  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 $\mathbb R^n$ \[\int_{\mathbb R^n} \int_{\mathbb R^n} f(x)
|x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p (\mathbb
R^n)}\, \|g\|_{L^r (\mathbb R^n)}\] 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
$\mathscr C_{n,p,r}$ and the existence of optimal functions for the above
inequality, which will shed light on some existing results in literature.
