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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.638/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |