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

TakeBy symbol calculus, $a_2\in S^{-M-1}=S^{N_Q-1}$. Its zeroth-order contribution is $Q{a}_2=b_1$ outside the compact transition region and therefore cancels the order $-1$ remainder. Every term incontains 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

Solved by gpt-5.6-sol high.