I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.

Riesz-Thorin gives us that there is also $F_p: L^p \rightarrow L^q$ for $p \in (1,2).$ Here, the dual map trick does not work, so this transform has a chance of being surjective. Since every $f \in L^q$ is also in $S'$ we can also define a promising candidate $F_{S'}^{-1}(f).$ Unfortunatly, this does not really tell me whether this $F_{S'}^{-1}(f) \in L^p$ again.

This raises the question whether $F_p$ is actually surjective or not?

Comment on the discussion below: Thanks to everybody participating in the disccusion. Actually this question came to my mind while I was thinking about this problem from PDEs, which would have an easy solution in this case. I have to admit that the fact that $L^p$ is not isomorphic to $L^q$ is indeed something I know, but I have never actually used it, as I am not primarily active in analysis. Probably I should give my questions more thought in the future.Sorry for any inconvenience my question caused.

in a different area, right? I mean, all my questions about finite group theory and group characters are the kinds of thing that algebraists might learn in "graduate education" but which I simply didn't $\endgroup$ – Yemon Choi May 12 '16 at 21:38