## Abstract

Let B be a homothecy invariant collection of convex sets in ℝ^{n}. Given a measure μ, the associated weighted geometric maximal operatorM_{B,μ} is defined by (Formula presented) It is shown that, provided μ satisfies an appropriate doubling condition with respect to B and ν is an arbitrary locally finite measure, the maximal operator M_{B,μ} is bounded on Lp(ν) for sufficiently large p if and only if it satisfies a Tauberian condition of the form (Formula presented) As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights A_{∞,B} for homothecy invariant Muckenhoupt bases B consisting of convex sets. Moreover, it is immediately seen that the strong maximal function M_{R,μ}, defined with respect to a product-doubling measure μ, is bounded on Lp(ν) for some p > 1 if and only if (Formula presented) holds for all ν-measurable sets E in ℝ^{n}. In addition, we discuss applications in differentiation theory, in particular proving that a μ-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate L∞(ν).

Original language | English |
---|---|

Pages (from-to) | 7999-8032 |

Number of pages | 34 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1 Nov 2015 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Automorphism
- Muckenhoupt weight
- Strong maximal function
- Tauberian condition