Improving bounds for singular operators via Sharp Reverse Hölder Inequality for $A_{\infty }$

Abstract

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for $A_{\infty }$ weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of Coifman-Fefferman type. $\Vert Tf{\Vert }_{L^p(w)}\le c_{n,w,p}\Vert Sf{\Vert }_{L^p(w)},$ 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 $T_b^k$. A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for $A_{\infty }$ weights involving Wilson's constant.

References in corpus (1)

• Quadratic $A_1$ bounds for commutators of singular integrals with BMO functions