\(L^p(\mathbb {R}^d)\) Boundedness for a Class of Nonstandard Singular Integral Operators

In this paper, let \(\Omega \) $\mathrm{\Omega }$ be homogeneous of degree zero which has vanishing moment of order one, A be a function on \(\mathbb {R}^d\) ${\mathbb{R}}^d$ such that \(\nabla A\in \textrm{BMO}(\mathbb {R}^d)\) $\mathrm{\nabla }A\in \text{BMO}({\mathbb{R}}^d)$ , we consider a class of nonstandard singular integral operators, \(T_{\Omega ,\,A}\) $T_{\mathrm{\Omega },A}$ , with rough kernel being of the form \( \frac{\Omega (x-y)}{\vert x-y\vert ^{d+1}}\big (A(x)-A(y)-\nabla A(y)(x-y)\big ) \) $\frac{\mathrm{\Omega }(x-y)}{{|x-y|}^{d+1}}(A(x)-A(y)-\mathrm{\nabla }A(y)(x-y))$ . This operator is closely related to the Calderón commutator. We prove that, under the Grafakos-Stefanov minimum size condition \(GS_{\beta }(S^{d-1})\) $G{S}_{\beta }(S^{d-1})$ with \(2<\beta <\infty \) $2<\beta <\infty$ for \(\Omega \) $\mathrm{\Omega }$ , \(T_{\Omega ,\,A}\) $T_{\mathrm{\Omega },A}$ is bounded on \(L^p(\mathbb {R}^d)\) $L^p({\mathbb{R}}^d)$ for p with \(1+1/(\beta -1)< p < \beta \) $1+1/(\beta -1)<p<\beta$ .