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

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Abstract

Let \(\mathbb M \) be a smooth connected manifold endowed with a smooth measure \(\mu \) and a smooth locally subelliptic diffusion operator \(L\) satisfying \(L1=0\), and which is symmetric with respect to \(\mu \). 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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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Fabrice Baudoin

  2. Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 Cours de la liberation, 33405 , Talence, France

    Michel Bonnefont

  3. Dipartimento d’Ingegneria Civile e Ambientale (DICEA), Università di Padova, via Trieste 63, 35131 , Padova, Italy

    Nicola Garofalo

Authors
  1. Fabrice Baudoin
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  2. Michel Bonnefont
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  3. Nicola Garofalo
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Corresponding author

Correspondence to Nicola Garofalo.

Additional information

F. Baudoin is supported in part by NSF Grant DMS 0907326. N. Garofalo is supported in part by NSF Grant DMS-1001317.

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Baudoin, F., Bonnefont, M. & Garofalo, N. A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality. Math. Ann. 358, 833–860 (2014). https://doi.org/10.1007/s00208-013-0961-y

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  • DOI: https://doi.org/10.1007/s00208-013-0961-y

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