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

Understand

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.