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\begin{document}
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{\bf \LARGE  Geometric maximal functions, oscillatory integrals, and non-linear evolution equations}

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Terence Tao (UCLA)\\

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{\tt www.math.ucla.edu/$\sim$tao/preprints }

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Abstract:

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We discuss the interconnections between the study of geometric combinatorial problems (such as the Kakeya problem), the study of oscillatory integrals (including the restriction and Bochner-Riesz conjectures), and the study of non-linear dispersive equations.  These interconnections have only been systematically exploited in the last ten years, and much progress on these problems has already been obtained as a consequence.
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\noindent
The Kakeya conjecture

\bi
\item In 1917 S. Kakeya posed the following problem: what is the least amount of area necessary to rotate a unit line segment (a ``needle'') continuously by $2\pi$ in the plane?  This became known as the \emph{Kakeya needle problem}.

\item Rotating around the center requires $\pi/4$ units of area, while a more sophisticated ``three-point U-turn'' can trim this to $\pi/8$.

\item In 1925, Besicovitch showed that one can achieve this rotation using arbitrarily small amounts of area.  There were two basic ideas in the proof:

\item Firstly, one can translate a needle for arbitrarily small area, by sliding very far away, doing a slight rotation, sliding back, and undoing the rotation.

\item Secondly, one can find a collection of rectangles whose long side is 1, whose orientations cover all possible directions, and whose union has arbitrarily small area (the \emph{Besicovitch set construction}).

\psfig{figure=besicovitch.eps,height=5in,width=5in}

\item In general dimension $\R^n$, define a \emph{Besicovitch set} to be a set which contains a unit line segment in every direction.  The construction of Besicovitch thus shows that such sets can have measure zero when $n=2$.

\item However, Besicovitch sets in $\R^2$ must still have Hausdorff and Minkowski dimension exactly 2 (Drury, 1971).  The key observation is that two line segments with different orientations can intersect in at most one point.

\item This problem can be discretized.  For any $0 < \delta \ll 1$, define a \emph{$\delta$-Besicovitch set} to be a set which contains a $1 \times \delta$ tube in every direction.  Besicovitch's construction gives $\delta$-Besicovitch sets in $\R^2$ with area about $\log\log(1/\delta)/\log(1/\delta)$, but Drury's argument shows that such sets can never have area less than about $1/\log(1/\delta)$.

\item The \emph{Kakeya conjecture} states that for any dimension $n$, Besicovitch sets must always have Hausdorff and Minkowski dimension $n$.  On the discretized level, this means that $\delta$-Besicovitch sets have volume at least $C_\eps \delta^\eps$ for any $\eps > 0$.

\item In two dimensions, this conjecture is solved, but in higher dimensions it is still open.  Much recent progress has been made by Bourgain, Wolff, and others.  For instance, when $n \geq 3$ the best lower bound on the Minkowski dimension is
$$\max(\frac{n+2}{2} + 10^{-10}, \frac{4n+3}{7})$$ 
(Katz-Laba-T 1999, Katz-T 1999, Laba-T 2000), while for the Hausdorff dimension the best bound is
$$\max(\frac{n+2}{2}, \frac{6n+5}{11})$$ 
(Wolff 1995, Katz-T 1999).

\item There are two types of arguments used to prove these bounds.  One is geometric and makes use of incidence geometry facts (e.g. if three lines intersect each other at different points, they must lie in a plane, and each plane contains only a one-parameter set of directions).  The other is arithmetic and makes use of work on arithmetic progressions, sum-sets and difference sets, in particular the work of Heath-Brown, Szemeredi, Gowers, and Bourgain.  Some of the more recent results have managed to combine the two arguments, but we are still far from settling the problem.

\item There are more quantitative versions of the Kakeya conjecture which involve the \emph{Kakeya maximal function}
$$ f^*(\omega) := \sup_{l // \omega} \int_l f$$
where $\omega$ ranges over directions, and $l$ ranges over lines parallel to $\omega$.  We will not discuss this here.

