i'm lost in the following question: why $(1*\delta')*H$ and why $1*(\delta' * H)$ are well defined? Where $*$ is the product of convolution and $H$ is the function of Heaviside.
Thank you in advance
In both cases the convolution inside parentheses is well-defined (as a convolution of a compactly supported distribution $\delta' \in \mathcal E'$ with a another distribution).
After that you need to prove that the value of the convolution inside parentheses is also compactly supported.