On the Boundedness of Non-standard Rough Singular Integral Operators

Abstract

Let $\Omega$ be a homogeneous function 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

$$\begin{aligned} T_{\Omega ,\,A}f(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\big (A(x)-A(y)-\nabla A(y)(x-y)\big )f(y)\textrm{d}y, \end{aligned}$$

where A is a function defined on ${\mathbb {R}}^d$ with derivatives of order one in ${\textrm{BMO}}({\mathbb {R}}^d)$. 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 results essentially improve the previous known results given by Hofmann (Stud Math 109:105–131, 1994) 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 (Bull Lond Math Soc 35:759–769, 2003) for the endpoint weak $L\log L$ type estimates when $\Omega \in \textrm{Lip}_{\alpha }({\mathbb {S}}^{d-1})$ for some $\alpha \in (0,\,1]$.