Lemma: Massless retarded convolution in four dimensions

With the normalization $\square {E_0}^{\mathrm{R}}=\mathrm{id}$,

The second equality is an equality of the already defined retarded convolution distributions.

Proof. The first formula is the retarded massless wave kernel in the declared normalization.

Let $\rho =|\mathbf{\xi }|$ and take the Fourier transform only in the spatial variables. The first kernel has transform

Its causal time convolution is

At $\rho =0$ the continuous value is $\Theta (t)t^3/6$.

The spatial Fourier transform of $\Theta (t)1_{\{|\mathbf{x}|\le t\}}/(8\pi )$ is the same expression, by radial integration over the ball of radius $t$. The formulas define locally tempered distributions, so equality of their spatial Fourier transforms proves the result.