Distributional conventions

Distributions and tests use a complex-bilinear pairing. Tensor components are taken in fixed finite-dimensional bases. Local coordinate-density conversions are included in smooth coefficients.

For a compact $L$ and a test supported in $L$, write

For tensor-valued tests take the maximum over components.

The strong distribution topology is generated by

where $\mathcal{B}$ is a bounded test set. Such a set has common compact support and bounded derivatives of every order.

For a closed cone $\Gamma \subset T^{\ast }U\setminus 0$, the normal topology on ${{\mathcal{D}}^{\prime }}_{\Gamma }(U)$ additionally uses

where $(\mathrm{supp}\chi \times V)\cap \Gamma =\varnothing$. The Fourier convention is

with inverse factor $(2\pi {)}^{-\dim x}$.

We use normal-topology pullback continuity, fixed-factor product continuity under the nonopposition condition, and proper-support pushforward continuity BDH (Proposition 5.1 and Theorems 6.1, 6.3). Their wavefront criteria are the standard distributional calculus Hormander (Section 8.2).