Open Access
2016 Geometric Characterizations of $p$-Poincaré Inequalities in the Metric Setting
Estibalitz Durand-Cartagena, Jesus A. Jaramillo, Nageswari Shanmugalingam
Publ. Mat. 60(1): 81-111 (2016).


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 $Q-1<p\leq Q$.


Download Citation

Estibalitz Durand-Cartagena. Jesus A. Jaramillo. Nageswari Shanmugalingam. "Geometric Characterizations of $p$-Poincaré Inequalities in the Metric Setting." Publ. Mat. 60 (1) 81 - 111, 2016.


Published: 2016
First available in Project Euclid: 22 December 2015

zbMATH: 1334.31007
MathSciNet: MR3447735

Primary: 31E05 , 46E35
Secondary: 30L10

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

Rights: Copyright © 2016 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.60 • No. 1 • 2016
Back to Top