Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces

2011/11/30 by Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré · 2 citations
Mathematics · #math.AP #math.CA #math.FA #math.MG #msc:52C23 #msc:49J52 #msc:49Q20 #msc:58J35 #msc:35K90 #msc:31C25

paper · pdf · doi:10.4171/rmi/746

published as Rev. Mat. Iberoam. 29 (2013), 969-996 · (v3) A simpler axiomatization of weak gradients, still equivalent to all other ones, has been proposed (Lemma 2.1 and Remark 4.10). The density of Lipschitz functions has been slightly enforced (Sect. 8.3). Formula (4.7) and Proposition 5.4 (of v2) removed, more details added (end of page 18). Minor typos

arxiv created 2012/05/16 · arxiv updated 2014/09/16

Abstract

We compare several notion of weak (modulus of) gradient in metric measure spaces. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independenly of doubling and Poincaré assumptions on the metric measure space.

Cited by

• Sobolev and BV spaces on metric measure spaces via derivations and integration by parts
• A new characterization of Sobolev spaces on Lipschitz differentiability spaces

Related

• Potpourri, 9
• A beginner's guide to analysis on metric spaces
• Characterisation of upper gradients on the weighted Euclidean space and applications
• Another introduction to the geometry of metric spaces
• Comparison Between Teichmuller and Lipschitz Metrics