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In this paper, we give necessary conditions and sufficient conditionsrespectively for the boundedness of the singular integral operator on theweighted Morrey spaces. We observe the phenomenon unique to the case of Morreyspaces; the $A_q$-theory by Muckenhoupt and Wheeden does not suffice. We alsodiscuss the boundedness of the commutators. The difference between the maximaloperator and the singular integral operators will be clarified. Further, theproperty of the sharp maximal operator is investigated.

The singular integral operator and its commutator on weighted Morrey spaces