Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 327 / 1 / b / ii / Solution

The scaling property of the Fourier transform gives, first for Schwartz functions and then by duality,If the homogeneous distribution $u$ has degree $\sigma$, thenPutting $s=1/t$ yields ${\delta }_s\hat{u}=s^{-n-\sigma }\hat{u}$. Thus the Fourier transform of a homogeneous distribution has degree $\boxed{-n-\sigma }$.