$H^1$ and BMO spaces for exponentially decreasing measures on homogeneous trees

TL;DR

This paper studies Hardy and BMO spaces on homogeneous trees with exponentially decreasing measures, establishing interpolation and boundedness results for integral operators linking these function spaces.

Contribution

It introduces atomic Hardy and BMO spaces for exponential measures on trees and proves interpolation and boundedness results, extending classical analysis to this setting.

Findings

Hardy and BMO spaces are well-defined for these measures

Interpolation results connect Hardy, BMO, and L^p spaces

Boundedness of integral operators is established

Abstract

We consider a family of measures on a $q$-homogeneous tree that decrease exponentially with respect to the distance from the origin. Such measures are doubling with respect to the Gromov distance. We define atomic Hardy and BMO spaces for that measures, and we prove interpolation results regarding such spaces. As a consequence we have boundedness results for integral operators involving Hardy, BMO, and $L^p$ spaces.