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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 82, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/82\hfil Cauchy-problem for gKP-II equations]
{On the Cauchy-problem for generalized
Kadomtsev-Petviashvili-II equations}

\author[A. Gr{\"u}nrock\hfil EJDE-2009/82\hfilneg]
{Axel Gr{\"u}nrock}

\address{Axel Gr{\"u}nrock\newline
Rheinische Friedrich-Wilhelms-Universit\"at Bonn, 
Mathematisches Institut \newline
Beringstra{\ss}e 1, 53115 Bonn, Germany}
\email{gruenroc@math.uni-bonn.de}

\thanks{Submitted April 9, 2009. Published July 10, 2009.}
\thanks{Supported by the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich 611}
\subjclass[2000]{35Q53}
\keywords{Cauchy-problem; local well-posedness; generalized KP-II equations}

\begin{abstract}
 The Cauchy-problem for the generalized
 Kadomtsev-Petviashvili-II equation
 $$
 u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,
 $$
 is shown to be locally well-posed in almost
 critical anisotropic Sobolev spaces. The proof
 combines local smoothing and maximal function
 estimates as well as bilinear refinements of
 Strichartz type inequalities via multilinear
 interpolation in $X_{s,b}$-spaces.
\end{abstract}

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

\section{Introduction}

Inspired by the work of Kenig and Ziesler \cite{KZa,KZb},
 we consider the Cauchy problem
\begin{equation}
 \label{CP}
u(x,y,0)=u_0(x,y), \quad (x,y) \in \mathbb{R}^2
\end{equation}
for the generalized Kadomtsev-Petviashvili-II equation
(for short: gKP-II)
\begin{equation}
 \label{gKP-II}
u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x,
\end{equation}
where $l \ge 3$ is an integer. Concerning earlier results on related
problems for this equation we
refer to the works of Saut \cite{S93}, I{\'o}rio and Nunes \cite{IN},
and Hayashi, Naumkin, and Saut \cite{HNS}.

For the Cauchy data we shall assume $u_0 \in H^{(s)}$, where for
$s=(s_1,s_2,\varepsilon)$ the Sobolev type space $H^{(s)}$
is defined by its norm in the following way: Let $\xi:=(k,\eta)\in \mathbb{R}^2$
denote the Fourier variables corresponding
to $(x,y) \in \mathbb{R}^2$ and
$\langle D_x\rangle ^{\sigma_1} = \mathcal{F} ^{-1}\langle k\rangle ^{\sigma_1}\mathcal{F}$,
$\langle D_y\rangle ^{\sigma_2} = \mathcal{F} ^{-1}\langle \eta\rangle ^{\sigma_2}\mathcal{F}$,
as well as $\langle D_x^{-1}D_y\rangle ^{\sigma_3}
= \mathcal{F} ^{-1}\langle k^{-1}\eta\rangle ^{\sigma_3}\mathcal{F}$, where $\mathcal{F}$
denotes the Fourier transform and
$\langle x\rangle^{\sigma}= (1 + x^2)^{\frac{\sigma}{2}}$. Setting
$$
\|u_0\|_{\sigma_1,\sigma_2,\sigma_3}:= \|\langle D_x
\rangle ^{\sigma_1}\langle D_y\rangle ^{\sigma_2}\langle D_x^{-1}D_y
\rangle ^{\sigma_3} u_0\|_{L^2_{xy}}
$$
we define
$$
\|u_0\|_{H^{(s)}}:= \|u_0\|_{s_1+2s_2+ \varepsilon,0,0}+\|u_0\|_{s_1,s_2,\varepsilon}.
$$
Almost the same data spaces are considered in \cite{KZa},
\cite{KZb},  the only new element here is the additional parameter
$s_2$, which will play a major role only for powers $l \ge 4$.


Using the contraction mapping principle we will prove a  local
well-posedness result for \eqref{CP}, \eqref{gKP-II} with regularity
assumptions on the data as weak as possible. Two types of
estimates will be be involved in the proof: On the one hand there
is the combination of local smoothing effect (\cite{KZa}, see also
\cite{IMS}) and maximal function estimate (proven in
\cite{KZa,KZb}), which has been used in \cite{KZa}, a strategy
that had been developed in \cite{KPV93} in the context of
generalized KdV equations. On the other hand we rely on Strichartz
type estimates (cf. \cite{S93}) and especially on the bilinear
refinement thereof taken from \cite{HT,H}, see also
\cite{IM01,Ta,Tz,T&T01}. A similar bilinear estimate involving
$y$-derivatives (to be proven below) will serve to deal with the
$\langle D_y\rangle$ containing part of the norm. To make these two
elements meet, we use Bourgain's Fourier restriction norm method
\cite{B93}, especially the following function spaces of
$X_{s,b}$-type. The basic space $X_{0,b}$ is given as usual by the
norm
$$
\|u\|_{X_{0,b}} :=\|\langle\tau - \phi(\xi)\rangle^b \mathcal{F} u\|_{L^2_{\xi \tau}},
$$
where $\phi(\xi)=k^3- \frac{\eta^2}{k}$ is the phase function of the
linearized KP-II equation. According to the data
spaces chosen above we shall also use
$$
\|u\|_{X_{\sigma_1, \sigma_2,\sigma_3;b}} :=\|\langle D_x
 \rangle ^{\sigma_1}\langle D_y\rangle ^{\sigma_2}\langle D_x^{-1}D_y
 \rangle ^{\sigma_3}u\|_{X_{0,b}}
$$
as well as
$$
\|u\|_{X_{(s),b}} :=\|u\|_{X_{s_1+2s_2+\varepsilon,0,0;b}}
+\|u\|_{X_{s_1,s_2,\varepsilon;b}} .
$$
Finally the time restriction norm spaces $X_{(s),b}(\delta)$
constructed in the usual manner will become our solution spaces.
Now we can state the main result of this note.

