\documentstyle[amssymb,amsfonts,psfig,12pt]{article}
\input MathMacro.tex 
\newcommand{\eqn}[1]{(\ref{#1})}
\newcommand{\is}{$I^{s}\;$}
\newcommand{\mbf}[1]{\mbox{\boldmath ${#1}$}}
\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{  \end{center}}
\newcommand{\page}{\vfil \break}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{  \end{itemize}}
\newcommand{\benum}{\begin{enumerate}}
\newcommand{\eenum}{  \end{enumerate}}
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{  \end{description}}
\def\rp{{r^\prime}}
\def\rpt{{\tilde r^\prime}}
\def\rtil{{\tilde r}}
\def\qtil{{\tilde q}}
\def\kp{{k^\prime}}
\def\kt{{\tilde{k}}}
\def\qp{{q^\prime}}
\def\qpt{{\tilde q^\prime}}
\def\R{{\hbox{\bf R}}}
\def\rn{{\R^n}}
\def\Z{{\hbox{\bf Z}}}
\def\eps{{\varepsilon}}
\def\implies{{\Rightarrow}}
\def\RR{{\frak R}}
% \newcommand{\lesssim}{\mbox{$\,\stackrel{\textstyle{<}}{\sim}\,$}}

\renewcommand {\theequation}{\thesection.\arabic{equation}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]

\begin{document}
\Large
\vskip 1in
\bc
{\bf \LARGE   A bilinear approach to the \\ ~ \\ Restriction and Kakeya conjectures
}\\
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
Terence Tao \\
\bigskip
{\it Mathematics Department \\ \bigskip UCLA \\ }  
\bigskip
With Ana Vargas and Luis Vega
\bigskip
\ 
\bigskip
{\tt www.math.ucla.edu/$\sim$tao/preprints}
\ec

\page

\noindent
{\LARGE
The purpose of this talk is to:
}
\bigskip
\bi
 \item Describe the linear restriction conjecture, and its bilinear
variant

 \item Discuss the equivalences between the linear and bilinear versions,
and the extra estimates available in the bilinear setting

 \item Show how the bilinear perspective on the restriction (and Kakeya) conjectures can be used to yield new progress on the linear restriction conjecture

\ei

\page

\noindent
{
Let $n \geq 2$, and let $S$ be a smooth compact hypersurface in $\R^n$ with surface
measure $d\sigma$.  Suppose that $f$ is an $L^p$ measure on $S$.  We ask whether
the Fourier transform $\widehat{fd\sigma}$ of $f$ is necessarily in an $L^q$ class.
More precisely, we ask whether one has the a priori estimate
$$ \| \widehat{fd\sigma} \|_{L^q(\R^n)} \lesssim \|f\|_{L^p(S)}$$
for test functions $f$.  If this is the case we say that $R^*(p \to q)$ holds.
\bi
 \item This is trivial if $q=\infty$.  Also, if it is true for some $(p,q)$ it is
also true for higher $p$ and higher $q$ by Sobolev and H\"older inequalities.
 \item If $S$ is flat then no further estimates are possible, because
$\widehat{fd\sigma}$ will not decay in the normal direction.  However, for
curved surfaces more estimates exist.
 \item These estimates are important to PDE: if $S$ is the light cone (or
sphere, or parabola, or cubic), then $\widehat{fd\sigma}$ is a solution to the
wave equation (or Helmholtz, or Schr\"odinger equation, or KdV).
 \item For simplicity we restrict ourselves to the paraboloid $x_n = \frac{1}{2} |\underline{x}|^2$, $|x| \lesssim 1$ in $\R^n$ for $n=2,3$, and ignore epsilon
losses in the $p$, $q$ indices.
\ei
}
\page

\bigskip
\noindent
Necessary conditions on $p,q$
\bi
 \item If one takes $f=1$, then $\widehat{fd\sigma}$ decays like $|x|^{-(n-
1)/2}$,
and so one needs $q > 2n/(n-1)$.
 \item If one takes $f$ to be a bump function on a cap (e.g. $|\underline{x}| \lesssim \delta$), then $\widehat{fd\sigma}$ is about $\delta^{n+1}$ on a $\delta^{-1} \times \delta^{-2}$
tube, and so one needs $q \geq (n+1)/(n-1) p^\prime$.  (The ``Knapp example'').
 \item For the paraboloid, this condition also can be obtained from scaling considerations
(using the natural parabolic scaling associated to $S$).
 \item The {\it restriction conjecture} states that these two conditions are
also sufficient for $R^*(p \to q)$.
 \item When $q = 2 p^\prime$ we say that the estimate $R^*(p \to q)$ is
{\it sharp}.  Sharp estimates have the advantage of being scale-invariant, and so
(for instance) can be proven for the entire paraboloid and not just a compact
subset.
\ei

\page
\noindent

The two-dimensional case

\bi

\item When $n=2$, the restriction conjecture has been proved.  We
sketch the argument of Fefferman (ignoring epsilons and logarithmic
divergences for simplicity).

