  We study the $L^p$ boundedness of Riesz transform as well as the reverse
inequality on Riemannian manifolds and graphs under the volume doubling
property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz
transform is then bounded on $L^p$ for $1 \textless{} p \textless{} 2$, which
shows that Gaussian estimates of the heat kernel are not a necessary condition
for this.In the particular case of Vicsek manifolds and graphs, we show that
the reverse inequality does not hold for $1 \textless{} p \textless{} 2$. This
yields a full picture of the ranges of $p\in (1,+\infty)$ for which
respectively the Riesz transform is $L^p$ -bounded and the reverse inequality
holds on $L^p$ on such manifolds and graphs. This picture is strikingly
different from the Euclidean one.
