Proposition: Restricted extension on homogeneous and compact sources

On ${\mathcal{D}}_{\mathrm{int}}\oplus {\mathcal{H}}_T$, define

This is a well-defined real linear extension of the compact-interior operator. It gives the unique smooth zero-data solution of

It extends uniquely by complex linearity and componentwise.

On the witness,

Proof. The direct-sum decomposition is unique, so the definition is unambiguous. The compact part has the equation and initial data from (eq:compactGreen). The homogeneous part has them by direct differentiation. The sum therefore solves the stated Cauchy problem. Lemma lem:uniqueness proves uniqueness.

Both witness sources belong to the complexification of ${\mathcal{H}}_T$. The first formula in (eq:restricted-witness) is (eq:Vlambda). For the second, put $H^{\mu }=\sigma {V_{\lambda }}^{0\mu }$. Then ${F_{\mathrm{co}}}^{\mu }=(H^{\mu }{)}^{\prime }$ and $H^{\mu }(0)=0$, so

by integration by parts. Taking the negative gives (eq:Xcand).