Let $\Omega$ be a bounded and Lipschitz domain and $F$ be the superposition/Nemytskii operator defined for functions $u\colon \Omega \to \mathbb{R}$ via $$F(u)(x) = f(u(x))$$ where $f\colon \mathbb{R} \to \mathbb{R}$ is $C^1$ with bounded derivative. I'm looking for results regarding the Frechet differentiability of $F\colon H^{1}(\Omega) \to L^2(\Omega)$.
I think the above hypotheses on $f$ are enough but I can't find a proof to cite anywhere. Does anyone know a reference? Also, I'm interested in cases replacing $H^1$ with a fractional Sobolev space, if anyone knows of a suitable reference.
All books I've seen thus far deal with mappings between $L^p$ spaces or between Sobolev spaces with no mixing.