Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

By Fabrice Baudoin, Michel Bonnefont. 2011

80 citations recorded by the source corpus.

Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ which is symmetric with respect to $\mu$. We assume that $L$ satisfies a generalized curvature dimension inequality as introduced by Baudoin-Garofalo \cite{BG1}. Our goal is to discuss functional inequalities for $\mu$ like the Poincar\'e inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/426579463d840bdfe34b26ead0415addf0018e46

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “th non-negative Ricci curvature and more generally, the RCD(0,+∞) spaces (see [50]); • Carnot groups (see [16]); • Complete sub-Riemannian manifolds with generalized non-negative Ricci curvature (see [14, 19]); • On non-compact metric graphs with ﬁnite number of edges, the weak Bakry-Emery curvature”
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations (2014, background): “In a joint program with Nicola Garofalo it has been proved in a very general and abstract framework that the generalized curvature dimension inequality implies several results: • Li-Yau type gradient bounds for the heat kernel and associated scale invariant parabolic Harnack inequalities [13]; • Upp”
A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality (2010, background): “clusion with T in place of t. The arbitrariness of T > 0 ﬁnishes the proof. A ﬁrst notable consequence of Proposition 2.4 is the following reverse log-Sobolev inequality which was also observed in [BB2]. Corollary 2.5. Let ε > 0 and f ∈ Aε. For x ∈ M, t > 0 one has tPtf(x)Γ(lnPtf)(x)+ρ2t2Pt”