The <symbol class> $\operatorname{Sym}(X,\mathbb R^n;N)=S^N_{1,0}$ consists of $c\in C^\infty(X\times\mathbb R^n)$ such that, for every compact $K\subset X$ and all multi-indices $\alpha,\beta$,
$$
\sup_{x\in K}|D_x^\alpha D_\lambda^\beta c(x,\lambda)|\leq C_{K,\alpha,\beta}\langle\lambda\rangle^{N-|\beta|}.
$$
The defining estimate gives
$$
D_x^\alpha D_\lambda^\beta c_i\in\operatorname{Sym}(X,\mathbb R^n;N_i-|\beta|).
$$
The <Leibniz rule> gives
$$
c_1c_2\in\operatorname{Sym}(X,\mathbb R^n;N_1+N_2),
$$
and the triangle inequality gives
$$
c_1+c_2\in\operatorname{Sym}(X,\mathbb R^n;\max\{N_1,N_2\}).
$$
These are the basic rules of <symbol calculus>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 3 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-327/3/c/i/solution