Mathematica does not give an answer to this one either:
InverseFourierTransform[\[Omega]^(1/4), \[Omega], t]
although it does for InverseFourierTransform[\[Omega], \[Omega], t]
and for InverseFourierTransform[\[Omega]^2, \[Omega], t]
. Anyone is expert in Fourier transforms of distributions?