Improving Bounds for Singular Operators via Sharp Reverse Höolder Inequality for

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:

$$ \parallel T\;f \parallel_{L^p}(w)\;\leq \; c_{n,w,p}\parallel S\;f \parallel _{L^p}(w),$$

that can be understood as a way to control T by S.

We will focus on a

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

$$ [w]A_{\infty}\; := \; {\rm sup_Q}\frac{1}{w(Q)}\int_{Q}{M({w_\mathcal{X}}_Q)} .$$

We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators. In the same spirit as in [10], 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 Hölder Inequality for

$$ A_{\infty}$$

weights involving Wilson’s constant from [10].