AN EXTENDED DISCRETE HARDY-LITTLEWOOD-SOBOLEV INEQUALITY

摘要

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: μ = n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: μ = 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.