Sharp asymptotics for higher-order Hardy constants on lattices

Abstract

We study the optimal constants in higher-order Hardy inequalities on the lattice $\mathbb{Z}^d$. For each fixed $\ell \in \mathbb{N}$, we prove that the optimal constant $\mathcal{C}_\text{opt}^\ell(d)$ in $$ \sum_{n \in \mathbb{Z}^d} |\Delta^{\ell/2}u(n)|^2 \geq \mathcal{C}_\text{opt}^\ell(d)\sum_{n \in \mathbb{Z}^d} \frac{|u(n)|^2}{|n|^{2\ell}}. $$ satisfies $$ \lim_{d\rightarrow\infty}\frac{\mathcal{C}_\text{opt}^\ell(d)}{d^\ell} =2^\ell. $$ The proof is based on a Fourier reduction to a family of singular Hardy inequalities on the flat torus, involving the weight \[ \omega(x)^{-2\ell}, \qquad \omega(x)^2=\sum_{j=1}^d\sin^2\left(\frac{x_j}{2}\right), \] and zero average condition on admissible functions. We establish these torus inequalities by combining a ground state representation formula with a weighted integrated Bochner identity in an iterative scheme. The method yields explicit constants, defined recursively in the order $\ell$, and requires only the classical unweighted Poincar\'e inequality on the torus. The appearance of the limiting constant $2^\ell$ is particularly striking, as it suggests that, in the high dimensional regime, the optimizers are localized near the unit sphere $\{n\in\mathbb{Z}^d:|n|=1\}$ in $\mathbb{Z}^d$.