\ei

\page

A link with algebra?

\bi

\item One can phrase finite field analogues of the Kakeya problem: in a finite geometry $F^n$, a Besicovitch set is a set which contains a line in every direction.  The Kakeya conjecture then states that such sets have cardinality at least $c_\eps |F|^{n-\eps}$ for any $\eps > 0$.

\item Progress on this problem and the continuous analogue have mostly been equivalent.  However, there are some intriguing features in the finite field case.  For instance, if $n=3$ and $F$ contains a subfield $G$ of cardinality $|G|=|F|^{1/2}$, then the Heisenberg group
$$ \{ (x,y,z) \in F^3: Im(z) = Im(x \overline{y}) \},$$
where $y \to \overline{y}$ is the automorphism fixing $G$ and $Im(z) := (z + \overline{z})/2$, has cardinality $|F|^{5/2}$ and is almost a counterexample to the Kakeya conjecture in that it contains a large number ($|F|^2$) of distinct lines (which however are not all parallel).

\item It is not known if the analogous situation occurs in the continuous case.  A good test question is the \emph{Erd\"os ring problem}: does $\R$ contain a subring of Hausdorff dimension exactly 1/2?  Such a ring would lead to some interesting near-counterexamples to Kakeya as well as other geometric measure theory problems such as the Falconer distance problem and the Furstenburg set problem (Katz-T., 2000).

\item I think these rings do not exist, and that the reason has to do with the fact that $\R$ is an ordered field.  Unfortunately we know of no way of combining the order theory of $\R$ effectively with the other arithmetic and geometric tools at our disposal.  

\item A possible attack route comes from applying inequalities from additive-multiplicative combinatorics, such as the inequality 
$$ \max(|A+A|, |A \cdot A|) \geq c |A|^{5/4}$$
whenever $A$ is a finite set of integers (Elekes, 1998).  However the arguments used to prove these types of inequalities do not seem to adapt well to the continuous setting.

\ei


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Kakeya and Fourier summation

\bi

\item The Kakeya problem is related to several problems in harmonic analysis and PDE involving oscillatory integrals.  An example arises from the theory of Fourier summation in higher dimensions.

\item If $f$ is a test function on $\R^n$, the \emph{Fourier transform} $\hat f$ is defined by
$$ \hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\ dx.$$
One has the \emph{Fourier inversion formula}
$$ f(x) = \int_{\R^n} \hat f(\xi) e^{2\pi i x \cdot \xi}\ dx.$$

\item Now suppose that $f$ is a much rougher function, for instance it is only in the Lebesgue space $L^p$ for some $1 < p < \infty$.  One can ask to what extent the above formulae are still valid.  A natural guess is that
$$ \int_{|\xi| < R} \hat f(\xi) e^{2\pi i x \cdot \xi}\ dx \to f$$
as $R \to \infty$ in some sense.

\item This convergence is true in the weak sense, but one might ask for stronger convergence, such as $L^p$ convergence or pointwise convergence a.e.  (Uniform convergence is easily shown to fail).

\item The problem can be reduced to the study of the \emph{disk multiplier} $S^0$, defined by
$$ \widehat{S^0 f}(\xi) = \chi_{|\xi| < 1} f(\xi).$$
It is easy to show that one has $L^p$ convergence if and only if $S^0$ is bounded on $L^p$.  (a.e. convergence is related to a maximal version of the disk multiplier which won't be discussed here).

\item In one dimension, it is a classical result of Riesz that $S^0$ is bounded on $L^p$ for all $1 < p < \infty$.  However, in 1971, C. Fefferman used Besicovitch sets to prove the surprising

\item {\bf Theorem.}  If $n > 2$ and $p \neq 2$, then $S^0$ is unbounded on $L^p$.

