We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$ By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,

"Let $F$ be a function on $\mathbb C$ and if $F(f)\in A^{1}(\mathbb T)$ whenever $f\in A^{1}(\mathbb T)$($F(f)$ is the composition of $F$ and f)(i.e, $F$ operates in $A^{1}(\mathbb T)$), then $F$ must be real-analytic on $\mathbb R^{2}.$"

My Question is: If $F$ operates in $A^{q}(\mathbb T), (1< q \leq \infty) $. Then what can we say about $F$ ? If it is well-known, and not trivial, references would be o.k. for me.

[We note, $q=2$, it is just $L^{2}(\mathbb T)$ we may not expect the exact analogue]