Take
$$
\boxed{a_2=a\,b_1}.
$$
By <symbol calculus>, $a_2\in S^{-M-1}=S^{N_Q-1}$. Its zeroth-order contribution is $Qa_2=b_1$ outside the compact transition region and therefore cancels the order $-1$ remainder. Every term in
$$
Q(x,\lambda+D_x)a_2-Qa_2
$$
contains at least one $\lambda$ derivative of $Q$ and has order at most $(M-1)+(-M-1)=-2$. Absorbing these terms and the smoothing cutoff contribution into $b_2$ gives
$$
\boxed{Q(x,\lambda+D_x)(a_1+a_2)=1-b_2},\qquad b_2\in S^{-2}.
$$

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 / v / Solution

 Solution

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