Show that if $f\in L^p(\mathbb{R^d})$ and $\phi\in\ S(\mathbb{R^d})$ then $f*\phi\in\mathbb{C^\infty}$, where $S(\mathbb{R^d})$ is the Schwartz class.
How does one prove this rigorously? I have always taken this fact for granted; that convolution of two functions is at least as smooth as either of them. But I cannot get started as to how to prove it in detail.
Let $1/p + 1/q = 1$ (allowing in particular $p=1,q=\infty$ or $p=\infty,q=1$). $$(f * \phi)(x) = \int_{\mathbb R^d} f(t) \phi(x-t) \; dt $$ (which makes sense since $\phi \in L^q$ Taking derivatives of this, for any multi-index $\alpha$, you should get
$$ \partial^\alpha (f * \phi)(x) = \int_{\mathbb R^d} f(t) \partial^\alpha \phi(x-t) \; dt $$
There is actually a use of the Lebesgue Dominated Convergence Theorem here, if you do this carefully.