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

True. Multiplication by a Schwartz function is a bounded map $H^s({\mathbb{R}}^n)\to H^s({\mathbb{R}}^n)$ for every real $s$.

True. If $s>t$, then $⟨\xi {⟩}^{2t}\le ⟨\xi {⟩}^{2s}$, so $H^s({\mathbb{R}}^n)\subset H^t({\mathbb{R}}^n)$ continuously.

False. Ellipticity concerns the principal part and large frequencies; lower-order terms may create nonzero real roots. For example, $P(\xi )={\xi }^2-1$ is elliptic in one dimension but vanishes at $\xi =\pm 1$.

True. Repeated use of the Leibniz rule and the polynomial Taylor formula gives

True. The Sobolev embedding theorem gives $H^s({\mathbb{R}}^n)\hookrightarrow C^0({\mathbb{R}}^n)$ when $s>n/2$.

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 $t$.

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 $t$.

True. Fourier transformation turns $D^{\alpha }$ into multiplication by ${\xi }^{\alpha }$, andHence $D^{\alpha }:H^s\to H^{s-|\alpha |}$ is bounded.