Fatou theorem of p-harmonic functions on trees

We study bounded p-harmonic functions u defined on a directed tree T with branching order k(1 < p < ∞ and K = 2, 3, . . .). Denote by BV(u) the set of paths on which u has finite variation and 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 ℱ (u) are equal over all bounded p-harmonic functions on T (with p and k fixed); the infimum d(k, p) is attained and is strictly between 0 and 1 expect when p = 2 or k = 2.