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.

Abstract (VO)
Résumé

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.

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.

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

Zbl

DOI : 10.5802/crmath.638

Classification : 42B20, 42B25
Keywords: Matrix weights, quantitative bounds, endpoint estimates
Mots-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

@article{CRMATH_2024__362_G10_1253_0,
     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},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.638},
     zbl = {07939456},
     language = {en},
}

TY  - JOUR
AU  - Andrei K. Lerner
AU  - Kangwei Li
AU  - Sheldy Ombrosi
AU  - Israel P. Rivera-Ríos
TI  - On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 1253
EP  - 1260
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.638
LA  - en
ID  - CRMATH_2024__362_G10_1253_0
ER  -

%0 Journal Article
%A Andrei K. Lerner
%A Kangwei Li
%A Sheldy Ombrosi
%A Israel P. Rivera-Ríos
%T On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities
%J Comptes Rendus. Mathématique
%D 2024
%P 1253-1260
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.638
%G en
%F CRMATH_2024__362_G10_1253_0

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/

Bibliographie
Cité par

[1] Marcela Caldarelli; Israel P. Rivera-Ríos A sparse approach to mixed weak type inequalities, Math. Z., Volume 296 (2020) no. 1-2, pp. 787-812 | DOI | MR | Zbl

[2] David Cruz-Uribe; Joshua Isralowitz; Kabe Moen; Sandra Pott; Israel P. Rivera-Ríos Weak endpoint bounds for matrix weights, Rev. Mat. Iberoam., Volume 37 (2021) no. 4, pp. 1513-1538 | DOI | MR | Zbl

[3] David Cruz-Uribe; José M. Martell; Carlos Pérez Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., Volume 2005 (2005) no. 30, pp. 1849-1871 | DOI | MR | Zbl

[4] Komla Domelevo; Stefanie Petermichl; Sergei Treil; Alexander Volberg The matrix $A_{2}$ conjecture fails, i.e. $3 / 2 > 1$ (2024)

[5] Tuomas P. Hytönen; Stefanie Petermichl; Alexander Volberg The sharp square function estimate with matrix weight, Discrete Anal., Volume 2019 (2019), 2, 8 pages | DOI | MR | Zbl

[6] Tuomas P. Hytönen The sharp weighted bound for general Calderón–Zygmund operators, Ann. Math., Volume 175 (2012) no. 3, pp. 1473-1506 | DOI | MR | Zbl

[7] Joshua Isralowitz; Kabe Moen Matrix weighted Poincaré inequalities and applications to degenerate elliptic systems, Indiana Univ. Math. J., Volume 68 (2019) no. 5, pp. 1327-1377 | DOI | MR | Zbl

[8] Joshua Isralowitz; Sandra Pott; Israel P. Rivera-Ríos Sharp $A_{1}$ weighted estimates for vector-valued operators, J. Geom. Anal., Volume 31 (2021) no. 3, pp. 3085-3116 | DOI | MR | Zbl

[9] Kangwei Li; Sheldy Ombrosi; Carlos Pérez Proof of an extension of E. Sawyer’s conjecture about weighted mixed weak-type estimates, Math. Ann., Volume 374 (2019) no. 1-2, pp. 907-929 | DOI | MR | Zbl

[10] Benjamin Muckenhoupt; Richard L. Wheeden Some weighted weak-type inequalities for the Hardy–Littlewood maximal function and the Hilbert transform, Indiana Univ. Math. J., Volume 26 (1977) no. 5, pp. 801-816 | DOI | MR | Zbl

[11] Fedor Nazarov; Stefanie Petermichl; Sergei Treil; Alexander Volberg Convex body domination and weighted estimates with matrix weights, Adv. Math., Volume 318 (2017), pp. 279-306 | DOI | MR | Zbl

[12] Sheldy Ombrosi; Carlos Pérez; Jorgelina Recchi Quantitative weighted mixed weak-type inequalities for classical operators, Indiana Univ. Math. J., Volume 65 (2016) no. 2, pp. 615-640 | DOI | MR | Zbl

[13] Eric Sawyer A weighted weak type inequality for the maximal function, Proc. Am. Math. Soc., Volume 93 (1985) no. 4, pp. 610-614 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique