# Hardy spaces on weighted homogeneous trees

## TL;DR

This paper develops an atomic Hardy space on an infinite weighted homogeneous tree with exponential growth, addressing challenges due to the non-doubling measure and exploring properties like interpolation and singular integral boundedness.

## Contribution

It introduces a new Hardy space framework on non-doubling, exponentially growing trees and analyzes its key functional properties.

## Key findings

- Constructed an atomic Hardy space H^1 on the tree
- Studied real interpolation properties of H^1
- Established boundedness of singular integrals on H^1

## Abstract

We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure \mu. The metric measure space V,d,\mu) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H^1 on (V,d,\mu) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H^1.