Geometric characterizations of -Poincaré inequalities in the metric setting

  1. Durand-Cartagena, Estibalitz
  2. Jaramillo, Jesus A.
  3. Shanmugalingam, Nageswari
Journal:
Publicacions matematiques

ISSN: 0214-1493

Year of publication: 2016

Volume: 60

Issue: 1

Pages: 81-111

Type: Article

DOI: 10.5565/10.5565-PUBLMAT_60116_04 DIALNET GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Publicacions matematiques

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.

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