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

By Fabrice Baudoin, Michel Bonnefont, Nicola Garofalo. 2010

102 citations recorded by the source corpus.

Let $$\mathbb M $$M be a smooth connected manifold endowed with a smooth measure $$\mu $$μ and a smooth locally subelliptic diffusion operator $$L$$L satisfying $$L1=0$$L1=0, and which is symmetric with respect to $$\mu $$μ. We show that if $$L$$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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/41250170920480b5455fc0f44901e666056adf86

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, background): “ach is totally different since it does not use the uniform doubling condition on the volume of the metric balls which is a key assumption in that work. We should also mention that in the sequel paper [14] we have in fact established a uniform global doubling condition under non negative lower bound ”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “ition 2.13) and use it in establishing fundamental properties of BV. Some examples where one has 2-Poincar´e and weak Bakry-Emery but where the validity of a 1-Poincar´e inequality´ is unknown are in [15]. Our approach can also be compared to the measure-valued upper gradient structure introduced in”
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations (2014, background): “ty implies several results: • Li-Yau type gradient bounds for the heat kernel and associated scale invariant parabolic Harnack inequalities [14]; • Upper and lower Gaussian bounds for the heat kernel [11, 12, 14]; • Boundedness of the Riesz transform [15]; • Sobolev embeddings and isoperimetric ineq”