Convolution of functions not in $L^1$

Question

Is it possible to have two functions strictly outside $L^{1}(\mathbb R)$, $f,g: \int_{-\infty}^{\infty}|f(x)|dx=\int_{-\infty}^{\infty}|g(x)|dx=\infty$, and such that their convolution $f\ast g (x)=\int_{-\infty}^{\infty}f(y)g(x-y)dy$ is well defined for a reasonable set of values of $x$?

Answer 1

Yes. If $f\in L^p$ and $g\in L^q$, $1/p+1/p=1$, then $f\ast g$ is defined for all $x\in \mathbb{R}$ and $$ \|f\ast g\|_1\le\|f\|_p\,\|g\|_q. $$