A look into some of the fine properties of functions with bounded $\mathscr A$-variation

Abstract

We establish certain fine properties for functions of bounded $\mathscr A$-variation known in the classical $BV$ setting. Here, $\mathscr A$ is a $k$th order constant-coefficient homogeneous linear differential operator with a finite-dimensional kernel (also known as a complex-elliptic operator). We prove that if $\mathscr Au$ can be represented by a finite Radon measure, then the potential $u$ has one-sided $L^p$-approximate limits on Lipschitz hypersurfaces, and, more generally, on countably rectifiable sets of codimension one. We use this to give pointwise characterizations of the (functional) interior and exterior traces. We also establish a quantitative scale-dependent continuity result, which allows us to prove that the Lebesgue discontinuity set has zero $(n-1)$-dimensional Riesz capacity. Lastly, we introduce a decomposition that reduces the complexity of analyzing $k$th-order operators to that of first-order methods and allows us to establish the $k$th order $L^p$-differentiability of $BV^{\mathscr A}$ maps.