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\begin{document}
\Large
\vskip 1in
\bc
{\bf \LARGE   Some connections between Oscillatory \\ ~ \\ Integrals
}\\
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
Terence Tao \\
\bigskip
{\it Mathematics Department \\ \bigskip UCLA \\ }  
\bigskip
{\tt www.math.ucla.edu$\backslash\sim$tao$\backslash$ preprints}
\ec

\page

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

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

 \item Discuss the analogous situation for parabolic Bochner-Riesz and
restriction

 \item Show that the Bochner-Riesz conjecture is equivalent to the weak-type endpoint Bochner-Riesz conjecture

 \item Show that the Kakeya and Nikodym conjectures are equivalent

 \item Discuss analogues of these results for the local smoothing conjecture

\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, 1936)
 \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, T., $\ldots$)
 \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 $p^\prime \geq 4 - \frac{2}{11}$.  (Wolff, 1995)
 \item Critical case is $p^\prime = 3$.
\ei

\bigskip\noindent
Denote by $BR(p,\alpha)$ the H\"ormander-type estimate
$$ \| \int e^{-2\pi i R |x-y|} a(x,y) f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime} + \alpha} \|f\|_p$$
where the bump function $a$ is supported away from $x=y$.
The Bochner-Riesz conjecture is equivalent to $BR(p,\alpha)$ holding
for all $1 \leq p \leq 2n/(n+1)$ and $\alpha > 0$.  (Carleson, Sj\"olin)

\ 

\noindent
$BR(p,\alpha)$ fails when $\alpha < 0$.  (We will address the
endpoint estimate $BR(p,0)$ later).
\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:
$$ \|\RR f\|_p = \| \hat f|_{S^{n-1}}\|_p \lesssim \|f\|_{L^p(\R^n)}.$$ 
\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 (Fefferman), but
such estimates are not available near the critical case.
\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; for $n=3$, Sj\"olin, 1972)
 \item True for $p^\prime > 4 - \frac{2}{15}$.  (Bourgain, 1991)
 \item True for $p^\prime > 4 - \frac{2}{11}$.  (Wolff, 1995; sharpened
by Moyua-Vargas-Vega, 1996)
 \item Critical case is $p^\prime = 3$.
\ei

\noindent
Denote by $R(p,\alpha)$ the H\"ormander-type estimate
$$ \| \int e^{-2\pi i R \langle \frac{x}{|x|},y\rangle} a(x,y) f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime} + \alpha} \|f\|_p,$$
where the bump function $a$ is supported away from $x=0$.  $R(p,0)$
is equivalent to the restriction conjecture for the exponent $p$, while
$R(p,\alpha)$ is equivalent to the localized restriction theorem
$$ \| \hat f|_{S^{n-1}}\|_p \lesssim R^\alpha \|f\|_{L^p(B(0,R))}$$ 
whenever $f$ is supported on $B(0,R)$.  Localized restriction conjectures
were used by Bourgain as stepping stones to the full restriction conjecture.

\page

\noindent
{\LARGE \noindent
Connection between Bochner-Riesz and Restriction
}

\bigskip\noindent
The oscillatory integral versions of $BR(p,\alpha)$ and $R(p,\alpha)$ are
clearly related:
$$ \| \int e^{-2\pi i R |x-y|} a(x,y) f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime} + \alpha} \|f\|_p$$
$$ \| \int e^{-2\pi i R \langle \frac{x}{|x|},y\rangle} a(x,y) f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime} + \alpha} \|f\|_p.$$
The phase function for $R(p,\alpha)$ is essentially the linearization of 
that of $BR(p,\alpha)$:
$$ R|x-y| = R|x| - R \langle \frac{x}{|x|},y\rangle + O(1)\quad \hbox{ for }
|x| \sim 1, |y| \lesssim R^{-1/2}.$$
From this we can obtain the relationship
$$ BR(p,\alpha) \Rightarrow R(p,2\alpha).$$
Unfortunately, the global restriction conjecture requires $\alpha = 0$,
and so this is not a complete proof that Bochner-Riesz implies restriction.
To finish the proof we need the following ``epsilon-removal'' result:
$$ [ R(p_0,\epsilon)\quad \forall \epsilon > 0 ] \Rightarrow
[ R(p,0)\quad \forall 1 \leq p < p_0 ].$$
Combining the two implications we see that the full Bochner-Riesz 
conjecture implies the full
restriction conjecture.

