## Abstract

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1, 1)-Poincaré inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball.

Original language | English |
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Pages (from-to) | 497-530 |

Number of pages | 34 |

Journal | Revista Matematica Iberoamericana |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Dirichlet form
- Doubling measure
- Functions of bounded variation
- Gauss-Green theorem
- Green function
- Harmonic function
- Inequality
- Metric space
- Minkowski content
- Newtonian space
- Perimeter measure
- Poincaré
- Singular function