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\begin{document}

\title{Three-dimensional restriction theorems}

\author{Terence Tao}
\address{Department of Mathematics, UCLA, Los Angeles, CA 90024}
\email{tao@@math.ucla.edu}

\author{Ana Vargas}
\address{Departamento de Matem\'aticas, Universidad Aut\'onoma
de Madrid, 28049 Madrid (Spain).}
\email{ana.vargas@@uam.es}

\author{Luis Vega}
\address{Departamento de Matem\'aticas, Universidad del
Pa\'is Vasco, Apartado 644, 48080, Bilbao (Spain).}
\email{mtpvegol@@lg.ehu.es}

\subjclass{42B10, 42B25}

\begin{abstract}
We summarize our recent work on linear and bilinear restriction conjectures
in three dimensions.
\end{abstract}

\maketitle

\section{Elliptic surfaces}

Let $S$ be the unit sphere or compact subset of the paraboloid in $\R^3$, and let
$\R^*(p \to q)$ denote the adjoint restriction estimate
$$ \| \widehat{f d\sigma} \|_q \lesssim \|f\|_p$$
where $d\sigma$ is surface measure on $S$.  We also let $\R^*(p \times p \to q/2)$ denote
the bilinear version
$$ \| \widehat{f d\sigma} \widehat{g d\sigma} \|_{q/2} \lesssim \|f\|_p \|g\|_p$$
where $f$ and $g$ have unit-seperated supports.  

\begin{conjecture} (Restriction conjecture) $R^*(p \to q)$ holds whenever
$q \geq 2 p^\prime$, $q > 3$.    
\end{conjecture}

\begin{conjecture} (Bilinear restriction conjecture) $R^*(p \times p \to q/2)$ holds
whenever $q \geq \frac{5}{3} p^\prime$,
$q \geq 3$, $\frac{5}{2q} + \frac{1}{p} \leq 2$.
In other words, we conjecture $R^*(3 \times 3 \to 3/2)$ and $R^*(2 \times 2 \to 5/3)$.
\end{conjecture}

These exponents are best possible.

Existing progress on these conjectures:

\begin{itemize}
\item (Tomas-Stein) The restriction conjecture is true whenever $q \geq 4$.
\item (Bourgain, Wolff, MVV) The restriction conjecture is true whenever $p > 7/3$ and $q > 4 - 2/11$.
\item (Bourgain, MVV) We have $R^*(p \times p \to 2)$ if and only if $p \geq 12/7$.  In particular, the bilinear restriction conjecture is true whenever
$q \geq 4$.
\end{itemize}

$R^*(p \to q)$ trivially implies $R^*(p \times p \to q/2)$.  Conversely,
if $R^*(p \times p \to q/2 - \eps)$ holds and $q \geq 2p^\prime$, then
$R^*(p \to q)$ holds.  In particular, the bilinear restriction conjecture
implies the restriction conjecture.  The proof involves a Whitney decomposition
of $\widehat{fd\sigma} \widehat{fd\sigma}$, parabolic scaling and the inequality
$$ \|\sum_k f_k\|_p \lesssim (\sum_k \|f_k\|_p^{p^*})^{1/p^*}$$
which holds whenever $f_k$ have frequency supports in disjoint rectangles
and $1 \leq p \leq \infty$; here $p^* = \min(p,p^\prime)$.

In \cite{tvv:bilinear} we used bilinear versions of arguments in \cite{borg:kakeya,borg:stein,
wolff:kakeya} to prove the following:

\begin{itemize}
\item The restriction conjecture is true whenever $q > 4 - 5/27$.
\item The restriction conjecture is true whenever $p > 170/77$ and $q > 4 - 2/9$.
\item We have $R^*(2 \times 2 \to q)$ for all $q > 2 - 5/69$.
\end{itemize}

In \cite{tv:cone} we combined the above techniques with the ideas from \cite{borg:cone} and
the x-ray estimate in \cite{wolff:xray} to improve these results to

\begin{itemize}
\item The restriction conjecture is true whenever $q > 4 - 8/31$.  
\item The restriction conjecture is true whenever $p > 26/11$ and $q > 4 - 2/7$.
\item We have $R^*(2 \times 2 \to q)$ for all $q > 2 - 2/17$.
\end{itemize}

To the best of our knowledge, these results (together with what can
be obtained by interpolation) are currently the best known
restriction theorems.

As an application, we show that the solution to the free Schr\"odinger
equation converges a.e. to its initial data whenever the data is in 
$H^s$ for $s > 15/32$.  This improves upon the earlier result of
\cite{vargas:restrict}, which showed it for $s > (164 + \sqrt{2})/339$; 
for comparison, the classical 
results were for $s > 1/2$.

\section{Conic surfaces}

We now repeat the preceding discussion, but with $S$ replaced by a small
subset of the light cone in $\R^3$.

