Optimal Poincaré-Hardy-type Inequalities on Manifolds and Graphs

2025/01/30 by Florian Fischer, Christian Rose, Fischer, Florian +1 · 2 citations
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2501.18379

Abstract

We review a method to obtain optimal Poincaré-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.

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