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\markboth{\hfil Regularity bounds on Zakharov system evolutions \hfil
 EJDE--2002/75}
{EJDE--2002/75\hfil James Colliander \& Gigliola Staffilani \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 75, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Regularity bounds on Zakharov system \\
  evolutions
%  
\thanks{ {\em Mathematics Subject Classifications:} 35Q55.
\hfil\break\indent
{\em Key words:} initial value problems, bilinear estimates, Zakharov
system, weak turbulence.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted March 15, 2002. Published August 20, 2002.
 \hfil\break\indent
J.C. was supported in part by NSF Grant DMS 0100595 and NSERC grant
\hfil\break\indent RGPIN 250233-03.\hfil\break\indent
G.S. was supported in part by NSF Grant DMS 0100375 and by
grants from \hfil\break\indent
Hewlett and Packard and the Sloan Foundation.
} }
\date{}


\author{James Colliander \& Gigliola Staffilani}
\maketitle

\begin{abstract}
  Spatial regularity properties of certain global-in-time
  solutions of the Zakharov system are established.
  In particular, the evolving solution $u(t)$ is shown to
  satisfy an estimate
  $\|u(t)\|_{H^s} \leq C |t|^{(s-1)+}$,
  where $H^s$ is the standard spatial Sobolev norm. The proof
  is an adaptation of earlier work on the nonlinear Schr\"odinger
  equation which reduces matters to bilinear estimates.
\end{abstract}


\newcommand{\Lp}[2]{\| #2 \|_{L^#1}}
\newcommand{\Hsub}[2]{\| #2 \|_{H_#1}}
\newcommand{\Hsup}[2]{\| #2 \|_{H^#1}}
\newcommand{\X}[3]{\| #3 \|_{X_{#1,#2}}}
\newcommand{\qsubpm}[1]{|\lambda_ #1 \pm |k_ #1|^2|}

\newcommand{\ctermpm}[3]{\frac {(1+|k_{#1}|)^{#2}c (k_#1,\lambda_ #1)}
                        {(1+|\lambda_ #1\pm |k_ #1 |^2|)^{#3}}}

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

\section{Introduction}


We consider the initial value problem for the Zakharov system
\cite{Z} on $\mathbb{R}^2$
\begin{equation}
 \begin{matrix}
i{u_t} + \Delta u =nu, & u : \mathbb{R}^2 \times [-T_*, T^*] \to \mathbb{C} , \\
\mathop{\square} n = \Delta  |u|^2,
& n: \mathbb{R}^2 \times [ - T_*, T^*]\to \mathbb{R} , \\
(u , n , \dot{n} )(0) = (\phi, a , b).
\end{matrix}
\label{ZS}
\end{equation}
Suppose $b$ is such that there exists $V: \mathbb{R}^2 \to \mathbb{R}^2$ with
$b = \nabla \cdot V$. Then the Zakharov system may be rewritten in Hamiltonian
form with Hamiltonian
\begin{equation}
H(u, \bar{u} , n , V) = \int_{\mathbb{R}^2} |\nabla u |^2
+ \frac12  ( n^2 + |V|^2) + n |u|^2 dx.
\label{Hamiltonian}
\end{equation}
For initial data $\phi$ small enough in $L^2$ we can conclude from conservation
of \eqref{Hamiltonian} that
\begin{equation}
\Hsub 1 {{u(t), n(t) , \dot{n} (t) }} =
\| u(t) , n(t) , \dot{n}(t) \|_{H^1 \times L^2 \times \widehat{H}^{-1}}
 \leq C \Hsub 1 {\phi, a , b }
\label{aprioriHone}
\end{equation}
where $H_1 := H^1 \times L^2 \times \widehat{H}^{-1}$ and
$\widehat{H}^{-1}$ is defined by $\| b \|_{\widehat{H}^{-1}} =
\Lp 2 V $.
Local wellposedness of \eqref{ZS} for data $(\phi, a , b ) \subset H_1$ was
established in \cite{BouCol97,Thesis}, with the lifetime of existence
satisfying
\begin{equation}
T > {{ \|{\phi, a , b }\| }_{H_1}^{-\alpha}} \quad\mbox{for some } \alpha > 0.
\label{lifetime}
\end{equation}
The regularity requirements for the local results in
\cite{BouCol97,Thesis}
have subsequently been improved in \cite{GTV}. Hence, for data implying
a priori $H_1$ control \eqref{aprioriHone}, the local result may be iterated
to prove the existence of global-in-time solutions of \eqref{ZS}. In fact,
global solutions of the initial value problem \eqref{ZS} had been shown to
exist earlier  \cite{AA} ($d=1$)  \cite{SS} ($d=2$) using energy methods in spaces requiring more regularity than $H_1$.

