The singular integral operator and its commutator on weighted Morrey spaces

In this paper, we give necessary conditions and sufficient conditions respectively for the boundedness of the singular integral operator on the weighted Morrey spaces. We observe the phenomenon unique to the case of Morrey spaces; the Aq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_q$$\end{document}-theory by Muckenhoupt and Wheeden does not suffice. We also discuss the boundedness of the commutators. The difference between the maximal operator and the singular integral operators will be clarified. Further, the property of the sharp maximal operator is investigated.