# 数学代写|自旋几何代写spin geometry代考|MATH6209

## 数学代写|自旋几何代写spin geometry代考|Pseudodifierential Operators

The concept of a pseudodifferential operator has its roots in the following observation. Let $P=\sum A^a(x) D^\alpha$ be a differential operator on $\mathbb{R}^n$ acting on functions $u$ with, say, compact support. By Fourier Inversion (2.3) any such $u$ can be written as
$$u(x)=(2 \pi)^{-\pi / 2} \int e^{i\langle x, \xi\rangle} \hat{u}(\xi) d \xi .$$
Applying $P$ we find that
$$P u(x)=(2 \pi)^{-\pi / 2} \int e^{i\langle x, \xi\rangle} p(x, \xi) \hat{\imath}(\xi) d \xi$$
where
$$p(x, \xi) \equiv \sum_{|a| \leqq m} A^a(x) \xi^{q a}$$
is the (total) symbol of $\boldsymbol{P}$. Replacing $p$ by a more general function of $x$ and $\xi$ defines a pseudodifferential operator. Note that in (3.2) the order of $P$ corresponds to the degree of $p$ as a polynomial in $\xi$. In the general case one must be careful with growth in the $\xi$-variable.

Definition 3.1. Fix $m \in \mathbb{R}$. A smooth (matrix-valued) function $p(x, \xi)$ on $\mathbb{R}^n \times \mathbb{R}^n$ is said to be a symbol of order $m$ if for each $\alpha, \alpha^{\prime}$ there is a constant $C_{a x^{\prime}}$ such that
$$\left|D_x^a D_{\xi}^{a^{\prime}} p(x, \xi)\right| \leqq C_{a x}(1+|\xi|)^{m-\left|a^{\prime}\right|}$$
for all $x, \xi$. Let $S y m^m$ denote the space of these symbols.
Proposition 3.2. To each $p \in \mathrm{Sym}^m$ the formula (3.1) defines a linear operator $P: \mathscr{S} \rightarrow \mathscr{S}$. If $p$ has compact $x$-support, this operator has a continuous extension $P: L_{s+m}^2 \rightarrow L_s^2$ for all $s$.

## 数学代写|自旋几何代写spin geometry代考|Elliptic Operators and Parametrices

Recall that a differential operator $P: \Gamma(E) \rightarrow \Gamma(F)$ over a compact manifold is called elliptic if its principal symbol $\sigma_{\xi}(P)$ is invertible at all nonzero cotangent vectors $\xi$. In this section we shall prove the fundamental result that modulo infinitely smoothing operators, an elliptic operator is invertible.

To this end we consider the “local” case of pseudodifferential operators on $\mathbb{R}^n$ which map $\mathbb{C}^k$-valued functions to themselves.

DEFINITION 4.1 An operator $P \in \Psi D O_m$ with symbol $p$ is said to be elliptic if there exists a constant $c>0$ such that for all $|\xi| \geqq c$ the matrix inverse of $p(x, \xi)$ exists and satisfies
$$\left|p(x, \xi)^{-1}\right| \leqq c(1+|\xi|)^{-m} .$$
For example, an operator $P$ whose symbol is of the form $p(|\xi|) I \mathrm{~d}$, where $p(t)$ is a polynomial with constant positive coefficients, is elliptic.

REMARK 4.2. It is straightforward to verify that if $P: \Gamma(E) \rightarrow \Gamma(F)$ is an elliptic differential operator, then the local representations of $P$ in a good presentation of $E$ and $F$ are elliptic in the sense of 4.1.

Theorem 4.3. Let $P \in \Psi D O_m$ be elliptic. Then there exists an operator $Q \in \Psi D O_{-m}$, unique up to equivalence, such that
$$P Q=\mathrm{Id}-S^{\prime} \text { and } Q P=\mathrm{Id}-S$$
where $S$ and $S^{\prime}$ are infinitely smoothing operators.
Proof. Let $p$ be the symbol of $P$ and let $c$ be the constant in Definition 4.1. Set $q_0(x, \xi)=\chi(|\xi|) p(x, \xi)^{-1}$ where $\chi: \mathbb{R}^{+} \rightarrow[0,1]$ is a smooth function with $\chi(t)=0$ for $t \leq c$ and $\chi(t)=1$ for $t \geqq 2 c$.

## 数学代写|自旋几何代写spin geometry代考|Pseudodifierential Operators

$$u(x)=(2 \pi)^{-\pi / 2} \int e^{i(x, \xi\rangle} \hat{u}(\xi) d \xi .$$

$$P u(x)=(2 \pi)^{-\pi / 2} \int e^{i\langle x, \xi\rangle} p(x, \xi) \hat{\imath}(\xi) d \xi$$

$$p(x, \xi) \equiv \sum_{|a| \leqq m} A^a(x) \xi^{q a}$$

$$\left|D_x^a D_{\xi}^{a^{\prime}} p(x, \xi)\right| \leqq C_{a x}(1+|\xi|)^{m-\left|a^{\prime}\right|}$$

## 数学代写|自旋几何代写spin geometry代考|Elliptic Operators and Parametrices

$$\left|p(x, \xi)^{-1}\right| \leqq c(1+|\xi|)^{-m}$$

$$P Q=\mathrm{Id}-S^{\prime} \text { and } Q P=\mathrm{Id}-S$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: