## Abstract

Given any quasi-Banach function space X over ℝ^{n}, an index αX is defined that coincides with the upper Boyd index ᾱX when the space X is rearrangement-invariant. This new index is defined by means of the local maximal operator mλf. It is shown then that the Hardy-Littlewood maximal operator M is bounded on X if and only if αX < 1 providing an extension of the classical theorem of Lorentz and Shimogaki for rearrangement-invariant X. As an application, it is shown a new characterization of the Muckenhoupt A_{p} class of weights: u ∈ A_{p} if and only if for any ε > 0 there is a constant C such that for any cube Q and any measurable subset E ∪ Q, |E|/|Q|log^{ε} (|Q|/|E|) ≤ c (u(E)/u(Q))^{1/p} The case ε = 0 is false corresponding to the class A_{P,1}. Other applications are given, in particular within the context of the variable L^{p} spaces.

Original language | English |
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Pages (from-to) | 2697-2722 |

Number of pages | 26 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 6 |

DOIs | |

State | Published - 2007 |

## Keywords

- Maximal operators
- Muckenhoupt weights
- Rearrangement-invariant spaces

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