Lemma: The advanced kernel, conditionally

Let $m\ge 0$, $P_m=\square +m^2$, $\omega =\sqrt{{\rho }^2+m^2}$. The tempered distribution ${E^{\mathrm{A}}}_m$ with ${\mathcal{F}}_{\mathbf{x}}{E^{\mathrm{A}}}_m(t,\mathbf{p})=-\Theta (-t)\sin (\omega t)/\omega$ satisfies $P_m{E}_m^{\mathrm{A}}={\delta }_0$, $\mathrm{supp}{E^{\mathrm{A}}}_m\subseteq J^{-}$, and is the unique fundamental solution of $P_m$ supported in a half-space $\{t\le c\}$; for $m=0$, ${E^{\mathrm{A}}}_0(t,\mathbf{x})=R(-t,\mathbf{x})$. For $f\in {C^{\infty }}_c$, ${E^{\mathrm{A}}}_m\ast f\in C^{\infty }$ with support in $\mathrm{supp}f+J^{-}$.

Proof. XVFpatch (Lemma lem:advanced): the transform solves $({{\partial }_t}^2+{\omega }^2)(\cdot )=\delta (t)$ with no boundary terms at $t=0$; the support follows from Paley–Wiener–Schwartz Horm (Thm. 7.3.1) applied to the entire function $\sin (\omega t)/\omega$ of $\mathbf{p}\in {\mathbb{C}}^3$ with exponential type $|t|$; uniqueness is the convolution argument of XVFpatch (Lemma lem:R(iii)) with $J^{-}$.