On the sharpness of some quantitative Muckenhoupt-Wheeden inequalities
| Autor: | Lerner, Andrei, Li, Kangwei, Ombrosi, Sheldy, Rivera-Ríos, Israel P. |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In a recent work by Cruz-Uribe et al. was obtained that \[|\{x\in{\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>\alpha\}|\lesssim\frac{[w]_{A_1}^2}{\alpha}\int_{{\mathbb{R}^d}}|f|dx\] both in the matrix and scalar settings, where $G$ is either the Hardy-Littlewood maximal function or any Calder\'on-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. Comment: 8 pages, 3 figures. Introduction and title changed after referee's report. Main results remain the same. To appear in Comptes Rendus Math\'ematique |
| Databáze: | arXiv |
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