Test functions (like p, smooth and compactly supported) belong to the Schwartz space.
I am following a proof of the elliptic regularity theorem, and I am having a few troubles in understanding where some of the workings come from.
I want to first prove that sing supp (u * v) is a subset of sing supp u + sing supp v.
The proof goes as follows:
Choose functions p (infinitely differentiable) and psi (infinitely differentiable with compact support) such that p = 1 on a neighbourhood of sing supp u and psi = 1 on a neighbourhood of sing supp v. Then:
(u * v) = (pu + (1 - p)u) * (psiv + (1 - psi)v) = pu * psiv + pu * (1 - psi)v + (1 - p)u * psiv + (1 - p)u * (1 - psi)v.
Each of the convolutions other than pu * psiv has at least one infinitely differentiable factor, so is infinitely differentiable.
Then, apparently, it follows that:
sing supp (u * v) is a subset of supp pu + supp psiv which is a subset of supp p + supp psi.
How does this follow? I cannot see it.