Matrix-valued modified logarithmic Sobolev inequality for sub-Laplacian on SU (2)

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 p t at time t satisfies matrix -valued modified log-Sobolev inequality with constants in order O ( t - 1 ). (c) 2024 Published by Elsevier Inc.