The <Leibniz rule> writes every derivative of $a_1a_2$ as a finite sum of products of derivatives of the factors. Multiplying their symbol bounds adds the orders, so
$$
\boxed{a_1a_2\in
\operatorname{Sym}(X,\mathbb R^k;N_1+N_2)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 327 / 3 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-327/3/a/ii/solution