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arXiv · 2501.18379

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

Florian Fischer·Christian Rose
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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.

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Florian Fischer, Christian Rose. 2025-01-30. Optimal Poincar\'e-Hardy-type Inequalities on Manifolds and Graphs. https://arxiv.org/abs/2501.18379

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