\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 68, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE--2003/68\hfil Kadomtsev-Petviashvili equation]
{Global solution for the Kadomtsev-Petviashvili equation (KPII)
in anisotropic Sobolev spaces of negative indices} 

\author[Pedro Isaza J. \& Jorge Mej\'{\i}a L.\hfil EJDE--2003/68\hfilneg]
{Pedro Isaza J. \& Jorge Mej\'{\i}a L.}  % in alphabetical order

\address{Pedro Isaza J.\newline
Escuela de Matem\'aticas\\
Universidad Nacional de Colombia\newline
A. A. 3840 Medell\'{\i}n, Colombia}
\email{pisaza@perseus.unalmed.edu.co}

\address{Jorge Mej\'{\i}a L.\newline
Escuela de Matem\'aticas\\
Universidad Nacional de Colombia\newline
 A. A. 3840 Medell\'{\i}n, Colombia}
\email{jemejia@perseus.unalmed.edu.co}


\date{}
\thanks{Submitted September 13, 2002. Published June 13, 2003.}
\thanks{Partially supported by grant 1118-05-11411 from Colciencias, Colombia.}
\subjclass[2000]{35Q53, 37K05}
\keywords{Nonlinear dispersive equations, global solutions, almost 
\hfill\break\indent conservation laws}


\begin{abstract}
 It is proved that the Cauchy problem for the Kadomtsev-Petviashvili
 equation (KPII) is globally well-posed for initial data in anisotropic
 Sobolev spaces $H^{s0}(\mathbb{R}^2)$ with $s>-1/14$.
 The extension of a local solution to a solution in an arbitrary interval
 is carried out by means of an almost conservation property of the 
 $H^{s0}$ norm of the solution.
\end{abstract}

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


\section{Introduction}

In this article, we consider the initial-value (IVP) problem for
the Kadomtsev-Petviashvili Equation (KP-II):
\begin{equation} \label{e1.1}
\begin{gathered}
\partial_tu+\partial_x^3u+ \partial_x^{-1}\partial_y^2u+u\partial_xu =0\\
u(x,y,0)=u_0(x,y)\,,
\end{gathered}
\end{equation}
with initial data $u_0$ in anisotropic Sobolev spaces with
negative indices. It is known that problem (1.1) is locally
well-posed for initial data in the anisotropic Sobolev spaces
$H^{s_1s_2}(\mathbb{R}^2)$ with $s_1>-\frac13$ and $s_2\geq0$ (see
\cite{IM} and \cite{TT}).

For $s_1\geq0$, the local result and the conservation law for the  $L^2$ norm
show that  $(1.1)$ is globally well-posed. Using the high-low frequency technique,
introduced by Bourgain \cite{B}, the global result for (1.1)  was proved in 
\cite{IM} when $u_0\in H^{s0}(\mathbb{R}^2)$ with $s>-\frac1{64}$.

In this paper, we apply a modification of this technique, proposed in
\cite{CKSTT} for the  Korteweg-de Vries Equation, KdV, to extend the global result
for the KP-II equation mentioned above to indices $s$ with $s\in(-\frac1{14},0)$.
The solution in any time interval $[0,T]$ is obtained from the local solutions
by means of an iterative process  in a finite number of steps.
Such process is possible because we  have in hand a norm which allows us to
control the size of the solution at any instant $t$ in an adequate way.
This norm, equivalent to the usual norm of $H^{s0}$, is essentially the $L^2$
norm for  frequencies below a chosen  parameter $N$.
Thus, when $N$ is sufficiently large, we can take the advantages of the $L^2$
conservation norm.

To establish our results in a precise manner, we  present several definitions
and introduce some notation.
Our initial data will be in the anisotropic Sobolev space
\[
H^{s0}=H^{s0}(\mathbb{R}^2):=\{u\in S'(\mathbb{R}^2) : \|u\|_s^2:=
\int_{\mathbb{R}^2}{\langle\xi\rangle}^{2s}|\widehat u(\zeta)|^2\,d\zeta<\infty\}\,,
\]
where $s<0$, $S'(\mathbb{R}^2)$ is the space of tempered distributions in
$\mathbb{R}^2$, $\widehat u$ is the  Fourier transform of $u$ in the space variables, $\zeta=(\xi,\eta)$ is the variable in the frequency space, with $\xi$ and $\eta$ corresponding to the space variables $x$ and $y$, respectively, and the symbol  $\langle\cdot\rangle$ stands for
$1+|\cdot|$.

For $N\in\mathbb{N}$ we define in $H^{s0}$ an equivalent norm $\|\cdot\|_{sN}$
by $\|u\|_{sN}:=\|I_Nu\|_{L^2}$, where $(I_Nu)\widehat{\;}(\zeta):=M(\xi)\widehat u(\zeta)$ and
\[
M(\xi):=M_N(\xi):=\begin{cases}1,  &\text{if $|\xi|\leq N$},\\
\frac{|\xi|^s}{N^s},   &\text{if $|\xi|>N$.}
\end{cases}
\]
It is easily seen that
\[\|u\|_s\leq\|u\|_{sN}\leq CN^{|s|}\|u\|_s\,.\]

The solutions of the problem will be in the space
\[
X_{s\gamma\varepsilon}:=\{u\in S'(\mathbb{R}^3)\mid \|u\|_{s\gamma\varepsilon}^2
:=\int_{\mathbb{R}^3}{\langle\xi\rangle}^{2s}{\langle\sigma\rangle}^{2\gamma}
{\langle\theta\rangle}^{2\varepsilon}|\widehat u(\lambda)|^2\,d\lambda
 <\infty\}\,,
\]
where $\gamma\in\mathbb{R}$, $\varepsilon>0$, $\widehat u$ is the  Fourier
transform of $u$ in the space-time variables, $\lambda=(\zeta,\tau)=(\xi,\eta,\tau)$
is the variable in the frequency space with $\xi$ and $\eta$ as before, and $\tau$
corresponding to the time variable $t$;
$\sigma:=\tau-m(\zeta)\equiv \tau-\xi^3+\frac{\eta^2}{\xi}$, and
$\theta:=\frac{\sigma}{1+{|\xi|}^3}$. We observe that
$m(\zeta)=\xi^3-\frac{\eta^2}{\xi}$ is the symbol associated to the linear part
of the KP-II equation.

For $N\in\mathbb{N}$ we consider in $X_{s\gamma\varepsilon}$ the equivalent norm
$\|\cdot\|_{s\gamma\varepsilon N}$ defined by
\[
\|u\|_{s\gamma\varepsilon N}:=\|I_Nu\|_{0\gamma\varepsilon}\,,
\]
where $(I_Nu)\widehat{\;\;}(\lambda)=M(\xi)\widehat u(\lambda)$.
If $\gamma>\frac12$, then  $X_{s\gamma\varepsilon}$ is continuously embedded
in $C_b(\mathbb{R}_t;H^{s0})$, the space of continuous bounded functions from the
variable $t$ to  $H^{s0}$.

