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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \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} \bibitem{BBC2} J. A. Barcel\'o, J. M. Bennett, and A. Carbery, 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}