Example for Young's inequality for convolution

Question

Good day everyone, I'm looking for $f,g \in L^1$ such that $\|f*g\|_1<\|f\|_1\cdot \|g\|_1$. Since the Young's inequality $\|f*g\|_1\le\|f\|_1\cdot \|g\|_1$ ist nor an equality, I suppose that such $f,g$ should exist. But I cannot find them. Thanks for help.

Answer 1

Let $f_n=n\left(\chi_{[0,1/n]}-\chi_{[-1/n,0]}\right)$. If $g\in C_c(\Bbb R)$ then $\lim_{n\to\infty}||f_n*g||_1=0$.

Proof: Since $g$ is uniformly continuous it's easy to show that $f_n*g\to0$ uniformly as $n\to\infty$. Also there exists a compact set $K$ such that every $f_n*g$ is supported in $K$.