$\quad$ I've heard that the regularity of convolution of two functions is more affected by the function that is more regular. As examples, here are some I know.
- if $f\in\mathcal L^1$ and $g\in\mathcal L^1$, then $f*g\in\mathcal L^1$.
- if $f\in\mathcal L^1$, $g\in\mathcal L^0$ and $g$ bounded, then $f*g$ is uniformly continuous.
- if $f\in\mathcal L^1$ and $g\in C^\infty$, then $f*g\in C^\infty$.
$\quad$ So it seems that as long as convolution exists, its regularity is not too bad. Is there, I wonder, a more general, rigorous conclusion? The statement "the regularity of convolution depends more on the function with better regularity" seems not rigorous.
$\quad$ This is an open question, and any discussion is welcome.
$\quad$ By the way, this seems to have a lot to do with the idea of partial differential equations, and anything about PDEs is also welcome to be discussed.