In my course, there is a proposition that states: For $p\in[1,+\infty]$ and $(f,g)\in L{p'}\times L^p$ where $1/p+1/p'=1$, $f\star g$ (convolution) has an element which is uniformly continuous.
In the proof, they use a result that they do not prove and that find challenging to prove myself. I would like to have some hints, thanks !
Here is the result they use:
$\phi\in L^\infty(\mathbb{R^d})$ has en element which is uniformly continuous if and only if $\Vert\tau_h\phi-\phi\Vert_{L^\infty}\to0$ when $\Vert h\Vert$ goes to $0$. $\tau_h$ is the translation operator.
Thanks !
Edit: First implication is done (left to right), but I do not know where to start for the other one.