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

True. The Fourier transform of a compactly supported distribution is a smooth function of at most polynomial growth. A sufficiently negative Sobolev weight makes its square integrable, so every $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^n)$ belongs to $H^t({\mathbb{R}}^n)$ for some $t$.