A T(1) Theorem for Singular Radon Transforms

The purpose of this paper is to extend and clarify the methods of the papers [G1] and [G2] by using the methods of [G3], incorporating ideas from Carnot-Caratheodory geometry such as those of the fundamental paper [NSW]. We will thereby prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe [DJ]. As in [G1] and [G2], we will associate a singular integral operator to a singular Radon transform. The crux of this paper consists of showing that the difference of the singular integral operator and the singular Radon transform is bounded on L, analogous to [G1] [G2] and their predecessors. Consequently, in some sense this T (1) theorem for singular Radon transforms follows from ”lifting” to the traditional T (1) theorem for the associated singular integral operator.