AI 中文总结
该研究建立了一类带粗糙核、满足Hölder条件的核及平均函数的奇异积分弱型$(1,1)$有界性判据，推广了已有结果，得到了一大类满足该有界性的奇异积分算子。
AI 中文摘要
本文建立了如下粗糙奇异积分弱型$(1,1)$有界性的一个判据：$$T_{Ω,K,a}f(x)={\rm p.v.}\int_{\mathbb{R}^n}Ω(x-y)K(x,y)m_{x,y}a f(y)dy,$$其中$m_{x,y}a=\int_0^1a(sx+(1-s)y)ds$，满足$a\in L^1(\mathbb{R}^n)$且$\hat{a}\in L^1(\mathbb{R}^n)$；$Ω$是零次齐次函数，在$\mathbb{S}^{n-1}$上可积，满足抵消条件$\int_{\mathbb{S}^{n-1}}Ω(θ)dσ(θ)=0$；$K$是定义在$\mathbb{R}^n\times\mathbb{R}^n\setminus \{(x,x):x\in\mathbb{R}^n\}$上的可测函数，满足Hölder条件。在假设$Ω\in L\log L(\mathbb{S}^{n-1})$且算子$T_{Ω, K}f(x)={\rm p.v.}\int_{\mathbb{R}^n}Ω(x-y)K(x,y)f(y)dy$在$L^2(\mathbb{R}^n)$上有界的前提下，我们证明了$T_{Ω,K,a}$的弱型$(1,1)$有界性。作为若干应用，我们得到了一大类具有弱型$(1,1)$有界性的奇异积分算子。本文的主要结果本质上拓展并推广了一些已知结论。
英文摘要
In this paper we establish a criterion on weak type $(1,\,1)$ bound of the following rough singular integral $$T_{Ω,K,a}f(x)={\rm p.v.}\int_{\mathbb{R}^n}Ω(x-y)K(x,y)m_{x,y}a f(y)dy,$$ where $m_{x,y}a=\int_0^1a(sx+(1-s)y)ds$ with $a\in L^1(\mathbb{R}^n)$ and $\hat{a}\in L^1(\mathbb{R}^n)$, $Ω$ is homogeneous of degree zero, integrable in $\mathbb{S}^{n-1}$ and satisfies the cancellation condition $\int_{\mathbb{S}^{n-1}}Ω(θ)dσ(θ)=0$ and $K$ is a measurable function defined on $\mathbb{R}^n\times\mathbb{R}^n\setminus \{(x,x):x\in\mathbb{R}^n\}$ and satisfies a Hölder condition. By assuming that $Ω\in L\log L(\mathbb{S}^{n-1})$ and the operator $T_{Ω, K}f(x)={\rm p.v.}\int_{\mathbb{R}^n}Ω(x-y)K(x,y)f(y)dy$ is bounded on $L^2(\mathbb{R}^n)$, we prove the weak type (1,1) bound of $T_{Ω,K,a}$. As several applications, we obtain a large class of singular integral operators which possess weak type $(1,\,1)$ bound. The main results of this paper essentially extend and generalize some known ones.
