Lemma: Sum-zero calculus

1. $\mathrm{WF}({{\mathsf{E}}_0}^{\mathrm{R}})\subset {\Sigma }_2$; $\mathrm{WF}({{\Delta }_{\mathrm{R}}}^{\mathrm{gr}})\subset {\Sigma }_2$; $\mathrm{WF}(\delta (y-y^{\prime }))\subset {\Sigma }_2$.

2. If $\mathrm{WF}(u)\subset {\Sigma }_n$, then $\mathrm{WF}(Du)\subset {\Sigma }_n$ for every differential operator $D$ with smooth coefficients on $M^n$, and $\mathrm{WF}(\psi u)\subset {\Sigma }_n$ for smooth $\psi$.

3. If $u\in {\mathcal{D}}^{\prime }(M^2)$ with $\mathrm{WF}(u)\subset {\Sigma }_2$, then the product $v(x,y,y^{\prime }):=u(x,y)\delta (y-y^{\prime })$ exists and $\mathrm{WF}(v)\subset {\Sigma }_3$.

4. If $A\in {\mathcal{D}}^{\prime }(X\times Z)$ is compactly supported with $\mathrm{WF}(A)\subset {\Sigma }_2$, and $B\in {\mathcal{D}}^{\prime }(Z\times Y)$ with $\mathrm{WF}(B)\subset {\Sigma }_2$, then the composition $A\circ B(x,y)=\int A(x,z)B(z,y)\mathrm{d}z$ exists and $\mathrm{WF}(A\circ B)\subset {\Sigma }_2$.

5. ${\Sigma }_n\cap ({\overline{V}}_{+}^n\cup {\overline{V}}_{-}^n)=\varnothing$.

Proof. (a) By xe:algebra, every element of $\mathrm{WF}({{\mathsf{E}}_0}^{\mathrm{R}})$ has the form $(x,\xi ;y,-\xi )$, $\xi \ne 0$; ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ is a constant matrix times ${{\mathsf{E}}_0}^{\mathrm{R}}$. $\delta (y-y^{\prime })=(2\pi {)}^{-4}\int e^{i(y-y^{\prime })\cdot \eta }\mathrm{d}\eta$ has $\mathrm{WF}=\{(y,\eta ;y,-\eta ):\eta \ne 0\}$. (b) $\mathrm{WF}(Du)\subset \mathrm{WF}(u)$ and $\mathrm{WF}(\psi u)\subset \mathrm{WF}(u)$. (c) $\mathrm{WF}(u\otimes 1_{y^{\prime }})\subset \{(x,\xi ;y,\eta ;y^{\prime },0):(x,\xi ;y,\eta )\in \mathrm{WF}(u)\}$ and $\mathrm{WF}(1_x\otimes \delta )\subset \{(x,0;y,\eta ;y^{\prime },-\eta ):\eta \ne 0\}$ Hor (Thm. 8.2.9). A sum of an element of the first set and an element of the second is $(\xi ,{\eta }_1+{\eta }_2,-{\eta }_2)$ with $\xi +{\eta }_1=0$; it vanishes only if $\xi ={\eta }_1={\eta }_2=0$, which is excluded. So the product exists Hor (Thm. 8.2.10), and its wavefront set is contained in the set of such sums with one summand possibly zero; every such sum has total covector $\xi +{\eta }_1+{\eta }_2-{\eta }_2=0$. (d) With ${\mathrm{WF}}^{\prime }(A)=\{(x,\xi ;z,-\zeta ):(x,\xi ;z,\zeta )\in \mathrm{WF}(A)\}=\{(x,\xi ;z,\xi )\}$ and likewise for $B$, the exceptional sets ${\mathrm{WF}}_X(A)=\{(x,\xi ):(x,\xi ;z,0)\in \mathrm{WF}(A)\}$ and ${\mathrm{WF}}_Z(A),{\mathrm{WF}}_Z(B),{\mathrm{WF}}_Y(B)$ are empty, because a zero second covector forces a zero first covector. Since $A$ has compact support, Hor (Thm. 8.2.14) applies and gives ${\mathrm{WF}}^{\prime }(A\circ B)\subset {\mathrm{WF}}^{\prime }(A)\circ {\mathrm{WF}}^{\prime }(B)=\{(x,\xi ;y,\xi )\}$, i.e.\ $\mathrm{WF}(A\circ B)\subset {\Sigma }_2$. (e) ${\overline{V}}_{+}$ is a closed convex cone with ${\overline{V}}_{+}\cap (-{\overline{V}}_{+})=\{0\}$. If ${\xi }_i\in {\overline{V}}_{+}$ and $\sum {\xi }_i=0$, then ${\xi }_1=-{\sum }_{i\ge 2}{\xi }_i\in {\overline{V}}_{+}\cap (-{\overline{V}}_{+})$, so ${\xi }_1=0$; inductively all ${\xi }_i=0$, contradicting $({\xi }_i)\ne 0$. Same for ${\overline{V}}_{-}$.