An upper gradient approach to weakly differentiable cochains

作者：Rajala Kai; Wenger Stefan^*

来源：Journal de Mathematiques Pures et Appliquees, 2013, 100(6): 868-906.

DOI：10.1016/j.matpur.2013.03.006

• Cochains
• Metric currents
• Upper gradients
• Continuity estimates

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)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchheim%26apos;s theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen-Koskela%26apos;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.

• 出版日期2013-12