*preprint*

**Inserted:** 24 feb 2021

**Year:** 2019

**Abstract:**

These lecture notes contain an extended version of the material presented in
the C.I.M.E. summer course in 2017, aiming to give a detailed introduction to
the metric Sobolev theory.
The notes are divided in four main parts. The first one is devoted to a
preliminary study of the underlying topological, metric, and measure-theoretic
aspects of a general extended metric-topological measure space $\mathbb
X=(X,\tau,\mathsf d,\mathfrak m)$.
The second part is devoted to the construction of the Cheeger energy,
initially defined on a distinguished unital algebra of Lipschitz functions.
The third part deals with the basic tools needed for the dual
characterization of the Sobolev spaces: the notion of $p$-Modulus of a
collection of (nonparametric) rectifiable arcs and its duality with the class
of nonparametric dynamic plans, i.e.~Radon measures on the space of rectifiable
arcs with finite $q$-barycentric entropy with respect to $\mathfrak m$.
The final part of the notes is devoted to the dual*weak formulation of the
Sobolev spaces $W^{1,p}(\mathbb X)$ in terms of nonparametric dynamic plans and
to their relations with the Newtonian spaces $N^{1,p}(\mathbb X)$ and with the
spaces $H^{1,p}(\mathbb X)$ obtained by the Cheeger construction. In
particular, when $(X,\mathsf d)$ is complete, a new proof of the equivalence
between these approaches is given by a direct duality argument.
A substantial part of these Lecture notes relies on well established
theories. New contributions concern the extended metric setting, the role of
general compatible algebras of Lipschitz functions and their density w.r.t.~the
Sobolev energy, a compactification trick, the study of reflexivity and
infinitesimal Hilbertianity inherited from the underlying space, and the use of
nonparametric dynamic plans for the definition of weak upper gradients.
*