Posted on Categories:Measure Theory and Fourier Analysis, 傅里叶分析, 数学代写

数学代写|傅里叶分析代写Fourier Analysis代考|Singular Integrals with Even Kernels

avatest™

avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

数学代写|傅里叶分析代写Fourier Analysis代考|Singular Integrals with Even Kernels

Since a general integrable function $\Omega$ on $\mathbf{S}^{n-1}$ with mean value zero can be written as a sum of an odd and an even function, it suffices to study singular integral operators $T_{\Omega}$ with even kernels. For the rest of this section, fix an integrable even function $\Omega$ on $\mathbf{S}^{n-1}$ with mean value zero. The following idea is fundamental in the study of such singular integrals. Proposition 4.1.16 implies that
$$T_{\Omega}=-\sum_{j=1}^n R_j R_j T_{\Omega} .$$
If $R_j T_{\Omega}$ were another singular integral operator of the form $T_{\Omega_j}$ for some odd $\Omega_j$, then the boundedness of $T_{\Omega}$ would follow from that of $T_{\Omega_j}$ via the identity (4.2.22) and Theorem 4.2.7. It turns out that $R_j T_{\Omega}$ does have an odd kernel, but it may not be integrable on $\mathbf{S}^{n-1}$ unless $\Omega$ itself possesses an additional amount of integrability. The amount of extra integrability needed is logarithmic, more precisely of this sort:
$$c_{\Omega}=\int_{\mathbf{S}^{n-1}}|\Omega(\theta)| \log ^{+}|\Omega(\theta)| d \theta<\infty .$$
Observe that
$$|\Omega|_{L^1} \leq c_{\Omega}+2 \omega_{n-1} \leq C_n\left(c_{\Omega}+1\right),$$
which says that the norm $|\Omega|_{L^1}$ is always controlled by a dimensional constant multiple of $c_{\Omega}+1$. The following theorem is the main result of this section.

数学代写|傅里叶分析代写Fourier Analysis代考|Maximal Singular Integrals with Even Kernels

We have the corresponding theorem for maximal singular integrals.
Theorem 4.2.11. Let $\Omega$ be an even integrable function on $\mathbf{S}^{n-1}$ with mean value zero that satisfies (4.2.23). Then the corresponding maximal singular integral $T_{\Omega}^{(* *)}$, defined in (4.2.4), is bounded on $L^p\left(\mathbf{R}^n\right)$ for $1<p<\infty$ with norm at most a dimensional constant multiple of $\max \left(p^2,(p-1)^{-2}\right)\left(c_{\Omega}+1\right)$.

Proof. For $f \in L_{\mathrm{loc}}^1\left(\mathbf{R}^n\right)$, define the maximal function of $f$ in the direction $\theta$ by setting
$$M_\theta(f)(x)=\sup {a>0} \frac{1}{2 a} \int{|r| \leq a}|f(x-r \theta)| d r .$$
In view of Exercise 4.2.6(a) we have that $M_\theta$ is bounded on $L^p\left(\mathbf{R}^n\right)$ with norm at $\operatorname{most} 3 p(p-1)^{-1}$.

Fix $\Phi$ a smooth radial function such that $\Phi(x)=0$ for $|x|<1 / 4, \Phi(x)=1$ for $|x|>3 / 4$, and $0 \leq \Phi(x) \leq 1$ for all $x$ in $\mathbf{R}^n$. For $f \in L^p\left(\mathbf{R}^n\right)$ and $0<\varepsilon<N<\infty$ we introduce the smoothly truncated singular integral
$$\widetilde{T}{\Omega}^{(\varepsilon, N)}(f)(x)=\int{\mathbf{R}^n} \frac{\Omega\left(\frac{x-y}{|x-y|}\right)}{|x-y|^n}\left(\Phi\left(\frac{x-y}{\varepsilon}\right)-\Phi\left(\frac{x-y}{N}\right)\right) f(y) d y$$
and the corresponding maximal singular integral operator
$$\tilde{T}{\Omega}^{(* *)}(f)=\sup {0<N<\infty 0<\varepsilon<N} \sup {\Omega}\left|\widetilde{T}{\Omega}^{(\varepsilon, N)}(f)\right| .$$
For $f$ in $L^p\left(\mathbf{R}^n\right)$ (for some $1<p<\infty$ ), we have
\begin{aligned} & -\int_{\frac{N}{4} \leq|y| \leq N} \frac{\Omega\left(\frac{y}{|y|}\right)}{|y|^n} \Phi\left(\frac{y}{N}\right) f(x-y) d y \mid \ & \leq \sup {0<\varepsilon{\frac{\varepsilon}{4} \leq|y| \leq \varepsilon} \frac{\left|\Omega\left(\frac{y}{|y|}\right)\right|}{|y|^n}|f(x-y)| d y+\int_{\frac{N}{4} \leq|y| \leq N} \frac{\left|\Omega\left(\frac{y}{|y|}\right)\right|}{|y|^n}|f(x-y)| d y\right] \ & \leq \sup {0<\varepsilon{\mathbf{S}^{n-1}}|\Omega(\theta)|\left[\frac{4}{\varepsilon} \int_{\frac{\varepsilon}{4}}^{\varepsilon}|f(x-r \theta)| d r+\frac{4}{N} \int_{\frac{N}{4}}^N|f(x-r \theta)| d r\right] d \theta \ & \leq 16 \int_{\mathbf{S}^{n-1}}|\Omega(\theta)| M_\theta(f)(x) d \theta . \ & \end{aligned}

