Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.

Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and that its density is in $L^\infty$.

What does this imply about the derivative of $f$? For example, about its Cantor part?

**Update 1:** Is it true that $\mathrm{div}\, f$ is equal to the trace of $D_S f$ (that is to the trace of the rank-one matrix $M$ such that $D_S f = M|D_S f|$)? Why?

**Update 2:** What is the form of the rank-one matrix $M$ such that $D_S f = M|D_S f|$? That is, what are its entries in general? What are its entries in the case $\mathrm{div} \, f$ is absolutely continuous with respect to the Lebesgue measure?