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

TL;DR

This paper explores recent advances in bounding singular operators using a sharp Reverse H"older Inequality for $A_{ olinebreak}_ olinebreak ext{infty}$ weights, emphasizing quantitative constants and sharp weighted inequalities.

Contribution

It provides improved bounds for singular operators, leveraging a refined Reverse H"older Inequality and analyzing constants in weighted norm inequalities, including for commutators.

Findings

Improved bounds for singular operators using $A_{ ext{infty}}$ weights.

Quantitative analysis of constants in weighted inequalities.

Sharp estimates for commutators and higher order operators.

Abstract

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.