The Sobolev embedding constant on Lie groups

In this paper we estimate the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with a left invariant measure. The bound that we obtain, up to a constant depending only on the group and its sub-Riemannian structure, reduces to the best known bound for the classical inhomogeneous Sobolev embedding constant on ${\mathbb{R}}^d$. As an application, we prove local and global Moser–Trudinger inequalities.