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Optimal Poincar\'e-Hardy-type Inequalities on Manifolds and Graphs
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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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Cited by 1 Pith paper
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An optimal fractional Hardy inequality on the discrete half-line
For σ in (0,1], the paper constructs an explicit optimal Hardy weight W^op_σ for (-Δ_N)^σ, with W^op_σ(n) approximately n^{-2σ} and an upper bound C_σ for the best constant in the n^{-2σ} Hardy inequality.
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