preprint

Inserted: 21 nov 2025

Year: 2025

Abstract:

We study the predual of the space of functions of bounded variation defined over a metric measure space $({\rm X},{\sf d},\mathfrak m)$ with $\mathfrak m$ finite. More specifically, for any exponent $p\in(1,\infty)$ we construct an isometric predual of the space ${\rm BV}_p({\rm X})$ of $p$-integrable functions of bounded variation, which we equip with the norm $\
f\
_{{\rm BV}_p({\rm X})}:=\
f\
_{L^p({\rm X})}+
Df
({\rm X})$. Moreover, we prove that the standard BV space ${\rm BV}({\rm X}):={\rm BV}_1({\rm X})$, which fails to have a predual for some choices of the metric measure space, does have a predual in the case where $({\rm X},{\sf d},\mathfrak m)$ is a PI space (i.e. a doubling metric measure space supporting a weak $(1,1)$-Poincaré inequality) of finite diameter. Along the way, we also develop a basic theory of BV functions in the setting of extended metric-topological measure spaces, which is of independent interest.