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\RequirePackage[warning,log]{snapshot}
\controldates{26-AUG-2004,26-AUG-2004,26-AUG-2004,26-AUG-2004}
 
\documentclass{era-l}
\issueinfo{10}{11}{}{2004}
\dateposted{August 31, 2004}
\pagespan{97}{102}
\PII{S 1079-6762(04)00134-9}


\copyrightinfo{2004}{American Mathematical Society}
\revertcopyright

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\supp}{supp}

\begin{document}

\title{A trilinear restriction problem\linebreak[1] for the
paraboloid in $\mathbb{R}^{3}$}

\author{Jonathan Bennett}
\address{School of Mathematics, JCMB, Kings Buildings,
Mayfield Road, Edinburgh, EH9 3JZ, Scotland}
\email{J.Bennett@ed.ac.uk}
\thanks{The author was supported by an EPSRC
Postdoctoral Fellowship.}

\subjclass[2000]{Primary 42B10}
\date{December 18, 2003}
\revdate{July 16, 2004}
\keywords{Multilinear estimates, Fourier extension operator}

\commby{Yitzhak Katznelson}

\begin{abstract}
We establish a sharp trilinear inequality for the extension operator
associated to the paraboloid in $\mathbb{R}^{3}$. Our proof relies
on a recent generalisation of the classical Loomis--Whitney
inequality.
\end{abstract}

\maketitle
\section{Introduction}

Let $S$ be the paraboloid in $\mathbb{R}^{3}$
given by
$$
\{(\xi,|\xi|^{2}):\xi\in\mathbb{R}^{2}\},
$$
and let $d\sigma$ be the measure supported on $S$ given by
$$
\int \phi \:d\sigma=\int_{\mathbb{R}^{2}}\phi(\xi,|\xi|^{2})d\xi.
$$
For $g\in L^{1}(d\sigma)$, 
we define the extension operator applied to $g$
to be
$$
\widehat{gd\sigma}(x)=\int e^{-ix\cdot y}g(y)d\sigma(y).
$$

The classical restriction conjecture (for the paraboloid in
this case)
proposes the exponents $p$ and $q$ for which this operator is bounded
from $L^{p}(d\sigma)$ to $L^{q}(\mathbb{R}^{3})$---see \cite{St}.

It has long been known  
that certain $L^{p}-L^{q}$ estimates of this
type have what are often referred to as
``bilinear improvements''. 
In particular, the range of $p$'s
and $q$'s for which the bilinear operator
\begin{equation*}
(f,g)\mapsto \widehat{fd\sigma}\:\widehat{gd\sigma}
\end{equation*}
maps $L^{p}\times L^{p}$ to $L^{q}$, for $f$ and $g$ satisfying a certain
``support 
separation'' condition, is wider than that which is directly predicted
by H\"older's inequality and the restriction conjecture. For a
detailed description of these notions see \cite{TVV}.

The purpose of this note is to bring to light certain natural
trilinear estimates in this context.

