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- Title
p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples.
- Authors
Durand-Cartagena, Estibalitz; Shanmugalingam, Nageswari; Williams, Alex
- Abstract
We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, ), we obtain a geometric property implied by the support of a p-Poincaré inequality, and demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith-Zhong (Ann Math 167(2):575-599, ). We also show that the persistence of Poincaré inequality under measured Gromov-Hausdorff limits fails for ∞-Poincaré inequality.
- Subjects
MATHEMATICS; POINCARE conjecture; GEOMETRY; ALGEBRA; SPACES of measures
- Publication
Mathematische Zeitschrift, 2012, Vol 271, Issue 1/2, p447
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-011-0871-1