Title: The two-weight inequality for Calder\’on-Zygmund operators with applications and results on two weight commutators of maximal functions on spaces of homogeneous type.

Abstract: For (X,d,w) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and w is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,w ).  Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon–Zygmund operator T from L^{2}(u) to L^{2}(v) in terms of the A_{2} condition and two testing conditions. The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

We also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.