On singular integrals associated to surfaces

This paper is devoted to studying the singular integral with rough kernel associated to surfaces, which contain many classical surfaces as model examples. Also, the kernel of our operator lacks smoothness on the unit sphere as well as in the radial direction. We obtain the $L^p$ boundedness of the singular integral under a sharp size condition on its kernels in an extrapolation argument. In addition, the corresponding results for maximal truncated singular integral operators are also established.