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\begin{document}
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{\bf \LARGE  $L^p$ improving estimates for averages along curves}

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Terence Tao (UCLA), Jim Wright (U. Edinburgh)\\

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{\tt www.math.ucla.edu/$\sim$tao/preprints} (in preparation)

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\noindent

Introduction

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\item Let $M$, $N$ be smooth $n$-dimensional manifolds.  Suppose that we have a smooth assignment $x \mapsto \gamma_x$ mapping points in $N$ to curves in $M$.  We can then form the averaging operator
$$ Tf(x) := \int_{\gamma_x} f.$$
This operator takes functions on $M$ and returns functions on $N$.

\item To avoid degenerate situations we assume that the dual curve assignment $y \mapsto \gamma^*_y$ defined by
$$ x \in \gamma^*_y \iff y \in \gamma_x$$
is also a smooth assignment from points to curves.

\item The oldest example of such an operator is the \emph{Radon transform} 
$$ Rf(l) := \int_l f$$
where $l$ ranges over lines in $\R^2$.  Other examples include convolution with the circle, helix, or other curves.


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Interesting questions:

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\item Invertibility: given $Tf$, can you reconstruct $f$?  More of an algebraic problem than an analytic one.

\item $L^2$ smoothing: Does $T$ map $L^2$ to $L^2_\alpha$?  FIO techniques, singularity theory. 

\item $L^p$ smoothing: Does $T$ map $L^p$ to $L^p_\alpha$?  Very difficult in general; related to local smoothing for the wave equation and similar hard problems.

\item $L^p$ improving: Does $T$ map $L^p$ to $L^q$ for some $q > p$?  Easier than the smoothing question, because no derivatives are involved.

\item This talk is concerned only with the last question.

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\item Work by Seeger, Greenleaf, Oberlin, and others on the $L^p$ improving problem obtained sharp results in low dimensions $n \leq 4$ and partial results in higher dimensions. Techniques: oscillatory integrals, $TT^*$ method, special properties of $L^2$ and $L^4$.
 
\item In 1998, Christ introduced a very different combinatorial approach (more similar in spirit to work on the Kakeya problem) to tackle this problem.  In the model case of convolution with a generic polynomial curve in $\R^n$, he obtained the sharp $(L^p, L^q)$ estimates up to endpoints.

\item We've managed to generalize Christ's technique to arbitrary smooth families of curves, and in all cases we obtain sharp results except for endpoints.

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Push-forwards and pull-backs 

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\item A fundamental role is played by the $n+1$-dimensional manifold 
$$ \Sigma := \{ (y,x) \in M \times N: y \in \gamma_x \}.$$
The manifold $\Sigma$ has the obvious projections $\pi_M: \Sigma \to M$ and $\pi_N: \Sigma \to N$ down to $M$ and $N$.  Our hypotheses imply that these projections are submersions.

\item A function on $M$ can be pulled back to $\Sigma$ by the pullback map $\pi_{M}^*$, and conversely a function on $\Sigma$ can be pushed forward to $M$ by the push-forward operator $\pi_{M,*}$, providing we define some smooth measures on $M$ and $\Sigma$.  

\item The averaging operator $T$ can then be factorized as
$$ T = \pi_{N,*} \pi_M^*.$$

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

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Vector fields

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\item Now we remove $M$ and $N$ and work exclusively in $\Sigma$.

\item The fibers of $\pi_M$ are one-dimensional curves in $\Sigma$.  One can then create a non-vanishing vector field $X_1$ which is a section of the fiber bundle.  We can essentially write 
$$ \pi_M^* \pi_{M,*} f(x) \sim \int e^{t_1 X_1} f(x)\ dt_1.$$
In other words, it is an averaging operator along the vector field $X_1$.

\item The fibers of $\pi_N$ similarly generate a vector field $X_2$.  The question is then: what is the $(L^p(\Sigma), L^q(\Sigma))$? properties of the compound averaging operator
$$ Af := \int\int e^{t_2 X_2} e^{t_1 X_1} f\ dt_1 dt_2?$$

\item This problem can easily be localized to a small neighbourhood of a point in $\Sigma$, call it 0.  It is then clear that the Lie algebra generated by $X_1$ and $X_2$ at 0 will play a crucial role.  Indeed, if the Lie algebra at 0 fails to span the entire tangent space, then it is easy to see that there are no $L^p$ improving estimates.

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Two-parameter Carnot-Caratheodory balls

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\item On our manifold $M$ there are only two structures, the vector fields $X_1$ and $X_2$.  These generate a very natural \emph{two-parameter} metric structure.

