Metric currents and the Poincaré inequality

Fässler, Katrin
Department of Mathematics, University of Fribourg, Switzerland

Orponen, Tuomas
Department of Mathematics and Statistics, University of Helsinki, Finland
Published in:
 Calculus of Variations and Partial Differential Equations.  2019, vol. 58, no. 2, p. 69
English
We show that a complete doubling metric space (X,d,μ) supports a weak 1Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points s,t∈X . This notion was introduced by S. Semmes in the 90’s, and has been previously known to be a sufficient condition for the weak 1Poincaré inequality. Our argument passes through the intermediate notion of a generalised pencil of curves (GPC). A GPC joining s and t is a normal 1current T, in the sense of Ambrosio and Kirchheim, with boundary ∂T=δt−δs , support contained in a ball of radius ∼d(s,t) around {s,t} , and satisfying ∥T∥≪μ , withd∥T∥/dμ(y)≲d(s,y)/μ(B(s,d(s,y)))+d(t,y)/μ(B(t,d(t,y))).We show that the 1Poincaré inequality implies the existence of GPCs joining any pair of points in X. Then, we deduce the existence of PCs from a recent decomposition result for normal 1currents due to Paolini and Stepanov.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/307599
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