Let
T
f
=
∑
I
ε
I
⟨
f
,
h
I
+
⟩
h
I
−
Tf =\sum _{I} \varepsilon _I \langle f,h_{I^+}\rangle h_{I^-}
. Here,
|
ε
I
|
=
1
\lvert \varepsilon _I\rvert =1
, and
h
J
h_J
is the Haar function defined on dyadic interval
J
J
. We show that, for instance,
‖
T
‖
L
2
(
w
)
→
L
2
(
w
)
≲
[
w
]
A
2
+
.
\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
A_2
characteristic for the weight
w
w
. This is an instance of a one-sided
A
2
A_2
conjecture. Our proof of this fact is difficult, as the very quick known proofs of the
A
2
A_2
theorem do not seem to apply in the one-sided setting.