\item Roughly speaking, the idea for $p > 2$ is as follows.  Using a slight variant of the Besicovitch set construction, one can find a large number of $N \times N^2$ rectangles $R_i$ with distinct orientations, such that the $R_i$ are disjoint but their shifts $\tilde R_i$ have large overlap:

\psfig{figure=fefferman.eps,height=6in,width=5in}

\item On each rectangle $R_i$, one constructs a ``wave packet'' $\psi_i$ such that $S^0 \psi_i$ is large on $\tilde R_i$.  One then applies the disk multiplier $S^0$ to a suitable linear combination of $\psi_i$.  The overlap of the $\tilde R_i$ make the $L^p$ norm large.

\item The $p < 2$ argument is dual and works in reverse (swap $R_i$ and $\tilde R_i$).

\item Thus Fourier summation need not converge in $L^p$.  This can be rectified by introducing a weight in the Fourier integral:
$$ \int_{|\xi| < R} (1 - \frac{|\xi|}{R})^\delta \hat f(\xi) e^{2\pi i x \cdot \xi}\ dx \to f$$
The \emph{Bochner-Riesz conjecture} asserts that these weighted Fourier integrals converge in $L^p$ if $\delta > 0, n |\frac{1}{p} - \frac{1}{2} - 2| - \frac{1}{2}$, or if $(\delta,p) = (0,2)$.  (These conditions are known to be necessary).

\item If the Kakeya conjecture was false, then one could exploit the counterexample to give a counterexample to the Bochner-Riesz conjecture.  Thus the Bochner-Riesz conjecture implies the Kakeya conjecture.  (Bourgain, unpublished; T., 1998.  Heuristically this implication dates back to the 1970s).

\item The converse implication is not known; we do not know how to convert a complete proof of the Kakeya conjecture to that of the Bochner-Riesz conjecture.  However, in 1991 Bourgain introduced an argument which uses partial progress on Kakeya to give partial progress on Bochner-Riesz, basically by decomposing an arbitrary function $f$ into wave packets $\psi$ of the type used above.  

\item However, progress is still slow.  The Bochner-Riesz conjecture is proven in two dimensions (Carleson, Sj\"olin, 1972), but in three dimensions it is only solved for $p > 4 - 8/31$ or $p' > 4 - 8/31$ (T., Vargas, 1998).

\item The Besicovitch set has been used as a counterexample in many other contexts as well.  For instance, in 1993 Bourgain used the Besicovitch set and the ``short sum'' construction to disprove the strongest form of Montgomery's conjecture on the distribution of large values of Dirichlet series.  (A less strong version of Montgomery's conjecture is still open, and would imply the Kakeya conjecture just like the Bochner-Riesz conjecture does).  

\item We now show some connections between the Kakeya problem and the linear wave equation.

\ei

\page

Local smoothing for the wave equation

\begin{itemize}

\item Consider the free wave equation in $\R \times \R^n$ with initial position $f$ and initial velocity zero:
$$ u_{tt} - \Delta u = 0; \quad u(0,x) = f(x); \quad u_t(0,x) = 0$$

\item One can solve for $u$ in terms of $f$:
$$ u(t) = \cos(t \sqrt{-\Delta}) f.$$ 

\item We are interested in how the size and smoothness of $u$ is determined by that of $f$ and $g$.

\item A basic estimate is the \emph{energy estimate}
$$ \| u(t) \|_{L^2(\R^n)} \leq \| f \|_{L^2(\R^n)}$$
or more generally
$$ \| \nabla^k u(t) \|_{L^2(\R^n)} \leq \| \nabla^k f \|_{L^2(\R^n)}$$
for any $t \in \R$, $k \geq 0$.

\item This estimate only gives $L^2$ information, and does not reveal whether a solution $u(t)$ focuses or disperses as time progresses.  To obtain such information we need to augment the above $L^2$ bounds with $L^p$ estimates.

