So, there are no general expressions for large-frequency asymptotics of Fourier transforms, but I am interested in any techniques that would allow to establish upper bounds on the rate of decrease of Fourier transforms (I am talking about cases when the decrease is faster than any inverse power of the frequency).

For example, one may have a function $f(t)$ and its Fourier transform $f(\omega)$ that cannot be evaluated explicitly, $f(\omega)=\frac{1}{2\pi}\int dt e^{i\omega t} f(t)$. It is known that if $f(t)$ is infinitely differentiable, then $f(\omega)$ decreases faster than $O(1/\omega^n)$ for any $n$ at large $\omega$. How could one go about proving more refined bounds on the asymptotic decrease? For example, what usable conditions could one set up for $f(t)$ to ensure that $f(\omega)$ decreases faster than $O(e^{-a\omega^2})$ for some $a$ at large $\omega$?