Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, {equation}∑i,j,i≠ j{fᵢgⱼ}{ i-jn-α}≤ Cr,s,α |f|l^r |g|l^s,{equation}where i,j∈ Zⁿ, $r,s>1$, $0 2$.