> c [w] _{A_p} ^{-1} $, which then means that the choice of $ \eta = \eta (r)$ in is at least as big is $ c [w] _{A_p} ^{-1} $. Then, our bound is $$\label{e:sharp?}
\langle T_P ( \sigma  f), g w \rangle _{} \lesssim [w] ^{1+\frac 1p }  _{A_p}\lVert f\rVert _{L ^{p} (\sigma )} \lVert g\rVert _{L ^{p'} (w)}, \qquad 2 < p < \infty .$$ We have no reason to believe that this estimate is sharp.

Random Hilbert Transforms
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The discrete Hilbert transform $$H f (x) = \sum_{n\neq 0} \frac {f (x-n)} {n}$$ satisfies a sparse bound: For all finitely supported functions $ f$ and $ g$, there is a sparse operator $ \Lambda $ so that $$\label{e:dHS}
\lvert \langle H f, g \rangle\rvert \lesssim \Lambda _{1,1} (f,g).$$ This is a consequence of the main results of Theorem \[t:czo\]. Recalling the definition of $ H _{\alpha } $ in , we see that $ \mathbb E H_ \alpha  f  = Hf$, so it remains to consider the difference $$\begin{aligned}
 H_ \alpha  f (x) - H f (x) &:= \sum_{k=1} ^{\infty } 
 \sum_{ n \;:\; 2 ^{k-1} \leq \lvert  n\rvert < 2 ^{k} }  \frac { X_n 