
CalderonZygmund theory of singular integrals.
1. CalderonZygmund decomposition Title: Understanding the CalderonZygmund Decomposition and Bounded Singular Integrals 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 techniques to analyze each part. The scheme is roughly as follows. Given a function $f$ an…

The large sieve and the BombieriVinogradov theorem
1.Motivation Large sieve: A Philosophy Reflecting a Large Group of Inequalities The large sieve is a philosophy that reflects a large group of inequalities which are very effective in controlling some linear sums or square sums of correlations of arithmetic functions. This idea could have originated in harmonic analysis, relying almost entirely on almost orthogonality. One fundamental example is the estimate of the quality: $$\sum_{n\leq x}\Lambda(n)\overline{\chi(n)}$$ One naive idea to con…

Linear metric on F2, free group with two generator.
Title: Constructing a Metric by Pullback on a Linear Normalized Space In this article, we aim to construct a metric by pulling back a metric on a suitable linear normalized space $H$ that we carefully constructed. We begin by defining the generators of the free group $F_2$ as $a$ and $b$. Step 1: Constructing the Linear Normalized Space $H We construct the linear normalized space $H$, which is spanned by the basis $\Lambda = \Lambda_a \cup \Lambda_b$. Here, $\Lambda_a$ and $\Lambda_b$ are define…

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 $\mathbb T$, we know $latex h(f)>\alpha$, this is just mean $\forall k\in \mathbb Z$ , assume $latex  \hat f(k)\leq e^{k\alpha}$ , $g=log(f)$ for which we assume $latex g$ is a lifting of f use the inverse of ramification map , $M_{n\times n} \to M_{n\times n} , A \to e^A$. Then exists $\beta=c(\alpha)>0$ such that , $\forall k \in \mathbb Z$, $ \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 $\Delta u=0$ in $\Omega$ , $\forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity: $\frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$ I need to point out first, this property is not difficult to proof by standard integral by part meth…

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 a compact onedimensional dynamic system. We focus on $S_1$, it does not mean $S_1$ is the only compact onedimensional system, but it is a typical example. Let $T: S_1 \rightarrow S_1$. If $T$ is a homomorphism, then $T$ stays the order of $S_1$ (by continuity and the zero point theorem). This just means: (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 expanding rate of the dynamic sys…

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