On the Optimality and Decay of $p$-Hardy Weights on Graphs

TL;DR

This paper constructs optimal Hardy weights for subcritical energy functionals on graphs, establishing their decay properties and implications for null-criticality, with applications to inequalities and principles in graph analysis.

Contribution

It introduces a method to construct optimal Hardy weights on graphs and links their decay properties to null-criticality, advancing understanding of energy functionals on graphs.

Findings

Optimal Hardy weights are constructed for subcritical energy functionals.

Decay conditions of Hardy weights are established in terms of integrability.

Null-criticality implies optimality near infinity.

Abstract

We construct optimal Hardy weights to subcritical energy functionals $h$ associated with quasilinear Schr\"odinger operators on locally finite graphs. Here, optimality means that the weight $w$ is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional $h-w$ is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle and a Rellich-type inequality.