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

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-α}}\leq C_{r,s,α} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in {\mathbb{Z}}^n$, $r,s>1$, $0<\alpha <n$, and $\frac{1}{r}+\frac{1}{s}+\frac{n-\alpha }{n}\ge 2$. Indeed, we can prove that the best constant is attainable in the supercritical case $\frac{1}{r}+\frac{1}{s}+\frac{n-\alpha }{n}>2$.

Cited by in corpus (1)

• Structure and $K$-theory of ${\ell }^p$ uniform Roe algebras