On the equivalence of BV notions in metric measure spaces

Abstract

The aim of the paper is to compare in detail several notions of $BV$ space of functions of bounded variation in metric measure spaces $({\rm X},\mathsf{d},\mathfrak{m})$. Informally, they can be grouped in two classes, either by a relaxation procedure starting from a class of nicer functions (and with different notions of pseudo-gradient in the relaxation procedure) or by requiring good behaviour along a rich class of absolutely continuous curves. In the second approach, richness can be understood according to the notion of approximation modulus of [O. Martio, Adv. Calc. Var., 9 (2016)] or according to the notion of test plan introduced in [L. Ambrosio, N. Gigli, and G. Savaré, Invent. Math., 195 (2014)]. Extending [L. Ambrosio and S. Di Marino, J. Funct. Anal., 266 (2014)], we prove that all these approaches are isometrically equivalent in any locally complete metric measure space.