Gaussian heat kernel bounds through elliptic Moser iteration

Frédéric Bernicot; Thierry Coulhon; Dorothee Frey

doi:10.1016/j.matpur.2016.03.019

Gaussian heat kernel bounds through elliptic Moser iteration

Frédéric Bernicot
, Thierry Coulhon
, Dorothee Frey

Research output: Contribution to journal › Article › peer-review

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 characterisation of (P₂) in terms of harmonic functions.

Original language	English
Pages (from-to)	995-1037
Number of pages	43
Journal	Journal des Mathematiques Pures et Appliquees
Volume	106
Issue number	6
DOIs	https://doi.org/10.1016/j.matpur.2016.03.019
Publication status	Published - 1 Dec 2016
Externally published	Yes

Keywords

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

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10.1016/j.matpur.2016.03.019

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title = "Gaussian heat kernel bounds through elliptic Moser iteration",

abstract = "On a doubling metric measure space endowed with a “carr{\'e} du champ”, we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincar{\'e} inequalities (Pp). We show that the combination of (Gp) and (Pp) 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 Lp H{\"o}lder regularity for a semigroup and on a characterisation of (P2) in terms of harmonic functions.",

keywords = "De Giorgi property, Gradient estimates, Heat kernel lower bounds, H{\"o}lder regularity of the heat semigroup, Poincar{\'e} inequalities",

author = "Fr{\'e}d{\'e}ric Bernicot and Thierry Coulhon and Dorothee Frey",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier Masson SAS",

year = "2016",

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doi = "10.1016/j.matpur.2016.03.019",

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journal = "Journal des Mathematiques Pures et Appliquees",

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TY - JOUR

T1 - Gaussian heat kernel bounds through elliptic Moser iteration

AU - Bernicot, Frédéric

AU - Coulhon, Thierry

AU - Frey, Dorothee

PY - 2016/12/1

Y1 - 2016/12/1

N2 - On a doubling metric measure space endowed with a “carré du champ”, we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincaré inequalities (Pp). We show that the combination of (Gp) and (Pp) 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 Lp Hölder regularity for a semigroup and on a characterisation of (P2) in terms of harmonic functions.

AB - On a doubling metric measure space endowed with a “carré du champ”, we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincaré inequalities (Pp). We show that the combination of (Gp) and (Pp) 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 Lp Hölder regularity for a semigroup and on a characterisation of (P2) in terms of harmonic functions.

KW - De Giorgi property

KW - Gradient estimates

KW - Heat kernel lower bounds

KW - Hölder regularity of the heat semigroup

KW - Poincaré inequalities

U2 - 10.1016/j.matpur.2016.03.019

DO - 10.1016/j.matpur.2016.03.019

M3 - Article

AN - SCOPUS:84962861250

SN - 0021-7824

VL - 106

SP - 995

EP - 1037

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 6

ER -

Gaussian heat kernel bounds through elliptic Moser iteration

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Keywords

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