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# 数学代写|常微分方程代考Ordinary Differential Equations代写|MA26600 The Analytic Wave-Front Set of a Distribution

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Analytic Wave-Front Set of a Distribution

In the real-analytic context the next definition is central to the whole idea of lifting the analysis of distributions to phase-space, in other words, of microlocalization. We introduce the concept of analytic wave-front set of a distribution (cf. Definition 2.1.5) through the FBI transform. Later (in Ch. 7), we shall introduce the analytic wave-front set of a hyperfunction $u$ (called its essential singular support in [SatoKawai-Kashiwara, 1973] and numerous other texts) using the representation of $u$ as a sum of boundary values (Definition 7.4.7).

We use standard terminology: a set $\mathcal{U} \subset \mathbb{R}^n \times\left(\mathbb{R}^n \backslash{0}\right)$ is said to be conic when $(x, \xi) \in \mathcal{U} \Longrightarrow \forall \lambda>0,(x, \lambda \xi) \in \mathcal{U}$. The transform $F_\kappa$ is defined in (3.4.8).

Definition 3.5.1 We say that $v \in \mathcal{E}^{\prime}\left(\mathbb{R}^n\right)$ is microanalytic at a point $\left(x^{\circ}, \xi^{\circ}\right) \in$ $\mathbb{R}^n \times\left(\mathbb{R}^n \backslash{0}\right)$ if there is a conic neighborhood $\mathcal{U}$ of $\left(x^{\circ}, \xi^{\circ}\right)$ in $\mathbb{R}^n \times\left(\mathbb{R}^n \backslash{0}\right)$ such that $\left|F_\kappa v(x, \xi)\right| \lesssim \mathrm{e}^{-c|\xi|}$ for some positive numbers $\kappa, c$, and all $(x, \xi) \in \mathcal{U}$.
Let $\Omega \subset \mathbb{R}^n$ be an open set. We say that a distribution $u \in \mathcal{D}^{\prime}(\Omega)$ is microanalytic at a point $\left(x^{\circ}, \xi^{\circ}\right) \in \Omega \times\left(\mathbb{R}^n \backslash{0}\right)$ if there is a $v \in \mathcal{E}^{\prime}\left(\mathbb{R}^n\right)$ equal to $u$ in some neighborhood of $x^{\circ}$ and microanalytic at $\left(x^{\circ}, \xi^{\circ}\right)$. The (closed) subset of $\Omega \times\left(\mathbb{R}^n \backslash{0}\right)$ consisting of the points $\left(x^{\circ}, \xi^{\circ}\right)$ at which $u$ is not microanalytic is called the analytic wave-front set of $u$ and shall be denoted by $W F_{\mathrm{a}}(u)$.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Analytic wave-front sets and holomorphic extension

In this subsection we relate the representation of a distribution $u$ as the sum of boundary values of holomorphic functions in a wedge (Theorem 3.3.6) to the analytic wave-front set of $u$. If $\Gamma \subset \mathbb{R}^n \backslash{0}$ is a cone we denote by $\Gamma^{\circ}$ its polar (sometimes called its dual), i.e., the set $\left{\xi \in \mathbb{R}^n ; \forall y \in \Gamma, y \cdot \xi \geq 0\right} ; \Gamma^{\circ}$ is a closed and convex cone in $\mathbb{R}^n$ (with $0 \in \Gamma^{\circ}$, obviously); $\Gamma^{\circ}$ is also the polar of the convex hull of $\Gamma$, i.e., the intersection of all the convex cones containing $\Gamma$. We also recall the notation (3.3.1): $W_\delta(U, \Gamma)={z=x+i y \in U+i \Gamma ;|y|<\delta}$.

