Borderline weak‐type estimates for singular integrals and square functions

Domingo-Salazar, Carlos; Lacey, Michael T.; Rey, Guillermo

doi:10.1112/blms/bdv090

Cited by 40 publications

(45 citation statements)

References 19 publications

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“…Now we turn to our contribution. Our approach exploits sparse domination and ideas from [25] that can be traced back to [8]. In the case of commutators our approach is inspired by [23] as well.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A sparse approach to mixed weak type inequalities

Caldarelli

Rivera-Rı́os

2019

Math. Z.

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In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that uv ∈ A∞ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in [8] and extended in [23] and [25]Sawyer also conjectured that (1.1) should hold as well for the Hilbert transform. Cruz-Uribe, Martell and Pérez [7] generalized (1.1) to higher dimensions and actually proved that Sawyer's conjecture holds for Calderón-Zygmund operators via the following extrapolation argument.Theorem C. Assume that for every w ∈ A ∞ and some 0 < p < ∞,Then for every u ∈ A 1 and everyThe conditions on the weights in that extrapolation result lead them to conjecture that (1.1), and consequently the corresponding estimate for Calderón-Zygmund operators should hold as well with u ∈ A 1 and v ∈ A ∞ . That conjecture was positively answered recently in [24] where several quantitative estimates were provided as well. At this point we would like to mention, as well, a recent generalization provided for Orlicz maximal operators in [2].In [7], besides the aforementioned results, it was shown that (1.1) holds if u ∈ A 1 and v ∈ A ∞ (u) (see Section 2.2 for the precise definition of A p (u)). The advantage of that condition is that the product uv is an A ∞ weight. Over the past few years, there have been new contributions under those assumptions such as [3] for the case of fractional integrals and related operators, [28,29] for related quantitative estimates and [26] for multilinear extensions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The purpose of the following sections we will be settling (3.1) for the operators in the main theorems. To achieve in that task we will rely upon sparse domination results, and more in particular we will use suitable splittings of the sparse families involved in the spirit of [8,23,25].…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

A sparse approach to mixed weak type inequalities

Caldarelli

Rivera-Rı́os

2019

Math. Z.

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“…Endpoint estimates. In this subsection we present some quantitative endpoint estimates that can be obtained following ideas in [10,24]. For the sake of clarity in this case we will present different statements for T and T m b with m a positive integer.…”

Section: )mentioning

confidence: 99%

“…The proofs that we present in this section will follow the strategy outlined in [10] and generalized in [24]. Let A be a Young function satisfying…”

Section: Proofs Of Endpoint Estimatesmentioning

confidence: 99%

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Sparse and weighted estimates for generalized Hörmander operators and commutators

Ibañez-Firnkorn

Rivera-Rı́os

2019

Monatsh Math

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In this paper a pointwise sparse domination for generalized Hörmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in [24]. Relying upon that sparse domination a number of quantitative estimates are derived. Some of them are improvements and complementary results to those contained in a series of papers due to M. Lorente, J. M. Martell, C. Pérez, S. Riveros and A. de la Torre [30,29,28]. Also the quantitative endpoint estimates in [24] are extended to iterated commutators. Other results that are obtained in this work are some local exponential decay estimates for generalized Hörmander operators in the spirit of [34] and some negative results concerning Coifman-Fefferman estimates for a certain class of kernels satisfying particular generalized Hörmander conditions.2000 Mathematics Subject Classification. 42B20, 42B25.

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Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

Ibañez‐Firnkorn,

Rivera‐Ríos

2023

Mathematische Nachrichten

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In this paper, we provide mixed weak‐type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente et al. One of the main novelties is that, besides obtaining estimates in the Euclidean setting, results are provided as well in spaces of the homogeneous type, being the first mixed weak‐type estimates that we are aware of in that setting.

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Borderline weak‐type estimates for singular integrals and square functions

Cited by 40 publications

References 19 publications

A sparse approach to mixed weak type inequalities

A sparse approach to mixed weak type inequalities

Sparse and weighted estimates for generalized Hörmander operators and commutators

Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

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