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 (**):