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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:

  1. 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 $r$, $$\mu(B) \lesssim_d r^{d-1}.$$
  2. 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 $|\nu|(X)$.
  3. 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).
  4. 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).
  5. 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:

  1. 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.
  2. Are there more recent (say since 2015) interesting results on this problem?
  3. 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?
  4. 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.

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    $\begingroup$ Apparently $BV$ denotes functions of bounded variation here. The same symbols are used for various other things in other contexts, so it would be best to spell that out. $\endgroup$ – Neil Strickland Apr 27 at 8:02

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