A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

Fabrice Baudoin; Michel Bonnefont; Nicola Garofalo

doi:10.1007/s00208-013-0961-y

A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

Fabrice Baudoin
, Michel Bonnefont
, Nicola Garofalo^*

^*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

51 Citations (Scopus)

Abstract

Let M be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1 = 0, and which is symmetric with respect to μ. We show that if L satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold: • The volume doubling property; • The Poincaré inequality; • The parabolic Harnack inequality. The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

Original language	English
Journal	Mathematische Annalen
Volume	358
Issue	3-4
Pages (from-to)	833-860
Number of pages	28
ISSN	0025-5831
DOIs	https://doi.org/10.1007/s00208-013-0961-y
Publication status	Published - Apr 2014
Externally published	Yes

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@article{77141a2aaf584fdf8708b463fcbc08ab,

title = "A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincar{\'e} inequality",

abstract = "Let M be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1 = 0, and which is symmetric with respect to μ. We show that if L satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold: • The volume doubling property; • The Poincar{\'e} inequality; • The parabolic Harnack inequality. The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.",

author = "Fabrice Baudoin and Michel Bonnefont and Nicola Garofalo",

year = "2014",

month = apr,

doi = "10.1007/s00208-013-0961-y",

language = "English",

volume = "358",

pages = "833--860",

journal = "Mathematische Annalen",

issn = "0025-5831",

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A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality. / Baudoin, Fabrice; Bonnefont, Michel; Garofalo, Nicola.
In: Mathematische Annalen, Vol. 358, No. 3-4, 04.2014, p. 833-860.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

TY - JOUR

T1 - A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

AU - Baudoin, Fabrice

AU - Bonnefont, Michel

AU - Garofalo, Nicola

PY - 2014/4

Y1 - 2014/4

N2 - Let M be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1 = 0, and which is symmetric with respect to μ. We show that if L satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold: • The volume doubling property; • The Poincaré inequality; • The parabolic Harnack inequality. The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

AB - Let M be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1 = 0, and which is symmetric with respect to μ. We show that if L satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold: • The volume doubling property; • The Poincaré inequality; • The parabolic Harnack inequality. The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

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U2 - 10.1007/s00208-013-0961-y

DO - 10.1007/s00208-013-0961-y

M3 - Journal article

AN - SCOPUS:84897597016

SN - 0025-5831

VL - 358

SP - 833

EP - 860

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -

A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

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