Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Question

Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert f\Vert_p \Vert g\Vert_q\ge\Vert f\ast g\Vert_\infty$$

I have no idea how to prove the convolution is essentially bounded but the second requirement reminds me Hölder's inequality but $\Vert f\Vert_1\not\geq\Vert f\Vert_\infty$ so also here I'll be glad for a hint.

Answer 1

You need to use Hölder's inequality. Here is a start

$$ (f*g)(x) = \int_{-\infty}^{\infty} f(x-t) g(t) dt \implies |(f*g)(x)| \leq \int_{-\infty}^{\infty} |f(x-t)| |g(t)| dt \leq ||f||_p||g||_q , $$

which proves boundedness. It easily follows that $f*g\in L^{\infty}$.