Liangbing Luo : Logarithmic Sobolev Inequalities on Non-isotropic Heisenberg Groups

Posted by Cheng Ouyang , part of the Statistics and Data Science Seminar.

At

Feb. 16, 2022, 4 p.m.

In

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Abstract

In this talk, I will discuss 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, I will talk about logarithmic Sobolev inequalities on non-isotropic Heisenberg groups and discuss the dimension (in)dependence of the constants. 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. Moreover, I will mention the application of these results to an infinite-dimensional Heisenberg group.