\item It suffices to prove the estimate $R^*(4 \to 4)$, since
the other estimates in the restriction conjecture can be obtained by interpolation and
H\"older's inequality.

\item We have to prove
$$ \| \widehat{fd\sigma} \|_4 \lesssim \|f\|_4$$
when $d\sigma$ is surface measure on an arc of the parabola in $\R^2$.  
 \item We square both sides and apply Plancherel's theorem:
$$ \| fd\sigma * fd\sigma \|_2 \lesssim \|f\|_4 \|f\|_4.$$
\item Now we use the geometric fact that every point $\xi \in \R^2$ is the sum
of at most one pair of points $\omega_1$, $\omega_2$ in $S$.  In fact, we can
say that
$$fd\sigma * fd\sigma(\omega_1 + \omega_2) \sim f(\omega_1) f(\omega_2) / |\omega_1 - \omega_2|.$$
So after a change of variables we have
$$ \| fd\sigma * fd\sigma\|_2^2 \sim \int\int |f(\omega_1)|^2 |f(\omega_2)|^2 \frac{d\omega_1 d\omega_2}{|\omega_1 - \omega_2|},$$
and estimate follows from fractional integration.
\ei

\page

\bi
\item We have just proved the estimate
$$ \| \widehat{fd\sigma} \widehat{fd\sigma} \|_2 \lesssim \|f\|_4 \|f\|_4.$$
This is obviously equivalent to the bilinear estimate
$$ \| \widehat{fd\sigma} \widehat{gd\sigma} \|_2 \lesssim \|f\|_4 \|g\|_4.$$
Without any further conditions on $f$ and $g$, this estimate is sharp
(just take $f=g=$ the Knapp example).
\item However, if we make the restriction that $f$ and $g$ are
supported on {\it disjoint} (unit-separated) portions on the
parabola, then one can do better.  Indeed, by repeating Fefferman's
argument we have
$$ \| \widehat{fd\sigma} \widehat{gd\sigma} \|_2^2 \lesssim
\int\int |f(\omega_1)|^2 |g(\omega_2)|^2 \frac{d\omega_1 d\omega_2}{|\omega_1 - \omega_2|}$$
and since $|\omega_1 - |\omega_2| \sim 1$ we thus have
$$ \| \widehat{fd\sigma} \widehat{gd\sigma} \|_2 \lesssim \|f\|_2 \|g\|_2$$
which is a superior estimate.
\ei

\page

Definition: We let $R^*(p \times p \to q)$ denote the estimate
$$ \| \widehat{fd\sigma} \widehat{gd\sigma} \|_q \lesssim \|f\|_p \|g\|_p.$$
whenever $f$, $g$ lie on unit-separated subsets of $S$.

\bi
\item By H\"older's inequality, $R^*(p \to q)$ implies $R^*(p \times p \to q/2)$,
however the converse is not true because of the separation condition.
On the previous slide we have proved $R^*(2 \times 2 \to 2)$, despite
the fact that the corresponding linear estimate $R^*(2 \to 4)$ is false.

\item Fortunately, a satisfactory converse can be proven:
If $R^*(p \times p \to q/2 - \eps)$ holds and $q \leq (n+1)/(n-1) p^\prime$,
then $R^*(p \to q)$ holds.  Recall that $q \leq (n+1)/(n-1) p^\prime$
was a necessary condition for $R^*(p \to q)$ (but not
for $R^*(p \times p \to q/2)$).

\item Thus the bilinear estimates include the linear estimates as a subset, but are
not 
subject to the scaling condition (aka the Knapp example) and so encompass a richer set of estimates.

\ei

\page

Bilinear restriction estimates, $n=3$.

