A decomposition theorem for singular integral operators on spaces of homogeneous type

Let (X,d,μ) be a space of homogeneous type. Under the assumption μ({x})=0 for all x∈X, we prove a decomposition theorem for singular integral operators on (X,d,μ). Isotropic Haar expansion gives a representation of the integral operator as a series of simple shifts and rearrangements plus two paraproducts. This yields a UMD-valued T(1) theorem on spaces of homogeneous type.