Suppose that $X$ is a square integrable random-variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. It's characteristic function/Fourier transform is defined to be $$ \mathfrak{F}[X](\gamma)\triangleq \int_{\omega \in \Omega} e^{-\gamma X(\omega)} \mathbb{P}(d\omega) . $$
For any fixed $\gamma \in \mathbb{R}$, when is the map \begin{align} L^2(\Omega,\mathcal{F},\mathbb{P})\rightarrow \mathbb{R} \\ X \mapsto \mathfrak{F}[X](\gamma), \end{align} Gateau differentiable (for what X)? Moreover, what is its Gateau derivative?