Suppose I have the Fourier transform $\hat f(\omega):= \int_{\mathbb R} f(t) e^{i\omega t} \text dt$ of a certain function $f\in L^2(\mathbb R)$. I don't necessarily have $f$ in closed form but I have $\hat f$ in closed form. I am interested in the tail behavior of $\hat f$, i.e.
$$ \bar{\hat{f}}(\omega, x) := \int_x^{\infty} f(t) e^{i\omega t} \text dt $$ for fixed $\omega$ and large $x >> 0$
I assume this has already been studied extensively but a Google search on "Fourier transform tail bound/behavior" did not return much. Could anyone point me in the right direction for theorems/upper bounds?
Thank you! p.