An Extended Discrete Hardy-Littlewood-Sobolev Inequality

TL;DR

This paper extends the Hardy-Littlewood-Sobolev inequality to the discrete critical case with a finite domain, establishing sharp constants, uniqueness, symmetry, and monotonicity of optimizers.

Contribution

It derives a finite form of the discrete HLS inequality at the critical case, providing sharp constants and analyzing the properties of optimizers.

Findings

Sharp estimate for the best constant obtained

Uniqueness and symmetry of the optimizer proved

Monotonicity of the optimizer established

Abstract

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.