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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 41, pp. 1--18.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/41\hfil Global solutions with infinite energy]
{Global solutions with infinite energy for the one-dimensional Zakharov 
system}
\author[H. Pecher\hfil EJDE-2005/41\hfilneg]
{Hartmut Pecher}

\address{Fachbereich Mathematik und Naturwissenschaften\\
 Bergische Universit\"at Wuppertal\\
 Gau{\ss}str.  20, D-42097 Wuppertal, Germany}
\email{Hartmut.Pecher@math.uni-wuppertal.de}

\date{}
\thanks{Submitted January 11, 2005. Published April 5, 2005.}
\subjclass[2000]{35Q55, 35L05}
\keywords{Zakharov system; global solutions; Fourier restriction norm method}

\begin{abstract}
 The one-dimensional Zakharov system is shown to have a unique global
 solution for data without finite energy. The proof uses the ``I-method''
 introduced by Colliander, Keel, Staffilani, Takaoka, and Tao in
 connection with a refined bilinear Strichartz estimate.
\end{abstract}

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

\section{Introduction}

Consider the (1+1)-dimensional Cauchy problem for the Zakharov system
\begin{gather}
 iu_t + u_{xx} =  nu \label{1} \\
n_{tt}-n_{xx} =  (|u|^2)_{xx} \label{2}\\
 u(0)  =  u_0, \quad
n(0)  =  n_0, \quad n_t(0)  =  n_1 \label{3}
 \end{gather}
where $u$ is a complex-valued and $n$ a real-valued function
defined for $(x,t) \in \mathbb{R}\times\mathbb{R}^+$.
This Zakharov system was introduced in \cite{Z} to describe Langmuir
turbulence in a plasma.

Our main result is the existence of a unique global solution for
data without finite energy, more precisely we assume
$ u_0\in H^s(\mathbb{R})$, $ n_0 \in H^{s-1}(\mathbb{R}) $, $ A^{-1/2}n_1
\in H^{s-1}(\mathbb{R}) $, where $ 1>s>5/6 $, $A:=-\frac{d^2}{dx^2}$.

This result can be proven by using the conservation laws, namely conservation
of $\|u(t)\| $ and
$$
E(u,n):=\|u_x(t)\|^2 + \frac{1}{2}(\|n(t)\|^2 + \|A^{-1/2}n_t(t)\|^2)
+ \int_{-\infty}^{\infty}
n(t)|u(t)|^2\, dx
$$
although under our assumptions these quantities are not finite, in general.

Results of this type were given in various situations in the previous years
in the framework of the Fourier restriction norm method in most of the 
applications.
One approach is to use Bourgain's trick to split
the data into high and low frequency parts. He used it to prove global
well-posedness for the (2+1)-and (3+1)-dimensional Schr\"odinger equations
with rough data without finite energy \cite{B1,B2}
and for the wave equation \cite{B3}. Later it was also used for other model
equations
\cite{CST,FLP,KT,KPV,P2}. Concerning the problem
at hand the author had been able to show global
well-posedness for data $(u_0,n_0,n_1)\in H^s \times L^2 \times \dot{H}^{-1} $
for $1>s>9/10$ \cite{P1}.
Remark here that no data $n_0 \not\in L^2$ were admissible because in such a 
case
the nonlinear part of $n(t)$ could not be shown to belong to $L^2$ which is
necessary for this method. In contrast, the approach
here allows data $n_0 \not\in L^2$ and also less regular data $u_0$.

Another approach was initiated by Colliander, Keel, Staffilani, Takaoka and 
Tao
in \cite{CKSTT3}, called the I-method. The main idea is to use a modified
energy functional which is also defined for less regular
functions and not strictly conserved. When one is able to control its growth
in time explicitly this allows
to iterate a modified local existence theorem to continue the solution to any
time $T$ and moreover to estimate its growth in time. This method was
successfully applied by these authors to several equations
which have a scaling invariance with sometimes even optimal global
well-posedness results. It was used in \cite{CKSTT3} to improve Bourgain's
global well-posedness results \cite{B1,B2} for the
(2+1)- and (3+1)-dimensional Schr\"odinger equation with a further improvement
in \cite{CKSTT6}. Later it was applied to the (1+1)-dimensional derivative
Schr\"odinger equation \cite{CKSTT1} with an (almost)
optimal result in \cite{CKSTT2} and to the KdV and modified KdV equation
with also optimal results in some cases \cite{CKSTT4,CKSTT5}.

Although in our situation such a scaling argument does not work we are able
to suitably modify the method to prove the above mentioned global existence
result for the Zakharov system.

The paper is organized as follows. We transform the system in the usual way
into a first order system.
Then we apply the multiplier $I_N$ for given $s<1$ and $N>>1$ to it, where
$ \widehat{I_N f}(\xi):=m_N(\xi)\widehat{f}(\xi) $. Here $m_N(\xi)$ is a 
smooth,
radially symmetric, and nonincreasing function of $|\xi|$, defined by
$ m_N(\xi) = 1 $ for $| \xi|\le N$ and $m_N(\xi) = (\frac{N}{|\xi|})^{1-s}$ 
for
$ |\xi| \ge 2N $. We drop $N$ from the notation for short and remark that
$I:H^s \to H^1$ is a smoothing operator in the following sense:
$$
\|u\|_{X^{m,b}_{\varphi}} \le c \|Iu\|_{X^{m+1-s,b}_{\varphi}}
\le c N^{1-s}\|u\|_{X^{m,b}_{\varphi}}
$$
Here we used the $X^{m,b}_{\varphi}$-spaces which are defined as
follows: For an equation of the form
$iu_t-\varphi(-i\partial_x)u= 0$, where $\varphi$ is a measurable function, 
let
$X^{m,b}_{\varphi}$ be the completion of 
$\mathcal{S}(\mathbb{R}\times\mathbb{R})$
with respect to
\begin{align*}
\|f\|_{X^{m,b}_{\varphi}}
:=& \|\langle\xi\rangle^m\langle\tau\rangle^b \mathcal{F}
(e^{it\varphi(-i\partial_x)}f(x,t))\|_{L^2_{\xi \tau}} \\
 =& \|\langle\xi\rangle^m\langle\tau+\varphi(\xi)\rangle^b
\widehat{f}(\xi,\tau)\|_{L^2_{\xi\tau}}
\end{align*}
For $\varphi(\xi) = \pm | \xi|$ we use the notation $X^{m,b}_{\pm}$ and
for $\varphi(\xi) = |\xi|^2$ simply $X^{m,b}$. For a given time interval
$I$ we define
$\|f\|_{X^{m,b}(I)} = \inf_{\tilde{f}_{|I} = f}\|\tilde{f}\|_{X^{m,b}} $
and similarly $\|f\|_{X^{m,b}_{\pm}(I)} $.

For the modified (by $I$ multiplied) Zakharov system we then prove a local
existence theorem by using the
precise estimates given by \cite{GTV} for the standard Zakharov system in
connection with an interpolation
type lemma in \cite{CKSTT5}. Our aim is to extract a factor $T^{\delta}$ with
maximal $\delta$ from the
nonlinear estimates in order to give an optimal lower bound for the local
existence time $T$ in terms of
the norms of the data. Because the difference of the differentiability classes
of the data is maximal
(=1), one is forced also to use here the auxiliary spaces
$Y^m_{\varphi}$ (cf. \cite{GTV}), defined by
\begin{align*}
\|f\|_{Y^m_{\varphi}}
:=&  \|\langle\xi\rangle^m \langle\tau\rangle^{-1} \mathcal{
F}(e^{-it\varphi(-i\partial_x)}f(x,t))\|_{L^2_{\xi}L^1_{\tau}} \\
 =&  \|\langle\xi\rangle^m\langle\tau +\varphi(\xi)\rangle^{-1}
\widehat{f}(\xi,\tau)\|_{L^2_{\xi}L^1_{\tau}}
\end{align*}
As is typical for the $I$-method one then has to consider in
detail the modified energy functional $E(Iu,In)$ and to control
its growth in time in dependence of the time interval and the
parameter $N$ (cf. the definition of $I$ above). The increment of
the energy has to be small for small time intervals and large $N$.
Because the modified energy functional is somehow close to the
original one here some sort of cancellation helps. An important
tool is also a refined Strichartz estimate for the product of a
wave and a Schr\"odinger part along the lines of Bourgain's
improvements for the simpler pure Schr\"odinger case (cf. Lemma
\ref{LemmaA}). This estimate for the modified energy functional
can also control the growth of the corresponding norms of the
solution of the problem during its time evolution. One iterates
the local existence theorem with time steps of equal length in
order to reach any given fixed time $T$. To achieve this one has
to make the process uniform which can be done if $s$ is close
enough to $1$ (namely $s>5/6$).

We collect some elementary facts about the spaces
$X^{m,b}_{\varphi}$ and $Y^m_{\varphi}$.
The following interpolation property is well-known:\\
$ X^{(1-\Theta)m_0+\Theta m_1,(1-\Theta)b_0+\Theta b_1}_{\varphi} =
(X^{m_0,b_0}_{\varphi},X^{m_1,b_1}_{\varphi})_{[\Theta]} $ for
$ \Theta \in [0,1] $.

