to see that the hypotheses on $ w$, namely , imply that for some choice of $ r > 1+ \alpha $, we have $$\label{e:ww}
w \in A_p, \quad w \in RH _{r}, \quad  \sigma = w ^{1- p'} \in RH _{r}.$$

Recall that $ v \in A_q \cap RH _{s} $ if and only if $ v ^{s} \in A _{s(q-1)+1}$. Now, by assumption, $ w  ^{1+ \alpha }\in A _{ (1+ \alpha ) (p-1)+1}$. So, there is a $ t>1$ so that $  w  ^{t(1+ \alpha )}\in A _{ (1+ \alpha ) (p-1)+1}$, and the $ A_q$ classes increase in $ q$, so we conclude that $ w \in A_p \cap  RH _{r} $, for a $ r > 1+ \alpha $.

The second hypothesis is $ w  \in A _{1 + \frac 1 {(1+\alpha) (p'-1)}} $. This is equivalent to $$(w ^{(1-p')}) ^{ 1+\alpha } \in A _{ (1+\alpha ) (p'-1) +1 }.$$ Now, $ w ^{1- p'} = \sigma $ is the dual weight. So by the argument in the previous paragraph, $ \sigma \in RH _{r}$, for some $ r > 1+ \alpha $. So the proof is complete.

Sparse Bounds and Weighted Inequalities {#s:wtd}
