Preprint

Inserted: 23 jun 2026
Last Updated: 23 jun 2026

Year: 2026

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.

Keywords: functions of bounded variation, metric measure spaces, Modulus of curves