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Logarithmic Sobolev inequalities on homogeneous spaces

Liangbing Luo, Lehigh University

Abstract: The logarithmic Sobolev inequality has been first introduced and studied by L. Gross on a Euclidean space, and since then it found many applications. The fact that the logarithmic Sobolev constant often does not depend on the dimension makes it applicable in infinite-dimensional settings. In this talk, we consider sub-Riemannian manifolds which are homogeneous spaces equipped with a natural sub-Riemannian structure. In such a setting, the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. Logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such homogeneous spaces are studied. We show that the logarithmic Sobolev constant only depends on the Lie group that acts transitively on such a homogeneous space but the constant is independent of the action of its isotropy group. In some concrete settings, this method will allow us to track the (in)dependence of the logarithmic Sobolev constant on the geometry of the underlying space, especially the dimension-independence of the constant. Several examples will be provided.

Tea and refreshments available at 3:00 p.m. in the Assmus Conference Room (CU 212).

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