Hardy inequalities for inverse square potentials with countable number of singularities

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.