\begin{theorem}  \label{lwp}
 Let $s_1 > 1/2$, $s_2 \ge \frac{l-3}{2(l-1)}$ and $0 < \varepsilon \le \min{(s_1,1)}$. Then for $s=(s_1,s_2,\varepsilon)$ and
 $u_0 \in H^{(s)}$ there exist $\delta = \delta(\|u_0\|_{H^{(s)}})>0$ and $b > \frac12$ such that there is a unique
 solution $u \in X_{(s),b}(\delta)$ of \eqref{CP}, \eqref{gKP-II}.
 This solution is persistent and the flow map
 $S: u_0 \mapsto u$, $H^{(s)}\rightarrow X_{(s),b}(\delta)$
 is locally Lipschitz.
\end{theorem}

The lower bounds on $s_1$, $s_2$, and $\varepsilon$ are optimal  (except
for endpoints) in the sense that scaling considerations strongly
suggest the necessity of the condition $s_1 + 2s_2 + \varepsilon \ge
\frac12 + \frac{l-3}{l-1}$. Moreover, for $l=3$ $C^2$-illposedness
is known for $s_1 < \frac12$ or $\varepsilon < 0$, see
\cite[Theorem 4.1]{KZa}. The affirmative result in \cite{KZa} concerning the
cubic gKP-II equation, that was local well-posedness for $s_1>
\frac34$, $s_2=0$, and $\varepsilon > \frac12$ is improved here by
$\frac34$ derivatives. Some effort was made to keep the number of
$y$-derivatives as small as possible\footnote{For $l\ge 5$ the use
of local smoothing effect and maximal function estimate
\emph{alone} yields LWP for $s_1 > \frac{l-3}{l-1}$, $s_2 =0$, and
$\varepsilon > \frac12$, which is optimal from the scaling point of view,
too. This result may be seen as essentially contained in
\cite[Theorem 2.1 and Lemma 3.2]{KZa} plus \cite[proofs of Theorem
2.10 and Theorem 2.17]{KPV93}. This variant always requires
$\frac12 +$ derivatives in $y$.}, but we shall not attempt to give
evidence for the necessity of the individual lower bounds on
$s_1$, $s_2$, and $\varepsilon$, respectively.

By the general arguments concerning the Fourier restriction norm
method introduced in \cite{B93} and further developed in
\cite{GTV97, KPV96}, matters reduce to proving the following
multilinear estimate.

\begin{theorem}  \label{multest}
 Let $s_1 > \frac12$, $s_2 \ge \frac{l-3}{2(l-1)}$ and
 $0 < \varepsilon \le \min{(s_1,1)}$. Then there exists $b'> - \frac12$, such that
 for all $b > \frac12$ and all $u_1,\dots,u_l \in X_{(s),b}$
 supported in $\{|t| \le 1\}$ the estimate
 $$
\|\partial_x \prod_{j=1}^l u_j\|_{X_{(s),b'}} \lesssim
\prod_{j=1}^l\|u_j\|_{X_{(s),b}}
 $$
 holds.
\end{theorem}

