Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and let $L^p_{\operatorname{loc}}$ denote the space of measurable functions that are locally $p$-integrable on $\mathbb R^n$, with $1 \leqslant p \leqslant \infty$. Moreover, let $C^k$ denote the space of functions defined on $\mathbb R^n$ that are $k$ times continuously differentiable, where $k \geqslant 1$ is an integer. As usual, define $C^\infty$ to be the intersection of all $C^k$ and let $C_c$ denote the space of continuous functions on $\mathbb R^n$ with compact support. Furthermore, let $C_c^\infty = C_c \cap C^\infty$. Finally, denote by $S$ the usual Schwartz space of functions that consists of all the functiosn in $C^\infty$ which, togheter with their derivatives, vanish at infinity faster than any power of $|x|$.
I am interested in knowing if the following two claims are true (I highly suspect they are but can't find a specific reference for such results).
Claim $1$. Given two functions $f \in L^p_{\operatorname{loc}}$ and $\phi \in S$, there holds $f \ast \phi \in C^\infty.$
Claim $2$. Given two functions $f \in L^p_{\operatorname{loc}}$ and $\phi \in S$ such that both $f$ and $\phi$ have compact support, there holds $f \ast \phi \in C^\infty_c.$
My investigation. As I've already indicated above, I highly believe that both these results are true, altought I wasn't able to find a proper reference for them. Despite this, I found some similar results in the literature. For example, in the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations", written by Brezis, the author proves the following results (cf. Proposition $4.20$):
If $f \in C^\infty_c$ and $g \in L^1_{\operatorname{loc}}$, then $f \ast g \in C^\infty$.
The last result is interesting because it allows us to conclude that if $f \in C^\infty_c$ and $g \in L^p_\text{loc}$, then $f \ast g \in C^\infty$ (recall the inclusion $L^p_{\text{loc}} \subset L^1_{\text{loc}}$, for every $1 \leqslant p \leqslant \infty$). This is similar to Claim $1$, but not yet what I wish.
The statement that makes me believe that the claims above are true is one provided by Folland in his famous book "Real Analysis: Modern Techniques and Their Applications" which says the following:
One of the most important properties of convolution is that, roughly speaking, $f \ast g$ is at least as smooth as either $f$ or $g$.
I wonder if someone can confirm the validity of my claims above and provide a reference that explicitly states this results.
Thanks for any help in advance.