Theorem: Green identities and spectral exactness

For every restricted ledger extension $B$:

1. Compactly supported smooth spacetime sections belong continuously to ${{\mathcal{D}}_{\mathrm{tc}}}_B$, and

   $${G_{\pm }}^B:{{\mathcal{D}}_{\mathrm{tc}}}_B⟶{\mathcal{E}}_B$$

   are continuous.

2. With $P_B={{\partial }_t}^2+B$,

   $$P_B{G}_{\pm }^{B}f={G_{\pm }}^B{P}_{B}f=f(f\in {{\mathcal{D}}_{\mathrm{tc}}}_B).$$

   On compactly supported smooth sections, $P_B=P_m$.

3. $G_{+}^{B}f$ vanishes before the time support of $f$, and $G_{-}^{B}f$ vanishes after it.

4. The sequence

   $$0⟶{{\mathcal{D}}_{\mathrm{tc}}}_B\stackrel{P_B}{\to }{{\mathcal{D}}_{\mathrm{tc}}}_B\stackrel{E_B}{\to }{\mathcal{E}}_B\stackrel{P_B}{\to }{\mathcal{E}}_B⟶0$$

   is exact.

5. In the Hermitian spacetime pairing,

   $$⟨g,G_{+}^{B}f⟩=⟨G_{-}^{B}g,f⟩,⟨g,E_{B}f⟩=-⟨E_{B}g,f⟩(f,g\in {{\mathcal{D}}_{\mathrm{tc}}}_B).$$

   These are formal adjoint identities on the stated test domain, not assertions of self-adjointness or skew-adjointness of unbounded spacetime operators.

6. Every ${G_{\pm }}^{B}f$ is smooth on the regular spacetime $M$.

Proof. The spatial differential expression maps compactly supported smooth sections to themselves. Every such section therefore lies in every power domain of $B$. The same statement with compact-support seminorm estimates proves the asserted continuous inclusion.

On compact time intervals the multipliers $w_t$ and ${\partial }_t{w}_t$ are bounded uniformly on the spectrum; higher time derivatives are controlled by finitely many additional graph seminorms. Differentiation under the integral in (eq:Green) consequently gives

on ${{\mathcal{H}}^{\infty }}_B$. Duhamel differentiation proves the right inverse identities. Integration by parts in $s$, using compact time support, proves the left inverse identities. The support-in-time assertions are immediate.

For completeness, the homogeneous Cauchy problem is unique in ${\mathcal{E}}_B$: applying bounded spectral projections of $B$ reduces it to a bounded-operator Cauchy problem; the projections tend strongly to the identity. Its solution with data $u_0,u_1\in {{\mathcal{H}}^{\infty }}_B$ is

If $P_{B}u=0$ and $u$ has compact time support, then $u=G_{+}^B{P}_{B}u=0$, so the first arrow in (eq:exact) is injective. The inverse identities give $E_B{P}_B=0$ and $P_B{E}_B=0$. If $E_{B}f=0$, then $u=G_{+}^{B}f=G_{-}^{B}f$ has time support in the compact interval spanning ${\mathrm{supp}}_{t}f$, and $P_{B}u=f$. This proves exactness at the second space.

If $P_{B}u=0$, choose $\chi \in C^{\infty }(\mathbb{R})$ equal to zero for $t<0$ and one for $t>1$, and set

Cauchy uniqueness gives

hence $E_{B}f=u$. Finally, for arbitrary $v\in {\mathcal{E}}_B$,

is a smooth spectral-domain solution of $P_{B}u=v$. This proves the last surjectivity.

Since $W_B(t{)}^{\ast }=W_B(t)$ and $W_B(-t)=-W_B(t)$, Fubini's theorem on the compact time supports of the paired tests gives the adjoint identities.

Interior elliptic regularity gives continuous local embeddings

Together with the spectral time regularity, these imply smoothness on $M$.