Optimal Poincaré-Hardy inequalities on graphs

We review a method to obtain optimal Poincaré-Hardy inequalities on the hyperbolic spaces by Berchio, Ganguly and Grillo. Then we show how to transfer the basic idea to the discrete setting. This yields optimal Poincaré-Hardy-type inequalities on model graphs which include fast enough growing trees and anti-trees. Moreover, this method yields optimal weights which are larger outside of a ball than the optimal weights constructed via the Fitzsimmons ratio of the square root of the minimal positive Green's function. Joint work with Christian Rose.