A Bellman function counterexample to the $A_1$ conjecture: the blow-up of the weak norm estimates of weighted singular operators

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the $A_1$ conjecture, thus strengthening the counterexamples built by Reguera–Thiele. We show a certain logarithmic blow-up for the end-point estimates of important weighted singular operators.