When   we use the norms $\|\cdot\|_{sN}$ and $\|\cdot\|_{s\gamma\varepsilon N}$
in $H^{s0}$ and $X_{s\gamma\varepsilon}$,  we will refer to these spaces as
$H_N^{s0}$ and $X_{s\gamma\varepsilon N}$, respectively.

For $T>0$ and $\gamma>\frac12$ we define $X_{s\gamma\varepsilon N}[0,T]$ as
the set of all restrictions to $[0,T]$ of the elements of $X_{s\gamma\varepsilon}$
 with norm defined by
\[
\|u\|_{X_{s\gamma\varepsilon N}[0,T]}:=\inf \{\|v\|_{s\gamma\varepsilon N} :
 v|_{[0,T]}=u\}\,.
\]
In our exposition we will make use of the space
$H^{\infty}(\mathbb{R}^2):=\cap_{s\in\mathbb{R}}H^s(\mathbb{R}^2)$, where
$H^s(\mathbb{R}^2)$ is the classical Sobolev space of  $L^2$ type defined by
\[
H^s(\mathbb{R}^2):=\{u\in S'(\mathbb{R}^2): \int_{\mathbb{R}^2}
\langle\zeta\rangle^{2s}|\widehat u(\zeta)|^2\,d\zeta<\infty\}\,.
\]

Our concept of solution comes from  Duhamel's formula for problem (1.1).
Formally, $u$ is a solution of (1.1) in $[0,T]$ if  for $t\in[0,T]$
\[
u(t)=W(t)u_0-\frac12\int_0^tW(t-t')\partial_x(u(t'))^2dt'\,,\tag{1.2}
\]
where $\{W(t)\}$ is the group associated to the linear part of KP-II equation.
This is:
\[
[W(t)u_0]\widehat{\quad}(\zeta):=e^{itm(\zeta)}\widehat{u_0}(\zeta)\,.
\]
In order to stay in the  context of the spaces $X_{s\gamma\varepsilon}$,
we multiply the right side of  (1.2) by $\Psi(T^{-1}t)$, where
 $\Psi\in C_0^{\infty}(\mathbb{R}_t)$, $\Psi\geq0$, $\Psi\equiv 1$ in $[0,1]$ and
$\mathop{\rm supp} \Psi\subset[-1,2]$. In this way, we consider the integral
equation
\[
u(t)=\Psi(T^{-1}t)W(t)u_0-\frac12\Psi(T^{-1}t)
\int_0^tW(t-t')\partial_x(u(t')^2)\,dt'\,,\quad t\in\mathbb{R}\,.\tag{1.3}
\]
We can see  that, formally, for $t\in[0,T]$ expressions (1.3) and (1.2) coincide.
By a direct calculation, it can be easily established  that
\[
\|\Psi(T^{-1}\cdot_t)W(\cdot_t)u_0\|_{s\gamma\varepsilon N}\leq C_T\|u_0\|_{sN}\,,
\tag{1.4}
\]
where $C_T$ depends on $T$ but not on  $N$.

For $\gamma>\frac12$ and $f\in S(\mathbb{R}^3)\cap X_{s(\gamma-1)\varepsilon}$,
where $S(\mathbb{R}^3)$ is the space of Schwartz functions in $\mathbb{R}^3$,
we define:
\[
G_T(f)(t):=\frac12\Psi(T^{-1}t)\int_0^tW(t-t')f(t')dt'\,,
\]
where $f(t):=f(\cdot_x,\cdot_y,t)$. Following a procedure similar to that in 
the proof in \cite[Lemma 3.3]{KPV1}, it can be seen that
\[
\|G_T(f)\|_{s\gamma\varepsilon N}\leq C_T\|f\|_{s(\gamma-1)\varepsilon N}\,.
\tag{1.5}
\]
Therefore, since $S(\mathbb{R}^3)\cap X_{s(\gamma-1)\varepsilon}$ is dense in
$X_{s(\gamma-1)\varepsilon}$, $G_T$ has a unique continuous extension, which we
denote again by $G_T$, to the space $X_{s(\gamma-1)\varepsilon}$.

In the study of the nonlinear part of the equation, an important role is played by
the bilinear form $\partial_x(uv)$; more precisely, we have the following result
whose proof will be given in section 2.

\begin{lemma} \label{lm1}
Let $s\in(-\frac13,0)$. For $\gamma>\frac12$ and $\varepsilon>\frac16$ such that
$(\frac13+s)-(\gamma-\frac12)-(\varepsilon-\frac16)\geq0$,
$(\frac12+s)-3(\varepsilon-\frac16)\geq0$, and
$\frac13(\frac12+s)-(\gamma-\frac12)\geq0$, it follows that
\[
\|\partial_x(uv)\|_{s(\gamma-1)\varepsilon N}\leq\overline C
\|u\|_{s\gamma\varepsilon N}\|v\|_{s\gamma\varepsilon N}
\quad\forall u,v\in X_{s\gamma\varepsilon}\,,\tag{1.6}
\]
with $\overline C$ independent  of $N$.
\end{lemma}

Estimates (1.4), (1.5), and (1.6) allow us to define the concept of solution for
(1.1):

\subsection*{Definition} 
For $u_0\in H^{s0}$, $T>0$, $s$, $\gamma$, and $\varepsilon$ as in Lemma
\ref{lm1},
we say that $u\in X_{s\gamma\varepsilon N}[0,T]$
is a solution of the IVP (1.1) in the interval $[0,T]$ if there is an extension
 $v\in X_{s\gamma\varepsilon N}$ of  $u$, such that
\[
u(t)=W(t)u_0-G_T(\partial_xv^2)(t)\quad\forall \,t\in[0,T]\,.
\]


It was proved in \cite{IM} that (1.1) is locally well-posed for initial data
$u_0$ in $H^{s0}$ with $s>-\frac13$. More precisely, the theorems of  existence
 and  uniqueness of local solutions were proved there. The proofs of continuous
dependence on the initial data and of regularity follow the same procedure
applied in \cite{IMS} for the corresponding proofs in the case $s>0$.