\begin{theorem}\label{thm1}
Suppose $P_{1}, P_{2}, P_{3}\in S$ are such that the normals
to $S$ at these points span $\mathbb{R}^{3}$.
Then there exist neighbourhoods $U_{1}, U_{2}, U_{3}\subset S$
of $P_{1}, P_{2}, P_{3}$ respectively, and a constant
$C$ such that
\begin{equation*}
\|\widehat{fd\sigma}\:\widehat{gd\sigma}\:\widehat{hd\sigma}\|_{
L^{2}(\mathbb{R}^{3})}
\leq C
\|f\|_{4/3}\|g\|_{4/3}\|h\|_{4/3}
\end{equation*}
for all $f,g,h \in L^{4/3}(d\sigma)$ satisfying
$$
\supp(f)\subset U_{1},\;\;\;\;\;\supp(g)\subset U_{2},\;\;\;\;\;
\mbox{ and }\;\;\;\supp(h)\subset U_{3}.
$$
\end{theorem}
\subsection*{Remarks}
\begin{enumerate}
\item
The exponent $4/3$ on the right hand side is sharp 
given the exponent $2$ on the left. Naturally, the estimate
fails to hold if the points $P_{1}, P_{2}, P_{3}$ are 
separated in a more naive way. In particular, if we merely
ask that the points $P_{1}, P_{2}, P_{3}$ are distinct, the best
$L^{2}$ estimate possible is with $4/3$ replaced by $18/13$. This
can be seen as a consequence of the bilinear analysis in \cite{TVV}.
Furthermore, if we place no restriction at all on the points $P_{1}, P_{2},
P_{3}$, the best $L^{2}$ estimate is with $18/13$ replaced by
$3/2$. This follows easily from the existing linear restriction theory
(see \cite{St}).
\item
On a technical level, our approach is related to the 12/7 bilinear
restriction inequality of Moyua, Vargas and Vega \cite{MVV} (see also
\cite{TVV}). 
Inherent in their estimate is a bound for a certain
linear Radon transform in the plane. In the trilinear setting
matters are different partly because the Radon-like transforms that
arise are bilinear.
\item
Theorem \ref{thm1} was originally inspired by a multilinear inequality
for certain spherical averages of the extension
operator (this time associated to the sphere)---see \cite{BBC2}. 
\item
It seems plausible that
multilinear restriction 
estimates of this nature 
might have a role to play in proving new linear restriction
theorems in dimensions 3 and above. One only needs to glance at
\cite{TVV} to imagine this.
\end{enumerate}

The key ingredient in our proof of Theorem \ref{thm1} is the following
generalisation of the classical Loomis--Whitney inequality
(see \cite{bcw} for a proof of this).

\begin{lemma}\label{LW}
If $\pi_{1},\pi_{2},\pi_{3}:\mathbb{R}^{3}\rightarrow
\mathbb{R}^{2}$ are submersions in a neighbourhood of $x_{0}\in\mathbb{R}^{3}$ 
such that the kernels of
$d\pi_{1}(x_{0})$, $d\pi_{2}(x_{0})$, 
and $d\pi_{3}(x_{0})$ span $\mathbb{R}^{3}$,
then for all cut-off functions $a$ supported in a sufficiently small
neighbourhood of $x_{0}$,
there is a constant $C$ such that
\begin{equation*}
\int_{\mathbb{R}^{3}}
f(\pi_{1}(x))g(\pi_{2}(x))
h(\pi_{3}(x))a(x)dx
\leq C\|f\|_{2}\|g\|_{2}\|h\|_{2}
\end{equation*}
for all $f,g,h\in L^{2}(\mathbb{R}^{2})$.
\end{lemma}
\section{The proof of Theorem \ref{thm1}}

Let $u,v,w\in\mathbb{R}^{2}$ be such that
$P_{1}=(u,|u|^{2})$, $P_{2}=(v,|v|^{2})$ and 
$P_{3}=(w,|w|^{2})$. It is easily seen that the hypothesis on the
points
$P_{1}$, $P_{2}$ and $P_{3}$ is equivalent to the non-colinearity of
the points $u$, $v$ and $w$ in $\mathbb{R}^{2}$.
By Plancherel's theorem, symmetry and multilinear interpolation 
it suffices to prove that there
exist neighbourhoods $\Omega_{1}, \Omega_{2},
\Omega_{3}\subset\mathbb{R}^{2}$
of $u$, $v$, $w$, and a constant $C$ such that
\begin{eqnarray}\label{both}
\begin{aligned}
&\int_{(\mathbb{R}^{2})^{6}}
f_{1}(x)g_{1}(y)h_{1}(z)f_{2}(x')g_{2}(y')h_{2}(z')\\
&\qquad\qquad\times\delta(|x|^{2}+|y|^{2}+|z|^{2}-|x'|^{2}-|y'|^{2}-|z'|^{2})\\
&\qquad\qquad\quad\times\delta(x+y+z-x'-y'-z')\;
dx\:dy\:dz\:dx'dy'dz'\\
&\qquad\leq C\left\{\begin{array}{ll}
\|f_{1}\|_{2}\|g_{1}\|_{2}\|h_{1}\|_{2}
\|f_{2}\|_{1}\|g_{2}\|_{1}\|h_{2}\|_{1}\\[1ex]
\|f_{1}\|_{2}\|g_{1}\|_{2}\|h_{1}\|_{1}
\|f_{2}\|_{1}\|g_{2}\|_{1}\|h_{2}\|_{2}
\end{array}
\right.
\end{aligned}
\end{eqnarray}
for all 
$$
\supp(f_{i})\subset \Omega_{1},\;\;\;\;\;\supp(g_{i})\subset 
\Omega_{2},\;\;\;\;\;
\mbox{ and }\;\;\;\supp(h_{i})\subset \Omega_{3}.
$$
(Note that
$f_{1}$, $g_{1}$, $h_{1}$,
$f_{2}$, $g_{2}$ and $h_{2}$ are now functions on $\mathbb{R}^{2}$
rather than $S$.)

