An upper gradient approach to weakly differentiable cochains

TL;DR

This paper introduces a new concept of weakly differentiable cochains in metric measure spaces, generalizing classical inequalities and relating capacity, modulus, and Hausdorff dimension.

Contribution

It defines weakly differentiable cochains in metric spaces using upper gradients and proves continuity estimates in Lie groups, extending Morrey-Sobolev inequalities.

Findings

Continuity estimates for cochains with $p$-integrable upper gradients.

Generalization of Morrey-Sobolev inequality to Lie groups.

Relations between capacity, modulus, and Hausdorff dimension.

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 $p$-integrable upper gradient in $n$-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.