I need to prove/disprove the following existence theorem:
$$ \forall \epsilon > 0, \quad \exists f\in L^1(\mathbb R), g\in L^p(\mathbb R) : \quad (1-\epsilon)\|f\|_1\|g\|_p < \|f\ast g\|_p, $$ where $$ 1\leq p \leq \infty, \quad f\ast g (x) \equiv \int_{\mathbb R} f(y)g(x-y)\,dy. $$
What comes to my mind is Young's inequality for convolutions, $\|f\ast g\|_p \leq \|f\|_1\|g\|_p$. But, alas, I'm unable to make any use of it here, not even sure whether it may be helpful here.
What could be a possible strategy to tackle this problem? Maybe, I should somehow approximate the functions $f$ and $g$ by sequences of (smooth, Schwartz, ... ) functions? Will appreciate any clarifying hint, or reference to the literature! Thanks a lot!