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

Abstract: 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 Calder\'{o}n--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 $B\subset X$, we have the following testing conditions, with $\mathbf{1}{B}$ taken as the indicator of $B$ \begin{equation*} \Vert T(u\mathbf{1}{B})\Vert {L^{2}(B, v)}\leq \mathcal{T}\Vert 1{B}\Vert {L^{2}(u)}, \end{equation*} \begin{equation*} \Vert T^{\ast }(v\mathbf{1}{B})\Vert {L^{2}(B, u)}\leq \mathcal{T}\Vert 1{B}\Vert _{L^{2}(v)}. \end{equation*} The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.