*Published Paper*

**Inserted:** 14 oct 2010

**Last Updated:** 2 sep 2013

**Journal:** Math. Z.

**Volume:** 271

**Pages:** 447--467

**Year:** 2012

**Abstract:**

We point out some of the differences between the consequences of $p$-Poincaré inequality and that of infinity-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of the infinity-Poincaré inequa\-lity given in DJS, we give a geometric implication of infinity-Poincaré 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 infinity-Poincaré inequality, but support no finite $p$-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for infinity-Poincaré inequality in the spirit of Keith-Zhong KZ. We also show that the persistence of Poincaré inequality under measured Gromov-Hausdorff limits fails for infinity-Poincaré inequality.

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