Sobolev algebras through heat kernel estimates

Frédéric Bernicot; Thierry Coulhon; Dorothee Frey

doi:10.5802/jep.30

Sobolev algebras through heat kernel estimates
[Algèbres de Sobolev via des estimations du noyau de la chaleur]

Frédéric Bernicot ¹ ; Thierry Coulhon ² ; Dorothee Frey ³

¹ CNRS - Université de Nantes, Laboratoire Jean Leray 2 rue de la Houssinière, 44322 Nantes cedex 3, France
² PSL Research University 75005 Paris, France
³ Mathematical Sciences Institute, The Australian National University Canberra ACT 0200, Australia

Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 99-161.

Résumé
Abstract

Sur un espace métrique mesuré doublant $(M, d, μ)$ equipé d’un « carré du champ », soit $ℒ$ le générateur markovien associé et ${\dot{L}}_{α}^{p} (M, ℒ, μ)$ l’espace de Sobolev homogène correspondant, d’ordre $0 < α < 1$ dans $L^{p}$ , $1 < p < + \infty$ , avec la norme $∥ ℒ^{α / 2} {f ∥}_{p}$ . Nous donnons des conditions suffisantes sur le semi-groupe de la chaleur ${(e^{- t ℒ})}_{t > 0}$ pour garantir que les espaces ${\dot{L}}_{α}^{p} (M, ℒ, μ) \cap L^{\infty} (M, μ)$ sont des algèbres pour le produit ponctuel. Deux approches sont développées, une première utilisant des paraproduits (basée sur l’extrapolation pour obtenir leur bornitude) et une seconde basée sur des fonctionnelles quadratiques géométriques (basée sur la notion d’oscillation). Des règles de composition et de paralinéarisation sont aussi obtenues. En comparaison avec les résultats précédents ([29, 11]), les améliorations principales consistent dans le fait que nous n’avons plus à imposer d’inégalité de Poincaré ou de bornitude $L^{p}$ des transformées de Riesz, mais seulement des bornitudes $L^{p}$ du gradient du semi-groupe. Comme conséquence, nous obtenons la propriété d’algèbre de Sobolev pour $p \in (1, 2]$ , sous la seule hypothèse d’estimations gaussiennes pour le noyau de la chaleur.

On a doubling metric measure space $(M, d, μ)$ endowed with a “carré du champ”, let $ℒ$ be the associated Markov generator and ${\dot{L}}_{α}^{p} (M, ℒ, μ)$ the corresponding homogeneous Sobolev space of order $0 < α < 1$ in $L^{p}$ , $1 < p < + \infty$ , with norm $∥ ℒ^{α / 2} {f ∥}_{p}$ . We give sufficient conditions on the heat semigroup ${(e^{- t ℒ})}_{t > 0}$ for the spaces ${\dot{L}}_{α}^{p} (M, ℒ, μ) \cap L^{\infty} (M, μ)$ to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and a second one through geometrical square functionals (relying on sharp estimates involving oscillations). A chain rule and a paralinearisation result are also given. In comparison with previous results ([29, 11]), the main improvements consist in the fact that we neither require any Poincaré inequalities nor $L^{p}$ -boundedness of Riesz transforms, but only $L^{p}$ -boundedness of the gradient of the semigroup. As a consequence, in the range $p \in (1, 2]$ , the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.

Reçu le : 2015-05-03
Accepté le : 2016-02-20
Publié le : 2016-03-07

MR Zbl

DOI : 10.5802/jep.30

Classification : 46E35, 22E30, 43A15
Keywords: Sobolev space, algebra property, heat semigroup
Mot clés : Espace de Sobolev, propriété d’algèbre, semi-groupe de la chaleur

Affiliations des auteurs :

Frédéric Bernicot ¹ ; Thierry Coulhon ² ; Dorothee Frey ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fr\'ed\'eric Bernicot and Thierry Coulhon and Dorothee Frey},
     title = {Sobolev algebras through heat kernel estimates},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     publisher = {ole polytechnique},
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Frédéric Bernicot; Thierry Coulhon; Dorothee Frey. Sobolev algebras through heat kernel estimates. Journal de l’École polytechnique — Mathématiques, Tome 3 (2016), pp. 99-161. doi : 10.5802/jep.30. https://jep.centre-mersenne.org/articles/10.5802/jep.30/

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