Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in $L^1(\mathbb{R}^n)$ such that the distributional gradient if a vector valued finite measure)?



Dual or pre-dual of BV
Depending on the locally convex topology you take on $BV\_c(\mathbb{R}^n)$ (BV functions with essentially compact support in $\mathbb{R}^n$), its dual is given by so called charges (or strong charges) as defined in "De Pauw, Pfeffer - Distributions for which div v = F has a continuous solution - Comm. Pure Appl. Math., 61(2), 2008. 230-260". An electronic version of this paper is here:


<http://webusers.imj-prg.fr/~thierry.de-pauw/preprints/div.v.equals.F.pdf>


A charge is a particular kind of distribution and its collection is given the structure of a Fréchet space. Moreover, a distribution in $\mathcal{D}'(\mathbb{R}^n)$ is a charge if and only if it can be realized as the distributional divergence of a continuous vector field, i.e. an element of $C(\mathbb{R}^n,\mathbb{R}^n)$. This result is also stated in Theorem 11.3.8 of the book "The Divergence Theorem and Sets of Finite Perimeter" by Pfeffer (some attention is needed since what is called a charge in this book is called a strong charge in the paper mentioned above). I hope this goes in the direction you are looking for...


