I'm going to let $BV := BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the problem of characterizing $BV^*$, the dual of $BV$, which is still an open problem.

Here are a few of the partial results that are out there:

- Williams and Ziemer: A positive measure $\mu$ is in $BV^*$ if and only if, for any open ball $B \subset \mathbb{R}^d$ of radius $r$, $$\mu(B) \lesssim_d r^{d-1}.$$
- Phuc and Torres: A signed measure $\nu$ belongs to $BV^*$ if and only if there exists a bounded vector field $\mathbf{F} \in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$ such that $$\mathrm{div} \ \mathbf{F} = \nu.$$ Additionally, they showed that $\nu \in BV^*$ if and only if, for any open or closed $X \subset \mathbb{R}^d$ with smooth boundary, $$|\nu(X)| \lesssim \mathcal{H}^{d-1}(\partial X).$$ Note that this is $|\nu(X)|$ as opposed to $|\nu|(X)$.
- There are a smattering of results regarding $BV_{\frac{d}{d-1}}^*$, where $BV_{\frac{d}{d-1}}$ denotes the space of all $u \in L^\frac{d}{d-1}$ whose distributional gradient $\nabla u$ is a vector-valued measure on $\mathbb{R}^d$ satisfying $|\nabla u|(\mathbb{R}^d) < +\infty.$ Said results are of interest because $BV^*$ is isometrically isomorphic to $BV_\frac{d}{d-1}^*$ (Phuc and Torres).
- De Pauw proved some results on $SBV^*$ including an independence result (representing the action of a continuous linear functional on the jump part of the distributional gradient of a $BV$ function as the flux of a $\mathcal{H}^{d-1}$-measurable vector field through the jump set is independent of ZFC).
- Phuc and Torres showed that $\mu \in BV(\Omega)^*$ if and only if $$|\mu(X)|\lesssim \mathcal{H}^{d-1}(\partial X)$$ for all smooth open $X \subset \mathbb{R}^d$, where $\mu$ is extended by zero to $\mathbb{R}^d-\Omega$.

Monica Torres has an interesting survey article [1] discussing the problem, and includes a discussion of most of the above results.

Now to my questions:

- What is still unknown regarding the problem? Based on what little I know, I might guess that characterizing the elements of $BV^*$ that aren't measures is at least partially open.
- Are there more recent (say since 2015) interesting results on this problem?
- Has $SBV^*$ been characterized completely? I don't think the De Pauw paper did this, but I could be wrong as I haven't worked through it that far (this isn't the primary area I work in). If not, what's left? If so, references?
- I'd love to hear any other thoughts: approaches being taken to resolve the remaining open questions, the particular obstructions to extending current results, who is doing interesting work in this area, etc.

**Reference**

[1] Monica Torres, "On the dual of BV". (English) in Galaz-García, Fernando (ed.) et al., *Contributions of Mexican mathematicians abroad in pure and applied mathematics. Second meeting “Matemáticos Mexicanos en el Mundo”, Centro de Investigación en Matemáticas, Guanajuato, Mexico, December 15–19, 2014*. Providence, RI: American Mathematical Society (AMS); México: Sociedad Matemática Mexicana, Contemporary Maththematics vol. 709, 115-129 (2018), DOI: 10.1090/conm/709/14296, MR3826951, Zbl 1427.46019.