Suppose $f \in L^p(\mathbb{R}^n), g \in L^q(\mathbb{R}^n)$.
I would like to show that $|| f*g||_{L^{\infty}} \leq ||f||_{L^p}||g||_{L^q}$ for $\frac{1}{p}+\frac{1}{q}=1.$
my main idea was to use Holder's inequality. This means that I have
$||fg||_{L^1} \leq ||f||_{L^p} ||g||_{L^q}$ Now I need to show $||f*g||_{L^\infty} \leq || fg||_{L^1}$
Is this the case? Why is this true?
Fix a point $x$ and define $G(y) = g(x-y)$. Observe that $\|G\|_q = \|g\|_q$. Then $$|f \ast g(x)| \le \int |f(y)g(x-y)| \, dy = \int |f(y) G(y)| \, dy = \|fG\|_1.$$ Now apply Holder's inequality.