The logarithmic Sobolev inequality is regarded as the limiting case of Sobolev inequality. Firstly, a brief introduction to the logarithmic Sobolev inequality on the Euclidean space (including the Riemannian manifolds) will be given. Next, I will introduce some results in recent years about the logarithmic Sobolev inequality on the Heisenberg group which is a non-trivial example of sub-Riemannian manifolds. Then, I will talk about the tensorization of the logarithmic Sobolev inequalities and the dimension independence of the constant in the inequality which implies some further application to some infinite-dimensional sub-Riemannian manifolds.