\item Such estimates exist, but they lose a lot of regularity.  For instance, one has the \emph{decay estimate}
$$ \| u(t) \|_{L^\infty(\R^n)} \leq C t^{-(n-1)/2} \sum_{k=0}^s \| \nabla^s f \|_{L^1(\R^n)}$$
for all $t > 0$, provided that $s > (n+1)/2$.  This estimate has many uses, such as establishing global well-posedness and scattering for non-linear wave equations with small data.

\item The decay estimate requires so many derivatives because waves can focus at a point, creating a large $L^\infty$ norm even if the initial data has small $L^1$ norm.

\item However, one can obtain estimates requiring far fewer derivatives if one averages in time.  Heuristically, this is because a wave can only focus at a point for very short periods of time before it disperses again.  This effect is known as \emph{local smoothing}.

\item The first estimates of this type to be discovered were the \emph{Strichartz estimates}, and grew out of the work on such problems as the Bochner-Riesz problem discussed earlier.  A typical estimate is
$$ \| u \|_{L^4(\R \times \R^3)} \leq C \| \nabla^{1/2} f \|_{L^2(\R^3)}.$$
(Strichartz, 1977) This particular estimate can be used to establish global well-posedness and scattering for the non-linear wave equation $u_{tt} - \Delta u = F(u)$ for small, low regularity initial data whenever the non-linearity $F$ is at most cubic.

\item Further estimates of Strichartz type were developed over the decades; the final outstanding endpoint estimate was settled recently (Keel, T. 1997).

\item However, even Strichartz estimates lose some derivatives, because of travelling wave counterexamples and the fact that the order of integrability on the left-hand side is higher than that of the right.  In 1991 Sogge conjectured an estimate of this type, namely
$$ \| u \|_{L^4([1,2] \times \R^2)} \leq C_\eps \| (1 + \sqrt{-\Delta})^\eps f \|_{L^4(\R^2)}$$
for all $\eps > 0$.  This estimate is only known for $\eps > 5/44$ (Wolff, 1999, and Vargas, T., 1998).  The argument again requires some estimates related to the Kakeya conjecture (but with line segments replaced by light rays).  The idea is to resolve $u$ into a linear combination of travelling waves, each of which is concentrated on the neighbourhood of a light ray.

\item Unfortunately, one must always lose at least an epsilon of derivatives in $L^p$ for $p \neq 2$ (Wolff, 1996).  This example is again based on the Besicovitch set, and involves placing a ``wave train'' as initial data on each of the rectangles $R_i$.  These wave trains will interact with large $L^p$ norm for all times $1 \leq t \leq 2$, so that averaging in time has no appreciable effect.

\psfig{figure=smoothing-counter.eps,height=6in,width=5in}

\item Variations of this example show that the local smoothing conjecture (in general dimensions) implies the Bochner-Riesz and Kakeya conjectures.

\item Similar situations occur for other dispersive equations (Schr\"odinger, KdV, etc.) but will not be discussed here.

\end{itemize}

\page

Non-linear wave equations

\begin{itemize}

\item Estimates for the free wave equation are important to the analytic study of non-linear wave equations.  A typical example is the cubic non-linear wave equation
$$ u_{tt} - \Delta u = u^3; \quad u(0,x) = \eps f(x); \quad u_t(0,x) = \eps g(x)$$
where $\eps > 0$ is a small parameter.

\item As it turns out, the solution to this equation depends analytically on the parameter $\eps$, at least for small $\eps$ and initial data $f$, $g$ which is not too rough.  One can then expand $u$ as a power series
$$ u = \eps u_1 + \eps^3 u_3 + \eps^5 u_5 + \ldots$$
where $u_1$ is the free solution
$$ (u_1)_{tt} - \Delta u_1 = 0; \quad u_1(0,x) = f(x); \quad (u_1)_t(0,x) = g(x)$$
and $u_3$ is obtained by solving the inhomogeneous wave equation
$$ (u_3)_{tt} - \Delta u_3 = (u_1)^3; \quad u_3(0,x) = 0; \quad (u_3)_t(0,x) = 0.$$
More complicated formulas exist for $u_5$, $u_7$, etc.

