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

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 $B{V}^{\mathbb{A}}$-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.

23 pages