Theorem: Exact surface support for the flat realization

For $f\in {C_c}^{\infty }(M_{+}^{\circ })$,

Proof. The lower inclusion follows from Theorem thm:sphere. For the upper inclusion, every source $a=-f\mathrm{d}\varphi$ has support in $F$. The advanced representation above gives $\mathrm{supp}b\cap M_{+}\subset {\mathcal{N}}_F$. The local differential operator $C^{\ast }$ does not enlarge this support. The surface term has support in $\{0\}\times {\mathcal{S}}_F$.

The scalar derivative has support in $F$, which is separated from $\Sigma$. Thus all first-derivative distributions are supported in the fixed closed set ${\mathcal{N}}_F\cup F$, whose intersection with $\Sigma$ is $\{0\}\times {\mathcal{S}}_F$. Their union and its closure cannot acquire another surface point.