A_p$, there is a $ r = r ([w] _{A_p}) >1$ so that $ w ^{r} \in A_p$.

5.  We have $ w \in A_p \cap RH_r $ if and only if $ w ^{r} \in A _{r (p-1)+1}$, by [@MR1018575].

This inequality is rephrased in the self-dual way, namely setting $ \sigma = w ^{1-p'}$, it is equivalent to show that for $ k\in \mathbb Z $, $$\label{e:ss} 
\sum_{ \substack{Q  \in \mathcal D \\ \lvert  Q\rvert  =  2 ^{nk} }  }
\langle  f    \sigma  \rangle _{Q,r} \langle g w  \rangle _{Q,r} \lvert  Q\rvert 
\lesssim 
[w] ^{ \frac1p } _{A_p}  [ \sigma ] _{RH_r} [ w ] _{RH_r} \lVert f\rVert _{L ^{p} (\sigma ) } \lVert g\rVert _{L ^{p'} (w)}.$$

Fix the integer $ k$. We can assume that for $ \lvert  Q\rvert= 2 ^{nk} $, if $ f$ is not zero on $ Q$, then $ f \mathbf 1_{3Q \setminus Q} \equiv 0$, and we assume the same for $ g$. Then, set $$f' =  \sum_{ Q  \in \mathcal D\;:\; \lvert  Q\rvert  =  2 ^{nk} }    \mathbf 1_{Q}  
\Bigl[
\frac 1 { \sigma (Q) } \int _{Q}  \lvert  f\rvert ^{r}  \; d \sigma 
\Bigr] ^{1/r}$$ and likewise for $ 