Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$

TL;DR

This paper proves a sharp reversed Hardy--Littlewood--Sobolev inequality on the half space for nonnegative functions in certain Lebesgue spaces with parameters p, r, and λ, and discusses extremal functions and estimates.

Contribution

It establishes the first sharp reversed Hardy--Littlewood--Sobolev inequality on the half space with new methods and analyzes extremal functions and constants.

Findings

Proved the sharp reversed inequality on the half space.

Derived estimates for the best constant _{n,p,r}.

Demonstrated existence of extremal functions.

Abstract

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$ \[ \int_{\mathbb R_+^n} \int_{\partial \mathbb R_+^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\partial \mathbb R_+^n)} \, \|g\|_{L^r(\mathbb R_+^n)} \] for any nonnegative functions $f\in L^p(\partial \mathbb R_+^n)$, $g\in L^r(\mathbb R_+^n)$, and $p,r\in (0,1)$, $\lambda > 0$ such that $(1-1/n)1/p + 1/r -(\lambda-1) /n =2$. Some estimates for $\mathscr C_{n,p,r}$ as well as the existence of extrema functions for this inequality are also considered. New ideas are also introduced in this paper.