Optimal Hardy Inequality for Fractional Laplacians on the Lattice

TL;DR

The paper resolves the optimal Hardy inequality constant for the fractional Laplacian on the lattice ${\mathbb Z}^d$ by constructing a parametric family of Hardy weights $w_{\sigma,\alpha}$ via the discrete Riesz kernels ${\kappa}_{\alpha}$. A criticality analysis shows a dichotomy: positive criticality for $\alpha<\alpha_0$, null-criticality at $\alpha=\alpha_0$, and subcriticality for $\alpha>\alpha_0$, with optimality at the threshold established through ground-state constructions and null-sequences. The main result identifies the optimal leading term $\frac{c_{d,\sigma}}{|x|^{2\sigma}}$ in $w_{\sigma,\alpha_0}$, providing an explicit constant $c_{d,\sigma}$ and connecting to the classical case when $\sigma=1$ and $d\ge3$. An application yields a Landis-type unique continuation at infinity, illustrating the practical impact of the optimal Hardy weight in discrete settings.

Abstract

We study the fractional Hardy inequality on the integer lattice. We prove null-criticality of the Hardy weight and hence optimality of the constant. More specifically, we present a family of Hardy weights with respect to a parameter and show that below a certain threshold the Hardy weight is positive critical while above the threshold it is subcritical. In particular, the Hardy weight at the threshold is optimal in the sense that any larger weight would fail to be a Hardy weight and the Hardy inequality does not allow for a minimizer. A crucial ingredient in our proof is an asymptotic expansion of the fractional discrete Riesz kernel.

Theorems & Definitions (25)

• Theorem 1.1
• Remark
• Theorem 2.1: Fractional Laplacian
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1: Theorems 2.15 and 2.16 of hake2025optimal
• Theorem 3.2: Positive critical weights