BMO spaces on weighted homogeneous trees

TL;DR

This paper develops a theory of BMO spaces on weighted homogeneous trees with exponential growth, establishing duality with Hardy spaces and analyzing maximal functions in this non-doubling setting.

Contribution

It introduces a BMO space on weighted homogeneous trees with exponential growth and proves its duality with a Hardy space, extending classical harmonic analysis to non-doubling spaces.

Findings

BMO() is identified as the dual of Hardy space H^1()

Properties of the sharp maximal function related to BMO() are investigated

The space  is non-doubling with exponential growth, challenging classical theory

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 $BMO(\mu)$.