Optimal Poincaré-Hardy-type Inequalities on Manifolds and Graphs
Florian Fischer, Christian Rose
TL;DR
The paper develops a unified framework for optimal Poincaré-Hardy-type inequalities on both manifolds and graphs by constructing a positive radial function $u$ with $-\Delta u\ge0$ and forming the Hardy weight $W=-\Delta\sqrt{u}/\sqrt{u}$, with optimality guaranteed by the Agmon-Allegretto-Piepenbrink theorem and a Khasminskii-type criterion. On manifolds, this yields explicit weights on hyperbolic spaces $W(r)=\lambda_0(\mathbb{H}^d)+\frac{1}{4r^2}+\frac{(d-1)(d-3)}{4\sinh^2 r}$ and generalises to model and harmonic manifolds, including Damek–Ricci spaces, via radial reductions with $u(r)=r/f(r)$ and curvature data. The discrete theory extends the method to homogeneous regular trees and weakly spherically symmetric graphs, producing explicit optimal weights $w_0$ and $w_\gamma$ in terms of area and curvature data, and showing these weights may exceed the Fitzsimmons ratio at infinity. Collectively, the results sharpen spectral and functional-analytic inequalities for Schrödinger operators on spaces with negative or controlled curvature, with potential applications to analysis on networks and curved spaces.
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.