\item Of course, one must establish that this series actually converges.  It is here that the $L^p$ estimates discussed earlier come into play.  These estimates can also be used to discover further properties of the solution, such as uniqueness, persistence of regularity, scattering, decay estimates, etc.

\item The above equation is called \emph{semi-linear} because the highest order terms are linear, and the non-linear terms contain no derivatives.  Thanks in large part to Strichartz and local smoothing estimates, the theory of semi-linear wave equations is now almost complete.  For instance, in almost all cases one knows the minimal regularity on the initial data one needs in order for the above Cauchy problem to be well-posed.

\item Our theory is less satisfactory in the \emph{semi-linear with derivatives} case, in which the non-linearity is allowed to contain first-order derivatives.  Examples include the wave map equation
$$ \phi_{tt} - \Delta u = - \phi (|\phi_t|^2 - |\nabla \phi|^2)$$
or the Yang-Mills equation
$$ D_\alpha D^\alpha A = 0,$$
where $D_\alpha = \partial_\alpha + [A_\alpha, \cdot]$ are covariant derivatives, and one has chosen an appropriate gauge.

\item For these equations, the Strichartz estimates are no longer as useful because they lose too many derivatives.  Local smoothing estimates have also been unsuccessful.  

\item However, it was realized in the last decade that one needs \emph{multi-linear} estimates rather than linear estimates in order to correctly analyze these equations.  A typical estimate is
$$ \| u_t v_t - \nabla u \cdot \nabla v \|_{L^2(\R \times \R^3)}
\leq C \| \nabla^2 f_1 \|_{L^2(\R^3)} \| \nabla f_2 \|_{L^2(\R^3)}$$
(Klainerman, Machedon, 1993), where
$$ u_{tt} - \Delta u = 0; u(0,x) = f_1(x); u_t(0,x) = 0$$
$$ v_{tt} - \Delta v = 0; v(0,x) = f_2(x); v_t(0,x) = 0.$$
This estimate can be used to show, for instance, that the wave map equation is analytically well-posed in three dimensions if the initial data has two derivatives in $L^2$.

\item Bilinear $L^2$ estimates for the homogeneous (Foschi, Klainerman, 1999) and inhomogeneous (T., 2000) wave equation have recently been completely classified.  These estimates have led to near-optimal regularity results for semi-linear wave equations with derivatives (much work by Klainerman-Machedon and others; parallel work by Bourgain, Kenig-Ponce-Vega and others for Schr\"odinger, KdV, etc.)  Interesting problems still remain, especially in the very difficult critical regularity case, at which point analyticity of the solution breaks down.  (For instance, it is not known whether singularities can form for two-dimensional wave maps; the difficulty here is that the energy class is critical).

\item Bilinear $L^2$ estimates are usually proven using Plancherel's theorem.  Bilinear $L^p$ estimates are more difficult, with the first non-trivial result being quite recent (Bourgain, 1995; Vargas, T., 1998).  In the pure space-time norm case, however, this problem has also been recently solved (Wolff, 1999; T., 1999).  These new estimates may have application to the critical regularity problem.  Again, these estimates rely on Kakeya type information concerning the combinatorics of light rays.

\item Very recently (Tataru, Bahouri-Chemin, Klainerman, 1999-2000), some of these techniques have begun to be extended to the \emph{quasi-linear} case, in which the spacetime metric is allowed to vary in a non-linear fashion depending on the data.  An extremely difficult instance of this occurs in the Einstein equations in general relativity.

\item These estimates also have application to controlling other types of behaviour of PDEs.  For instance, one can control the movement of energy from one frequency range to another over long periods of time, and thus show that no blowup occurs for such equations as the KdV equation even when the energy and $L^2$ norm are infinite (Colliander, Keel, Staffilani, Takaoka, T., 2000).
These estimates may also play a part in solving the \emph{soliton conjecture}, which states that dispersive equations of KdV type always disperse into travelling waves (solitons) even when the equation is not completely integrable.

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