Weighted Estimates for One Sided Martingale Transforms

Let $ Tf =sum_{ I} varepsilon_I langle f,h_{I^+}rangle h_{I^-}$. Here, $ lvert varepsilon _Irvert=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.