Preduals of metric BV spaces

We study the predual of the space of functions of bounded variation defined over a metric measure space $(\mathrm{X},\mathsf{d},\mathfrak{m})$ with $\mathfrak{m}$ finite. More specifically, for any exponent $p\in (1,\mathrm{\infty })$ we construct an isometric predual of the space ${\mathrm{B}\mathrm{V}}_p(\mathrm{X})$ of $p$-integrable functions of bounded variation, which we equip with the norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathrm{B}\mathrm{V}}_p(\mathrm{X})}:=\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathrm{X})}+\mathrm{\mid }Df\mathrm{\mid }(\mathrm{X})$. Moreover, we prove that the standard BV space $\mathrm{B}\mathrm{V}(\mathrm{X}):={\mathrm{B}\mathrm{V}}_1(\mathrm{X})$, which fails to have a predual for some choices of the metric measure space, does have a predual in the case where $(\mathrm{X},\mathsf{d},\mathfrak{m})$ 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.