Gaussian heat kernel bounds through elliptic Moser iteration

On a doubling metric measure space endowed with a “carré du champ”, we consider LpLp estimates (Gp)(Gp) of the gradient of the heat semigroup and scale-invariant LpLp Poincaré inequalities (Pp)(Pp). We show that the combination of (Gp)(Gp) and (Pp)(Pp) for p≥2p≥2 always implies two-sided Gaussian heat kernel bounds. The case p=2p=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 LpLp Hölder regularity for a semigroup and on a characterisation of (P2)(P2) in terms of harmonic functions.