We'll outline the proof in the next few transparencies.

\page
{\LARGE Step I: Phase linearization: $BR(p,\alpha) \Rightarrow R(p,2\alpha)$}

\bi 
\item If we apply the above approximation
to $BR(p,\alpha)$, rescale and eliminate trivial factors, we obtain
$$ \| \int e^{-2\pi i (R \langle \frac{x}{|x|},y\rangle + O(1))} a(x,y) f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime} + 2\alpha} \|f\|_p,$$
which would be $R(p,2\alpha)$ but for the $O(1)$ phase error.

\item To finish this step, we use the principle that oscillatory
integral estimates are stable under smooth $O(1)$ perturbations of
the phase or amplitude.  (Proof: First 
consider multiplication by $\exp(2\pi i (k\cdot x + l \cdot y))$,
then use Fourier series to handle the general case).  [This observation
has been made by several authors, including Christ, Bourgain,
T., $\ldots$]

\item One can recast this argument in terms of the original Bochner-Riesz
and restriction operators.  The key observation is that
$$ S^{\delta(p) + \alpha} f(x) \sim \frac{e^{\pm 2\pi i |x|}}{|x|^{\frac{n}{p} + \alpha}} \RR f(\pm \frac{x}{|x|})$$
when $x$ is far from the support of $f$.  
\ei

\page

{\LARGE
Step II: Reduction to sparse sets
}

\bi
\item Assume $R(p_0,\eps)$ holds for all $\eps > 0$.  We wish to show that
$R(p,0)$ holds for all $1 \leq p < p_0$.

\item By interpolation theory it suffices to check the following estimate
for characteristic functions:
$$ \| \RR \chi_E \|_{p_0} \lesssim |E|^{\frac{1}{p_0} + \eps}.$$
By the uncertainty principle we may assume $E$ is the union of $1/100$-cubes.  
(cf. the stability lemma)

\item If $E$ were supported in a ball of radius $R$, then we could
employ the hypothesis $R(p_0,\eps)$ and obtain
$$ \| \RR \chi_E \|_{p_0} \lesssim |E|^{\frac{1}{p_0}} R^\eps.$$
So if $E$ had a diameter of $O(|E|^N)$ for some $N$, we would be done.

\item So, the bad case occurs when $E$ is thinly spread over a huge ball.
For example, $E$ could be contained in an $R^C$-seperated set of $R$-balls.
The following decomposition lemma shows
that this is essentially the only case.

\item {\bf Definition.} A collection of $R$-balls with cardinality $M$ is said to 
be {\it sparse} if they are $R^C M^C$-seperated.  ($C$ is a fixed large 
constant).

\item {\bf Lemma.} If $E$ is the union of $1/100$-cubes and $N \gg 1$, then there exists
$O(N|E|^{1/N})$ sparse collections of balls (with radii $O(|E|^{C^N})$)
which cover $E$. 

\item This lemma is proven by the usual Calder\'on-Zygmund type
stopping-time arguments.  For each $x \in \R^n$, let $k(x) \in \Z^+$ be the 
first number such that
$$ |E \cap B(x,|E|^{C^k})| \leq |E|^{k/N}.$$
The sets $E_k = \{x \in E: k(x) = k\}$ can be covered by $|E|^{1/N}$
sparse collections of $|E|^{C^{k-1}}$ balls, with the cardinality 
of each collection at most $|E|$.

\item In order to use the lemma to finish the proof, we need to bootstrap
the estimate
$$ \| \RR \chi_E \|_{p_0} \lesssim |E|^{\frac{1}{p_0}} R^\eps$$
to the case when $E$ is not supported on a single $R$-ball, but rather on
a sparse collection of $R$-balls.  This will be done in the next transparency.

