Sparse bounds for maximal rough singular integrals via the Fourier transform

Abstract

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Quantitative matrix weighted estimates for certain singular integral operators

“…Note that this convex body domination result was sketched in [9,Remark 6.6]. Here we provide a full proof of this result that has interest on its own.…”

“…In this paper our purpose is to provide strong type and and endpoint weak type quantitative estimates for vector valued extensions of rough singular integrals and L r ′ -Hörmander operators. Some results had already been obtained for maximal rough singular integrals in [9]. In the case of L r ′ -Hörmander operators we are not aware of any result in this direction.…”

“…A key ingredient in our proofs is convex body domination for all the aforementioned operators. We will as well provide convex body domination results for both L r ′ -Hörmander operators and rough singular integrals, relying upon ideas of [36,9,24,30].…”

“…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.…”

Sparse bounds for the bilinear spherical maximal function

“…It reduces the study of operators of various natures to simpler dyadic operators that are localized and of averaging type. Sparse domination has becoming a leading method in deducing sharp weighted norm inequalities (for operators such as Calderón-Zygmund operators [CAR16,Lac17,Ler13], rough singular integrals [CACDPO17,DPHL20], the spherical maximal function [Lac19], Bochner-Riesz multipliers [BBL17,LMR19], to name a few, and even non-integral operators [BFP16]). In addition to weighted estimates (which in particular includes the unweighted L p space estimates), sparse bounds are also known to imply (often sharp) weak type endpoint estimates.…”