Calculus of Variations and Geometric Measure Theory

E. Durand-Cartagena - N. Shanmugalingam - A. Williams

$p$-Poincaré inequality vs. infinity-Poincaré inequality; some counter-examples

created by durandcar on 14 Oct 2010
modified by shanmugal on 02 Sep 2013


Published Paper

Inserted: 14 oct 2010
Last Updated: 2 sep 2013

Journal: Math. Z.
Volume: 271
Pages: 447--467
Year: 2012


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.