A simple proof of the $A_2$ conjecture

TL;DR

This paper presents a straightforward proof of the $A_2$ conjecture, simplifying previous approaches by eliminating the need for Haar shift operators and utilizing local mean oscillation decomposition.

Contribution

The paper introduces a simplified proof of the $A_2$ conjecture that also provides a new proof of the two-weight conjecture, avoiding complex operator notions.

Findings

Simplified proof of the $A_2$ conjecture

Elimination of Haar shift operator in the proof

New proof of the two-weight conjecture

Abstract

We give a simple proof of the $A_2$ conjecture proved recently by T. Hyt\"onen. Our proof avoids completely the notion of the Haar shift operator, and it is based only on the "local mean oscillation decomposition". Also our proof yields a simple proof of the "two-weight conjecture" as well.