Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space ${\mathbb{R}}_{+}^n$

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 ${\mathbb{R}}_{+}^n$ \[ \int_{\mathbb R_+^n} \int_{\partial \mathbb R_+^n} f(x) |x-y|^λg(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\partial \mathbb R_+^n)} \, \|g\|_{L^r(\mathbb R_+^n)} \] 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 ${\mathcal{C}}_{n,p,r}$ as well as the existence of extrema functions for this inequality are also considered. New ideas are also introduced in this paper.

31 pages, 0 figure. arXiv admin note: substantial text overlap with arXiv:1508.02041. To appear in International Mathematics Research Notices (IMRN)

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)

• Sharp reversed Hardy-Littlewood-Sobolev inequality on ${\mathbb{R}}^n$
• Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space ${\mathbb{R}}_{+}^n$