The linear estimates on the cone are completely known:

\begin{theorem} \cite{barcelo} $R^*(p \to q)$ holds if and only if
$q \geq 3 p^\prime$, $q > 4$.    
\end{theorem}

However, the exact range of bilinear estimates are still open:

\begin{conjecture} $R^*(p \times p \to q/2)$ holds
whenever $q \geq \frac{5}{3} p^\prime$, $q \geq 10/3$.
In other words, we conjecture $R^*(2 \times 2 \to 5/3)$.
\end{conjecture}

Thee exponents are sharp.

It is easy to show (by a variant of an argument of Sj\"olin) that $R^*(2 \times 2 \to 2)$.
By Radon transform $L^p$ mapping properties one can improve this to $R^*(12/7 \times 12/7 \to 
2)$; this is the same argument which gave the corresponding estimate for elliptic surfaces.

In \cite{borg:cone} the estimate $R^*(p \times p \to 2)$ was proven for all $p > 2-\tau$ for some $\tau > 0$ (we conservatively estimate $\tau = 13/2408$).  In \cite{tv:cone} we
prove this estimate for all $p > 2-1/20$.

In \cite{borg:cone} Bourgain observed that this estimate could be used
to improve Mockenhaupt's $1/8$ cone multiplier result on $L^4$.  Indeed we can lower the
order of the cone multiplier from $1/8 + \eps$ to $1/8 - 1/238 + \eps$.

Similar considerations apply to Sogge's local smoothing conjecture, the $1/8 + \eps$ loss
in derivatives for the wave equation from $L^4_x$ to $L^4_{x,t}$ can be improved
to $1/8 - 1/238 + \eps$.  As in \cite{borg:cone}, the idea is to first improve
the square function estimate in \cite{mock:cone}.

One can also improve the sharp $(L^p, L^q)$ local smoothing estimates for the
wave equation of Schlag and Sogge, with $q = 3p^\prime$ from $q \geq 5$
to $q \geq 5 - 1/20$.





\begin{thebibliography}{10}

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B. Barcelo, \emph{On the restriction of the Fourier transform to
a conical surface}, Trans. Amer. Math. Soc. \textbf{292} (1985),
321--333.

\bibitem{borg:kakeya}
J. Bourgain, \emph{Besicovitch-type maximal operators and
applications to Fourier analysis}, Geom. and Funct. Anal. \textbf{22}
(1991), 147--187.

\bibitem{borg:16-9}
J. Bourgain, \emph{On the restriction and multiplier problem
in $\R^3$}, Lecture notes in Mathematics, no. 1469.  Springer Verlag, 1991.

\bibitem{borg:schrodinger}
J. Bourgain, \emph{A remark on Schrodinger operators},
Israel J. Math. 77 (1992), 1--16.

\bibitem{borg:cone}
J. Bourgain, \emph{Estimates for cone multipliers},
Operator Theory: Advances and Applications, \textbf{77} (1995), 41--60.

\bibitem{borg:stein}
J. Bourgain, \emph{Some new estimates on oscillatory integrals},
Essays in Fourier Analysis in honor of E.~M. Stein, Princeton University
Press (1995), 83--112.

\bibitem{mock:cone}
G. Mockenhaupt, \emph{A note on the cone multiplier}, Proc. AMS \textbf{117}
(1993), 145--152.

\bibitem{vargas:restrict}
A. Moyua, A. Vargas, L. Vega, \emph{Schr\"odinger Maximal Function
and Restriction Properties of the Fourier transform}, International Math.
Research Notices \textbf{16} (1996).

\bibitem{vargas:2}
A. Moyua, A. Vargas, L. Vega, \emph{Restriction theorems and Maximal
operators related to oscillatory integrals in $\R^3$}, to appear, Duke
Math. J.

\bibitem{schlag:circular}
W. Schlag, C.~D. Sogge, \emph{Local smoothing estimates
related to the circular maximal theorem}, Math. Res. Lett.
\textbf{4} (1997): 1--15.

\bibitem{sogge:smoothing}
C.~D. Sogge, \emph{Propogation of singularities and maximal functions in the
plane}, Invent. Math. \textbf{104} (1991), 349--376.

\bibitem{tvv:bilinear}
T. Tao, A. Vargas, L. Vega, \emph{A bilinear approach to the
restriction and Kakeya conjectures}, to appear, JAMS.

\bibitem{tv:cone}
T. Tao, A. Vargas, \emph{A bilinear approach to cone multipliers
and applications}, in preparation.

\bibitem{wolff:kakeya}
T.~H. Wolff, \emph{An improved bound for Kakeya type maximal functions},
Revista Mat. Iberoamericana. \textbf{11} (1995). 651--674.

\bibitem{wolff:xray}
T.~H. Wolff, \emph{A mixed norm estimate for the x-ray transform}, to 
appear in Revista Mat. Iberoamericana.

\end{thebibliography}

\end{document}
\endinput