Sharp asymptotics for higher-order Hardy constants on lattices

We study the optimal constants in higher-order Hardy inequalities on the lattice ${\mathbb{Z}}^d$. For each fixed $\mathrm{\ell }\in \mathbb{N}$, we prove that the optimal constant ${\mathcal{C}}_{\text{opt}}^{\mathrm{\ell }}(d)$ in $$\sum _{n\in {\mathbb{Z}}^d}\mathrm{\mid }{\mathrm{\Delta }}^{\mathrm{\ell }\mathrm{/}2}u(n){\mathrm{\mid }}^2\ge {\mathcal{C}}_{\text{opt}}^{\mathrm{\ell }}(d)\sum _{n\in {\mathbb{Z}}^d}\frac{\mathrm{\mid }u(n){\mathrm{\mid }}^2}{\mathrm{\mid }n{\mathrm{\mid }}^{2\mathrm{\ell }}}\mathrm{.}$$ satisfies $$\underset{d\to \mathrm{\infty }}{\lim }\frac{{\mathcal{C}}_{\text{opt}}^{\mathrm{\ell }}(d)}{d^{\mathrm{\ell }}}=2^{\mathrm{\ell }}\mathrm{.}$$ The proof is based on a Fourier reduction to a family of singular Hardy inequalities on the flat torus, involving the weight $$\omega (x{)}^{-2\mathrm{\ell }},\omega (x{)}^2=\sum _{j=1}^d{\sin }^2\left(\frac{x_j}{2}\right),$$ 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 $\mathrm{\ell }$, and requires only the classical unweighted Poincaré inequality on the torus. The appearance of the limiting constant $2^{\mathrm{\ell }}$ is particularly striking, as it suggests that, in the high dimensional regime, the optimizers are localized near the unit sphere $\{n\in {\mathbb{Z}}^d:\mathrm{\mid }n\mathrm{\mid }=1\}$ in ${\mathbb{Z}}^d$.