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We study the slicing and fine properties of functions in$\mathrm{BV}^{\mathcal A}$, the space of functions with bounded $\mathcalA$-variation. Here, $\mathcal A$ is a homogeneous linear differential operatorwith constant coefficients (of arbitrary order). Our main result is thecharacterization of all $\mathcal A$ satisfying the following one-dimensionalstructure theorem: every $u \in \mathrm{BV}^{\mathcal A}$ can be sliced intoone-dimensional $\mathrm{BV}$-sections. Moreover, decomposing $\mathcal A u$into an absolutely continuous part $\mathcal A^a u$, a Cantor part $\mathcalA^c u$ and a jump part $\mathcal A^j u$, each of these measures can berecovered from the corresponding classical $D^a,D^c$ and $D^j$ $BV$-derivativesof its one-dimensional sections. By means of this result, we are able toanalyze the set of Lebesgue points as well as the set of jump points wherethese functions have approximate one-sided limits. Thus, proving a structureand fine properties theorem in $\mathrm{BV}^{\mathcal A}$. Our results extendmost of the classical fine properties of $\mathrm{BV}$ (and all of those knownfor $\mathrm{BD}$). In particular, we establish a slicing theory and fineproperties for $\mathscr {BV}^k, \mathrm{BD}^k$ and a whole class of$\mathrm{BV}^{\mathcal A}$-spaces that is not covered by the existing theory.

Slicing and fine properties for functions with bounded $\mathcal A$-variation