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

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.

(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

References in corpus (1)

• Metric measure spaces with Riemannian Ricci curvature bounded from below

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