## Abstract

Let w denote a weight in ℝ^{n} which belongs to the Muckenhoupt class A_{∞} and let M_{w} denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure w(x)dx. The sharp Tauberian constant of M_{w} with respect to α, denoted by C_{w}(α), is defined by (Formula presented.). In this paper, we show that the Solyanik estimate (Formula presented.) holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator M and a weight w: (Formula presented.). We show that we have (Formula presented.) if and only if w ∈ A_{∞}. As a corollary of our methods we obtain a quantitative embedding of A_{∞} into A_{p}.

Original language | English |
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Pages (from-to) | 924-946 |

Number of pages | 23 |

Journal | JOURNAL OF GEOMETRIC ANALYSIS |

Volume | 26 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2016 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Doubling measure
- Halo function
- Maximal function
- Muckenhoupt weights
- Tauberian conditions