### 数学代写|偏微分方程代写partial difference equations代考|Math462

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|偏微分方程代写partial difference equations代考|The wave-front set of a distribution

Let $\Omega \subset \mathbb{R}^n$ be an open set and let $x^{\circ} \in \Omega, \xi^{\circ} \in \mathbb{R}^n \backslash{0}$ be arbitrary. By a cone in $\mathbb{R}^n \backslash{0}$ we shall always mean a set invariant under all dilations $\xi \mapsto \lambda \xi, \lambda>0$ (i.e., a cone with vertex at the origin).
Lemma 2.1.4 Let $u \in \mathcal{D}^{\prime}(\Omega)$ have the following property:
(NWF) There exist an open set $U \subset \subset \Omega$ containing $x^{\circ}$ and $\varphi \in C_c^{\infty}(\Omega), \varphi(x)=1$ for every $x \in U$, and an open cone $\Gamma \subset \mathbb{R}^n \backslash{0}$ containing $\xi^{\circ}$ such that
$$\forall m \in \mathbb{Z}{+}, \sup {\xi \in \Gamma}\left((1+|\xi|)^m|\overline{(\varphi u)}(\xi)|\right)<+\infty .$$
Then, if $\Gamma^{\prime} \subset \mathbb{R}^n \backslash{0}$ is an open cone such that $\Gamma^{\prime} \cap \mathbb{S}^{n-1} \subset \subset \Gamma$, we have
$$\forall m \in \mathbb{Z}{+}, \sup {\xi \in \Gamma^{\infty}}\left((1+|\xi|)^m|\widehat{(\psi u)}(\xi)|\right)<+\infty$$
for every $\psi \in C_c^{\infty}(U)$
Proof Let $\varphi$ and $\psi$ be as in the statement; we have $\psi u=\psi \varphi u$ and therefore
$$\widehat{(\psi u)}(\xi)=(2 \pi)^{-n} \int \widehat{\psi}(\xi-\eta) \widehat{(\varphi u)}(\eta) \mathrm{d} \eta .$$
Here we shall use the notation, for $k \in \mathbb{Z}{+}$, $$|\psi|_k=\sup {\xi \in \mathbb{R}^n}\left((1+|\xi|)^k|\widehat{\psi}(\xi)|\right)$$
as well as
$$|\varphi u|{k, \Gamma}=\sup {\xi \in \Gamma}\left((1+|\xi|)^k|\overline{(\varphi u)}(\xi)|\right) .$$
Using the self-evident inequality $(1+|\xi|)^m \leq(1+|\eta|)^m(1+|\xi-\eta|)^m$ we get, for $\xi \in \Gamma^{\prime}$

## 数学代写|偏微分方程代写partial difference equations代考|Action of diferential operators on distributions

The action of a linear PDO on a distribution $u$ in $\Omega$ is defined by transposition:
$$\langle P(x, \mathrm{D}) u, \varphi\rangle=\left\langle u, P(x, \mathrm{D})^{\top} \varphi\right\rangle, \varphi \in \mathcal{C}{\mathrm{c}}^{\infty}(\Omega) .$$ When $u \in C^{\infty}(\Omega)$, (2.1.6) simply reflects integration by parts. Likewise, $$\langle P(x, \mathrm{D}) u, \bar{\varphi}\rangle=\left\langle u, \overline{P(x, \mathrm{D})^* \varphi}\right\rangle, \varphi \in C{\mathrm{c}}^{\infty}(\Omega) .$$
It follows directly from (2.1.6) that the inclusion (1.3.2), $\operatorname{supp} P(x, \mathrm{D}) f \subset$ supp $f$, remains valid when $f \in \mathcal{D}^{\prime}(\Omega)$. It is also obvious that
$$\text { singsupp } P(x, \text { D) } f \subset \operatorname{singsupp} f \text {, }$$
and if the coefficients of $P(x, \mathrm{D})$ are real-analytic, that
$$\text { singsupp }{\mathrm{a}} P(x, \mathrm{D}) f \subset \text { singsupp }{\mathrm{a}} f \text {. }^2$$
In other words, differential operators “decrease” the singular supports, just like they decrease the supports.

Every linear PDO maps $\mathcal{D}^{\prime}(\Omega)$ linearly and continuously into itself, and $\mathcal{E}^{\prime}(\Omega)$ into itself. In particular, $P(x, \mathrm{D}$ ) acts in the distribution sense (often called “the weak sense”) on a function $f \in L_{\text {loc }}^1(\Omega)$ :
$$\langle P(x, \mathrm{D}) f, \varphi\rangle=\int f P(x, \mathrm{D})^{\top} \varphi \mathrm{d} x, \varphi \in C_{\mathrm{c}}^{\infty}(\Omega) .$$
Actually [cf. (2.1.5)], every distribution $u \in \mathcal{D}^{\prime}(\Omega)$ can be represented locally as a finite sum of derivatives of continuous functions.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The wave-front set of a distribution

(NWF) 存在一个开集 $U \subset \subset \Omega$ 含有 $x^0$ 和 $\varphi \in C_c^{\infty}(\Omega), \varphi(x)=1$ 每一个 $x \in U$ ，和一个开雉 $\Gamma \subset \mathbb{R}^n \backslash 0$ 含有 $\xi^{\circ}$ 这样
$$\forall m \in \mathbb{Z}+, \sup \xi \in \Gamma\left((1+|\xi|)^m|\overline{(\varphi u)}(\xi)|\right)<+\infty$$

$$\forall m \in \mathbb{Z}+, \sup \xi \in \Gamma^{\infty}\left((1+|\xi|)^m|\widehat{(\psi u)}(\xi)|\right)<+\infty$$

$$\widehat{(\psi u)}(\xi)=(2 \pi)^{-n} \int \widehat{\psi}(\xi-\eta) \widehat{(\varphi u)}(\eta) \mathrm{d} \eta .$$

$$|\psi|_k=\sup \xi \in \mathbb{R}^n\left((1+|\xi|)^k|\widehat{\psi}(\xi)|\right)$$

$$|\varphi u| k, \Gamma=\sup \xi \in \Gamma\left((1+|\xi|)^k|\overline{(\varphi u)}(\xi)|\right)$$

## 数学代写|偏微分方程代写partial difference equations代考|Action of diferential operators on distributions

$$\langle P(x, \mathrm{D}) u, \varphi\rangle=\left\langle u, P(x, \mathrm{D})^{\top} \varphi\right\rangle, \varphi \in \mathcal{C c}^{\infty}(\Omega) .$$

$$\langle P(x, \mathrm{D}) u, \bar{\varphi}\rangle=\left\langle u, \overline{P(x, \mathrm{D})^* \varphi}\right\rangle, \varphi \in C \mathrm{c}^{\infty}(\Omega) .$$

$$\text { singsupp a } P(x, \mathrm{D}) f \subset \text { singsupp a } f .{ }^2$$

$$\langle P(x, \mathrm{D}) f, \varphi\rangle=\int f P(x, \mathrm{D})^{\top} \varphi \mathrm{d} x, \varphi \in C_{\mathrm{c}}^{\infty}(\Omega) .$$

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