In this paper, the authors discuss the weighted $L^p$ boundedness for the rough Marcinkiewicz integrals associated to surfaces. More precisely, the kernel of our operator lacks smoothness not only on the unit sphere, but also in the radial directions. Moreover, the surface is defined by using a differentiable function with monotonicity and some properties on the positive real line. The results given in this paper improve and extend some known results.
Yong Ding. Qingying Xue. Kôzô Yabuta. "Boundedness of the Marcinkiewicz integrals with rough kernel associated to surfaces." Tohoku Math. J. (2) 62 (2) 233 - 262, 2010. https://doi.org/10.2748/tmj/1277298647