If $u$ is a solution of $iu_t+\varphi(-i\partial_x)u = 0$ with
$u(0)=f$ and $\psi$ is a cutoff function in
$C^{\infty}_0(\mathbb{R})$ with $\mathop{\rm supp} \psi \subset (-2,2)$,
$\psi \equiv 1$ on $[-1,1]$ , $ \psi(t) = \psi(-t) $, $
\psi(t)\ge 0 $ , $\psi_{\delta}(t):=\psi(\frac{t}{\delta})$,
$ 0<\delta \le 1$, we have for $b>0$:
$$
\|\psi_1 u\|_{X^{m,b}_{\varphi}} \le c \|f\|_{H^m}
$$
If $v$ is a solution of the problem $iv_t +\varphi(-i\partial_x)v = F $,
$ v(0)=0 $, we have for $b'+1 \ge b \ge 0 \ge b' > -1/2$
$$
\|\psi_{\delta} v \|_{X^{m,b}_{\varphi}} \le c \delta^{1+b'-b}
 \|F\|_{X^{m,b'}_{\varphi}}
$$
and, if $b'+1 \ge b \ge 0 \ge b'$, we have
$$
\|\psi_{\delta} v \|_{X^{m,b}_{\varphi}} \le c (\delta^{1+b'-b}
\|F\|_{X^{m,b'}_{\varphi}} +
\delta^{\frac{1}{2}-b} \|F\|_{Y^m_{\varphi}})
$$
(for a proof cf. \cite{GTV}, Lemma 2.1). Moreover, if
$w(t) = \int_0^t e^{i(t-s)\varphi(-i\partial_x)}F(s) \, ds $ we have by
 \cite[Lemma 2.2]{GTV}, especially (2.35), for $\delta \le 1 $
\begin{equation}
\label{b}
\|w\|_{C^0([0,\delta],H_x^1)} \le c \|F\|_{Y^1_{\varphi}[0,\delta]}
\end{equation}
Finally, if $1/2 > b > b' \ge 0$ , $ m \in \mathbb{R} $, we have
the embedding
\begin{equation}
\|f\|_{X^{m,b'}_{\varphi}[0,\delta]} \le c \delta^{b-b'} 
\|f\|_{X^{m,b}_{\varphi}[0,\delta]}
\end{equation}
For the convenience of the reader we repeat the proof of \cite{G}, Lemma 1.10. 
The claimed estimate is an
immediate consequence of the following

\begin{lemma} \label{lem0.1}
For $1/2 > b > b' \ge 0$ , $ 0 < \delta \le 1 $ , $ m \in
\mathbb{R} $ the following estimate holds:
$$
\| \psi_{\delta} f \|_{X^{m,b'}_{\varphi}} \le c \delta^{b-b'}
\|f\|_{X^{m,b}_{\varphi}}
$$
\end{lemma}

\begin{proof} The following Sobolev multiplication rule holds:
$$
\|fg\|_{H^{b'}_t} \le c \|f\|_{H^{\frac{1}{2}-(b-b')}_t} \|g\|_{H^b_t}
$$
This rule follows easily by the Leibniz rule for fractional derivatives, using 
$J^s := \mathcal{
F}^{-1}\langle\tau\rangle^s \mathcal{F}$:
\begin{align*}
\|fg\|_{H^{b'}_t} & \le  c(\|(J^{b'}f)g\|_{L^2_t} + \|f(J^{b'}g)\|_{L^2_t}) \\
& \le  c(\|J^{b'}f\|_{L^p_t} \|g\|_{L^{p'}_t} + \|f\|_{L^{q'}_t}
\|J^{b'}g\|_{L^q_t})
\end{align*}
with $\frac{1}{p}=b$, $ \frac{1}{p'} = \frac{1}{2}-b $,
$ \frac{1}{q'}=b-b' $, $ \frac{1}{q} =
\frac{1}{2} - (b-b') $. Sobolev's embedding theorem gives the claimed result.
Consequently we get
$$
\|\psi_{\delta}g\|_{H^{b'}_t} \le c 
\|\psi_{\delta}\|_{H^{\frac{1}{2}-(b-b')}_t}
\|g\|_{H^b_t} \le c \delta^{b-b'} \|g\|_{H^b_t}
$$
and thus
\begin{align*}
\|\psi_{\delta}f\|_{X^{m,b'}_{\varphi}}
&= \|e^{it\varphi(-i\partial_x)}\psi_{\delta}f\|_{H^m_x\otimes H^{b'}_t}\\
& \le  c \delta^{b-b'}\|e^{it\varphi(-i\partial_x)}f\|_{H^m_x \otimes H^b_t}\\
& =  c\delta^{b-b'}\|f\|_{X^{m,b}_{\varphi}}
\end{align*}
Fundamental are the following linear Strichartz type estimates for the
Schr\"odinger equation (cf. e.g. \cite[Lemma 2.4]{GTV}):
$$
 \|e^{it\partial^2_x} \psi\|_{L^q_t(I,L_x^r(\mathbb{R}))}
\le c \|\psi\|_{L^2_x(\mathbb{R})}
$$
and
$$
\|u\|_{L^q_t(I,L_x^r(\mathbb{R}))} \le c \|u\|_{X^{0,\frac{1}{2}+}(I)}
$$
if $ 0 \le \frac{2}{q} = \frac{1}{2} - \frac{1}{r} $, especially
$$
\|u\|_{L^6_{xt}} \le c \|u\|_{X^{0,\frac{1}{2}+}}
$$
which by interpolation with the trivial case $\|u\|_{L^2_{xt}} = 
\|u\|_{X^{0,0}}$
gives:
$$
\|u\|_{L^p_{xt}} \le c \|u\|_{X^{0,\frac{3}{2}(\frac{1}{2}-\frac{1}{p})+}}
$$
if $ 2 < p \le 6 $. For the wave equation we only use
$ \|n_{\pm}\|_{L^{\infty}_t L^2_x} \le c
\|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}}$.
\end{proof}
We use the notation $\langle \lambda \rangle := (1+\lambda^2)^{1/2} $.
Let $a \pm$ denote a number
slightly larger (resp., smaller) than $a$.

\section{Local existence}

The system (\ref{1})--(\ref{3}) has the following two quantities conserved:
$ M:=\|u(t)\|$ and
$$
E(u,n):= \|A^{1/2}u(t)\|^2 + 1/2(\|n(t)\|^2 + \|V(t)\|^2)
+ \int_{-\infty}^{+\infty} n(t)|u(t)|^2 \, dx
$$
where $ V_x := -n_t$ and $ A:=-\frac{d^2}{dx^2}$ . \\
The system (\ref{1})--(\ref{3}) is now transformed into a first order system
in $t$ as follows: with $ n_{\pm}:= n \pm iA^{-1/2}n_t $ , i.e.
$ n=\frac{1}{2}(n_++n_-) ,$ $2iA^{-1/2}n_t=n_+-n_-$, and
$\overline{n_+} = n_- $ this gives
\begin{gather} \label{1'}
iu_t + u_{xx}  =  \frac{1}{2}(n_+ + n_-)u \\
\label{2'}
in_{\pm t} \mp A^{1/2}n_{\pm} =  \pm A^{1/2}(|u|^2) \\
\label{3'}
u(0) = u_0\,, \quad n_{\pm}(0) =
 n_{\pm 0} :=  n_0 \pm iA^{-1/2}n_1\,.
\end{gather}
The energy is given by
$$
E(u,n_+)  = \|A^{1/2}u\|^2 + \frac{1}{2} \|n_+\|^2
+ \frac{1}{2} \int (n_++\overline{n_+})|u|^2 \, dx
$$
By Gagliardo-Nirenberg,
\begin{align*}
\int n|u|^2 \, dx &\le \frac{1}{4}\int n^2 \, dx + c \int |u|^4 \, dx \\
& \le  \frac{1}{4} \|n\|^2 + c \|u_x\| \|u\|^3 \\
& \le  \frac{1}{4}(\|n\|^2 + \|u_x\|^2) + c_0 \|u\|^6
\end{align*}
This implies
\begin{equation}\label{*}
\|A^{1/2}u\|^2 + \|n\|^2 + \|V\|^2 \le c(E+\|u\|^6) = c_0(E+M^6)
\end{equation}
and
\begin{equation}\label{*'}
E \le c_0(\|A^{1/2}u\|^2 + \|n\|^2 + \|V\|^2 +M^6)
\end{equation}
We want to apply the I-method (for the definition of I see the introduction).
A crucial role is played by the modified energy $E(Iu,In_+)$ for the system
\begin{gather} \label{1''}
iIu_t + Iu_{xx}  =  \frac{1}{2} I[(n_+ +n_-)u] \\
\label{2''}
iIn_{\pm t} \mp A^{1/2}In_{\pm}  =  \pm IA^{1/2}(|u|^2) \\
\label{3''}
Iu(0) = Iu_0 , \quad  In_{\pm} (0)  =  In_{\pm 0} = I(n_0 \pm iA^{-1/2}n_1),
\end{gather}
namely
$$
E(Iu,In_+) := \|Iu_x\|^2 + \frac{1}{2} \|In_+\|^2
+ \frac{1}{2} \int I(n_+ + \overline{n_+})|Iu|^2 \,dx
$$
 which is not conserved but its growth is controllable.
An elementary but lengthy calculation shows
\begin{equation}
\begin{aligned}
 \frac{d}{dt} & E(Iu,In_+)  \\
 = & \mathop{\rm Re} \langle I(n_+ + \overline{n_+}) Iu - I((n_+ +
\overline{n_+})u),Iu_t\rangle
+ \mathop{\rm Re} \langle In_+,iA^{1/2}(|Iu|^2 - I(|u|^2))\rangle
\end{aligned} \label{4}
\end{equation}
If $I= id $ this again shows the conservation of $E(u,n_+)$.

Before considering this modified energy in detail we give a local existence 
result for the system
\eqref{1''}--\eqref{3''}, which essentially uses the bilinear estimates given 
by
\cite{GTV} for their local existence result of the Zakharov system.

\begin{prop} \label{Prop1}
Assume $ s \ge 1/2 $. Let $(u_0,n_{+0},n_{-0}) \in H^s \times H^{s-1} \times 
H^{s-1} $ be given. Then
there exists a positive number $\delta \sim \frac{1}{(\|Iu_0\|_{H^1} + 
\|In_{+0}\|_{L^2} +
\|In_{-0}\|_{L^2})^{4+}} $ such that the system
\eqref{1''}--\eqref{3''} has a unique local solution in the time interval
$[0,\delta]$ with the property (dropping from now on $[0,\delta]$ from the
notation):
$$
\|Iu\|_{X^{1,\frac{1}{2}}} + \|In_+\|_{X^{0,\frac{1}{2}+}_+} + 
\|In_-\|_{X^{0,\frac{1}{2}+}_-} \le
c(\|Iu_0\|_{H^1} + \|In_{+0}\|_{L^2} + \|In_{-0}\|_{L^2})
$$
This solution also belongs to $C^0([0,\delta],H^1_x(\mathbb{R}))$ and
$$
\|Iu\|_{C^0([0,\delta],H^1_x(\mathbb{R}))} \le c ( \|Iu_0\|_{H^1} +  
\|In_{+0}\|_{L^2} + \|In_{-0}\|_{L^2})
$$
\end{prop}

