## Publicacions Matemàtiques

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

#### 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

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