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 Coifman and Weiss, i.e. d is quasi metric on X μ positive measure satisfying doubling condition. Suppose that u v are two locally finite Borel measures (X,d,μ). Subject to pair weights side condition, we characterize boundedness Calderón–Zygmund operator T from L2(u) L2(v) terms A2 condition testing conditions. For every cube B⊂X, have following conditions, with 1B taken as indicator B‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u),‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes corona decompositions originating work Nazarov, Treil Volberg, along pivotal