\paragraph{Remark.}
The initial value problem \eqref{ZS} has solutions which
blow up in finite time \cite{GM1,GM2}. At the present time, there is no
criteria known which identifies those initial data leading to blow up
and those leading to global-in-time solutions. This paper provides
regularity bounds on those global solutions obtained by iterating the
local wellposedness result using a priori $H_1$ control. \smallskip

Let $\square^{-1} F$ denote the solution of the inhomogeneous wave equation
with zero data,
\begin{equation}
\begin{gathered}
\mathop{\square} n = F, \\
(n , \dot{n} )(0) = (0 , 0).
\end{gathered}
\label{inhomog}
\end{equation}
Let $W(a,b)$ denote the solution of
\begin{equation}
 \begin{gathered}
\mathop{\square} n = 0 \\
(n , \dot{n} )(0) = (a , b).
\end{gathered}\label{homog}
\end{equation}
Note that $\square^{-1} F$ and $W(a,b)$ may be explicitly represented using the
Fourier transform. The (formal) solution of the second equation in \eqref{ZS}
is
\begin{equation}
n = W(a,b) + \square^{-1} ({ \Delta {{|u|}^2} }).
\label{nsolved}
\end{equation}
Substituting this expression for $n$ into the first equation in \eqref{ZS}
gives
\begin{equation}
\begin{gathered}
u_t = i \Delta u - i W(a,b) u - i\,\square^{-1} \Delta ( {{|u|}^2} ) u, \\
u (0) = \phi .
\end{gathered}
\label{nremoved}
\end{equation}
Note that the regularity properties of the data $a,b$ and of $u$,
inferred from solving
\eqref{nremoved}, determine the regularity properties of $n$ through
\eqref{nsolved}.

Let $\mathcal{S}$ denote the Schwarz class. Consider initial data $\phi, a \in \mathcal{S}, ~ b \in \mathcal{S} \bigcap
{{\widehat{H}}^{-1}}$ implying a priori $H_1$ control \eqref{aprioriHone}.
How do the regularity properties of the global solution $(u(t), n(t))$
behave as $t \to \infty$? In particular, can we describe, or
at least bound from above and below, $\Hsub s {{u(t) , n(t) , \dot{n} (t) }} $
for $s \gg 1$ as $t \to \infty$? These estimates quantify the shift of the conserved $L^2$ mass in frequency space. In particular, the upper bounds we obtain in this paper limit the rate of transfer from low frequencies to high frequencies. By the note following
\eqref{nremoved}, it suffices to understand $\Hsup s {u(t) }$.

The local result for \eqref{nremoved} implies
\begin{equation}
\sup_{t \in [0,T]} \Hsup s {u(t) } \leq \Hsup s {\phi} + C \Hsup s {\phi}
\label{givesexp}
\end{equation}
which iterates to give an exponential bound $\Hsup s {u(t) } \leq C^{|t|}.$
Bourgain observed that a slight improvement of \eqref{givesexp}
\begin{equation}
\sup_{t \in [0,T]} \Hsup s {u(t) } \leq \Hsup s {\phi} + C {{ \| \phi \|}_{
H^s}^{1 - \delta}}, ~0 < \delta < 1,
\label{givespoly}
\end{equation}
implies the polynomial bound
$\Hsup s {u(t) } \leq C |t|^{1/\delta}$. This observation was
exploited in \cite{B4} to prove polynomial bounds on high Sobolev norms
for solutions of the nonlinear Schr\"odinger equation and certain
nonlinear wave equations.

Staffilani \cite{Staff97,StaffQ97} improved the degree of the
polynomial upper bound using a different approach to prove \eqref{givespoly}
in the case of the nonlinear Schr\"odinger equation. The crucial bilinear
estimate used in this approach has recently been improved \cite{CDKS}
giving a slightly better polynomial estimate. This paper adapts the arguments
from \cite{StaffQ97} for the nonlinear Schr\"odinger equation to
prove similar polynomial bounds on high Sobolev norms for the global
solutions of \eqref{ZS} constructed in \cite{BouCol97,Thesis}.

\begin{theorem}
\label{zspoly}
Assume $(\phi, a ,b ) \in \mathcal{S} \times \mathcal{S}
\times ( \mathcal{S} \cap {{\widehat{H}}^{-1}})$.
Global solutions of \eqref{ZS} satisfying \eqref{aprioriHone} also satisfy
\begin{equation}
\Hsup s {u(t)} \leq C {{|t|}^{(s-1)+}} .
\end{equation}

\end{theorem}

The question of lower bounds showing growth of high Sobolev norms
remains a fascinating open question. For a more thorough discussion, including
model equations other than the Zakharov system, see the book of
Bourgain \cite{BAMS}.


