An elementary proof of the $A_2$ Bound

TL;DR

This paper provides an elementary proof of the $A_2$ bound for martingale transforms, establishing sharp weighted inequalities through a simple method that extends to vector-valued and continuous cases.

Contribution

It introduces a straightforward proof technique for $A_2$ bounds, simplifying previous complex arguments and extending results to broader settings.

Findings

Martingale transforms are dominated by positive sparse operators.

The method yields sharp $A_p$ bounds for martingale transforms.

The proof extends to vector-valued and continuous cases.

Abstract

A martingale transform $ T$, applied to an integrable locally supported function $ f$, is pointwise dominated by a positive sparse operator applied to $ \lvert f\rvert $, the choice of sparse operator being a function of $ T$ and $ f$. As a corollary, one derives the sharp $ A_p$ bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-$L^1$ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the $ A_2$ bounds in that setting.