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Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$ (1510.04680v2)

Published 15 Oct 2015 in math.AP and math.FA

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|{Lp(\partial \mathbb R+n)} \, |g|{Lr(\mathbb R+n)} ] for any nonnegative functions $f\in Lp(\partial \mathbb R_+n)$, $g\in Lr(\mathbb R_+n)$, 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.

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