Sobolev embedding constants and Moser--Trudinger Inequalities on Lie groups.

In this paper we prove a precise estimate of the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with relatively invariant measures. Such an estimate appears to be new even in the case of the classical inhomogeneous Sobolev spaces on $\mathbb{R}^d$. As an application, we prove local and global Moser--Trudinger inequalities.