On the boundedness of non-standard rough singular integral operators

Abstract

Let $\Omega$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator $$T_{\Omega,\,A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y){\rm d}y,$$ where $A$ is a function defined on $\mathbb{R}^d$ with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that $T_{\Omega,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $\Omega\in L(\log L)^{2}(\mathbb{S}^{d-1})$. These resuts essentially improve the previous known results given by Hofmann for the $L^p$ boundedness of $T_{\Omega,\,A}$ under the condition $\Omega\in L^{q}(\mathbb {S}^{d-1})$ $(q>1)$, Hu and Yang for the endpoint weak $L\log L$ type estimates when $\Omega\in {\rm Lip}_{\alpha}(\mathbb{S}^{d-1})$ for some $\alpha\in (0,\,1]$. Quantitative weighted strong and endpoint weak $L\log L$ type inequalities are proved whenever $\Omega\in L^{\infty}(\mathbb {S}^{d-1})$. The analysis of the weighted results relies heavily on two bilinear sparse dominations of $T_{\Omega,\,A}$ established herein.