Fatou theorem of p-harmonic functions on trees

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 $\kappa = 2$.

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