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 \(\alpha \in (0,\,1]\).