Wolff 的 Kakeya maximal estimate：X-ray 估计与三维障碍

旧博客原文

原题：Kakeya conjecture (Tomas Wolff 1995)

There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the result established by Tomas Wolff in 1995 is almost the best result in $R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate.

For $f\in L_{loc}^1(R^d)$ ,for $0<\delta<1$ :

$f_{\delta}^*:P^{d-1}\longrightarrow R$ . $f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$ .

$T$ is varise in all cylinders with length 1.radius $\delta$ .axis in the $e$ direction.

$f_{\delta}^{**}:R^d\longrightarrow R$ . $f^{**}_{\delta}(x)=sup_{T}\frac{1}{|T|}\int_{T}|f|$ .

$T$ varise in cylinders contains x,length 1,radius $\delta$ .

Keeping this two maximal function in mind,we give the statement of the Kakeya maximal function conjecture:

$||M_{\delta}f||_d\leq C_{\epsilon} \delta^{-\epsilon}||f||_d$

Where $M_{\delta}=f_{\delta}^*$ or $M_{\delta}=f_{\delta}^{**}$ .

Because we have the obviously $1-\infty$ estimate:

$||f^*_{\delta}||_{\infty}\leq\frac{||f||_1}{|T|}=\delta^{1-d}||f||_1$ .

$||f^{**}_{\delta}||_{\infty}\leq\frac{||f||_1}{|T|}=\delta^{1-d}||f||_1$ .

So by the Riesz-Thorin interpolation we have:

$||M_{\delta}f||_{q}\leq C_{\epsilon}\delta^{-(\frac{d}{p}-1+\epsilon)}||f||_p$ . (*)

for $1\leq p\leq d,q\leq(d-1)p'$ .the task is establish (*) for $(p,q)$ as large as posible in the range.

for the 2 dimension case,the result is well know.the key estimate is:

$\sum_{j}|T_i\cap T_j|\leq log(\frac{1}{\delta})|T_i|$

for $d\geq 3$ case,the main result of Wolff is:

$||M_{\delta}f||_q\leq C_{\epsilon}\delta^{-(\frac{d}{p}-1+\epsilon)}||f||_p$

hold for $p=\frac{d+2}{2}.q=(d-1)p'$ . $M_{\delta}=f_{\delta}^*$ or $f_{\delta}^{**}$ .

Now we sketch the proof.

prove $f_{\delta}^*,f_{\delta}^{**}$ cases together.

We can make some reduction:

the first one is we can assume the sup of $f$ is in a fix compact set.

the second is instead of consider $f_{\delta}^{**}$ ,we can consider $f_{\delta}^{***}(x)=\sup_{T}\frac{1}{|T|}\int_T|f|$ .

where $T$ varies in all cylinder with radius $\delta$ ,length 1,axis $\frac{\pi}{100}$ with a fix direction.

the first reduction is obvious(why?)

the second reduction rely on a observe:

$||f_{\delta}^{***}||_q\leq A(\delta)||f||_p$ $\Longrightarrow$ $||f_{\delta}^{**}||_q\leq CA(\delta)||f|_p$

this is just finite cover by rotation of the coordinate and triangle inequality.

now we begin to establish a frame and put the two situations $f_{\delta}^*,f_{\delta}^{***}$ into it.

Let $M(d,1)$ be all line in $R^d$ .

then $M(d,1)=R^d\times S^{d-1}/\sim$ is a $2d-2$ dim manifold.

$M(d,1)\longrightarrow P^{d-1}$

$l \longrightarrow e_l$

$e_l$ is the line parallel to $l$ .and the middle point is original.

$dist(l_1,l_2)\sim \theta(l_1,l_2)+d_{mis}(l_1,l_2)$ .

Wolf axiom:

$(A,d)$ metric space.

$\mu(D(\alpha,\delta)) \sim \delta^m$ . $\alpha\in A$ . $\delta \leq diam(A)$ .

for certain $m\in R^+$ .

$\forall \alpha\in A$ . $F_{\alpha} \subset M(d,1)$ is given.and $\bar{\cup_{\alpha}F_{\alpha}}$ is compact.

$d(\alpha,\beta)\lesssim inf_{l\in F_{\alpha};m\in F_{\beta}}dist(l,m)$ for all $\alpha,\beta \in A$ .

If $f:R^d\longrightarrow R$ then we define $M_{\delta}f:A\longrightarrow R$ by

$M_{\delta}f(\alpha)=\sup_{l\in F(\alpha)}\frac{1}{|T_{l}^{\delta}|}\int_{|T^{\delta}_l|}|f|$ .

Property (**):

If $l_0\in \cup_{\alpha F_{\alpha}}$ . $\Pi$ is a 2-plane.containing $l_0$ and if $\sigma \geq \delta$ and if $\{\alpha_j\}_{j=0}^N$ is a $\delta-seperated$ subset of $A$ and for each j,there is $l_j\in F_{\alpha_j}$ with $dist(l_j,M(\Pi,l))<\delta$ and $dist(l,l_0)<\sigma$ .then