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

The <symbol class> $\operatorname{Sym}(X,\mathbb R^n;N)=S^N_{1,0}$ consists of $c\in C^\infty(X\times\mathbb R^n)$ such that, for every compact $K\subset X$ and all multi-indices $\alpha,\beta$,
$$
\sup_{x\in K}|D_x^\alpha D_\lambda^\beta c(x,\lambda)|\leq C_{K,\alpha,\beta}\langle\lambda\rangle^{N-|\beta|}.
$$
The defining estimate gives
$$
D_x^\alpha D_\lambda^\beta c_i\in\operatorname{Sym}(X,\mathbb R^n;N_i-|\beta|).
$$
The <Leibniz rule> gives
$$
c_1c_2\in\operatorname{Sym}(X,\mathbb R^n;N_1+N_2),
$$
and the triangle inequality gives
$$
c_1+c_2\in\operatorname{Sym}(X,\mathbb R^n;\max\{N_1,N_2\}).
$$
These are the basic rules of <symbol calculus>.

Solved by gpt-5.6-sol high.

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

{title2=$S^m_{1,0}(X\times\mathbb R^n)$}
{wiki}

A smooth function $a(x,\xi)$ belongs to $S^m_{1,0}$ when, on every compact $K\subset X$,
$$
|D_x^\alpha D_\xi^\beta a(x,\xi)|\leq C_{K,\alpha,\beta}\langle\xi\rangle^{m-|\beta|}.
$$


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