The proofs of the two inequalities in \eqref{both} follow the
same general scheme. We begin with the second as it is slightly more
straightforward algebraically. 
It should be remarked that in order to prove Theorem \ref{thm1} for
characteristic functions it is enough to obtain just one of these
inequalities.

Since $h_{1}$, $f_{2}$ and $g_{2}$
are controlled in $L^{1}$, we may suppose that $h_{1}=\delta_{z}$,
$f_{2}=\delta_{x'}$ and $g_{2}=\delta_{y'}$ for some $(z,x',y')$
in a sufficiently small neighbourhood of $(w,u,v)$.
Writing $X=x-x'$ and $Y=y-y'$, the left hand side of the
above becomes
\begin{eqnarray*}
\begin{aligned}
\int &f_{1}(X+x')g_{1}(Y+y')h_{2}(X+Y+z)\\
&\;\;\;
\;\;\;\;\;\times\delta\bigl(|X+x'|^{2}+|Y+y'|^{2}+|z|^{2}
-|x'|^{2}-|y'|^{2}-|X+Y+z|^{2}\bigr)\;dX\:dY\\
&=\frac{1}{2}
\int f_{1}(X+x')g_{1}(Y+y')h_{2}(X+Y+z)\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\times\delta\bigl((x'-z)\cdot X+(y'-z)\cdot Y
-X\cdot Y\bigr)\;dX\:dY.
\end{aligned}
\end{eqnarray*}
By the translation invariance of $L^{p}$-norms, 
it suffices to prove that
\begin{eqnarray}\label{last}
\begin{aligned}
\int f(X)g(Y)h(X+Y)\delta\bigl((x'-z)\cdot X+(y'-z)\cdot Y
-X\cdot Y\bigr)\;dX\:dY
\end{aligned}
\end{eqnarray}
is bounded by $C\|f\|_{2}\|g\|_{2}\|h\|_{2}$, for all
$f$, $g$, and $h$ supported in sufficiently small
neighbourhoods of the origin.

By translation (or Galilean) invariance, scaling and a rotation, 
we may suppose that $z=0$,
$x'=e_{1}$ and $y'=a$, where 
$a_{2}>c$ for some constant $c>0$ depending on $u$, $v$, and $w$.
Since
$$
e_{1}\cdot X+a\cdot Y
-X\cdot Y=(1-Y_{1})\left(X_{1}-
\frac{X_{2}Y_{2}-a\cdot Y}{1-Y_{1}}\right),
$$
we are reduced to proving that for some neighbourhood $U$ of the
origin in $\mathbb{R}^{3}$,
\begin{equation}\label{enough}
\int_{U} f(\pi_{1}(X_{2},Y))g(\pi_{2}(X_{2},Y))
h(\pi_{3}(X_{2},Y))\:dX_{2}\:dY
\leq C\|f\|_{2}\|g\|_{2}\|h\|_{2},
\end{equation}
where\footnote{In deriving this representation of the trilinear form 
we have used the fact that for $Y$ in a sufficiently
small neighbourhood of the origin, $1-Y_{1}$ is bounded
away from $0$.}
$$
\pi_{1}(X_{2},Y)=\left(\frac{X_{2}Y_{2}-a\cdot
Y}{1-Y_{1}},X_{2}\right),
\;\;\;\;
\pi_{2}(X_{2},Y)=Y
$$
and $\pi_{3}=\pi_{1}+\pi_{2}$.
(As may be expected, there are other parametrisations
that may be chosen here.)

