integers $ k$, and all functions $ f, g$ supported on an interval $ I$ of length $ 2 ^{k}$, we have $$\label{e:calmost}
\lvert  \langle   T_k f, g \rangle\rvert \lesssim 
2 ^{-  \eta k   }  \langle f  \rangle _{I,r} \langle g \rangle _{I,r} \lvert  I\rvert .$$

This follows from Lemma \[l:almost\] by interpolation. The relevant interpolation parameter $ \theta _0$ at which we have only an epsilon loss in the interpolated estimate is given by $$\begin{aligned}
(1-\theta_0)\alpha & = \theta_0 \frac {1- \alpha } 2 , 
\\
\textup{and then} \quad 
\frac 1 {r_0} &= \frac {1-\theta _0} 1 + \frac {\theta _0} 2 . \end{aligned}$$ We see that $ r_0 = 1+ \alpha $. And so we conclude that for $ r_0 = 1+ \alpha < r < 2$, we have the required gain in the interpolated bound, which proves the Corollary.

We now turn to the weighted inequalities of Corollary \[c:R\].

For the deterministic Hilbert transform, we have the sharp bound of Petermichl [@MR2354322], namely $$\lVert H \;:\; \ell ^{p} (w) \mapsto \ell ^{p} (w)\rVert \lesssim [w] _{A_p} ^{ \max \{1, \frac 1 {p-1}\}}.$$ So, it remains to bound the terms in . By Proposition \[p:scale\], we then need 