A <phase function> is a real smooth function $\Phi$ on $X\times(\mathbb R^k\setminus\{0\})$, positively homogeneous of degree one in $\theta$, with $d_{x,\theta}\Phi\ne0$. The <symbol class>
$$
\operatorname{Sym}(X,\mathbb R^k;N)
$$
consists of smooth amplitudes satisfying, on each compact $K\subset X$,
$$
|D_x^\alpha D_\theta^\beta a(x,\theta)|
\leq C_{K,\alpha,\beta}\langle\theta\rangle^{N-|\beta|}.
$$

To define the <oscillatory integral>, insert a cutoff $\chi(\varepsilon\theta)$ equal to one near zero and set
$$
\langle I_\Phi(a),f\rangle
=\lim_{\varepsilon\downarrow0}
\int_X\int_{\mathbb R^k}
e^{i\Phi(x,\theta)}a(x,\theta)f(x)
\chi(\varepsilon\theta)\,d\theta\,dx.
$$
Repeated integration by parts with an operator $L$ satisfying $Le^{i\Phi}=e^{i\Phi}$ makes the integral absolutely convergent after enough iterations and shows that the limit defines a distribution.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 327 / 3 / a / Solution

{wiki}

A phase function is a real smooth function $\Phi(x,\theta)$ on $X\times(\mathbb R^k\setminus\{0\})$ that is positively homogeneous of degree one in $\theta$ and has nonvanishing total differential. Its critical points in the frequency variable determine where an associated oscillatory integral may be singular.


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 Incoming links: Phase function