A famous factorization theorem of Cohen states that for any locally compact group $G$, $$L^1(G)=L^1(G)*L^1(G).$$ I want to know if analogous results exist for the class of smooth functions when $G$ is a Lie group. For instance, is it true that $$C^\infty(G)=C^\infty(G)*C^\infty(G)?$$
(Feel free to impose any simplifying assumption such as compactness of $G$, compactly supported functions, or $$C^\infty(G)=\text{Span } C^\infty(G)*C^\infty(G)$$ etc.)
It would be great if someone could point out some of the known positive or negative results or open problems that might exist. References are also highly appreciated.
Added in Edit:
- While browing the web, I came across the thesis of Marc Palm, where he mentions on page 65 that
Every smooth function is the convolution product of smooth functions, briefly denoted by $$C_c^\infty(G)=C_c^\infty(G)*C_c^\infty(G).$$
This is clearly stronger than the D-M result, and in contrast, in my opinion, to what Paul writes below in the comments:
A technical point is that finite sums are necessary in that context, so not every test function is exactly a single convolution of two. Does Marc really mean $C_c^\infty(G)=\text{Span }C_c^\infty(G)*C_c^\infty(G)$?
- I also came across this post by Marc Palm where he gives an answer to my question below (in the comments).
