Weighted Estimates for One Sided Martingale Transforms

Let $Tf =\sum_{ I} \varepsilon_I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $\lvert \varepsilon _I\rvert=1$, and $h_J$ is the Haar function defined on dyadic interval $J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} . \end{equation*} Above, we use the one sided $A_2$ characteristic for the weight $w$. This is an instance of a one sided $A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $A_2$ theorem do not seem to apply in the one sided setting.