\(p\)-Poincaré inequality versus \(\infty \)-Poincaré inequality: some counterexamples
This paper deals with the \(p\)-Poincaré inequality, \(1\leq p\leq\infty\), in the setting of the doubling metric space \(X=(X,d,\mu)\), where \(\mu\) is a doubling measure. The notion of \(p\)-thick quasiconvex spaces is introduced for \(1\leq p\leq\infty.\) In particular, \(X\) is thick quasiconvex (cf. [\textit{D. Cartagena} et al., Mich. Math. J. 61, No. 1, 63--85 (2012)] if it is \(\infty\)-thick quasiconvex. In [loc. cit.], it is shown that a complete connected doubling metric measure space supports an \(\infty\)-Poincaré inequality if and only if it is \(\infty\)-thick quasiconvex. Here, it is proved that, when \(p\) is finite, \(X\) \(p\)-thick quasiconvex does not imply that \(X\) supports a weak \(p\)-Poincaré inequality (Example 2). Nevertheless, \( p\)-thick quasiconvexity for some finite \(p\) guarantees a weak \(\infty\)-Poincaré inequality (Lemma 2). An example (Example 3) which shows that the \(\infty\)-Poincaré inequality does not persist under Gromov-Hausdorff convergence is also presented. Examples 2 and 3 are closely related to the Sierpinski carpet. Section 4 is devoted to doubling weights in \(\mathbb R^n\). Some open problems are mentioned in the last section of the paper.
scientific article; zbMATH DE number 6050615
scientific article