Characterizations of signed measures in the dual of $BV$ and related isometric isomorphisms

Abstract

We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and $BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved by constructing a locally integrable function $f$ such that $f$ belongs to $BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dot W^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dot W^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a bounded open set $Ω$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(Ω)^*$. One of the goals of this paper is to make precise the definition of $BV_0(Ω)$, which is the space of functions of bounded variation with zero trace on the boundary of $Ω$. We show that the measures in $BV_0(Ω)^*$ coincide with the measures in $W^{1,1}_0(Ω)^*$. Finally, the class of finite measures in $BV(Ω)^*$ is also characterized.