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

TL;DR

This paper investigates Bakry-Emery estimates for the Laplace-Beltrami operator on totally geodesic foliations, showing that weaker curvature conditions can still imply key functional inequalities and convergence properties.

Contribution

It demonstrates that under a bracket generating condition, weaker curvature bounds suffice for important inequalities, and establishes uniform curvature-dimension inequalities for converging Riemannian metrics.

Findings

Weaker curvature conditions imply Wang-Harnack and log-Sobolev inequalities.

Under certain assumptions, the generalized curvature dimension inequality holds uniformly.

Convergence of Riemannian metrics preserves curvature-dimension inequalities.

Abstract

We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the $\Gamma_2$ operator may not be bounded from below but the horizontal Bakry-Emery 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-Garofalo is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one.