A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L²(u) to L²(v) in terms of the A₂ condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1_B taken as the indicator of B ‖T(u1_B)‖_L²(B,v)≤T‖1_B‖_L²(u), ‖T^⁎(v1_B)‖_L²(B,u)≤T‖1_B‖_L²(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.