Gaussian heat kernel bounds through elliptic Moser iteration

On a doubling metric measure space endowed with a “carré du champ”, we consider L^p estimates (G_p) of the gradient of the heat semigroup and scale-invariant L^p Poincaré inequalities (P_p). We show that the combination of (G_p) and (P_p) for p≥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 [37]. This relies in particular on a new notion of L^p Hölder regularity for a semigroup and on a characterisation of (P₂) in terms of harmonic functions.