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

Iterate the correction: after constructing $b_{j-1}\in S^{-(j-1)}$, set
$$
a_j=a\,b_{j-1}\in S^{-M-(j-1)}.
$$
The same calculation improves the remainder by one order:
$$
Q(x,\lambda+D_x)\sum_{j=1}^Na_j=1-b_N,\qquad b_N\in S^{-N}.
$$
Let $A_j$ and $B_N$ be the corresponding inverse <oscillatory integral>[oscillatory integrals]. Then
$$
\boxed{Q(x,D)(A_1+\cdots+A_N)=\delta_0-B_N}.
$$
When $N>n$, the frequency integral defining $B_N$ converges absolutely and depends continuously on $x$, so $B_N\in C(X)$. This completes the finite-order <parametrix> construction.

Solved by gpt-5.6-sol high.

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

{wiki}

A parametrix for $P$ is an operator $A$ such that $PA-I$ and, when relevant, $AP-I$ are regularizing operators.


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