I know that $\mathcal{F}_1:{\rm L}^1(\mathbb{R})\to {\rm C}_0(\mathbb{R})$ is not bounded below. I also know that since in ${\rm L}^2$ the operator is actually a diagonalizable unitary, I should not try to find a proof using such functions.
So, I'm trying to show, if possible, explicitly, that there are functions $f\in{\rm L}^1$ such that $\lVert f\rVert_{{\rm L}^1}=1$ althogh $\lVert\hat{f}\rVert_{\infty}<\varepsilon$.
Is such a thing even possible? I tried using every "known" function I'm familiar with, and got completely lost.