\section{Reduction to bilinear estimate}

Our goal is to bound $\Hsup s {u(t)}$ for $u$, the solution of
\eqref{nremoved}, with $t \in [0,T]$ and $T$ as in \eqref{lifetime}.
Since $\Lp 2 {u(t)} = \Lp 2 {\phi}$ for all $t$, it suffices to bound
$\Lp 2 {B^s u(t) }$ where $B = {\sqrt{-\Delta}}$. Let's assume $s = 2 m$
for $1 \ll m \in \mathbb{N}$ to avoid certain technical issues involving fractional
derivatives below. Let $\langle \cdot, \cdot \rangle$ denote the
standard $L^2_x$
inner product, $\langle f , g \rangle = \int_{\mathbb{R}^2}  f {\bar{g}} dx$.
By the fundamental theorem of calculus,
\begin{equation}
{{\| B^s u(t) \|}_{L^2}^2} = {{\| B^s u(0) \|}_{L^2}^2} +
\int_0^t \frac{d}{d\sigma} \langle B^s u(\sigma ) , B^s u(\sigma) \rangle
d\sigma .
\label{one}
\end{equation}
We calculate
\begin{equation}
I = 2 \Re \int_0^t \langle B^s \dot{u} (\sigma ) , B^s u(\sigma ) \rangle
d \sigma .
\label{two}
\end{equation}
Now, using the equation \eqref{nremoved}, we find
\begin{equation} \label{three}
\begin{aligned}
I =& - 2 \Im \int_0^t \langle B^s \Delta u (\sigma ), B^s u(\sigma ) \rangle
d \sigma \\
&+2 \Im \int_0^t \langle B^s [ W(a,b)(\sigma) u(\sigma)] ,
B^s u(\sigma) \rangle d \sigma \\
&+ 2 \Im \int_0^t \langle B^s [ \square^{-1} \Delta
({{|u|}^2}) u (\sigma )], B^s u(\sigma ) \rangle d \sigma .
\end{aligned}
\end{equation}
Denote the three terms on the right-side of \eqref{three} by $I_1 + I_2 + I_3$.
Upon writing $- \Delta = B^2$ and integrating by parts, the first term $I_1$
is seen to have a real integrand so this term is zero. The term $I_2$
involves $B^s (W(a,b) u )$. Various terms arise from the Leibniz rule
for differentiating a product. The most dangerous of these is $W(a,b) B^s u$
but, since $W(a,b) $ is a real-valued function, this term leads to a purely
real integrand in \eqref{three} and so disappears. Hence, the term $I_2$
leads to a sum of terms of the form
\begin{equation}
C \Im \int_0^t \langle [B^{s_1} W(a,b)(\sigma)][B^{s_2} u(\sigma )] ,
B^s u(\sigma ) \rangle d \sigma,
\label{four}
\end{equation}
where $s = s_1 + s_2 , ~ 1 \leq s_1 \leq s, ~0 \leq s_2 \leq s-1, ~
s_1, s_2 \in \mathbb{N}$,

We can multiply by a smooth cutoff function in time $\psi_T \thicksim
\chi_{[0,T]}$ and estimate these terms via the H\"older inequality by
\begin{equation}
{{\| B^{s_1} W(a,b) \|}_{L^2_{x,t \in [0,T]}}} {{\| B^{s_2} u \|}_{L^4_{xt}}}
{{\| B^s u \|}_{L^4_{xt}}}.
\label{five}
\end{equation}
The Strichartz estimate for the paraboloid and properties of $X_{s,b}$ spaces
\cite{B1} imply for $ b = \frac12+$,
\begin{equation}
{{\| B^{\tilde{s}} u \|}_{L^4_{xt}}} \leq C_T \X {\tilde{s}} b u .
\label{six}
\end{equation}
Here the space $X_{s,b}$ is defined using the norm
$$ \| u \|_{X_{s,b}} =  \Big( \int (1+|k|)^{2s} (1+|\lambda + |k|^2|)^{2b}
| \widehat{u} (k , \lambda )|^2 dk d\lambda\Big)^{1/2}.$$
The local wellposedness result \cite{BouCol97,GTV} gives
\begin{equation}
\X {\tilde{s}} b u
\leq C \Hsup {{\tilde{s}}} {u(0)}.
\label{seven}
\end{equation}
Therefore, the second term $I_2$ in \eqref{three} is estimated by a sum
of terms of the form
\begin{equation}
{{\| a, b \|}_{H^{s_1} \times H^{{s_1} - 1}}} \Hsup {{s_2}} {u(0)}
\Hsup s {u(0)} .
\label{eight}
\end{equation}
The first factor is bounded by a constant which depends upon the initial
data $a,b$. The second factor may be interpolated between $\Hsup 1 {\phi} $
and $\Hsup s {\phi} $ which leads to the bound
\begin{equation}
|I_2 | \leq C \sum_{0 \leq s_2 \leq s-1 }
{{\| u(0) \|}_{H^s}^{1 + {\frac{{s_2} -1}{s-1}}}} \leq C
{{\| u(0) \|}_{H^s}^{2 - {\frac{1}{s-1}}}}.
\label{nine}
\end{equation}