数学代写|傅里叶分析代写Fourier Analysis代考|Singular Integrals with Even Kernels

$$T_{\Omega}=-\sum_{j=1}^n R_j R_j T_{\Omega} .$$

$$c_{\Omega}=\int_{\mathbf{S}^{n-1}}|\Omega(\theta)| \log ^{+}|\Omega(\theta)| d \theta<\infty .$$

$$|\Omega|{L^1} \leq c{\Omega}+2 \omega_{n-1} \leq C_n\left(c_{\Omega}+1\right),$$

数学代写|傅里叶分析代写Fourier Analysis代考|Maximal Singular Integrals with Even Kernels

4.2.11.定理设$\Omega$为$\mathbf{S}^{n-1}$上的偶可积函数，其均值为零，满足(4.2.23)。则对应的极大奇异积分$T_{\Omega}^{(* *)}$，定义在(4.2.4)中，对于$1<p<\infty$有界于$L^p\left(\mathbf{R}^n\right)$，其范数至多为$\max \left(p^2,(p-1)^{-2}\right)\left(c_{\Omega}+1\right)$的一个维度常数倍。

$$M_\theta(f)(x)=\sup {a>0} \frac{1}{2 a} \int{|r| \leq a}|f(x-r \theta)| d r .$$

$$\widetilde{T}{\Omega}^{(\varepsilon, N)}(f)(x)=\int{\mathbf{R}^n} \frac{\Omega\left(\frac{x-y}{|x-y|}\right)}{|x-y|^n}\left(\Phi\left(\frac{x-y}{\varepsilon}\right)-\Phi\left(\frac{x-y}{N}\right)\right) f(y) d y$$

$$\tilde{T}{\Omega}^{(* *)}(f)=\sup {0<N<\infty 0<\varepsilon<N} \sup {\Omega}\left|\widetilde{T}{\Omega}^{(\varepsilon, N)}(f)\right| .$$

\begin{aligned} & -\int_{\frac{N}{4} \leq|y| \leq N} \frac{\Omega\left(\frac{y}{|y|}\right)}{|y|^n} \Phi\left(\frac{y}{N}\right) f(x-y) d y \mid \ & \leq \sup {0<\varepsilon{\frac{\varepsilon}{4} \leq|y| \leq \varepsilon} \frac{\left|\Omega\left(\frac{y}{|y|}\right)\right|}{|y|^n}|f(x-y)| d y+\int_{\frac{N}{4} \leq|y| \leq N} \frac{\left|\Omega\left(\frac{y}{|y|}\right)\right|}{|y|^n}|f(x-y)| d y\right] \ & \leq \sup {0<\varepsilon{\mathbf{S}^{n-1}}|\Omega(\theta)|\left[\frac{4}{\varepsilon} \int_{\frac{\varepsilon}{4}}^{\varepsilon}|f(x-r \theta)| d r+\frac{4}{N} \int_{\frac{N}{4}}^N|f(x-r \theta)| d r\right] d \theta \ & \leq 16 \int_{\mathbf{S}^{n-1}}|\Omega(\theta)| M_\theta(f)(x) d \theta . \ & \end{aligned}

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。