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

Abstract

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 1B taken as the indicator of B‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u),‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.