Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group

TL;DR

This paper proves the existence of maximizers for sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group using concentration compactness, and also provides upper bounds for the sharp constants involved.

Contribution

It offers a new proof of maximizer existence on the Heisenberg group and generalizes previous results to all admissible cases, along with bounds for the sharp constants.

Findings

Existence of maximizers established for all admissible cases.

Provided upper bounds for the sharp constants in the inequalities.

Generalized previous special-case results to broader settings.

Abstract

In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. On one hand, we apply the concentration compactness principle to prove the existence of the maximizers. While the approach here gives a different proof under the special cases discussed in a recent work of Frank and Lieb, we generalize the result to all admissible cases. On the other hand, we provide the upper bounds of sharp constants for these inequalities.