Fatou theorem of p-harmonic functions on trees
Robert Kaufman and Jang-Mei Wu
Source: Ann. Probab.
Volume 28, Number 3
(2000), 1138-1148.
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 $\kappa = 2$.
Primary Subjects: 31C20, 31C45
Secondary Subjects: 31A20, 60G42
Keywords: Fatou Theorem; trees; $p$-harmonic functions; dimension; entropy
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1019160328
Mathematical Reviews number (MathSciNet):
MR1797306
Digital Object Identifier: doi:10.1214/aop/1019160328
Zentralblatt MATH identifier:
01905943
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