Matrix-valued modified logarithmic Sobolev inequality for sub-Laplacian on [formula omitted]

Abstract

We prove that the canonical sub-Laplacian on SU(2) admits a modified log-Sobolev inequality on matrix-valued functions, independent of the matrix sizes. This establishes the first example of a matrix-valued modified log-Sobolev inequality for a sub-Laplacian. We also show that on Lie groups the heat kernel measure pt at time t satisfies matrix-valued modified log-Sobolev inequality with constants in order O(t−1).