It remains to bound $I_3$. Since differentiation in $x$ commutes with
$\square^{-1}\Delta$, the Leibniz rule shows
\begin{equation}
I_3 = 2 \Im \sum_{\begin{smallmatrix}
  0\leq \sigma_1 , \sigma_2, \sigma_3 \leq s\\
  \sigma_1 + \sigma_2 + \sigma_3 = s \end{smallmatrix}}
c_{\sigma_1 , \sigma_2 , \sigma_3 }
\int_0^t \langle \square^{-1}\Delta ( B^{\sigma_1} u B^{\sigma_2} {\bar{u}} )
B^{\sigma_3} u ,
B^s u \rangle d \tau.
\label{ten}
\end{equation}
In the case $\sigma_3 = s$, the resulting integrand is purely real so this
term disappears. Consider first those terms with $\sigma_3 \leq s - 2$
and after treating these we will consider the terms with $\sigma_3 = s-1$.
Since we are interested in proving a local-in-time estimate, we can
insert a smooth cutoff $\psi_T \thicksim \chi_{[0,T]}$ and wish to
bound
\begin{equation}
\label{eleven}
| \psi_T (t) \int_0^t \square^{-1}\Delta ( B^{\sigma_1} u B^{\sigma_2} {\bar{u}} )
B^{\sigma_3} u B^s {\bar{u}} d\tau |.
\end{equation}
A formal ``integration by parts'' (which is justified rigorously in the
next section when we define $(\square^{-1}\Delta)^{1/2}$) allows us to bound by
\begin{equation}
\label{twelve}
| \psi_T (t) \int_0^t (\square^{-1}\Delta)^{1/2}( B^{\sigma_1} u B^{\sigma_2} {\bar{u}} )
(\square^{-1}\Delta)^{1/2}( B^{\sigma_3} u B^s {\bar{u}}) d\tau |
\end{equation}
and Cauchy-Schwarz reduces matters to controlling
\begin{equation}
\|(\square^{-1}\Delta)^{1/2}( B^{\sigma_1} u B^{\sigma_2} \bar{u})\|_{L^2_{xT}}
\|(\square^{-1}\Delta)^{1/2}( B^{\sigma_3} u B^s \bar{u})\|_{L^2_{xT}} .
\label{thirteen}
\end{equation}

\begin{proposition}
\label{bilinearestimate}
Let $0 \leq s_1 \in \mathbb{N}$ and $1 \leq s_2 \in \mathbb{N},~ s_1 \leq s_2.$ For
$b > 1/2 $,
\begin{equation}
\label{propstatement}
\| (\square^{-1}\Delta)^{1/2}([B^{s_1} u_1 ][ B^{s_2} {\overline{u_2}} ]) \|
_{L^2_{xT}}
\leq C \X {{s_1}+1} b {u_1}  \X {s_2 - \frac12  } b {u_2} .
\end{equation}
The estimate is also valid if the complex conjugation is moved from
$u_2$ to $u_1$ on the left-side of \eqref{propstatement}.
\end{proposition}

Suppose the proposition is true. The bilinear expressions in \eqref{thirteen}
are estimated by
\begin{equation}
\X {\sigma_1 + 1} b {u}  \X {\sigma_2 - \frac12 } b u
\X {\sigma_3 + 1 } b u  \X {s - \frac12  } b u .
\label{fifteen}
\end{equation}
Using the local result we know $ \X {\tilde{s}} b u \leq
C \Hsup {{\tilde{s}}} {u(0)} $
and upon interpolating the various $H^{\tilde{s}}$
norms between $H^1$ and $H^s$ (using \eqref{aprioriHone}) bounds \eqref{fifteen}
by
\begin{equation}
C {{\| u(0) \|}_{H^s}^{ \frac{\sigma_1 + \sigma_2 - \frac12  - 1 + \sigma_3 + s
- \frac12  - 1 }{s-1}}} .
\label{sixteen}
\end{equation}
Recalling that $\sigma_1 + \sigma_2 + \sigma_3 = s,$ the exponent simplifies
to $2 - \frac{1}{s-1}$, just as in \eqref{nine}.

