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\begin{document}
\Large
\vskip 1in
\bc
{\bf \LARGE   Some connections between Oscillatory \\ ~ \\ Integrals
and other Oscillatory Integrals}\\
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
Terence Tao \\
\bigskip
{\it Mathematics Department \\ \bigskip UCLA }  
\ec

\page

\noindent
{\LARGE
The purpose of this talk is to:
}
\bi
 \item Briefly survey the known results concerning the (spherical)
Bochner-Riesz and restriction conjectures

 \item Show that the Bochner-Riesz conjecture implies the restriction conjecture

 \item Present related results connecting these conjectures to their
parabolic analogues

 \item (If time permits) talk about the ``geometric" or ``non-oscillatory"
counterparts to these statements.
\ei

\page

\bigskip
\noindent
{
Let $n \geq 2$.
The (spherical) Bochner-Riesz conjecture states that the Fourier multipliers
$$ \widehat{S^\delta f}(\xi) = (1 - |\xi|^2)^\delta_+ \hat f(\xi)$$
are bounded on $L^p$ whenever $\delta > n|1/p - 1/2| - 1/2, 0$.
}
 
\noindent
\bi
 \item These operators have applications to the convergence of 
spherically summed Fourier series, and discrete versions of these 
multipliers appear in number theory.  
 \item The multiplier is singular on the sphere and so cannot be treated
by classical multiplier theorems.  
 \item Apart from the trivial case $p = 2$, $\delta = 0$, the conditions
on $\delta$ are necessary. (Herz, Fefferman)
 \item It suffices to consider $1 \leq p \leq 2n/(n+1)$.  The critical case
is $2n / (n+1)$.
 \item The case $n=2$ is very well studied, and most questions concerning
Bochner-Riesz multipliers and related operators have been completely settled.
(Carleson, Sj\"olin, Carbery, Cordoba, Seeger, \ldots).  However these 
techniques do not seem to extend to general $n$.
\ei

\page

\bigskip
\noindent
{\LARGE Summary of progress on the conjecture, for n=3}
\bi
 \item True for $p^\prime = \infty$.  (Bochner)
 \item True for $p^\prime \geq 6$. (Fefferman, Stein, 1970)
 \item True for $p^\prime \geq 4$. (Stein, Tomas, 1975; refined and extended by Strichartz, Greenleaf, Christ, Sogge, Tao, \ldots)
 \item True for $p^\prime \geq 4 - \frac{8}{75}$.  (Bourgain, 1991)
 \item True for $p^\prime \geq 4 - \frac{2}{15}$.  (Bourgain, 1995)
 \item True for $p^\prime \geq 4 - \frac{2}{9}$.  (Wolff, 1995)
 \item Critical case is $p^\prime = 3$.
\ei

\page

\noindent
The (spherical) Restriction conjecture states that whenever $f \in L^p(\R^n)$
for $1 \leq p < 2n/(n+1)$, then $\hat f$ has a well-defined restriction
to the sphere:
$$ \|Rf\|_p = \| \hat f|_{S^{n-1}}\|_p \lesssim \|f\|_{L^p(\R^n)}.$$ 
\bi
 \item The conjecture is known to fail for the critical exponent
$p = 2n/(n+1)$.
 \item The exponent $p$ on the left-hand side is negotiable.  If the
conjecture was true, one could use interpolation to replace the left-hand
exponent by $q$ for any $1 \leq q \leq \frac{n-1}{n+1}p^\prime$.
 \item Progress on the restriction conjecture has paralleled the progress
on the Bochner-Riesz conjecture.  For example, the $n=2$ case is completely
solved.  A restriction estimate with $q=2$ is known to imply the
corresponding Bochner-Riesz estimate, by Plancherel's theorem, but
such estimates are not available near the critical case.
 \item The restriction conjecture and the Bochner-Riesz conjecture
can be reduced to very similar oscillatory integral problems of H\"ormander
type.  However, the restriction problem is simpler
because the associated phase function is linear in the second variable.
\ei

\page

\bigskip
\noindent
{\LARGE Summary of progress on the conjecture, for n=3}
\bi
 \item True for $p^\prime = \infty$.  (Riemann-Lebesgue)
 \item True for $p^\prime \geq 6$. (Stein, 1967)
 \item True for $p^\prime > 4$. (Tomas, 1975)
 \item True for $p^\prime \geq 4$. (Stein, 1975)
 \item True for $p^\prime > 4 - \frac{2}{15}$.  (Bourgain, 1991)
 \item True for $p^\prime > 4 - \frac{2}{9}$.  (Wolff, 1995)
 \item Critical case is $p^\prime = 3$.
\ei

