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

Estibalitz Durand-Cartagena, Jesus A. Jaramillo, and Nageswari Shanmugalingam

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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

Publ. Mat., Volume 60, Number 1 (2016), 81-111.

First available in Project Euclid: 22 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31E05: Potential theory on metric spaces 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 30L10: Quasiconformal mappings in metric spaces

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


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.

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