### Abstract

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_{p}(R^{d}) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^{d} > 1) on the variable Lebesgue spaces L_{p}(·)(R^{d}), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_{s} ^{γδ}, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_{s} ^{γδ} is bounded on the space L_{p}(·)(R^{d}) (or the maximal operator is of weak-type (p(·), p(·))).

Original language | English |
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Pages (from-to) | 1079-1101 |

Number of pages | 23 |

Journal | Czechoslovak Mathematical Journal |

Volume | 66 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2016 |

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### Keywords

- Besicovitch’s covering theorem
- maximal operator
- strong-type inequality
- variable Lebesgue space
- weak-type inequality
- γ-rectangle

### ASJC Scopus subject areas

- Mathematics(all)