\ei

\page


{\LARGE Step III: Bootstrapping $R(p_0,\alpha)$ to sparse sets
}

\bi
\item $R(p_0,\alpha)$ states that
$$ \| \RR f \|_{p_0} \lesssim R^\alpha \|f\|_{p_0}$$
whenever $f$ is supported on a ball of radius $R$.  To finish the proof
we need to bootstrap this to
$$ \| \sum_i \RR f_i\|_{p_0} \lesssim R^\alpha \| \sum_i f_i \|_{p_0}$$
whenever the $f_i$ are supported on different $R$-balls in
a sparse collection.

\item The idea is to exploit the fact that $\hat f_i$ and $\hat f_j$
behave ``independently'' on the sphere.  More precisely,
we have the quasi-orthogonality estimate
$$ \|  \sum_i \RR f_i\|_p \lesssim (\sum_i \| \tilde \RR f_i \|_p^p)^{1/p}$$
for all $1 \leq p \leq 2$, where $\tilde \RR$ is a technical perturbation
of $\RR$.  $p=1$ is just the triangle inequality, and $p=2$ will follow
from almost orthogonality and sparseness.  This finishes the proof.
\ei

\page
{\LARGE Remarks}
\bi

\item The epsilon-removal argument actually shows that
$$ R(p, \alpha) \Rightarrow R(p - \frac{C}{\log(1/\alpha)}, 0).$$
Thus, this argument only gives a meaningful restriction theorem for
exponentially small $\alpha$.  (This may be compared with the implication
$$ R(p, \alpha) \Rightarrow R(2 + \frac{1}{\frac{n+1}{2p^\prime} - \alpha},0)$$
(Bourgain), which is used in all the recent progress on the restriction
conjecture.)

\item The argument can be generalized to arbitrary oscillatory integrals
(at least, if the amplitudes are smooth and compact).  The phases
$|x-y|$ and $\langle \frac{x}{|x|},y\rangle$ can be replaced by
more general functions $\phi(x,y)$ and $y \cdot \nabla_y \phi(x,0)$.
\ei

\page

{\LARGE The morals of the story}

\bi

\item The first moral is that any optimal or near-optimal estimate can
be scaled into an optimal or near-optimal linear (or polynomial) approximation
of that estimate.  The idea is to restrict the operator near a critical 
counterexample, perform the linear (or polynomial) approximation,
and rescale.

Examples:
$$
\left.\begin{array}{ccc}
\hbox{spherical Bochner-Riesz } &\Rightarrow &\hbox{ spherical restriction}\\
\hbox{parabolic Bochner-Riesz } &\Rightarrow &\hbox{ parabolic restriction}\\
\hbox{spherical restriction } &\Rightarrow &\hbox{ parabolic restriction}\\
\hbox{Nikodym maximal } &\Rightarrow &\hbox{ Kakeya maximal}\\
\hbox{Local smoothing } &\Rightarrow &\hbox{ conic hyperplane Radon}\\
\end{array}\right.
$$

\item The second moral is that any epsilons in a near-optimal estimate
can usually be removed by ``interpolating'' with a trivial estimate (e.g.
$R(1,0)$).

Examples:
$$
\left.\begin{array}{ccc}
R(p_0,\eps) &\Rightarrow & R(p,0) \\
BR(p_0,\eps) &\Rightarrow & BR(p,0) \\
K(p_0,\eps) &\Rightarrow & K(p,0) \\
N(p_0,\eps) &\Rightarrow & N(p,0) \\
\end{array}\right.
$$
\ei
\page

{\LARGE Parabolic Bochner-Riesz and restriction}

\bi

\item Analogues of the spherical Bochner-Riesz and restriction
conjectures exist for paraboloids.  As H\"ormander problems they
can be written as
$$ \| \int e^{-2\pi i R |\underline x- \underline y|^2/(x_n - y_n)} a(x,y) 
f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime} + \alpha} \|f\|_p$$
$$ \| \int e^{-2\pi i R (\langle \underline{x},\underline{y}\rangle
+ |\underline{x}|^2 y_n)} a(x,y) f(y)\ dy\|_p \lesssim 
R^{-\frac{n}{p^\prime}} \|f\|_p$$
respectively.

\item These problems are closely related to Strichartz estimates for
the Schr\"odinger equation.

\item The linearization and epsilon-removal argument extends easily
to the parabolic situation.  Thus the parabolic Bochner-Riesz conjecture
implies the parabolic restriction conjecture.

