Sharp reversed Hardy-Littlewood-Sobolev inequality on R^n

This is the first in our series of papers that concerns Hardy-Littlewood-Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbf R^n$
\[
\int_{\mathbf R^n} \int_{\mathbf R^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\mathbf R^n)}\, \|g\|_{L^r(\mathbf R^n)}
\]for any non-negative functions $f\in L^p(\mathbf R^n)$, $g\in L^r(\mathbf R^n)$, 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.