  In this expository article we collect and discuss some recent results on
different consequences of a Sharp Reverse H\"older Inequality for $A_{\infty}$
weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of
Coifman-Fefferman type. $ \|Tf\|_{L^p(w)}\le c_{n,w,p} \|Sf\|_{L^p(w)}, $ that
can be understood as a way to control $T$ by $S$. We will focus on a
\emph{quantitative} analysis of the constants involved and show that we can
improve classical results regarding the dependence on the weight $w$ in terms
of Wilson's $A_{\infty}$ constant $
[w]_{A_{\infty}}:=\sup_Q\frac{1}{w(Q)}\int_Q M(w\chi_Q).$ We will also exhibit
recent improvements on the problem of finding sharp constants for weighted norm
inequalities involving several singular operators. We obtain mixed
$A_{1}$--$A_{\infty}$ estimates for the commutator $[b,T]$ and for its higher
order analogue $T^k_{b}$. A common ingredient in the proofs presented here is a
recent improvement of the Reverse H\"older Inequality for $A_{\infty}$ weights
involving Wilson's constant.