\begin{proof}
We use the corresponding integral equations to define a mapping $S=(S_0,S_1)$ 
by
\begin{gather*}
S_0(Iu(t))  =  Ie^{it\partial^2_x}u_0 + \frac{1}{2}\int_0^t
e^{i(t-s)\partial^2_x}I(u(s)((n_+(s)+n_-(s)))\, ds \\
S_1(In_{\pm}(t))  =  Ie^{itA^{1/2}}n_{\pm 0} \pm i \int_0^t
e^{\mp i(t-s)A^{1/2}}A^{1/2}I(|u(s)|^2)\, ds
\end{gather*}
We use \cite[Lemma 4.3]{GTV}  to conclude for $s \ge 1/2 $,
$$
 \|n_{\pm}u\|_{X^{s,-\frac{1}{2}}} \le c 
\|n_{\pm}\|_{X^{s-1,\frac{3}{8}+}_{\pm}}
\|u\|_{X^{s,\frac{3}{8}+}}\,.
$$
Similarly \cite[Lemma 4.5]{GTV}  shows
$$
\|n_{\pm}u\|_{Y^s} \le c \|n_{\pm}\|_{X^{s-1,\frac{3}{8}+}_{\pm}}
\|u\|_{X^{s,\frac{3}{8}+}}\,.
 $$
Finally, \cite[Lemma 4.4 ]{GTV} shows that for $s \ge 0$,
$$
\|A^{1/2}(|u|^2)\|_{X^{s-1,-\frac{1}{2}+}_{\pm}}
\le c \|u\|_{X^{s,\frac{1}{4}++}}^2
$$
These estimates imply similar estimates including the I-operator by the
interpolation lemma of \cite{CKSTT5}, namely
$$
\|I(n_{\pm}u)\|_{X^{1,-\frac{1}{2}}} + \|I(n_{\pm}u)\|_{Y^1} \le c
\|In_{\pm}\|_{X^{0,\frac{3}{8}+}_{\pm}} \|Iu\|_{X^{1,\frac{3}{8}+}}
$$
and
$$
\|IA^{1/2}(|u|^2)\|_{X^{0,-\frac{1}{2}+}_{\pm}}
\le c \|Iu\|_{X^{1,\frac{1}{4}++}}^2
$$
where $c$ is independent of $N$.

The same estimates also hold true for functions defined on $[0,\delta]$ only, 
and for such functions we
can also use the embedding (5). This gives
$$
\|I(n_{\pm}u)\|_{X^{1,-\frac{1}{2}}} + \|I(n_{\pm}u)\|_{Y^1} \le c
\|In_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}} \|Iu\|_{X^{1,\frac{1}{2}}}
\delta^{\frac{1}{4}-}
$$
and
$$
\|IA^{1/2}(|u|^2)\|_{X^{0,-\frac{1}{2}+}_{\pm}} \le c 
\|Iu\|_{X^{1,\frac{1}{2}}}^2
\delta^{\frac{1}{2}-}
$$
Using these estimates the integral equations lead to (remark here that one 
needs the space $Y^1$, cf.
\cite[Lemma 2.1]{GTV}):
\begin{gather*}
 \|S_0(Iu)\|_{X^{1,\frac{1}{2}}}
\le  c \|Iu_0\|_{H^1}+ c (\|In_+\|_{X^{0,\frac{1}{2}+}_+} +
\|In_-\|_{X^{0,\frac{1}{2}+}_-}) \|Iu\|_{X^{1,\frac{1}{2}}} 
\delta^{\frac{1}{4}-}\,,
\\
 \|S_1(In_{\pm})\|_{X^{0,\frac{1}{2}+}_{\pm}}  \le  c \|In_{\pm 0}\|_{L^2} + c
\|Iu\|_{X^{1,\frac{1}{2}}}^2 \delta^{\frac{1}{2}-}
 \end{gather*}
 The standard contraction argument gives the existence of a unique solution
on $[0,\delta]$ with
 $$
\|Iu\|_{X^{1,\frac{1}{2}}} + \|In_+\|_{X^{0,\frac{1}{2}+}_+}
+ \|In_-\|_{X^{0,\frac{1}{2}+}_-}
  \le 2c(\|Iu_0\|_{H^1} + \|In_{+0}\|_{L^2} + \|In_{-0}\|_{L^2})
$$
 provided
$$
c \delta^{\frac{1}{4}-}(\|Iu_0\|_{H^1} + \|In_{+0}\|_{L^2}
+ \|In_{-0}\|_{L^2}) <1
$$
Concerning the property $Iu \in C^0([0,\delta],H^1_x)$ we refer to
\cite[Lemma 2.2]{GTV},  (use the first
integral equation and $I(un_{\pm}) \in Y^1$). Moreover (\ref{b}) gives
\begin{align*}
\|Iu\|_{C^0([0,\delta],H^1)}
& \le  \|Iu_0\|_{H^1} + c (\|I(n_+u)\|_{Y^1} + \|I(n_-u)\|_{Y^1}) \\
& \le  \|Iu_0\|_{H^1} + c \delta^{\frac{1}{4}-} 
(\|In_+\|_{X^{0,\frac{1}{2}+}_+} +
\|In_-\|_{X^{0,\frac{1}{2}+}_-}) \|Iu\|_{X^{1,\frac{1}{2}}} \\
& \le  \|Iu_0\|_{H^1} + c
\frac{(\|Iu_0\|_{H^1}+\|In_{+0}\|_{L^2}+\|In_{-0}\|_{L^2})^2}{\|Iu_0\|_{H^1}+
\|In_{+0}\|_{L^2}+\|In_{-0}\|
_{L^2}} \\
& \le  c(\|Iu_0\|_{H^1}+\|In_{+0}\|_{L^2}+\|In_{-0}\|_{L^2})
\end{align*}
\end{proof}

\section{A bilinear Strichartz estimate}

\begin{lemma} \label{Lemma2.1}
$$
 \|(D^{1/2}_x u) n_{\pm}\|_{L^2_{xt}} \le c 
\|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}}
\|u\|_{X^{0,\frac{1}{2}+}} $$
\end{lemma}

\begin{proof} We split the domain of integration into two parts
(a) $ \mathop{\rm supp } \widehat{u} \subset \{ |\xi|\ge 2 \} $
(b) $ \mathop{\rm supp } \widehat{u} \subset \{ |\xi| \le 2 \} $

(a) We assume $ \mathop{\rm supp} \widehat{m_{\pm}} \subset \{ \xi \ge 0 \} $
(the other part $ \mathop{\rm supp}\widehat{m_{\pm}} \subset \{ \xi \le 0 \} $
can be treated similarly) and
$ \mathop{\rm supp} \widehat{v} \subset \{|\xi| \ge 2 \} $. In this region
we have
\begin{equation} \label{***}
\begin{aligned}
\, & \|e^{it\partial_x^2}D^{1/2}_x v e^{\pm it|\partial_x|} 
m_{\pm}\|_{L^2_{xt}}^2 \\
& =  \int d\xi dt \Big| \int_{ \xi = \xi_1 + \xi_2 ,  \xi_2 \ge 0 , |\xi_1| 
\ge
2}e^{-it\xi_1^2\pm it|\xi_2|} \widehat{v}(\xi_1)
\widehat{m_{\pm}}(\xi_2)|\xi_1|^{1/2} d\xi_1 \Big|^2 \\
& =  \int d\xi dt \int_{\xi=\xi_1+\xi_2=\eta_1+\eta_2 ,  \eta_2,\xi_2\ge 0 ,
|\xi_1|,|\eta_1|\ge 2 } e^{-it(\xi_1^2\pm |\xi_2| - \eta_1^2 \mp |\eta_2|)}
\widehat{v}(\xi_1)\overline{\widehat{v}(\eta_1)} \\
&\quad \times \widehat{m_{\pm}}(\xi_2) \overline{\widehat{m_{\pm}}(\eta_2)}
|\xi_1|^{1/2} |\eta_1|^{1/2}d\xi_1 d\eta_1  \\
& =  \int d\xi \int d\xi_1 d\eta_1 \delta
(P(\eta_1))\widehat{v}(\xi_1)\overline{\widehat{v}(\eta_1)}\widehat{m_{\pm}}(\xi
_2)\overline{\widehat{m_{\pm}
}(\eta_2)} |\xi_1|^{1/2}|\eta_1|^{1/2}
\end{aligned}
\end{equation}
with
\begin{align*}
P(\eta_1) & :=  \xi_1^2 \pm |\xi_2| - \eta_1^2 \mp |\xi - \eta_1| = \xi_1^2
\pm |\xi_2| - \eta_1^2 \mp |\eta_2| \\
& =  \xi_1^2 \pm \xi_2 - \eta_1^2 \mp \eta_2 = \xi_1^2 \pm (\xi -\xi_1)
- \eta_1^2 \mp (\xi - \eta_1) \\
& =  \xi_1^2 - \eta_1^2 \mp (\xi_1 - \eta_1) = (\xi_1 - \eta_1)
[(\xi_1+\eta_1) \mp 1]
\end{align*}
This function has the simple zeroes $\eta_1 = \xi_1$ and $\eta_1 = \pm 1 - 
\xi_1 $.
Moreover $ P'(\eta_1) = -2\eta_1 \pm 1$ , thus $ |P'(\eta_1)| \sim |\eta_1| $
in our region $|\eta_1| \ge 2$. Using the
well-known identity
$$
\int \delta(P(\eta_1))f(\eta_1)\, d\eta_1 = \sum \frac{f(x_k)}{|P'(x_k)|}
$$
where $x_k$ denotes the simple zeroes of $P$, we remark that in our case for 
the zeroes we have $|\eta_1|
\sim |\xi_1| $, and therefore the factor $|\xi_1|^{1/2} |\eta_1|^{1/2}$ 
cancels with
$|P'(x_k)|$. Thus we can estimate (\ref{***}) using Schwarz' inequality by
\begin{align*}
&c \int d\xi \int d\xi_1 
|\widehat{v}(\xi_1)\overline{{\widehat{v}(\xi_1)}}\widehat{m_{\pm}}(\xi -
\xi_1) \overline{\widehat{m_{\pm}}(\xi - \xi_1)}| \\
& +  c\int d\xi \int d\xi_1 |\widehat{v}(\xi_1)\overline{{\widehat{v}(\pm 1
-\xi_1)}}\widehat{m_{\pm}}(\xi - \xi_1) \overline{\widehat{m_{\pm}}(\xi -(\pm 
1 - \xi_1))}| \\
& \le  c \|\widehat{v}\|_{L^2}^2 \|\widehat{m_{\pm}}\|_{L^2}^2 = c 
\|v\|_{L^2}^2 \|m_{\pm}\|_{L^2}^2
\end{align*}
and the claimed estimate follows directly in the region
$\mathop{\rm supp} \widehat{u} \subset \{|\xi|\ge 2\}$ (cf.
e.g. \cite[Lemma 1.4]{P2}, \cite[Lemma 2.1]{G},  \cite[Section 3]{KS}).

