Hardy and Rellich inequality on lattices

Abstract

In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice ${\mathbb{Z}}^d$ as $d\to \infty$. In the process, we proved some Hardy-type inequalities for the operators ${\Delta }^m$ and $\nabla ({\Delta }^m)$ for non-negative integers $m$ on a $d$ dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as $d$ and $d^2$ respectively as $d\to \infty$.

minor changes: some references and remarks added

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