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

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-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.

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References in corpus (3)

• Sharp weighted estimates for classical operators
• Non-probabilistic proof of the A_2 theorem, and sharp weighted bounds for the q-variation of singular integrals
• A pointwise estimate for positive dyadic shifts and some applications

Cited by in corpus (5)

• Bilinear Calderón-Zygmund theory on product spaces
• Off-diagonal estimates for bi-commutators
• Dyadic representation theorem using smooth wavelets with compact support
• Sharp $L^p$ estimates of powers of the complex Riesz transform
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting