Optimal Hardy Inequality for Fractional Laplacians on the Lattice

TL;DR

This paper establishes the optimal Hardy inequality for fractional Laplacians on the integer lattice, identifying critical weights and thresholds, and providing asymptotic analysis of the fractional discrete Riesz kernel.

Contribution

It introduces a family of Hardy weights on the lattice, proves their criticality or subcriticality depending on parameters, and demonstrates the optimality of the Hardy constant.

Findings

Hardy weight is null-critical at the optimal constant

Below a threshold, Hardy weights are positive critical

At the threshold, Hardy weights are subcritical

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.