Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$.
Let $x=(x_1,x_2,\ldots,x_n)$ and $y=(x_1+h,x_2,\ldots,x_n)$, $h \ne 0$.
The first step is to show $$ \partial_{x_1} \left(f \ast g\right)= f \ast \left(\partial_{x_1}g\right), $$ through the dominated convergence theorem.
Is it possible to bound $$ \int_{\Bbb R^n} \left|f(t)\right| \frac{\left|g(x-t)-g(y-t)\right|}{|x-y|}dt $$ with the maximal function of $g$?
Or resorting to the mean value theorem $$ g(x-t)-g(y-t)=\int_0^1 Jg\left(x+t(y-x)\right)dt \cdot (y-x) $$ is the only way, where $Jg$ is the Jacobian matrix of $g$.