\page

\bigskip
\noindent
{\LARGE \noindent
Proof of Bochner-Riesz $\Rightarrow$ Restriction
}

\bigskip\noindent
Denote by $BR(p,\alpha)$ the statement that the Bochner-Riesz
conjecture is satisfied for exponent $p$ with
$\delta = n(1/p - 1/2) - 1/2 + \alpha$, and denote by $R(p,0)$ the 
restriction conjecture for exponent $p$.  We claim that
$$ [ BR(p_0,\epsilon)\quad \forall \epsilon > 0 ] \Rightarrow
[ R(p,0)\quad \forall 1 \leq p < p_0 ].$$
In particular, the full Bochner-Riesz conjecture implies the full
restriction conjecture.

\bigskip\noindent
We will need a ``localized'' form $R(p,\alpha)$ of the
restriction conjecture, namely the estimate
$$ \| \hat f|_{S^{n-1}}\|_p \lesssim R^\alpha \|f\|_{L^p(B(0,R))}$$ 
whenever $f$ is supported on $B(0,R)$.  Localized restriction conjectures
were used by Bourgain as stepping stones to the full restriction conjecture.

\bigskip\noindent
$R(p,\epsilon)$ is strictly weaker than $R(p,0)$, even if $\epsilon$ can
be made arbitrarily small.  Passing from the local estimate to the global
estimate is the most difficult part of the proof.

\page
{\LARGE There are three steps to the proof:}

\bi
\item The first step of the proof is to show that $BR(p_0,\varepsilon) \Rightarrow
R(p_0,2\varepsilon)$ for $\varepsilon > 0$.  This will follow from
a scaling argument; at large distances, the Bochner-Riesz operator resembles
a rescaled, partly degenerate version of the restriction operator.  Because
we are near the critical index $\varepsilon = 0$, the contribution of the 
large-scale regions of space is non-trivial.  Thus the Bochner-Riesz
estimate implies the local restriction estimate.

\item The local restriction estimate gives us control of the restriction
operator on large balls.  The second step of the proof
is to bootstrap this estimate so that we may control the operator on
any sufficiently sparse collection of large balls.

\item The third step will be to decompose a certain exceptional set
into the finite union of sparse collections of large balls, allowing
us to use the estimate from the second step.  This will be done
using a variant of the Calder\'on-Zygmund decomposition.  There
is a small cost, however; the exponent $p$ will be worsened by
a small amount $O(1/\log(1/\varepsilon))$. 
\ei


\page
{\LARGE
Proof of the first part
}

\bi
\item Assume $BR(p_0,\varepsilon)$ holds, and set $\delta = n(1/p_0 - 1/2) - 1/2
+ \varepsilon$.  Let $f$ be a function supported on $B(0,R)$.

\item The Bochner-Riesz operator can be rewritten as
$$ S^\delta f(x) = \int_{|y| \leq R} f(y) K^\delta (x-y)\ dy,$$
where we have the asymptotic expansion
$$ K^\delta(x-y) \sim \frac{e^{\pm 2\pi i |x-y|}}{|x-y|^{\frac{n}{p_0} + \varepsilon}}.$$

\item Restrict $x$ to the range $|x| \sim R^2$.  By applying a Taylor series
expansion to $|x-y|$ we can approximate the kernel by
$$ K^\delta(x-y) \sim e^{\pm 2\pi i |x|} e^{\mp 2\pi i \langle \frac{x}{|x|},
y\rangle} R^{-\frac{2n}{p_0} - 2\varepsilon}.$$
But this means that $S^\delta f$ is essentially a rescaled version of the 
restriction operator:
$$ S^\delta f(x) \sim e^{\pm 2\pi i |x|} R^{-\frac{2n}{p_0} - 2\varepsilon} Rf(\pm \frac{x}{|x|}),$$
from which the localized restriction estimate follows by rescaling.

\item Remark: The proof uses an easy but useful lemma that states that the 
kernel
of a linear operator can always be adjusted by a $C^\infty$ bump function
without losing any regularity properties.  