Now, consider a term in \eqref{ten} with $\sigma_3 = s-1 $. Evidently,
$\sigma_1 = 1,~ \sigma_2 = 0$ or $\sigma_1 = 0,~\sigma_2 = 1$. In this
case, we apply Cauchy-Schwarz directly to the term as it appears in
\eqref{ten} to bound by
\begin{equation}
\label{seventeen}
\| \square^{-1}\Delta ( B^{\sigma_1 } u B^{\sigma_2} {\bar{u}} ) \|_{L^2_{xT}}
\| B^{\sigma_3} u B^s {\bar{u}} \|_{L^2_{xT}} .
\end{equation}
The second factor is readily estimated using the Bourgain's refinement of the
Strichartz inequality \cite{BRefine} to give
\begin{equation}
\label{eighteen}
\X {\sigma_3 + \frac12  +} b u  \X {s - \frac12  } b u \leq
C {{\| u \|}_{{X_{s-\frac12  + , b}}}^2}.
\end{equation}
The first factor in \eqref{seventeen} is bounded using a variant of
Proposition \ref{bilinearestimate}.

\begin{proposition}
\label{nregularity}
Let $0 \leq s_1 \in \mathbb{N},~1 \leq s_2 \in \mathbb{N},~ s_1 \leq s_2$. For $b > 1/2 $,
\begin{equation}
\label{nineteen}
\| \square^{-1}\Delta ( [B^{s_1} u_1 ] [ B^{s_2} {\overline{u_2}} ] ) \|_{L^2_{xT}}
\leq C \X {s_1 + 1 } b {u_1 }  \X {s_2} b u .
\end{equation}
The estimate is also valid if the complex conjugation is moved from
$u_2$ to $u_1$ on the left-side of \eqref{nineteen}.
\end{proposition}

Combining \eqref{eighteen}, \eqref{nineteen} shows \eqref{seventeen}
may be bounded using $\sigma_1 = 1, \sigma_2 = 0$ or $\sigma_1 = 0, \sigma_2
= 1$ and $\sigma_3 = s-1$,
\begin{equation*}
\X 1 b u  \X 1 b u  {{\| u \|}_{{X_{s-\frac12  + , b}}}^2}
\leq C  {{\| u \|}_{{X_{s-\frac12  + , b}}}^2} .
\end{equation*}
The local result and interpolation bounds this by
\begin{equation*}
C{{\| u(0) \|}_{H^s}^{ 2 ( \frac{s - \frac12  - 1 }{s-1} )+}} =
C{{\| u(0) \|}_{H^s}^{ 2- \frac{1}{s-1} +}}
\end{equation*}
which (up to the $+$) is the same as in \eqref{nine}, \eqref{sixteen}.

Summarizing, the two Propositions above show that the integral term in
\eqref{one} is bounded by
\begin{equation*}
C{{\| u(0) \|}_{H^s}^{ 2- \frac{1}{s-1} +}}.
\end{equation*}
We may assume that $\Lp 2 { {B^s} u(t) } \geq \Lp 2 { {B^s} u(0) }$ for
otherwise $\eqref{givespoly}$ is automatic. Therefore, we can divide
\eqref{one} through by $\Lp 2 {B^s u(t)} $ and with $L^2$ conservation
observe \eqref{givespoly} holds with $\delta = \frac{1}{s-1}-$
proving Theorem \ref{zspoly}.

The next section establishes the Propositions and defines $(\square^{-1}\Delta)^{1/2}$ used
in the treatment of $\sigma_3 \leq s-2$ terms above.



\section{Bilinear Estimates}

In this section, we present a proof of Proposition \ref{bilinearestimate}.
Along the way we will observe explicit properties of the operator
$\square^{-1}\Delta$ which allow us to justify step \eqref{twelve} in the previous
section. Proposition \ref{nregularity} will follow from modifications of
the proof of Proposition \ref{bilinearestimate}.

