A <phase function> is a real <smooth function> $\Phi$ on $X\times(\mathbb R^k\setminus\{0\})$ that is <positively homogeneous> of degree one in $\theta$ and has nonzero total differential $d_{x,\theta}\Phi$. The <symbol class>
$$
\operatorname{Sym}(X,\mathbb R^k;N)
$$
consists of smooth amplitudes for which, for every compact $K\subset X$ and all <multi-indices> $\alpha,\beta$,
$$
|D_x^\alpha D_\theta^\beta a(x,\theta)|
\leq C_{K,\alpha,\beta}\langle\theta\rangle^{N-|\beta|}.
$$

Choose a <smooth cutoff function> $\chi$ equal to one near zero. The associated <oscillatory integral> is defined on a <test function> $f$ by
$$
\langle I_\Phi(a),f\rangle
=\lim_{\varepsilon\downarrow0}
\int_X\int_{\mathbb R^k}
e^{i\Phi(x,\theta)}a(x,\theta)f(x)
\chi(\varepsilon\theta)\,d\theta\,dx.
$$
On the compact $x$-support of $f$, use an integration-by-parts operator $L$ satisfying $Le^{i\Phi}=e^{i\Phi}$. Repeated application of its formal adjoint lowers the effective symbol order until the integral is absolutely convergent. The resulting bounds involve only finitely many derivatives of $f$, prove that the limit is independent of $\chi$, and give the seminorm estimate required for
$$
\boxed{I_\Phi(a)\in\mathcal D'(X)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 3 / a / Solution

A smooth cutoff function is a smooth function used to localize an argument, typically equal to one on one region and zero outside a slightly larger region. Such functions can be built from <bump functions> and smooth <partition of unity>[partitions of unity].

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