Lp boundedness of some rough operators with different weights

Abstract

In this paper we prove that the maximal operator $M_{\Omega }$, the singular integral operator $T_{\Omega }$, and the maximal singular integral operator $T_{\Omega }^{*}$ with rough kernels are all bounded operators from $L^p\left(v\right)$ to $L^p\left(u\right)$ for the weight functions pair $(u,v)$. Here the kernel function $\Omega$ satisfies a size condition only; that is, $\Omega \in L^q\left(S^{n-1}\right),$$q>1$, but has no smoothness on $S^{n-1}$.