Help on (ratio) Tauberian Theorems

Hello all, I have encountered a problem where my friend advised me to look into using Tauberian theorems.

It is something highly specific and I would appreciate some input, like existing results, literatures etc. If you

know the answer, that will be fantastic!

Here is the problem:

I have a real-valued, positive function defined on . I want to study the behavior of

when . Sadly, I don't have the explicit form of .

But I have the Fourier transform of ( ) and ( ) and they are related in this way:

My question is: is it possible to show the the ratio grows at most in polynomial rate for large

by showing the the ratio behaves like some polynomial (or rational function) for small ?

From reading the literature, it looks like the ratio Tauberian theorem is the "closest" tool to solve the problem,

but it is based on Laplace transform and usually deals with monotone function defined on the positive axis.

So I am kind of lost and have little clue on how to extend the Tauberian stuff to my problem.

Thank you ahead, any suggestion will be highly appreciated!