## Abstract

Let φ: [ 0, ∞) → [ 0, ∞) be a Young's function satisfying the Δ_{2} -condition and let M_{B} be the geometric maximal operator associated to a homothecy invariant basis B acting on measurable functions on R^{n}. Let Q be the unit cube in R^{n} and let L^{φ}(Q) be the Orlicz space associated to φ with the norm given by {equation presented} We show that M_{B} satisfies the weak type estimate {equation presented} for all measurable functions f on R^{n} and α > 0 if and only if M_{B} satisfies the weak type estimate {equation presented} for all measurable functions f supported on Q and α > 0. As a consequence of this equivalence, we prove that if φ satisfies the above conditions and B is a homothecy invariant basis differentiating integrals of all measurable functions f on R_{n} such that {equation presented}, then the associated maximal operator M_{B} satisfies both of the above weak type estimates.

Original language | English |
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Pages (from-to) | 607-613 |

Number of pages | 7 |

Journal | Georgian Mathematical Journal |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2024 |

## Keywords

- Weak-type inequality
- differentiation of integrals
- maximal operator