Hardy and Rellich inequality on lattices

TL;DR

This paper investigates how the best constants in discrete Hardy and Rellich inequalities on lattices grow with dimension, providing asymptotic behavior and new Hardy-type inequalities for certain operators.

Contribution

It establishes the asymptotic growth of the sharp constants in Hardy and Rellich inequalities on lattices and introduces new Hardy-type inequalities for specific operators.

Findings

Sharp constant in Hardy inequality grows linearly with dimension d.

Sharp constant in Rellich inequality grows quadratically with d.

New Hardy-type inequalities for operators Δ^m and ∇(Δ^m) on a d-dimensional torus.

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$.