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

Genggeng Huang; Congming Li; Ximing Yin

doi:10.3934/dcds.2015.35.935

Article Contents

2015, Volume 35, Issue 3: 935-942. Doi: 10.3934/dcds.2015.35.935

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

1.
Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai

Received: January 31, 2014

Revised: March 31, 2014

Published: February 2015

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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 $\frac {1} {r} + \frac {1} {s} + \frac {n-\alpha}{n} \geq 2$. Indeed, we prove that the best constant is attainable in the supercritical case $\frac {1}{r} + \frac {1} {s} + \frac {n-\alpha}{n} > 2$.

Keywords:

Discrete Hardy-Littlewood-Sobolev inequality,
best constant,
concentration compactness,
existence,
supercritical.

Mathematics Subject Classification: Primary: 35A23.

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References

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