w \rangle _{Q}}.$$

\[p:scale\] Fix an integer $ k$, and $ 1< r < 2$. We have the bound below for all $ w\in A_p$, where $ r \leq p \leq r' = \frac r {r-1}$. $$\label{e:scale}
\sum_{ Q \in \mathcal D \;:\; \lvert  Q\rvert ={2 ^{nk}} } 
\langle f \rangle _{Q,r} \langle g \rangle _{Q,r} \lvert  Q\rvert 
\lesssim [w] _{A_p} ^{1/p} [w] _{RH_r} [ \sigma ] _{RH_r} \lVert f\rVert _{L ^{p} (w)} \lVert g\rVert _{L ^{p'} (w)}$$ where $ \sigma = w ^{1 - p'}$ is the ‘dual’ weight to $ w$.

Let us recall these well known facts.

1.  We always have $ [w] _{A_p}, [w] _{RH_r} \geq 1$.

2.  For $ w\in A_p$ and $ \sigma = w ^{1-p'}$, the weight $ \sigma $ is locally finite, its ‘dual’ weight is $ w$, and $ [\sigma ] _{A _{p'} } = [w ] _{A_p} ^{p'-1}$.

3.  For every $ w \in A _{p}$ there is a $ r = r ([w] _{A_p}) >1$ so that $ w \in RH _{r}$. (In particular, we can take $ r$ so that $ r -1 \simeq [w] _{A_p} ^{-1} $, by [@MR3092729]\*[Thm 2.3]{}. )

4.  For every $ w \in 