Geometric characterizations of -Poincaré inequalities in the metric setting
- Durand-Cartagena, Estibalitz
- Jaramillo, Jesus A.
- Shanmugalingam, Nageswari
ISSN: 0214-1493
Year of publication: 2016
Volume: 60
Issue: 1
Pages: 81-111
Type: Article
More publications in: Publicacions matematiques
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Abstract
We prove that a locally complete metric space endowed with a doubling measure satisfies an ∞-Poincar´e 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 R satisfying an ∞-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 ≤ Q.