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The Hardy Inequality (HI) for potentials with countably many singularities ofthe 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: itis well-known that in all the existing results about the HI with finite numberof singularities the best constants converge to 0 with the number $n$ ofsingularities going to infinity.  In this note we provide an example of nontrivial HI in right cylinders offixed radius $R>0$ in $\mathbf{R}^d$, for a potential $V$ defined above havingthe singularities $\{a_k\}_{k\in \mathbf{Z}}$ uniformly distributed on the axisof the cylinders. For this example we prove that an upper bound for the Hardyconstant is $(d-2)^2/4$, the clasical Hardy constant in $\mathbf{R}^d$corresponding to one singular potential. We also prove positive lower bounds ofthe Hardy constant which allow to deduce that the asymptotic behavior as $R\to0$ of the Hardy constant coincides with $(d-2)^2/4$. The proof of the mainresult 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.

Hardy inequalities for inverse square potentials with countable number of singularities