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

By Luigi Ambrosio, Nicola Gigli, Giuseppe Savar'e. 2011

251 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d77b42dc5c3dd832b07c8b19620241b5a995a641

How researchers see this paper

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In Applied Mathematics

1. Nicola Gigli
2. Elia Brué
3. Daniele Semola
4. C. Ketterer
5. Filippo Santambrogio
6. Renjin Jiang
7. Hui-Chun Zhang
8. Huaiqian Li

In Geometry and Topology

1. Nicola Gigli
2. Filippo Santambrogio
3. Elia Brué
4. C. Ketterer
5. Andrzej Swiech
6. Renjin Jiang
7. Daniele Semola
8. Hui-Chun Zhang

Citations in the corpus

• Heat Kernel Bounds on Metric Measure Spaces and Some Applications (2014, background): “ argument that, for each f ∈ S2(X) there exists a unique minimal G in the µ-a.e. sense such that (2.2) holds. We then denote the minimalG by |∇f| and call it the minimal weak upper gradient following [6]. The inhomogeneous Sobolev space W1,2(X) is defined as S2(X)∩L2(X) equipped with the norm kfkW1,2”
• Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “We continue with a brief discussion about Sobolev functions and Sobolev spaces over an arbitrary metric measure space, referring to [3, 7, 9, 39] for a more detailed discussion about this topic.”
• Independence on p of weak upper gradients on RCD spaces (2014, background): “1 Introduction There is a large literature concerning the definition of the Sobolev spaceW (X, d,m) of real valued functions defined on a metric measure space (X, d,m), we refer to [7] and [3] for historical comments and a presentation of the various - mostly equivalent - approaches.”
• Cones over metric measure spaces and the maximal diameter theorem (2013, methodology): “ric measure spaces that are mainly due to Hajlasz and Koskela [29]. In section 4.3 we present the first order calculus for metric measure spaces that was developed by Ambrosio, Gigli and Savar´e, e.g. [6] and allows us to give the definition of Riemannian curvature bounds in section 4.4. Finally, in s”
• Hamilton Jacobi equations on metric spaces and transport-entropy inequalities (2012, unknown): “6 has also been obtained by Ambrosio, Gigli and Savaré in their recent paper [1] (see also [2]), with a very similar proof.”
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces (2014, background): “Notice that the Sobolev space W 1,2(X) coincides with the Sobolev spaces based on upper gradients introduced by Cheeger [14] and Shanmugalingam [40]; see Ambrosio, Gigli and Savaré [5].”