Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 327 / 2 / b / vi / Solution

False. The constant distribution $u=1$ is tempered, but its Fourier transform is a multiple of the Dirac delta function, which is not an $L^2$ function after multiplication by any Sobolev weight. Thus $1\notin H^t({\mathbb{R}}^n)$ for every $t$.