Improved Hardy–Littlewood–Sobolev Inequality on \({\mathbb{S}^n}\) under Constraints

Abstract

In this paper, we establish an improved Hardy–Littlewood–Sobolev inequality on \({\mathbb{S}^n}\) under higher-order moments constraint. Moreover, by constructing precise test functions, using improved Hardy–Littlewood–Sobolev inequality on \({\mathbb{S}^n}\), we show such inequality is almost optimal in critical case. As an application, we give a simpler proof of the existence of the maximizer for conformal Hardy–Littlewood–Sobolev inequality.