To prepare for the proof of Theorem \ref{multest} let us first
recall  those estimates for free solutions $W(t)u_0$ of the
linearized KP-II equation, which we take over from the literature.
First we have the local smoothing estimate from \cite[Lemma
3.2]{KZa}: For $0 \le \lambda \le 1$,
\begin{equation}
 \label{ls}
\|D_x^{(1-\lambda)}(D_x^{-1}D_y)^{\lambda}W(t)u_0\|_{L_x^{\infty}
 L_{yt}^2}\lesssim \|u_0\|_{L^2_{xy}}
\end{equation}
and hence by the transfer principle \cite[Lemma 2.3]{GTV97} for
$b> \frac12$,
\begin{equation}
  \label{lsx}
 \|D_x^{(1-\lambda)}(D_x^{-1}D_y)^{\lambda}u\|_{L_x^{\infty}L_{yt}^2}
 \lesssim \|u\|_{X_{0,b}}.
\end{equation}
Interpolation with the trivial case $L^2_{xyt}=X_{0,0}$ and
duality give for $0\le \theta \le 1$,
 $\frac{1}{p_{\theta}}=\frac{1-\theta}{2}$,
\begin{equation}
  \label{lsxtheta}
 \|[D_x^{(1-\lambda)}(D_x^{-1}D_y)^{\lambda}]^{\theta}u\|_{L_x^{p_{\theta}}L_{yt}^2}\lesssim \|u\|_{X_{0,\theta b}}
\end{equation}
as well as
\begin{equation}
  \label{lsxtheta'}
 \|[D_x^{(1-\lambda)}(D_x^{-1}D_y)^{\lambda}]^{\theta}u\|_{X_{0,-\theta b}}
 \lesssim \|u\|_{L_x^{p'_{\theta}}L_{yt}^2}.
\end{equation}
(For H\"older exponents $p$ we will always have 
$\frac1p + \frac1{p'}=1$.) To complement the local smoothing effect, we shall
use the maximal function estimate
\begin{equation}
 \label{max}
\|W(t)u_0\|_{L_x^4L_{yT}^{\infty}}\lesssim \|\langle D_x
 \rangle^{\frac34 +}\langle D_x^{-1}D_y\rangle^{\frac12 +} u_0\|_{L^2_{xy}}
\end{equation}
due to Kenig and Ziesler \cite[Theorem 2.1]{KZa}, which is
probably the hardest part of the whole story. The capital $T$ here
indicates, that this estimate is only valid local in time, the
$+$-signs at the exponents on the right denote positive numbers,
which can be made arbitrarily small at the cost of the implicit
constant but independent of other parameters. (This notation will
be used repeatedly below.) The transfer principle implies for $u$
supported in $\{|t|\le 1\}$
\begin{equation}
 \label{maxx}
\|u\|_{L_x^4L_{yt}^{\infty}}\lesssim \|\langle D_x\rangle^{\frac34 +}
\langle D_x^{-1}D_y\rangle^{\frac12 +} u\|_{X_{0,b}},
\end{equation}
where $b > \frac12$. The Strichartz type estimate
\begin{equation}
 \label{str}
\|W(t)u_0\|_{L_{xyt}^4}\lesssim \|u_0\|_{L^2_{xy}},
\end{equation}
taken from \cite[Proposition 2.3]{S93}, becomes
\begin{equation}
 \label{strx}
\|u\|_{L_{xyt}^4}\lesssim \|u\|_{X_{0,b}}, \quad b > \frac12.
\end{equation}
For its bilinear refinement
\begin{equation}
 \label{bilstr}
\|uv\|_{L_{xyt}^2}\lesssim \|D_x^{1/2}u\|_{X_{0,b}}\|D_x^{-\frac12}v\|_{X_{0,b}}, \quad b > \frac12,
\end{equation}
and the dualized version thereof
\begin{equation}
 \label{bilstr'}
\|D_x^{1/2}(uv)\|_{X_{0,-b}}\lesssim \|D_x^{1/2}u\|_{X_{0,b}}\|v\|_{L_{xyt}^2}, \quad b > \frac12,
\end{equation}
we refer to \cite[Theorem 3.3 and Proposition 3.5]{H}. In order to
estimate the $X_{s_1,s_2,\varepsilon;b'}$-norm of the nonlinearity the
following bilinear estimate involving $y$-derivatives will be
useful. We introduce the bilinear pseudodifferential operator
$M(u,v)$ in terms of its Fourier transform
$$
\widehat{M(u,v)}(\xi):= \int_{\xi=\xi_1+\xi_2}|k_1\eta
-k \eta_1|^{1/2}\widehat{u}(\xi_1)\widehat{v}(\xi_2)d\xi_1
$$
and define the auxiliary space $\widehat{L}_x^pL^q_{yt}$ by
$\|f\|_{\widehat{L}_x^pL^q_{yt}}:= \|\mathcal{F}_x f\|_{L_x^{p'}L^q_{yt}}$,
where $\mathcal{F}_x$ denotes the partial Fourier transform with respect to
the first space variable $x$ only. Then we have:

\begin{lemma}\label{lbily}
 \begin{equation}\label{bily}
  \|M(W(t)u_0,W(t)v_0)\|_{\widehat{L}_x^1L^2_{yt}} \lesssim \|D_x^{1/2}u_0\|_{L^2_{xy}}\|D_x^{1/2}v_0\|_{L^2_{xy}}.
 \end{equation}
\end{lemma}

