Differentiation in $x$ does not change the allowed growth order in $\theta$, while every $\theta$-derivative lowers it by one. More precisely, for further multi-indices $\gamma,\delta$,
$$
|D_x^\gamma D_\theta^\delta(D_x^\alpha D_\theta^\beta a)|
=|D_x^{\alpha+\gamma}D_\theta^{\beta+\delta}a|
\leq C(1+|\theta|)^{N-|\beta|-|\delta|}.
$$
Thus
$$
\boxed{D_x^\alpha D_\theta^\beta a
\in\operatorname{Sym}(X,\mathbb R^k;N-|\beta|)}.
$$


Solution

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

 Solution

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