Logarithmic Sobolev inequalities on homogeneous spaces

Abstract

We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a natural sub-Riemannian structure induced by a transitive action by a Lie group. In such a setting, the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such homogeneous spaces. We show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a homogeneous space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we are able to show that the logarithmic Sobolev constants is independent of the dimension of the underlying spaces in several examples. We illustrate the results by considering the Grushin plane, non-isotropic Heisenberg groups, Heisenberg-like groups, Hopf fibration, $\operatorname{SO}(3)$, $\operatorname{SO}(4)$, and compact Heisenberg manifolds.