### 数学代写|偏微分方程代写partial difference equations代考|MATH4310

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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|Sobolev Spaces

Possibly the most important scales of distribution spaces consist of the Sobolev spaces. In this text we will solely make use of the Sobolev spaces based on $L^2$, which we shall denote by $H^s\left(\mathbb{R}^n\right)$ with $s \in \mathbb{R}: H^s\left(\mathbb{R}^n\right)$ is the linear space of tempered distributions $u$ whose Fourier transform $\widehat{u}$ is a square-integrable function in $\mathbb{R}^n$ with respect to the density $\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$. The Hermitian product
$$(u, v)s=(2 \pi)^{-n} \int{\mathbb{R}^n} \widehat{u}(\xi) \overline{\widehat{v}(\xi)}\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$$ defines a Hilbert space structure on $H^s\left(\mathbb{R}^n\right)$; we use the notation $|u|_s=\sqrt{(u, u)s}$. We have $H^0\left(\mathbb{R}^n\right)=L^2\left(\mathbb{R}^n\right)$; if $s^{\prime}{s^{\prime}} \leq|u|_{s^s}$. All the Hilbert spaces $H^s\left(\mathbb{R}^n\right)$ are isomorphic: it is immediate to see that the operators
$$\left(1-\Delta_x\right)^{t / 2} \varphi(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi}\left(1+|\xi|^2\right)^{t / 2} \widehat{\varphi}(\xi) \mathrm{d} \xi, t \in \mathbb{R},$$
form a group of (continuous linear) automorphisms of $\mathcal{S}\left(\mathbb{R}^n\right) ;(2.2 .2)$ extends as an isometry of $H^s\left(\mathbb{R}^n\right)$ onto $H^{s-t}\left(\mathbb{R}^n\right)$, whatever the real numbers $s, t$.

We mention a useful inequality, valid for all $s, t \in \mathbb{R}$ such that $a=s-t>0$, all $\varepsilon>0$ and $u \in H^s\left(\mathbb{R}^n\right)$
$$|u|_t^2 \leq \varepsilon|u|_s^2+\frac{1}{4 \varepsilon}|u|_{t-a}^2,$$
a direct consequence of the inequality $A^t \leq \varepsilon A^s+\frac{1}{4 \varepsilon} A^{t-a}, A=1+|\xi|^2$.

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

We must now introduce distributions $F(x, y)$ on products $\Omega_1 \times \Omega_2$ with $\Omega_1 \subset$ $\mathbb{R}^{n_1}, \Omega_2 \subset \mathbb{R}^{n_2}$ open sets. Distributions belonging to $\mathcal{D}^{\prime}\left(\Omega_1 \times \Omega_2\right)$ are often referred to as kernels or distribution kernels. We can regard the product of two test-functions $\varphi \in C_{\mathrm{c}}^{\infty}\left(\Omega_1\right)$ and $\psi \in C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$ as an element of $C_{\mathrm{c}}^{\infty}\left(\Omega_1 \times \Omega_2\right)$, denoted by $\varphi \otimes \psi$, and evaluate $F \in \mathcal{D}^{\prime}\left(\Omega_1 \times \Omega_2\right)$ on it. Fixing $\psi$ defines a distribution in $\Omega_1$ :
$$C_{\mathrm{c}}^{\infty}\left(\Omega_1\right) \ni \varphi \mapsto\langle F, \varphi \otimes \psi\rangle \in \mathbb{C} .$$
To emphasize this partial action it is convenient to adopt the “Volterra notation”: to write $\int F(x, y) \psi(y)$ d $y$ rather than $\langle F(x, y), \psi(y)\rangle$. (Keep in mind, however, that $\int$ does not stand for a true integral!) In passing we point out that the Fubini formula is always true in distribution theory: $$\int\left(\int F(x, y) \psi(y) \mathrm{d} y\right) \varphi(x) \mathrm{d} x=\int\left(\int F(x, y) \varphi(x) \mathrm{d} x\right) \psi(y) \mathrm{d} y .$$
The map
$$C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \ni \psi \mapsto \mathfrak{I}F \psi(x)=\int F(x, y) \psi(y) \mathrm{d} y \in \mathcal{D}^{\prime}\left(\Omega_1\right)$$ is linear and continuous. The Schwartz Kernel Theorem states that, actually, every continuous linear map $C{\mathrm{c}}^{\infty}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$ is of the kind (2.3.1), and that the correspondence between continuous linear maps and distribution kernels is one-toone. This is a very special property of $\mathcal{D}^{\prime}$, obviously false for any infinite-dimensional Banach space (but true for $\mathcal{E}^{\prime}, C^{\infty}, C_{\mathrm{c}}^{\infty}$, if properly reformulated).

The composition $A_{1,2} \circ A_{2,3}$ of two linear operators $A_{1,2}: C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$, $A_{2,3}: C_{\mathrm{c}}^{\infty}\left(\Omega_3\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_2\right)$, puts requirements of regularity and support on the factors. For instance, we might require that $A_{2,3}$ maps $C_{\mathrm{c}}^{\infty}\left(\Omega_3\right)$ into $C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$, or else that $A_{1,2}$ extend as a continuous linear operator $\mathcal{D}^{\prime}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$, which is equivalent to requiring that the transpose $A_{1,2}^{\top}$ maps $C_{\mathrm{c}}^{\infty}\left(\Omega_1\right)$ into $C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$. These concerns are addressed in Definitions $2.3 .1$ and $2.3 .6$ below.

# 偏微分方程代写

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

$$(u, v) s=(2 \pi)^{-n} \int \mathbb{R}^n \widehat{u}(\xi) \overline{\hat{v}(\xi)}\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$$

$$\left(1-\Delta_x\right)^{t / 2} \varphi(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi}\left(1+|\xi|^2\right)^{t / 2} \widehat{\varphi}(\xi) \mathrm{d} \xi, t \in \mathbb{R}$$

$$|u|t^2 \leq \varepsilon|u|_s^2+\frac{1}{4 \varepsilon}|u|{t-a}^2,$$

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

$$C_{\mathrm{c}}^{\infty}\left(\Omega_1\right) \ni \varphi \mapsto\langle F, \varphi \otimes \psi\rangle \in \mathbb{C} .$$

$$\int\left(\int F(x, y) \psi(y) \mathrm{d} y\right) \varphi(x) \mathrm{d} x=\int\left(\int F(x, y) \varphi(x) \mathrm{d} x\right) \psi(y) \mathrm{d} y .$$

$$C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \ni \psi \mapsto \Im F \psi(x)=\int F(x, y) \psi(y) \mathrm{d} y \in \mathcal{D}^{\prime}\left(\Omega_1\right)$$

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