Lemma: The kernel R

1. (eq:R) defines the positive measure $⟨R,\varphi ⟩=\frac{1}{4\pi }{{\int }_0}^{\infty }t{\int }_{S^2}\varphi (t,t\omega )\mathrm{d}\omega \mathrm{d}t$, equal to $\Theta (t)\delta (t-|\mathbf{x}|)/(4\pi |\mathbf{x}|)$; it is tempered, with $\mathrm{supp}R=\partial J^{+}$.

2. ${\mathcal{F}}_{\mathbf{x}}R(t,\mathbf{p})=\Theta (t)\sin (\rho t)/\rho$.

3. $\square R={\delta }_0$. If $R^{\prime }\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ satisfies $\square R^{\prime }={\delta }_0$ and $\mathrm{supp}{R}^{\prime }\subseteq \{t\ge c\}$ for some $c\in \mathbb{R}$, then $R^{\prime }=R$. In particular $R$ is the unique fundamental solution supported in $J^{+}$.

4. If $E:\mathcal{D}({\mathbb{R}}^4)\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ is linear with $\square E\varphi =\varphi$ and $\mathrm{supp}E\varphi \subseteq \{t\ge c_{\varphi }\}$ for every $\varphi$, then $E\varphi =R\ast \varphi$.

In particular (eq:R) agrees with XIV-I's pole chart Ipole, which characterizes $R$ by the transform in (ii).

Proof. (i) In polar coordinates $\mathbf{x}=r\omega$, for $t>0$ the composite $\delta (t^2-r^2)$ as a distribution in $r>0$ equals $\delta (r-t)/(2t)$, so $\frac{1}{2\pi }\int \Theta (t)\delta (t^2-r^2)\varphi r^2\mathrm{d}r\mathrm{d}\omega \mathrm{d}t=\frac{1}{4\pi }{{\int }_0}^{\infty }t{\int }_{S^2}\varphi (t,t\omega )\mathrm{d}\omega \mathrm{d}t$; the same computation with $\delta (t-r)/(4\pi r)$ gives the same value. The mass of $R$ in the ball of radius $\varrho$ is at most ${\varrho }^2/2$, so $R$ is tempered. The support is the cone $t=r$.

(ii) For $\psi \in \mathcal{S}({{\mathbb{R}}^4}_{(t,\mathbf{p})})$, by (i) and Fubini, $⟨R,{\mathcal{F}}_{\mathbf{x}}\psi ⟩={{\int }_0}^{\infty }\int \psi (t,\mathbf{p})\left[\frac{t}{4\pi }{\int }_{S^2}{e}^{-it\mathbf{p}\cdot \omega }\mathrm{d}\omega \right]{\mathrm{d}}^3\mathbf{p}\mathrm{d}t$, and ${\int }_{S^2}{e}^{-it\mathbf{p}\cdot \omega }\mathrm{d}\omega =4\pi \sin (\rho t)/(\rho t)$.

(iii) Let $u(t)=\sin (\rho t)/\rho$; $u(0)=0$, $u^{\prime }(0)=1$, $u^{\prime \prime }+{\rho }^{2}u=0$. Then $({{\partial }_t}^2+{\rho }^2)(\Theta u)=\Theta (u^{\prime \prime }+{\rho }^{2}u)+u(0){\delta }^{\prime }+u^{\prime }(0)\delta =\delta (t)$ for every $\mathbf{p}$, including $\rho =0$ where $u=t$. Since ${\mathcal{F}}_{\mathbf{x}}(\square R)=({{\partial }_t}^2+{\rho }^2){\mathcal{F}}_{\mathbf{x}}R$, $\square R={\delta }_0$. Uniqueness: $v=R-R^{\prime }$ satisfies $\square v=0$ and $\mathrm{supp}v\subseteq \{t\ge c\}$. The addition map is proper on $J^{+}\times \{t\ge c\}$: if $x\in J^{+}$, $y^0\ge c$ and $x+y$ lies in a compact set, then $x^0\le C-c$, $|\mathbf{x}|\le x^0$, so $x$ and hence $y$ are bounded. Thus $R\ast v$ is defined and $v={\delta }_0\ast v=(\square R)\ast v=\square (R\ast v)=R\ast (\square v)=0$ Horm (4.2).

(iv) $E\varphi -R\ast \varphi$ solves $\square (\cdot )=0$ with support in a half-space $\{t\ge c\}$; apply the argument of (iii).