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

TL;DR

This paper characterizes all signed measures in the dual of certain BV spaces, establishes isometric isomorphisms between these duals, and resolves longstanding questions about measure representations and boundary conditions in BV spaces.

Contribution

It provides a comprehensive characterization of measures in the duals of BV and related spaces, clarifies their relationships, and addresses open problems in measure representation and boundary trace definitions.

Findings

Measures in $BV_{n/(n-1)}$ dual coincide with those in $ abla W^{1,1}$ dual.

Established isometric isomorphisms between dual spaces of BV and Sobolev spaces.

Resolved an open issue by constructing a measure in $BV^*$ with non-integrable absolute value.

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 $\Omega$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(\Omega)^*$. One of the goals of this paper is to make precise the definition of $BV_0(\Omega)$, which is the space of functions of bounded variation with zero trace on the boundary of $\Omega$. We show that the measures in $BV_0(\Omega)^*$ coincide with the measures in $W^{1,1}_0(\Omega)^*$. Finally, the class of finite measures in $BV(\Omega)^*$ is also characterized.