# On smoothness of a function and decay of its Fourier transform

I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here.

I am interested in a quantitative version of the principle that smoothness of a function on the real line is connected with decay of its Fourier transform. Namely, smoothness can be measured by the continuity modulus $\omega$: $$|f(x)-f(y)|\le \omega(|x-y|),$$ and for the decay of the Fourier transform of $f$ we consider estimates of the type $$|\hat f(x)|\le M(|x|).$$ Given $\omega$, for which the former estimate is fulfilled, what is $M$ in the latter estimate? And vice versa, given $M$, what is $\omega$?

I am interested in results where $\omega$ and $M$ are more or less arbitrary under the assumption that they are reasonably regular. I was not able to find an answer elsewhere.

New contributor
Durac is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.