Hardy spaces on homogeneous trees with flow measures

TL;DR

This paper investigates Hardy spaces on homogeneous trees with non-doubling exponential measures, showing that certain classical characterizations via heat, Poisson semigroups, and Riesz transforms do not hold in this setting.

Contribution

It demonstrates the failure of maximal and Riesz transform characterizations of Hardy spaces on homogeneous trees with non-doubling measures, highlighting limitations of classical harmonic analysis tools.

Findings

Maximal characterizations via heat and Poisson semigroups fail.

Riesz transform characterization of Hardy spaces fails.

Classical harmonic analysis tools are limited in this setting.

Abstract

We consider a homogeneous tree endowed with a nondoubling flow measure $\mu$ of exponential growth and a probabilistic Laplacian $\mathcal{L}$ self-adjoint with respect to $\mu$. We prove that the maximal characterization in terms of the heat and the Poisson semigroup of $\mathcal{L}$ and the Riesz transform characterization of the atomic Hardy space introduced in a previous work fail.