\begin{proof}\footnote{It is not apparent from the proof, but there
is a very simple idea behind Lemma \ref{lbily}. If we take the
partial Fourier transform $\mathcal{F}_x W(t)u_0 (k)=
e^{itk^3}e^{i\frac{t}{k}\partial^2_{y}}\mathcal{F}_xu_0(k)$ with respect to
the $x$-variable only, we obtain a free solution of the linear
Schr\"odinger equation with rescaled time variable
$s=\frac{t}{k}$, multiplied by a phase factor of size one. So any
space time estimate for the Schr\"odinger equation should give a
corresponding estimate for linearized KP-type equations. This idea
was exploitet in \cite{GPS08}, \cite{G09} to obtain suitable
substitutes for Strichartz type estimates for semiperiodic and
periodic problems. From this point of view Lemma \ref{lbily}
corresponds to the one-dimensional estimate
$$
\|D_y^{1/2}(e^{it\partial_y^2}u_0
e^{-it\partial_y^2}v_0)\|_{L^2_{yt}} \lesssim
\|u_0\|_{L^2_y}\|v_0\|_{L^2_y}.
$$}
 We have
\begin{align*}
 & \widehat{M(W(t)u_0,W(t)v_0)}(\xi,\tau)\\
&=  \int_{\xi=\xi_1+\xi_2}|k_1\eta -k \eta_1|^{1/2}\delta(\tau
-k_1^3-k_2^3+ \frac{\eta_1^2}{k_1}+\frac{\eta_2^2}{k_2})
\widehat{u_0}(\xi_1)\widehat{v_0}(\xi_2)d\xi_1.
\end{align*}
Because of $\frac{\eta_1^2}{k_1}+\frac{\eta_2^2}{k_2}
=\frac{\eta^2}{k}+\frac{k}{k_1k_2}(\eta_1-\frac{k_1}{k}\eta)^2$ and
with $a=\tau -k_1^3-k_2^3+ \frac{\eta^2}{k}$ as well as
$g(\eta_1)=\frac{k}{k_1k_2}(\eta_1-\frac{k_1}{k}\eta)^2-a$ this
equals
$$
\int_{\xi=\xi_1+\xi_2}|k_1\eta -k \eta_1|^{1/2}\delta(g(\eta_1))
\widehat{u_0}(\xi_1)\widehat{v_0}(\xi_2)d\eta_1 dk_1.
$$
The zero's of $g$ are $\eta_1^{\pm}
=\frac{k_1}{k}\eta\pm \sqrt{\frac{k_1k_2a}{k}}$ and for the derivative
we have
$|g'(\eta_1)|=\frac{2}{|k_1k_2|}|k_1\eta -k \eta_1|$. So we get the
two contributions
$$
I^{\pm}(\xi,\tau)=\frac12 \int_{k_1+k_2=k}\frac{|k_1k_2|}{|k_1\eta
-k \eta_1^{\pm}|^{1/2}}
\widehat{u_0}(k_1,\eta_1^{\pm})\widehat{v_0}(k_2,\eta-\eta_1^{\pm}) dk_1.
$$
By Minkowski's integral inequality
$$
\|I^{\pm}(\xi,\cdot)\|_{L^2_{\tau}} \lesssim \int_{k_1+k_2=k}|k_1k_2|
\||k_1\eta -k \eta_1^{\pm}|^{-\frac12}
\widehat{u_0}(k_1,\eta_1^{\pm})\widehat{v_0}(k_2,\eta
-\eta_1^{\pm})\|_{L^2_{\tau}}dk_1,
$$
where, with $\lambda:=\sqrt{\frac{k_1k_2a}{k}}$, the square of
the last $L^2_{\tau}$-norm equals
\begin{align*}
 & \int |k_1\eta -k \eta_1^{\pm}|^{-1}|\widehat{u_0}(k_1,\frac{k_1}{k}\eta\pm \lambda)
\widehat{v_0}(k_2,\frac{k_2}{k}\eta\mp \lambda)|^2 d\tau \\
&\le \frac{2}{|k_1k_2|}\int|\widehat{u_0}(k_1,\frac{k_1}{k}\eta\pm
\lambda) \widehat{v_0}(k_2,\frac{k_2}{k}\eta\mp \lambda)|^2
d\lambda,
\end{align*}
since $d\tau =2\frac{\lambda k}{|k_1k_2|}d \lambda
= \mp\frac{2}{k_1k_2}(k_1\eta -k \eta_1^{\pm})d \lambda$. This gives
$$
\|I^{\pm}(\xi,\cdot)\|_{L^2_{\tau}} \lesssim
\int_{k_1+k_2=k}|k_1k_2|^{1/2}
\Big(
\int|\widehat{u_0}(k_1,\frac{k_1}{k}\eta\pm \lambda)
\widehat{v_0}(k_2,\frac{k_2}{k}\eta\mp \lambda)|^2 d \lambda
\Big)^{1/2} dk_1.
$$
Using Parseval and again Minkowski's
inequality for the $L^2_{\eta}$-norm we arrive at
\begin{align*}
 & \|\mathcal{F}_x M(W(t)u_0,W(t)v_0)(k)\|_{L^2_{yt}}\\
&\lesssim  \int_{k_1+k_2=k}|k_1k_2|^{1/2}
\Big( \int|\widehat{u_0}(k_1,\frac{k_1}{k}\eta\pm \lambda)
\widehat{v_0}(k_2,\frac{k_2}{k}\eta\mp \lambda)|^2 d \lambda d\eta \Big)^{1/2} dk_1 \\
&=  \int_{k_1+k_2=k}|k_1k_2|^{1/2} \|\widehat{u_0}(k_1,\cdot)
 \|_{L^2_{\eta}}\|\widehat{v_0}(k_2,\cdot)\|_{L^2_{\eta}}dk_1 \\
&\lesssim \|D_x^{1/2}u_0\|_{L^2_{xy}}\|D_x^{1/2}v_0\|_{L^2_{xy}}
\end{align*}
by Cauchy-Schwarz and a second application of Parseval's identity.
\end{proof}

\begin{corollary} Let $b> 1/2$. Then
 \begin{gather}\label{bilyx}
  \|M(u,v)\|_{\widehat{L}_x^1L^2_{yt}} \lesssim \|D_x^{1/2}u\|_{X_{0,b}}\|D_x^{1/2}v\|_{X_{0,b}}
\\ \label{bilyx'}
  \|D_x^{-\frac12}M(u,v)\|_{X_{0,-b}}\lesssim \|u\|_{\widehat{L}_x^{\infty}L^2_{yt}}\|D_x^{1/2}v\|_{X_{0,b}}.
 \end{gather}
\end{corollary}

\begin{proof}
 Lemma \ref{lbily} implies \eqref{bilyx} via the transfer principle,
\eqref{bilyx'} is then obtained by duality. In fact, if
we fix $v$ and consider the linear map $M_v(u):=M(u,v)$, then its
adjoint is given by $M^*_v=M_{\overline{v}}$, and we have
$\|\overline{v}\|_{X_{0,b}}=\|v\|_{X_{0,b}}$.
\end{proof}

