The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is dense in $C_0$. Everything I have read/heard says that the range is "difficult to describe".

But, since $\mathcal{F}$ is injective and continuous, the image of $L^1$ must be Borel inside of $C_0$. Is anything else known about its descriptive complexity? If not, might this be an example of a natural set of high Borel rank?

This question may be relevant. Thanks for any insight/references.