Representation of singular integrals by dyadic operators, and the [formula omitted] theorem

Abstract

This exposition presents a self-contained proof of the A2 theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space L2(w). The strategy of the proof is a streamlined version of the author’s original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David–Journé T(1) theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided.