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 μ 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 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 $B\subset X$, we have the following testing conditions, with ${\mathbf{1}}_B$ taken as the indicator of B${\Vert T(u{\mathbf{1}}_B)\Vert }_{L^2(B,v)}\le \mathcal{T}{\Vert 1_B\Vert }_{L^2(u)},$${\Vert T^{⁎}(v{\mathbf{1}}_B)\Vert }_{L^2(B,u)}\le \mathcal{T}{\Vert 1_B\Vert }_{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.

让 $(X,d,\mu )$是 Coifman 和 Weiss 意义上的齐次型空间，即d是X上的拟度量，μ是满足加倍条件的正测度。假设u和v是两个局部有限正 Borel 测度$(X,d,\mu )$. 根据满足边条件的权重对，我们描述了 Calderón-Zygmund 算子T的有界性，从${升}^2(你)$ 到 ${升}^2(v)$ 就 ${\mathrm{一种}}_2$条件和两个测试条件。对于每个立方体$乙\subset X$，我们有以下测试条件，其中 ${\mathbf{1}}_{乙}$作为B的指标${\Vert 吨(你{\mathbf{1}}_{乙})\Vert }_{{升}^2(乙,v)}\le \mathcal{吨}{\Vert 1_{乙}\Vert }_{{升}^2(你)},$${\Vert {吨}^{⁎}(v{\mathbf{1}}_{乙})\Vert }_{{升}^2(乙,你)}\le \mathcal{吨}{\Vert 1_{乙}\Vert }_{{升}^2(v)}.$该证明使用源自 Nazarov、Treil 和 Volberg 工作的停止立方体和电晕分解，以及关键的边条件。