Optimal Poincar\'e-Hardy-type Inequalities on Manifolds and Graphs

TL;DR

This paper reviews methods for deriving optimal Poincaré-Hardy inequalities on hyperbolic spaces, extends these to graphs, and introduces new inequalities with larger weights at infinity, enhancing understanding of geometric and graph structures.

Contribution

It presents a unified method to obtain optimal inequalities on manifolds and graphs, including new results on spherically symmetric graphs and trees.

Findings

Derived optimal inequalities on hyperbolic spaces and graphs.

Established larger weights at infinity than classical methods.

Provided a new proof for inequalities on homogeneous regular trees.

Abstract

We review a method to obtain optimal Poincar\'e-Hardy-type inequalities on the hyperbolic spaces, and discuss briefly generalisations to certain classes of Riemannian manifolds. Afterwards, we recall a corresponding result on homogeneous regular trees and provide a new proof using the aforementioned method. The same strategy will then be applied to obtain new optimal Hardy-type inequalities on weakly spherically symmetric graphs which include fast enough growing trees and anti-trees. In particular, this yields optimal weights which are larger at infinity than the optimal weights classically constructed via the Fitzsimmons ratio of the square root of the minimal positive Green's function.