Convolutions of $L^p$ functions

Question

Denote by $(fg)$ the convolution of $f$ and $g$. If $g$ is square integrable and $(fg)$ is square integrable for every square integrable $f$ can we conclude that $g $ is integrable? This is a converse to the well-known fact that if $f$ is square integrable and $g$ is integrable then $(fg)$ is square integrable.