A <parametrix> for $P(D)$ is a distribution $E$ for which
$$
P(D)E=\delta_0+r
$$
with $r\in C^\infty$. Choose a smooth cutoff $\chi$ that vanishes on a large ball containing every real zero of $P$ and equals one outside a slightly larger ball. Ellipticity makes
$$
q(\lambda)=\frac{\chi(\lambda)}{P(\lambda)}
$$
a <symbol class>[symbol] of order $-N$. For $E=\mathcal F^{-1}q$,
$$
P(D)E=\mathcal F^{-1}\chi=\delta_0-\mathcal F^{-1}(1-\chi).
$$
Since $1-\chi$ is smooth and compactly supported, its inverse <Fourier transform> is smooth. Thus $E$ is a parametrix.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

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