second main result.

\begin{proof}[Proof of Theorem \ref{teorp1}]
Now let $u_0 \in \dot{H}^s$, with $0\le -s<1/2$. For $0<T<1$, define a ball
\begin{align*}
Z_{a, T}=\{w \in C([0,T], \dot{H}^s); \quad
| | |w| | |_{T} \leq a \},
\end{align*}
where
$$
| | |w| | |_{T}= \|w\|_{\dot{H}^s}+\|w_x\|_{L_{T}^{2}L_{x}^{p_1}}+
T^{\{\frac{q_1-2}{2q_1(k+2)}-\frac{1}{2}-\frac{s}{k+2}\}}
\|w\|_{L_{T}^{2}L_{x}^{q_1}},
$$
 $-1/s\le p_1<\infty$, $q_1\ge 2$, $1/p_1+1/q_1=(1-2s)/2$ and $p_1$
is chosen as in inequality (\ref{x32}) of Corollary \ref{corx2}.
Using  Corollary \ref{corx2} and Proposition \ref{propx0} we obtain
\begin{align}\label{inqf1}
| | |V_k(t)u_0| | |_{T} \le c\|D_x^s (u_0)\|_{L^2}.
\end{align}
Also, using the inequality (\ref{hls}) we obtain
\begin{equation}
\begin{aligned}
\int_0^T \|D_x^s (vv_x)\|_{L_x^2}\leq & \,c\int_0^T \|vv_x\|_{
L_x^{2/(1-2s)}} \\
\leq & \,c\|v_x\|_{L_T^2 L_x^{p_1}}\,\|v\|_{L_T^{2}L_x^{q_1}} \\
\leq & \, cT^{\big\{-\frac{q_1-2}{2q_1(k+2)}+\frac{1}{2}
+\frac{s}{k+2}\big\}}a^2.
\end{aligned} \label{inqf2}
\end{equation}

Now, define an  application
\begin{align*}
\Psi(v)(t)= V_k(t)u_{0}-\int_{0}^{t}V_k(t-\tau)v v_x(\tau) d\tau,
\end{align*}
where $V_k(t)$ is the evolution operator defined in (\ref{gV}) with
$\Phi(\xi)=|\xi|^k-|\xi|^{k+2}$.
With the help of the inequalities (\ref{inqf1}) and
(\ref{inqf2}), it can be shown that the application $\Psi$ maps $Z_{a,T}$
into $Z_{a,T}$ and is a contraction considering $a=2c\|D_x^s(u_0)\|_{L^2}$,
and $cT^{\big\{-\frac{q_1-2}{2q_1(k+2)}+\frac{1}{2}+\frac{s}{k+2}\big\}}a<1/2$.
The rest of the proof follows from a standard argument.
\end{proof}

\begin{remark} \rm
If we use the inequality (\ref{hls1}) we can also take the following space
in the proof of the Theorem \ref{teorp1}
\begin{align*}
Z_{a, T}=\{w \in C([0,T], \dot{H}^s); \quad
| | |w| | |_{T} \leq a \},
\end{align*}
 where
 \begin{align*}
 | | |w| | |_{T}&= \|w\|_{\dot{H}^s}+\|w_x\|_{L_{T}^{2}L_{x}^{p_1}}
 +\|w_x\|_{L_{T}^{2}L_{x}^{2}}+
T^{\{\frac{q_1-2}{2q_1(k+2)}-\frac{1}{2}-\frac{s}{k+2}\}}
\|w\|_{L_{T}^{2}L_{x}^{q_1}} \\
&+T^{\{-\frac{1}{2}-\frac{s}{k+2}\}}\|w\|_{L_{T}^{2}L_{x}^{2}},
\end{align*}
 $-1/s<p_1<\infty$, $q_1 \geq 2$, $1/p_1+1/q_1=1/2$ and $p_1$ is chosen as in Corollary
\ref{corx2} inequality (\ref{x32}). By Corollary \ref{corx2} and Proposition
\ref{propx0} we obtain
\[
| | |V_k(t)u_0| | |_{T} \le c\|D_x^s (u_0)\|_{L^2}.
\]
Then using (\ref{hls1}),
\begin{align*}
\int_0^T \|D_x^s (vv_x)\|_{L_x^2}
\leq & c\int_0^T (\|vv_x\|_{L_x^1}+\|vv_x\|_{L_x^2})\\
\leq & \,c\|v_x\|_{L_T^2 L_x^{2}}\,\|v\|_{L_T^{2}L_x^{2}}
+c\|v_x\|_{L_T^2 L_x^{p_1}}\,\|v\|_{L_T^{2}L_x^{q_1}}\\
\leq & \, c \, T^{\big\{\frac{1}{2}+\frac{s}{k+2}\big\}}a^{2}
 +cT^{\big\{-\frac{q_1-2}{2q_1(k+2)}+\frac{1}{2}+\frac{s}{k+2}\big\}}a^2 \\
\leq & \, cT^{\big\{-\frac{q_1-2}{2q_1(k+2)}+\frac{1}{2}
 +\frac{s}{k+2}\big\}}a^2.
\end{align*}
\end{remark}

\subsection*{Acknowledgments}
 The authors would like to thank the anonymous referee for his/her comments
that helped us improve this article.

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\end{document}
