On Phase-Space Orthogonality for Higher-Order Distributions: Obstructions and Algebraic Resolutions

Abstract

The extension of microlocal defect measures to the $L^p$-$L^q$ framework necessitates the use of distributions which possess strictly positive finite order. A topological $\eps$-bounding definition of orthogonality for such distributions was recently introduced by Antonić, Mitrović, and Perić. Building on their framework, we systematically map the obstructions that arise when this natural approach is transported to strictly positive order, and we provide a complementary algebraic resolution. We first establish that while smooth spatial multipliers provide a robust framework for separating order-zero Radon measures, the partition-of-unity route to phase-space geometric separation breaks down for higher-order distributions: the separating frequency multiplier suffers an $\mathcal{O}(δ^{-κ})$ derivative divergence when frequency supports touch, which we prove defeats the a priori bound at every order $κ\ge 1$ and produces an explicit non-orthogonality already at order one. Transitioning to algebraic phase-space projectors reveals a second severe obstruction: it is mathematically impossible to separate scalar distributions algebraically, as one-dimensional smooth projectors enforce topological triviality. We demonstrate that the resolution is achieved exclusively within vector-valued PDE systems. Rather than relying on arbitrarily constructed, non-physical geometric scalar cut-offs $χ(x,ξ)$ to separate distinct modes, we redefine orthogonality canonically via zero-order algebraic matrix projectors. This replaces artificial geometric separation with the intrinsic spectral polarization of the governing system, yielding a coordinate-free microlocal orthogonality for systems of higher-order distributions.