Reversed Hardy-Littlewood-Sobolev inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$

TL;DR

This paper establishes a reversed Hardy-Littlewood-Sobolev inequality on the Heisenberg group and CR sphere, providing explicit bounds, proving extremal functions' existence, and introducing a rearrangement-free method applicable to related inequalities.

Contribution

It introduces a roughly reversed HLS inequality on $\, ext{Heisenberg}\, ext{group}$ and $ ext{CR sphere}$, with explicit bounds and extremal functions, using a novel rearrangement-free approach.

Findings

Established a reversed HLS inequality on $\, ext{Heisenberg}\, ext{group}$ and $ ext{CR sphere}$.

Provided explicit lower bounds for the sharp constant.

Proved existence of extremal functions using subcritical and compactness techniques.

Abstract

This paper is mainly devoted to the study of the reversed Hardy-Littlewood-Sobolev (HLS) inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$. First, we establish the roughly reversed HLS inequality and give a explicitly lower bound for the sharp constant. Then, the existence of the extremal functions with sharp constant is proved by subcritical approach and some compactness techniques. Our method is rearrangement free and can be applied to study the classical HLS inequality and other similar inequalities.