Fatou theorem of p-harmonic functions on trees
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 $BV(u)$ …