Existence and uniqueness of continuous scalar Green maps on smooth globally hyperbolic spacetimes

(Retarded construction.) Fix the smooth time $t$ of [F1]. For nonzero compact test $f$, choose $a<\min t(\mathrm{supp}f)$ and solve $Pu=f$ with zero data on $S_a$ using [F1]. Below that slice the source and both data vanish, so uniqueness and its time-reversed support assertion give $u=0$. Above it the one-sided support assertion gives $\mathrm{supp}u\subseteq J^{+}(\mathrm{supp}f)$. Define $E_{+}f=u$ and $E_{+}0=0$. A different sufficiently early slice gives the same solution: both solutions have zero data on a still earlier common slice and the same source, so [F1] uniqueness identifies them. Reversing time constructs $E_{-}f$ from zero data on any slice later than the support and gives the past support.

(Two-sided inverse identities and uniqueness.) The construction gives $P{E}_{\pm }f=f$. If $k$ is compact, locality gives $\mathrm{supp}Pk\subseteq \mathrm{supp}k$. The compact smooth function $k$ and $E_{+}(Pk)$ both solve $Pu=Pk$ and have zero data on a common slice earlier than $\mathrm{supp}k$; uniqueness gives $E_{+}Pk=k$. The advanced identity is time reversed. If any other smooth retarded solution of $Pu=f$ has support in $J^{+}(\mathrm{supp}f)$, it has zero data on that earlier slice and equals $E_{+}f$ by [F1]. The same holds for advanced solutions, proving uniqueness of the maps. Common early or late slices for a finite list of tests and uniqueness give additivity and real homogeneity, hence real linearity.

(Continuity in the actual topologies.) Fix compact input support $K$ and compact output set $L$ with a specified derivative order. Choose $a<\min t(K)$ and a finite slab containing $S_a$ and $L$; if $K$ is empty the restriction is zero. Step 1.1 constructs every $E_{+}f$ for $\mathrm{supp}f\subseteq K$ from zero data on this same $S_a$. The estimate in [F1] therefore bounds the selected output seminorm by a constant times finitely many fixed-$K$ input derivative seminorms. This proves continuity of $E_{+}$ restricted to each fixed-compact test space. The LF topology in [F2] is the locally convex inductive-limit topology of these spaces, so a linear map into the locally convex space $C^{\mathrm{\infty }}(M)$ is continuous precisely when all such restrictions are continuous. Thus $E_{+}$ is continuous; the late-slice estimate gives continuity of $E_{-}$. No sequential-continuity substitute or unsupported closed-graph argument is used.

(The remaining interface conditions.) For compact smooth $f$ and arbitrary smooth $u$, the coordinate divergence expression for ${\mathrm{\square }}_g$ and integration by parts give ${\int }_{M}f(-{\mathrm{\square }}_{g}u){\mu }_g={\int }_M{g}^{ab}({\mathrm{\partial }}_{a}f)({\mathrm{\partial }}_{b}u){\mu }_g={\int }_M(-{\mathrm{\square }}_{g}f)u{\mu }_g\mathrm{.}$ These identities are justified by localizing the compact support to finitely many charts using [F2]: the extra partition derivatives sum to zero, and each chart integration has compact support. Multiplication by the real $V$ is symmetric, proving formal self-adjointness. Arbitrary smooth spacelike Cauchy-surface uniqueness is the last conclusion of [F1]. Together with steps 1.1–3.1 these establish every condition in the defined Green-data interface for the full stated class of backgrounds.