This question is a simplified version of something I am facing in a research problem and wondering if I'm missing something very elementary that lets me prove things immediately. It's elementary that if we have a function $f$ that $ (1 + \omega^2)^{k/2} \hat{f}(\omega) \in L^2$ if and only if $f \in H^k(\mathbb{R})$. I am interested in similar statements about $f \in H^k([0, T])$ using Laplace bounds.
In particular let's say $f: \mathbb{R} \to \mathbb{C}$ satisfies $f(t) = 0$ for $t < 0$. We can take Laplace transforms: let $s = \sigma + i\omega$ with $\sigma > 0$. Also assume that I have Laplace bounds on $F(s)$ the Laplace transform of $f$, say that $s^k F(s)$ is in $L^2(\mathbb{R})$ as a function of $\omega$ (i.e. on a complex line) for any integer $k \ge 0$ that we'd like. Is there a way to see that this immediately implies that $f \in H^p([0, T])$ for some $p$?
(I realize it's very tempting to use the property $\mathcal{L}\lbrace f'\rbrace(s) = sF(s) - f(0)$ but we do not know that $f$ is differentiable.)