In order to prove \eqref{enough} we appeal to Lemma \ref{LW}. After a 
straightforward computation we see that $\pi_{1}$, $\pi_{2}$, and
$\pi_{3}$ are submersions in a neighbourhood of $0$, and furthermore,
$$
\ker d\pi_{1}(0)=\left<(0,-a_{2},a_{1})\right>,
$$
$$
\ker d\pi_{2}(0)=\left<(1,0,0)\right>,
$$
and
$$
\ker
d\pi_{3}(0)=\left<(1-a_{1},-a_{2},a_{1}-1)\right>.
$$
Since the determinant of the above three generators is equal to $a_{2}>c>0$,
\eqref{enough} follows.

We now turn to the proof of the first inequality in \eqref{both}.
Since $f_{2}$, $g_{2}$, and $h_{2}$
are controlled in $L^{1}$, we may suppose that
$f_{2}=\delta_{x'}$, $g_{2}=\delta_{y'}$ and $h_{2}=\delta_{z'}$
for some $(x',y',z')$
in a sufficiently small neighbourhood of $(u,v,w)$.
Again, by Galilean invariance, 
scaling, and a rotation, we may suppose that
$z'=0$, $x'=e_{1}$ and $y'=a$, where $a_{2}>c$ for some constant $c>0$ 
depending on $u$, $v$, and $w$. 
Writing $X=x-x'$ and $Y=y-y'$, the left hand side of \eqref{both} becomes
\begin{eqnarray*}
\begin{aligned}
\int &f_{1}(X+x')g_{1}(Y+y')h_{1}(-X-Y+z')\\
&\;\;\;\;\;\times\delta\bigl(|X+x'|^{2}+|Y+y'|^{2}+|X+Y-z'|^{2}
-|x'|^{2}-|y'|^{2}-|z'|^{2}\bigr)\;dX\:dY\\
&=\frac{1}{2}
\int f_{1}(X+x')g_{1}(Y+y')h_{1}(-X-Y)\\
&\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\times\delta\bigl(|X+(e_{1}+Y)/2|^{2}
-|e_{1}+Y|^{2}/4+|Y|^{2}+a\cdot Y\bigr)\;dX\:dY.
\end{aligned}
\end{eqnarray*}
It thus suffices to prove that
\begin{eqnarray*}
\begin{aligned}
\int &f(X)g(Y)h(X+Y)\\
&\;\;\;\;\;\times\delta\bigl(|X+(e_{1}+Y)/2|^{2}
-|e_{1}+Y|^{2}/4+|Y|^{2}+a\cdot Y\bigr)\;dX\:dY
\end{aligned}
\end{eqnarray*}
is bounded by 
$C\|f\|_{2}\|g\|_{2}\|h\|_{2}$, for all
$f$, $g$, and $h$ supported in sufficiently small
neighbourhoods of the origin.

