Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver’s metric derivations. The definition hereby given is shown to be equivalent to many others present in literature. Introduction In the last few years a great attention has been devoted to the theory of Sobolev spaces W 1,q on metric measure spaces (X, d,m), see for instance [10, 9, 6] for an overview on this subject and [3] for more recent developments. These definitions of Sobolev spaces usually come with a weak definition of modulus of gradient, in particular the notion of q-upper gradient has been introduced in [13] and used in [14] for a Sobolev space theory. Also, in [14] the notion of minimal q-upper gradient has been proved to be equivalent to the notion of relaxed upper gradient arising in Cheeger’s paper [7]. In [3] the definitions of q-relaxed slope and q-weak upper gradient are given and the minimal ones are seen to be equivalent to the ones in [14]. All of those approach give us a notion of modulus of the gradient instead of the gradient itself, and an integration by parts formula is present only in special cases, and moreover it is often only an inequality. In this paper we want to fill this gap, namely giving a definition of Sobolev spaces more similar to the classical one given with an integration by parts formula; in R this formula can be written as ∫