\documentstyle[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\R{{\hbox{\bf 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 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 $p 0$. \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}