On the sharpness of some quantitative Muckenhoupt-Wheeden inequalities
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866916204020498432 |
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| author | Lerner, Andrei Li, Kangwei Ombrosi, Sheldy Rivera-Ríos, Israel P. |
| author_facet | Lerner, Andrei Li, Kangwei Ombrosi, Sheldy Rivera-Ríos, Israel P. |
| contents | In a recent work by Cruz-Uribe et al. was obtained that \[|\{x\in{\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>α\}|\lesssim\frac{[w]_{A_1}^2}α\int_{{\mathbb{R}^d}}|f|dx\] 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 | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06718 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the sharpness of some quantitative Muckenhoupt-Wheeden inequalities Lerner, Andrei Li, Kangwei Ombrosi, Sheldy Rivera-Ríos, Israel P. Classical Analysis and ODEs Functional Analysis In a recent work by Cruz-Uribe et al. was obtained that \[|\{x\in{\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>α\}|\lesssim\frac{[w]_{A_1}^2}α\int_{{\mathbb{R}^d}}|f|dx\] 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. |
| title | On the sharpness of some quantitative Muckenhoupt-Wheeden inequalities |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2310.06718 |