The Sobolev embedding constant on Lie groups

TL;DR

This paper estimates the Sobolev embedding constant on noncompact Lie groups with sub-Riemannian structures, providing bounds that generalize classical Euclidean results and applying them to establish Moser--Trudinger inequalities.

Contribution

It introduces bounds for Sobolev embedding constants on Lie groups with sub-Riemannian geometry, extending classical Euclidean bounds to more general noncompact settings.

Findings

Derived bounds depend only on the group and its structure

Bound reduces to classical Euclidean Sobolev embedding constant

Established local and global Moser--Trudinger inequalities

Abstract

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.