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

Abstract

We characterize all (signed) measures in $B{V}_{\frac{n}{n-1}}({\mathbb{R}}^n{)}^{\ast }$, where $B{V}_{\frac{n}{n-1}}({\mathbb{R}}^n)$ 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 $B{V}_{\frac{n}{n-1}}({\mathbb{R}}^n{)}^{\ast }$ and $BV({\mathbb{R}}^n{)}^{\ast }$ 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 $BV({\mathbb{R}}^n{)}^{\ast }$ but $|f|$ does not. Moreover, we show that the measures in $B{V}_{\frac{n}{n-1}}({\mathbb{R}}^n{)}^{\ast }$ coincide with the measures in ${\dot{W}}^{1,1}({\mathbb{R}}^n{)}^{\ast }$, 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 $B{V}_0(\Omega {)}^{\ast }$. One of the goals of this paper is to make precise the definition of $B{V}_0(\Omega )$, which is the space of functions of bounded variation with zero trace on the boundary of $\Omega$. We show that the measures in $B{V}_0(\Omega {)}^{\ast }$ coincide with the measures in $W_0^{1,1}(\Omega {)}^{\ast }$. Finally, the class of finite measures in $BV(\Omega {)}^{\ast }$ is also characterized.

26 pages