\item Because of the algebraic properties of the two phases, it turns
out that one can reverse the implication (Carbery).  The main ideas
are a projective change of variables $x_n \to 1/x_n$ and an unfreezing
argument.

\ei

\page
\bi

\item It also turns out that there is an implication
between spherical restriction and parabolic restriction conjecture.
This is easiest to see in the context of $L^p \to L^q$ estimates
on the sharp line $q = (n-1)p^\prime/(n+1)$.

\item The idea is to write spherical restriction as
$$ \| \int e^{-2\pi i R (\langle \underline{x},\underline{y}\rangle
+ \phi(\underline{x}) y_n)} a(x,y) f(y)\ dy\|_q \lesssim 
R^{-\frac{n}{p^\prime}} \|f\|_p.$$
If $q = (n-1)p^\prime/(n+1)$ one can rescale as
$$ \| \int e^{-2\pi i R (\langle \underline{x},\underline{y}\rangle
+ \lambda^2 \phi(\underline{x}/\lambda) y_n)} a(x,y/\lambda) f(y)\ dy\|_q \lesssim 
R^{-\frac{n}{p^\prime}} \|f\|_p,$$
and if one lets $\lambda \to \infty$ one obtains the parabolic restriction
conjecture.
(cf. the osculation arguments of
Wolff).  One can also obtain a result for $(p,q)$ near
the sharp line
using the (parabolic) linearization and epsilon removal arguments.

\item At the endpoint $p=q=2n/(n+1)$ the estimate is on the sharp line.
Thus the full spherical restriction conjecture implies the full
parabolic restriction conjecture.

\ei

\page

{\LARGE Weak-type endpoint Bochner-Riesz}

\bi
\item
Let $1 \leq p < 2n/(n+1)$.  The Bochner-Riesz conjecture for $p$
states that $BR(p,\alpha)$ holds for all $\alpha > 0$, and is
equivalent to the $L^p$ boundedness of the multiplier operators
$S^\delta$ whenever
$$\delta > \delta(p) = \frac{n}{p} - \frac{n+1}{2}.$$

\item This conjecture is known to be sharp: $S^\delta$ cannot be bounded
on $L^p$ for $\delta \leq \delta(p)$ (Herz).  However, it is possible
that $S^{\delta(p)}$ is of weak-type $(p,p)$.  This is the weak-type
endpoint Bochner-Riesz conjecture.

\item Heuristically speaking, the endpoint conjecture is related to
the endpoint estimate $BR(p,0)$.  Thus one can hope to use epsilon-removal
techniques to prove the weak-type conjecture.

\item {\bf Theorem.}  If the Bochner-Riesz conjecture holds for 
some $1 < p_0 \leq 2n/(n+1)$, then the weak-type endpoint Bochner-Riesz
conjecture holds for all $1 \leq p < p_0$.

\ei

\page
\bigskip
\noindent
{\LARGE Summary of progress on the conjecture, for n=3}
\bi
 \item True for $p^\prime = \infty$.  (Christ, 1988)
 \item True for $p^\prime > 4$. (Christ, 1987)
 \item Full conjecture true for $n=2$.  (Seeger, 1996)
 \item True for $p^\prime \geq 4$. (T., 1996)
 \item True for $p^\prime > 4 - \frac{2}{11}$.  (T., 1997)
 \item The critical case is $p^\prime = 3$.
\ei

\bigskip\noindent
The weak-type endpoint Bochner-Riesz conjecture implies the
standard Bochner-Riesz conjecture by interpolation.  Thus
the two conjectures are actually equivalent.

\bigskip\noindent
Analogues of this result exist for more general
Riesz means (cf. Christ, Sogge, T.)

\page

{\LARGE Sketch of proof}

\bi

\item We use a standard dyadic decomposition
$$ S^{\delta(p)} f = \sum_{j>0} 2^{-j n/p} K_j * f + \hbox{error terms},$$
where $K_j(x) \sim e^{\pm 2\pi i |x-y|}$ for $|x-y| \sim 2^j$.