Now for fixed $Y$, $X$ lives on the circle given parametrically
by
$$
X=-\frac{1}{2}(e_{1}+Y)+r(Y)(\cos t, \sin t),
$$
where
$r(Y)^{2}=\tfrac{1}{4}|e_{1}+Y|^{2}-|Y|^{2}-a\cdot Y$ 
and $t\in\mathbb{R}$.
Observe that since we are only concerned with $X$ and $Y$ in a small
neighbourhood
of the origin, we need only consider $t$ 
in a small neighbourhood of $0$.
Hence we are reduced to proving that for some
neighbourhood $U$ of the origin in $\mathbb{R}^{3}$,
\begin{equation}\label{enough'}
\int_{U} f(\pi_{1}(Y,t))g(\pi_{2}(Y,t))
h(\pi_{3}(Y,t))\:dY\:dt
\leq C\|f\|_{2}\|g\|_{2}\|h\|_{2},
\end{equation}
where\footnote{In deriving this representation 
we have used the fact that for $Y$ in a
sufficiently small neigbourhood of the origin, $r(Y)$ is bounded away
from $0$.}
$$
\pi_{1}(Y,t)=Y,
$$
$$
\pi_{2}(Y,t)=-(e_{1}+Y)/2+r(Y)(\cos t,\sin t),
$$
and $\pi_{3}=\pi_{1}+\pi_{2}$.
After a 
straightforward computation we see that $\pi_{1}$, $\pi_{2}$, and
$\pi_{3}$ are submersions in a neighbourhood
of $0$, and furthermore,
$$
\ker d\pi_{1}(0)=\left<(0,0,1)\right>,
$$
$$
\ker d\pi_{2}(0)=\left<\left(-a_{2},a_{1},a_{1}\right)\right>,
$$
and
$$
\ker
d\pi_{3}(0)=\left<\left(-a_{2},a_{1}-1,
1-a_{1}\right)\right>.
$$
As before, the determinant of the above three generators is equal to 
$a_{2}>c>0$,
and so \eqref{enough'} follows.

\subsubsection*{Remark}
 
There is a minor technical issue in our argument that we have
glossed over here. It is of course important that the neighbourhoods
of the origin and the constant $C$ appearing in \eqref{last} 
may be chosen independently of $(z,x',y')$ belonging to a
sufficiently small neighbourhood of $(w,u,v)$. This detail may be easily
dealt with by appealing to a more quantative version of Lemma \ref{LW}
(such as that in \cite{bcw}).
On doing this, one may also quantify the constant $C$
and the neighbourhoods $U_{1}, U_{2}$, and $U_{3}$ appearing in the
statement of Theorem \ref{thm1}. 
These issues will be elucidated in a subsequent 
paper.

\section{The wider context}

The standard examples (see \cite{St}) in the context of the restriction
conjecture suggest the following $n$-linear conjecture in $n$ dimensions. 
Here $S$ will denote the paraboloid
$$
\{(\xi, |\xi|^{2}):\xi\in\mathbb{R}^{n-1}\}
$$ in $\mathbb{R}^{n}$, and $d\sigma$ the measure supported on $S$
given by
$$
\int \phi \:d\sigma=\int_{\mathbb{R}^{n-1}}\phi(\xi,|\xi|^{2})d\xi.
$$

\begin{conjecture}
Suppose $P_{1},\dots, P_{n}\in S$ are such that the normals
to $S$ at these points span $\mathbb{R}^{n}$.
Then there exist neighbourhoods $U_{1},\dots, U_{n}\subset S$
of $P_{1},\dots, P_{n}$ respectively, and a constant
$C$ such that
\begin{equation*}
\Bigl\|\prod_{j=1}^{n}\widehat{g_{j}d\sigma}\Bigr\|_{
L^{q/n}(\mathbb{R}^{n})}
\leq C
\prod_{j=1}^{n}\|g_{j}\|_{p}
\end{equation*}
for all $g_{j} \in L^{p}(d\sigma)$ satisfying
$$
\supp(g_{j})\subset U_{j},\;\;\;1\leq j\leq n,
$$
if and only if $q\geq \frac{2n}{n-1}$ and $p'\leq \frac{n-1}{n}q$.
\end{conjecture}

We remark that by 
interpolation and H\"older's inequality, this conjecture is equivalent 
to the inequality
\begin{equation}\label{equivmlrc}
\Bigl\|\prod_{j=1}^{n}\widehat{g_{j}d\sigma}\Bigr\|_{
L^{2/(n-1)}(\mathbb{R}^{n})}
\leq C
\prod_{j=1}^{n}\|g_{j}\|_{2}.
\end{equation}

