Subcritical Approach to Sharp Hardy-Littlewood-Sobolev Type Inequalities on the Upper Half Space

TL;DR

This paper establishes a reversed sharp Hardy-Littlewood-Sobolev inequality on the upper half space, classifies extremal functions, and computes best constants using a unified approach that extends to classical inequalities.

Contribution

It introduces a new uniform method to derive and classify sharp HLS inequalities and their extremal functions on the upper half space.

Findings

Reversed sharp HLS inequality established on the upper half space.

Extremal functions classified via the method of moving spheres.

Best constants for the inequalities computed.

Abstract

In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in \cite{HWY2008}) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.