In this paper we will obtain global solution for initial data in $H^{s0}$ with
 $s>-\frac1{14}$. In this case, the extension from a local solution  $u$ to a
solution in an arbitrary interval $[0,T]$ is carried out by keeping control of
the norm  $\|u(T)\|_{sN}$ with the use of the homogeneity properties of the KP
equation and the aid of an almost conservation law which uses a cancellation
effect expressed by the following estimate of a new bilinear form.

\begin{lemma} \label{lm2}
For $s\in(-\frac14,0)$, let $\gamma>\frac12$, and $\varepsilon>\frac16$ be chosen
to satisfy  the hypotheses of Lemma \ref{lm1}  and the condition
$(\frac12-\gamma)+\frac23(\frac14+s)>0$ (i.e., $2|s|<2-3\gamma$), then, for
$\alpha\in(2|s|,2-3\gamma)$ it follows that
\[
\|\partial_x[(I_Nu)(I_Nv)-I_N(uv)]\|_{0,\gamma-1,0}
\leq CN^{-\alpha}\|I_Nu\|_{0\gamma\varepsilon}\|I_Nv\|_{0\gamma\varepsilon}\,.
\tag{1.7}
\]
\end{lemma}

The proof of this lemma will be given in section 3.
Finally, in section 4 we will prove our  main result,
whose precise formulation is:

\begin{theorem} \label{thm1}
For $s\in(-\frac1{14},0)$, $T>0$, and $u_0\in H^{s0}$ with
$\partial_x^{-1}u_0\in S'(\mathbb{R}^2)$ (i.e.,
$\frac{\widehat{u_0}}{i\xi}\in S'(\mathbb{R}^2)$), there exists $N>0$ such that
problem (1.1) has  a solution in $X_{s\gamma\varepsilon N}[0,T]$.
\end{theorem}

Note that the role played by $N$ is merely technical and also that the
obtained solution is in
\[
X_{s\gamma\varepsilon}[0,T]:=\{v|_{[0,T]}\mid v\in X_{s\gamma\varepsilon}\}\,.
\]
Through this article, the letter $C$ will denote diverse  constants and the
notation $x\sim y$, for two variables  $x$ and $y$, will mean the existence
 of positive constants $C_1$ and $C_2$ such that $C_1|x|\leq|y|\leq C_2|x|$.

\section{Proof of  Lemma \ref{lm1}}

It was proved in  \cite{IM} that, under the hypotheses of Lemma \ref{lm1},  the following
estimate takes place:
\[
\|\partial_x(uv)\|_{s(\gamma-1)\varepsilon}\leq C\|u\|_{s\gamma\varepsilon}
\|v\|_{s\gamma\varepsilon}\,.\tag{2.1}
\]
If $\lambda_1=(\zeta_1,\tau_1)$, $\lambda_2=(\zeta_2,\tau_2)=(\zeta-\zeta_1,
\tau-\tau_1)=\lambda-\lambda_1$; ${\sigma_1}=\sigma(\zeta_1,\tau_1)$,
$\sigma_2=\sigma(\zeta_2,\tau_2)$;
${\theta_1}=\theta(\zeta_1,\tau_1)$, $\theta_2=\theta(\zeta_2,\tau_2)$,
estimate (2.1) is equivalent to the estimate
\[
\begin{aligned}
&\int_{\mathbb{R}^6}\widetilde{K}(\lambda,\lambda_1)f(\lambda_1)
g(\lambda_2)h(\lambda)\,d\lambda _1\,d\lambda \\
&:=\int_{\mathbb{R}^6}\frac{{|\xi|}{\langle\theta\rangle}^{\varepsilon}}
{{\langle\sigma\rangle^{1-\gamma}}{\langle\sigma_1\rangle}^{\gamma}
{\langle\sigma_2\rangle}^{\gamma}{\langle\theta_1\rangle}^{\varepsilon}
{\langle\theta_2\rangle}^{\varepsilon}}\frac{{\langle\xi\rangle}^s}
{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}f(\lambda_1)g(\lambda_2)h(\lambda)\,d
\lambda_1\,d\lambda\\
&\leq C{\|h\|\,\|f\|\,\|g\|}\,,
\end{aligned} \tag{2.2}
\]
where $f,\;g,\;h\geq0$ and $\|\cdot\|:=\|\cdot\|_{L_{\lambda}^2}$.
Using duality,  estimate $(1.6)$ is equivalent to:
\begin{align*}
&\int_{\mathbf{R}^6}\frac{{|\xi|}{\langle\theta\rangle}^{\varepsilon}}
{{\langle\sigma\rangle^{1-\gamma}}{\langle\sigma_1\rangle}^{\gamma}
{\langle\sigma_2\rangle}^{\gamma}{\langle\theta_1\rangle}^{\varepsilon}
{\langle\theta_2\rangle}^{\varepsilon}}\frac{M(\xi)}{M(\xi_1)M(\xi_2)}
f(\lambda_1)g(\lambda_2)h(\lambda)\,d\lambda_1\,d\lambda\\
&\leq \overline C{\|h\|\,\|f\|\,\|g\|}\,.\tag{2.3}
\end{align*}
In this way, to establish $(2.3)$ it suffices to prove that
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}\leq C\frac{{\langle\xi\rangle}^s}
{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\,,\tag{2.4}
\]
with $C$ independent of $N$. In the proof of (2.4) we will take into account that
\[
1\leq2^{|s|}\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\,,
\]
and besides that, for $s<0$, $M(\xi)\leq 1$.

By a symmetry argument it is sufficient to analyze the following cases:

\noindent(i) $|\xi_1|\leq\frac{N}2\land|\xi_2|\leq\frac{N}2$. Therefore,
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}=M(\xi)\leq 1\leq 2^{|s|}
\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\,.
\]
(ii) $|\xi_1|\geq\frac{N}2\land|\xi_2|\leq1$. Then,
${|\xi|}\sim|\xi_1|\sim{\langle\xi\rangle}\sim\langle\xi_1\rangle$ and
$\langle\xi_2\rangle\sim 1$. Hence,
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}\leq C\frac{\frac{{|\xi|}^s}{N^s}}
{\frac{|\xi_1|^s}{N^s}1}\leq C\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s
\langle\xi_2\rangle^s}\,.
\]
(iii) $\frac{N}2\leq|\xi_1|\leq N\land1\leq|\xi_2|\leq \frac{N}2$. Thus,
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}=M(\xi)\leq 1\leq  2^{|s|}
\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\,.
\]
(iv) $N\leq|\xi_1|\land1\leq|\xi_2|\leq \frac{N}2$. Then,
${|\xi|}\geq\frac{N}2$, ${|\xi|}\sim{\langle\xi\rangle}$, and
$|\xi_1|\sim\langle\xi_1\rangle$. Therefore,
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}\leq C\frac{\frac{{|\xi|}^s}{N^s}}
{\frac{|\xi_1|^s}{N^s}1}\leq C\frac{{\langle\xi\rangle}^s}
{\langle\xi_1\rangle^s}\leq C\frac{{\langle\xi\rangle}^s\langle\xi_2\rangle^{|s|}}
{\langle\xi_1\rangle^s}
=C\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\,.
\]
(v) $|\xi_1|\geq\frac{N}2\land|\xi_2|\geq\frac{N}2$. Thus,
$|\xi_1|\sim\langle\xi_1\rangle$ and $|\xi_2|\sim\langle\xi_2\rangle$. \smallskip


