## The Annals of Probability

### Fatou theorem of p-harmonic functions on trees

#### Abstract

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

#### Article information

Source
Ann. Probab., Volume 28, Number 3 (2000), 1138-1148.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160328

Digital Object Identifier
doi:10.1214/aop/1019160328

Mathematical Reviews number (MathSciNet)
MR1797306

Zentralblatt MATH identifier
1038.31007

#### Citation

Kaufman, Robert; Wu, Jang-Mei. Fatou theorem of p -harmonic functions on trees. Ann. Probab. 28 (2000), no. 3, 1138--1148. doi:10.1214/aop/1019160328. https://projecteuclid.org/euclid.aop/1019160328

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