\item Define the \emph{two-parameter Carnot-Caratheodory ball} $B(x; \delta_1, \delta_2)$ to be the set of all points obtainable from $x$ by applying the flow maps $e^{tX_1}$ and $e^{tX_2}$, as long as the total time spent flowing along $X_1$ does not exceed $\delta_1$, and the total time spent flowing along $X_2$ does not exceed $\delta_2$.  (When $\delta_1 = \delta_2$ this is essentially the standard Carnot-Cartheodory metric generated by $X_1$, $X_2$).

\item These balls provide natural test functions for the averaging operator $A$. 

\item The volume of the balls $B(0; \delta_1, \delta_2)$ have essentially a polynomial relationship with $\delta_1$ and $\delta_2$ as $\delta_1, \delta_2 \to 0$, with the exact relationship given by the nature of Lie algebra of $X_1$ and $X_2$ at 0 (or specifically, which combinations of Lie brackets are linearly dependent).  The relationship is a bit complicated, but straightforward, and can be proven rigorously using the Baker-Campbell-Hausdorff formula.

\item Combining these observations one gets some natural restrictions on the set of all possible $p$, $q$ for which we have an $(L^p, L^q)$ mapping property for the averaging operator $A$ - a certain convex polygon determined by the degeneracy of the Lie algebra.

\item Theorem [T., Wright]: In the interior of this polygon, the operator $A$ does indeed map $L^p$ to $L^q$.

\item In other words, up to epsilons, the two-parameter Carnot-Caratheodory balls provide the only obstructions to $L^p$ improving estimates.

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Idea 1: Iteration 

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\item The first ingredient in the proof is an iteration idea of Mike Christ.  Instead of proving bounds for the averaging operator $A$, we study instead a large power of $A$.  After some standard arguments, the problem is now as follows.

\item We need two parameters $0 < \lambda_1, \lambda_2  < 1$.  We adopt the notation that $X_i = X_1$ for $i$ odd and $X_i = X_2$ for $i$ even, and similarly for $\lambda_i$.

\item Start with a point $x_0$ in $\Sigma$ and Let $E_0 := \{x_0\}$, and write
$$ E_1 = \{ e^{tX_1} x_0: t \in T_1(x_0) \}$$
where $T_1(x_0)$ is a set of times of measure $\lambda_1$.  Then write
$$ E_2 = \{ e^{tX_2} x_1: x_1 \in E_1; t in T_2(x_1) \}$$
where $T_2(x_1)$ is a set of times of measure $\lambda_2$ for each $x_1$.  And so forth.

\item The set $E_i$ lives in an $i$-dimensional set for each $i$.  Thus one expects $E_n$ to be full dimensional.  The problem of obtaining $L^p$ estimates for $A$ turns out to be equivalent (up to endpoints) to obtaining lower bounds for $|E_n|$ in terms of $\lambda_1$ and $\lambda_2$.

\item Thus everything depends on the behaviour of the map
$$ \Phi: (t_1, \ldots, t_n) \mapsto e^{t_n X_n} \ldots e^{t_1 X_1} x_0.$$
Ideally we want $\Phi$ to be one-to-one and have Jacobian $J(\Phi)$ bounded away from zero.  

\item Unfortunately, $J(\Phi)$ degenerates at places.  However, the Jacobian cannot stay close to zero everywhere, and in fact there is some high-order derivative $\partial^\alpha J$ of the Jacobian which does not vanish at 0 (exactly which derivative depends on the degeneracy of the Lie algebra). 

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Idea 2: Widths

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\item Our main innovation over Christ's argument is to do some preparatory reductions on the sets $T_j(x_{j-1})$.  We need to control not just the measure of this set, but its shape.  The key lemma is

\item {\bf Lemma.}  Let $T \subset [0,1]$ be a set of measure $\lambda$.  Then there exists an interval $I \subset [0,1]$ of some length $\lambda \leq \delta \leq 1$ such that given any two points $x,y \in T$ selected in random, one has
$x,y \in I$ and $|x-y| \sim \delta$ with probability at least $\gtrsim 1/\log(1/\lambda)$.

\item This lemma is not hard to prove.  Basically, one sets $\sigma$ to be the median separation (or ``width'') of two random points in $T$.  

\item This allows one to localize the $T_j$ to a fixed interval, which basically localizes the iterated sets $E_1, E_2, \ldots$ to a two-parameter Carnot-Caratheodory ball.  We then rescale the ball to unit size and apply Christ's argument to obtain a near-optimal bound.

\item The large median separation of times allows one to avoid the bad set where $J(\Phi)$ is small with very high probability.

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\end{document}

