In the first part of the talk, I will present a novel proof of the equivalence of definitions of metric p-Sobolev space, based on a smooth analysis (involving cylindrical functions on Banach spaces) coupled with some classical duality techniques in Convex Analysis. In the second part of the talk, I will show that the strategy employed in the proof is a particular instance of a more general principle: the identification of plans with barycenter, Lipschitz derivations, and a suitable family of normal 1-currents. Time permitting, I will discuss the use of the above techniques also in the setting of extended metric measure spaces.
The talk is based on recent joint works with Luigi Ambrosio, Toni Ikonen, and Enrico Pasqualetto.