Improving Bounds for Singular Operators via Sharp Reverse Höolder 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∞ weights. For two given operators T and S, we study L p (w) bounds of Coifman-Fefferman type:, 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∞ constantWe 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∞ 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∞ weights involving Wilson's constant from [10].

Quantitative matrix weighted estimates for certain singular integral operators

“…Note that a quantitative version with c w = [w] A∞ was obtained in [38]. In the case of rough singular integrals the corresponding quantitative counterpart was settled in [32] and for L r ′ -Hörmander operators, for instance in [18].…”

Borderline Weighted Estimates for Commutators of Fractional Integrals

“…This lemma can be obtained with small and straightforward modifications in the proof of Lemma 3.1 in [21]. 15,16]). Let 0 < p < ∞, 0 < δ < 1 and w ∈ A ∞ , M d denotes the dyadic maximal function.…”

A lower bound for $A_p$ exponents for some weighted weak-type inequalities

“…These type of estimates in this context go back to the work of R. Bagby and D. Kurtz in the mid 80s (see [3] and [4]). The proof of the Lemma can be found in [16] in the context of A p weights and in [15] in the context of A ∞ weights. As a consequence we have the following.…”