Representation of singular integrals by dyadic operators, and the A_2 theorem

TL;DR

This paper provides a clear proof of the $A_2$ theorem, demonstrating that singular integral operators in weighted $L^2(w)$ spaces satisfy sharp norm inequalities, using a probabilistic dyadic representation approach.

Contribution

It introduces a streamlined probabilistic dyadic representation proof of the $A_2$ theorem, offering structural insights and connections to the $T(1)$ theorem, with minimal probability assumptions.

Findings

Proves the $A_2$ theorem with a probabilistic dyadic approach.

Establishes connections between singular integrals and the $T(1)$ theorem.

Provides a structurally informative proof with broad applicability.

Abstract

This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. 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-Journ\'e $T(1)$ 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.