Now we are prepared for the proof of the central multilinear estimate.

\begin{proof}[Proof of Theorem \ref{multest}] \hfill \\
1. We use \eqref{lsxtheta'} with $\lambda = 0$,
$\theta = \frac12$ and H\"older to obtain for $b_0 < - \frac14$
\begin{align*}
\|D_x^{1/2} \prod_{j=1}^l u_j\|_{X_{0,b_0}}
&\lesssim \|\prod_{j=1}^l u_j\|_{L_x^{\frac43}L^2_{yt}}\\
&\lesssim  \|u_1\|_{L_x^{4}L^2_{yt}}\|u_2\|_{L_x^{4}L^{\infty}_{yt}}\|u_3\|_{L_x^{4}L^{\infty}_{yt}}\prod_{j \ge 4}\|u_j\|_{L^{\infty}_{xyt}}.
\end{align*}
For the first factor we have by \eqref{lsxtheta}, again with
$\lambda = 0$, $\theta = \frac12$,
$$
\|u_1\|_{L_x^{4}L^2_{yt}} \lesssim \|D_x^{-\frac12}u_1\|_{X_{0,\frac14+}},
$$
while the second and third factor are estimated by \eqref{maxx}
$$
\|u_{2,3}\|_{L_x^{4}L^{\infty}_{yt}}\lesssim \|\langle D_x\rangle^{\frac34 +}
\langle D_x^{-1}D_y\rangle^{\frac12 +} u_{2,3}\|_{X_{0,b}},
$$
where $b > \frac12$. It is here that the time support assumption
is needed. For $j \ge 4$ we use Sobolev embeddings in all variables to obtain for $b > \frac12$
$$
\|u_j\|_{L^{\infty}_{xyt}} \lesssim \|\langle D_x\rangle^{\frac12 +}\langle
D_y\rangle^{\frac12 +} u_{j}\|_{X_{0,b}}.
$$
Summarizing we have for $b_0 < - 1/4$, $b > 1/2$,

\begin{equation}  \label{1}
\begin{aligned}
\|D_x^{1/2} \prod_{j=1}^l u_j\|_{X_{0,b_0}}
&\lesssim \|D_x^{-\frac12}u_1\|_{X_{0,b}}
\|\langle D_x\rangle^{\frac34 +}\langle D_x^{-1}D_y\rangle^{\frac12 +} u_{2}
 \|_{X_{0,b}} \\
&\times  \|\langle D_x\rangle^{\frac34 +}\langle D_x^{-1}D_y\rangle^{\frac12 +}
  u_{3}\|_{X_{0,b}} \prod_{j \ge 4}\|\langle D_x\rangle^{\frac12 +}
 \langle D_y\rangle^{\frac12 +} u_{j}\|_{X_{0,b}}.
\end{aligned}
\end{equation}

2. Combining the dual version \eqref{bilstr'} of the bilinear estimate
with H\"older's inequality, \eqref{bilstr} and
Sobolev embeddings we obtain for $b_1 < - 1/2$, $b > 1/2$,
\begin{equation} \label{2}
\begin{aligned}
&\|D_x^{1/2} \prod_{j=1}^l u_j\|_{X_{0,b_1}}\\
&\lesssim \|D_x^{1/2}u_3\|_{X_{0,b}}\|u_1u_2\|_{L^{2}_{xyt}}
 \prod_{j \ge 4}\|u_j\|_{L^{\infty}_{xyt}} \\
&\lesssim \|D_x^{-\frac12}u_1\|_{X_{0,b}}\|D_x^{1/2}u_2\|_{X_{0,b}}
 \|D_x^{1/2}u_3\|_{X_{0,b}}
\prod_{j \ge 4}\|\langle D_x\rangle^{\frac12 +}\langle D_y\rangle^{\frac12 +}
  u_{j}\|_{X_{0,b}}.
\end{aligned}
\end{equation}

3. Bilinear interpolation involving $u_2$ and $u_3$ gives
\begin{align*}
 \|D_x^{1/2} \prod_{j=1}^l u_j\|_{X_{0,b'}}
& \lesssim  \|D_x^{-\frac12}u_1\|_{X_{0,b}}
 \|\langle D_x\rangle^{\frac12 +\frac{\theta}{4} +}\langle D_x^{-1}
  D_y\rangle^{\frac{\theta}{2} +} u_{2}\|_{X_{0,b}} \\
& \times  \|\langle D_x\rangle^{\frac12 +\frac{\theta}{4} +}\langle D_x^{-1}
  D_y\rangle^{\frac{\theta}{2} +} u_{3}\|_{X_{0,b}}
 \prod_{j \ge 4}\|\langle D_x\rangle^{\frac12 +}\langle D_y
 \rangle^{\frac12 +} u_{j}\|_{X_{0,b}},
\end{align*}
where $0 < \theta \ll 1$ and $b' = \theta b_0 + (1-\theta)b_1$.
Now symmetrization via $(l-1)$-linear interpolation
among $u_2, \dots, u_l$ yields
\begin{equation}  \label{3}
\|D_x^{1/2} \prod_{j=1}^l u_j\|_{X_{0,b'}}  \lesssim  \|D_x^{-\frac12}u_1\|_{X_{0,b}}\prod_{j \ge 2}
\|\langle D_x\rangle^{\alpha_1 +}\langle D_y\rangle^{\alpha_2 +}\langle D_x^{-1}D_y\rangle^{\alpha_3 +} u_{j}\|_{X_{0,b}},
\end{equation}

