It is well known that an integrable function $u \colon \mathbb R^d \to \mathbb R$ is said to be of [bounded variation](https://en.wikipedia.org/wiki/Bounded_variation#BV_functions_of_several_variables) iff the distributional gradient $Du$ is (representable by) a finite Radon measure, still denoted by $Du$. 


Then it is also well known that the measure $Du$ can be decomposed into three terms, $D^{a}u, D^{j}u, D^c u$ resp. absolutely continuous part, jump part and Cantor part. Quite a plethora of fine results are present in the literature (see, e.g. the monography by Ambrosio-Fusco-Pallara also for the notation below, which is however quite standard). For instance, $D^au= \nabla u \mathscr L^d$ (being $\nabla u$ the approximate differential) and $D^j u = (u^+-u^-) \otimes \nu\_{J\_u} \mathscr H^{d-1}$.


On $D^c u$ little is present: it is always generically said that, as a measure, is something intermediate between the jump part $\mathscr H^{d-1}$ and the a.c. part $\mathscr L^d$. As an example, it is always considered the Cantor-Vitali staircase (the devil's function), whose derivative has only Cantor part and is exactly $\mathscr H^{\alpha}$, for $\alpha = \log\_3 2$ which is - incidentally - the Hausdorff dimension of the standard Cantor set. 


So here is my question: 



> 
> Is it always true that $D^c u$ is (absolutely continuous w.r.t.) $\mathscr H^\alpha$ for suitable $\alpha \in (d-1, d)$? In other words, is $D^c u$ always an Hausdorff measure (up to a density) restricted to some Cantor-like set of certain dimension $\alpha \in (d-1,d)$? 
> 
> 
> 


That should have something to do with densities and Besicovitch Theorem... but I do not know.



Structure of the Cantor part of the derivative of a BV function
Certainly not with a single $\alpha$, but it is tempting to decompose $D^cu$ into an integral $\int\_{d-1}^d\mu\_\alpha\ d\nu(\alpha)$ with $\mu\_\alpha$ having a density with respect to $\mathcal H^\alpha$ on an $\alpha$-dimensional set. I don't know if it is always possible.


