The subelliptic heat kernels on SL(2,R) and on its universal covering SL(2,R)~\widetilde{SL(2,R)}: integral representations and some functional inequalities

In this paper, we study a subelliptic heat kernel on the Lie group SL(2,R) and on its universal covering. The subelliptic structure on SL(2,R) comes from the fibration $SO(2) -> SL(2,R) -> H^2$ and it can be lifted to its universal covering. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small time of the heat kernels and give us a way to compute the subriemannian distances. Then, we establish some gradient estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincar\'e inequality that are valid for both heat kernels

Essential spectrum and Weyl asymptotics for discrete Laplacians

In this paper, we investigate spectral properties of discrete Laplacians. Our study is based on the Hardy inequality and the use of super-harmonic functions. We recover and improve lower bounds for the bottom of the spectrum and of the essential spectrum. In some situation, we obtain Weyl asymptotics for the eigenvalues. We also provide a probabilistic representation of super-harmonic functions. Using coupling arguments, we set comparison results for the bottom of the spectrum, the bottom of the essential spectrum and the stochastic completeness of different discrete Laplacians. The class of weakly spherically symmetric graphs is also studied in full detail

A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincar\'e inequality

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 \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincar\'e inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional 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 non negative, all Carnot groups with step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is non negative

INTERTWININGS AND GENERALIZED BRASCAMP-LIEB INEQUALITIES

International audienceWe continue our investigation of the intertwining relations for Markov semigroups and extend the results of [9] to multi-dimensional diffusions. In particular these formulae entail new functional inequalities of Brascamp-Lieb type for log-concave distributions and beyond. Our results are illustrated by some classical and less classical examples