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# 数学代写|傅里叶分析代写Fourier Analysis代考|The Space of Fourier Multipliers $\mathscr{M}_p\left(\mathbf{R}^n\right)$

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## 数学代写|傅里叶分析代写Fourier Analysis代考|The Space of Fourier Multipliers $\mathscr{M}_p\left(\mathbf{R}^n\right)$

We have now characterized all convolution operators that map $L^2$ to $L^2$. Suppose now that $T$ is in $\mathscr{M}^{p, p}$, where $1<p<2$. As discussed in Theorem 2.5.7, $T$ also maps $L^{p^{\prime}}$ to $L^{p^{\prime}}$. Since $p<2<p^{\prime}$, by Theorem 1.3.4, it follows that $T$ also maps $L^2$ to $L^2$. Thus $T$ is given by convolution with a tempered distribution whose Fourier transform is a bounded function.

Definition 2.5.11. Given $1 \leq p<\infty$, we denote by $\mathscr{M}p\left(\mathbf{R}^n\right)$ the space of all bounded functions $m$ on $\mathbf{R}^n$ such that the operator $$T_m(f)=(\widehat{f} m)^{\vee}, \quad f \in \mathscr{S},$$ is bounded on $L^p\left(\mathbf{R}^n\right)$ (or is initially defined in a dense subspace of $L^p\left(\mathbf{R}^n\right)$ and has a bounded extension on the whole space). The norm of $m$ in $\mathscr{M}_p\left(\mathbf{R}^n\right)$ is defined by $$|m|{\mathscr{M}p}=\left|T_m\right|{L^p \rightarrow L^p}$$
Definition 2.5.11 implies that $m \in \mathscr{M}p$ if and only if $T_m \in \mathscr{M}^{p, p}$. Elements of the space $\mathscr{M}_p$ are called $L^p$ multipliers or $L^p$ Fourier multipliers. It follows from Theorem 2.5.10 that $\mathscr{M}_2$, the set of all $L^2$ multipliers, is $L^{\infty}$. Theorem 2.5 .8 implies that $\mathscr{M}_1\left(\mathbf{R}^n\right)$ is the set of the Fourier transforms of finite Borel measures that is usually denoted by $\mathscr{M}\left(\mathbf{R}^n\right)$. Theorem 2.5.7 states that a bounded function $m$ is an $L^p$ multiplier if and only if it is an $L^{p^{\prime}}$ multiplier, and in this case $$|m|{\mathscr{M}p}=|m|{\mathscr{M}_{p^{\prime}}}, \quad 1<p<\infty$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Oscillatory Integrals

Oscillatory integrals have played an important role in harmonic analysis from its outset. The Fourier transform is the prototype of oscillatory integrals and provides the simplest example of a nontrivial phase, a linear function of the variable of integration. More complicated phases naturally appear in the subject; for instance, Bessel functions provide examples of oscillatory integrals in which the phase is a sinusoidal function.

In this section we take a quick look at oscillatory integrals. We mostly concentrate on one-dimensional results, which already require some significant analysis. We examine only a very simple higher-dimensional situation. Our analysis here is far from adequate.
Definition 2.6.1. An oscillatory integral is an expression of the form
$$I(\lambda)=\int_{\mathbf{R}^n} e^{i \lambda \varphi(x)} \psi(x) d x$$
where $\lambda$ is a positive real number, $\varphi$ is a real-valued function on $\mathbf{R}^n$ called the phase, and $\psi$ is a complex-valued and smooth integrable function on $\mathbf{R}^n$, which is often taken to have compact support.

## 数学代写|傅里叶分析代写Fourier Analysis代考|The Space of Fourier Multipliers $\mathscr{M}_p\left(\mathbf{R}^n\right)$

2.5.11.定义给定$1 \leq p<\infty$，我们用$\mathscr{M}p\left(\mathbf{R}^n\right)$表示$\mathbf{R}^n$上所有有界函数$m$的空间，使得算子$$T_m(f)=(\widehat{f} m)^{\vee}, \quad f \in \mathscr{S},$$在$L^p\left(\mathbf{R}^n\right)$上有界(或者在$L^p\left(\mathbf{R}^n\right)$的稠密子空间中初始定义并在整个空间上有界扩展)。$\mathscr{M}p\left(\mathbf{R}^n\right)$中$m$的范数由$$|m|{\mathscr{M}p}=\left|T_m\right|{L^p \rightarrow L^p}$$定义 定义2.5.11意味着$m \in \mathscr{M}p$当且仅当$T_m \in \mathscr{M}^{p, p}$。空间$\mathscr{M}_p$的元素称为$L^p$乘数或$L^p$傅里叶乘数。由定理2.5.10可知，所有$L^2$乘数的集合$\mathscr{M}_2$为$L^{\infty}$。定理2.5 .8表明$\mathscr{M}_1\left(\mathbf{R}^n\right)$是有限波雷尔测度的傅里叶变换的集合，通常用$\mathscr{M}\left(\mathbf{R}^n\right)$表示。定理2.5.7指出有界函数$m$是$L^p$乘子当且仅当它是$L^{p^{\prime}}$乘子，在这种情况下 $$|m|{\mathscr{M}p}=|m|{\mathscr{M}{p^{\prime}}}, \quad 1<p<\infty$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Oscillatory Integrals

2.6.1.定义振荡积分是这样的表达式
$$I(\lambda)=\int_{\mathbf{R}^n} e^{i \lambda \varphi(x)} \psi(x) d x$$

## MATLAB代写

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