# Hardy-Littlewood maximal operator

The *Hardy-Littlewood maximal operator* in ${\mathbb{R}}^{n}$ is an operator defined on ${L}_{\text{loc}}^{1}({\mathbb{R}}^{n})$ (the space of locally integrable functions in ${\mathbb{R}}^{n}$ with the Lebesgue measure^{}) which maps each locally integrable function $f$ to another function $Mf$, defined for each $x\in {\mathbb{R}}^{n}$ by

$$Mf(x)=\underset{Q}{sup}\frac{1}{m(Q)}{\int}_{Q}|f(y)|\mathit{d}y,$$ |

where the supremum is taken over all cubes $Q$ containing $x$.
This function is lower semicontinuous (and hence measurable), and it is called the *Hardy-Littlewood maximal function* of $f$.

The operator $M$ is sublinear, which means that

$$M(af+bg)\le |a|Mf+|b|Mg$$ |

for each pair of locally integrable functions $f,g$ and scalars $a,b$.

Title | Hardy-Littlewood maximal operator |
---|---|

Canonical name | HardyLittlewoodMaximalOperator |

Date of creation | 2013-03-22 13:27:30 |

Last modified on | 2013-03-22 13:27:30 |

Owner | azdbacks4234 (14155) |

Last modified by | azdbacks4234 (14155) |

Numerical id | 8 |

Author | azdbacks4234 (14155) |

Entry type | Definition |

Classification | msc 28A25 |

Classification | msc 28A15 |

Related topic | HardyLittlewoodMaximalTheorem |

Defines | Hardy-Littlewood maximal function |