If ${|\xi|}\leq1$, then ${\langle\xi\rangle}\sim1$ and
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}\leq C\frac1{N^{2|s|}|\xi_1|^s|\xi_2|^s}
\leq C\frac{{\langle\xi\rangle}^s}{N^{2|s|}\langle\xi_1\rangle^s\langle
\xi_2\rangle^s}\leq C\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s
\langle\xi_2\rangle^s}\,.
\]
If $1\leq{|\xi|}\leq N$, then $\frac{{|\xi|}}N\leq1$, $\frac{{|\xi|}^s}{N^s}\geq1$,
and ${|\xi|}\sim{\langle\xi\rangle}$. Therefore,
\begin{align*}
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}&\leq C\frac1{N^{2|s|}|\xi_1|^s|\xi_2|^s}\leq C\frac{\frac{{|\xi|}^s}{N^s}}{N^{2|s|}|\xi_1|^s|\xi_2|^s}\\
&\leq C\frac{{\langle\xi\rangle}^s}{N^{|s|}\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\leq C\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s\langle\xi_2\rangle^s}\,.
\end{align*}
If $N\leq{|\xi|}$, then
\[
\frac{M(\xi)}{M(\xi_1)M(\xi_2)}\leq C\frac{\frac{{|\xi|}^s}{N^s}}{N^{2|s|}|\xi_1|^s|
\xi_2|^s}\leq C\frac{{\langle\xi\rangle}^s}{N^{|s|}\langle\xi_1\rangle^s
\langle\xi_2\rangle^s}\leq C\frac{{\langle\xi\rangle}^s}{\langle\xi_1\rangle^s
\langle\xi_2\rangle^s}\,.
\]
Which completes the proof.

\section{ Proof of Lemma \ref{lm2}}

We begin this section by describing the procedure that leads to an almost
conservation law, according to which, for a solution $v$ of problem (1.1) in the
interval $[0,1]$, the size of  $\|v(1)\|_{sN}$ is controlled by the size
of $\|v(0)\|_{sN}$. In the description of this procedure, the necessity of
estimate (1.7) comes out in a natural way.

Let $v_0\in H^{\infty}(\mathbb{R}^2)$ with
$\partial_x^{-1}v_0\in H^{\infty}(\mathbb{R}^2)$. If $w\in X_{s\gamma\varepsilon N}$
is such that $v:=w|_{[0,1]}$ is a solution of the IVP  (1.1) with initial datum
$v_0$, then, from a regularity theorem for problem (1.1) (see  
\cite[Theorem IV]{IMS})
and from the application of standard techniques of the theory of semigroups,
we can conclude that $v(t)$ and $\partial_x^{-1}v(t)$ are in
$H^{\infty}(\mathbb{R}^2)$, and
\[
v'(t)+\partial_x^3v(t)+\partial_x^{-1}\partial_y^2v(t)+v(t)\partial_xv(t)
=0\quad\text{in}\quad H^{\infty}(\mathbb{R}^2),\;\forall t\in [0,1]\,.\tag{3.1}
\]
When we apply the operator $I_N$ to the equation above and take the inner product
in  $L^2(\mathbb{R}^2)$ with $I_Nv(t)$, we obtain
\[
\frac12\frac{d}{dt}\|I_Nv(t)\|_{L^2}^2+\frac12\int_{\mathbb{R}^2}
(\partial_xI_N(v(t)^2))I_Nv(t)dxdy=0\,.
\]
Taking into account that
$\frac12\int_{\mathbb{R}^2}(\partial_x(I_Nv(t))^2)I_Nv(t)dxdy=0$, and denoting by
$\chi$ the characteristic function of the interval $[0,1]$, an integration with
respect to $t$ in $[0,1]$ yields
\begin{align*}
\|I_Nv(1)\|_{L^2}^2&=\|I_Nv_0\|_{L^2}^2-\int_0^1\langle\partial_x[I_N(v(t)^2)
 -(I_Nv(t))^2],I_Nv(t)\rangle dt\\
&\leq \|I_Nv_0\|_{L^2}^2+|\int_{-\infty}^{+\infty}\langle\chi(t)
 \partial_x[I_N(w(t)^2)-(I_Nw(t))^2],\chi(t)I_Nw(t)\rangle dt|\\
&\leq |I_Nv_0\|_{L^2}^2+C\|\chi(\cdot_t)\partial_x[I_N(w^2)-(I_Nw)^2]\|_{0,
 \gamma^--1,0}\|\chi(\cdot_t)I_Nw\|_{0,1-\gamma^-\kern-2pt,0}\\
&\leq |I_Nv_0\|_{L^2}^2+C\|\partial_x[I_N(w^2)-(I_Nw)^2]\|_{0,\gamma-1,0}
 \|I_Nw\|_{0,(1-\gamma^-)^+,0}\,,\tag{3.2}
\end{align*}
where $\langle\cdot,\cdot\rangle$ is the inner product in $L^2(\mathbb{R}^2)$,
$\frac12<\gamma^-<\gamma$, and, in the last inequality we have used the following
lemma of technical character.

\begin{lemma} \label{lm3}
If $\overline\gamma\in(0,\frac12)$ and $\overline\gamma<\overline\gamma^+<\frac12$,
then
\begin{gather*}
\|\chi(\cdot_t)u\|_{0\overline\gamma0}\leq C\|u\|_{0\overline\gamma^+0},
\tag{3.3}\\
\|\chi(\cdot_t)u\|_{0,(-\overline\gamma^+),0}\leq C\|u\|_{0,-\overline\gamma,0}\,.
\tag{3.4}
\end{gather*}
\end{lemma}

