Sharp $ A_2$ Inequality for Haar Shift Operators

TL;DR

This paper provides a new, flexible proof for the sharp A_2 inequality of Haar shift operators, including the Hilbert transform, using a corona decomposition and a two-weight T1 theorem, simplifying previous Bellman function methods.

Contribution

It introduces a novel proof technique for the A_2 inequality that applies uniformly to all Haar shifts, avoiding Bellman functions.

Findings

Proved the A_2 bound for Haar shift operators using corona decomposition.

Extended the proof to the Hilbert transform as a corollary.

Demonstrated the flexibility of the new proof approach.

Abstract

As a corollary to our main theorem we give a new proof of the result that the norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2 characteristic of a weight to the first power, a theorem of one of us. This new proof begins as the prior proofs do, by passing to Haar shifts. Then, we apply a deep two-weight T1 theorem of Nazarov-Treil-Volberg, to reduce the matter to checking a certain carleson measure condition. This condition is checked with a corona decomposition of the weight. Prior proofs of this type have used Bellman functions, while this proof is flexible enough to address all Haar shifts at the same time.