Theorem 3.5.5 Let $\Omega \subset \mathbb{R}^n$ be an open set and $\Gamma \subset \mathbb{R}^n \backslash{0}$ a convex open cone. The following properties of a distribution $u \in \mathcal{D}^{\prime}(\Omega)$ are equivalent:
(a) whatever the open set $U \subset \subset \Omega$ and the open cone $\Gamma_* \subset \Gamma$ in $\mathbb{R}^n \backslash{0}$ such that $\Gamma_* \cap \mathbb{S}^{n-1} \subset \subset \Gamma$ the restriction of $u$ to $U$ is the boundary value of a function $h \in O_{\text {temp }}\left(\mathcal{W}\delta\left(U, \Gamma*\right)\right)$;
(b) $W F_a(u) \subset \Omega \times \Gamma^{\circ}$.
Proof I. (a) $\Longrightarrow$ (b). Let $\xi^{\circ} \in \mathbb{R}^n$ be such that $y^{\circ} \cdot \xi^{\circ}<0$ for some $y^{\circ} \in \Gamma$. Let $x^{\circ} \in U$ be arbitrary and select $\varphi \in C_{\mathrm{c}}^{\infty}(U), \varphi(x)=1$ if $\left|x-x^{\circ}\right|<\rho, \rho>0$. By Lemma $3.4 .5$ it suffices to show that $\left(x^{\circ}, \xi^{\circ}\right) \notin W F_{\mathrm{a}}(\varphi u)$. We have
\begin{aligned} F_K(\varphi u)(x, \xi) & =\int \mathrm{e}^{i \xi \cdot\left(x-x^{\prime}\right)-\left|\xi | x-x^{\prime}\right|^2} \varphi\left(x^{\prime}\right) u\left(x^{\prime}\right) \mathrm{d} x^{\prime} \ & =\lim {0{\mathbb{R}^n} \mathrm{e}^{i \xi \cdot\left(x-x^{\prime}\right)-\left|\xi | x-x^{\prime}\right|^2} \varphi\left(x^{\prime}\right) h\left(x^{\prime}+i t y^{\circ}\right) \mathrm{d} x^{\prime} . \end{aligned}

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Analytic

Wave-Front Set of a Distribution 的解折波前焦的概念 (参见定义 2.1.5) 。稍后 (在第 7 章)，我们将介络超函数的解析波前集 $u$ (在 [satoKawai-Kashiwara， 1973] 和许多其他文本中称为它的璐本单数支持) 使用表示 $u$ 作为边界值的总和（定义 7.4.7）。 义

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Analytic wavefront sets and holomorphic extension

$\$ \\begin {aligned } F_{-} \mathrm{K}(\backslash varphi u )(\mathrm{x}, \backslash \mathrm{xi}) \&=\backslash int \backslash \operatorname{mathrm}{\mathrm{e}} \wedge{\mathrm{i} \backslash \mathrm{xi} \backslash \mathrm{cdot} \backslash left (\mathrm{xx} \wedge{\backslash prime } \backslash right )-\backslash \sqrt{1}|\backslash \mathrm{xi}| \mathrm{xx} \wedge{\backslash prime } \backslash right \mid \wedge 2} \backslash varphi \backslash left (\mathrm{x} \wedge{\backslash prime } \backslash right ) u \backslash left (\mathrm{x} \wedge{\backslash prime } \backslash right ) \backslash mathrm {\mathrm{d}} \mathrm{x} \wedge{\backslash prime } \backslash \&=\backslash \lim {0{\backslash \operatorname{mathbb}{\mathrm{R}} \wedge \mathrm{n}} \backslash \operatorname{mathrm}{\mathrm{e}} \wedge{\mathrm{i} \backslash \mathrm{xi} \backslash \mathrm{cdot} \backslash left (\mathrm{xx} \wedge{\backslash prime } \backslash right )-\backslash left \backslash \backslash \mathrm{xi} \mid \mathrm{xx} \wedge{\backslash prime } \backslash right \mid \wedge 2} \backslash varphi \backslash left (x \wedge{\backslash prime } \backslash right ) h \backslash left (x \wedge{\backslash prime }+i t y \wedge{\backslash circ } \backslash right ) \backslash \operatorname{mathrm}{\mathrm{d}} x \wedge{\backslash prime } 。 \backslash 结束 { 对㐎 } \ \

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