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

الأصل · EN

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, equation ∑ᵢ,ⱼ,ᵢ≠ ⱼfᵢgⱼ i-jⁿ⁻α≤ Cᵣ,ₛ,α |f|ₗᵣ |g|ₗₛ, equationwhere i,j∈ Zⁿ, r,s>1, 0<α<n, and 1r+ 1s+ n-αn≥ 2. Indeed, we can prove that the best constant is attainable in the supercritical case 1r+ 1s+ n-αn> 2.