  On a doubling metric measure space endowed with a "carr\'e du champ", we
consider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup and
scale-invariant $L^p$ Poincar\'e inequalities $(P_p)$. We show that the
combination of $(G_p)$ and $(P_p)$ for $p\ge 2$ always implies two-sided
Gaussian heat kernel bounds. The case $p=2$ is a famous theorem of
Saloff-Coste, of which we give a shorter proof, without parabolic Moser
iteration. We also give a more direct proof of the main result in \cite{HS}.
This relies in particular on a new notion of $L^p$ H\"older regularity for a
semigroup and on a characterization of $(P_2)$ in terms of harmonic functions.
