## Abstract

Let P(ℝ^{n}) be the class of all exponents p for which the Hardy- Littlewood maximal operator M is bounded on L^{p}(·) (ℝ^{n}). A recent result by T. Kopaliani provides a characterization of P in terms of the Muckenhoupttype condition A under some restrictions on the behavior of p at infinity. We give a different proof of a slightly extended version of this result. Then we characterize a weak type (_{p}(·), _{p}(·)) property of M in terms of A for radially decreasing _{p}. Finally, we construct an example showing that _{p} ε P(ℝ^{n}) does not imply _{p}(·) - α ε P(ℝ^{n}) for all α < _{p-} - 1. Similarly, _{p} ε P(ℝ_{n}) does not imply α_{p}(·) ε P(ℝ^{n}) for all a > 1/_{p-}.

Original language | English |
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Pages (from-to) | 4229-4242 |

Number of pages | 14 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2010 |

## Keywords

- Maximal operator
- Variable L spaces

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