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

TL;DR

This paper compares different notions of weak gradients in metric measure spaces and proves the density of Lipschitz functions in energy without requiring doubling or Poincaré conditions, using optimal transportation tools.

Contribution

It establishes the density of Lipschitz functions in energy in metric measure spaces independently of doubling and Poincaré assumptions.

Findings

Lipschitz functions are dense in energy in metric measure spaces.

Weak gradient notions are comparable in these spaces.

Optimal transportation tools are used to prove density results.

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 Poincar\'e assumptions on the metric measure space.