Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

TL;DR

This paper establishes dimension-independent logarithmic Sobolev inequalities on non-isotropic Heisenberg groups and extends these results to an infinite-dimensional setting, advancing understanding of sub-Riemannian geometry and hypoelliptic operators.

Contribution

It introduces dimension-independent logarithmic Sobolev inequalities for non-isotropic Heisenberg groups and extends these to infinite-dimensional models, a novel advancement in sub-Riemannian analysis.

Findings

Logarithmic Sobolev constants are independent of dimension.

Comparison of inequalities between isotropic and non-isotropic groups.

Extension to infinite-dimensional Heisenberg groups.

Abstract

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modelled on an abstract Wiener space.