\bi 
\item In the bilinear setting, replacing $R^*(p \to q)$ by $R^*(p \times p
\to q/2)$, we expect a wider range of estimates because the Knapp counterexample
is much less of a nuisance.  Indeed, a few simple examples ("squashed
Knapp examples") lead one to expect the bilinear estimate to hold in a 
pentagonal region which is
obtained by interpolating the restriction conjecture estimates with
a bilinear estimate $R^*(2 \times 2 \to 5/3)$, which was first
conjectured by Machedon and Klainerman.

{
\centering\ \psfig{figure=bilinear.eps,height=3in,width=5in}
}
\ei
\page

Sketch of bilinear to linear implication
\bi 

\item Assume that we have the bilinear estimate $R^*(p \times p \to q/2 - \eps)$
$$ \| \widehat{fd\sigma} \widehat{gd\sigma} \|_{q/2 - \eps} \lesssim \|f\|_p \|g\|_p$$
for separated $f$, $g$, and we want to prove the estimate
$$ \| \widehat{fd\sigma} \|_{q} \lesssim \|f\|_{p}$$
(ignoring epsilons).  We square this as
$$ \| \widehat{fd\sigma} \widehat{fd\sigma} \|_{q/2} \lesssim \|f\|_p \|f\|_p.$$
\item Break up $f$ into two pieces $f_1$, $f_2$ supported on different
halves of $S$.  Roughly speaking, the contribution of
$$ \| \widehat{f_1d\sigma} \widehat{f_2d\sigma} \|_{q/2} $$
can be controlled by the bilinear hypothesis since $f_1$, $f_2$ are
essentially separated.  So we only need to deal with the diagonal
terms 
$$ \| \widehat{f_1d\sigma} \widehat{f_1d\sigma} \|_{q/2} $$
$$ \| \widehat{f_2d\sigma} \widehat{f_2d\sigma} \|_{q/2} $$
From Planchere's theorem these functions have disjoint Fourier supports and
so we can exploit some orthogonality.  To estimate each term,
we take advantage of the fact that the paraboloid is invariant under parabolic 
scaling to rescale $f_1$ and $f_2$ to live on the original surface $S$
(not just one half of $S$).  Then we iterate the entire process,
decomposing $f_1$, $f_2$, etc.  Providing we have the scaling condition
$q \leq (n+1)/(n-1) p^\prime$, this process converges (modulo a technicality which
requires the epsilon in the hypothesis).
\ei

\page

\bigskip
\noindent
Bilinear and linear restriction theorems in three dimensions
\bi
\item In three dimensions the restriction conjecture is not completely solved.
The ideal estimate would be $R^*(3 \to 3)$ (ignoring epsilons), but we
are some way from achieving that goal.  In the 1970s Tomas and Stein (with
independent work by Strichartz and Sj\"olin) proved the estimate $R^*(2 \to 4)$,
which is sharp with respect to the scaling condition.  This estimate was
not improved until the 1990s by Bourgain, Wolff, and Muyoa, Vargas, and Vega;
for instance, the latter three authors proved $R^*(7/3 \to 4 - 2/11 + \eps)$.
These estimates improved the $4$ exponent of Tomas-Stein, but worsened the
$2$ exponent; in particular the estimates were no longer sharp with respect
to scaling.
\item Our first  application of the bilinear hypothesis is to
show that there are improvements to Tomas-Stein which are sharp with respect
to scaling.
\ei

\page

First step: a bilinear improvement to Tomas-Stein

\bi  
\item The Tomas-Stein estimate $R^*(2 \to 4)$ is sharp in the sense
that the 2 exponent cannot be lowered without raising the 4 exponent,
as can be seen from scaling/Knapp.  However, the bilinear version
$R^*(2 \times 2 \to 2)$ of Tomas Stein is not sharp, and
it is possible to improve the indices.

\item Theorem (Muyoa, Vargas, Vega): $R^*(12/7 \times 12/7 \to 2)$.

