### Abstract

Kinnunen and Lindqvist proved in [9] that the local Hardy-Littlewood maximal function, defined in an open set Ω in the Euclidean space, is a bounded operator from W^{1,p}(Ω) to W^{1,p} (Ω), provided p > 1. In this note we give a simpler argument which leads to a slightly more general result. Furthermore, the argument also allows us to prove boundedness of the spherical maximal function in the Sobolev space W ^{1,p}(Ω), p > n/(n - 1). We also raise many questions concerning boundedness of maximal operators in Sobolev spaces.

Original language | English (US) |
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Pages (from-to) | 167-176 |

Number of pages | 10 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 29 |

Issue number | 1 |

State | Published - May 10 2004 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Hajłasz, P., & Onninen, J. (2004). On boundedness of maximal functions in Sobolev spaces.

*Annales Academiae Scientiarum Fennicae Mathematica*,*29*(1), 167-176.