\begin{proof}
We prove (3.3) only, since (3.4) follows from (3.3) by duality.
For $u\in S(\mathbb{R}^3)\cap X_{0\overline\gamma^+0}$,
\begin{align*}
\|\chi(\cdot)u\|_{0\overline\gamma0}^2
&=\iint{\langle\sigma\rangle}^{2\overline\gamma}|[\widehat\chi\ast_{\tau}
\widehat u(\zeta,\cdot_{\tau})](\tau)|^2d\tau d\zeta\\
&\leq C\iint|[\chi(\cdot_t)\widehat{u(\cdot_t)}(\zeta)]
\widehat{\quad}^t(\tau)|^2d\tau d\zeta\\&+C\iint|\tau|^{2\overline\gamma}|
[\widehat\chi\ast_{\tau}\widehat u(\zeta,\cdot_{\tau})](\tau+m(\zeta))|^2d\tau d\zeta
=:I+II\,.\tag{3.5}
\end{align*}
Using  Plancherel identity in the variable $t$ we obtain
\[
\begin{aligned}
I&\leq C\iint|\chi(t)\widehat{u(t)}(\zeta)|^2dtd\zeta
\leq C\iint|\widehat{u(t)}(\zeta)|^2dtd\zeta\\
&=C\iint|\widehat{u}(\zeta,\tau)|^2d\zeta d\tau
=\|u\|_{000}^2\leq\|u\|_{0\overline\gamma^+0}\,.
\end{aligned}\tag{3.6}
\]
To estimate  $II$ we use the following  Leibniz formula for fractional
derivatives proved in \cite{KPV2}:
If $\beta\in(0,1)$ and $1<q<\infty$, then
\[
\|D^\beta(fg)-fD^\beta g\|_{L^q(\mathbb{R})}\leq C\|g\|_{L^\infty(\mathbb{R})}\|D^\beta f\|_{L^q(\mathbb{R})}\,.
\tag{3.7}
\]
Therefore, an application of Plancherel identity and of estimate (3.7) with $q=2$
gives us
\[\begin{aligned}
II&= C\int\|D^{{\overline\gamma}}_t[e^{-i(\cdot_t)m(\zeta)}\chi(\cdot_t)
\widehat{u(\cdot_t)}(\zeta)]\|^2_{L^2_t}\,d\zeta\\
&\leq C\int \|D^{{\overline\gamma}}_t[e^{-i(\cdot_t)m(\zeta)}\widehat{u(\cdot_t)}
(\zeta)]\|^2_{L^2_t}\|\chi(\cdot_t)\|_{L^\infty_t}^2\,d\zeta
\\&+C\int \|e^{-i(\cdot_t)m(\zeta)}\widehat{u(\cdot_t)}
(\zeta)D^{{\overline\gamma}}_t\chi(\cdot_t)\|^2_{L_t^2}\,d\zeta\\
&\leq C\|u\|^2_{0{\overline\gamma} 0} +C\int\|e^{-i(\cdot_t)m(\zeta)}
\widehat{u(\cdot_t)}(\zeta)\|^2_{L^{2p}}\|D_t^{{\overline\gamma}}
\chi(\cdot_t)\|^2_{L^{2p'}}\,d\zeta\,,
 \end{aligned}\tag{3.8}
\]
where $p$ and $p'$ are conjugate exponents. If we choose $p$ in such a way that
$\frac12-\frac1{2p}={\overline\gamma}^+$, then
$H^{{\overline\gamma}^+}(\mathbb{R}_t)\hookrightarrow L^{2p}(\mathbb{R}_t)$.
Bearing in mind that the inverse Fourier transform  operator is bounded from
$L^{\frac{2p'}{2p'-1}}(\mathbb{R})$ to $L^{2p'}(\mathbb{R})$;  it follows from
(3.8) that
 \[
II\leq C\|u\|^2_{0{\overline\gamma}0}+C\int\|e^{-i(\cdot_t)m(\zeta)}
\widehat{u(\cdot_t)}(\zeta)\|^2_{H^{{\overline\gamma}^+}_t}\,d\zeta\;\cdot\;\|
(D_t^{{\overline\gamma}}\chi)\,\widehat{\,}\,{\,}^t\|^2_{L^{\frac{2p'}{2p'-1}}
(\mathbb{R}_\tau)}\,.
\]
Since  $\frac{2p'}{2p'-1}=\frac1{1-{\overline\gamma}^+}$ and
$|\widehat\chi(\tau)|\leq \frac C{\langle\tau\rangle}$, we have
\[
\|(D_t^{\overline\gamma}\chi)\,\widehat{\,}\,{\,}^t\|_{L^{\frac{2p'}{2p'-1}}}
\leq C\bigl(\int_{-\infty}^{+\infty}\frac{d\tau}
{\langle\tau\rangle^\frac{1-{\overline\gamma}}{1-{\overline\gamma}^+}}\bigr)
^{1-{\overline\gamma}^+}<\infty\,.
\]
Therefore,
\[
II\leq C\|u\|^2_{0{\overline\gamma}0}+C\|u\|^2_{0{\overline\gamma}^+0}\,.\tag{3.9}
\]
From (3.5), (3.6), and (3.9) we obtain (3.3)  for
$u\in {S}(\mathbb{R}^3)\cap X_{0{\overline\gamma}^+0}$. The result of the lemma
follows from a density argument.
\end{proof}


\begin{proof}[Proof of Lemma \ref{lm2}]
We use the notation introduced in the proof of Lemma \ref{lm1}. Reasoning by duality,
estimate (1.7) is equivalent to the estimate
\begin{align*}
&\iint K(\lambda,{\lambda_1})f({\lambda_1})g(\lambda_2)h(\lambda)\,d\lambda_1
\,d\lambda\\
&:=\int_{\mathbb{R}^3_\lambda}\int_{\mathbb{R}^3_{\lambda_1}}
\frac{{|\xi|}\quad\quad\quad\quad\quad({M(\xi_1)}{M(\xi_2)}-{M(\xi)})}
{{\langle\sigma\rangle}^{1-\gamma}{\langle\sigma_1\rangle}^\gamma{
\langle\theta_1\rangle}^\varepsilon{\langle\sigma_2\rangle}^\gamma
{\langle\theta_2\rangle}^\varepsilon{M(\xi_1)}{M(\xi_2)}}f({\lambda_1})
g(\lambda_2)h(\lambda)\,d\lambda_1\,d\lambda \\
&\leq CN^{-{\alpha}}{\|h\|\,\|f\|\,\|g\|}\,.\tag{3.10}
\end{align*}
For  $A\subseteq\mathbb{R}^6$, we will denote by $J_A$ the integral over the set  $A$
of the former integrand. By a symmetry argument, it suffices to show that
\[
J:=\sum_{i=1}^4 J_{A_i}\leq CN^{-{\alpha}}{\|h\|\,\|f\|\,\|g\|}\,,\tag{3.11}
\]
where
\begin{align*}
A_1&:=\{(\lambda,\lambda_1)\mid {|\xi_1|} \leq\frac N2       \land {|\xi_2|} \leq\frac N2\}\,,\\
A_2&:=\{(\lambda,\lambda_1)\mid {|\xi_1|} \geq\frac N2       \land {|\xi_2|} \leq 1\}\,,\\
A_3&:=\{(\lambda,\lambda_1)\mid {|\xi_1|} \geq \frac N2       \land 1<{|\xi_2|}<  N\}\,,\text{  and}\\
A_4&:=\{(\lambda,\lambda_1)\mid {|\xi_1|}  \geq \frac N2      \land {|\xi_2|}\geq  N\}\,.
\end{align*}

