BMO Spaces on Weighted Homogeneous Trees

We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space (V, d, μ) 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(μ) on (V, d, μ) and investigate some of its properties. We prove in particular that BMO(μ) can be identified with the dual of a Hardy space H1(μ) introduced in a previous work and we investigate the sharp maximal function related with BMO(μ).