Lemma: Advanced root

For compact smooth $f$, the root is smooth, satisfies $P_{m}F=f$, and is the unique advanced solution. For fixed compact source support and every compact $L\subset M$, each $C^r(L)$-seminorm of $F$ is bounded by finitely many compact smooth seminorms of $f$.

Proof. The spatial Fourier transform of $f$ decays rapidly and uniformly on its time support. After taking sufficiently many spatial derivatives of $f$, this decay dominates all differentiated integrands. Time derivatives introduce the source boundary terms and polynomially bounded oscillatory factors. The Fourier equation is

with zero future data.

Finite propagation and uniqueness follow from the Klein–Gordon energy identity on a backward causal domain. For $m=0$, a standard auxiliary local $L^2$-term can be added to the energy estimate; it does not alter the finite-propagation conclusion. These arguments also give the stated seminorm estimates.