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 $$\sum _{n\in {\mathbb{Z}}^d}|\hat{I}{”}^{\ell /2}u(n){|}^2\ge {\mathcal{C}}_{\text{opt}}^{\ell }(d)\sum _{n\in {\mathbb{Z}}^d}\frac{|u(n){|}^2}{|n{|}^{2\ell }}.$$ satisfies $$\underset{d\to \infty }{\lim }\frac{{\mathcal{C}}_{\text{opt}}^{\ell }(d)}{d^{\ell }}=2^{\ell }.$$ The proof is based on a Fourier reduction to a family of singular Hardy inequalities on the flat torus, involving the weight \[ ω(x)^{-2\ell}, \qquad ω(x)^2=\sum_{j=1}^d\sin^2\left(\frac{x_j}{2}\right), \] 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 Poincaré 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 ${\mathbb{Z}}^d$.

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