### Abstract

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided that ∫^{1}_{0}Θ ^{-1}(t^{1-s}h(t))dt< ∞ where Θ ^{-1} is the inverse of the function Θ(t)=Φ(t)/t, and s is the "upper dimension" of the metric measure space. This condition is a generalization of a well known condition in R ^{n} . For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.

Original language | English (US) |
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Pages (from-to) | 131-146 |

Number of pages | 16 |

Journal | Mathematische Zeitschrift |

Volume | 251 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2005 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Björn, J., & Onninen, J. (2005). Orlicz capacities and Hausdorff measures on metric spaces.

*Mathematische Zeitschrift*,*251*(1), 131-146. https://doi.org/10.1007/s00209-005-0792-y