Relationship of relationship

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…

There is a major open problem: Is there a polynomial $latex f(z)$ such that the julia set $latex T(J)$ of  map$latex T:z\longrightarrow f(z)$ satisfied the hausdorff dimension of J equal to 2?

this note will talk about the Ostrowski representation and approximation by continue fraction. As well-known,by the Weyl criterion,$latex \{n\alpha\}$ is uniformly distribution in $latex [0,1]$ iff $latex \alpha\in R-Q$. i.e. we have:$latex \forall 0\leq a\leq b\leq 1$,we have: $latex \lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$. but this will not give the effective version.i.e. we do not the the more information about the decay of $latex o(N)$. we will give a approach…

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…

Kakeya conjecture in $latex R^3$ is very subtle.in fact wolff stay the best(but not very difficult to get,just use the structure so-called hairbrush)result $latex \frac{5}{2}$ until the result of Katz and Tao $latex \frac{5}{2}+\epsilon$.Where $latex \epsilon$ is a constant independent with kakeya set.and in the article of Tao,they proved $latex \epsilon>\frac{1}{10^{10}}$. Two-dimensional case first we overview the case of dimension 2,these is the only case that is proved.and the key point i…

semyon dyatlov的一篇文章 semyon dyatlov的文章https://arxiv.org/pdf/1710.05430.pdf，用fractional uncertainly priciple导出了hyperbolic surface上测地线诱导的zeta函数在$latex Re(s)>1-\epsilon$只有有限个零点。   就我的理解，这件事情至少和3个事情有关系， 1.p-adic上的黎曼猜想，因为这篇文章的证明强烈依赖于markov性质，这和p adic的结构也很像，有可能可以利用p adic猜想的证明思路继续做一部分。   2.billiard的传播子，但是这里不一样，文章中的 Schottky groups本质上是对于算子的逆写成一种级数形式其中级数由Schottky group生成，但是对于billiard传播子的情况所有的涉及的热核或者波核的paramatrix不仅仅具备markov性质，起主导作用的却是某种需要X-ray估计的性质，级数和并不是对全空间求而是某种截断了的子空间里面，所以比这个证明要难。建立起billiar…

  1. the most natural problem in harmonic analysis may be: investigate for what pair $latex (p,q)$ we have : $latex L^p(R^n)\longrightarrow L^q(R^n)$ $latex \hat f(x)=\int_{R^n}e^{-2\pi ix\xi}f(\xi)d\xi$ is strong-$latex (p,q)$ bounded. obvious we have the paserval identity:$latex ||\hat f||_{2}=||f||_2$,and we have $latex ||\hat f||_{\infty}\leq||f||_{1}$. so by the Riesz-Thorin inteplotation theorem we have the Hausdorff-Young inequality: $latex \forall 1\leq p\leq 2,\frac{1}{p}+\frac{1}{…

Last year I read a nice blog articles Recent progress on the Kakeya conjecture and have several questions with this article. follows the proof strategy called Multiscale analysis,although we can use the estimate with large $ \delta_1$ to get estimate with small $ \delta_2$,(may be loss some $ \delta^c$ in the inequality in this way),but the main difficult is we should proof the new tubes with scales $ \delta_2$ is contains in the the olders.as soon as we proof this ,to obtain a lower b…

1. Let $ f,g$ be real-valued integrable functions on a measure space $ (X,B,\mu)$,and define: $ F_t=\{x\in X:f(x>t)\},G_t=\{x\in X:g(x)>t\}$. Prove: $ \int|f-g|d\mu=\int_{-\infty}^{\infty}\mu((F_t-G_t)\cup(G_T-F_t))dt$. proof: by Fubini theorem(cake representation theorem in fact): $ \int|f-g|=\int_{0}^{\infty}\mu(\{x||f-g|(x)>t\})dt\\ =\int_{-\infty}^{\infty}\mu(\{x|f(x)>t>g(x)\})+\mu(\{x|f(x)<t<g(x)\})dt\\ =\int_{-\infty}^{\infty}\mu((F_t-G_t)\cup(G_T-F_t))dt$. (there is a…

In this note we discuss the Schauder theory for uniformly elliptic linear equations and Sobelov inequality. the three main topics ars a priori estimate in Holder norms,regularity of arbitrary solutions and the solvability of the Dirichlet problem.Among these topics,a priori estimates are the most fundamental and the basis of the follows two.we will discuss both the interior Schauder estimate and global Schauder estimate. -Schauder Theory- 1. Interior Schauder Theory $ {\Omega} $ be a domain in $…