The Hölder regularity of harmonic function on bounded and unbounded p.c.f self-similar sets

In this paper, we prove a generalized reverse Hölder inequality of harmonic functions on cable systems induced by post-critically finite (p.c.f.) self-similar sets. Furthermore, we also establish the Hölder regularity of harmonic functions on both bounded and unbounded p.c.f. self-similar sets, which does not involve heat kernel estimates and resistance estimates.