We compare several notions of weak (modulus of) gradients in metric measure spaces and prove their equivalence. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independently of doubling and Poincaré assumptions on the metric measure space.
Cite this article
Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 969–996