Hardy inequalities for inverse square potentials with countable number of singularities

TL;DR

This paper establishes nontrivial Hardy inequalities for potentials with countably many singularities arranged along cylinders, showing the Hardy constant approaches the classical value as the cylinder radius shrinks.

Contribution

It provides the first example of Hardy inequalities with infinitely many singularities in cylinders, with bounds matching the classical Hardy constant in the limit.

Findings

Upper bound for Hardy constant is (d-2)^2/4.

Lower bounds show asymptotic behavior matches classical Hardy constant.

Hardy constant approaches (d-2)^2/4 as cylinder radius tends to zero.

Abstract

The Hardy Inequality (HI) for potentials with countably many singularities of the form $V=\sum_{k\in \mathbf{Z}}\frac{1}{|x-a_k|^2}$ is not a trivial issue. In principle, the more singular poles are, the less the Hardy constant is: it is well-known that in all the existing results about the HI with finite number of singularities the best constants converge to 0 with the number $n$ of singularities going to infinity. In this note we provide an example of nontrivial HI in right cylinders of fixed radius $R>0$ in $\mathbf{R}^d$, for a potential $V$ defined above having the singularities $\{a_k\}_{k\in \mathbf{Z}}$ uniformly distributed on the axis of the cylinders. For this example we prove that an upper bound for the Hardy constant is $(d-2)^2/4$, the clasical Hardy constant in $\mathbf{R}^d$ corresponding to one singular potential. We also prove positive lower bounds of the Hardy constant which allow to deduce that the asymptotic behavior as $R\to 0$ of the Hardy constant coincides with $(d-2)^2/4$. The proof of the main result lies on using a nice identity due to Allegretto and Huang (Theorems 1.1, 2.1 in reference [1]) for particularly well chosen test functions.