A two weight inequality for Calderon-Zygmund operators on spaces of homogeneous type with applications

Let (X, d, mu) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and mu is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on ( X, d, mu). 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. For every cube B subset of X, we have the following testing conditions, with 1(B) taken as the indicator of B parallel to T(u1(B))parallel to L-2(B,v) <= T parallel to 1(B)parallel to L-2(u), parallel to T*(v1(B))parallel to L-2(B,u) <= T parallel to 1(B)parallel to 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. (C) 2021 Elsevier Inc. All rights reserved.