On the sharpness of some quantitative Muckenhoupt–Wheeden inequalities

Lerner, Andrei K.; Li, Kangwei; Ombrosi, Sheldy; Rivera-Ríos, Israel P.

Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1253-1260

In the recent work [Cruz-Uribe et al. (2021)] it was obtained that

$$\left|\right\{x\in {ℝ}^d:w\left(x\right)|G\left(f{w}^{-1}\right)\left(x\right)|>\alpha \}|\lesssim \frac{{\left[w\right]}_{A_1}^2}{\alpha }{\int }_{{ℝ}^d}\left|f\right|\mathrm{d}x$$

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 ${\left[w\right]}_{A_1}$ 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

$$\left|\right\{x\in {ℝ}^d:w\left(x\right)|G\left(f{w}^{-1}\right)\left(x\right)|>\alpha \}|\lesssim \frac{{\left[w\right]}_{A_1}^2}{\alpha }{\int }_{{ℝ}^d}\left|f\right|\mathrm{d}x$$

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 ${\left[w\right]}_{A_1}$ 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.

Classification : 42B20, 42B25
Keywords: Matrix weights, quantitative bounds, endpoint estimates
Mots-clés : Poids matriciel, borne quantitative, estimations de type faible $(1,1)$

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