\documentstyle[amssymb,amsfonts,12pt,psfig]{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\rp{{r^\prime}} \def\rpt{{\tilde r^\prime}} \def\rtil{{\tilde r}} \def\qtil{{\tilde q}} \def\kp{{k^\prime}} \def\kt{{\tilde{k}}} \def\qp{{q^\prime}} \def\pp{{p^\prime}} \def\qpt{{\tilde q^\prime}} \def\R{{\hbox{\bf R}}} \def\C{{\hbox{\bf C}}} \def\rn{{\R^n}} \def\Z{{\hbox{\bf Z}}} \def\dimsup{{\overline{\rm dim}}} \def\eps{{\varepsilon}} \def\implies{{\Rightarrow}} \def\RR{{\frak R}} \def\emph#1{{\it #1}} % \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 $L^p$ improving estimates for averages along curves} \bigskip Terence Tao (UCLA), Jim Wright (U. Edinburgh)\\ \bigskip {\tt www.math.ucla.edu/$\sim$tao/preprints} (in preparation) \ec \page \noindent Introduction \bi \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. \ei \page Interesting questions: \bi \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. \ei \page \bi \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. \ei \page Push-forwards and pull-backs \bi \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 $$ T = \pi_{N,*} \pi_M^*.$$ \ei \psfig{figure=radon.eps,height=5in,width=6in} \page Vector fields \bi \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. \ei \page Two-parameter Carnot-Caratheodory balls \bi \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. \ei \page Idea 1: Iteration \bi \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). \ei \page Idea 2: Widths \bi \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. \ei \end{document}