Complex-ellipticity, dimensional estimates and plane wave rigidity in $BV^{\mathbb A}$

Abstract

We characterize complex-elliptic operators $\mathbb A(D)$ through a hierarchy of overdeterminacy ($\ell$-vanishing) quantifying the structural twisting of their symbols. This framework yields the optimal dimensional estimate for $BV^{\mathbb A}$-functions: a measure ${\mathbb A} u$ cannot concentrate on sets of dimension below $n-1$. Consequently, the jump part of ${\mathbb A} u$ is characterized as an $(n-1)$-dimensional surface measure with density given by the symbol and the two-sided traces. Building on this dimensional bound, we prove that measures satisfying $\frac{{\mathbb A} u}{|{\mathbb A} u|} \in \mathrm{span}\{P_0\}$ precisely decompose into finite sums of one-dimensional $BV$ profiles. Ultimately, these results reveal that complex-ellipticity strictly enforces a plane-wave structure on tangent measures.