\noindent (b) In the region
$ \mathop{\rm supp} \widehat{u} \subset \{ |\xi| \le 2 \} $ we have
\begin{align*}
\|(D_x^{1/2}u)n_{\pm}\|_{L^2_{xt}}
& \le  \|D_x^{1/2}u\|_{L^2_t L^{\infty}_x}
\|n_{\pm}\|_{L^{\infty}_t L^2_x} \le c \|D^{1/2}_x u\|_{L^2_t 
H^{\frac{1}{2}+}_x}
\|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}} \\
& \le  c \|u\|_{L^2_t L^2_x} \|n_{\pm}\|_{X_{\pm}^{0,\frac{1}{2}+}} \le  c 
\|u\|_{X^{0,\frac{1}{2}+}}
\|n_{\pm}\|_{X_{\pm}^{0,\frac{1}{2}+}}
\end{align*}
\end{proof}

\begin{lemma} \label{LemmaA}
$$ \|(D^{1/2}_x u) n_{\pm}\|_{L^2_{xt}} \le c 
\|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}}
\|u\|_{X^{0+,\frac{1}{2}}} \,.
$$
\end{lemma}

For the proof of the above lemma, we interpolate the estimate of the previous 
lemma with
\begin{align*}
\|(D^{1/2}_x u)n_{\pm}\|_{L^2_{xt}}
& \le  \|n_{\pm}\|_{L^{\infty}_t L^2_x} \|D^{1/2}_x
u\|_{L^2_t L^{\infty}_x} \le c \|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}}
\|u\|_{L^2_t H^{1+}_x} \\
& \le  c \|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}} \|u\|_{X^{1+,0}}\,.
\end{align*}

\begin{lemma} \label{LemmaA'}
$$
\|(D^{1/2}_x u) n_{\pm}\|_{L^{2+}_t L^2_x}
\le c \|n_{\pm}\|_{X^{0,\frac{1}{2}}_{\pm}}
\|u\|_{X^{0+,\frac{1}{2}}}\,.
$$
\end{lemma}

\begin{proof} By Sobolev's embedding theorem and Strichartz' estimate we have
\begin{align*}
\|(D_x^{1/2}u)n_{\pm}\|_{L^{4-}_tL^2_x}
& \le c \|n_{\pm}\|_{L^{\infty -}_tL^2_x} \|D^{1/2}_x u\|_{L^4_tL^{\infty}_x} 
\\
& \le c \|n_{\pm}\|_{X^{0,\frac{1}{2}-}_{\pm}} \|D^{1/2}_x
u\|_{L^4_tH_x^{\frac{1}{4}+,4}} \\
& \le  c \|n_{\pm}\|_{X^{0,\frac{1}{2}-}_{\pm}} \|u\|_{X^{\frac{3}{4}+,
\frac{3}{8}+}}
\end{align*}
Interpolation with Lemma \ref{Lemma2.1} gives the claimed result.
\end{proof}
A variant of this lemma is given next.

\begin{lemma} \label{LemmaB}
$$
\|(\widehat{D^{1/2}_x u}) \ast \widehat{n_{\pm}} \|_{L^2_{\xi}L^{2-}_{\tau}} 
\le c
\|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}} \|u\|_{X^{0+,\frac{1}{2}}}
 $$
\end{lemma}

\begin{proof} On the one hand, Lemma \ref{Lemma2.1} gives
\begin{equation}
\label{**''}
\|(\widehat{D^{1/2}_x u}) \ast \widehat{n_{\pm}} \|_{L^2_{\xi \tau}} \le c
\|n_{\pm}\|_{X^{0,\frac{1}{2}+}_{\pm}} \|u\|_{X^{0,\frac{1}{2}+}}\,.
\end{equation}
On the other hand Young's inequality shows
\begin{equation}
\label{*''}
\|(\widehat{D^{1/2}_x u}) \ast \widehat{n_{\pm}} 
\|_{L^2_{\xi}L^{\frac{4}{3}}_{\tau}} \le c
\|\widehat{n_{\pm}}\|_{L^2_{\xi}L^1_{\tau}}
\|\widehat{D^{1/2}_x u}\|_{L^1_{\xi} L^{\frac{4}{3}}_{\tau}}
\end{equation}
Now by Schwarz' inequality
\begin{align*}
 \|\widehat{n_{\pm}}\|_{L^2_{\xi}L^1_{\tau}}
& =  \|\int |\widehat{n_{\pm}}(\xi,\tau)| \langle \tau \pm
|\xi| \rangle ^{\frac{1}{2}+} \langle \tau \pm |\xi| \rangle ^{-\frac{1}{2}-}
d\tau \|_{L^2_{\xi}} \\
& \le  c \|\widehat{n_{\pm}}(\xi,\tau) \langle \tau \pm |\xi| \rangle 
^{\frac{1}{2}+} \|_{L^2_{\xi \tau}}
  =  c \|n_{\pm}\|_{X^{0,\frac{1}{2}+}}
\end{align*}
 and by H\"older's inequality in $\tau$ and Schwarz' inequality in $\xi$:
\begin{align*}
 \|\widehat{D^{1/2}_x u}\|_{L^1_{\xi}L^{\frac{4}{3}}_{\tau}}
& =  \|\widehat{D^{1/2}_x u}(\xi,\tau)
\langle \tau + \xi^2 \rangle ^{\frac{1}{4}+} \langle \tau + \xi^2 \rangle
^{-\frac{1}{4}-}\|_{L^1_{\xi}L^{\frac{4}{3}}_{\tau}} \\
& \le  c \|\widehat{D^{1/2}_x u}(\xi,\tau) \langle \tau + \xi^2 \rangle 
^{\frac{1}{4}+}
\|_{L^1_{\xi}L^2_{\tau}} \\
& =   c \|\widehat{D^{1/2}_x u}(\xi,\tau) \langle \tau + \xi^2 \rangle 
^{\frac{1}{4}+} \langle \xi
\rangle ^{\frac{1}{2}+}  \langle \xi \rangle 
^{-\frac{1}{2}-}\|_{L^1_{\xi}L^2_{\tau}} \\
& \le  c \|\widehat{D^{1/2}_x u}(\xi,\tau) \langle \tau + \xi^2 \rangle 
^{\frac{1}{4}+} \langle \xi
\rangle ^{\frac{1}{2}+}  \|_{L^2_{\xi \tau}} \\
& \le  c \|u\|_{X^{1+,\frac{1}{4}+}}
\end{align*}
Interpolating (\ref{*''}) and (\ref{**''}) we get the result.
\end{proof}

We also need a bilinear Strichartz' refinement for the pure Schr\"odinger 
problem.
We have the well-known

\begin{lemma} \label{LemmaC}
If $u_1$,$u_2$ fulfill $|\xi_1| \gg  |\xi_2| \ge 1$ for $\xi_i \in
\mathop{\rm supp} \widehat{u_i} $ $(i=1,2)$, the
following estimate holds:
$$
\|(D_x^{1/2}u_1) u_2\|_{L^2_{xt}} \le c \|u_1\|_{X^{0,\frac{1}{2}+}}
\|u_2\|_{X^{0,\frac{1}{2}+}}\,.
$$
\end{lemma}
The proof the lemma above can be found in \cite[Lemma 7.1]{CKSTT1}.

We also have the following variant.

\begin{lemma} \label{LemmaD}
Under the assumptions of Lemma \ref{LemmaC} we have:
$$
\|(D_x^{1/2}u_1) u_2\|_{L^{2+}_t L^2_x} \le c \|u_1\|_{X^{0+,\frac{1}{2}}}
\|u_2\|_{X^{0,\frac{1}{2}}}
$$
\end{lemma}

The proof is similar to that of Lemma \ref{LemmaA'}.

\noindent{\bf Remark:} All the estimates in this section remain true,
if any of the functions on the left-hand sides  of the
estimates are replaced by their complex conjugates.

\section{Estimates for the modified energy}

The main step towards global existence is an exact control of the increment
of the modified energy.

\begin{prop} \label{Prop3.1}
Let $(u,n_{\pm})$ be a solution of \eqref{1'}--\eqref{3'} on $[0,\delta]$
in the sense of Proposition \ref{Prop1}. Then the following estimate holds
(for $N \ge 1 $, $ s > 3/4 $):
\begin{align*}
 &|E(Iu(\delta),In_+(\delta)) - E(Iu(0),In_+(0))|  \\
&  \le   c( (N^{-\frac{1}{2}+} \delta^{\frac{1}{2}-} + N^{-\frac{3}{2}+} 
\delta^{0+})
\|In_+\|_{X^{0,\frac{1}{2}+}_+}   \|Iu\|_{X^{1,\frac{1}{2}}}^2 \\
&+ (N^{-3+} + N^{-1+}\delta^{\frac{1}{2}-})\|In_+\|_{X^{0,\frac{1}{2}+}_+} ^2
\|Iu\|_{X^{1,\frac{1}{2}}}^2)\,.
\end{align*}
\end{prop}

\begin{proof} Using (\ref{4}) and replacing $Iu_t$ by (\ref{1''}), we have to
show
\begin{equation} \label{5}
\Big|\int_0^{\delta} \int In_+ A^{1/2}(|Iu|^2 - I(|u|^2)) dxdt \Big|
 \le \frac{c}{N^{1-}}\delta^{\frac{1}{2}-} \|In_+\|_{X^{0,\frac{1}{2}+}_+}
\|Iu\|_{X^{1,\frac{1}{2}}}^2
\end{equation}
and
\begin{equation} \label{6}
\begin{aligned}
\, & \Big| \int_0^{\delta} \int (Iu)_{xx}(I(n_+u)-In_+Iu)dxdt\Big|   \\
& \le c(N^{-\frac{1}{2}+}\delta^{\frac{1}{2}-}
+ N^{-\frac{3}{2}+} \delta^{0+}) \|In_+\|_{X^{0,\frac{1}{2}+}_+}
\|Iu\|_{X^{1,\frac{1}{2}}}^2
\end{aligned}
\end{equation}
as well as
\begin{equation} \label{7}
\begin{aligned}
\, &\Big| \int_0^{\delta} \int I(n_+u)(I(n_+u)-In_+Iu)dxdt\Big| \\
&\le c (\frac{1}{N^{3-}} + N^{-1+} \delta^{\frac{1}{2}-}) \|In_+
\|_{X^{0,\frac{1}{2}+}_+}^2 \|Iu\|_{X^{1,\frac{1}{2}}}^2
\end{aligned}
\end{equation}
Here and in the sequel we assume, without loss of generality, that the Fourier
transforms of all these functions to be nonnegative,
ignore the appearance of complex conjugates, use dyadic decompositions
with respect to the frequencies $|\xi_j|
\sim N_j = 2^k $ $(k=0,1,2,\dots)$. To sum over the dyadic pieces at the end,
we need to have extra factors $N_j^{0-}$ everywhere.

