Hardy and BMO spaces on graphs, application to Riesz transform

TL;DR

This paper develops Hardy and BMO spaces on graphs with doubling volume properties, characterizes their duality, and applies these results to establish boundedness of the Riesz transform.

Contribution

It introduces Hardy and BMO spaces adapted to reversible random walks on graphs and characterizes their duality, extending harmonic analysis tools to graph settings.

Findings

Characterization of Hardy spaces on graphs with volume doubling.

Duality between Hardy spaces and BMO-type spaces on graphs.

Boundedness of the Riesz transform on these spaces.

Abstract

Let $\Gamma$ be a graph with the doubling property for the volume of balls and $P$ a reversible random walk on $\Gamma$. We introduce $H^1$ Hardy spaces of functions and $1$-forms adapted to $P$ and prove various characterizations of these spaces. We also characterize the dual space of $H^1$ as a $BMO$-type space adapted to $P$. As an application, we establish $H^1$-$H^1$ and $H^1$-$L^1$ boundedness of the Riesz transform.