Conventions, defining kernels, and exact adjoints

Pairings are complex bilinear and use coordinate densities. For a trivial finite-dimensional bundle $E$, $\mathcal{D}(X;E^{\ast })={C_c}^{\infty }(X;E^{\ast })$ has its usual test topology and ${\mathcal{D}}^{\prime }(X;E)=\mathcal{D}(X;E^{\ast }{)}^{\prime }$. A bounded test set has common compact support and bounds on every fixed derivative order. Strong distribution seminorms are ${\sup }_{\Phi \in \mathcal{A}}|⟨u,\Phi ⟩|$ for bounded $\mathcal{A}$.

Spacetime indices range over $\{0,1,2,3\}$ and are summed when repeated. Component labels $I,J,L,M$ range over $\{1,\dots ,10\}$; finite component contractions are understood where indicated. Source multiindices are explicitly summed. The internal array has components $(I,J)$ and the output has components $J$.

For compact $H$, $\Vert \Phi {\Vert }_{j,H}$ is the maximum over components and derivatives of order at most $j$. All estimates concern tests supported in the indicated compact set; enlarged fixed neighborhoods may be used for coefficient norms.

Define

The first formula fixes the cone vertex convention.