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. © 2011 Elsevier Inc. All rights reserved