A question on convolutions

Question

Let $f$ be an $L^2$ function on the line. If $f*g$ is an $L^2$ function for every $g$ in $L^2$ does it follows that $f$ is in $L^1$?

Answer 1

No. The Plancherel theorem shows that $f*g\in L^2$ for every $g\in L^2$ if $f$ is any function with a bounded Fourier transform, for example $f(t)=\sin(t)/t$.