  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.

Classical Analysis and ODEs