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

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse H"older Inequality for $A_{\mathrm{\infty }}$ weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of Coifman-Fefferman type. $\mathrm{\parallel }Tf{\mathrm{\parallel }}_{L^p(w)}\le c_{n,w,p}\mathrm{\parallel }Sf{\mathrm{\parallel }}_{L^p(w)},$ 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 $A_{\mathrm{\infty }}$ constant $[w{]}_{A_{\mathrm{\infty }}}:={\sup }_Q\frac{1}{w(Q)}{\int }_{Q}M(w{\chi }_Q)\mathrm{.}$ We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. We obtain mixed $A_1$–$A_{\mathrm{\infty }}$ estimates for the commutator $[b,T]$ and for its higher order analogue $T_b^k$. A common ingredient in the proofs presented here is a recent improvement of the Reverse H"older Inequality for $A_{\mathrm{\infty }}$ weights involving Wilson’s constant.