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.
this is a note after reading the blog:Heat flow and the zero of polynomial.
1.instead of consider the original version:
$ \partial_{zz}f(z,t)=\partial_tf(z,t)$.
consider the corresponding “equidistribution version” is also interesting:
$ \partial_{zz}f(z,t)=\theta(z,t)\partial_tf(z,t)$,especially $ \theta(z,t)=e^{2\pi i\alpha t},\alpha\in R-Q$.
2.
where $ f(z)=z^n+a_{n-1}z^{n-1}+…+a_1z+a_0$.
$ f(z,t)=\sum_{k=1}^n\sum_{0\leq m\leq k-2,2|k-m}\frac{k!}{m!(k-m)!}z^mt^{k-m}.$
$ =\sum_{k=1}^m\sum_{0\leq m\leq k-2,2|k-m}C_k^mt^{k-m})z^mt^{k-m}$
$ \sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_k^mt^{k-m})z^m$.
rescaling:
$ F_t:(z_1(t),…,z_n(t))\longrightarrow (\frac{z_1(t)}{t},…,\frac{z_n(t)}{t})$.
$ F_t\cdot f(z,t)=\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_{k}^mt^{k-n})z^m$.
$ \lim_{t\to \infty}F_t\cdot f(z,t)=\sum_{m=0,2|n-m}^{n-2}C_n^mz^m$.(*)
even term $ \longrightarrow$ constant.(after renormelization)
odd term $ \longrightarrow$ 0(invariant).so at least the sum zeros of is invarient.
by the algebraic fundamental theorem,we have n zero $ \{z_1,…,z_n\}$of (*).
until now,we already now if the n zeros is distinct,then because the energy is the energy is the same and the entropy is increase so $ \exists T>>0,\forall t_i,t_j>T$,$ \{t>T|z_i(t)\} \cap \{t>T|z_j(t)\}=\emptyset$.$ \lim_{t\to \infty}|z_i(t)|=\infty$ and $ \lim_{t\to \infty}arg(z_i(t))=z_i$.
but how to know the information of the change of direction at “blow up” time?
1.change direction only at $ blow up$.
2.energy invariant $ \sum_{1\leq i\neq j\leq n}\frac{1}{|x_i-x_j|^2}$.
3.general philosophy
deformation some function under some evolution equation, such like heat equation,wave equation,shrodinger equation.and there is some conversion thing under the equation,and some quantity that could calculate directly such like the trace of spectral.
4.difficultis
this philosophy could generate to the analytic function case,but to make the limit case(I only know how ti deal with this now)coverage.we need very good control on the coefficient.
and to investigate the change of direction at blow up point maybe we need some knowledge about the burid group.