p-Poincare ́ inequality vs. ∞-Poincare ́ inequality; some counter-examples

We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincare ́ inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincare ́ inequa-lity given in [DJS], we give a geometric implication of p-Poincare ́ inequality and show throughout examples that the characterization in the p finite case is not possible. The examples we give are metric measure spaces which are doubling and support an∞-Poincare ́ inequality, but support no finite p-Poincare ́ inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincare ́ inequality in the spirit of Keith-Zhong [KZ]. We also show that the persistence of Poincare ́ inequality under measured Gromov-Hausdorff limits fails for ∞-Poincare ́ inequality.