Sharp Hardy-Littlewood-Sobolev inequality on the upper half space

TL;DR

This paper establishes a sharp Hardy-Littlewood-Sobolev inequality on the upper half space, extending classical results to new geometric settings and classifying extremal functions for certain exponents.

Contribution

It proves the inequality on the upper half space and classifies extremal functions using the method of moving sphere, extending the inequality to a new geometric context.

Findings

Established sharp inequality on the upper half space.

Proved existence of extremal functions.

Classified extremal functions for certain exponents.

Abstract

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $\lambda=n-\alpha$ (that is for the case of $\alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.