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

Abstract

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