Lemma: Kernel lemma

Let ${\chi }_i$ denote the function $\chi$ of Lemma lem:spcompact in the time variable of slot $i$, with $T_0$ chosen so that $\mathrm{supp}w\subset \{|t_i|<T_0,i=1,2,3\}$. Then for every $w\in {\mathcal{E}}^{\prime }$,

Consequently

Proof. {${\mathcal{I}}_{\mathrm{sol}}\subset \ker$.} For $r\in {\mathcal{E}}^{\prime }$, ${{\mathsf{E}}^{\mathrm{R}}}_{\mu ,i}({\square }_i+{\mu }^2)r=r={{\mathsf{E}}^{\mathrm{A}}}_{\mu ,i}({\square }_i+{\mu }^2)r$, so ${\mathsf{E}}_{\mu ,i}({\square }_i+{\mu }^2)r=0$; the other slot operators then give ${\mathsf{W}}_{({\square }_i+{\mu }^2)r}=0$.

One slot. For $w$ compactly supported in the slot-$i$ direction, with $P_i$ the slot-$i$ operator (${\square }_i$ or $P_{m,3}$), write

The support of $(1-{\chi }_i){{\mathsf{E}}_{\mathsf{i}}}^{\mathrm{R}}w$ lies in $J^{+}(\mathrm{supp}w)\cap \{t_i\le T_0\}$ in slot $i$, and in the slot projections of $\mathrm{supp}w$ otherwise; it is compact. Likewise for ${\chi }_i{{\mathsf{E}}_{\mathsf{i}}}^{\mathrm{A}}w$. Hence

so $T_{i}w\in w+P_i{\mathcal{E}}^{\prime }$, $T_{i}w$ is compactly supported, and $T_{i}w$ depends on $w$ only through ${\mathsf{E}}_{i}w$.

Three slots. Apply (eq:Ti) successively: $w\equiv T_{1}w\equiv T_2{T}_{1}w\equiv T_3{T}_2{T}_{1}w$ modulo ${\mathcal{I}}_{\mathrm{sol}}$, each step legitimate because the previous output is compactly supported. Since ${\mathsf{E}}_j$, ${\chi }_j$ and $P_j$ commute with ${\mathsf{E}}_i$, ${\chi }_i$ and $P_i$ for $i\ne j$ (they act on different variables, and ${\chi }_i{\mathsf{E}}_{i}w$ is compactly supported in every slot other than $i$, so the slot-$j$ operators are defined on it),

which proves (eq:decomp). If ${\mathsf{W}}_w=0$ the bracket vanishes and $w\in {\mathcal{I}}_{\mathrm{sol}}$, proving (eq:kernel).