Yet another proof of the density in energy of Lipschitz functions (Q6593256)

The paper proves an identification of two different notions of Sobolev space over a metric measure space \((X,\mathsf{d},\mu)\). The two Sobolev spaces are:\N\begin{itemize}\N\item \(H^{1,p}(X,\mu)\), the first order Sobolev space defined via the Cheeger energy, an energy obtained with a relaxation procedure starting from the integral of the \(p\)-power of the asymptotic slope of a Lipschitz function.\N\item \(W^{1,p}(X,\mu)\), the first order Sobolev space defined using the notion of plans with barycenter, which is known to be equivalent to the Sobolev space defined using the (slightly more common) notion of test plans.\N\end{itemize}\NThis result is usually informally rephrased as ``Lipschitz functions are dense in energy'', and it was already known in the literature starting from the original proof given in [\textit{L. Ambrosio} et al., Rev. Mat. Iberoam. 29, No. 3, 969--996 (2013; Zbl 1287.46027)]. Nowadays, different proofs of this fact are available; the one presented in this paper is the first that relies on techniques of smooth analysis. Using the Kuratowski embedding the problem is reduced from metric measure spaces to Banach spaces, and then the study of cylindrical functions, using tools of convex analysis and Smirnov's superposition principle for normal 1-currents, allows to prove the result.\N\NIt is worth to mention that the proof requires only completeness and separability of the metric space \((X,\mathsf{d})\), endowed with a boundedly-finite Borel measure \(\mu\), and it is valid for every \(p\in (1,\infty)\) (see [\textit{S. Eriksson-Bique}, Calc. Var. Partial Differ. Equ. 62, No. 2, Paper No. 60, 23 p. (2023; Zbl 1514.46028)] for a similar result valid also for \(p=1\)).