On weak weighted estimates of martingale transform

TL;DR

This paper investigates weak type estimates for singular operators using Bellman functions, disproving the longstanding $A_1$ conjecture of Muckenhoupt, thus advancing understanding in weighted inequalities.

Contribution

It provides the first disproof of the $A_1$ conjecture for weighted estimates of martingale transforms using Bellman function techniques.

Findings

Disproved the $A_1$ conjecture for martingale transforms

Established new weak type estimates for singular operators

Extended Bellman function methods to weighted inequalities

Abstract

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture of Muckenhoupt, which stayed open after Muckenhoupt--Wheeden's conjecture was disproved by Reguera--Thiele.