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

The Sobolev space $H^s({\mathbb{R}}^n)$ consists of $u\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ for whichThe Local Sobolev space $H_{\mathrm{loc}}^s(X)$ consists of distributions $u$ such that $\chi u\in H^s({\mathbb{R}}^n)$ for every $\chi \in C_c^{\infty }(X)$.

If $P$ has degree $m$ and principal homogeneous part $P_m$, then $P(D)$ is an elliptic differential operator whenEquivalently, $|P(\xi )|\ge c|\xi {|}^m$ for all sufficiently large real $\xi$.

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.

The derivative hypothesis implies that $1/Q(\xi )$ is a symbol of order $-\delta N$ at high frequency. Choose cutoffs $\chi \prec \psi$ in $X$ and a high-frequency cutoff in $\xi$. The corresponding Fourier multiplier is a parametrix for $Q(D)$, and the symbol calculus, together with the product formula from part (b), gives the localized estimatefor some sufficiently negative $t$. The commutator terms contain derivatives $Q^{(\alpha )}$; the assumed factor $|\xi {|}^{-\delta |\alpha |}$ lowers their order and lets them be absorbed inductively. Therefore

If $Q(D)u$ is smooth, it belongs locally to $H^s$ for every $s$. Starting from the fact that every compactly supported distribution has some negative Sobolev order and repeatedly applying the gain $\delta N>0$ places $u$ in every local Sobolev space. The Sobolev embedding theorem then gives $u\in C^{\infty }(X)$. Thus $Q(D)$ is a hypoelliptic differential operator.

The heat operatoris hypoelliptic: its symbol $i\tau +|\xi {|}^2$ satisfies the derivative estimates that yield local regularity by the argument in part (c). It is not elliptic as an operator of total order two, because its principal symbol is $|\xi {|}^2$, which vanishes at every nonzero covector $(\tau ,0)$ with $\tau \ne 0$. Hence it is a hypoelliptic differential operator that is not an elliptic differential operator.