On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities
In the recent work [Cruz-Uribe et al. (2021)] it was obtained that \[ |\lbrace x\in {\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>\alpha \rbrace |\lesssim \frac{[w]_{A_1}^2}{\alpha }\int _{{\mathbb{R}^d}}|f|\,\mathrm{d} x \] both in the matrix and scalar settings, where $G$ is either the Hardy–Littlewood ma...
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| Language: | English |
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.638/ |
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| author | Lerner, Andrei K. Li, Kangwei Ombrosi, Sheldy Rivera-Ríos, Israel P. |
| author_facet | Lerner, Andrei K. Li, Kangwei Ombrosi, Sheldy Rivera-Ríos, Israel P. |
| author_sort | Lerner, Andrei K. |
| collection | DOAJ |
| description | In the recent work [Cruz-Uribe et al. (2021)] it was obtained that
\[ |\lbrace x\in {\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>\alpha \rbrace |\lesssim \frac{[w]_{A_1}^2}{\alpha }\int _{{\mathbb{R}^d}}|f|\,\mathrm{d} x \]
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. |
| format | Article |
| id | doaj-art-b0bf075f1f6c4fb49fa01a3af009c082 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-b0bf075f1f6c4fb49fa01a3af009c0822025-02-07T11:23:31ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G101253126010.5802/crmath.63810.5802/crmath.638On the sharpness of some quantitative Muckenhoupt–Wheeden inequalitiesLerner, Andrei K.0Li, Kangwei1Ombrosi, Sheldy2Rivera-Ríos, Israel P.3Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, IsraelCenter for Applied Mathematics, Tianjin University, Weijin Road 92, 300072 Tianjin, ChinaDepartamento de Análisis Matemático y Matemática Aplicada Universidad Complutense, Spain; Departamento de Matemática e Instituto de Matemática. Universidad Nacional del Sur - CONICET ArgentinaDepartamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada. Facultad de Ciencias. Universidad de Málaga (Málaga, Spain).In the recent work [Cruz-Uribe et al. (2021)] it was obtained that \[ |\lbrace x\in {\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>\alpha \rbrace |\lesssim \frac{[w]_{A_1}^2}{\alpha }\int _{{\mathbb{R}^d}}|f|\,\mathrm{d} x \] 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.638/Matrix weightsquantitative boundsendpoint estimates |
| spellingShingle | Lerner, Andrei K. Li, Kangwei Ombrosi, Sheldy Rivera-Ríos, Israel P. On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities Comptes Rendus. Mathématique Matrix weights quantitative bounds endpoint estimates |
| title | On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities |
| title_full | On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities |
| title_fullStr | On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities |
| title_full_unstemmed | On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities |
| title_short | On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities |
| title_sort | on the sharpness of some quantitative muckenhoupt wheeden inequalities |
| topic | Matrix weights quantitative bounds endpoint estimates |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.638/ |
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