An upper gradient approach to weakly differentiable cochains

2012/08/21 by Kai Rajala, Stefan Wenger, Rajala, Kai +1
Mathematics · #30L99 #46E35 #49J52 #49Q15 #53C65 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:30L99 #msc:46E35 #msc:49J52 #msc:49Q15 #msc:53C65

paper · pdf · doi:10.48550/arxiv.1208.4350

arxiv created 2012/08/21 · arxiv updated 2012/08/22

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.

Related

• Wolfe's theorem for weakly differentiable cochains
• Sobolev spaces via chains in metric measure spaces
• Cheeger's differentiation theorem via the multilinear Kakeya inequality
• Characterisation of upper gradients on the weighted Euclidean space and\n applications
• Dimension-independent functional inequalities by tensorization and projection arguments