Qualitative And Quantitative Analysis Of The Maximizing Pair For The Discrete Hardy-Littlewood-Sobolev Inequality

In this paper，we study the attainability of the best constant for the discrete Hardy-Littlewood-Sobolev inequality(DHLS for abbreviation).In the first part,we mainly focus on the following classical DHLS inequality: where i,j∈Zn,r,s>1,0<α<n,and1/r+1/s+n-α/n≥2.Indeed,we can prove that the maximizing pair(f,g)exist in the supercritical case1/r+1/s+n-α/n>2. In the second part,based on the idea from the the previous part,we further our research into the following doubly weighted Hardy—Littlewood—Sobolev inequality: where r,s>1,0<λ<n,α+β>0,1-1/r-λ/n<1-1/r-1/s<β/n<1-1/s and 1/r+1/s+λ+α+β≥2. We can also prove that the best constant is attainable in the supercritical case1/r+1/s+λ+α+β/n>2.