A characterization of two-weighted inequalities for singular operators and their commutators in generalized weighted Morrey spaces on spaces of homogeneous type

In this paper we give a characterization of two-weighted inequalities for singular operators and their commutators in generalized weighted Morrey spaces on spaces of homogeneous type Mp,?? (X). We prove the boundedness of the Calder?n-Zygmund singular operators T and their commutators [b,T] from the spaces Mp,?1??1 (X)to the spaces Mp,?2??2 (X), where 1< p < ? ,0 < ? < 1 and (?1,?2) ? ?p (X). Finally we give some examples for singular integral operators on Mp,?? (X) as applications of our results.