For the rest of this article,  we will use the notation
\[
L(\xi_1,\xi_2):=\frac{M(\xi_1)M(\xi_2)-M(\xi)}{M(\xi_1)M(\xi_2)}\,.\tag{3.12}
\]
{\it Estimate of $J_{A_1}$:}
For $(\lambda,\lambda_1)\in A_1$ we have that ${L(\xi_1,\xi_2)}=0$. Thus
\[ J_{A_1}=0\tag{3.13}.
\]
{\it Estimate of $J_{A_2}$:}
For $(\lambda,\lambda_1)\in A_2$, an application of the mean value theorem leads to
\begin{align*}
|{L(\xi_1,\xi_2)}|
&=\Bigl|\frac{{M(\xi_1)}-{M(\xi_1+\xi_2)}}{{M(\xi_1)}}\Bigr|
\leq\frac {N^{|s|}}{{M(\xi_1)}}
\Bigl|\frac 1{|{\xi_1}|^{|s|}}-\frac1{|{\xi_1}+{\xi_2}|^{|s|}}\Bigr|
\\
&\leq C\frac{N^{|s|}{|\xi_2|}}{{M(\xi_1)}{|\xi_1|}^{{|s|}+1}}\leq C\frac{N^{|s|}{|\xi_2|}}{\frac{N^{|s|}}{{|\xi_1|}^{|s|}}{|\xi_1|}^{{|s|}+1}}\\
&\leq\frac C{|\xi_1|}\leq\frac CN\leq\frac CN\frac{{\langle\xi_1\rangle}^{|s|}
{\langle\xi_2\rangle}^{|s|}}{{\langle\xi\rangle}^{|s|}}\,.
\end{align*}
In this way, taking into account the definition of the kernel $\widetilde K$ in
(2.2), it follows that
$K(\lambda,\lambda_1)\leq \frac CN\widetilde K(\lambda,\lambda_1)$ and therefore,
according to estimate (2.2),
\[
J_{A_2}\leq \frac CN{\|h\|\,\|f\|\,\|g\|}
\leq CN^{-{\alpha}}{\|h\|\,\|f\|\,\|g\|}\,.\tag{3.14}
\]
{\it Estimate of $J_{A_3}$:}
For $(\lambda,\lambda_1)\in A_3$,
\[
|{L(\xi_1,\xi_2)}|=\Bigl|\frac{{M(\xi_1)}-{M(\xi_1+\xi_2)}}{{M(\xi_1)}}\Bigr|\,.
\]
When $(\frac N2\leq{|\xi_1|}\leq N\;\land\;{|\xi|}\geq N)$ or $(N\leq{|\xi_1|}
\leq\frac{3N}2\;\land\;1\leq{|\xi_2|}\leq\frac N2)$ or
${|\xi_1|}\geq\frac{3N}2$, from an application of the mean value theorem we have
\[
|{L(\xi_1,\xi_2)}|\leq C\frac{N^{|s|}}{{M(\xi_1)}}
\frac{{|\xi_2|}}{{|\xi_1|}^{{|s|}+1}}\leq C\frac{|\xi_2|}{|\xi_1|}\,.
\]
If $\frac N2\leq{|\xi_1|}\leq N$ and ${|\xi|}\leq N$, then ${L(\xi_1,\xi_2)}=0$.
If $N\leq{|\xi_1|}\leq\frac{3N}2$ and $\frac N2\leq{|\xi_2|}\leq N$, then
\[
|{L(\xi_1,\xi_2)}|\leq 1+\frac{{M(\xi)}}{{M(\xi_1)}}\leq C
\leq C\frac{{|\xi_2|}}{{|\xi_1|}}\,,\]
since ${|\xi_2|}\sim{|\xi_1|}$.

Thus, for $(\lambda,\lambda_1)\in A_3$:
\[
|K(\lambda,\lambda_1)|\leq C\frac{{|\xi|}}{{\langle\sigma\rangle}^{1-\gamma}}\frac{|\xi_2|}{|\xi_1|}\frac1{{\langle\sigma_1\rangle}^\gamma{\langle\sigma_2\rangle}^\gamma}\,.\tag{3.15}
\]
According to our definitions we have
\[
{\sigma_1}+{\sigma_2}-\sigma=3\xi{\xi_1}{\xi_2}
+\frac{(\eta{\xi_1}-\eta_1\xi)^2}{\xi{\xi_1}{\xi_2}}\,.
\]
Therefore, $|\xi{\xi_1}{\xi_2}|\leq\max\{|\sigma|,{|\sigma_1|},{|\sigma_2|}\}$.
If ${|\sigma|}=\max\{|\sigma|,{|\sigma_1|},{|\sigma_2|}\}$, then
\begin{align*}
\frac{{|\xi|}}{{\langle\sigma\rangle}^{1-\gamma}}
\frac{{|\xi_2|}}{{|\xi_1|}}&\leq \frac{{|\xi|}^\gamma{|\xi_2|}^\gamma|\xi{\xi_1}
{\xi_2}|^{1-\gamma}}{{\langle\sigma\rangle^{1-\gamma}}{|\xi_1|}^{2-\gamma}}\\
&\leq C\frac{{|\xi|}^\gamma{|\xi_2|}^\gamma}{{|\xi_1|}^{2-\gamma}}
\leq C\frac{{|\xi_1|}^\gamma{|\xi_2|}^\gamma}{{|\xi_1|}^{2-\gamma}}
+C\frac{{|\xi_2|}^\gamma{|\xi_2|}^\gamma}{{|\xi_1|}^{2-\gamma}}\\
&\leq C\frac{N^\gamma}{N^{2-2\gamma}}+C\frac{N^{2\gamma}}{N^{2-\gamma}}
\leq CN^{-(2-3\gamma)}\,,
\end{align*}
 and from (3.15),
\[
|K(\lambda,\lambda_1)|\leq CN^{-(2-3\gamma)}
\frac1{{\langle\sigma_1\rangle}^\gamma{\langle\sigma_2\rangle}^\gamma}\,.
\tag{3.16}
\]
If ${|\sigma_1|}=\max\{|\sigma|,{|\sigma_1|},{|\sigma_2|}\}$, then
\begin{align*}
\frac{|\xi|}{{\langle\sigma_1\rangle}^\gamma}\frac{|\xi_2|}
{|\xi_1|}\frac1{\langle\sigma\rangle^{1-\gamma}}
&\leq \frac{{|\xi|}{|\xi_2|}}{{\langle\sigma_1\rangle}^{1-\gamma}{|\xi_1|}
{\langle\sigma\rangle}^\gamma}\leq\frac{{|\xi|}^\gamma{|\xi_2|}^\gamma
|\xi{\xi_1}{\xi_2}|^{1-\gamma}}{{\langle\sigma_1\rangle}^{1-\gamma}
{|\xi_1|}^{2-\gamma}{\langle\sigma\rangle}^\gamma}\\
&\leq C\frac{{|\xi|}^\gamma{|\xi_2|}^\gamma}{{|\xi_1|}^{2-\gamma}{\langle\sigma
\rangle}^\gamma}
\leq C\frac{N^{-(2-3\gamma)}}{{\langle\sigma\rangle}^\gamma}\,,
\end{align*}
and from (3.15)
\[
|K(\lambda,\lambda_1)|\leq CN^{-(2-3\gamma)}
\frac1{{\langle\sigma\rangle}^\gamma{\langle\sigma_2\rangle}^\gamma}\,.
\tag{3.17}
\]
If ${|\sigma_2|}=\max\{|\sigma|,{|\sigma_1|},{|\sigma_2|}\}$, then, as in the
former case,
\[
\frac{|\xi|}{{\langle\sigma_2\rangle}^\gamma}\frac{|\xi_2|}{|\xi_1|}
\frac1{\langle\sigma\rangle^{1-\gamma}}
\leq C\frac{N^{-(2-3\gamma)}}{{\langle\sigma\rangle}^\gamma}
\]
and from (3.15),
\[
|K(\lambda,\lambda_1)|\leq CN^{-(2-3\gamma)}
\frac1{{\langle\sigma\rangle}^\gamma{\langle\sigma_1\rangle}^\gamma}\,.
\tag{3.18}
\]

