
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…

An Fourier coefficient decay estimate.
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 , $latex \forall k \in \mathbb Z$, $latex \hat g(k) \leq e^{k\beta}$.

Geometric intuition of mean value property of nonlinear elliptic equation
I wish to gain some understanding of the MVP of nonlinear elliptic equation by geometric intuition. Linear elliptic equation case First of all, I have a very good geometric explain of the MVP of Laplace equation, i.e. MVP of laplace equation $latex \Delta u=0$ in $latex \Omega$ , $latex \forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity: $latex \frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$ I need to point out first, this property is not difficult to p…

Diophantine approximation of algebraic number
An important theorem in Diophantine approximation is the theorem of Liuoville: **Liuoville Theorem** If x is a algebraic number of degree $latex n$ over the rational number then there exists a constant $latex c(x) > 0$ such that:$latex \leftx{\frac {p}{q}}\right>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $latex p,q\in N^*$ where $latex q>0$. This theorem explain a phenomenon, the approximation of algebraic number by rational number could not be very well. Which was generated la…

Rotation number
Consider compact 1 dimension dynamic system. We focus on $latex S_1$, it does not mean $latex S_1$ is the only compact 1 dimensional system , but it is a typical example. $latex T: S_1\to S_1$. If $latex T$ is a homomorphism then $latex T$ stay the order of $latex S_1$ (by continuous and the zero point theorem). That is just mean: (may be do a reflexion $latex e^{2\pi i\theta}\to e^{2\pi i\theta}$). In the homomorphism case. We try to define the rotation number to describe the expending …

Eloquent theory
Consider matrix ODE: $latex \dot{\phi}(t)=A(t)\phi(t)$ Where $latex A(t)$ is a given periodic matrix with period $latex T$, i.e. $latex A(x)=A(x+T), \forall x\in R$. Then the solution $\phi(t)$ satisfied identity: $latex \phi(t+T)=\phi(t)\phi^{1}(0)\phi(T)$. This could be explained as $latex \phi^{1}\phi(T)=\int_{0}^T\phi(t)$. Now we consider to solve the equation: $latex e^{TB}=\phi^{1}(0)\phi(T)$. At least formally it could be solved: $latex B=\frac{1}{T}log(\frac{\phi(T)}{\phi(0)})$. (Unfo…