I was reviewing some lectures and saw that the teacher told us that, for a vector valued measure $\mu$, it's not true that they are the distributional derivative of some function $u$. It should hold:
\begin{equation}\mu=D u \text{ with } u \in L^1 _{loc} \iff \frac{\partial \mu_i}{\partial x_j}= \frac{\partial \mu_j}{\partial x_i} \; \forall i,j.\end{equation} In particular, he said that not all vector valued functions are representable as a sum of three components: a $W^{1,1} $part, an atomic part and a Cantor-like part. What is true, though, is that every measure (maybe Radon? I don't remember the exact hypotheses of Radon-Nykodim) can be decomposed in such a way. In dimension 1 it holds instead
$$u= u^1 + u^2 + u^3$$
with $u^1 \in W^{1,1}$, $u^2$ constant and composed of jumps only and $u^3$ a Cantor-like part.
The question is: how can I see the decomposition is not possible for vector valued functions? A example he gave is to try to decompose $u$ such that $u$ is $C^1$ in $\bar{B_1}$, not constant in $\partial B_1$ and $0$ in $\bar{B_1}^c$. Then you can't write $u$ as a decomposition like $u= u^1 + u^2 + u^3$ like before because, if my intuition is correct, the fact that $u$ is not constant in $\partial B_1$ prevents such a decomposition. In this case there's no Cantor part (not quite sure how to see this, nor about what the Cantor-part of a function should look like), so assume
$$ u= u^a + u^j $$
with $ u^a \in W^{1,1}$ and $u^j$ constant except for jump sets.
The idea I came up with (not sure about how rigorous/precise this is) is that if such a decomposition were possible, then we'd also have:
$$ Du = Du^a + Du^j$$
and since $u \equiv 0$ on $\bar{B_1} ^c$ we'd have $Du^j=0 $ on $\mathbb{R}^2 \setminus \bar{B_1}^c$. This means that $u^j$ is constant in that set because it is connected (it's here that the dimension greater than 1 comes into play) and so since $u^j \equiv 0 $ in $B_1$ we'd have $u= u^a + c$ for some constant $c$, which is absurd because I'd end up with $u \in W^{1,1}$.
Now, regardless of this example, do you have any suggestions on how to prove the iff at the beginning?