On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities

Andrei K. Lerner; Kangwei Li; Sheldy Ombrosi; Israel P. Rivera-Ríos

doi:10.5802/crmath.638

Article de recherche - Analyse harmonique

On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities
[Optimalité des inégalités quantitatives de Muckenhoupt–Wheeden]

Andrei K. Lerner ¹ ; Kangwei Li ² ; Sheldy Ombrosi ^3,⁴ ; Israel P. Rivera-Ríos ⁵

¹ Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel
² Center for Applied Mathematics, Tianjin University, Weijin Road 92, 300072 Tianjin, China
³ Departamento de Análisis Matemático y Matemática Aplicada Universidad Complutense, Spain
⁴ Departamento de Matemática e Instituto de Matemática. Universidad Nacional del Sur - CONICET Argentina
⁵ Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada. Facultad de Ciencias. Universidad de Málaga (Málaga, Spain).

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1253-1260.

Résumé
Abstract

Dans le récent travail [Cruz-Uribe et al. (2021)], il a été démontré

| {x \in ℝ^{d} : w (x) | G (f w^{- 1}) (x) | > α} | ≲ \frac{{[w]}_{A_{1}}^{2}}{α} \int_{ℝ^{d}} | f | d x

à la fois dans les contextes matriciel et scalaire, où $G$ est soit la fonction maximale de Hardy-Littlewood ou tout opérateur de Calderón-Zygmund. Dans cette note, nous démontrons que la dépendance quadratique par rapport à ${[w]}_{A_{1}}$ est optimale. Cela est réalisé en construisant une séquence de poids à valeurs scalaires avec des caractéristiques d’éclatements, de sorte que les bornes correspondantes à la transformation de Hilbert et la fonction maximale soient exactement quadratiques.

In the recent work [Cruz-Uribe et al. (2021)] it was obtained that

| {x \in ℝ^{d} : w (x) | G (f w^{- 1}) (x) | > α} | ≲ \frac{{[w]}_{A_{1}}^{2}}{α} \int_{ℝ^{d}} | f | d x

both in the matrix and scalar settings, where $G$ is either the Hardy–Littlewood maximal function or any Calderón–Zygmund operator. In this note we show that the quadratic dependence on ${[w]}_{A_{1}}$ is sharp. This is done by constructing a sequence of scalar-valued weights with blowing up characteristics so that the corresponding bounds for the Hilbert transform and maximal function are exactly quadratic.

Reçu le : 2024-03-12
Accepté le : 2024-04-14
Publié le : 2024-11-05

DOI : 10.5802/crmath.638

Classification : 42B20, 42B25
Keywords: Matrix weights, quantitative bounds, endpoint estimates
Mot clés : Poids matriciel, borne quantitative, estimations de type faible $(1,1)$

Affiliations des auteurs :

Andrei K. Lerner ¹ ; Kangwei Li ² ; Sheldy Ombrosi ^{3, 4} ; Israel P. Rivera-Ríos ⁵

¹ Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel
² Center for Applied Mathematics, Tianjin University, Weijin Road 92, 300072 Tianjin, China
³ Departamento de Análisis Matemático y Matemática Aplicada Universidad Complutense, Spain
⁴ Departamento de Matemática e Instituto de Matemática. Universidad Nacional del Sur - CONICET Argentina
⁵ Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada. Facultad de Ciencias. Universidad de Málaga (Málaga, Spain).

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Andrei K. Lerner and Kangwei Li and Sheldy Ombrosi and Israel P. Rivera-R{\'\i}os},
     title = {On the sharpness of some quantitative {Muckenhoupt{\textendash}Wheeden} inequalities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1253--1260},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2024},
     doi = {10.5802/crmath.638},
     language = {en},
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Andrei K. Lerner; Kangwei Li; Sheldy Ombrosi; Israel P. Rivera-Ríos. On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1253-1260. doi : 10.5802/crmath.638. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.638/

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