Preduals of metric BV spaces

TL;DR

This paper constructs explicit preduals for p-integrable BV spaces over metric measure spaces and shows the existence of a predual for the classical BV space in PI spaces, advancing the functional analysis of BV functions.

Contribution

It provides the first explicit construction of preduals for p-BV spaces and establishes the existence of a predual for BV spaces in PI spaces, extending the theory to extended metric-topological measure spaces.

Findings

Constructed isometric preduals for ${ m BV}_p({ m X})$ spaces.

Proved the existence of a predual for ${ m BV}({ m X})$ in PI spaces.

Developed foundational theory of BV functions in extended metric-topological measure spaces.

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 $(1,1)$-Poincar\'{e} 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.