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On a doubling metric measure space endowed with a "carr\'e du champ", weconsider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup andscale-invariant $L^p$ Poincar\'e inequalities $(P_p)$. We show that thecombination of $(G_p)$ and $(P_p)$ for $p\ge 2$ always implies two-sidedGaussian heat kernel bounds. The case $p=2$ is a famous theorem ofSaloff-Coste, of which we give a shorter proof, without parabolic Moseriteration. 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 asemigroup and on a characterization of $(P_2)$ in terms of harmonic functions.

Gaussian heat kernel bounds through elliptic Moser iteration