We start with (\ref{5}) which follows from
\begin{equation}
\begin{aligned}
\, &\int_0^{\delta} \int_* \widehat{n_+}(\xi_1,t)|\xi_2 + \xi_3|^{1/2} \Big|
\frac{m(\xi_2+\xi_3)-m(\xi_2)m(\xi_3)}{m(\xi_2)m(\xi_3)} \Big|
\widehat{u_2}(\xi_2,t)\widehat{u_3}(\xi_3,t) d\xi dt  \\
& \le  \frac{c}{N^{1-}} \delta^{\frac{1}{2}-} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
\|u_2\|_{X^{1,\frac{1}{2}}}\|u_3\|_{X^{1,\frac{1}{2}}}\,.
\end{aligned} \label{8}
\end{equation}
Here and in the sequel, * denotes integration over the set
$\sum_{i=1}^3 \xi_i = 0 $ (or $\sum_{i=1}^4 \xi_i = 0 $).
The symmetry in $\xi_2,\xi_3$ allows to assume $N_2 \ge N_3$, and moreover we
can assume $N_2 \ge N$, because otherwise the symbol is $\equiv 0$.
The condition $\sum_{i=1}^3 \xi_i =0$ implies $N_1 \le cN_2$.
Thus $N_2 \sim N_{max}$, where $ N_{max} := \max(N_1,N_2,N_3) $.

We have $ |\frac{m(\xi_2+\xi_3)-m(\xi_2)m(\xi_3)}{m(\xi_2)m(\xi_3)}|
\le \frac{c}{|m(\xi_2)||m(\xi_3)|} \le
c\left\langle(\frac{N_2}{N})^{1/2}\right\rangle $ and thus the bound
\begin{equation}
\begin{aligned}
\, & c \int_0^{\delta} \int_* \widehat{n_+}(\xi_1,t)|\xi_2|^{1/2}
\widehat{u_2}(\xi_2,t)\widehat{u_3}(\xi_3,t)d\xi dt
\langle(\frac{N_2}{N})^{1/2}\rangle \\
& \le  c \|n_+ D^{1/2}_x u_2\|_{L^2_{xt}} \|u_3\|_{L^2_{xt}}
\langle (\frac{N_2}{N})^\frac{1}{2} \rangle \\
& \le  c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{0+,\frac{1}{2}}}
\|u_3\|_{X^{0,0}} \langle (\frac{N_2}{N})^{1/2} \rangle \\
& \le  c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \frac{1}{N_2^{1-}}
\|u_2\|_{X^{1,\frac{1}{2}}} \frac{1}{N_3}
\delta^{\frac{1}{2}-} \|u_3\|_{X^{1,\frac{1}{2}}}
\langle(\frac{N_2}{N})^{1/2}\rangle \\
& \le  \frac{c}{N^{1-}} N_{max}^{0-} \delta^{\frac{1}{2}-} \|n_+\|_{X^{0,
\frac{1}{2}+}_+}
\|u_2\|_{X^{1,\frac{1}{2}}} \|u_3\|_{X^{1,\frac{1}{2}}}
\end{aligned}\label{9}
\end{equation}
by the bilinear Strichartz estimate. This implies (\ref{8}).
Next we prove (\ref{6}) which is implied by
\begin{equation}\label{10}
\begin{aligned}
\, & \int_0^{\delta} \int_* \Big| \frac{m(\xi_2 +\xi_3)
-m(\xi_2)m(\xi_3)}{m(\xi_2)m(\xi_3)} \Big|
\widehat{u_1}(\xi_1,t)\widehat{u_2}(\xi_2,t)\widehat{n_+}(\xi_3,t) \, d\xi dt  
\\
& \le   c(N^{-\frac{1}{2}+} \delta^{\frac{1}{2}-}
+ N^{-\frac{3}{2}+}\delta^{0+})
\|u_1\|_{X^{-1,\frac{1}{2}}} \|u_2\|_{X^{1,
\frac{1}{2}}} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
\end{aligned}
\end{equation}

\noindent{\bf Case 1:} $N_2 \sim N_3 \ge cN $. Then $N_1 \le cN_2$ as above.
The multiplier is estimated by
$ \frac{c}{m(\xi_2)^2} \le c(\frac{N_2}{N})^{\frac{1}{2}-\epsilon} $, so that
we get the bound
\begin{align*}
 & c \|n_+ D^{1/2}_x u_2\|_{L^2_{xt}} \frac{1}{N_2^{1/2}} \|u_1\|_{L^2_{xt}}
(\frac{N_2}{N})^{\frac{1}{2}-\epsilon} \\
 & \le  c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{0+,\frac{1}{2}}}
\frac{N_1}{N_2^{1/2}}
\delta^{\frac{1}{2}-} \|u_1\|_{X^{-1,\frac{1}{2}}} 
(\frac{N_2}{N})^{\frac{1}{2}
-\epsilon} \\
 & \le  c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{1,\frac{1}{2}}}
\frac{N_1}{N_2^{\frac{3}{2}-}}
\delta^{\frac{1}{2}-} \|u_1\|_{X^{-1,\frac{1}{2}}} 
(\frac{N_2}{N})^{\frac{1}{2}
-\epsilon}
\end{align*}
which implies (\ref{10}).

\noindent{\bf Case 2:} $N_1 \sim N_2 \ge cN$, thus $N_3 \le cN_1$.
The symbol is majorized by $\frac{c}{m(\xi_2)m(\xi_3)} \le
c\langle(\frac{N_2}{N})^{\frac{1}{2}-\epsilon}\rangle$, which can be handled
as in Case 1.

\noindent{\bf Case 3:} $N_1 \sim N_3 \ge cN$ , $N_2 << N_1 \sim N_3$.\\
{\bf Subcase a:} $ N_2 \le N $.
By the mean value theorem we have
$$
\Big| \frac{m(\xi_2 + \xi_3) - m(\xi_2)m(\xi_3)}{m(\xi_2)m(\xi_3)}\Big|
= \Big| \frac{m(\xi_2 +\xi_3) - m(\xi_3)}{m(\xi_3)}\Big|
\le c \Big|\frac{(\nabla m)(\xi_3)}{m(\xi_3)}\xi_2\Big|
\le c \frac{N_2}{N_3}
$$
and we get the bound
\begin{align*}
 & c \|n_+ D_x^{1/2}u_1\|_{L^2_{xt}} N_1^{-\frac{1}{2}} \|u_2\|_{L^2_{xt}}
\frac{N_2}{N_3}  \\
 & \le  c \|n_+\|_{X_+^{0,\frac{1}{2}+}} \|u_1\|_{X^{0+,\frac{1}{2}}} 
N_1^{-\frac{1}{2}} \|u_2\|_{X^{0,0}}
\frac{N_2}{N_3} \\
 & \le  c \|n_+\|_{X_+^{0,\frac{1}{2}+}} \|u_1\|_{X^{-1,\frac{1}{2}}}N_1^{1+} 
N_1^{-\frac{1}{2}} N_2^{-1}
\delta^{\frac{1}{2}-} \|u_2\|_{X^{1,\frac{1}{2}}} \frac{N_2}{N_3} \\
 & \le  c N_1^{-\frac{1}{2}+} \delta^{\frac{1}{2}-} 
\|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_1\|_{X^{-1,\frac{1}{2}}} \|u_2\|_{X^{1,\frac{1}{2}}}
\end{align*}
which implies (\ref{10}).

\noindent{\bf Subcase b:} $ |\xi_1| \sim |\xi_3| >> |\xi_2| \ge N $.
This is the technically most complicated region where we want to use algebraic
manipulations on the Fourier side with respect to $\tau$ and $\xi$ and have 
also
to take into account the characteristic function
$\psi(t)$ of the time interval $[0,\delta]$. The problem is that
$\widehat{\psi}(\tau) = \frac{1}{\sqrt{2\pi}}
\frac{e^{i\tau \delta}-1}{i \tau} \not\in L^1_{\tau}$, but fortunately
$\in L_{\tau}^{1+}$. We perform no dyadic decompositions at all here.

We estimate the multiplier by
$\frac{c}{|m(\xi_2)|} \le c|\xi_2|^{1/2}N^{-\frac{1}{2}}$. Thus our
aim is to give the following bound
\begin{equation}
\begin{aligned}
\, &\int_0^\delta \int_* \widehat{u_1}(\xi_1,t) |\xi_2|^{1/2}
\widehat{u_2}(\xi_2,t)\widehat{n_+}(\xi_3,t) d\xi dt\\
\, &\le c N^{-1+} \delta^{0+} \|u_1\|_{X^{-1,\frac{1}{2}}} 
\|u_2\|_{X^{1,\frac{1}{2}}}
\|n_+\|_{X_+^{\frac{1}{2}+}}
\end{aligned}
\label{***a}
\end{equation}
which would imply (\ref{10}).
Abusing notation we denote the Fourier transform with respect to $x$ and $t$
also by $\widehat{ }$. The
left-hand side is bounded by
\begin{equation}
\label{'}
\int_{**} \widehat{u_1}(\xi_1,\tau_1) |\widehat{\psi}(\tau_0)| |\xi_2|^{1/2}
\widehat{u_2}(\xi_2,\tau_2) \widehat{n_+}(\xi_3,\tau_3) \, d\xi d\tau
\end{equation}
Here ** denotes integration over $ \sum_{i=1}^3 \xi_i = \sum_{i=0}^3 \tau_i = 
0 $.
Remark again that
without loss of generality  $\widehat{u_1},\widehat{u_2},\widehat{n_+} \ge 0 
$.
The crucial algebraic inequality in our
region is the following:
$$
|\xi_1| \le c \Big( \langle \tau_1 + |\xi_1|^2\rangle^{1/2} + \langle \tau_2 +
|\xi_2|^2\rangle^{1/2} + \langle \tau_3 + |\xi_3|\rangle^{1/2} + 
|\tau_0|^{1/2}\Big)
$$
We consider 4 cases according to which of the terms on the right-hand side
is dominant.

