Kernels and supports

Write $D_1$ for the kernel of $d_1$ xe:tests, with $\mathrm{supp}{D}_1\subset K_f\times (J^{-}(K_f)\cap \{t\ge 0\})$, and set

indices suppressed; the $y$-slot of $N_0$ is covariant, paired with sources. Let $\mathfrak{B}({\partial }_y,{\partial }_{y^{\prime }})$ be the constant-coefficient bidifferential operator with ${\mathcal{G}}_2[u,u](y)=\mathfrak{B}({\partial }_y,{\partial }_{y^{\prime }})[u(y)u(y^{\prime })]{|}_{y^{\prime }=y}$, and $\mathfrak{T}$ the analogous operator with $T[\varphi ]=\mathfrak{T}[\varphi (y)\varphi (y^{\prime })]{|}_{y^{\prime }=y}$. For a differential operator $D$ in $y$, $D^{†}$ is its formal transpose.