The operator $\square^{-1}$ was defined as the mapping taking the inhomogeneity
$F$ to the solution of the linear initial value problem \eqref{inhomog}.
It can be explicitly represented using the Fourier transform as
\begin{multline}
\square^{-1} F (x,t) \\
= - \iint e^{i k \cdot x } \big[ e^{i \lambda t } - \frac12
( 1 + \frac{\lambda}{|k|} ) e^{i |k| t } - \frac12
( 1 - \frac{\lambda}{|k|} ) e^{-i |k| t }  \big]
\frac{ {\widehat{F}}(k , \lambda ) }{ (\lambda - |k|) (\lambda + |k|) } dk \,d\lambda
\label{bone}
\end{multline}
where ${\widehat{F}}$ denotes the space-time Fourier transform of $F$.
A Taylor series argument show that the apparent singularities along $\lambda
\pm |k| = 0$ do not occur and that
\begin{equation}
| {\widehat{\square^{-1} \Delta F}} (k, \lambda) |
\leq C | {\widehat{F}} (k , \lambda ) | \frac{ |k| }{ ( 1 + |\lambda \pm
|k| |) } + | {\widehat{F}} (k , \lambda ) |_{\{\lambda = \mp |k|\} }  |
\frac{ |k| }{ ( 1 + |\lambda \pm |k| |) } .
\label{btwo}
\end{equation}
 From an $L^2$ point-of-view, it is therefore natural to define for
real numbers $\alpha$,
\begin{multline}
{\widehat{ {{(\square^{-1}\Delta )}^\alpha} } } ( k, \lambda ) \\
=| {\widehat{F}} (k , \lambda ) | {\big(\frac{ |k| }{ ( 1 + |\lambda \pm
|k| |) } \big)^\alpha}
+ \big| {\widehat{F}} (k , \lambda ) |_{\{\lambda = \mp |k| \}}  \big|
{\big(\frac{ |k| }{ ( 1 + |\lambda \pm |k| |) } \big)^\alpha} .
\label{bthree}
\end{multline}
In particular, we have defined the operator
$\left( \square^{-1}\Delta \right)^{1/2}$
which appears in the statement of Proposition \ref{bilinearestimate}.
For two functions of space-time, $F,G$, which are cutoff to $t \in [0,T]$,
consider the expression (analagous to \eqref{eleven})
\begin{equation}
\iint {(\square^{-1}\Delta ) (F )} G\, dx\, dt
= \iint {\widehat{ {{ (\square^{-1}\Delta ) (F ) }} }}
{\widehat{G}}\, dk \,d\lambda.
\label{bfour}
\end{equation}
We insert \eqref{btwo} and take the absolute value under the integral
sign. Then, upon writing
$$ \frac{|k|}{(1 + |\lambda \pm |k|\,| )} =
\frac{|k|^{1/2}}{(1 + |\lambda \pm |k|\,| )^{1/2}}
\frac{|k|^{1/2}}{(1 + |\lambda \pm |k|\,| )^{1/2}}$$
we observe that
\begin{equation}
\Big| \iint {(\square^{-1}\Delta ) (F )} G\, dx\, dt \Big|
\leq \Big| \iint (\square^{-1}\Delta )^{1/2} \widetilde{F} \cdot
(\square^{-1}\Delta )^{1/2} \widetilde{G}\, dx\, dt\Big|.
\label{bfive}
\end{equation}
where $\widetilde{F} (x,t) = \int e^{i(kx + \lambda t)} | \widehat{F} (k,\lambda)|
dk d\lambda$ and $\widetilde{G}$ is similarly defined. For proving $L^2$-type
estimates, the distinction between $F$ and $\widetilde{F}$ is
unimportant.  In particular, the ``integration by parts''
step \eqref{twelve} is validated.

Now that $(\square^{-1}\Delta)^{1/2}$ has been given a precise meaning, we turn our attention
to proving the inequality \eqref{propstatement}

\paragraph{Proof of Proposition \ref{bilinearestimate}}
Since ${\widehat{\square^{-1}\Delta}} (k, \lambda ) \thicksim
\frac{|k|}{\lambda \pm |k| }$, we see that $\square^{-1}\Delta$ can be as bad as
one derivative in $x$. Therefore, the number of derivatives on both
the left-side and right-side of \eqref{propstatement} is $s_1 + s_2 + \frac{1}{2}$.
The desired estimate \eqref{propstatement} may be reexpressed using duality
and certain renormalizations as
\begin{multline}
  \label{reexpressed}
  \int_* d(k, \lambda) \big( \frac{|k|}{(1+|\lambda \pm |k||)}\big)^{1/2}
 \ctermpm 1 {-1} b  \ctermpm 2 {\frac12 } b \\
 \leq \|d\|_{L^2} \| c_1\|_{L^2} \|c_2\|_{L^2}
\end{multline}
where $\int_*$ is shorthand for
$\int_{\begin{smallmatrix}k=k_1+k_2\\
\lambda=\lambda_1+\lambda_2\end{smallmatrix}}$
and without loss of generality we
assume $d, c_1, c_2 \geq 0$. The choices of $\pm$ in \eqref{reexpressed}
are assumed to be independent
in the following analysis. In fact, this is only the first contribution
arising from the right-side of \eqref{bthree}. The other ``on-light-cone''
piece may be similarly estimated. We analyze \eqref{reexpressed} in cases
depending upon the size of $|k_1|, |k_2|$.

