## Abstract

Let A∞+ denote the class of one-sided Muckenhoupt weights, namely all the weights w for which M^{+}: L^{p}(w) → L^{p} ^{,} ^{∞}(w) for some p> 1 , where M^{+} is the forward Hardy–Littlewood maximal operator. We show that w∈A∞+ if and only if there exist numerical constants γ∈ (0 , 1) and c> 0 such that (Formula presented.) for all measurable sets E⊂ R. Furthermore, letting (Formula presented.) Cw+(α):=sup0<w(E)<+∞1w(E)w({x∈R:M+1E(x)>α})we show that for all w∈A∞+ we have the asymptotic estimate Cw+(α)-1≲(1-α)1c[w]A∞+ for α sufficiently close to 1 and c> 0 a numerical constant, and that this estimate is best possible. We also show that the reverse Hölder inequality for one-sided Muckenhoupt weights, previously proved by Martín-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of A∞+. Our methods also allow us to show that a weight w∈A∞+ satisfies w∈Ap+ for all p>ec[w]A∞+.

Original language | English |
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Pages (from-to) | 151-161 |

Number of pages | 11 |

Journal | Collectanea Mathematica |

Volume | 69 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Muckenhoupt weight
- One sided maximal function
- Tauberian condition