I'm currently stumbling over a criteria of integrability.
Let's suppose we have a real-valued function $h$ defined on the positive real axis which is integrable. We then define the function $K :\mathbb{R}_+^* \to \mathbb{C}$ ; $x \to \int_0^\infty i h(s) e^{ixh(s)}ds$
I am trying to characterize the class of functions $h$ that makes $\vert K \vert^2$ integrable.
It is easy to find that $\{ h_\alpha : t \to e^{-\alpha t} \}_{\alpha > 0} $ belong to this class because after a change of variable (there are $\mathcal{C}^1$ diffeomorphisms ), the expression is explicit. In such a case, we have $K(x)=\frac{e^{ix}-1}{\alpha x}$
Does anyone of you has a hint (for example an inequality I should know) ?