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Article Contents

# 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, $$\sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-\alpha}}\leq C_{r,s,\alpha} |f|_{l^r} |g|_{l^s},$$where $i,j\in \mathbb Z^n$, $r,s>1$, $0 < \alpha < n$, and $\frac {1} {r} + \frac {1} {s} + \frac {n-\alpha}{n} \geq 2$. Indeed, we prove that the best constant is attainable in the supercritical case $\frac {1}{r} + \frac {1} {s} + \frac {n-\alpha}{n} > 2$.
Mathematics Subject Classification: Primary: 35A23.

 Citation:

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