\subsection*{Remarks}
\begin{enumerate}
\item
The known linear and bilinear restriction theory 
clearly implies progress on the above conjecture, just by H\"older's
inequality. As may be expected,
the non-trivial exponents obtained in this way 
lie away from the sharp line $p'=\frac{n-1}{n}q$. The purpose of Theorem 1 is to
provide
a non-trivial point on this line in three dimensions. 
\item
The above conjecture may in fact be made for quite general smooth
codimension 1
submanifolds of $\mathbb{R}^{n}$. The proof of
Theorem \ref{thm1} 
may also be
adapted
to this general context. The details of this will be made explicit 
in a subsequent paper.
\item
By a standard Rademacher function argument, the 
above conjecture \eqref{equivmlrc} implies a certain multilinear Kakeya-type
estimate, which we now describe. 
Merely for expositional convenience we replace the paraboloid
with the unit sphere here.
Suppose
that the vectors $\omega_{1},\dots, \omega_{n}
\in\mathbb{S}^{n-1}$ span $\mathbb{R}^{n}$, then
there exist neighbourhoods $U_{1},\dots, U_{n}\subset
\mathbb{S}^{n-1}$ of
$\omega_{1},\dots, \omega_{n}$ respectively, and a constant $C$ 
such that
$$
\;\;\;\;\;\;\;\;\;\;\;\;\Bigl\|\sum_{T_{1}\in\mathbb{T}_{1}}\chi_{T_{1}}
\sum_{T_{2}\in\mathbb{T}_{2}}\chi_{T_{2}}\cdot\cdot\cdot
\sum_{T_{n}\in\mathbb{T}_{n}}\chi_{T_{n}}\Bigr\|_{L^{1/(n-1)}(\mathbb{R}^{n})}
\leq C\sum_{T_{1}\in\mathbb{T}_{1}}|T_{1}|\cdot\cdot\cdot
\sum_{T_{n}\in\mathbb{T}_{n}}|T_{n}|
$$
for all families $\mathbb{T}_{1}$,\dots,$\mathbb{T}_{n}$ of
$\delta\times\cdot\cdot\cdot\times\delta\times 1$-tubes 
in $\mathbb{R}^{n}$ such that their directions belong to
$U_{1},\dots,U_{n}$ respectively. Here $C$ should be independent of the
small parameter $0<\delta\leq 1$.
We refer the reader to \cite{TVV} for a discussion of the linear and 
bilinear Kakeya phenomena.
\item
It might be interesting to establish
whether Theorem \ref{thm1} may be generalised to 
$n\geq 4$. It is not immediately clear how our approach may be
extended since the multilinear Radon transforms that arise
cannot be dealt with directly by the natural higher-dimensional version
of Lemma \ref{LW}. 
\end{enumerate}

\section*{Acknowledgements}
We would like to thank Tony Carbery, Susana Guti\'errez and Jim Wright
for a number of very useful discussions on the subject of this note.

\bibliographystyle{amsplain}
\begin{thebibliography}{36}
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A multilinear extension inequality in $\mathbb{R}^{n}$,
\textit{Bull. London Math. Soc.} {\bf 36 (3)}
(2004),
407--412. \MR{2038728}

\bibitem{bcw} J. M. Bennett, A. Carbery, and J. Wright, A
generalisation of the Loomis--Whitney inequality in $\mathbb{R}^{n}$,
in preparation.

\bibitem{MVV} A. Moyua, A. Vargas, L. Vega, Restriction
theorems and maximal operators related to oscillatory integrals in
$\mathbb{R}^{3}$, \textit{Duke Math. J.} {\bf 96 (3)} (1999), 547--574.
\MR{1671214 (2000b:42017)}

\bibitem{St} E. M. Stein, Harmonic Analysis, Princeton University Press,
Princeton, NJ, 1993.
\MR{1232192 (95c:42002)}

\bibitem{TVV} T. Tao, A. Vargas, L. Vega, 
A bilinear approach to the restriction and Kakeya
conjectures,
\textit{J. Amer. Math. Soc.} {\bf 11} (1998), 967--1000.
\MR{1625056 (99f:42026)}
\end{thebibliography}
\end{document}