Hardy and BMO spaces on graphs, application to Riesz transform

Let $\mathrm{\Gamma }$ be a graph with the doubling property for the volume of balls and $P$ a reversible random walk on $\mathrm{\Gamma }$. 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.