Lemma: Actual conormal singularities

For every $a$ as above,

If $b(0,{\mathbf{x}}_0)\ne 0$, then

In particular, the hypersurface delta term cannot be cancelled by the step-function term in (eq:q-adjoint).

Proof. Both coefficients in (eq:q-adjoint) are smooth. The standard wavefront bounds for a smooth multiple of a step or delta distribution therefore give (eq:q-wf-upper); see BDHwavefront.

For the lower bound, choose a tensor component for which $\mathsf{B}(b(0,{\mathbf{x}}_0))\ne 0$, possible by Lemma lem:B-injective. Locally that component is

with $r,v$ smooth.

After compact localization, Fourier transformation in the normal variable has the form

at fixed tangential frequency $\mathbf{\zeta }$, in either normal direction. For any localization nonzero at $(0,{\mathbf{x}}_0)$, the localized $v$ is not identically zero, so its tangential Fourier transform is nonzero at some fixed $\mathbf{\zeta }$. Thus rapid decay fails in every conic neighbourhood of each nonzero normal covector. The step term has one lower normal Fourier order and cannot cancel the nonzero leading term.