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# 数学代写|傅里叶分析代写Fourier Analysis代考|Sufficient Conditions for $L^p$ Boundedness of Singular Integrals

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Sufficient Conditions for $L^p$ Boundedness of Singular Integrals

We first note that under conditions (4.4.1), (4.4.2), and (4.4.3), there exists a tempered distribution $W$ that coincides with $K$ on $\mathbf{R}^n \backslash{0}$. Indeed, condition (4.4.3) implies that there exists a sequence $\delta_j \downarrow 0$ such that
$$\lim {j \rightarrow \infty} \int{\delta_j<|x| \leq 1} K(x) d x=L$$

exists. Using (4.3.8), we conclude that there exists such a tempered distribution $W$. Note that we must have $|L| \leq A_3$.
We observe that the difference of two distributions $W$ and $W^{\prime}$ that coincide with $K$ on $\mathbf{R}^n \backslash{0}$ must be supported at the origin.

Theorem 4.4.1. Assume that $K$ satisfies (4.4.1), (4.4.2), and (4.4.3), and let $W$ be a tempered distribution that coincides with $K$ on $\mathbf{R}^n \backslash{0}$. Then we have
$$\sup {0<\varepsilon{\xi \neq 0}\left|\left(K \chi_{\varepsilon<|\cdot|<N}\right)(\xi)\right| \leq 15\left(A_1+A_2+A_3\right) \text {. }$$
Thus the operator given by convolution with $W$ maps $L^2\left(\mathbf{R}^n\right)$ to itself with norm at most $15\left(A_1+A_2+A_3\right)$. Consequently, it also maps $L^1\left(\mathbf{R}^n\right)$ to $L^{1, \infty}\left(\mathbf{R}^n\right)$ with bound at most a dimensional constant multiple of $A_1+A_2+A_3$ and $L^p\left(\mathbf{R}^n\right)$ to itself with bound at most $C_n \max \left(p,(p-1)^{-1}\right)\left(A_1+A_2+A_3\right)$, for some dimensional constant $C_n$, whenever $1<p<\infty$.

Proof. Let us set $K^{(\varepsilon, N)}(x)=K(x) \chi_{\varepsilon<|x|<N}$. If we prove (4.4.4), then for all $f$ in $\mathscr{S}\left(\mathbf{R}^n\right)$ we will have the estimate
$$\left|f * K^{\left(\delta_j, j\right)}\right|_{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|_{L^2}$$
uniformly in $j$. Using this, (4.3.9), and Fatou’s lemma, we obtain that
$$|f * W|_{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|_{L^2},$$
thus proving the second conclusion of the theorem.
Let us now fix a $\xi$ with $\varepsilon<|\xi|^{-1}<N$ and prove (4.4.4). Write $\widehat{K^{(\varepsilon, N)}}(\xi)=$ $I_1(\xi)+I_2(\xi)$, where
\begin{aligned} & I_1(\xi)=\int_{\varepsilon<|x|<|\xi|^{-1}} K(x) e^{-2 \pi i x \cdot \xi} d x \ & I_2(\xi)=\int_{|\xi|^{-1}<|x|<N} K(x) e^{-2 \pi i x \cdot \xi} d x \end{aligned}

## 数学代写|傅里叶分析代写Fourier Analysis代考|An Example

We now give an example of a distribution that satisfies conditions (4.4.1), (4.4.2), and (4.4.3).

Example 4.4.2. Let $\tau$ be a nonzero real number and let $K(x)=\frac{1}{\mid x^{n+i \tau}}$ defined for $x \in \mathbf{R}^n \backslash{0}$. For a sequence $\delta_k \downarrow 0$ and $\varphi$ a Schwartz function on $\mathbf{R}^n$, define
$$\langle W, \varphi\rangle=\lim {k \rightarrow \infty} \int{\delta_k \leq|x|} \varphi(x) \frac{d x}{|x|^{n+i \tau}},$$
whenever the limit exists. We claim that for some choices of sequences $\delta_k, W$ is a well defined tempered distribution on $\mathbf{R}^n$. Take, for example, $\delta_k=e^{-2 \pi k / \tau}$. For this sequence $\delta_k$, observe that
$$\int_{\delta_k \leq|x| \leq 1} \frac{1}{|x|^{n+i \tau}} d x=\omega_{n-1} \frac{1-\left(e^{-2 \pi k / \tau}\right)^{-i \tau}}{-i \tau}=0,$$
and thus
$$\langle W, \varphi\rangle=\int_{|x| \leq 1}(\varphi(x)-\varphi(0)) \frac{d x}{|x|^{n+i \tau}}+\int_{|x| \geq 1} \varphi(x) \frac{d x}{|x|^{n+i \tau}},$$
which implies that $W \in \mathscr{S}^{\prime}\left(\mathbf{R}^n\right)$, since
$$|\langle W, \varphi\rangle| \leq C\left[|\nabla \varphi|_{L^{\infty}}+||x| \varphi(x)|_{L^{\infty}}\right]$$
If $\varphi$ is supported in $\mathbf{R}^n \backslash{0}$, then
$$\langle W, \varphi\rangle=\int K(x) \varphi(x) d x$$
Therefore $W$ coincides with the function $K$ away from the origin. Moreover, (4.4.1) and (4.4.2) are clearly satisfied for $K$, while (4.4.3) is also satisfied, since
$$\left|\int_{R_1<|x|<R_2} \frac{1}{|x|^{n+i \tau}} d x\right|=\omega_{n-1}\left|\frac{R_1^{-i \tau}-R_2^{-i \tau}}{-i \tau}\right| \leq \frac{2 \omega_{n-1}}{|\tau|}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Sufficient Conditions for $L^p$ Boundedness of Singular Integrals

$$\lim {j \rightarrow \infty} \int{\delta_j<|x| \leq 1} K(x) d x=L$$

$$\sup {0<\varepsilon{\xi \neq 0}\left|\left(K \chi_{\varepsilon<|\cdot|<N}\right)(\xi)\right| \leq 15\left(A_1+A_2+A_3\right) \text {. }$$

$$\left|f * K^{\left(\delta_j, j\right)}\right|{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|{L^2}$$

$$|f * W|{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|{L^2},$$

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