BMO Spaces on Weighted Homogeneous Trees

Abstract

We consider an infinite homogeneous tree $${\mathcal {V}}$$ endowed with the usual metric d defined on graphs and a weighted measure $$\mu $$ . The metric measure space $$({\mathcal {V}},d,\mu )$$ is nondoubling and of exponential growth, hence the classical theory of Hardy and BMO spaces does not apply in this setting. We introduce a space $$BMO(\mu )$$ on $$({\mathcal {V}},d,\mu )$$ and investigate some of its properties. We prove in particular that $$BMO(\mu )$$ can be identified with the dual of a Hardy space $$H^1(\mu )$$ introduced in a previous work and we investigate the sharp maximal function related with $$BMO(\mu )$$ .