\noindent {\bf Region 1:} $ \langle \tau_1 + |\xi_1|^2 \rangle^{1/2} $ 
dominant.
We get the following bound for (\ref{'}):
\begin{align*}
& c \int_{**} \langle \tau_1 + |\xi_1|^2 \rangle^{1/2} |\xi_1|^{-1}
\widehat{u_1}(\xi_1,\tau_1) |\widehat{\psi}(\tau_0)||\xi_2|^{1/2} 
\widehat{u_2}(\xi_2,\tau_2)
\widehat{n_+}(\xi_3,\tau_3)\, d\xi d\tau  \\
& \le  c \|u_1\|_{X^{-1,\frac{1}{2}}}
 \|\mathcal{F}^{-1}(|\widehat{\psi}|) (D^{1/2}_x u_2) n_+ \|_{L^2_{xt}} \\
& \le  c \|u_1\|_{X^{-1,\frac{1}{2}}} \| 
\mathcal{F}^{-1}(|\widehat{\psi}|)\|_{L_t^{\infty -}} \|(D^{1/2}_x
u_2) n_+\|_{L^{2+}_t L^2_x} \\
& \le  c \delta^{0+} \|u_1\|_{X^{-1,\frac{1}{2}}} 
\|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_2\|_{X^{0+,\frac{1}{2}}} \\
& \le  c \delta^{0+} N^{-1+} \|u_1\|_{X^{-1,\frac{1}{2}}} 
\|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_2\|_{X^{1,\frac{1}{2}}}
\end{align*}
by Lemma \ref{LemmaA'} and by Hausdorff-Young, which gives
$$
 \| \mathcal{F}^{-1}(|\widehat{\psi}|)\|_{L^{\infty-}_t}
 \le c \|\widehat{\psi}\|_{L^{1+}_{\tau}} \le c
\delta^{0+} $$
as one easily calculates.

\noindent{\bf Region 2:} $ \langle \tau_2 + |\xi_2|^2 \rangle^{1/2} $ 
dominant.
As before, we estimate (\ref{'}) by
\begin{align*}
& c \int_{**}  |\xi_1|^{-1} \widehat{u_1}(\xi_1,\tau_1)
|\widehat{\psi}(\tau_0)|\langle \tau_2 +
|\xi_2|^2 \rangle^{1/2}|\xi_2|^{1/2} \widehat{u_2}(\xi_2,\tau_2)
\widehat{n_+}(\xi_3,\tau_3)\, d\xi d\tau  \\
& \le  c \|u_2\|_{X^{\frac{1}{2},\frac{1}{2}}} 
\|\mathcal{F}^{-1}(|\widehat{\psi}|) (D^{-1}_x u_1) n_+
\|_{L^2_{xt}} \\
& \le  c \|u_2\|_{X^{\frac{1}{2},\frac{1}{2}}} \| 
\mathcal{F}^{-1}(|\widehat{\psi}|)\|_{L_t^{\infty -}}
\|(D^{-1}_x u_1) n_+\|_{L^{2+}_t L^2_x} \\
& \le  c \delta^{0+} \|u_2\|_{X^{\frac{1}{2},\frac{1}{2}}} 
\|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_1\|_{X^{-\frac{3}{2}+,\frac{1}{2}}} \\
& \le  c \delta^{0+} N^{-\frac{1}{2}} \|u_2\|_{X^{1,\frac{1}{2}}} 
\|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_1\|_{X^{-1,\frac{1}{2}}} N^{-\frac{1}{2}+}\,,
\end{align*}
where we have used Lemma \ref{LemmaA'} again.

\noindent{\bf Region 3:} $ \langle \tau_3 + |\xi_3|^2 \rangle^{1/2} $ 
dominant.
Using Lemma \ref{LemmaD}, we control (\ref{'}) by:
\begin{align*}
& c \int_{**}  |\xi_1|^{-1} \widehat{u_1}(\xi_1,\tau_1)
|\widehat{\psi}(\tau_0)| |\xi_2|^{1/2}
\widehat{u_2}(\xi_2,\tau_2) \langle \tau_3 + |\xi_3|^2 \rangle^{\frac{1}{2}+}
\widehat{n_+}(\xi_3,\tau_3)\, d\xi d\tau  \\
& \le  c \|n_+\|_{X_+^{0,\frac{1}{2}+}} \|\mathcal{F}^{-1}(|\widehat{\psi}|) 
(D^{-1}_x u_1) ( D_x^{1/2}
u_2) \|_{L^2_{xt}} \\
& \le  c \|n_+\|_{X_+^{0,\frac{1}{2}+}} \| 
\mathcal{F}^{-1}(|\widehat{\psi}|)\|_{L_t^{\infty -}} \|(D^{-1}_x
u_1) ( D_x^{1/2} u_2)\|_{L^{2+}_t L^2_x} \\
& \le  c \delta^{0+} \|n_+\|_{X_+^{0,\frac{1}{2}+}} 
\|u_1\|_{X^{-\frac{3}{2}+,\frac{1}{2}}} \|
u_2\|_{X^{\frac{1}{2},\frac{1}{2}}} \\
& \le  c \delta^{0+} \|n_+\|_{X_+^{0,\frac{1}{2}+}} N^{-\frac{1}{2}+} 
\|u_1\|_{X^{-1,\frac{1}{2}}}
\|u_2\|_{X^{1,\frac{1}{2}}} N^{-\frac{1}{2}}\,.
\end{align*}

\noindent{\bf Region 4:} $ |\tau_0|^{1/2} $ dominant.
The upper bound for (\ref{'}) here is
\begin{align*}
& c \int_{**}  |\xi_1|^{-1} \widehat{u_1}(\xi_1,\tau_1) |\tau_0|^{1/2}
|\widehat{\psi}(\tau_0)| |\xi_2|^{1/2} \widehat{u_2}(\xi_2,\tau_2)
\widehat{n_+}(\xi_3,\tau_3)\, d\xi d\tau  \\
& \le   c \|\widehat{D_x^{-1}u_1}\|_{L^2_{\xi_1} L^{1+}_{\tau_1}} \| 
|\tau|^{1/2}
|\widehat{\psi}| \ast \widehat{D_x^{1/2} u_2} \ast \widehat{n_+} \|_{L^2_{\xi}
 L^{\infty -}_{\tau}}
\end{align*}
by H\"older. The first factor is estimated as follows by H\"older
with respect to $\tau_1$:
\begin{eqnarray*}
\|\widehat{D_x^{-1} u_1}\|_{L^2_{\xi_1} L^{1+}_{\tau_1}} & = & \| 
\widehat{D_x^{-1} u_1} \, \langle \tau_1
+ \xi_1^2 \rangle^{1/2} \langle \tau_1 + \xi_1^2 \rangle^{-\frac{1}{2}} 
\|_{L^2_{\xi_1}
L^{1+}_{\tau_1}} \\
& \le & \| \widehat{D_x^{-1} u_1} \, \langle \tau_1 + \xi_1^2 \rangle^{1/2}  
\|_{L^2_{\xi_1
\tau_1}} \le c \|u_1\|_{X^{-1,\frac{1}{2}}}\,.
\end{eqnarray*}
The second factor is bounded by Young's inequality by
\begin{align*}
c \| |\tau|^{1/2} |\widehat{\psi}| \|_{L^{2+}_{\tau}} \| \widehat{D^{1/2}_x 
u_2}
\ast \widehat{n_+} \|_{L^2_{\xi} L^{2--}_{\tau}}
& \le   \delta^{0+} \|n_+\|_{X^{0,\frac{1}{2}+}_+} 
\|u_2\|_{X^{0+,\frac{1}{2}}}\\
&\le  c \delta^{0+} N^{-1+} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
 \|u_2\|_{X^{1,\frac{1}{2}}}\,.
\end{align*}
Here we used Lemma \ref{LemmaB} and the bound $ \| |\tau|^{1/2} 
|\widehat{\psi}|
\|_{L^{2+}_{\tau}} \le c \delta^{0+} $, which is easily checked.
Thus we get (\ref{***a}) in all regions.

Finally we prove (\ref{7}). It is implied by
\begin{equation}
\begin{aligned}
\, &\int_0^{\delta} \int_* \Big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} 
\cdot
\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)} \Big|\\
\,&\times \widehat{n_+}(\xi_1,t)\widehat{u_2}(\xi_2,t)
\widehat{n_+}(\xi_3,t)\widehat{u_4}(\xi_4,t) d\xi dt  \\
\, & \le  c(\frac{1}{N^{3-}} + N^{-1+}\delta^{\frac{1}{2}-}) 
\|n_+\|_{X^{0,\frac{1}{2}+}_+}^2
\|u_2\|_{X^{1,\frac{1}{2}}}\|u_4\|_{X^{1,\frac{1}{2}}}\,.
\end{aligned}\label{11}
\end{equation}

\noindent{\bf Case 1:} $N_1 \sim N_3 \ge cN ,  N_1 \sim N_3 >> N_2,N_4 $\\
Subcase a: $N_4 \le N$
\begin{gather*}
\big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{m(\xi_2)} \le c 
\langle(\frac{N_2}{N})^{\frac{1}{2}-\epsilon}\rangle
\\
\big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
= \big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)}{m(\xi_3)}\big|
\le \big| \frac{(\nabla m)(\xi_3)}{m(\xi_3)} \xi_4 \big|
\le \frac{cN_4}{N_3}
\end{gather*}
by the mean value theorem. Thus we get the bound
\begin{align*}
&c\frac{N_4}{N_3} \|n_+\|_{L^{\infty}_t L^2_x} \|u_2\|_{L^2_t L^{\infty}_x} 
\|n_+ D^{1/2}_x
u_4\|_{L^2_{xt}} \frac{1}{N_4^{1/2}}
\langle(\frac{N_2}{N})^{\frac{1}{2}-\epsilon}\rangle  \\
& \le  c\frac{N_4}{N_3} \|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_2\|_{L^2_t H^{\frac{1}{2}+}_x}
\|n_+\|_{X^{0,\frac{1}{2}+}_+} \| u_4\|_{X^{0+,\frac{1}{2}}} 
\frac{1}{N_4^{1/2}}
\langle(\frac{N_2}{N})^{\frac{1}{2}-\epsilon}\rangle \\
& \le  c\frac{N_4}{N_3} \|n_+\|_{X_+^{0,\frac{1}{2}+}}
\|u_2\|_{X^{1,\frac{1}{2}}}\frac{1}{N_2^{\frac{1}{2}-}} \delta^{\frac{1}{2}-}
\|n_+\|_{X^{0,\frac{1}{2}+}_+} \| u_4\|_{X^{1,\frac{1}{2}}}
\frac{1}{N_4^{\frac{3}{2}-}}
\langle(\frac{N_2}{N})^{\frac{1}{2}-\epsilon}\rangle
\end{align*}
which implies (\ref{11}).

