Riesz transform on graphs under subgaussian estimates

Let $\Gamma$ be a doubling graph satisfying some pointwise subgaussian estimates of the Markov kernel. We introduce a space $H^1(\Gamma)$ of functions and a space $H^1(T_\Gamma)$ of 1-forms and give various characterizations of them. We prove the $H^1$-boundedness of the Riesz transform, from which we deduce the $L^p$ boundedness of the Riesz transform for any $p\in (1,2)$. In a previous work, we showed a $H^1_w$-boundedness of the Riesz transform under weaker assumptions, but the $L^p$ boundedness was not established.