\item This is obvious when the
multiplier is of the form $e^{2\pi i \langle x,\xi\rangle} e^{2\pi i
\langle y, \eta \rangle}$, and one uses the Fourier series decomposition for the general
case.

\ei

\page

{\LARGE
Proof of the second part
}

\bi
\item Assume $R(p_0,\varepsilon)$ holds for all $\varepsilon > 0$.  This means 
$$ \| Rf\|_{p_0} \lesssim R^\epsilon \|f\|_{p_0}$$
as long as $f$ is supported on a ball of radius $R$.

\item If $f_i$ and $f_j$ are two functions supported
on very widely seperated balls (seperation $\gg R^2$), then the Fourier 
transforms $\hat f_i$ and $\hat f_j$ are almost orthogonal on the sphere, and 
behave
like independent random variables.  This suggests that we should be
able to bootstrap the above hypothesis and control $Rf$ whenever $f$
is supported on a sufficiently sparse collection of balls.

\item Fix a $\delta > 0$, $\lambda \gg 1$ to be chosen later.  It turns out that
if a collection of disjoint balls $B(x_i,R)$ satisfies sparseness condition
$$ \#( \{ x_i \} \cap B(x, \lambda^{N\delta} R^2) ) \lesssim \lambda^\delta$$
for all $x$ and some large but fixed $N$, then one has
$$ \|Rf\|_{p_0} \lesssim R^\delta R^\epsilon \|f\|_{p_0}$$
whenever $f$ is supported on $\bigcup_i B(x_i,R)$.

\item The estimate is a routine exercise in harmonic analysis, using
standard techniques (interpolation with $L^2$ estimates, Plancherel's
theorem, and almost orthogonality).  The decay of $\widehat{d\sigma}$,
the Fourier transform of surface measure of the sphere, plays a minor
but necessary role.

\ei

\page


{\LARGE
Proof of the third part
}

\bi
\item We now switch to the adjoint setting.  Let $\phi$ be 
a bounded function
on the unit sphere, and let $E$ be the exceptional
set
$$ E = \{ |R^* \phi| \geq \frac{1}{\lambda} \} = \{x \in \R^n:
|\widehat{\phi d\sigma}| \geq \frac{1}{\lambda}\}.$$
By standard reduction techniques (rotation theory, real interpolation) the
restriction theorem $R(p,0)$ for $p<p_0$ will follow from the weak-type
estimate
$$ |E| \lesssim \lambda^{p_0^\prime + \epsilon}$$
for all $\epsilon > 0$.

\item In contrast, the results from the second part of the proof 
imply the weaker, localized estimate
$$ |E \cap \bigcup_i B(x_i,R)| \lesssim R^\varepsilon 
\lambda^{p_0^\prime} \lambda^{\delta} $$
for all sparse collections of balls $\{B(x_i,R)\}$ and all $\varepsilon > 0$.
\ei

\page


\bi

\item However, we may use the standard Calder\'on-Zygmund stopping time
argument and break up $E$ into sparse sets, by starting at the scale $R \sim 1$
and moving from scale $R$ to scale $\lambda^{N\delta} R^2$
once the sparse sets at scale $R$ have been exhausted.

\item This can be thought of as an exponentially low-dimensional
Calder\'on-Zygmund decomposition; we only require the mass of $E$ on the
exceptional balls $B(x,R)$ to be comparable to some power of $\log R$.

\item Technically, one has to consider the small-scale behaviour
$R \ll 1$ as well, but this is easily dealt with because
frequency is localized.

\item $E$ cannot be too big (because of known restriction theorems such as
the Tomas-Stein result), so the Calder\'on-Zygmund procedure halts after
a finite number of steps $O(1/\delta)$.  Applying our estimate to each
sparse collection, we obtain
$$ |E| \lesssim \lambda^{p_0^\prime + \delta + \varepsilon 2^{1/\delta}}$$
for all $\varepsilon > 0$, which gives the result by choosing $\delta$
appropriately.\hfill //

