Gaussian heat kernel bounds through elliptic Moser iteration

Frédéric Bernicot, Thierry Coulhon, Dorothee Frey

Research output: Contribution to journalArticleScientificpeer-review

17 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)995-1037
Number of pages43
JournalJournal de Mathematiques Pures et Appliquees
Issue number6
Publication statusPublished - 2016


  • Heat kernel lower bounds
  • Hölder regularity of the heat
  • semigroup
  • Gradient estimates
  • Poincaré inequalities
  • De Giorgi property


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