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

TL;DR

This paper proves the existence of a maximizing pair for the best constant in a discrete Hardy-Littlewood-Sobolev inequality under supercritical conditions, advancing understanding of discrete integral inequalities.

Contribution

It establishes the attainability of the best constant in the discrete Hardy-Littlewood-Sobolev inequality for supercritical parameters, which was previously unresolved.

Findings

Best constant is attainable in supercritical case

Provides conditions for existence of maximizing pairs

Advances discrete inequality theory

Abstract

In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in \mathbb Z^n$, $r,s>1$, $0<\alpha<n$, and $\frac 1r+\frac 1s+\frac {n-\alpha}n\geq 2$. Indeed, we can prove that the best constant is attainable in the supercritical case $\frac 1r+\frac 1s+\frac {n-\alpha}n> 2$.