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 (**):