\(p\)-Poincaré inequality versus \(\infty \)-Poincaré inequality: some counterexamples
type: Item
instance of: scholarly article
title: \(p\)-Poincaré inequality versus \(\infty \)-Poincaré inequality: some counterexamples
zbMATH Open document ID: 1255.46015
author: Estibalitz Durand-Cartagena
author: Nageswari Shanmugalingam
author: Alex Williams
published in: Mathematische Zeitschrift
publication date: 2012-06-26T00:00:00Z
review text: 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.
reviewed by: Denise Huet
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 30L10
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 31E05
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 6050615
zbMATH Keywords: Poincaré inequalities
zbMATH Keywords: doubling metric measure spaces
zbMATH Keywords: \({{\mathcal{A}}_{\infty}}\) weights
zbMATH Keywords: \(p\)-thick convexity
MaRDI profile type: Publication
full work available at URL: s00209-011-0871-1
OpenAlex ID: W2044004866
Wikidata QID: Q109994480
cites work: The boundary correspondence under quasiconformal mappings
cites work: Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces
cites work: Q4552627
cites work: Admissible measures in one dimension
cites work: Nonlinear potential theory on metric spaces
cites work: A course in metric geometry
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Pointwise Lipschitz functions on metric spaces
cites work: The \(\infty \)-Poincaré inequality on metric measure spaces
cites work: Q4431283
cites work: Q4255465
cites work: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
cites work: Sobolev met Poincaré
cites work: Lectures on analysis on metric spaces
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Q5294569
cites work: A differentiable structure for metric measure spaces
cites work: Modulus and the Poincaré inequality on metric measure spaces
cites work: Geometric implications of the Poincaré inequality
cites work: Q4238869
cites work: The Poincaré inequality is an open ended condition
cites work: Q4328334
cites work: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
cites work: Q4528988
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: Q3142876
cites work: Lectures on \(n\)-dimensional quasiconformal mappings
DOI: 10.1007/S00209-011-0871-1
description: scientific article; zbMATH DE number 6050615