An upper gradient approach to weakly differentiable cochains
Abstract
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchheim's theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen-Koskela's concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with -integrable upper gradient in -dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey-Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.
Cite
@article{arxiv.1208.4350,
title = {An upper gradient approach to weakly differentiable cochains},
author = {Kai Rajala and Stefan Wenger},
journal= {arXiv preprint arXiv:1208.4350},
year = {2012}
}