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 L 2 ( u ) to L 2 ( v ) in terms of the A 2 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 ( u 1 B ) ‖ L 2 ( B , v ) ≤ T ‖ 1 B ‖ L 2 ( u ) , ‖ T ⁎ ( v 1 B ) ‖ L 2 ( B , u ) ≤ T ‖ 1 B ‖ L 2 ( v ) . The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.