The \(\infty \)-Poincaré inequality on metric measure spaces
type: Item
instance of: scholarly article
title: The \(\infty \)-Poincaré inequality on metric measure spaces
zbMATH Open document ID: 1275.46018
author: Estibalitz Durand-Cartagena
author: Jesús A. Jaramillo
author: Nageswari Shanmugalingam
published in: Michigan Mathematical Journal
publication date: 2013-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.mmj/1331222847
review text: A metric measure space \((X,d,\mu)\) is said to support a weak \(\infty\)-Poincaré inequality if there exist constants \(C>0\) and \(\lambda \geq1\) such that, for every Borel measurable function \(u:X\rightarrow \mathbb R \cup \{\infty\}\) and every upper gradient \(g:X\rightarrow [0,\infty]\) of \(u\), the pair \((u,g)\) satisfies the inequality \[ \frac {1}{\mu(B(x,r))} \int_{B(x,r)}|u-u_{B(x,r)}|\, d\mu\leq Cr\|g\|_{L^{\infty}(B(x,\lambda r))} \] for each ball \(B(x,r)\subset X\). The \(\infty\)-Poincaré inequality is a weaker condition than any \(p\)-Poincaré inequality with finite \(p\geq 1\), but it still gives reasonable information on the geometry of the space. The main result of this paper states that a connected complete doubling metric measure space supports a weak \(\infty\)-Poincaré inequality if and only if it is thick quasiconvex, i.e., every pair of points can be connected by a ``thick'' family of quasiconvex curves in the sense that the \(\infty\)-modulus of this family is positive. The authors also prove that supporting an \(\infty\)-Poincaré inequality is also equivalent to the condition that \(LIP^\infty(X)=N^{1,\infty}(X)\).
reviewed by: Riikka Korte
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 30L99
zbMATH DE Number: 6141588
zbMATH Keywords: metric measure spaces
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: quasiconvexity
zbMATH Keywords: thick quasiconvex
MaRDI profile type: Publication
OpenAlex ID: W1975514418
Wikidata QID: Q109994460
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DOI: 10.1307/MMJ/1331222847
description: scientific article; zbMATH DE number 6141588