The boundedness of some operators with rough kernel on the weighted Morrey spaces

Abstract

Let $\Omega \in L^q(S^{n-1})$ with $1<q\le \infty$ be homogeneous of degree zero and has mean value zero on $S^{n-1}$. In this paper, we will study the boundedness of homogeneous singular integrals and Marcinkiewicz integrals with rough kernel on the weighted Morrey spaces $L^{p,\kappa }(w)$ for $q^{\prime }\le p<\infty$(or $q^{\prime }<p<\infty$) and $0<\kappa <1$. We will also prove that the commutator operators formed by a $BMO({\mathbb{R}}^n)$ function $b(x)$ and these rough operators are bounded on the weighted Morrey spaces $L^{p,\kappa }(w)$ for $q^{\prime }<p<\infty$ and $0<\kappa <1$.

17 pages