Proposition: Tensor-product definition of the internal products

Let $R\in {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$. Both products in (eq:internalproducts), understood as distributions in the independent variables $(a,b,z)$, are defined by tensor-product pullback, including at $a=z$.

Proof. The first linear coordinate change is

Its inverse is

Thus $F_1$ is a diffeomorphism. The first product means

The tensor product exists, and pullback by a diffeomorphism is defined.

For the second placement use

with inverse $b=z+r$ and $a=z-u$. The same argument applies. Additional residual variables may be carried along by the identity map. Finite differentiation and multiplication by fixed smooth weights preserve existence.