Sharp reversed Hardy-Littlewood-Sobolev inequality on ${\mathbb{R}}^n$

Abstract

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space ${\mathbb{R}}^n$ \[\int_{\mathbb R^n} \int_{\mathbb R^n} f(x) |x-y|^λg(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p (\mathbb R^n)}\, \|g\|_{L^r (\mathbb R^n)}\] for any nonnegative functions $f\in L^p({\mathbb{R}}^n)$, $g\in L^r({\mathbb{R}}^n)$, and $p,r\in (0,1)$, $\lambda >0$ such that $1/p+1/r-\lambda /n=2$. We will also explore some estimates for ${\mathcal{C}}_{n,p,r}$ and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.

24 pages, 0 figure, to appear in Israel Journal of Mathematics

References in corpus (2)

• Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications
• Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space ${\mathbb{R}}_{+}^n$

Cited by in corpus (2)

• Reverse Hardy-Littlewood-Sobolev inequalities
• Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds