Convolution of compactly supported distributions

Question

Is the convolution of two compactly supported distributions of non disjoint supports always defined? If yes is it itself compactly supported?

Answer 1

Yes, it is. If $T,S$ are compactly supported with compact supports $K_T$, $K_S$ respectively, then $T\ast S$ is also compactly supported, and supp$(T\ast S)\subset K_T+K_S:=\{t+s: t\in K_T,\; s\in K_S\}$. Refernce: Schwartz's books on distribution theory.