Preduals of metric BV spaces
Enrico Pasqualetto
TL;DR
The work addresses the existence of isometric preduals for metric BV spaces, constructing a dual framework for BV_p(X) via a quotient of a tensor-product space by a divergence-consistency subspace. It proves BV_p(X) ≈ (W_q(X)/V_q(X))^* in locally compact settings and extends to noncompact spaces through Gelfand compactification, with BV on hat X matching BV on X. It shows BV equals BV_* under mild conditions, enabling weak-* convergence theory for BV; and in bounded PI spaces, BV(X) admits a predual via isoperimetric embeddings, while p=1 cases can fail in general. These results provide a robust functional-analytic foundation for BV on nonsmooth spaces and facilitate weak-* compactness in variational analysis.
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.