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 $BMO(\mu )$.

14 pages