\item The proof begins like the Fefferman argument: to prove
$$ \| \widehat{fd\sigma} \widehat{gd\sigma} \|_{2} \lesssim \|f\|_{12/7} \|f\|_{12/7}$$
it suffices by Plancherel to prove
$$ \| fd\sigma * gd\sigma\|_{2} \lesssim \|f\|_{12/7} \|f\|_{12/7}$$
Because we are now in three dimensions instead of two,
the convolution is a bit more complicated than before:
$$ fd\sigma * gd\sigma(\xi) = \int_{\Gamma_\xi} f(\eta) g(\xi - \eta)\ d\eta$$
where $\Gamma_\xi$ is some ellipse depending on $\xi$ in a way which
satisfies the cinematic curvature condition.
\item The estimate then follows from the well-developed theory of
$L^p,L^q$ estimates for Radon transforms.
\ei

\page

Second step: using the linear-bilinear equivalence

\bi

 \item By interpolating the bilinear improvement of Tomas-Stein with
other improvements to Tomas-Stein, we can obtain new
sharp restriction theorems.
 \item We convert the linear improvements to Tomas-Stein to bilinear improvements,
then by interpolation we obtain bilinear improvements on the line of scaling.
Then we apply the bilinear-linear equivalence.
\ei
{
\centering\ \psfig{figure=bilinear.eps,height=3in,width=5in}
}

\page

More new restriction theorems

\bi
\item  In addition to obtaining new restriction theorems from
interpolation, we have also proven new restriction theorems which
are not interpolants.

\item The restriction theorem $R^*(7/3 \to 2 - 2/11)$ obtained by the work of
Bourgain, Wolff, Muyoa, Vargas, and Vega is based on a complex
argument by Bourgain, which combines a variant of the Tomas-Stein argument
with a version of Cordoba's proof of the Bochner-Riesz conjecture in
two dimensions.  It requires two estimates as input: a "local" restriction
theorem, which was obtained in the above work by interpolating
the Tomas-Stein estimate with an estimate coming from the Sobolev trace lemma,
and a Kakeya estimate (of which the best known result in three dimensions
is due to Wolff).

\item By converting Bourgain's argument to a bilinear argument we can 
improve the local restriction theorem imput by replacing
Tomas-Stein with the bilinear improvement to Tomas-Stein.  

\item One can also find bilinear improvement's to Wolff's Kakeya result
by inspecting the Wolff's proof.  Also there is an x-ray estimate of Wolff
which is slightly stronger than the Kakeya result, and which can also be used
as a substitute for Kakeya in this argument when combined with
the bilinear improvements to Tomas Stein.

\item  Combining all these improvements together we have managed
to improve the restriction theorem exponents from 
$R^*(7/3 \to 2 - 2/11 + \eps)$ to $R^*(26/11 \to 2 - 2/7 + \eps)$
\ei

\page
Restriction theorems for the cone

\bi

\item The above restriction theorems were stated for paraboloids, but also
apply to any surface of nonzero curvature.  The situation becomes
more subtle when one of the principal curvatures is allowed to vanish.

\item In the case of the cylinder there are no non-trivial linear restriction
theorems, and the best possible bilinear restriction theorem is
$R^*(2 \times 2 \to 2)$, which is fairly easy to prove.

\item In the case of the cone, the best possible linear theorem
is $R^*(4 \to 4)$, which is also fairly easy to prove.  One
can also show $R^*(2 \times 2 \to 2)$, but this is not sharp,
in 1995 Bourgain improved this to $R^*(2 \times 2 \to 2 - 13/2408 + \eps)$.
Machedon and Klainerman have conjectured that $R^*(2 \times 2 \to 5/3)$;
this is best possible.

\item The problem for the cone is more difficult than that for the 
surfaces with non-zero curvature, partly because of the lack of
Kakeya estimates, and partly because the Fourier transform of the cone
decays more slowly than that of the paraboloid or similar surfaces.

\item By recasting Bourgain's arguments using our techniques,
we have managed to improve the cone restriction theorem to
$R^*(2 \times 2 \to 2 - 8/121 + \eps)$.

\ei

\page

Further consequences

\bi
\item The bilinear estimates proved above can be used to improve
progress on several open problems in harmonic analysis and PDE.  For instance:

\item We can prove pointwise convergence of the solution of the free
Schr\"odinger equation in $R^{2+1}$ to its initial data if the data is in $H^s$
for $s > 1/2 - 1/32$.  This improves upon previous work of
Muyoa, Vargas, and Vega.

\item We can obtain some null form estimates for the wave equation for $L^p$
in $R^{2+1}$
for $2 > p > 2 - 8/121$, of the type studied by Klainerman, Machedon,
Foschi, Selberg, and others.  Previously the only results were for
$p \geq 2$.

\item We can improve the "$1/8$" square function estimate of Mockenhaupt
for the cone; this leads to some improvements on the local smoothing
conjecture, the Bochner-Riesz conjecture for the cone, and the
smoothing conjecture for convolution with the helix.

\ei

\end{document}
