The multivariable <Leibniz rule> writes each derivative of $a_1a_2$ as a finite sum of products
$$
(D_x^{\alpha_1}D_\theta^{\beta_1}a_1)
(D_x^{\alpha_2}D_\theta^{\beta_2}a_2),
\qquad
\beta_1+\beta_2=\beta.
$$
Each term is bounded by a constant times
$$
(1+|\theta|)^{N_1-|\beta_1|}
(1+|\theta|)^{N_2-|\beta_2|}
=(1+|\theta|)^{N_1+N_2-|\beta|}.
$$
Hence
$$
\boxed{a_1a_2\in\operatorname{Sym}(X,\mathbb R^k;N_1+N_2)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 327 / 2 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-327/2/ii/solution