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

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 $d_{x,\theta }\Phi \ne 0$. The symbol classconsists of smooth amplitudes satisfying, on each compact $K\subset X$,

To define the oscillatory integral, insert a cutoff $\chi (\epsilon \theta )$ equal to one near zero and setRepeated integration by parts with an operator $L$ satisfying $L{e}^{i\Phi }=e^{i\Phi }$ makes the integral absolutely convergent after enough iterations and shows that the limit defines a distribution.