\ei

\page

{\LARGE
Bochner-Riesz and restriction for paraboloids
}

\bi

\item The above argument can be generalized to other oscillatory integrals.
Basically, any oscillatory integral estimate
$$ \|\int e^{i \lambda \phi(x,y)} a(x,y) f(y)\ dy \|_{L^{p_0}(\R^m)} \lesssim \lambda^{-n/p_0^\prime + 
\varepsilon} \|f\|_{L^{p_0}(\R^n)}$$
implies an estimate for the microlocalized restriction operators
associated to the oscillatory integral: for all $y_0 \in \R^n$ and $p < p_0$,
$$ \| \hat f( \phi_y(\cdot, y_0) )\|_{L^p(\R^m)} \lesssim \|f\|_{L^p(\R^n)}.$$

\item In particular, the parabolic Bochner-Riesz conjecture implies the
parabolic restriction conjecture.  (These conjectures are formed by
replacing the sphere by the paraboloid in the obvious manner).  Similarly for
paraboloids replaced by light cones.  Restriction theorems for these
surfaces are related to Strichartz estimates for the Schr\"odinger
equation and wave equation.

\ei
\page
\bi

\item By a freezing argument and a projective change of variables,
the parabolic restriction conjecture can be shown to imply the parabolic
Bochner-Riesz conjecture (Carbery, 1992).  Thus the two conjectures
are equivalent.

\item An easy rescaling argument also shows that the spherical restriction
conjecture also implies the parabolic restriction conjecture.  This 
comes from the fact that a spherical cap of radius $\delta$ looks like
a $\delta \times \delta^2$ rescaled version of the paraboloid.  Although
this argument is sharp at the critical index $p = 2n/(n+1)$, it loses a lot 
of exponents away from this index.

\ei

\page

{\LARGE
Connection with geometrical maximal operators
}

\bi
\item The above oscillatory integrals are known to be closely related to
geometrical conjectures, in particular the Kakeya conjecture [a set
with a line segment in every direction must have full dimension] and
the Nikodym conjecture [a set with a line segment concurrent with every
point must have full dimension].  These geometric conjectures can be thought
of as the ``non-oscillatory'' versions of the oscillatory integral conjectures.

\item In particular, both the spherical and parabolic restriction conjectures
are known to imply the Kakeya conjecture (Fefferman, Cordoba).  The 
spherical and 
parabolic Bochner-Riesz conjectures are related to the Nikodym conjecture, 
but the connection is not as sharp.

\ei
\page
\bi

\item In the reverse direction, Bourgain (1991) has shown that progress
on the Kakeya and Nikodym conjectures implies a small but non-trivial
amount of progress on the restriction and Bochner-Riesz conjectures.
However it is not known whether the full Kakeya/Nikodym conjectures
imply the full restriction or Bochner-Riesz conjectures.

\item The equivalences between the oscillatory integral conjectures
leads us to suspect similar equivalences between the geometric conjectures.
This is indeed the case: the Nikodym and Kakeya conjectures are equivalent.

\ei

\page

{\LARGE Equivalence of Nikodym and Kakeya}

\bi

\item Kakeya $\Rightarrow$ Nikodym: This is the non-oscillatory version
of Carbery's argument for parabolic restriction $\Rightarrow$
parabolic Bochner-Riesz.  Let $E$ be a Nikodym set, containing a line
segment concurrent with every point.  In particular $E$ contains a line
segment concurrent with every point in a fixed hyperplane.  By a projective
change of variables we may move this hyperplane to the hyperplane at infinity.
Now $E$ has a line segment in every direction, and we use the Kakeya
hypothesis.

\item Nikodym $\Rightarrow$ Kakeya: This is the non-oscillatory version
of the Bochner-Riesz $\Rightarrow$ restriction argument.  Let $E$ be
a Kakeya set, containing a line segment in every direction.  We consider
the $\delta$-thickened neighbourhood $E_\delta$ of $E$, containing a
$1 \times \delta$ tube in every direction.  We now observe that the
dilate $\delta E_\delta$ is a $\delta^2$-thickened neighbourhood
of a Nikodym set, containing a $\delta \times \delta^2$ tube concurrent
with every point in the unit annulus.  The claim then follows by a 
dimension counting argument.

\ei
\page

{\LARGE In summary, the known equivalences are}

\bigskip

\centerline{Spherical Bochner-Riesz $\Rightarrow$ Spherical restriction
$\Rightarrow$}

\bigskip

\centerline{
$\Rightarrow$ Parabolic Restriction $\iff$ Parabolic Bochner-Riesz $\Rightarrow$}

\bigskip

\centerline{ $\Rightarrow$ Kakeya $\iff$ Nikodym.}



\end{document}
