Publicacions Matemàtiques

Geometric Characterizations of $p$-Poincaré Inequalities in the Metric Setting

Estibalitz Durand-Cartagena, Jesus A. Jaramillo, and Nageswari Shanmugalingam

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove that a locally complete metric space endowed with a doubling measure satisfies an $\infty$-Poincaré inequality if and only if given a null set, every two points can be joined by a quasiconvex curve which "almost avoids" that set. As an application, we characterize doubling measures on ${\mathbb R}$ satisfying an $\infty$-Poincaré inequality. For Ahlfors $Q$-regular spaces, we obtain a characterization of $p$-Poincaré inequality for $p>Q$ in terms of the $p$-modulus of quasiconvex curves connecting pairs of points in the space. A related characterization is given for the case $Q-1<p\leq Q$.

Article information

Source
Publ. Mat. Volume 60, Number 1 (2016), 81-111.

Dates
First available in Project Euclid: 22 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.pm/1450818484

Mathematical Reviews number (MathSciNet)
MR3447735

Subjects
Primary: 31E05: Potential theory on metric spaces 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 30L10: Quasiconformal mappings in metric spaces

Keywords
$p$-Poincaré inequality metric measure space thick quasiconvexity quasiconvexity singular doubling measures in ${\mathbb R}$ $\operatorname{Lip}$-$\operatorname{lip}$ condition

Citation

Durand-Cartagena, Estibalitz; Jaramillo, Jesus A.; Shanmugalingam, Nageswari. Geometric Characterizations of $p$-Poincaré Inequalities in the Metric Setting. Publ. Mat. 60 (2016), no. 1, 81--111.https://projecteuclid.org/euclid.pm/1450818484


Export citation