Showing $f$ is an $L^p$ function if $f$ is "self-convoluted.

Question

If $f$ is $L^2(\mathbb{R})$ and $f=f*f$, show that $f$ is $L^p$ for $2\leq p\leq \infty$.

Answer 1

Young's inequality says

$$ \|f\|_p = \|f*f\|_p \leq \|f\|_2\|f\|_q $$

provided $1\leq p,q \leq \infty$ and

$$ \frac1q + \frac12 = \frac1p + 1 $$

If you let $q = 2$, $p = \infty$, you get $f \in L^\infty$ and you're done.