Verification of the reproduced displays

Throughout, $\eta =\mathrm{diag}(+,-,-,-)$, $\square ={\eta }^{\lambda \kappa }{\partial }_{\lambda }{\partial }_{\kappa }={{\partial }_t}^2-\Delta$ on ${\mathbb{R}}^4$, $J^{+}=\{(t,\mathbf{x}):t\ge |\mathbf{x}|\}$, $J^{-}=-J^{+}$, $\partial J^{+}=\{t=|\mathbf{x}|\}$, and ${\mathcal{F}}_{\mathbf{x}}$ the partial Fourier transform in $\mathbf{x}$, ${\mathcal{F}}_{\mathbf{x}}\varphi (t,\mathbf{p})=\int e^{-i\mathbf{p}\cdot \mathbf{x}}\varphi (t,\mathbf{x}){\mathrm{d}}^3\mathbf{x}$, extended to ${\mathcal{S}}^{\prime }({\mathbb{R}}^4)$ as an isomorphism; $\rho =|\mathbf{p}|$.