\noindent\textbf{Case 1.} $|k_1|, |k_2| \leq 10$.
We may ignore $|k|^\frac12  (1+|k_1|)^{-1} (1+|k_2|)^{\frac12 }$ and then drop the
(potentially helpful) wave remnant $(1+|\lambda \pm |k||)^{\frac12 }$ to bound the
left-side of \eqref{reexpressed} by
\begin{equation}
  \label{reduced}
  \int_* d(k, \lambda) \frac{c_1 ( k_1, \lambda_1 ) }{ {{(1+|\lambda_1 \pm |k_1|^2|)}^b} }
 \frac{c_2 ( k_2, \lambda_2 ) }{ {{(1+|\lambda_2 \pm |k_2|^2|)}^b} }.
\end{equation}
Fourier transform properties show this equals
$\langle \widehat{ \mathcal{D} },
\widehat{\mathcal{C}_1} * \widehat{\mathcal{C}_2} \rangle
= \langle \mathcal{D} ,
\mathcal{C}_1 \mathcal{C}_2 \rangle $ where
$\mathcal{D}$, $\mathcal{C}_1$, $\mathcal{C}_2$ are functions of space and time
whose Fourier transforms are $d$,
$\frac{c_1 ( k_1, \lambda_1 ) }{ {{(1+|\lambda_1 \pm |k_1|^2|)}^b} }$,
 $\frac{c_2 ( k_2, \lambda_2 ) }{ {{(1+|\lambda_2 \pm |k_2|^2|)}^b} }$, respectively.
By H\"older's inequality, we can estimate by $\| \mathcal{D} \|_{L^2_{xT}} {\| {\mathcal{C}}_1
\|}_{L^4_{xT}} {\| {\mathcal{C}}_2 \|}_{L^4_{xT}} $ and obtain \eqref{reexpressed}
in this case using Plancherel and the Strichartz inequality for the paraboloid
as written in \cite{B1},
\begin{equation}
{{\Big\| \int  \frac{a(k, \lambda)}{ {{(1+ |\lambda +|k|^2| )}^b} } dk d\lambda \Big\|}_{
L^4 (\mathbb{R}^2_x \times \mathbb{R}_t )}} \leq {{\| a \|}_{L^2_{k\lambda}}},
\quad b > \frac12.
\label{bstrichartz}
\end{equation}
The standard steps going from\eqref{reduced} through $L^2 L^4 L^4$ to
\eqref{reexpressed} will be omitted from the discussion below.

\noindent\textbf{Case 2.} $|k_1 | \gtrsim |k_2 |,~ |k_1| \gtrsim 10$.
The case defining conditions imply $|k| \lesssim |k_1|.$ We again ignore the wave
remnant and use $(1+|k_1|)^{-1}$ to cancel away $|k|^\frac12 $ and $(1+|k_2|)^\frac12 $.
We again encounter \eqref{reduced} and complete this case with the $L^2
L^4 L^4$ H\"older argument using \eqref{bstrichartz}.

\noindent\textbf{Case 3.}
$|k_1| \ll |k_2|,~|k_2| \gtrsim 10 \implies |k_2| \thicksim |k|$.
The numerator $(1+|k_1|)^{-1}$ is not helpful in this case so we exploit the
denominators in \eqref{reexpressed} to cancel $|k|^\frac12 $ and $(1+|k_2|)^\frac12 $.
Since $k = k_1 + k_2, ~\lambda = \lambda_1 + \lambda_2$, the triangle inequality implies
\begin{equation}
\max ( |\lambda \pm |k|\,| , \qsubpm 2  , \qsubpm 1 ) \geq | \pm |k|
+ {{|k_2 |}^2} - {{|k_1 |}^2} | \thicksim {{|k_2 |}^2 } \thicksim {{|k|}^2}.
\label{denoms}
\end{equation}
\textbf{Case 3.A.} $|\lambda \pm |k|\,| $ is the max in \eqref{denoms}.
We use the large denominator to cancel $|k|^\frac12 $ and $(1+|k_2|)^\frac12 $ and
proceed as with \eqref{reduced}.

\noindent\textbf{Case 3.B.} $\qsubpm 2$ is the max in \eqref{denoms}.
Most of the large denominator is used to cancel away $|k|^\frac12 $ and
$(1+|k_2|)^\frac12 $ and we need to control
\begin{equation*}
  \int_* \frac{d(k,\lambda)}{(1 + |\lambda \pm |k|\,| )^\frac12 } \frac{c_1 (k_1 , \lambda_1)}{(1+
\qsubpm 1 )^b} \frac{c_2 (k_2 , \lambda_2)}{(1+
\qsubpm 2 )^{b-\frac12 }} .
\end{equation*}
Since $b> \frac12 $, so that $b - \frac12  > 2\delta > 0$, and $\qsubpm 2 \gtrsim |\lambda \pm |k|\,|$, we can write
\begin{equation}
  \label{stepping}
  \int_*
\frac{ (1+|k|)^{-\delta} d(k , \lambda )}{(1+|\lambda \pm |k||)^{\frac12 +}}
\ctermpm 1 {-\delta} b  c_2 (k_2 , \lambda_2 ).
\end{equation}
Let $\mathcal{D} (k , \lambda) = \frac{ (1+|k|)^{-\delta} d(k , \lambda )}{(1+|\lambda \pm |k||)^{\frac12 +}}$ and $\mathcal{C} = \ctermpm 1 {-\delta} b$. Then \eqref{stepping}
may be expressed as $\langle \mathcal{D} * \mathcal{C} , c_2 \rangle$ and Cauchy-Schwarz reduces matters to controlling $\| \mathcal{D} * \mathcal{C} \|_{L^2_{xT}}$.
This is accomplished in the following lemma.

