Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation

Fabrice Baudoin; Michel Bonnefont

doi:10.1016/j.na.2015.06.025

Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation

^*Corresponding author for this work

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

5 Citations (Scopus)

Abstract

We study Bakry-Émery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the F² operator may not be bounded from below but the horizontal Bakry-Émery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang-Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin and Garofalo (2015) is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.

Original language	English
Article number	10576
Journal	Nonlinear Analysis: Theory, Methods & Applications
Volume	126
Pages (from-to)	159-169
Number of pages	11
ISSN	0362-546X
DOIs	https://doi.org/10.1016/j.na.2015.06.025
Publication status	Published - Oct 2015
Externally published	Yes

Keywords

Curvature dimension inequalities
Laplacian
Riemannian foliation

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10.1016/j.na.2015.06.025

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title = "Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation",

abstract = "We study Bakry-{\'E}mery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the F2 operator may not be bounded from below but the horizontal Bakry-{\'E}mery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang-Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin and Garofalo (2015) is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.",

keywords = "Curvature dimension inequalities, Laplacian, Riemannian foliation",

author = "Fabrice Baudoin and Michel Bonnefont",

note = "Publisher Copyright: {\textcopyright} 2015 Elsevier Ltd.",

year = "2015",

month = oct,

doi = "10.1016/j.na.2015.06.025",

language = "English",

volume = "126",

pages = "159--169",

journal = "Nonlinear Analysis: Theory, Methods & Applications",

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Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation. / Baudoin, Fabrice; Bonnefont, Michel.
In: Nonlinear Analysis: Theory, Methods & Applications, Vol. 126, 10576, 10.2015, p. 159-169.

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

TY - JOUR

T1 - Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation

AU - Baudoin, Fabrice

AU - Bonnefont, Michel

PY - 2015/10

Y1 - 2015/10

N2 - We study Bakry-Émery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the F2 operator may not be bounded from below but the horizontal Bakry-Émery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang-Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin and Garofalo (2015) is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.

AB - We study Bakry-Émery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the F2 operator may not be bounded from below but the horizontal Bakry-Émery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang-Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin and Garofalo (2015) is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.

KW - Curvature dimension inequalities

KW - Laplacian

KW - Riemannian foliation

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U2 - 10.1016/j.na.2015.06.025

DO - 10.1016/j.na.2015.06.025

M3 - Journal article

AN - SCOPUS:84940447645

SN - 0362-546X

VL - 126

SP - 159

EP - 169

JO - Nonlinear Analysis: Theory, Methods & Applications

JF - Nonlinear Analysis: Theory, Methods & Applications

M1 - 10576

ER -

Curvature-dimension estimates for the Laplace-Beltrami operator of a totally geodesic foliation

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