BMO spaces on weighted homogeneous trees

We consider an infinite homogeneous tree $\mathcal V$ endowed with the usual metric $d$ defined on graphs and a weighted measure $μ$. The metric measure space $(\mathcal V,d,μ)$ 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 $(\mathcal V,d,μ)$ and investigate some of its properties. We prove in particular that $BMO(μ)$ can be identified with the dual of a Hardy space $H^1(μ)$ introduced in a previous work and we investigate the sharp maximal function related with $BMO(μ)$.

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