\item By real interpolation techniques and the Calder\'on-Zygmund
decomposition (Seeger), it suffices to
show the strong-type estimate
$$ \| \sum_{J} 2^{-j n/p} K_j * f_J \|_p \lesssim (\sum_J \|f_J\|_p^p)^{1/p}$$
where $f_J$ are any functions supported on $J$,
$2^j$ is the sidelength of $J$, and $J$ ranges over all dyadic
cubes with $j > 0$.
By the uncertainty principle we may assume the $f_J$ are constant on 
$1/100$-cubes.

\item The Bochner-Riesz hypothesis 
states that $BR(p_0,\eps)$ holds for some $p_0 > p$.  This
gives us an estimate
$$ \|K_j * f_J\|_p \lesssim 2^{\eps j} \|f_J\|_{p_0}.$$
If $f_J$ is dispersed over a set of size at least $2^{C \eps j}$, then
the $L^{p_0}$ norm is smaller than the $L^p$ norm of $f$, and one can obtain
the strong type estimate relatively easily.

\item If $f_J$ is concentrated in a ball of radius $2^{C \eps j}$,
then $K_j * f_J$ looks like the restriction operator applied to $f_J$,
and are almost orthogonal as $J$ ranges over all dyadic cubes.  So
one can use quasi-orthogonality techniques to obtain the estimate.

\item The bad case is when $f_J$ is concentrated on a set of
measure $2^{C \eps j}$ but is very thinly spread over $J$.  Then
one uses the sparse Calder\'on-Zygmund decomposition to reduce 
matters essentially to the case when each $f_J$ is
supported on a sparse collection of relatively small balls.  One then
uses quasi-orthogonality techniques to obtain the estimate.

\ei
\page

{\LARGE Kakeya and Nikodym conjectures}

\bi
\item A Kakeya set (or Besicovitch set) is a set in $\R^n$ that contains a 
unit line segment in each direction in $S^{n-1}$. 
A Nikodym set is a set such that every point in $\R^n$ is incident
to a line that contains a unit line segment in the set. 
It is conjectured that both types of sets must have full dimension $n$.

\item Let $K(p,\alpha)$ denote the estimate
$$ \| \sup_{T // \frac{x}{|x|}} \int_T a(x,y) f(y)\ dy \|_p \lesssim R^{-\frac{n}{p^\prime}+\alpha} \|f\|_p,$$
where $T$ ranges over all $1 \times 1/R$ tubes parallel to $x/|x|$.  The
Kakeya maximal conjecture states that $K(p,\alpha)$ for all 
$1 \leq p \leq n$ and $\alpha > 0$.  $K(p,\eps)$ implies that
Kakeya sets have dimension at least $p$.

\item The Nikodym maximal conjecture is similar but involves the estimate
$N(p,\alpha)$:
$$ \| \sup_{x \in T} \int_T a(x,y) f(y)\ dy \|_p \lesssim R^{-\frac{n}{p^\prime}+\alpha} \|f\|_p.$$
\ei

\page
\bigskip
\noindent
{\LARGE Summary of progress on the conjectures, for n=3}
\bi
 \item True for $p = 1$.  (Trivial)
 \item True for $p \leq 2$. (Drury, 1985)
 \item True for $p \leq 7/3$. (Bourgain, 1991) (Schlag, 1997)
 \item True for $p \leq 5/2$.  (Wolff, 1995)
 \item The critical case is $p = 3$.
\ei

\bigskip\noindent
The Nikodym and Kakeya conjectures are the non-oscillatory counterparts
of the (spherical/parabolic) Bochner-Riesz and restriction conjectures.
The restriction conjectures are known to imply the Kakeya conjecture
(Fefferman, Cordoba).  In the reverse direction, it is possible to
use progress on Kakeya and Nikodym to obtain a small amount of
progress on the Bochner-Riesz and restriction conjectures. (Bourgain)

\bigskip\noindent
The situation is more complicated in curved space.  (Sogge, Minicozzi)

\bigskip\noindent
It turns out that the Kakeya and Nikodym conjectures are equivalent,
at least on the maximal function level:
$$ K(p,\alpha) \implies N(p,\alpha) \implies K(p,2\alpha).$$

\page

{\LARGE Sketch of equivalence}

\bi
\item If $x \sim 1$ then an $1 \times 1/R$ tube
containing $x$ is essentially equivalent to a tube 
parallel to $x/|x|$, when one restricts oneself to
the region $|y| \lesssim R^{-1/2}$.  (cf. the connection between Bochner-Riesz
and restriction).  If one applies this observation and rescales one obtains
$N(p,\alpha) \implies K(p,2\alpha)$.

