Harmonic analysis

There is some gap, in fact I can improve half of the argument of Discrete harmonic function , the pdf version is Discrete harmonic function in Z^n, but I still have some gap to deal with the residue half… 1. The statement of result First of all, we give the definition of discrete harmonic function. Definition 1 (Discrete harmonic function) We say a function $ {f: {\mathbb Z}^n \rightarrow {\mathbb R}}$ is a discrete harmonic function on $ {{\mathbb Z}^n}$ if and only if for any $ {(x_1,&#8…

The pdf version is Uncertainty principle. The nice note of terrence tao seems given a nice answer for the problem below. 1. Introduction Is there a Brunn-Minkowski inequality approach to the phenomenon charged by uncertainty principle? More precisely, is it possible to say some thing about the Gaussian distribution $ G(x)=e^{-|x|^2} \ \ \ \ \ (1) $   to be the best choice that $ {\|\hat G-G\|_2} $ arrive minimum? Remark 1 Or some other suitable distance space on reasonable function (ma…

I already understand this material 3days ago but it is a little difficult for me to type the latex…   1. Introduction There is two space to understand a function’s behaviour, the physics space and the frequency space (Why thing going like this? Why there is such a duality?). Namely, we have: $latex \displaystyle \hat f(\xi)=\int_{{\mathbb R}^d}e^{2\pi i\xi x}f(x)dx \ \ \ \ \ (1)&fg=000000$   The key point is, waves is a parameter group of scaling of definition of a cons…

1. Introduction We have talked about a very basic result in singular integral, i.e. if we have an additional condition, i.e. $latex {q-q}&fg=000000$ bounded condition, then by interpolation theorem we only need to establish the weak $latex {1-1}&fg=000000$ bound then we establish the $latex {p-p}&fg=000000$ bound of $latex {T}&fg=000000$, $latex {\forall 1< p< q }&fg=000000$. The category of of singular integral is very general, in fact the singular integral we interest…

1. Calderon-Zygmund decomposition The Calderon-Zygmund decomposition is a key step in the real variable analysis of singular integrals. The idea behind this decomposition is that it is often useful to split an arbitrary integrable function into its “small” and “large” parts, and then use different technique to analyze each part. The scheme is roughly as follows. Given a unction $latex { f}&fg=000000$ and an altitude $latex { \alpha}&fg=000000$, we write $latex { f…

  Motivation and Cotlar’s lemma We always need to consider a transform $latex T$ on Hilbert space $latex l^2(\mathbb Z)$ (this is a discrete model), or a finite dimensional space $latex V$. If under a basis $latex T$ is given by a diagonal matrix this story is easy, $latex \displaystyle A = \begin{pmatrix} \Lambda_1 & 0 & \ldots & 0 \\ 0 & \Lambda_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & \Lambda_n \end{pmatrix} \…

f is a matrix value analytic function on $latex \mathbb T$, we know $latex h(f)>\alpha$, this is just mean $latex \forall k\in \mathbb Z$ , assume $latex | \hat f(k)|\leq e^{-|k|\alpha}$ , $latex g=log(f)$ for which we assume $latex g$ is a lifting of f use the inverse of ramification map , $latex M_{n\times n} \to M_{n\times n} , A \to e^A$. Then exists $latex \beta=c(\alpha)>0$ such that , $latex \forall k \in \mathbb Z$, $latex |\hat g(k)| \leq e^{-k\beta}$.

The technique that transform a problem which is in a linear setting to a multilinear setting is very powerful. such like: 1.The renormalization technique in complex dynamic system, and the generalization this is mainly the Ostrowoski representation,and something else. 2.Fouriour analysis   this can be view when it is difficult to investigate a quality about a function $latex f$, it is always easier to take charge with some some part of $latex f$, in this case is given by $latex \hat f(\xi),…

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…

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…