Complete Modified Logarithmic Sobolev inequality for sub-Laplacian on $SU(2)$

TL;DR

This paper establishes a matrix-valued modified log-Sobolev inequality for the sub-Laplacian on SU(2), providing a novel example of such an inequality with uniform bounds independent of matrix dimension, and analyzes heat kernel measures on Lie groups.

Contribution

It proves the first example of a sub-Laplacian with a matrix-valued modified log-Sobolev inequality that is uniform across all matrix dimensions.

Findings

Uniform modified log-Sobolev inequality for sub-Laplacian on SU(2)

Matrix-valued constants for heat kernel measures scale as O(t^{-1})

First example of matrix-valued modified log-Sobolev inequality for sub-Laplacian

Abstract

We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform modified log-Sobolev inequality for all its matrix-valued functions, independent of the matrix dimension. This is the first example of sub-Laplacian that a matrix-valued modified log-Sobolev inequality has been obtained. We also show that on Lie groups, the heat kernel measure $p_t$ at time $t$ admits matrix-valued modified log-Sobolev constants of order $O(t^{-1})$.