### Abstract

We show that if X is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of X with zero p-capacity are precisely the p-polar sets; that is, a relatively compact subset of a domain in X is of zero p-capacity if and only if there exists a p-superharmonic function whose set of singularities contains the given set. In addition, we prove that if X is a p-hyperbolic metric space, then the p-superharmonic function can be required to be p-superharmonic on the entire space X. We also study the the following question: If a set is of zero p-capacity, does there exist a p-superharmonic function whose set of singularities is precisely the given set?

Original language | English |
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Pages (from-to) | 11-37 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Minimizers
- Polar sets
- Variational integrals
- Zero capacity sets

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## Cite this

*Transactions of the American Mathematical Society*,

*358*(1), 11-37. https://doi.org/10.1090/S0002-9947-05-04085-7