Properties of convolutions w.r.t. continuity and partial differentiability

Question

Is there some good summary of properties of convolutions available out there? I'm interested in continuity and partial differentiability topics, like, when exactly do we have $f*g$ is continuous at $x$ if $g$ is, when do we have $\partial_j(f*g)(x)$ exists and equals $f*\partial_j g(x)$, etc.?

Edit: I postet an answer with all conditions I could derive for myself so far.

Answer 1

These are the conditions I (think I) could prove to be sufficient...

...for $f*g\in C^0$:

• $f\in L^1$ and $g\in C^0$ is bounded.
• $f\in L^1_{\rm loc}$ and $g\in C^0_c$.

...for $\partial_j(f*g)(x)$ exists and equals $f*\partial_j g(x)$ (assuming $\partial_j g$ exists everywhere and ${\rm dom}(f*g)$ contains some line segment around $x$ in direction $j$):

• $f\in L^1$ and $\partial_jg$ is bounded.
• $f\in L^1_{\rm loc}$ and $\partial_j g$ is bounded with compact support.