with $b,b'$ as before, $\alpha_1 = \frac12 + \frac{\theta}{2(l-1)}$,
$\alpha_2 = \frac{l-3}{2(l-1)}$, and
$\alpha_3 = \frac{\theta}{l-1}$. Using
$\langle \eta \rangle \le \langle k \rangle\langle k^{-1}\eta\rangle$, we may replace
$\alpha_2 +$ by $\alpha_2 $ in \eqref{3}. Now,
for given $s_1 > \frac12$ and $\varepsilon > 0$ we choose $\theta$ close
enough to zero, so that $\alpha_1 < s_1$
and $\alpha_3 < \varepsilon$, and $b_0$ (respectively $b_1$) close enough
to $-\frac14$ (respectively to $-\frac12$), so that $b' > - \frac12$.
Then, assuming by symmetry that $u_1$ has the largest frequency
with respect to the $x$-variable (i. e. $|k_1| \ge |k_{2, \dots , l}|$),
we obtain
\begin{equation} \label{4}
\|\partial_x \prod_{j=1}^l u_j\|_{X_{s_1+2s_2+\varepsilon,0,0;b'}}
\lesssim \|u_1\|_{X_{s_1+2s_2+\varepsilon,0,0;b}}
\prod_{j \ge 2}\|u_{j}\|_{X_{s_1,s_2,\varepsilon;b}}.
\end{equation}

4. The same upper bound holds for
$\|\partial_x \prod_{j=1}^l u_j\|_{X_{s_1,s_2,\varepsilon;b'}}$,
if $|\eta|\le |k|$ (or even if $|\eta|\le |k|^2$), where
$|k|$ (respectively $|\eta|$) are the frequencies in $x$
(respectively in $y$) of the whole product.
In the case where $|k|\le 1$ - assuming $\varepsilon \le \min{(s_1,1)}$
and $b'$ sufficiently close to $-\frac12$, the estimate
\begin{equation}  \label{5}
\|\partial_x \prod_{j=1}^l u_j\|_{X_{s_1,s_2,\varepsilon;b'}}
\lesssim\prod_{j=1}^l \|u_j\|_{X_{s_1,s_2,\varepsilon;b}}
\end{equation}
is easily derived by a combination of the standard Strichartz type
estimate \eqref{strx} and Sobolev embeddings. So we may
henceforth assume $|k|\ge1$, $|\eta|\ge 1$, and $|k^{-1}\eta|\ge 1$.

One last simple observation concerning the estimation
of $\|\partial_x \prod_{j=1}^l u_j\|_{X_{s_1,s_2,\varepsilon;b'}}$:
If we assume in addition to $|k_1| \ge |k_{2, \dots , l}|$ that
$u_1$ has a large frequency with respect to $y$; i.e.,
$|\eta| \lesssim |\eta_1|$, then from \eqref{3} we also obtain \eqref{5}.

5. It remains to estimate
$\|\partial_x \prod_{j=1}^l u_j\|_{X_{s_1,s_2,\varepsilon;b'}}$ in the case
where $|k_1| \ge |k_{2, \dots , l}|$ (symmetry assumption as before)
and $|\eta_1|\ll|\eta|$. By symmetry among $u_{2,\dots,l}$ we may assume in addition
that $|\eta_2| \ge |\eta_{1,3,\dots,l}|$ and hence that $|k_2|\ll|k|$,
because otherwise previous arguments apply with $u_1$
and $u_2$ interchanged. For this distribution of frequencies the
symbol of the Fourier multiplier $M(u_1u_3 \cdots u_l,u_2)$
becomes
$$
|k\eta_2-k_2\eta|^{1/2} \sim |k\eta_2|^{1/2} \gtrsim |k\eta|^{1/2}.
$$
Now let $P(u_1,\dots,u_l)$ denote the projection in Fourier space
on $\{|\eta_2| \ge |\eta_{1,3,\dots,l}|\}\cap
\{\langle k_2 \rangle \ll|k|\lesssim k_1\}$. Then by \eqref{bilyx'} we obtain
for $b>\frac12$, $b_1 < -\frac12$
\begin{align*}
& \|D_y^{1/2}P(u_1,\dots,u_l)\|_{X_{0,b_1}}\\
&\lesssim \|D_x^{1/2}u_2\|_{X_{0,b}}\|u_1u_3 \cdots
 u_l\|_{\widehat{L}_x^{\infty}L^2_{yt}}\\
&\lesssim \|D_x^{1/2}u_2\|_{X_{0,b}}\|\langle D_x \rangle^{\frac12 +}(u_1u_3)\|_{L^2_{xyt}}\prod_{j\ge 4}
  \|u_j\|_{\widehat{L}_x^{\infty}L^{\infty}_{yt}}\\
&\lesssim \| D_x ^{0 +}u_1\|_{X_{0,b}}\|\langle D_x
 \rangle^{\frac12 +}u_2\|_{X_{0,b}}\|\langle D_x \rangle^{\frac12 +}u_3\|_{X_{0,b}}
\prod_{j\ge 4} \|\langle D_x \rangle^{\frac12 +}\langle D_y
 \rangle^{\frac12 +}u_j\|_{X_{0,b}},
\end{align*}
where besides Sobolev type inequalities we have used \eqref{bilstr}
in the last step. Interpolation with \eqref{2} gives
for $0 \le \lambda \le 1$
\begin{align*}
&\|D_x^{\frac{\lambda}{2}}D_y^{\frac{1-\lambda}{2}}
P(u_1,\dots,u_l)\|_{X_{0,b_1}}\\
&\lesssim  \|D_x^{-\frac{\lambda}{2}+}u_1\|_{X_{0,b}}
\|\langle D_x \rangle^{\frac12 +}u_2\|_{X_{0,b}} \|\langle D_x \rangle^{\frac12 +}
 u_3\|_{X_{0,b}} \prod_{j\ge 4} \|\langle D_x \rangle^{\frac12 +}
 \langle D_y \rangle^{\frac12 +}u_j\|_{X_{0,b}}.
\end{align*}

