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Pesudo differential opertor and singular integral
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…

Symplectic geometry
1. Introduction This is the first note of a series of notes concert on semiclassical analysis. Given the basic material on symplectic geometry. Including the following material, The case at a point, or we can look it as the case in $latex {{\mathbb R}^{2n}}&fg=000000$. The standard material in symplectic geometry, i.e. Hamiltonian mechanics, two approach, global one concentrating on lie derivative, and a locally one concentrating on the power of Darboux theorem, i.e. the existence of a canon…

Some interesting problems
There are some interesting problem, I post them at there in case I forget them. Excuse me if they are trivial, I have not took enough time to consider them about I think they are valuable to be consider. Problem 1: This problem is stated by graph coloring. there are two prat of it, in fact the first part I heard from someone else and I try to generate it to high dimension. there are finite lines $latex \{l_i\}_{i\in I}, l_i\subset \mathbb R^2$, crossing each other and the is a set $latex J$ of c…

A glimpse to the general theory
1. Introduction We have talked about a very basic result in singular integral, i.e. if we have an additional condition, i.e. $latex {qq}&fg=000000$ bounded condition, then by interpolation theorem we only need to establish the weak $latex {11}&fg=000000$ bound then we establish the $latex {pp}&fg=000000$ 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…

Periodic orbits and Sturm–Liouville theory
I thinks there is some problem related to the solution of a 2 order differential equation given by SturmLiouville system which is nontrivial. It is wellknow that the power of SturmLiouville theory see wiki, is due to it is some kind of “spectral decomposition” in the solution space. Two kind of problem is interesting, one is the eigenvalue estimate, both upper bound and lower bound, this already investigated in ESTIMATING THE EIGENVALUES OF STURMLIOUVILLE. PROBLEMS BY APPROXIMAT…

CalderonZygmund theory of singular integrals.
1. CalderonZygmund decomposition The CalderonZygmund 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…

The large sieve and the BombieriVinogradov theorem
1.Motivation Large sieve a philosophy reflect as a large group of inequalities which is very effective on controlling some linear sum or square sum of some correlation of arithmetic function, some idea of which could have originated in harmonic analysis, merely rely on almost orthogonality. One fundamental example is the estimate of the quality, $latex \sum_{n\leq x}\Lambda(n)\overline{\chi(n)}$ One naive idea of control this quality is using Cauchyschwarz inequality. But stupid use this we…

Linear metric on F2, free group with two generator.
I may have made a stupid mistake, but if not, we could construct a metric by pullback a metric on a suitable linear normalized space $latex H$ which we carefully constructed. Let we define the generators of free group $latex F_2$ by $latex a,b$. Step 1. Constructed the linear normalized space $latex H$. the space $latex H$ was spanned by basis $latex \Lambda=\Lambda_a \coprod \Lambda_b$, $latex \Lambda_a, \Lambda_b$ are defined by look at the Cayley graph of $latex F_2$, there is a lot of vertic…

Almost orthogonality
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} \…

The correlation of Mobius function and nilsequences in short interval
I wish to establish the following estimate: Conjecture :(correlation of Mobius function and nilsequences in short interval) $ \lambda(n)$ is the liouville function we wish the following estimate is true. $ \int_{0\leq x\leq X}\sup_{f\in \Omega^m}\sum_{x\leq n\leq x+H}\lambda(n)e^{2\pi if(x)}dx =o(XH)$. Where we have $ H\to \infty$ as $ x\to \infty$, $ \Omega^m=\{a_mx^m+a_{m1}x^{m1}+…+a_1x+a_0  a_m,…,a_1,a_0\in [0,1]\}$ is a compact space. I do not know how to prove this but thi…