Lemma: Properties of

Let ${\Delta }_{\mathrm{F}}={\Delta }_{+}+i{{\Delta }_S}^{\mathrm{A}}={\Delta }_{+}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, regarded as the Schwartz kernel in ${\mathcal{D}}^{\prime }(M\times M)$.

1. ${{\Delta }_{\mathrm{F}}}^{\mathsf{T}}={\Delta }_{\mathrm{F}}$, i.e.\ ${\Delta }_{\mathrm{F}}(x,y)={\Delta }_{\mathrm{F}}(y,x)$.

2. $(P_m\otimes 1){\Delta }_{\mathrm{F}}=-i{\delta }_{\Delta }$, where ${\delta }_{\Delta }$ is the diagonal delta, $⟨{\delta }_{\Delta },\psi ⟩=\int \psi (x,x){\mathrm{d}}^{4}x$. Equivalently $P_m({\Delta }_{\mathrm{F}}c)=-ic$ for $c\in \mathcal{D}$.

3. For $c\in \mathcal{D}$, both ${\Delta }_{+}c$ and ${\Delta }_{\mathrm{F}}c$ are smooth functions.

4. $\mathrm{WF}({\Delta }_{\mathrm{F}})\supset \{(x,x;\xi ,-\xi ):x\in M,\xi \ne 0\}$.

5. ${\Delta }_{\mathrm{F}}-{\Delta }_{+}=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ and ${\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathsf{T}}=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

Proof. (i) Using $({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{)}^{\mathsf{T}}={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$ and ${\Delta }_{+}^{\mathsf{T}}={\Delta }_{+}-i{\Delta }_S$,

(ii) ${\Delta }_{+}$ is a bisolution, so $(P_m\otimes 1){\Delta }_{+}=0$; $P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c=c$ for all $c$ says $(P_m\otimes 1){{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}={\delta }_{\Delta }$ for the kernel of ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. Hence $(P_m\otimes 1){\Delta }_{\mathrm{F}}=-i{\delta }_{\Delta }$.

(iii) ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is smooth by the Green operator property. For ${\Delta }_{+}c$: since $\mathrm{WF}({\Delta }_{+})$ contains no point $(x,\xi ;y,0)$ with $\xi \ne 0$, the kernel maps $\mathcal{D}$ into $C^{\infty }$ Hormander (Thm. 8.2.13).

(iv) ${\delta }_{\Delta }$ is the pullback of ${\delta }_0\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ under the submersion $(x,y)\mapsto x-y$, so $\mathrm{WF}({\delta }_{\Delta })=\{(x,x;\xi ,-\xi ):\xi \ne 0\}$ Hormander (Thm. 8.2.4). A differential operator with smooth coefficients does not enlarge the wavefront set Hormander ((8.1.11)), so $\mathrm{WF}({\delta }_{\Delta })=\mathrm{WF}((P_m\otimes 1){\Delta }_{\mathrm{F}})\subset \mathrm{WF}({\Delta }_{\mathrm{F}})$.

(v) The first is the definition. For the second, ${\Delta }_{\mathrm{F}}-{\Delta }_{+}^{\mathsf{T}}={\Delta }_{+}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{\Delta }_{+}+i{\Delta }_S=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}+i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}=-i{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.