Yet another interpolation, now with \eqref{1},
gives for $0< \theta \ll 1$, $b'=\theta b_0 + (1-\theta)b_1$,
$s_x=\frac12 (\lambda(1-\theta)+\theta)$ and
$s_y=\frac12(1-\theta)(1-\lambda)$
\begin{align*}
&\|D_x^{s_x}D_y^{s_y}P(u_1,\dots,u_l)\|_{X_{0,b'}}  \\
&\lesssim  \|D_x^{-s_x+}u_1\|_{X_{0,b}}
\|\langle D_x \rangle^{\frac12 +\frac{\theta}{4}+}\langle D_x^{-1}D_y
\rangle^{\frac{\theta}{2} +}u_2\|_{X_{0,b}}
\|\langle D_x \rangle^{\frac12 +\frac{\theta}{4}+}\langle D_x^{-1}D_y
\rangle^{\frac{\theta}{2} +}u_3\|_{X_{0,b}}\\
&\quad \times\prod_{j\ge 4} \|\langle D_x \rangle^{\frac12 +}
\langle D_y \rangle^{\frac12 +}u_j\|_{X_{0,b}}.
\end{align*}
The next step is to equidistribute the $\langle D_y \rangle$'s on $u_2,\dots,u_l$.
Here we must be careful, because the symmetry
between $u_2$ and $u_3,\dots,u_l$ was broken.
But since $u_2$ has the largest $y$-frequency we may first shift a
$\langle D_y \rangle^{\frac{l-3}{2(l-1)}+}$ onto $u_2$ and then interpolate
among $u_3,\dots,u_l$ in order to obtain
\begin{equation}
\begin{aligned}\label{6}
&\|D_x^{s_x}D_y^{s_y}P(u_1,\dots,u_l)\|_{X_{0,b'}} \\
& \lesssim  \|D_x^{-s_x+}u_1\|_{X_{0,b}}
  \prod_{j\ge 2} \|\langle D_x \rangle^{\beta_1 +}\langle D_y
  \rangle^{\beta_2 +}\langle D_x^{-1}D_y \rangle^{\beta_3 +}u_j\|_{X_{0,b}},
\end{aligned}
\end{equation}
where $\beta_1 = \frac12 + \frac{\theta}{4}$,
$\beta_2=\frac{l-3}{2(l-1)}$ and $\beta_3=\frac{\theta}{2}$.
Again we may replace $\beta_2 +$ by $\beta_2$. Now \eqref{6}
is applied to
$\langle D_x \rangle^{s_1 }\langle D_y \rangle^{s_2 }\langle D_x^{-1}D_y \rangle^{\varepsilon}
\partial_x P(u_1,\dots,u_l)$, where we can shift the
$\langle D_x \rangle^{s_1 }$ partly from the product to $u_1$ and the
$\langle D_y \rangle^{s_2 }$ partly to $u_2$. Moreover, since
$|k_2| \lesssim |k|$ and $|\eta| \lesssim |\eta_2|$ we have
$|k^{-1}\eta|\lesssim|k_2^{-1}\eta_2|$, so that a
$\langle D_x^{-1}D_y \rangle^{\varepsilon - \theta}$ may be thrown from the product
onto $u_2$.
The result is
\begin{align*}
&\|\partial_x P(u_1,\dots, u_l)\|_{X_{s_1,s_2,\varepsilon;b'}}\\
&\lesssim \|D_x^{s_1+1-\theta-2s_x+}u_1\|_{X_{0,b}}
 \|\langle D_x \rangle^{\beta_1 +}\langle D_y \rangle^{\beta_2 + s_2 -s_y
 + \theta}\langle D_x^{-1}D_y \rangle^{\varepsilon - \theta + \beta_3 +}u_2\|_{X_{0,b}}\\
&\times  \prod_{j\ge 3}\|\langle D_x \rangle^{\beta_1 +}\langle D_y
 \rangle^{\beta_2 }\langle D_x^{-1}D_y \rangle^{\beta_3 +}u_j\|_{X_{0,b}}.
\end{align*}
Here $\beta_2 \le s_2$ and by choosing
 $\theta < \min{(\varepsilon,\frac{2}{3(l-1)}, s_1-\frac12)}$ and $\lambda$ such
that $s_y=\beta_2+\theta$
we can achieve that
\begin{itemize}
 \item $s_1+1-\theta-2s_x < s_1 + 2 s_2 + \varepsilon$,
 \item $\beta_2 + s_2 -s_y + \theta \le s_2$,
 \item $\varepsilon - \theta + \beta_3 < \varepsilon$,
\end{itemize}
as well as $\beta_1 < s_1$, $\beta_3 < \varepsilon$. This gives
$$
\|\partial_x P(u_1,\dots, u_l)\|_{X_{s_1,s_2,\varepsilon;b'}}\lesssim
\|u_1\|_{X_{s_1+2s_2+\varepsilon,0,0;b}}
\prod_{j \ge 2}\|u_{j}\|_{X_{s_1,s_2,\varepsilon;b}}
$$
as desired.
\end{proof}

