The A_2 theorem: Remarks and complements

TL;DR

This paper surveys various approaches to the $A_2$ theorem, introduces new streamlined results and proofs, and discusses the sharpness of bounds and testing conditions in harmonic analysis.

Contribution

It provides a mini-survey biased towards the corona approach, introduces a streamlined Lerner's local oscillation formula, and offers new proofs and results on $A_p$--$A_inf$ testing and two-weight $T1$ theorems.

Findings

Streamlined form of Lerner's local oscillation formula

Sharpness of linear-in-complexity weak $(1,1)$ bound for dyadic shifts

New proofs of $A_p$--$A_inf$ testing conditions and two-weight $T1$ theorem

Abstract

I give a mini-survey of several approaches to the $A_2$ theorem, biased towards the "corona" rather than the "Bellman" side of the coin. There are two new results (a streamlined form of Lerner's local oscillation formula, and the sharpness of the linear-in-complexity weak $(1,1)$ bound for dyadic shifts) and two new proofs of known results (the $A_p$--$A_\infty$ testing conditions, and the two-weight $T1$ theorem for positive dyadic operators).