Let us  denote by  $A_{30}$, $A_{31}$, and $A_{32}$ the subsets of $A_{3}$
corresponding to each one of the former cases respectively. Then, from (3.16)
we have
\begin{align*}
J_{A_{30}}&\leq CN^{-(2-3\gamma)}\int_{\mathbb{R}^3}h(\lambda)
\int_{\mathbb{R}^3}\frac{f({\lambda_1})}{{\langle\sigma_1\rangle}^\gamma}
\frac{g(\lambda_2)}{{\langle\sigma_2\rangle}^\gamma}\,d\lambda_1\,d\lambda\\
&\leq CN^{-(2-3\gamma)}\|h\|\|FG\| \\
&\leq CN^{-(2-3\gamma)}\|h\|\|F\|_{L^4_{xyt}}\|G\|_{L^4_{xyt}}\,,
\end{align*}
where $\widehat F(\lambda):=\frac{f(\lambda)}{{\langle\sigma\rangle}^\gamma}$
and $\widehat G(\lambda):=\frac{g(\lambda)}{{\langle\sigma\rangle}^\gamma}$.

Using the Strichartz inequality
(see \cite{T}, Proposition 2.1 and \cite[Lemma 3.3]{G})
\[
\|F\|_{L^4_{xyt}}\leq C\|f\|\quad\text{if } \gamma>\frac12\,,\tag{3.19}
\]
it follows that
\[ J_{A_{30}}\leq CN^{-(2-3\gamma)}{\|h\|\,\|f\|\,\|g\|}\,.
\]
In a similar way, from (3.17) and (3.18),
\[ J_{A_{31}}+J_{A_{32}}\leq CN^{-(2-3\gamma)}{\|h\|\,\|f\|\,\|g\|}\,.
\]
In this manner,
\[J_{A_{3}}\leq C N^{-{\alpha}}{\|h\|\,\|f\|\,\|g\|}\,.\tag{3.20}
\]
{\it Estimate of $J_{A_4}$:}
To estimate $J_{A_4}$ we require the following lemma.



\begin{lemma} \label{lm4}
If $\overline s\in(-\frac14,0)$, $\gamma>\frac12$ satisfy
$(\frac12-\gamma)+\frac23(\frac14+\overline s)>0$, and
$\varepsilon>\frac16$, then, for $u$ and $v$ such that
$\mathop{\rm supp}\widehat u$,
$\mathop{\rm supp}\widehat v\subseteq\{\lambda\mid{|\xi|}\geq 1\}$,
 the following inequality holds
 \[\|\partial_x(uv)\|_{0(\gamma-1)0}\leq C\|u\|_{\overline s\gamma\epsilon}
 \|v\|_{\overline s\gamma\epsilon}.
\tag{3.21}
\]
\end{lemma}

The proof of this lemma is a direct adaptation of \cite[lemma 3.1]{IM},
with the exception that in this case it is not necessary to consider the set
$\Omega_6$  used there for the study of low frequencies and that demands the
 condition $\varepsilon>\frac16+\frac{2|\overline s|}{3}$, which we have
weakened here.

Inequality (3.21) is equivalent to the estimate
\[
\begin{aligned}
&\iint_{\{(\lambda,\lambda_1)\mid{|\xi_1|},{|\xi_2|}\geq1\}}
{|\xi|}{\langle\sigma\rangle}^{\gamma-1}\langle\xi_1
\rangle^{|\overline s|}\langle\xi_2\rangle^{|\overline s|} h(\lambda)
\frac{\overline f({\lambda_1})}{{\langle\sigma_1\rangle}^\gamma{
\langle\theta_1\rangle}^\varepsilon}
\frac{\overline g(\lambda_2)}{{\langle\sigma_2\rangle}^\gamma
{\langle\theta_2\rangle}^\varepsilon}\,d\lambda_1\,d\lambda\\
&\leq C\|h\|\|\overline f\|\|\overline g\|,
\end{aligned} \tag{3.22}
\]
which we will use now to estimate  $J_{A_4}$.
For $(\lambda,\lambda_1)\in A_4$,
\begin{align*}
|{L(\xi_1,\xi_2)}|&\leq 1+\frac{M(\xi)}{{M(\xi_1)}{M(\xi_2)}}
\leq C\frac{{\langle\xi_1\rangle}^{\frac{{\alpha}}2}{\langle\xi_2
\rangle}^{\frac{{\alpha}}2}}{N^{\frac{{\alpha}}2} N^{\frac{{\alpha}}2}}
+\frac C{\frac{N^{|s|}}{{|\xi_1|}^{|s|}}\frac{N^{|s|}}{{|\xi_2|}^{|s|}}}\\
&\leq C\frac{{\langle\xi_1\rangle}^{\frac\alpha 2}{\langle\xi_2\rangle}
^{\frac\alpha 2}}{N^{\alpha}}+C\frac{{\langle\xi_1\rangle}^{|s|}
{\langle\xi_2\rangle}^{|s|}}{N^{|s|} N^{|s|}}\cdot
\frac{{\langle\xi_1\rangle}^{\frac\alpha 2-{|s|}}{\langle\xi_2
\rangle}^{\frac\alpha 2-{|s|}}}{N^{\frac\alpha 2-{|s|}}N^{\frac\alpha 2-{|s|}}}\\
&\leq C\frac{{\langle\xi_1\rangle}^{\frac\alpha 2}{\langle\xi_2
\rangle}^{\frac\alpha 2}}{N^{\alpha}},
\end{align*}
since $\frac\alpha 2-{|s|}\geq0$.

Therefore, using (3.22) with $\overline s=-\frac {\alpha} 2$
(which is possible since if  $2{|s|}<{\alpha}<2-3\gamma$, then
${|s|}<\frac {\alpha} 2<1-\frac32\gamma$ and thus,
$-\frac\alpha 2\in(-\frac14,0)$ and
$(\frac12-\gamma)+\frac23(\frac14-\frac\alpha 2)=\frac13(2-3\gamma-{\alpha})>0$),
it follows that
\[
J_{A_4}\leq CN^{-{\alpha}}{\|h\|\,\|f\|\,\|g\|}.
\tag{3.23}
\]
The statement of Lemma \ref{lm2} follows now from (3.11), (3.13), (3.14), (3.20), and (3.23).
\end{proof}

\section{Proof of Theorem \ref{thm1}}

In our proof we will make use of the homogeneity properties of the KPII equation
which we now describe as:
For $\delta>0$ we define
\begin{align*}
u_{0\delta}(x,y)&:=\delta^\frac23u_0(\delta^\frac13 x,\delta^\frac23 y)\,,\\
u_\delta(x,y,t)&:=\delta^\frac23u(\delta^\frac13x,\delta^\frac23y,\delta t).\\
\end{align*}
Then $u\in {X_{s\gamma\epsilon {\scriptscriptstyle N}}}[0,T]$ is a solution of
the IVP (1.1) in $[0,T]$ with initial datum $u_0$ if and only if
$u_\delta\in {X_{s\gamma\epsilon {\scriptscriptstyle N}}}[0,T/\delta]$ is a
solution of (1.1) in $[0,T/\delta]$ with initial datum $u_{0\delta}$.
We also observe that,
\[
\|u_{0\delta}\|_{s{\scriptscriptstyle N}}\leq CN^{|s|}
\delta^{(\frac16-\frac{|s|} 3)}\|u_0\|_s. \tag{4.1}
\]

