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

Language


Classification

Mathematics

License

License undefined

Identifiers


Persistent URL

https://folia.unifr.ch/unifr/documents/307599
Statistics
Document views: 15
File downloads: