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Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. (English) Zbl 1304.26020

Summary: In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \[ \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{|i-j|^{n-{\alpha}}}\leq C_{r,s,{\alpha}} |f|_{l^r} |g|_{l^s}, \] 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 \(\frac {1}{r} + \frac {1} {s} + \frac {n-\alpha}{n} > 2\).

MSC:

26D15 Inequalities for sums, series and integrals
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References:

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