On the Boundedness of Non-standard Rough Singular Integral Operators

Let \(\Omega \) $\mathrm{\Omega }$ be a homogeneous function of degree zero, have vanishing moment of order one on the unit sphere \(\mathbb {S}^{d-1}\) ${\mathbb{S}}^{d-1}$ ( \(d\ge 2\) $d\ge 2$ ). In this paper, our object of investigation is the following rough non-standard singular integral operator \(\begin{aligned} T_{\Omega ,\,A}f(x)=\mathrm{p.\,v.}\int _{{\mathbb {R}}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\big (A(x)-A(y)-\nabla A(y)(x-y)\big )f(y)\textrm{d}y, \end{aligned}\) $$\begin{array}{c}\hfill T_{\mathrm{\Omega },A}f(x)=\mathrm{p}.\mathrm{v}.{\int }_{{\mathbb{R}}^d}\frac{\mathrm{\Omega }(x-y)}{{|x-y|}^{d+1}}(A(x)-A(y)-\mathrm{\nabla }A(y)(x-y))f(y)\text{d}y,\end{array}$$ where A is a function defined on \({\mathbb {R}}^d\) ${\mathbb{R}}^d$ with derivatives of order one in \({\textrm{BMO}}({\mathbb {R}}^d)\) $\text{BMO}({\mathbb{R}}^d)$ . We show that \(T_{\Omega ,\,A}\) $T_{\mathrm{\Omega },A}$ enjoys the endpoint \(L\log L\) $LlogL$ type estimate and is \(L^p\) $L^p$ bounded if \(\Omega \in L(\log L)^{2}({\mathbb {S}}^{d-1})\) $\mathrm{\Omega }\in L{(logL)}^2({\mathbb{S}}^{d-1})$ . These results essentially improve the previous known results given by Hofmann (Stud Math 109:105–131, 1994) for the \(L^p\) $L^p$ boundedness of \(T_{\Omega ,\,A}\) $T_{\mathrm{\Omega },A}$ under the condition \(\Omega \in L^{q}({\mathbb {S}}^{d-1})\) $\mathrm{\Omega }\in L^q({\mathbb{S}}^{d-1})$ \((q>1)\) $(q>1)$ , Hu and Yang (Bull Lond Math Soc 35:759–769, 2003) for the endpoint weak \(L\log L\) $LlogL$ type estimates when \(\Omega \in \textrm{Lip}_{\alpha }({\mathbb {S}}^{d-1})\) $\mathrm{\Omega }\in {\text{Lip}}_{\alpha }({\mathbb{S}}^{d-1})$ for some \(\alpha \in (0,\,1]\) $\alpha \in (0,1]$ .