Spaces and topologies

Use

Let $U⋐M^4$ be open. Let $E_{\mathrm{out}}$ be a fixed finite-dimensional complex vector space and set

All component pairings are complex-bilinear, with a fixed trivialization and the displayed flat coordinate densities.

The test space is

For $\mathrm{supp}\Phi \subset L⋐U$, define

No restriction is made to factorized tests, selected witness families, or normal annuli.

Strong convergence in ${\mathcal{T}}^{\prime }$ means uniform convergence on bounded subsets of $\mathcal{T}$. A bounded subset has common compact support and bounded derivative seminorms of every order.

When wavefront-constrained spaces are used, the normal topology means the strong distributional seminorms together with the localized Fourier seminorms outside the declared cone BDH (Section 3).

For local coordinate Fourier transforms, fix

where $w\cdot \xi$ is the coordinate covector pairing. Minkowski index raising is separately performed with $\eta$. An identification with another convention must record the corresponding sign conversion.