Gaussian heat kernel bounds through elliptic Moser iteration

Abstract

Abstract. 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 characterization of (P 2 ) in terms of harmonic functions.

Riesz transforms through reverse Hölder and Poincaré inequalities

“…The result was shown in [12,Proposition 5.4] in the case N = 1. The proof for general N ∈ N is similar, we give a proof here for the sake of completeness.…”

“…Following [12,Theorem 6.3], we know that if L and Γ satisfy the conservation property, the combination (G p ) with (P p ) for some p > 2 implies (P 2 ), and so implies (R q ) for every q ∈ (2, p) by [7].…”

“…The proof of this statement in [12,Theorem 6.3] relies on the self-adjointness of the operator, which is used through the finite propagation speed property to deduce a perfectly localised Caccioppoli inequality. So as far as we know, the assumptions are actually weaker than those in [7].…”

“…We follow the same scheme as in [12,Lemma 4.6], where a proof was given in the particular setting of a Dirichlet form. We explain here how can we extend it to the current more general framework.…”

Hardy spaces for semigroups with Gaussian bounds

“…We refer the reader to Grigor'yan [16], Hebisch and Saloff-Coste [19], Saloff-Coste [28], Gyrya and Saloff-Coste [17], Bernicot, Coulhon, and Frey [2], and references therein. Here we present a short alternative proof of (4.3).…”