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In this expository article we collect and discuss some recent results ondifferent 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 ofCoifman-Fefferman type. $ \|Tf\|_{L^p(w)}\le c_{n,w,p} \|Sf\|_{L^p(w)}, $ thatcan 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 canimprove classical results regarding the dependence on the weight $w$ in termsof Wilson's $A_{\infty}$ constant $[w]_{A_{\infty}}:=\sup_Q\frac{1}{w(Q)}\int_Q M(w\chi_Q).$ We will also exhibitrecent improvements on the problem of finding sharp constants for weighted norminequalities involving several singular operators. We obtain mixed$A_{1}$--$A_{\infty}$ estimates for the commutator $[b,T]$ and for its higherorder analogue $T^k_{b}$. A common ingredient in the proofs presented here is arecent improvement of the Reverse H\"older Inequality for $A_{\infty}$ weightsinvolving Wilson's constant.

Improving bounds for singular operators via Sharp Reverse H\"older Inequality for $A_{\infty}$