\begin{lemma}
\label{waveschrodingerlemma}
For $b=\frac12 +$ and a fixed small $\delta > 0$
\begin{multline}
\label{waveschro}
\int_{*, |k_2|, |k_1| \geq 10} f(k, \lambda) \frac{ (1+|k_2|)^{-\delta} d(k_2 ,
\lambda_2 )}{(1+|\lambda_2 \pm |k_2||)^{\frac12 +}} \ctermpm 1 {-\delta} b\\
\leq \Lp 2 f \Lp 2 d \Lp 2 {c_1}.
\end{multline}
\end{lemma}

\paragraph{Proof.}
Since $\frac12 +$ and $b$ exceed $\frac12 $, the estimate \eqref{waveschro}
may be reduced to the ``on-curve'' setting using parabolic (for $c_1$) and
light-cone (for $d$) level set decompositions (see, for example, \cite{CDKS}).
 This reduces considerations to showing that
\begin{multline}
  \label{oncurve}
  \int_{|k_2| \geq |k_1| \geq 1} f(x_1 + k_2 ,
  \pm |k_1|^2 \pm |k_2| ) |k_2|^{-\delta}
\psi (k_2) |k_1|^{-\delta} \phi (k_1) dk_1 dk_2\\
\leq {{\| f \|}_{L^2_{k\lambda}}} {{\| \psi \|}_{L^2_k}} {{\| \phi \|}_{L^2_k}} .
\end{multline}
Consider the piece of the integration on the left-side of \eqref{oncurve}
arising from $\{ k_2 : |k_2| \thicksim K_2 (\mathrm{dyadic})\} \times
\{ k_1 : |k_1| \thicksim K_1 (\mathrm{dyadic})\}$. We make a change of variables,
where superscripts refer to vector components,
$u^1 = k_1^1 + k_2^1,~u^2 = k_1^2 + k_2^2, ~ v = \pm |k_1|^2 \pm |k_2|$ and we
assume that the component $k_1^2$ of $k_1$ satisfies $k_1^2  \thicksim |k_1| \thicksim K_1.$ (This may be accomplished by cutting in pie slices and making a rotation
of coordinates if necessary.) This change of variables followed by Cauchy-Schwarz
shows \eqref{oncurve} is bounded by
\begin{equation}
\label{cov}
  {{\| f \|}_{L^2_{k\lambda}}} K_1^{-\delta} K_2^{-\delta}
\int_{|k_1^1| \leq K_1} \Big( \int_{|k_i| \thicksim K_i}
|\psi(k_2)|^2 |\phi(k_1)|^2 \frac{1}{|J|} dk_1^2 dk_2^1 dk_2^2 \Big)^{1/2}
dk_1^1
\end{equation}
where the Jacobian is
\begin{equation}
  |J| = |2k_1^2 \pm 1| \thicksim K_1.
\end{equation}
We apply Cauchy-Schwarz in $k_1^1$ and pick up an extra factor of
$K_1^\frac12 $ which is cancelled away by the Jacobian factor.
The small prefactors $K_i^{-\delta}$ allow us to sum over large dyadic
scales thereby proving \eqref{oncurve} and the lemma.

The lemma shows that \eqref{stepping} is bounded as claimed in
\eqref{reexpressed}.

\noindent\textbf{Case 3.C.}
 $\qsubpm 1 $ is the max in \eqref{denoms}.
This case follows with a modification of the argument for Case 3.B.

The proof of Proposition \ref{nregularity} follows the same case structure
as the proof of Proposition \ref{bilinearestimate}. The only difference is
in the accounting of the extra $\frac12$ derivative in both sides of
\eqref{nineteen} in comparison with \eqref{propstatement}.

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\noindent\textsc{James Colliander }\\
 Department of Mathematics\\
 University of Toronto\\
 Toronto, ON M5R 1W8 CANADA\\
 e-mail: colliand@math.toronto.edu \smallskip

\noindent\textsc{Gigliola Staffilani }\\
Department of Mathematics\\
Massachusets Institute of Technology\\
77 Massachusets Avenue\\ 
Cambridge, MA 02139-4307 USA\\
e-mail: gigliola@math.mit.edu


\enddocument