### Abstract

We study bounded p-harmonic functions u defined on a directed tree T with branching order k(1 < p < ∞ and K = 2, 3, . . .). Denote by BV(u) the set of paths on which u has finite variation and ℱ (u) the set of paths on which u has a finite limit. Then the infimum of dim BV(u) and the infimum of dim ℱ (u) are equal over all bounded p-harmonic functions on T (with p and k fixed); the infimum d(k, p) is attained and is strictly between 0 and 1 expect when p = 2 or k = 2.

Original language | English (US) |
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Pages (from-to) | 1138-1148 |

Number of pages | 11 |

Journal | Annals of Probability |

Volume | 28 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2000 |

### Keywords

- Dimension
- Entropy
- Fatou theorem
- P-harmonic functions
- Trees

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Kaufman, R., & Wu, J. M. (2000). Fatou theorem of p-harmonic functions on trees.

*Annals of Probability*,*28*(3), 1138-1148. https://doi.org/10.1214/aop/1019160328