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 rightarrow 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 rightarrow infty .