Let $C:=\max\{C_1,\overline C\}$, where $C_1$ is the constant in estimates  (1.4) and (1.5) corresponding to $T=1$, and $\overline C$ is the constant in (1.6). Let $R:=\frac1{8C^2}$. If $v_0\in H^{s0}$ with $\|v_0\|_{s{\scriptscriptstyle N}}\leq R$, then from (1.4), (1.5), and (1.6) we can conclude that the operator $\Phi_{v_0}$ defined by
\[ \Phi_{v_0}(w):=\Psi(\cdot_t)W(\cdot_t)v_0-G_1(\partial_x w^2)\]
maps the closed ball $\overline{B(0,2CR)}$ of
${X_{s\gamma\epsilon {\scriptscriptstyle N}}}$ into itself and is a contraction.
Therefore, $\Phi_{v_0}$ has a unique fixed point $w$ in this ball.
Thus, $w|_{{}_{[0,1]}}\in{X_{s\gamma\epsilon {\scriptscriptstyle N}}}[0,1]$ is
a solution in $[0,1]$ of (1.1) with initial datum $v_0$.

Let $u_0\in H^{s0}$ with $\partial_x^{-1}u_0\in {S'}(\mathbb{R}^2)$ and suppose
that  $N$ and $\delta>0$ are such that
\[
\|u_{0\delta}\|_{s{\scriptscriptstyle N}}\leq CN^{|s|}
\delta^{(\frac16-\frac{|s|}3)}\|u_0\|_s=\frac R4. \tag{4.2}
\]
(We are supposing without loss of generality that  $\|u_0\|_s\not=0$).
Since $\|u_{0\delta}\|_{s{\scriptscriptstyle N}}\leq R$, the operator
$\Phi_{u_{0\delta}}$ has a fixed point
$w\in{X_{s\gamma\epsilon {\scriptscriptstyle N}}}$ in $\overline{B(0,2CR)}$.
Let $v:=w|_{{}_{[0,1]}}$. We can take a sequence  $\{v_{0n}\}$ in
$H^\infty(\mathbb{R}^2)$ with $\partial^{-1}_xv_{0n}\in H^\infty(\mathbb{R}^2)$,
such that $v_{0n}\to u_{0\delta}$ in $H^{s0}$ and such that
$\|v_{0n}\|_{s{\scriptscriptstyle N}}\leq\frac R2$.

For each  $n$, let $w_n\in{X_{s\gamma\epsilon {\scriptscriptstyle N}}}$ be the
fixed point in $\overline{B(0,2CR)}$ of $\Phi_{v_{0n}}$ and set
$v_n:=w_n|_{{}_{[0,1]}}$.


Since the local problem is locally well-posed in  $X_{s\gamma\varepsilon}$ for initial data in $H^{s0}$ and the norms $\|\;\|_s$ and $\|\;\|_{s\gamma\varepsilon}$ are equivalent to $\|\;\|_{s{\scriptscriptstyle N}}$ and $\|\;\|_{s\gamma\varepsilon {\scriptscriptstyle N}}$, respectively, we have that
\[
v_n\to v\quad\text{in}\; {X_{s\gamma\epsilon {\scriptscriptstyle N}}}[0,1].
\tag{4.3}
\]
Since $v_{0n}\in H^\infty(\mathbb{R}^2)$ and
$\partial_x^{-1}v_{0n}\in H^\infty(\mathbb{R}^2)$,  from (3.1) and (3.2)
it follows that
\[
\|I_Nv_n(1)\|_{L^2}^2\leq\|I_Nv_{0n}\|_{L^2}^2+C\|\partial_x[I_N(w_n^2)
-(I_Nw_n)^2]\|_{0(\gamma-1)0}\|I_Nw_n\|_{0\gamma0}\,,
\]
and from Lemma \ref{lm2} we obtain that
\begin{align*}
\|I_Nv_n(1)\|_{L^2}^2&\leq \frac {R^2}4
+CN^{-{\alpha}}\|I_Nw_n\|^2_{0\gamma\varepsilon}\|I_Nw_n\|_{0\gamma\varepsilon}
\\
&\leq \frac{R^2}4+CN^{-{\alpha}}\|w_n\|^3_{s\gamma\varepsilon N}\\
&\leq\frac {R^2}4+CN^{-{\alpha}}8C^3R^3\,.
\end{align*}
From (4.3) and the  immersion of ${X_{s\gamma\epsilon {\scriptscriptstyle N}}}[0,1]$
in $C([0,1];H^{s0}_N)$, we conclude that
\[
\|I_Nv(1)\|^2_{L^2}\leq\frac{R^2}4+CN^{-{\alpha}}\,, \tag{4.4}
\]
where $C$ is independent of  $N$ and $\delta$.

In virtue of (4.4), for $k\in\mathbb{N}$, we can obtain a solution of problem (1.1)
with initial datum $u_{0\delta}$ in the interval $[0,k]$ whenever
$(k-1)CN^{-{\alpha}}\leq \frac34 R^2$.

The largest  $k$ with this property is the integer $\overline k$ for which
$(\overline k-1)CN^{-{\alpha}}\leq \frac 34 R^2<\overline k CN^{-{\alpha}}$.
If we wish to have a solution with initial datum $u_0$ in the interval $[0,T]$,
it suffices to have $\overline k>\frac T\delta$. We know from (4.2) that
$\delta^{\frac16-\frac{|s|} 3}=CN^{-{|s|}}$; or, which is the same,
$\delta=CN^{\frac{-6{|s|}}{1-2{|s|}}}$ and
$T/\delta=CTN^{\frac{6{|s|}}{1-2{|s|}}}$. Now, since $\overline k>CN^{\alpha}$,
if we choose  $N$ in such a way that $CN^{\alpha}>CTN^{\frac{6{|s|}}{1-2{|s|}}}$,
then we will have a solution of the problem with initial datum $u_{0\delta}$
in the interval $[0,\frac T\delta]$. This is possible if
\[
{\alpha}>\frac{6{|s|}}{1-2{|s|}}\,. \tag{4.5}
\]

If $s$ is such that $\frac{6{|s|}}{1-2{|s|}}< {2-3\gamma}$, then we can find
${\alpha}$ which satisfies (4.5) and the hypotheses of Lemma \ref{lm2}.
This last inequality is satisfied by an allowed value of  $\gamma>\frac12$
if $\frac{6{|s|}}{1-2{|s|}}<\frac12$; i.e. for $s\in(-\frac 1{14},0)$.

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\end{document}
)