Gaussian heat kernel bounds through elliptic Moser iteration

Journal article (2016)

Authors

Frédéric Bernicot Université de Nantes
Thierry Coulhon PSL Research University, Chimie ParisTech, CNRS, Institut de Recherche de Chimie Paris
D. Frey Australian National University, Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
Heat kernel lower bounds Hölder regularity of the heat Semigroup Gradient estimates Poincaré inequalities De Giorgi property
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Published Date

2016

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

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.