Let $h\in C^\infty(\mathbb{R}/\mathbb{Z})$ be a bounded exceptionally well behave function and let the sequence $\{a_n\}\in\mathbb{C}^\mathbb{N}$ be the Fourier coefficients for $h$. My goal is to effectively bound the sum $$A(h)=\sum_{n\in\mathbb{N}}|a_n|$$
Specifically as $h$ approaches the indicator function of an interval I want to bound how this blows up. Let $\mathcal{S}_{n,p}(h) = ||h^{(n)}||_p$ where $h^{(n)}$ is $n$th derivative of $h$ and $||\cdot||_p$ is the normal $L^p$ norm ($1\leq p\leq\infty$). One can somewhat easily show that $A(h)\ll\mathcal{S}_{2,2}(h)$ by the Fourier expansion of $h^{(2)}$ which is simply $\{-n^2a_n\}$. We then have very crudely $$|a_nn^2|\leq(\sum_n|n^4a_n^2|)^{1/2}=\mathcal{S}_{2,2}(h)$$ so $a_n\leq \mathcal{S}_{2,2}(h)/n^2$. However I suspect a better bound would be something like $\mathcal{S}_{1,\infty}(h)$ but this doesn't seem so easy to show.
I suppose that in general how various norms on the sequence $\{a_n\}$ translate into (or are bounded by) Sobolev norms on $h$ is very well understood but this doesn't seem to be mentioned in any of the resources I have looked at.