Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

Abstract

Given 1 ≤ q < p < ∞, quantitative weighted L p estimates, in terms of Aq weights, for vector valued maximal functions, Calderón-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular operators will rely upon suitable convex body domination results, which in the case of commutators will be provided in this work, obtaining as a byproduct a new proof for the scalar case as well.

Quantitative matrix weighted estimates for certain singular integral operators

“…A 1 and A q estimates. Our results here are the counterparts for rough singular integrals and L r ′ -Hörmander operators of the results obtained in [21]. In these cases we recover as well the optimal estimates known in the scalar case (see [32]).…”

“…Convex body domination was introduced by Nazarov, Petermichl, Treil and Volberg who settled in [36] a "pointwise" domination result for Calderón-Zygmund operators (see [8] for a "bilinear" version of that result). Those techniques where also explored for commutators in [7,21,20] and the idea of relying upon convex bodies to control maximal rough singular integrals was exploited by Di Plinio, Hytönen and Li [9]. We shall begin borrowing some definitions from the latter.…”

“…In these cases we recover as well the optimal estimates known in the scalar case (see [32]). We remit the interested reader to [21], to read about further references about the motivation to study this kind of estimates.…”

“…The following Lemma can be derived from the arguments given for the proof of Lemma 2 in [21]. We will provide a proof here for reader's convenience.…”

“…In the case of estimates in terms of the A 1 constant, the best dependences have been retrieved for several operators in [21].…”