It is well known that an integrable function $ u \colon \mathbb R^d \to \mathbb R$ is said to be of bounded variation 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 am not sure.