Proposition: Off-diagonal singularity and scaling

On the region $\mathbf{x}\ne \mathbf{y}$,

In relative variables $(\sigma ,\mathbf{r})=(t-u,\mathbf{x}-\mathbf{y})$, the principal kernel

is homogeneous of degree $-2$ and has scaling degree two at the origin.

Proof. For $\mathbf{x}\ne \mathbf{y}$, the spatial factor in (eq:instantaneous) is smooth and nowhere zero. Multiplication by it does not change the wavefront set of $\delta (t-u)$. In local coordinates $(t-u,u,\mathbf{x},\mathbf{y})$, this is the tensor product of a one-dimensional delta with a smooth nonvanishing factor. Its nonzero covectors are exactly the conormal covectors to $t=u$, proving (eq:WF).

Under positive dilation of all four relative coordinates,

The distribution is nonzero, so its scaling degree is exactly two. The smooth nonzero background coefficient preserves the leading scaling degree.