Lemma: Boundary injectivity and noncancellation

The map $v\mapsto \mathcal{B}(v)$ is injective. In the present signature,

If $b(0,{\mathbf{x}}_{\ast })\ne 0$, every nonzero normal covector at $(0,{\mathbf{x}}_{\ast })$ belongs to the wavefront set of at least one component of $q_a$.

Proof. The displayed components prove injectivity.

For the wavefront assertion, choose a component whose delta coefficient is nonzero at the point. After localization that component has the form

with $v,r$ smooth and $v$ not identically zero near the point. At fixed tangential Fourier frequency the delta term has a nonzero normal-frequency coefficient of order zero. The step term is of order at most $|\tau {|}^{-1}$. It cannot cancel the delta coefficient in either nonzero normal direction.