Lemma: Properties of

The completion distribution

has the following properties:

1. ${{\Delta }_{\mathrm{F}}}^{\mathrm{T}}={\Delta }_{\mathrm{F}}$;

2. $P_{m,x}{\Delta }_{\mathrm{F}}=-i{\delta }_{\mathrm{diag}}$;

3. ${\Delta }_{\mathrm{F}}c$ is smooth for $c\in \mathcal{D}$;

4. $$\mathrm{WF}({\Delta }_{\mathrm{F}})\supset \{(x,x;\xi ,-\xi ):x\in \hat{M},\xi \ne 0\};$$
5. $${\Delta }_{\mathrm{F}}-{\Delta }_{+}=-i{\mathsf{E}}_{\mathrm{A}},{\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathrm{T}}=-i{\mathsf{E}}_{\mathrm{R}}.$$

The transpose symmetry in (i) is not a Hermiticity assertion.

Proof. Use ${{\mathsf{E}}_{\mathrm{A}}}^{\mathrm{T}}={\mathsf{E}}_{\mathrm{R}}$ and ${\Delta }_{+}^{\mathrm{T}}={\Delta }_{+}-i({\mathsf{E}}_{\mathrm{A}}-{\mathsf{E}}_{\mathrm{R}})$:

The bisolution equation and $P_m{\mathsf{E}}_{\mathrm{A}}=I$ give (ii). The no-zero-slot wavefront condition implies that ${\Delta }_{+}$ maps compact smooth tests to smooth functions; adding $-i{\mathsf{E}}_{\mathrm{A}}c$ gives (iii).

The wavefront set of the diagonal delta is

A differential operator with smooth coefficients does not enlarge wavefront sets. Hence (ii) implies

which proves (iv). These are standard kernel and wavefront facts Hormander (Chapter 8). The last identities follow directly from the definition and (eq:Hadamard).