Journal article

An upper gradient approach to weakly differentiable cochains

  • Rajala, Kai University of Jyväskylä, Department of Mathematics and Statistics, Finland
  • Wenger, Stefan Université de Fribourg, Mathématiques, Switzerland
Published in:
  • Journal de Mathématiques Pures et Appliquées. - 2013, vol. 100, no. 6, p. 868–906
English 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–Kirchheimʼs theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen–Koskelaʼs 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.
Faculté des sciences et de médecine
Département de Mathématiques
  • English
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