Fatou theorem of \(p\)-harmonic functions on trees.
The authors study bounded \(p\)-harmonic functions \(u\) defined on a directed tree \(T\) with branching order \(\kappa\) (\(1<p<\infty\) and \(\kappa=2,3,\dots\)). Denote by \(BV(u)\) the set of paths on which \(u\) has finite variation and \(\mathcal{F}(u)\) the set of paths on which \(u\) has a finite limit. Then the infimum of dim \(BV(u)\) and the infimum of dim \(\mathcal{F}(u)\) are equal over all bounded \(p\)-harmonic functions on \(T\) (with \(p\) and \(\kappa\) fixed); the infimum \(d(\kappa,p)\) is attained and is strictly between 0 and 1 except when \(p=2\) or \(\kappa=2\).
scientific article; zbMATH DE number 1905943
scientific article