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 $$T_{\Omega ,A}f(x)=p.v.{\int }_{{\mathbb{R}}^d}\frac{\Omega (x-y)}{|x-y{|}^{d+1}}(A(x)-A(y)-\nabla A(y)(x-y))f(y)\mathrm{d}y,$$ where $A$ is a function defined on ${\mathbb{R}}^d$ with derivatives of order one in $\mathrm{BMO}({\mathbb{R}}^d)$. 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 $\Omega \in {\mathrm{Lip}}_{\alpha }({\mathbb{S}}^{d-1})$ 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.

49 pages