terms on the right in . Now, it is well known [@MR3085756] that $$\Lambda _{1} (   f, g ) \lesssim 
[w] ^{\max \{ 1, \frac {1} {p-1} \}} _{A_p} \lVert f\rVert _{L ^{p} (w ) } \lVert g\rVert _{L ^{p'} (w)}.$$ Indeed, this is a key part of the proof of the $ A_2$ Theorem by sparse operators.

So, it remains to consider the second term on the right in . For each $ k \in \mathbb N $, we have by Proposition \[p:scale\], $ k\in \mathbb Z $, $$\sum_{ Q  \in \mathcal D\;:\; \lvert  Q\rvert  =  2 ^{nk} }  
\langle  f     \rangle _{Q,r} \langle g \rangle _{Q,r} \lvert  Q\rvert 
\lesssim 
 [w] _{A_p} ^{1/p} [w] _{RH_r} [ \sigma ] _{RH_r}
 \lVert f\rVert _{L ^{p} (w ) } \lVert g\rVert _{L ^{p'} (w)} .$$ As we recall in § \[s:wtd\], there is a $ r = r ([w] _{A_p}) >1$ so that $  [w] _{RH_r}  [ \sigma ] _{RH_r} < 4 $. And so the proof of the Corollary is complete.

Indeed, it is easy enough to make this step quantitative. For $ 2 < p < \infty $, the choice of $ r$ can be taken to satisfy $ r-1 