Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope

By Luigi Ambrosio, Maria Colombo, Simone Di Marino. 2012

125 citations recorded by the source corpus.

In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheeger's work because no Poincar\'e inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2b2d180e9b4902efc0926d45047de071136c893e

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In Applied Mathematics

Citations in the corpus

Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, background): “3] (see also [1] for a more detailed proof): for every f ∈ L(2)(X,m) with Ch(f) <∞ there exist fn ∈ Lipb(X) ∩ L(2)(X,m) such that fn → f, |Dfn| → |Df |w strongly in L(2)(X,m).”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, background): “We recall the following result, proved in [4], about reflexivity and separability of W 1,2(M):”
A new optimal transport distance on the space of finite Radon measures (2015, background): “, [2, 6] and the references therein) are less suitable for our purposes.”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, methodology): “this space is Banach, and reflexive if (X,d) is doubling (see [1]).”