we suspect that stronger results are possible. For instance, this Conjecture would imply nearly sharp $ A_p$ bounds, for all $ 1< p < 2$.

\[j:oscillatory\] For $ 1 < r < \infty $, the operator $ T_P$, where $ P = P (y)$ is of degree $ d$, for each bounded compactly supported function $ f$, there is a sparse operator $ \Lambda _{1,r}$ so that $$\lvert  \langle T _{P} f ,g  \rangle\rvert  \lesssim \Lambda _{1,r} (f,g).$$

It seems likely that the weak type argument of Chanillo and Christ [@MR883667] would establish the Conjecture for $ r=2$. Also see [@160901564].

We turn to weighted inequalities for *discrete random Hilbert transforms* acting on functions on $ \ell ^2 (\mathbb Z )$. Define a sequence of Bernoulli rvs $ \{ X _{n} \;:\; n\neq 0\}$ with $ \mathbb P (X_n =1) = \lvert  n\rvert ^{- \alpha } $, where $ 0 \leq \alpha < 1$. Then, the set $ \{n \;:\; X_n=1\}$ is a.s. infinite, by the Borel-Cantelli Lemma. Then, we consider the random Hilbert transform, and maximal function below. $$\begin{aligned}
  \label{e:Ha}
H _{\alpha } f (x)& = \sum_{n\neq 0}  \frac {X_n} {n ^{1- \alpha }} f (x-n). 
\\ \label{e:Ma}
M 