Geometric characterizations of -Poincaré inequalities in the metric setting

• Autores: Estibalitz Durand Cartagena, Jesús Angel Jaramillo Aguado , Nageswari Shanmugalingam
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 60, Nº 1, 2016, págs. 81-111
• Idioma: inglés
• DOI: 10.5565/10.5565-PUBLMAT_60116_04
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• Resumen

  • 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.