# Hardy-Littlewood maximal theorem

There is a constant $K>0$ such that for each Lebesgue integrable^{} function $f\in {L}^{1}({\mathbb{R}}^{n})$, and each $t>0$,

$$m(\{x:Mf(x)>t\})\le \frac{K}{t}{\parallel f\parallel}_{1}=\frac{K}{t}{\int}_{{\mathbb{R}}^{n}}|f(x)|\mathit{d}x,$$ |

where $Mf$ is the Hardy-Littlewood maximal function of $f$.

Remark. The theorem holds for the constant $K={3}^{n}$.

Title | Hardy-Littlewood maximal theorem |
---|---|

Canonical name | HardyLittlewoodMaximalTheorem |

Date of creation | 2013-03-22 13:27:33 |

Last modified on | 2013-03-22 13:27:33 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 4 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 28A15 |

Classification | msc 28A25 |

Related topic | HardyLittlewoodMaximalOperator |

Defines | Hardy-Littlewood theorem |