\noindent Subcase b: $|\xi_1| \sim |\xi_3| \ge cN ,  |\xi_1| \sim |\xi_3| >> 
|\xi_2|,|\xi_4| \, , \,
|\xi_2|,|\xi_4| \ge N$.
In this case we avoid any dyadic decomposition and estimate as follows:
\begin{gather*}
\big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{|m(\xi_2)|} \le c (\frac{|\xi_2|}{N})^{\frac{1}{2}-}, \\
\big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_4)|} \le c(\frac{|\xi_4|}{N})^{\frac{1}{2}-}
\end{gather*}
Thus we get the bound
\begin{align*}
&\int_0^{\delta} \int_* \frac{|\xi_2|^{\frac{1}{2}-}}{N^{\frac{1}{2}-}}
\frac{|\xi_4|^{\frac{1}{2}-}}{N^{\frac{1}{2}-}}
\widehat{n_+}(\xi_1,t)(|\xi_2|^{\frac{1}{2}-}\widehat{u_2}(\xi_2,t))\\
&\times \frac{1}{|\xi_2|^{\frac{1}{2}-}}
\widehat{n_+}(\xi_3,t)(|\xi_4|^{\frac{1}{2}-}\widehat{u_4}(\xi_4,t))
\frac{1}{|\xi_4|^{\frac{1}{2}-}} \,d\xi dt \\
& \le  \frac{c}{N^{1-}} \|n_+\|_{L^{\infty}_t L^2_x} \|D^{\frac{1}{2}-}_x u_2
\|_{L^2_t L^{\infty}_x}
\|n_+\|_{L^{\infty}_t L^2_x} \|D^{\frac{1}{2}-}_x u_4\|_{L^2_t L^{\infty}_x} 
\\
& \le  \frac{c}{N^{1-}} \|n_+\|_{X_+^{0,\frac{1}{2}+}} 
\|u_2\|_{X^{1,\frac{1}{2}}}
\delta^{\frac{1}{2}-}
\|n_+\|_{X_+^{0,\frac{1}{2}+}} \|u_4\|_{X^{1,\frac{1}{2}}} 
\delta^{\frac{1}{2}-}
\end{align*}
which is sufficient, because no dyadic decomposition was performed.

\noindent Subcase c: $|\xi_1| \sim |\xi_3| \ge cN ,
|\xi_1| \sim |\xi_3| \gg |\xi_2|,|\xi_4| \, , \,  |\xi_4|
\ge N \ge |\xi_2|$.
We again perform no dyadic decomposition and estimate as follows:
\begin{gather*}
 \big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big| \le c,  \\
\big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_4)|} \le c(\frac{|\xi_4|}{N})^{1/2}
\end{gather*}
Thus we get the bound
\begin{align*}
& \int_0^{\delta} \int_*  \frac{|\xi_4|^{1/2}}{N^{1/2}}
\widehat{n_+}(\xi_1,t)\widehat{u_2}(\xi_2,t))
\widehat{n_+}(\xi_3,t)(|\xi_4|^{1/2}\widehat{u_4}(\xi_4,t))\frac{1}{|\xi_4|^{1
/2}} \, d\xi
dt \\
& \le  \frac{c}{N^{1/2}} \|n_+\|_{L^{\infty}_t L^2_x} \|u_2\|_{L^2_t 
L^{\infty}_x} \|n_+
D^{1/2}_x u_4\|_{L^2_{xt}} \\
& \le  \frac{c}{N^{1/2}} \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{L^2_t 
H^{\frac{1}{2}+}_x}
\|n_+\|_{X^{0,\frac{1}{2}+}_+} \| u_4\|_{X^{0+,\frac{1}{2}}} \\
& \le  \frac{c}{N^{1/2}} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
\delta^{\frac{1}{2}-}\|u_2\|_{X^{1,\frac{1}{2}}} 
\|n_+\|_{X^{0,\frac{1}{2}+}_+} \frac{1}{N^{1-}} \|
u_4\|_{X^{1,\frac{1}{2}}} \\
& \le  \frac{c}{N^{\frac{3}{2}-}} \delta^{\frac{1}{2}-} 
\|n_+\|_{X^{0,\frac{1}{2}+}_+}^2
\|u_2\|_{X^{1,\frac{1}{2}}} \| u_4\|_{X^{1,\frac{1}{2}}}
\end{align*}

\noindent {\bf Case 2:} $N_2 \sim N_4 \ge cN ,  N_2 \sim N_4 \gg N_1,N_3 $.
We have
\begin{gather*}
\big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{|m(\xi_1)|}  \le c
\langle(\frac{N_1}{N})^{1/2}\rangle \le c (\frac{N_4}{N})^{1/2}\,,\\
\big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_3)|}\le c \langle(\frac{N_3}{N})^{1/2}\rangle
\le c (\frac{N_2}{N})^{1/2}\,.
\end{gather*}
This gives the bound
\begin{align*}
& c \|n_+ D^{1/2}_x u_2\|_{L^2_{xt}} N_2^{-1/2} \|n_+ D^{1/2}_x u_4
\|_{L^2_{xt}} N_4^{-1/2}
\big(\frac{N_4}{N}\big)^{1/2} \big(\frac{N_2}{N}\big)^{1/2}  \\
& \le  \frac{c}{N} \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{0+,\frac{1}{2}}}
\|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_4\|_{X^{0+,\frac{1}{2}}} \\
& \le  \frac{c}{N} N_2^{-1+} N_4^{-1+} \|n_+\|_{X^{0,\frac{1}{2}+}_+}^2
 \|u_2\|_{X^{1,\frac{1}{2}}} \|u_4\|_{X^{1,\frac{1}{2}}}
\end{align*}
which implies (\ref{11}).

\noindent {\bf Case 3:} $N_1 \sim N_2 \ge cN \, , \, N_1 \sim N_2 \gg
 N_3,N_4 $.
Using
\begin{gather*}
 \big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{|m(\xi_1)|^2}  \le c (\frac{N_1}{N})^{1/2}\,,\\
 \big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_3)m(\xi_4)|}  \le c \langle(\frac{N_3}{N})^{1/2}\rangle
\langle(\frac{N_4}{N})^{1/2}\rangle\,,
\end{gather*}
we get the bound
\begin{align*}
& c \|n_+ D^{1/2}_x u_2\|_{L^2_{xt}} N_2^{-1/2} \|n_+\|_{L^{\infty}_t L^2_x}
 \|u_4\|_{L^2_t L^{\infty}_x} \langle (\frac{N_3}{N})^{1/2}\rangle
\langle (\frac{N_4}{N})^{1/2}\rangle (\frac{N_1}{N})^{1/2}  \\
&\le  c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{0+,\frac{1}{2}}}
N_2^{-1/2} \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_4\|_{L^2_t H^{\frac{1}{2}+}_x}
\langle(\frac{N_3}{N})^{1/2}\rangle \langle (\frac{N_4}{N})^{1/2}\rangle
(\frac{N_1}{N})^{1/2} \\
& \le  c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{1,\frac{1}{2}}}
N_2^{-\frac{3}{2}+} \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_4\|_{X^{1,\frac{1}{2}}}
\delta^{\frac{1}{2}-} N_4^{-\frac{1}{2}+}  \\
&\quad \times \langle (\frac{N_2}{N})^{1/2}\rangle
\langle (\frac{N_4}{N})^{1/2}\rangle (\frac{N_1}{N})^{1/2} \\
&\le  c N^{-\frac{3}{2}+} (N_1N_2N_3N_4)^{0-} \delta^{\frac{1}{2}-}
\|n_+\|_{X^{0,\frac{1}{2}+}_+}^2 \|u_2\|_{X^{1,\frac{1}{2}}}
\|u_4\|_{X^{1,\frac{1}{2}}}
\end{align*}

\noindent{\bf Case 4:} $N_2 \sim N_3 \ge cN ,  N_2 \sim N_3 \gg N_1,N_4$.
Using
\begin{gather*}
 \big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{|m(\xi_1)|}  \le c \langle(\frac{N_1}{N})^{1/2}\rangle \,,\\
\big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_4)|} \le c \langle(\frac{N_4}{N})^{1/2}\rangle
\end{gather*}
we get the bound
\begin{align*}
& c \|n_+ D^{1/2}_x u_2\|_{L^2_{xt}} N_2^{-1/2} \|n_+\|_{L^{\infty}_t L^2_x}
 \|u_4\|_{L^2_t L^{\infty}_x} \langle (\frac{N_1}{N})^{1/2}\rangle
\langle (\frac{N_4}{N})^{1/2}\rangle  \\
&  \le   c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{1,\frac{1}{2}}}
 N_2^{-\frac{3}{2}+} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
\|u_4\|_{X^{1,\frac{1}{2}}}\delta^{\frac{1}{2}-} N_4^{-\frac{1}{2}+} \\
&\quad\times \langle (\frac{N_2}{N})^{1/2}\rangle
\langle(\frac{N_4}{N})^{1/2}\rangle \\
&  \le   c N^{-\frac{3}{2}+} (N_1N_2N_3N_4)^{0-} \delta^{\frac{1}{2}-}
\|n_+\|_{X^{0,\frac{1}{2}+}_+}^2
\|u_2\|_{X^{1,\frac{1}{2}}} \|u_4\|_{X^{1,\frac{1}{2}}}
\end{align*}

\noindent{\bf Case 5:} $N_3 \sim N_4\ge cN ,  N_3\sim N_4>> N_1,N_2$.
Using
\begin{gather*}
 \big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{|m(\xi_1)m(\xi_2)|}
\le c \langle(\frac{N_1}{N})^{1/2}\rangle
\langle(\frac{N_2}{N})^{1/2}\rangle \,,\\
 \big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_4)|^2} \le c (\frac{N_4}{N})^{1/2}\,,
\end{gather*}
we get the bound
\begin{align*}
 & c  \|n_+\|_{L^{\infty}_t L^2_x} \|u_2\|_{L^2_t L^{\infty}_x}
\|n_+ D^{1/2}_x u_4\|_{L^2_{xt}}
N_4^{-1/2}\langle (\frac{N_1}{N})^{1/2}\rangle \langle
(\frac{N_2}{N})^{1/2}\rangle (\frac{N_4}{N})^{1/2}  \\
&  \le   c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{1,\frac{1}{2}}}
N_2^{-\frac{1}{2}+}\delta^{\frac{1}{2}-} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
\|u_4\|_{X^{1,\frac{1}{2}}} N_4^{-\frac{3}{2}+}\\
&\quad \times
\langle (\frac{N_1}{N})^{1/2}\rangle \langle
(\frac{N_2}{N})^{1/2}\rangle (\frac{N_4}{N})^{1/2} \\
&  \le   c N^{-\frac{3}{2}+} (N_1N_2N_3N_4)^{0-} \delta^{\frac{1}{2}-}
\|n_+\|_{X^{0,\frac{1}{2}+}_+}^2
\|u_2\|_{X^{1,\frac{1}{2}}} \|u_4\|_{X^{1,\frac{1}{2}}}
\end{align*}