\item A Nikodym set has a line segment collinear with every point in $\R^n$.  
Projectively, a Kakeya set has a line segment collinear with every point 
on the hyperplane at infinity.  Thus, by a projective transformation
$x_n \to 1/x_n$ and an unfreezing argument, every Nikodym set can be
converted into a Kakeya set (cf. the argument by Carbery).  Thus the 
Kakeya set conjecture implies the Nikodym set conjecture.

\item The maximal operator analogue of this argument
yields the implication $K(p,\alpha) \implies N(p,\alpha)$.

\ei

\page

{\LARGE Local smoothing for the wave equation}

\bi

\item The local smoothing conjecture states that, if $u$ is
the solution to the homogeneous wave equation with initial data $u(0)=f$,
$u_t(0) = 0$, then 
$$ \|u \chi_{[1,2]}(t) \|_{L^{2n/(n-1)}_{t,x}} \lesssim \|\nabla^\eps f\|_{2n/(n-1)}.$$

\item H\"older's inequality and Plancherel's theorem can be used to show that
the local smoothing conjecture 
implies the square-function version of the Bochner-Riesz conjecture (Sogge).  
By Sobolev imbedding this implies the maximal Bochner-Riesz conjecture
for $p>2$, which implies the Bochner-Riesz conjecture (Stein, Weiss).  (The
maximal Bochner-Riesz conjecture is partially false for $p<2$).

\item If $F(x,t)$ is a function supported on a fixed compact set in $R^{n+1}$,
we define the conic hyperplane Radon transform $RF(\omega,s)$ for
$\omega \in S^{n-1}$, $s \in R$ by
$$ RF(\omega,s) = \int F(x, s + \langle \omega,x\rangle)\ dx.$$
If one linearizes the (adjoint) local smoothing conjecture one obtains the following
estimate on $R$:
$$ \|\nabla^{\frac{n-1}{2} - \eps} RF \|_{2n/(n+1)} \lesssim 
\|F\|_{L^{2n/(n+1)}_{x,t}}.$$

\item Similarly, if one linearizes the square function Bochner-Riesz
conjecture, one obtains
$$ \|\nabla^{\frac{n-1}{2} - \eps} RF \|_{2n/(n+1)} \lesssim 
\|F\|_{L^{2n/(n+1)}_x L^2_t}.$$

\item Either of these estimates implies the spherical restriction conjecture.
The idea is to apply the estimate to a function of the form
$$ F(x,t) = f(x/\lambda) e^{{}^{2\pi i \lambda t}} a(x,t)$$
to obtain a localized restriction conjecture, and then use epsilon-removal
techniques.

\ei

\page

{\LARGE
Summary of known equivalences}

\bigskip

$$
\left.\begin{array}{cccc}
& \hbox{Local smoothing } & \Rightarrow & \hbox{conic Radon}\\
&\Downarrow&&\Downarrow\\
&\hbox{Square fn. spherical BR} &\Rightarrow& L^2\ \hbox{conic Radon}\\
&\Downarrow&&\Downarrow\\
&\hbox{Maximal spherical BR} &&\Downarrow\\
&\Downarrow&&\Downarrow\\
&\hbox{Weak-type spherical BR}&&\hbox{Spherical}\\
&\hbox{Spherical BR} &\Rightarrow&\hbox{restriction}\\
&&&\Downarrow\\
&\hbox{Weak-type parabolic BR} &&\hbox{Parabolic}\\
&\hbox{Parabolic BR} &\iff&\hbox{restriction}\\
&&&\Downarrow\\
&\hbox{Nikodym maximal}&\iff&\hbox{Kakeya maximal}\\
&\Downarrow&&\Downarrow\\
&\hbox{Nikodym set}&\Leftarrow&\hbox{Kakeya set}
\end{array}\right.
$$

\end{document}