\begin{thebibliography}{00}

\bibitem{B93} Bourgain, J.:
Fourier transform restriction phenomena for certain lattice
subsets and applications to nonlinear evolution equations,
GAFA 3 (1993), 107 - 156 and 209 - 262

\bibitem{GTV97} Ginibre, J., Tsutsumi, Y., Velo, G.:
On the Cauchy-Problem for the Zakharov-System,
J. of Functional Analysis 151 (1997), 384 - 436

\bibitem{GPS08} Gr\"unrock, A., Panthee, M., Silva, J.:
On KP-II type equations on cylinders. Ann. I. H. Poincare - AN (2009), doi:10.1016/j.anihpc.2009.04.002

\bibitem{G09} Gr\"unrock, A.:
Bilinear space-time estimates for linearised KP-type equations on
the three-dimensional torus with applications. J. Math. Anal. Appl. 357 (2009) 330-339

\bibitem{HT} Hadac, M.:
On the local well-posedness of the Kadomtsev-Petviashvili II equation.
Thesis, Universit\"at Dortmund, 2007

\bibitem{H} Hadac, M.:
Well-posedness for the Kadomtsev-Petviashvili equation (KPII) and
generalisations, Trans. Amer. Math. Soc., S 0002-9947(08)04515-7, 2008

\bibitem{HNS} Hayashi, N., Naumkin, P. I., and Saut, J.-C.:
Asymptotocs for large time of global solutions to the generalized
Kadomtsev-Petviashvili equation, Commun. Math. Phys. 201, 577-590 (1999)

\bibitem{IN} I\'orio, R. J., and Nunes, W. V. L.:
On equations of KP-type, Proc. Royal Soc. Edin., 128A (1998), 725-743

\bibitem{IM01} Isaza, P., Mejia, J.:
Local and global Cauchy problems for the Kadomtsev-Petviashvili (KPII)
equation in Sobolev spaces of negative indices,
Comm. Partial Differential Equations, 26 (2001), 1027--1054.

\bibitem{IMS} Isaza, P., Mejia, J., and Stallbohm, V.:
Regularizing effects for the linearized Kadomtsev-Petviashvili
(KP) equation, Revista Colombiana de Matem\'aticas 31, 37 - 61 (1997)

\bibitem{KPV93}Kenig, C. E., Ponce, G., Vega, L.:
Wellposedness and scattering results for the generalized Korteweg-de
Vries equation via the contraction principle, CPAM 46 (1993), 527 - 620

\bibitem{KPV96}Kenig, C. E., Ponce, G., Vega, L.:
Quadratic forms for the 1 - D semilinear Schr\"odinger equation,
Transactions of the AMS 348 (1996), 3323 - 3353

\bibitem{KZa} Kenig, C. E., Ziesler, S. N.:
Local well posedness for modified Kadomstev-Petviashvili equations.
Differential Integral Equations 18 (2005), no. 10, 1111-1146

\bibitem{KZb} Kenig, C. E., Ziesler, S. N.:
Maximal function estimates with applications to a modified
Kadomstev-Petviashvili equation Commun. Pure Appl. Anal. 4 (2005), no. 1, 45-91

\bibitem{S93} Saut, J.-C.:
Remarks on the generalized Kadomtsev-Petviashvili equations,
Indiana Univ. Math. J. 42 (1993), no. 3, 1011--1026

\bibitem{Ta} Takaoka, H.:
Well-posedness for the Kadomtsev-Petviashvili II equation,
Adv. Differential Equations, 5 (10-12),
 1421-1443, 2000

\bibitem{T&T01} Takaoka, H., Tzvetkov, N.:
On the local regularity of the Kadomtsev-Petviashvili-II equation,
IMRN (2001), no. 2, 77--114

\bibitem{Tz} Tzvetkov, N.:
Global low regularity solutions for Kadomtsev-Petviashvili equation,
Differential Integral Equations, 13 (10-12), 1289-1320, 2000

\end{thebibliography}

\end{document}