\noindent{\bf Case 6:} $N_1 \sim N_4\ge cN ,  N_1\sim N_4>> N_2,N_3$.
\begin{align*}
\big| \frac{m(\xi_1 + \xi_2)}{m(\xi_1)m(\xi_2)} \big|
\le \frac{c}{|m(\xi_2)|}  \le c \langle(\frac{N_2}{N})^{1/2}\rangle \,,\\
\big|\frac{m(\xi_3 + \xi_4) - m(\xi_3)m(\xi_4)}{m(\xi_3)m(\xi_4)}\big|
\le \frac{c}{|m(\xi_3)|}\le c \langle(\frac{N_3}{N})^{1/2}\rangle
\end{align*}
we get the bound
\begin{align*}
 & c  \|n_+\|_{L^{\infty}_t L^2_x} \|u_2\|_{L^2_t L^{\infty}_x}
\|n_+ D^{1/2}_x u_4\|_{L^2_{xt}}
N_4^{-1/2}\langle (\frac{N_2}{N})^{1/2}\rangle \langle
(\frac{N_3}{N})^{1/2}\rangle  \\
&  \le   c \|n_+\|_{X^{0,\frac{1}{2}+}_+} \|u_2\|_{X^{1,\frac{1}{2}}}
N_2^{-\frac{1}{2}+}\delta^{\frac{1}{2}-} \|n_+\|_{X^{0,\frac{1}{2}+}_+}
\|u_4\|_{X^{1,\frac{1}{2}}} N_4^{-\frac{3}{2}+}
 \langle (\frac{N_2}{N})^{1/2}\rangle \langle
(\frac{N_3}{N})^{1/2}\rangle \\
&  \le   c N^{-\frac{3}{2}+} (N_1N_2N_3N_4)^{0-} \delta^{\frac{1}{2}-}
\|n_+\|_{X^{0,\frac{1}{2}+}_+}^2
\|u_2\|_{X^{1,\frac{1}{2}}} \|u_4\|_{X^{1,\frac{1}{2}}}
\end{align*}
The remaining cases where at least three factors have equivalent frequencies
$\ge cN$ are similar or
easier to handle so that (\ref{11}) is proved in all possible situations.
The proof of the proposition is complete.
\end{proof}

\section{The global existence result}

\begin{theorem}
Let $ 1 > s > 5/6 $. The Zakharov system
\eqref{1}--\eqref{3} has a unique global solution for
data $ u_0 \in H^s(\mathbb{R})$ , $n_0 \in H^{s-1}(\mathbb{R}) $,
$ A^{-1/2}n_1 \in  H^{s-1}(\mathbb{R}) $. More precisely, for any
$T> 0$ there exists a unique solution
$$
(u,n,A^{-1/2}n_t) \in X^{s,\frac{1}{2}}[0,T] \times 
\widetilde{X}^{s-1,\frac{1}{2}+}[0,T] \times
\widetilde{X}^{s-1,\frac{1}{2}+}[0,T]
$$
where $ \widetilde{X}^{s-1,\frac{1}{2}+}[0,T] := X_+^{s-1,\frac{1}{2}+}[0,T] + 
X_-^{s-1,\frac{1}{2}+}[0,T] $. This solution
satisfies
\begin{gather*}
 (u,n,A^{-1/2}n_t) \in C^0([0,T],H^s(\mathbb{R}) \times H^{s-1}(\mathbb{R})
\times  H^{s-1}(\mathbb{R}))\,,\\
\|u(t)\|_{H^s} + \|n(t)\|_{H^{s-1}} + \|A^{-1/2} n_t(t)\|_{H^{s-1}}
\le c (1+t)^{\frac{2(1-s)}{6s-5}+}
\end{gather*}
\end{theorem}

\begin{proof} The data satisfy the estimates
\begin{gather*}
\|Iu_0\|_{H^1}  \le  c N^{1-s}\|u_0\|_{H^s} \\
\|In_{\pm 0}\|_{L^2}  \le  c N^{1-s} (\|n_0\|_{H^{s-1}} +\|A^{-1/2}
n_1\|_{H^{s-1}})\,.
\end{gather*}
We use our local existence theorem on $[0,\delta]$, where
$ \delta \sim \frac{1}{N^{4(1-s)+}} $ and conclude
\begin{equation}
\begin{aligned}
\, &\|Iu\|_{X^{1,\frac{1}{2}}[0,\delta]} + 
\|In_+\|_{X^{0,\frac{1}{2}+}_+[0,\delta]} +
\|In_-\|_{X^{0,\frac{1}{2}+}_-[0,\delta]}  \\
& \le  c(\|Iu_0\|_{H^1} + \|In_+\|_{L^2} + \|In_-\|_{L^2}) \le c_2 N^{1-s}
\end{aligned} \label{**}
\end{equation}
 From (\ref{*'}) we get
$$
E(Iu_0,In_{+0}) \le c_0 (\|Iu_0\|_{H^1}^2 + \|In_{+0}\|_{L^2}^2
+ \|Iu_0\|_{L^2}^6) \le \overline{c} N^{2(1-s)}
$$
and from (\ref{*})
\begin{gather*}
 \|A^{1/2}Iu_0\|_{L^2}^2 + \|In_+\|_{L^2}^2 + \|In_-\|_{L^2}^2
\le \widehat{c} N^{2(1-s)} \,, \
\|Iu_0\|_{L^2} \le M
\end{gather*}
with $ \widehat{c} = \widehat{c}(\overline{c}) $. Thus the constant in 
(\ref{**}) depends only on
$\overline{c}$ and $M$, i.e. $c_2 = c_2(\overline{c},M)$.\\
To reapply the local existence result with time intervals of equal length we 
need a uniform bound
of the solution at time $t=\delta$ and $t=2\delta$ etc. which follows from a 
uniform control over the
energy by (\ref{*}). The increment of the energy is controlled by Proposition
\ref{Prop3.1} and
(\ref{**}) as follows:
\begin{align*}
& |E(Iu(\delta),In_+(\delta)) - E(Iu_0,In_{+0})|  \\
& \le  c[(N^{-\frac{1}{2}+} \delta^{\frac{1}{2}-} + N^{-\frac{3}{2}+})
\|In_+\|_{X^{0,\frac{1}{2}+}_+[0,\delta]} 
\|Iu\|_{X^{1,\frac{1}{2}}[0,\delta]}^2 \\
&\quad + (N^{-3+} + N^{-1+} \delta^{\frac{1}{2}-})
\|In_+\|_{X^{0,\frac{1}{2}+}_+[0,\delta]}^2
\|Iu\|_{X^{1,\frac{1}{2}}[0,\delta]}^2] \\
& \le  c((N^{-\frac{1}{2}+} \delta^{\frac{1}{2}-} + N^{-\frac{3}{2}+}) 
N^{3(1-s)}
+(N^{-3+} + N^{-1+} \delta^{\frac{1}{2}-}) N^{4(1-s)})
\end{align*}
Using the definition of $\delta$ we arrive at
\begin{align*}
&|E(Iu(\delta),In_+(\delta)) - E(Iu_0,In_{+0})| \\
& \le  c_3 ((N^{-\frac{1}{2}+} N^{-2(1-s)+}+ N^{-\frac{3}{2}+})
N^{3(1-s)} + (N^{-3+} + N^{-1+} N^{-2(1-s)+})N^{4(1-s)}) \\
& \le  c_3( N^{-\frac{1}{2}+} N^{-2(1-s)+} N^{3(1-s)} + N^{-1+}
N^{-2(1-s)+} N^{4(1-s)})
\end{align*}
where $ c_3 = c_3(\overline{c},M) $. This is easily seen to be bounded by
$ \overline{c}N^{2(1-s)}$ (for large $N$).

The number of iteration steps to reach the given time $T$ is
$\frac{T}{\delta} \sim T N^{4(1-s)+} $. This
means that in order to give a uniform bound of the energy of the iterated
solutions, namely by $2\overline{c}N^{2(1-s)}$, from the last inequality
the following condition has to be fulfilled:
$$
c_3(N^{-\frac{1}{2}+}N^{-2(1-s)}N^{3(1-s)} + N^{-1+}N^{-2(1-s)+}N^{4(1-s)})
TN^{4(1-s)+}   < \overline{c} N^{2(1-s)}
$$
where $c_3 = c_3(2\overline{c},2M)$ (recall here that the initial energy
is bounded by $\overline{c}N^{2(1-s)}$).
This can be fulfilled for $N$ sufficiently large provided the following
conditions hold:
\begin{gather*}
-\frac{1}{2}-2(1-s)+3(1-s)+4(1-s)<2(1-s)  \Longleftrightarrow  s > 5/6 \\
-1-2(1-s)+4(1-s)+4(1-s)<2(1-s)  \Longleftrightarrow  s > 3/4
\end{gather*}
So here is the point where the decisive bound on $s$ appears.
A uniform bound of the energy implies by (\ref{*}) uniform control of
$$
\|A^{1/2}Iu(t)\| + \|In(t)\| + \|A^{-1/2}In_t(t)\| \le c N^{1-s}
$$
Moreover $ \|Iu(t)\| \le \|u(t)\| = \|u_0\| $, thus
$$
\|u(t)\|_{H^s} + \|n(t)\|_{H^{s-1}} + \|A^{-1/2}n_t(t)\|_{H^{s-1}} \le c 
N^{1-s}
$$
Now, one can directly give a bound on the growth of the solution as follows.
The most restrictive
condition on $N$ comes from the inequality
$$
c_3 T N^{-\frac{1}{2}+} N^{-2(1-s)} N^{3(1-s)} N^{4(1-s)+}
< \overline{c} N^{2(1-s)} \Longleftrightarrow
N > c T^{\frac{2}{6s-5}+}
$$
This implies
$$
\sup_{0\le t \le T} (\|u(t)\|_{H^s} + \|n(t)\|_{H^{s-1}} + 
\|A^{-1/2}n_t(t)\|_{H^{s-1}}) \le c
(1+T)^{\frac{2(1-s)}{6s-5}+}\,.
$$
\end{proof}

\subsection*{Acknowledgement}
The author wants to thank Axel Gr\"unrock for his very helpful discussions.

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\end{document}