A criterion on weak type $(1,1)$ bound of rough singular integrals

Abstract

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.