Characterising Sobolev inequalities by controlled coarse homology and applications for hyperbolic spaces. (English) Zbl 1401.53035
The main result of the present paper establishes a Sobolev inequality characterisation for the vanishing of a fundamental class in the controlled coarse homology of Nowak-Špakula. The abstract setting corresponds to quasiconvex uniform spaces that support a local weak \((1,1)\)-Poincaré inequality. This result is applied in the framework of visual Gromov hyperbolic spaces and Carnot groups.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
30L99 | Analysis on metric spaces |
58J32 | Boundary value problems on manifolds |